Sample records for accurate mathematical representation

  1. Facilitating Mathematical Practices through Visual Representations

    ERIC Educational Resources Information Center

    Murata, Aki; Stewart, Chana

    2017-01-01

    Effective use of mathematical representation is key to supporting student learning. In "Principles to Actions: Ensuring Mathematical Success for All" (NCTM 2014), "use and connect mathematical representations" is one of the effective Mathematics Teaching Practices. By using different representations, students examine concepts…

  2. Improving students’ mathematical representational ability through RME-based progressive mathematization

    NASA Astrophysics Data System (ADS)

    Warsito; Darhim; Herman, T.

    2018-01-01

    This study aims to determine the differences in the improving of mathematical representation ability based on progressive mathematization with realistic mathematics education (PMR-MP) with conventional learning approach (PB). The method of research is quasi-experiments with non-equivalent control group designs. The study population is all students of class VIII SMPN 2 Tangerang consisting of 6 classes, while the sample was taken two classes with purposive sampling technique. The experimental class is treated with PMR-MP while the control class is treated with PB. The instruments used are test of mathematical representation ability. Data analysis was done by t-test, ANOVA test, post hoc test, and descriptive analysis. The result of analysis can be concluded that: 1) there are differences of mathematical representation ability improvement between students treated by PMR-MP and PB, 2) no interaction between learning approach (PMR-MP, PB) and prior mathematics knowledge (PAM) to improve students’ mathematical representation; 3) Students’ mathematical representation improvement in the level of higher PAM is better than medium, and low PAM students. Thus, based on the process of mathematization, it is very important when the learning direction of PMR-MP emphasizes on the process of building mathematics through a mathematical model.

  3. Characterizing Interaction with Visual Mathematical Representations

    ERIC Educational Resources Information Center

    Sedig, Kamran; Sumner, Mark

    2006-01-01

    This paper presents a characterization of computer-based interactions by which learners can explore and investigate visual mathematical representations (VMRs). VMRs (e.g., geometric structures, graphs, and diagrams) refer to graphical representations that visually encode properties and relationships of mathematical structures and concepts.…

  4. Pattern of mathematic representation ability in magnetic electricity problem

    NASA Astrophysics Data System (ADS)

    Hau, R. R. H.; Marwoto, P.; Putra, N. M. D.

    2018-03-01

    The mathematic representation ability in solving magnetic electricity problem gives information about the way students understand magnetic electricity. Students have varied mathematic representation pattern ability in solving magnetic electricity problem. This study aims to determine the pattern of students' mathematic representation ability in solving magnet electrical problems.The research method used is qualitative. The subject of this study is the fourth semester students of UNNES Physics Education Study Program. The data collection is done by giving a description test that refers to the test of mathematical representation ability and interview about field line topic and Gauss law. The result of data analysis of student's mathematical representation ability in solving magnet electric problem is categorized into high, medium and low category. The ability of mathematical representations in the high category tends to use a pattern of making known and asked symbols, writing equations, using quantities of physics, substituting quantities into equations, performing calculations and final answers. The ability of mathematical representation in the medium category tends to use several patterns of writing the known symbols, writing equations, using quantities of physics, substituting quantities into equations, performing calculations and final answers. The ability of mathematical representations in the low category tends to use several patterns of making known symbols, writing equations, substituting quantities into equations, performing calculations and final answer.

  5. The Microevolution of Mathematical Representations in Children's Activity.

    ERIC Educational Resources Information Center

    Meira, Luciano

    1995-01-01

    Discusses children's design of mathematical representations on paper. Suggests that the design of displays during problem solving shapes one's mathematical activity and sense making in crucial ways, and that knowledge of mathematical representations is not simply recalled and applied to problem solving, but also emerges out of one's interactions…

  6. Indicators that influence prospective mathematics teachers representational and reasoning abilities

    NASA Astrophysics Data System (ADS)

    Darta; Saputra, J.

    2018-01-01

    Representational and mathematical reasoning ability are very important ability as basic in mathematics learning process. The 2013 curriculum suggests that the use of a scientific approach emphasizes higher order thinking skills. Therefore, a scientific approach is required in mathematics learning to improve ability of representation and mathematical reasoning. The objectives of this research are: (1) to analyze representational and reasoning abilities, (2) to analyze indicators affecting the ability of representation and mathematical reasoning, (3) to analyze scientific approaches that can improve the ability of representation and mathematical reasoning. The subject of this research is the students of mathematics prospective teachers in the first semester at Private Higher Education of Bandung City. The research method of this research was descriptive analysis. The research data were collected using reasoning and representation tests on sixty-one students. Data processing was done by descriptive analysis specified based on the indicators of representation ability and mathematical reasoning that influenced it. The results of this first-year study showed that students still had many weaknesses in reasoning and mathematical representation that were influenced by the ability to understand the indicators of both capabilities. After observing the results of the first-year research, then in the second and third year, the development of teaching materials with a scientific approach in accordance with the needs of prospective students was planned.

  7. Brain Correlates of Mathematical Competence in Processing Mathematical Representations

    PubMed Central

    Grabner, Roland H.; Reishofer, Gernot; Koschutnig, Karl; Ebner, Franz

    2011-01-01

    The ability to extract numerical information from different representation formats (e.g., equations, tables, or diagrams) is a key component of mathematical competence but little is known about its neural correlate. Previous studies comparing mathematically less and more competent adults have focused on mental arithmetic and reported differences in left angular gyrus (AG) activity which were interpreted to reflect differential reliance on arithmetic fact retrieval during problem solving. The aim of the present functional magnetic resonance imaging study was to investigate the brain correlates of mathematical competence in a task requiring the processing of typical mathematical representations. Twenty-eight adults of lower and higher mathematical competence worked on a representation matching task in which they had to evaluate whether the numerical information of a symbolic equation matches that of a bar chart. Two task conditions without and one condition with arithmetic demands were administered. Both competence groups performed equally well in the non-arithmetic conditions and only differed in accuracy in the condition requiring calculation. Activation contrasts between the groups revealed consistently stronger left AG activation in the more competent individuals across all three task conditions. The finding of competence-related activation differences independently of arithmetic demands suggests that more and less competent individuals differ in a cognitive process other than arithmetic fact retrieval. Specifically, it is argued that the stronger left AG activity in the more competent adults may reflect their higher proficiency in processing mathematical symbols. Moreover, the study demonstrates competence-related parietal activation differences that were not accompanied by differential experimental performance. PMID:22069387

  8. Mathematical Representation Ability by Using Project Based Learning on the Topic of Statistics

    NASA Astrophysics Data System (ADS)

    Widakdo, W. A.

    2017-09-01

    Seeing the importance of the role of mathematics in everyday life, mastery of the subject areas of mathematics is a must. Representation ability is one of the fundamental ability that used in mathematics to make connection between abstract idea with logical thinking to understanding mathematics. Researcher see the lack of mathematical representation and try to find alternative solution to dolve it by using project based learning. This research use literature study from some books and articles in journals to see the importance of mathematical representation abiliy in mathemtics learning and how project based learning able to increase this mathematical representation ability on the topic of Statistics. The indicators for mathematical representation ability in this research classifies namely visual representation (picture, diagram, graph, or table); symbolize representation (mathematical statement. Mathematical notation, numerical/algebra symbol) and verbal representation (written text). This article explain about why project based learning able to influence student’s mathematical representation by using some theories in cognitive psychology, also showing the example of project based learning that able to use in teaching statistics, one of mathematics topic that very useful to analyze data.

  9. Students’ mathematical representations on secondary school in solving trigonometric problems

    NASA Astrophysics Data System (ADS)

    Istadi; Kusmayadi, T. A.; Sujadi, I.

    2017-06-01

    This research aimed to analyse students’ mathematical representations on secondary school in solving trigonometric problems. This research used qualitative method. The participants were 4 students who had high competence of knowledge taken from 20 students of 12th natural-science grade SMAN-1 Kota Besi, Central Kalimantan. Data validation was carried out using time triangulation. Data analysis used Huberman and Miles stages. The results showed that their answers were not only based on the given figure, but also used the definition of trigonometric ratio on verbal representations. On the other hand, they were able to determine the object positions to be observed. However, they failed to determine the position of the angle of depression at the sketches made on visual representations. Failure in determining the position of the angle of depression to cause an error in using the mathematical equation. Finally, they were unsuccessful to use the mathematical equation properly on symbolic representations. From this research, we could recommend the importance of translations between mathematical problems and mathematical representations as well as translations among mathematical representaions (verbal, visual, and symbolic) in learning mathematics in the classroom.

  10. Mathematical formalisms based on approximated kinetic representations for modeling genetic and metabolic pathways.

    PubMed

    Alves, Rui; Vilaprinyo, Ester; Hernádez-Bermejo, Benito; Sorribas, Albert

    2008-01-01

    There is a renewed interest in obtaining a systemic understanding of metabolism, gene expression and signal transduction processes, driven by the recent research focus on Systems Biology. From a biotechnological point of view, such a systemic understanding of how a biological system is designed to work can facilitate the rational manipulation of specific pathways in different cell types to achieve specific goals. Due to the intrinsic complexity of biological systems, mathematical models are a central tool for understanding and predicting the integrative behavior of those systems. Particularly, models are essential for a rational development of biotechnological applications and in understanding system's design from an evolutionary point of view. Mathematical models can be obtained using many different strategies. In each case, their utility will depend upon the properties of the mathematical representation and on the possibility of obtaining meaningful parameters from available data. In practice, there are several issues at stake when one has to decide which mathematical model is more appropriate for the study of a given problem. First, one needs a model that can represent the aspects of the system one wishes to study. Second, one must choose a mathematical representation that allows an accurate analysis of the system with respect to different aspects of interest (for example, robustness of the system, dynamical behavior, optimization of the system with respect to some production goal, parameter value determination, etc). Third, before choosing between alternative and equally appropriate mathematical representations for the system, one should compare representations with respect to easiness of automation for model set-up, simulation, and analysis of results. Fourth, one should also consider how to facilitate model transference and re-usability by other researchers and for distinct purposes. Finally, one factor that is important for all four aspects is the regularity

  11. Communication and Representation as Elements in Mathematical Literacy

    ERIC Educational Resources Information Center

    Thompson, Denisse R.; Chappell, Michaele F.

    2007-01-01

    The process standards of communication and representation in the "Principles and Standards for School Mathematics" are critical tools to help students develop mathematical literacy. In the mathematics classroom, students need to be encouraged to use speaking, listening, reading, and writing to communicate their understanding of mathematics words,…

  12. Electromagnetic Concepts in Mathematical Representation of Physics.

    ERIC Educational Resources Information Center

    Albe, Virginie; Venturini, Patrice; Lascours, Jean

    2001-01-01

    Addresses the use of mathematics when studying the physics of electromagnetism. Focuses on common electromagnetic concepts and their associated mathematical representation and arithmetical tools. Concludes that most students do not understand the significant aspects of physical situations and have difficulty using relationships and models specific…

  13. Interactivity of Visual Mathematical Representations: Factors Affecting Learning and Cognitive Processes

    ERIC Educational Resources Information Center

    Sedig, Kamran; Liang, Hai-Ning

    2006-01-01

    Computer-based mathematical cognitive tools (MCTs) are a category of external aids intended to support and enhance learning and cognitive processes of learners. MCTs often contain interactive visual mathematical representations (VMRs), where VMRs are graphical representations that encode properties and relationships of mathematical concepts. In…

  14. Mexican High School Students' Social Representations of Mathematics, Its Teaching and Learning

    ERIC Educational Resources Information Center

    Martínez-Sierra, Gustavo; Miranda-Tirado, Marisa

    2015-01-01

    This paper reports a qualitative research that identifies Mexican high school students' social representations of mathematics. For this purpose, the social representations of "mathematics", "learning mathematics" and "teaching mathematics" were identified in a group of 50 students. Focus group interviews were carried…

  15. Mathematical representations in science: a cognitive-historical case history.

    PubMed

    Tweney, Ryan D

    2009-10-01

    The important role of mathematical representations in scientific thinking has received little attention from cognitive scientists. This study argues that neglect of this issue is unwarranted, given existing cognitive theories and laws, together with promising results from the cognitive historical analysis of several important scientists. In particular, while the mathematical wizardry of James Clerk Maxwell differed dramatically from the experimental approaches favored by Michael Faraday, Maxwell himself recognized Faraday as "in reality a mathematician of a very high order," and his own work as in some respects a re-representation of Faraday's field theory in analytic terms. The implications of the similarities and differences between the two figures open new perspectives on the cognitive role of mathematics as a learned mode of representation in science. Copyright © 2009 Cognitive Science Society, Inc.

  16. Reading Mathematics Representations: An Eye-Tracking Study

    ERIC Educational Resources Information Center

    Andrá, Chiara; Lindström, Paulina; Arzarello, Ferdinando; Holmqvist, Kenneth; Robutti, Ornella; Sabena, Cristina

    2015-01-01

    We use eye tracking as a method to examine how different mathematical representations of the same mathematical object are attended to by students. The results of this study show that there is a meaningful difference in the eye movements between formulas and graphs. This difference can be understood in terms of the cultural and social shaping of…

  17. Problem representation and mathematical problem solving of students of varying math ability.

    PubMed

    Krawec, Jennifer L

    2014-01-01

    The purpose of this study was to examine differences in math problem solving among students with learning disabilities (LD, n = 25), low-achieving students (LA, n = 30), and average-achieving students (AA, n = 29). The primary interest was to analyze the processes students use to translate and integrate problem information while solving problems. Paraphrasing, visual representation, and problem-solving accuracy were measured in eighth grade students using a researcher-modified version of the Mathematical Processing Instrument. Results indicated that both students with LD and LA students struggled with processing but that students with LD were significantly weaker than their LA peers in paraphrasing relevant information. Paraphrasing and visual representation accuracy each accounted for a statistically significant amount of variance in problem-solving accuracy. Finally, the effect of visual representation of relevant information on problem-solving accuracy was dependent on ability; specifically, for students with LD, generating accurate visual representations was more strongly related to problem-solving accuracy than for AA students. Implications for instruction for students with and without LD are discussed.

  18. Using Student Contributions and Multiple Representations To Develop Mathematical Language.

    ERIC Educational Resources Information Center

    Herbel-Eisenmann, Beth A.

    2002-01-01

    Describes a way to introduce and use mathematical language as an alternative to using vocabulary lists to introduce students to mathematical language in mathematics classrooms. Draws on multiple representations and student language. (YDS)

  19. Students’ Representation in Mathematical Word Problem-Solving: Exploring Students’ Self-efficacy

    NASA Astrophysics Data System (ADS)

    Sahendra, A.; Budiarto, M. T.; Fuad, Y.

    2018-01-01

    This descriptive qualitative research aims at investigating student represented in mathematical word problem solving based on self-efficacy. The research subjects are two eighth graders at a school in Surabaya with equal mathematical ability consisting of two female students with high and low self-efficacy. The subjects were chosen based on the results of test of mathematical ability, documentation of the result of middle test in even semester of 2016/2017 academic year, and results of questionnaire of mathematics word problem in terms of self-efficacy scale. The selected students were asked to do mathematical word problem solving and be interviewed. The result of this study shows that students with high self-efficacy tend to use multiple representations of sketches and mathematical models, whereas students with low self-efficacy tend to use single representation of sketches or mathematical models only in mathematical word problem-solving. This study emphasizes that teachers should pay attention of student’s representation as a consideration of designing innovative learning in order to increase the self-efficacy of each student to achieve maximum mathematical achievement although it still requires adjustment to the school situation and condition.

  20. Investigating Trigonometric Representations in the Transition to College Mathematics

    ERIC Educational Resources Information Center

    Byers, Patricia

    2010-01-01

    This Ontario-based qualitative study examined secondary school and college textbooks' treatment of trigonometric representations in order to identify potential sources of student difficulties in the transition from secondary school to college mathematics. Analysis of networks, comprised of trigonometric representations, identified numerous issues…

  1. An analysis of primary school students’ representational ability in mathematics based on gender perspective

    NASA Astrophysics Data System (ADS)

    Kowiyah; Mulyawati, I.

    2018-01-01

    Mathematic representation is one of the basic mathematic skills that allows students to communicate their mathematic ideas through visual realities such as pictures, tables, mathematic expressions and mathematic equities. The present research aims at: 1) analysing students’ mathematic representation ability in solving mathematic problems and 2) examining the difference of students’ mathematic ability based on their gender. A total of sixty primary school students participated in this study comprising of thirty males and thirty females. Data required in this study were collected through mathematic representation tests, interviews and test evaluation rubric. Findings of this study showed that students’ mathematic representation of visual realities (image and tables) was reported higher at 62.3% than at in the form of description (or statement) at 8.6%. From gender perspective, male students performed better than the females at action planning stage. The percentage of males was reported at 68% (the highest), 33% (medium) and 21.3% (the lowest) while the females were at 36% (the highest), 37.7% (medium) and 32.6% (the lowest).

  2. Is Mathematical Representation of Problems an Evidence-Based Strategy for Students with Mathematics Difficulties?

    ERIC Educational Resources Information Center

    Jitendra, Asha K.; Nelson, Gena; Pulles, Sandra M.; Kiss, Allyson J.; Houseworth, James

    2016-01-01

    The purpose of the present review was to evaluate the quality of the research and evidence base for representation of problems as a strategy to enhance the mathematical performance of students with learning disabilities and those at risk for mathematics difficulties. The authors evaluated 25 experimental and quasiexperimental studies according to…

  3. [Representation and mathematical analysis of human crystalline lens].

    PubMed

    Tălu, Stefan; Giovanzana, Stefano; Tălu, Mihai

    2011-01-01

    The surface of human crystalline lens can be described and analyzed using mathematical models based on parametric representations, used in biomechanical studies and 3D solid modeling of the lens. The mathematical models used in lens biomechanics allow the study and the behavior of crystalline lens on variables and complex dynamic loads. Also, the lens biomechanics has the potential to improve the results in the development of intraocular lenses and cataract surgery. The paper presents the most representative mathematical models currently used for the modeling of human crystalline lens, both optically and biomechanically.

  4. Mathematical Representation by Students in Building Relational Understanding on Concepts of Area and Perimeter of Rectangle

    ERIC Educational Resources Information Center

    Anwar, Rahmad Bustanul; Yuwono, Ipung; As'ari, Abdur Rahman; Sisworo; Dwi, Rahmawati

    2016-01-01

    Representation is an important aspect of learners in building a relational understanding of mathematical concepts. But the ability of a mathematical representation of students in building relational understanding is still very limited. The purpose of this research is to description of mathematical representation of students who appear in building…

  5. The Effects of Multiple Linked Representations on Student Learning in Mathematics.

    ERIC Educational Resources Information Center

    Ozgun-Koca, S. Asli

    This study investigated the effects on student understanding of linear relationships using the linked representation software VideoPoint as compared to using semi-linked representation software. It investigated students' attitudes towards and preferences for mathematical representations--equations, tables, or graphs. An Algebra I class was divided…

  6. The Design of Lessons Using Mathematics Analysis Software to Support Multiple Representations in Secondary School Mathematics

    ERIC Educational Resources Information Center

    Pierce, Robyn; Stacey, Kaye; Wander, Roger; Ball, Lynda

    2011-01-01

    Current technologies incorporating sophisticated mathematical analysis software (calculation, graphing, dynamic geometry, tables, and more) provide easy access to multiple representations of mathematical problems. Realising the affordances of such technology for students' learning requires carefully designed lessons. This paper reports on design…

  7. Mexican high school students' social representations of mathematics, its teaching and learning

    NASA Astrophysics Data System (ADS)

    Martínez-Sierra, Gustavo; Miranda-Tirado, Marisa

    2015-07-01

    This paper reports a qualitative research that identifies Mexican high school students' social representations of mathematics. For this purpose, the social representations of 'mathematics', 'learning mathematics' and 'teaching mathematics' were identified in a group of 50 students. Focus group interviews were carried out in order to obtain the data. The constant comparative style was the strategy used for the data analysis because it allowed the categories to emerge from the data. The students' social representations are: (A) Mathematics is…(1) important for daily life, (2) important for careers and for life, (3) important because it is in everything that surrounds us, (4) a way to solve problems of daily life, (5) calculations and operations with numbers, (6) complex and difficult, (7) exact and (6) a subject that develops thinking skills; (B) To learn mathematics is…(1) to possess knowledge to solve problems, (2) to be able to solve everyday problems, (3) to be able to make calculations and operations, and (4) to think logically to be able to solve problems; and (C) To teach mathematics is…(1) to transmit knowledge, (2) to know to share it, (3) to transmit the reasoning ability, and (4) to show how to solve problems.

  8. Development of Number Line Representations in Children With Mathematical Learning Disability

    PubMed Central

    Geary, David C.; Hoard, Mary K.; Nugent, Lara; Byrd-Craven, Jennifer

    2015-01-01

    Children with a mathematical learning disability (MLD, n = 19) and low achieving (LA, n = 43) children were identified using mathematics achievement scores below the 11th percentile and between the 11th and 25th percentiles, respectively. A control group of typically achieving (TA, n = 50) children was also identified. Number line and speed of processing tasks were administered in 1st and 2nd grade and a working memory battery in 1st grade. In both grades, the MLD children were less accurate in their number line placements and more reliant on a natural number-magnitude representational system to make these placements than were TA children. The TA children were more reliant on the school-taught linear system in both grades. The performance of the LA children was similar to that of the MLD children in first grade and to the TA children in second. The central executive component of working memory contributed to across-grade improvements in number line performance and to group differences in this performance. PMID:18473200

  9. The Concrete-Representational-Abstract Sequence of Instruction in Mathematics Classrooms

    ERIC Educational Resources Information Center

    Mudaly, Vimolan; Naidoo, Jayaluxmi

    2015-01-01

    The purpose of this paper is to explore how master mathematics teachers use the concrete-representational-abstract (CRA) sequence of instruction in mathematics classrooms. Data was collected from a convenience sample of six master teachers by observations, video recordings of their teaching, and semi-structured interviews. Data collection also…

  10. The written mathematical communication profile of prospective math teacher in mathematical proving

    NASA Astrophysics Data System (ADS)

    Pantaleon, K. V.; Juniati, D.; Lukito, A.; Mandur, K.

    2018-01-01

    Written mathematical communication is the process of expressing mathematical ideas and understanding in writing. It is one of the important aspects that must be mastered by the prospective math teacher as tool of knowledge transfer. This research was a qualitative research that aimed to describe the mathematical communication profile of the prospective mathematics teacher in mathematical proving. This research involved 48 students of Mathematics Education Study Program; one of them with moderate math skills was chosen as the main subject. Data were collected through tests, assignments, and task-based interviews. The results of this study point out that in the proof of geometry, the subject explains what is understood, presents the idea in the form of drawing and symbols, and explains the content/meaning of a representation accurately and clearly, but the subject can not convey the argument systematically and logically. Whereas in the proof of algebra, the subject describes what is understood, explains the method used, and describes the content/meaning of a symbolic representation accurately, systematically, logically, but the argument presented is not clear because it is insufficient detailed and complete.

  11. Effects of Computer-Based Visual Representation on Mathematics Learning and Cognitive Load

    ERIC Educational Resources Information Center

    Yung, Hsin I.; Paas, Fred

    2015-01-01

    Visual representation has been recognized as a powerful learning tool in many learning domains. Based on the assumption that visual representations can support deeper understanding, we examined the effects of visual representations on learning performance and cognitive load in the domain of mathematics. An experimental condition with visual…

  12. Developing Instruction Materials Based on Joyful PBL to Improve Students Mathematical Representation Ability

    ERIC Educational Resources Information Center

    Minarni, Ani; Napitupulu, E. Elvis

    2017-01-01

    Solving problem either within mathematics or beyond is one of the ultimate goal students learn mathematics. It is since mathematics takes role tool as well as vehicle to develop problem solving ability. One of the supporting components to problem solving is mathematical representation ability (MRA). Nowadays, many teachers and researchers find out…

  13. Relations of different types of numerical magnitude representations to each other and to mathematics achievement.

    PubMed

    Fazio, Lisa K; Bailey, Drew H; Thompson, Clarissa A; Siegler, Robert S

    2014-07-01

    We examined relations between symbolic and non-symbolic numerical magnitude representations, between whole number and fraction representations, and between these representations and overall mathematics achievement in fifth graders. Fraction and whole number symbolic and non-symbolic numerical magnitude understandings were measured using both magnitude comparison and number line estimation tasks. After controlling for non-mathematical cognitive proficiency, both symbolic and non-symbolic numerical magnitude understandings were uniquely related to mathematics achievement, but the relation was much stronger for symbolic numbers. A meta-analysis of 19 published studies indicated that relations between non-symbolic numerical magnitude knowledge and mathematics achievement are present but tend to be weak, especially beyond 6 years of age. Copyright © 2014 Elsevier Inc. All rights reserved.

  14. Why Use Multiple Representations in the Mathematics Classroom? Views of English and German Preservice Teachers

    ERIC Educational Resources Information Center

    Dreher, Anika; Kuntze, Sebastian; Lerman, Stephen

    2016-01-01

    Dealing with multiple representations and their connections plays a key role for learners to build up conceptual knowledge in the mathematics classroom. Hence, professional knowledge and views of mathematics teachers regarding the use of multiple representations certainly merit attention. In particular, investigating such views of preservice…

  15. Enhancing students’ mathematical representation and selfefficacy through situation-based learning assisted by geometer’s sketchpad program

    NASA Astrophysics Data System (ADS)

    Sowanto; Kusumah, Y. S.

    2018-05-01

    This research was conducted based on the problem of a lack of students’ mathematical representation ability as well as self-efficacy in accomplishing mathematical tasks. To overcome this problem, this research used situation-based learning (SBL) assisted by geometer’s sketchpad program (GSP). This research investigated students’ improvement of mathematical representation ability who were taught under situation-based learning (SBL) assisted by geometer’s sketchpad program (GSP) and regular method that viewed from the whole students’ prior knowledge (high, average, and low level). In addition, this research investigated the difference of students’ self-efficacy after learning was given. This research belongs to quasi experiment research using non-equivalent control group design with purposive sampling. The result of this research showed that students’ enhancement in their mathematical representation ability taught under SBL assisted by GSP was better than the regular method. Also, there was no interaction between learning methods and students prior knowledge in student’ enhancement of mathematical representation ability. There was significant difference of students’ enhancement of mathematical representation ability taught under SBL assisted by GSP viewed from students’ prior knowledge. Furthermore, there was no significant difference in terms of self-efficacy between those who were taught by SBL assisted by GSP with the regular method.

  16. Middle-Level Preservice Mathematics Teachers' Mental Representations of Classroom Floor Plans

    ERIC Educational Resources Information Center

    Matteson, Shirley M.; Ganesh, Bibi S.; Coward, Fanni L.; Patrick, Patricia

    2012-01-01

    This study reports the results of an innovative assignment in which preservice teachers' mental representations were examined through drawing floor plans of an "ideal middle-level mathematics classroom." The 41 middle-level mathematics preservice teachers created two floor plans, one at the beginning of the semester and the other for the…

  17. Accurate metacognition for visual sensory memory representations.

    PubMed

    Vandenbroucke, Annelinde R E; Sligte, Ilja G; Barrett, Adam B; Seth, Anil K; Fahrenfort, Johannes J; Lamme, Victor A F

    2014-04-01

    The capacity to attend to multiple objects in the visual field is limited. However, introspectively, people feel that they see the whole visual world at once. Some scholars suggest that this introspective feeling is based on short-lived sensory memory representations, whereas others argue that the feeling of seeing more than can be attended to is illusory. Here, we investigated this phenomenon by combining objective memory performance with subjective confidence ratings during a change-detection task. This allowed us to compute a measure of metacognition--the degree of knowledge that subjects have about the correctness of their decisions--for different stages of memory. We show that subjects store more objects in sensory memory than they can attend to but, at the same time, have similar metacognition for sensory memory and working memory representations. This suggests that these subjective impressions are not an illusion but accurate reflections of the richness of visual perception.

  18. Social Representations as Mediators of Practice in Mathematics Classrooms with Immigrant Students

    ERIC Educational Resources Information Center

    Gorgorio, Nuria; de Abreu, Guida

    2009-01-01

    This article suggests that a critical perspective of the notion of social representations can offer useful insights into understanding practices of teaching and learning in mathematics classrooms with immigrant students. Drawing on literature using social representations, previous empirical studies are revisited to examine three specific…

  19. Public Conceptions of Algorithms and Representations in the Common Core State Standards for Mathematics

    ERIC Educational Resources Information Center

    Nanna, Robert J.

    2016-01-01

    Algorithms and representations have been an important aspect of the work of mathematics, especially for understanding concepts and communicating ideas about concepts and mathematical relationships. They have played a key role in various mathematics standards documents, including the Common Core State Standards for Mathematics. However, there have…

  20. The enhancement of students’ mathematical representation in junior high school using cognitive apprenticeship instruction (CAI)

    NASA Astrophysics Data System (ADS)

    Yusepa, B. G. P.; Kusumah, Y. S.; Kartasasmita, B. G.

    2018-03-01

    This study aims to get an in-depth understanding of the enhancement of students’ mathematical representation. This study is experimental research with pretest-posttest control group design. The subject of this study is the students’ of the eighth grade from junior high schools in Bandung: high-level and middle-level. In each school, two parallel groups were chosen as a control group and an experimental group. The experimental group was given cognitive apprenticeship instruction (CAI) treatment while the control group was given conventional learning. The results show that the enhancement of students’ mathematical representation who obtained CAI treatment was better than the conventional one, viewed which can be observed from the overall, mathematical prior knowledge (MPK), and school level. It can be concluded that CAI can be used as a good alternative learning model to enhance students’ mathematical representation.

  1. Multi-representation ability of students on the problem solving physics

    NASA Astrophysics Data System (ADS)

    Theasy, Y.; Wiyanto; Sujarwata

    2018-03-01

    Accuracy in representing knowledge possessed by students will show how the level of student understanding. The multi-representation ability of students on the problem solving of physics has been done through qualitative method of grounded theory model and implemented on physics education student of Unnes academic year 2016/2017. Multiforms of representation used are verbal (V), images/diagrams (D), graph (G), and mathematically (M). High and low category students have an accurate use of graphical representation (G) of 83% and 77.78%, and medium category has accurate use of image representation (D) equal to 66%.

  2. Social Representations of High School Students about Mathematics Assessment

    ERIC Educational Resources Information Center

    Martínez-Sierra, Gustavo; Valle-Zequeida, María E.; Miranda-Tirado, Marisa; Dolores-Flores, Crisólogo

    2016-01-01

    The perceptions of students about assessment in mathematics classes have been sparsely investigated. In order to fill this gap, this qualitative study aims to identify the social "representations" (understood as the system of values, ideas, and practices about a social object) of high school students regarding "assessment in…

  3. How Young Children View Mathematical Representations: A Study Using Eye-Tracking Technology

    ERIC Educational Resources Information Center

    Bolden, David; Barmby, Patrick; Raine, Stephanie; Gardner, Matthew

    2015-01-01

    Background: It has been shown that mathematical representations can aid children's understanding of mathematical concepts but that children can sometimes have difficulty in interpreting them correctly. New advances in eye-tracking technology can help in this respect because it allows data to be gathered concerning children's focus of attention and…

  4. The unique and shared contributions of arithmetic operation understanding and numerical magnitude representation to children's mathematics achievement.

    PubMed

    Wong, Terry Tin-Yau

    2017-12-01

    The current study examined the unique and shared contributions of arithmetic operation understanding and numerical magnitude representation to children's mathematics achievement. A sample of 124 fourth graders was tested on their arithmetic operation understanding (as reflected by their understanding of arithmetic principles and the knowledge about the application of arithmetic operations) and their precision of rational number magnitude representation. They were also tested on their mathematics achievement and arithmetic computation performance as well as the potential confounding factors. The findings suggested that both arithmetic operation understanding and numerical magnitude representation uniquely predicted children's mathematics achievement. The findings highlight the significance of arithmetic operation understanding in mathematics learning. Copyright © 2017 Elsevier Inc. All rights reserved.

  5. Mathematics Teacher-Candidates' Performance in Solving Problems with Different Representation Styles: The Trigonometry Example

    ERIC Educational Resources Information Center

    Dündar, Sefa

    2015-01-01

    Using multiple representations of a problem can reveal the relationship between complex concepts by expressing the same mathematical condition differently and can contribute to the meaningful learning of mathematical concepts. The purpose of this study is to assess the performances of mathematics teacher-candidates on trigonometry problems…

  6. Relations of Different Types of Numerical Magnitude Representations to Each Other and to Mathematics Achievement

    ERIC Educational Resources Information Center

    Fazio, Lisa K.; Bailey, Drew H.; Thompson, Clarissa A.; Siegler, Robert S.

    2014-01-01

    We examined relations between symbolic and non-symbolic numerical magnitude representations, between whole number and fraction representations, and between these representations and overall mathematics achievement in fifth graders. Fraction and whole number symbolic and non-symbolic numerical magnitude understandings were measured using both…

  7. The application of brain-based learning principles aided by GeoGebra to improve mathematical representation ability

    NASA Astrophysics Data System (ADS)

    Priatna, Nanang

    2017-08-01

    The use of Information and Communication Technology (ICT) in mathematics instruction will help students in building conceptual understanding. One of the software products used in mathematics instruction is GeoGebra. The program enables simple visualization of complex geometric concepts and helps improve students' understanding of geometric concepts. Instruction applying brain-based learning principles is one oriented at the efforts of naturally empowering the brain potentials which enable students to build their own knowledge. One of the goals of mathematics instruction in school is to develop mathematical communication ability. Mathematical representation is regarded as a part of mathematical communication. It is a description, expression, symbolization, or modeling of mathematical ideas/concepts as an attempt of clarifying meanings or seeking for solutions to the problems encountered by students. The research aims to develop a learning model and teaching materials by applying the principles of brain-based learning aided by GeoGebra to improve junior high school students' mathematical representation ability. It adopted a quasi-experimental method with the non-randomized control group pretest-posttest design and the 2x3 factorial model. Based on analysis of the data, it is found that the increase in the mathematical representation ability of students who were treated with mathematics instruction applying the brain-based learning principles aided by GeoGebra was greater than the increase of the students given conventional instruction, both as a whole and based on the categories of students' initial mathematical ability.

  8. Mathematical representation of joint time-chroma distributions

    NASA Astrophysics Data System (ADS)

    Wakefield, Gregory H.

    1999-11-01

    Originally coined by the sensory psychologist Roger Shepard in the 1960s, chroma transforms frequency into octave equivalence classes. By extending the concept of chroma to chroma strength and how it varies over time, we have demonstrated the utility of chroma in simplifying the processing and representation of signals dominated by harmonically-related narrowband components. These investigations have utilized an ad hoc procedure for calculating the chromagram from a given time-frequency distribution. The present paper is intended to put this ad hoc procedure on more sound mathematical ground.

  9. Investigating the Representational Fluency of Pre-Service Mathematics Teachers in a Modelling Process

    ERIC Educational Resources Information Center

    Delice, Ali; Kertil, Mahmut

    2015-01-01

    This article reports the results of a study that investigated pre-service mathematics teachers' modelling processes in terms of representational fluency in a modelling activity related to a cassette player. A qualitative approach was used in the data collection process. Students' individual and group written responses to the mathematical modelling…

  10. Gender Representation on Journal Editorial Boards in the Mathematical Sciences.

    PubMed

    Topaz, Chad M; Sen, Shilad

    2016-01-01

    We study gender representation on the editorial boards of 435 journals in the mathematical sciences. Women are known to comprise approximately 15% of tenure-stream faculty positions in doctoral-granting mathematical sciences departments in the United States. Compared to this group, we find that 8.9% of the 13067 editorships in our study are held by women. We describe group variations within the editorships by identifying specific journals, subfields, publishers, and countries that significantly exceed or fall short of this average. To enable our study, we develop a semi-automated method for inferring gender that has an estimated accuracy of 97.5%. Our findings provide the first measure of gender distribution on editorial boards in the mathematical sciences, offer insights that suggest future studies in the mathematical sciences, and introduce new methods that enable large-scale studies of gender distribution in other fields.

  11. Gender Representation on Journal Editorial Boards in the Mathematical Sciences

    PubMed Central

    2016-01-01

    We study gender representation on the editorial boards of 435 journals in the mathematical sciences. Women are known to comprise approximately 15% of tenure-stream faculty positions in doctoral-granting mathematical sciences departments in the United States. Compared to this group, we find that 8.9% of the 13067 editorships in our study are held by women. We describe group variations within the editorships by identifying specific journals, subfields, publishers, and countries that significantly exceed or fall short of this average. To enable our study, we develop a semi-automated method for inferring gender that has an estimated accuracy of 97.5%. Our findings provide the first measure of gender distribution on editorial boards in the mathematical sciences, offer insights that suggest future studies in the mathematical sciences, and introduce new methods that enable large-scale studies of gender distribution in other fields. PMID:27536970

  12. The Role of Visual Representations for Structuring Classroom Mathematical Activity

    ERIC Educational Resources Information Center

    David, Maria Manuela; Tomaz, Vanessa Sena

    2012-01-01

    It is our presupposition that there is still a need for more research about how classroom practices can exploit the use and power of visualization in mathematics education. The aim of this article is to contribute in this direction, investigating how visual representations can structure geometry activity in the classroom and discussing teaching…

  13. From Number Lines to Graphs in the Coordinate Plane: Investigating Problem Solving across Mathematical Representations

    ERIC Educational Resources Information Center

    Earnest, Darrell

    2015-01-01

    This article reports on students' problem-solving approaches across three representations--number lines, coordinate planes, and function graphs--the axes of which conventional mathematics treats in terms of consistent geometric and numeric coordinations. I consider these representations to be a part of a "hierarchical representational…

  14. Analysis of Student Understanding of Science Concepts Including Mathematical Representations: Ph Values and the Relative Differences of pH Values

    ERIC Educational Resources Information Center

    Park, Eun-Jung; Choi, Kyunghee

    2013-01-01

    In general, mathematical representations such as formulae, numbers, and graphs are the inseparable components in science used to better describe or explain scientific phenomena or knowledge. Regardless of their necessity and benefit, science seems to be difficult for some students, as a result of the mathematical representations and problem…

  15. Improving of Junior High School Visual Thinking Representation Ability in Mathematical Problem Solving by CTL

    ERIC Educational Resources Information Center

    Surya, Edy; Sabandar, Jozua; Kusumah, Yaya S.; Darhim

    2013-01-01

    The students' difficulty which was found is in the problem of understanding, drawing diagrams, reading the charts correctly, conceptual formal mathematical understanding, and mathematical problem solving. The appropriate problem representation is the basic way in order to understand the problem itself and make a plan to solve it. This research was…

  16. Reading Visual Representations

    ERIC Educational Resources Information Center

    Rubenstein, Rheta N.; Thompson, Denisse R.

    2012-01-01

    Mathematics is rich in visual representations. Such visual representations are the means by which mathematical patterns "are recorded and analyzed." With respect to "vocabulary" and "symbols," numerous educators have focused on issues inherent in the language of mathematics that influence students' success with mathematics communication.…

  17. An Accurate Projector Calibration Method Based on Polynomial Distortion Representation

    PubMed Central

    Liu, Miao; Sun, Changku; Huang, Shujun; Zhang, Zonghua

    2015-01-01

    In structure light measurement systems or 3D printing systems, the errors caused by optical distortion of a digital projector always affect the precision performance and cannot be ignored. Existing methods to calibrate the projection distortion rely on calibration plate and photogrammetry, so the calibration performance is largely affected by the quality of the plate and the imaging system. This paper proposes a new projector calibration approach that makes use of photodiodes to directly detect the light emitted from a digital projector. By analyzing the output sequence of the photoelectric module, the pixel coordinates can be accurately obtained by the curve fitting method. A polynomial distortion representation is employed to reduce the residuals of the traditional distortion representation model. Experimental results and performance evaluation show that the proposed calibration method is able to avoid most of the disadvantages in traditional methods and achieves a higher accuracy. This proposed method is also practically applicable to evaluate the geometric optical performance of other optical projection system. PMID:26492247

  18. Limited knowledge of fraction representations differentiates middle school students with mathematics learning disability (dyscalculia) vs. low mathematics achievement

    PubMed Central

    Mazzocco, Michèle M. M.; Myers, Gwen F.; Lewis, Katherine E.; Hanich, Laurie B.; Murphy, Melissa M.

    2014-01-01

    Fractions pose significant challenges for many children, but for some children those challenges persist into high school. Here we administered a fractions magnitude comparison test to 122 children, from Grades 4 to 8, to test whether their knowledge of fractions typically learned early in the sequence of formal math instruction (e.g., fractions equivalent to “one-half,” and fraction pairs with common denominators) differentiates those with mathematical learning disability (MLD) versus low achievement (LA) or typical achievement (TA) in mathematics, and whether long term learning trajectories of this knowledge also differentiate these groups. We confirmed that although 4th graders with LA (n = 18) or TA (n = 93) are more accurate evaluating one-half vs. non-half fractions (until they reach ceiling performance levels on both types of fractions), children with MLD (n=11) do not show a one-half advantage until Grade 7 and do not reach ceiling performance even by Grade 8. Both the MLD and LA groups have early difficulties with fractions, but by Grade 5 the LA group approaches performance levels of the TA group and deviates from the MLD group. All groups showed a visual model advantage over Arabic number representation of fractions, but this advantage was short lived for the TA group (because ceiling level was achieved across formats), slightly more persistent for the LA group, and persisted through Grade 8 for children with MLD. Thus, difficulties with fractions persist through Grade 8 for many students, but the nature and trajectories of those difficulties varies across children with math difficulties (MLD or LA). PMID:23587941

  19. Refocusing on Oral Language and Rich Representations to Develop the Early Mathematical Understandings of Indigenous Students

    ERIC Educational Resources Information Center

    McDonald, Susan; Warren, Elizabeth; DeVries, Eva

    2011-01-01

    This article examines the nature of oral language and representations used by teachers as they instruct young Indigenous Australian students at the beginning of formal schooling during play-based activities in mathematics. In particular, the use of Standard Australian English (SAE), the mathematical register used, and the interplay with…

  20. Using Representations, Decomposition, and Approximations of Practices to Support Prospective Elementary Mathematics Teachers' Practice of Organizing Discussions

    ERIC Educational Resources Information Center

    Tyminski, Andrew M.; Zambak, V. Serbay; Drake, Corey; Land, Tonia J.

    2014-01-01

    This paper examines a series of instructional activities that provide prospective elementary teachers with an opportunity to engage in one of the more difficult practices to learn within mathematics teaching--organizing a mathematical discussion. Within a mathematics methods course, representations and decomposition of practice built from the Five…

  1. The Mental Representation of Integers: An Abstract-to-Concrete Shift in the Understanding of Mathematical Concepts

    ERIC Educational Resources Information Center

    Varma, Sashank; Schwartz, Daniel L.

    2011-01-01

    Mathematics has a level of structure that transcends untutored intuition. What is the cognitive representation of abstract mathematical concepts that makes them meaningful? We consider this question in the context of the integers, which extend the natural numbers with zero and negative numbers. Participants made greater and lesser judgments of…

  2. Using a Technology-Supported Approach to Preservice Teachers' Multirepresentational Fluency: Unifying Mathematical Concepts and Their Representations

    ERIC Educational Resources Information Center

    McGee, Daniel; Moore-Russo, Deborah

    2015-01-01

    A test project at the University of Puerto Rico in Mayagüez used GeoGebra applets to promote the concept of multirepresentational fluency among high school mathematics preservice teachers. For this study, this fluency was defined as simultaneous awareness of all representations associated with a mathematical concept, as measured by the ability to…

  3. Visual Representation in Mathematics: Special Education Teachers' Knowledge and Emphasis for Instruction

    ERIC Educational Resources Information Center

    van Garderen, Delinda; Scheuermann, Amy; Poch, Apryl; Murray, Mary M.

    2018-01-01

    The use of visual representations (VRs) in mathematics is a strongly recommended practice in special education. Although recommended, little is known about special educators' knowledge of and instructional emphasis about VRs. Therefore, in this study, the authors examined special educators' own knowledge of and their instructional emphasis with…

  4. Fast and accurate grid representations for atom-based docking with partner flexibility.

    PubMed

    de Vries, Sjoerd J; Zacharias, Martin

    2017-06-30

    Macromolecular docking methods can broadly be divided into geometric and atom-based methods. Geometric methods use fast algorithms that operate on simplified, grid-like molecular representations, while atom-based methods are more realistic and flexible, but far less efficient. Here, a hybrid approach of grid-based and atom-based docking is presented, combining precalculated grid potentials with neighbor lists for fast and accurate calculation of atom-based intermolecular energies and forces. The grid representation is compatible with simultaneous multibody docking and can tolerate considerable protein flexibility. When implemented in our docking method ATTRACT, grid-based docking was found to be ∼35x faster. With the OPLSX forcefield instead of the ATTRACT coarse-grained forcefield, the average speed improvement was >100x. Grid-based representations may allow atom-based docking methods to explore large conformational spaces with many degrees of freedom, such as multiple macromolecules including flexibility. This increases the domain of biological problems to which docking methods can be applied. © 2017 Wiley Periodicals, Inc. © 2017 Wiley Periodicals, Inc.

  5. Limited knowledge of fraction representations differentiates middle school students with mathematics learning disability (dyscalculia) versus low mathematics achievement.

    PubMed

    Mazzocco, Michèle M M; Myers, Gwen F; Lewis, Katherine E; Hanich, Laurie B; Murphy, Melissa M

    2013-06-01

    Fractions pose significant challenges for many children, but for some children those challenges persist into high school. Here we administered a fractions magnitude comparison test to 122 children, from Grades 4 to 8, to test whether their knowledge of fractions typically learned early in the sequence of formal math instruction (e.g., fractions equivalent to one-half, fraction pairs with common denominators) differentiates those with mathematics learning disability (MLD) versus low achievement (LA) or typical achievement (TA) in mathematics and whether long-term learning trajectories of this knowledge also differentiate these groups. We confirmed that although fourth graders with TA (n=93) were more accurate in evaluating "one-half" fractions than in evaluating "non-half" fractions (until they reached ceiling performance levels on both types of fractions), children with MLD (n=11) did not show a one-half advantage until Grade 7 and did not reach ceiling performance even by Grade 8. Both the MLD and LA groups had early difficulties with fractions, but by Grade 5 the LA group approached performance levels of the TA group and deviated from the MLD group. All groups showed a visual model advantage over Arabic number representation of fractions, but this advantage was short-lived for the TA group (because ceiling level was achieved across formats), whereas it was slightly more persistent for the LA group and persisted through Grade 8 for children with MLD. Thus, difficulties with fractions persist through Grade 8 for many students, but the nature and trajectories of those difficulties vary across children with math difficulties (MLD or LA). Copyright © 2013 Elsevier Inc. All rights reserved.

  6. Representations of numerical and non-numerical magnitude both contribute to mathematical competence in children.

    PubMed

    Lourenco, Stella F; Bonny, Justin W

    2017-07-01

    A growing body of evidence suggests that non-symbolic representations of number, which humans share with nonhuman animals, are functionally related to uniquely human mathematical thought. Other research suggesting that numerical and non-numerical magnitudes not only share analog format but also form part of a general magnitude system raises questions about whether the non-symbolic basis of mathematical thinking is unique to numerical magnitude. Here we examined this issue in 5- and 6-year-old children using comparison tasks of non-symbolic number arrays and cumulative area as well as standardized tests of math competence. One set of findings revealed that scores on both magnitude comparison tasks were modulated by ratio, consistent with shared analog format. Moreover, scores on these tasks were moderately correlated, suggesting overlap in the precision of numerical and non-numerical magnitudes, as expected under a general magnitude system. Another set of findings revealed that the precision of both types of magnitude contributed shared and unique variance to the same math measures (e.g. calculation and geometry), after accounting for age and verbal competence. These findings argue against an exclusive role for non-symbolic number in supporting early mathematical understanding. Moreover, they suggest that mathematical understanding may be rooted in a general system of magnitude representation that is not specific to numerical magnitude but that also encompasses non-numerical magnitude. © 2016 John Wiley & Sons Ltd.

  7. Research Findings' Impact on the Representation of Proportional Reasoning in Swedish Mathematics Textbooks

    ERIC Educational Resources Information Center

    Ahl, Linda Marie

    2016-01-01

    This article investigates the impact of research findings on the representation of proportional reasoning in two commonly used Swedish mathematics textbook series for grades 7-9. A research-based framework that identifies five learning goals for understanding of proportional reasoning was used to analyse the textbooks. The results brought to…

  8. Developing calculus textbook model that supported with GeoGebra to enhancing students’ mathematical problem solving and mathematical representation

    NASA Astrophysics Data System (ADS)

    Dewi, N. R.; Arini, F. Y.

    2018-03-01

    The main purpose of this research is developing and produces a Calculus textbook model that supported with GeoGebra. This book was designed to enhancing students’ mathematical problem solving and mathematical representation. There were three stages in this research i.e. define, design, and develop. The textbooks consisted of 6 chapters which each chapter contains introduction, core materials and include examples and exercises. The textbook developed phase begins with the early stages of designed the book (draft 1) which then validated by experts. Revision of draft 1 produced draft 2. The data were analyzed with descriptive statistics. The analysis showed that the Calculus textbook model that supported with GeoGebra, valid and fill up the criteria of practicality.

  9. Prospective Teachers' Representations for Teaching Note Values: An Analysis in the Context of Mathematics and Music

    ERIC Educational Resources Information Center

    Özgül, Ilhan; Incikabi, Lütfi

    2017-01-01

    In this study, the representations preferred by prospective teachers in the teaching of note values were determined and the accuracy of these representations was analyzed in the context of mathematics and music. The case study, one of the qualitative research designs, was used in the study. Study group of the research consisted of 113 pre-school…

  10. Investigating Mathematics Students' Use of Multiple Representations when Solving Linear Equations with One Unknown

    ERIC Educational Resources Information Center

    Beyranevand, Matthew L.

    2010-01-01

    Although it is difficult to find any current literature that does not encourage use of multiple representations in mathematics classrooms, there has been very limited research that compared such practice to student achievement level on standardized tests. This study examined the associations between students' achievement levels and their (a)…

  11. Representations of Numerical and Non-Numerical Magnitude Both Contribute to Mathematical Competence in Children

    ERIC Educational Resources Information Center

    Lourenco, Stella F.; Bonny, Justin W.

    2017-01-01

    A growing body of evidence suggests that non-symbolic representations of number, which humans share with nonhuman animals, are functionally related to uniquely human mathematical thought. Other research suggesting that numerical and non-numerical magnitudes not only share analog format but also form part of a general magnitude system raises…

  12. Using Integer Manipulatives: Representational Determinism

    ERIC Educational Resources Information Center

    Bossé, Michael J.; Lynch-Davis, Kathleen; Adu-Gyamfi, Kwaku; Chandler, Kayla

    2016-01-01

    Teachers and students commonly use various concrete representations during mathematical instruction. These representations can be utilized to help students understand mathematical concepts and processes, increase flexibility of thinking, facilitate problem solving, and reduce anxiety while doing mathematics. Unfortunately, the manner in which some…

  13. Promoting Decimal Number Sense and Representational Fluency

    ERIC Educational Resources Information Center

    Suh, Jennifer M.; Johnston, Chris; Jamieson, Spencer; Mills, Michelle

    2008-01-01

    The abstract nature of mathematics requires the communication of mathematical ideas through multiple representations, such as words, symbols, pictures, objects, or actions. Building representational fluency involves using mathematical representations flexibly and being able to interpret and translate among these different models and mathematical…

  14. Representations and Rafts

    ERIC Educational Resources Information Center

    Hartweg, Kimberly Sipes

    2011-01-01

    To build on prior knowledge and mathematical understanding, middle school students need to be given the opportunity to make connections among a variety of representations. Graphs, tables, algebraic formulas, and models are just a few examples of representations that can help students explore quantitative relationships. As a mathematics educator,…

  15. Strategies to Increase Representation of Students with Disabilities in Science, Technology, Engineering and Mathematics (STEM)

    ERIC Educational Resources Information Center

    White, Jeffry L.; Massiha, G. H.

    2015-01-01

    As a nation wrestles with the need to train more professionals, persons with disabilities are undereducated and underrepresented in science, technology, engineering, and mathematics (STEM). The following project was proposed to increase representation of students with disabilities in the STEM disciplines. The program emphasizes an integrated…

  16. The Representational Value of Hats

    ERIC Educational Resources Information Center

    Watson, Jane M.; Fitzallen, Noleine E.; Wilson, Karen G.; Creed, Julie F.

    2008-01-01

    The literature that is available on the topic of representations in mathematics is vast. One commonly discussed item is graphical representations. From the history of mathematics to modern uses of technology, a variety of graphical forms are available for middle school students to use to represent mathematical ideas. The ideas range from algebraic…

  17. Mathematization of experience in a grade 8 open-inquiry environment: An introduction to the representational practices of science

    NASA Astrophysics Data System (ADS)

    Roth, Wolff-Michael; Bowen, G. Michael

    The purpose of this classroom study was to investigate the use of mathematical representations in three Grade 8 general science classes that engaged in a 10-week open inquiry about the correlations between biological and physical variables in the environment. A constructivist perspective was used to design the study and to assemble the data sources. These data sources included videotapes of students in their work, audiotapes of teacher-student interactions and teacher interviews, the transcripts of these tapes, the students' field notebooks, field reports, special problem assignments, examinations, and the teachers' curriculum guides, field notes, and reflective journal. An interpretive method was used to construct assertions and the supporting data. In the setting provided, students increasingly used mathematical representations such as graphs and data tables to support their claims in a convincing manner; the use of abstract equations and percent calculations did not change over the course of the study. Representations such as graphs, maps, averages, and equations were not only useful as inscriptions (representations in some permanent medium, usually paper), but also as conscription devices in the construction of, and through which, students engaged each other to collaboratively construct meaning. This study demonstrates the use of representations as conscription devices, and illustrates how the use and understanding of inscriptions changes over time. Understanding representations as inscription and conscription devices focuses on the social aspects of knowing, which has important implications for teachers' conceptualization of learning and their organization of science classrooms.

  18. An Investigation of Pattern Problems Posed by Middle School Mathematics Preservice Teachers Using Multiple Representation

    ERIC Educational Resources Information Center

    Yilmaz, Yasemin; Durmus, Soner; Yaman, Hakan

    2018-01-01

    This study investigated the pattern problems posed by middle school mathematics preservice teachers using multiple representations to determine both their pattern knowledge levels and their abilities to transfer this knowledge to students. The design of the study is the survey method, one of the quantitative research methods. The study group was…

  19. Brain Activity Associated with Translation between Graphical and Symbolic Representations of Functions in Generally Gifted and Excelling in Mathematics Adolescents

    ERIC Educational Resources Information Center

    Waisman, Ilana; Leikin, Mark; Shaul, Shelley; Leikin, Roza

    2014-01-01

    In this study, we examine the impact and the interplay of general giftedness (G) and excellence in mathematics (EM) on high school students' mathematical performance associated with translations from graphical to symbolic representations of functions, as reflected in cortical electrical activity (by means of ERP--event-related…

  20. Pupils' Visual Representations in Standard and Problematic Problem Solving in Mathematics: Their Role in the Breach of the Didactical Contract

    ERIC Educational Resources Information Center

    Deliyianni, Eleni; Monoyiou, Annita; Elia, Iliada; Georgiou, Chryso; Zannettou, Eleni

    2009-01-01

    This study investigated the modes of representations generated by kindergarteners and first graders while solving standard and problematic problems in mathematics. Furthermore, it examined the influence of pupils' visual representations on the breach of the didactical contract rules in problem solving. The sample of the study consisted of 38…

  1. Generalized zeta function representation of groups and 2-dimensional topological Yang-Mills theory: The example of GL(2, #Mathematical Double-Struck Capital F#{sub q}) and PGL(2, #Mathematical Double-Struck Capital F#{sub q})

    DOE Office of Scientific and Technical Information (OSTI.GOV)

    Roche, Ph., E-mail: philippe.roche@univ-montp2.fr

    We recall the relation between zeta function representation of groups and two-dimensional topological Yang-Mills theory through Mednikh formula. We prove various generalisations of Mednikh formulas and define generalization of zeta function representations of groups. We compute some of these functions in the case of the finite group GL(2, #Mathematical Double-Struck Capital F#{sub q}) and PGL(2, #Mathematical Double-Struck Capital F#{sub q}). We recall the table characters of these groups for any q, compute the Frobenius-Schur indicator of their irreducible representations, and give the explicit structure of their fusion rings.

  2. Examining Fourth-Grade Mathematics Writing: Features of Organization, Mathematics Vocabulary, and Mathematical Representations

    ERIC Educational Resources Information Center

    Hebert, Michael A.; Powell, Sarah R.

    2016-01-01

    Increasingly, students are expected to write about mathematics. Mathematics writing may be informal (e.g., journals, exit slips) or formal (e.g., writing prompts on high-stakes mathematics assessments). In order to develop an effective mathematics-writing intervention, research needs to be conducted on how students organize mathematics writing and…

  3. Les Representations Graphiques Dans La Resolution De Problemes: Une Experience D'Entrainement D'Etudiants Dans Un Club Mathematique (Graphic Representations in Problem Solving: A Training Program for Students in a Mathematical Club).

    ERIC Educational Resources Information Center

    Callejo, Maria Luz

    1994-01-01

    Reports, in French, an investigation on the use of graphic representations in problem-solving tasks of the type in Spanish Mathematical Olympiads. Analysis showed that the choice and interpretation of the first graphic representation played a decisive role in the discovery of the solution. (34 references) (Author/MKR)

  4. Mapping of the Academic Production at Science and Mathematics Education Postgraduate about the Theory of Social Representations

    NASA Astrophysics Data System (ADS)

    Barbosa, José Isnaldo de Lima; Curi, Edda; Voelzke, Marcos Rincon

    2016-12-01

    The theory of social representations, appeared in 1961, arrived in Brazil in 1982, and since then has advanced significantly, been used in various areas of knowledge, assumed a significant role also in education. Thus, the aim of this article is to make a mapping of theses and dissertations in post-graduation programs, whose basic area is the Teaching of Science and Mathematics, and used as the theoretical foundation the theory of social representations, highlighted the social groups that are subject of this research. This is a documentary research, and lifting to the "state of knowledge" of two theses and 36 dissertations, defended in ten of the 37 existing programs in the basic area of Science and Mathematics Teaching, with the delimitation of academic masters and doctorates. The data collection was executed on December 2014 and was placed in the virtual libraries of these masters and doctoral programs, these elements were analysed according to some categories established after reading the summaries of the work, and the results showed that the theory of social representations has been used as a theoretical framework in various research groups, established in postgraduate programs in this area, for almost the entire Brazil. As for the subjects involved in this research, three groups were detected, which are: Middle school and high school students, teachers who are in full swing, spread from the early years to higher education, and undergraduates in Science and Mathematics.

  5. Objects, Signs, and Representations in the Semio-Cognitive Analysis of the Processes Involved in Teaching and Learning Mathematics: A Duvalian Perspective

    ERIC Educational Resources Information Center

    Iori, Maura

    2017-01-01

    In mathematical activities and in the analysis of mathematics teaching-learning processes, "objects," "signs", and "representations" are often mentioned, where the meaning assigned to those words is sometimes very broad, sometimes limited, other times intuitive, allusive, or not completely clear. On the other hand, as…

  6. VStops: A Thinking Strategy and Visual Representation Approach in Mathematical Word Problem Solving toward Enhancing STEM Literacy

    ERIC Educational Resources Information Center

    Abdullah, Nasarudin; Halim, Lilia; Zakaria, Effandi

    2014-01-01

    This study aimed to determine the impact of strategic thinking and visual representation approaches (VStops) on the achievement, conceptual knowledge, metacognitive awareness, awareness of problem-solving strategies, and student attitudes toward mathematical word problem solving among primary school students. The experimental group (N = 96)…

  7. The role of physical digit representation and numerical magnitude representation in children's multiplication fact retrieval.

    PubMed

    De Visscher, Alice; Noël, Marie-Pascale; De Smedt, Bert

    2016-12-01

    Arithmetic facts, in particular multiplication tables, are thought to be stored in long-term memory and to be interference prone. At least two representations underpinning these arithmetic facts have been suggested: a physical representation of the digits and a numerical magnitude representation. We hypothesized that both representations are possible sources of interference that could explain individual differences in multiplication fact performance and/or in strategy use. We investigated the specificity of these interferences on arithmetic fact retrieval and explored the relation between interference and performance on the different arithmetic operations and on general mathematics achievement. Participants were 79 fourth-grade children (M age =9.6 years) who completed a products comparison and a multiplication production task with verbal strategy reports. Performances on a speeded calculation test including the four operations and on a general mathematics achievement test were also collected. Only the interference coming from physical representations was a significant predictor of the performance across multiplications. However, both the magnitude and physical representations were unique predictors of individual differences in multiplication. The frequency of the retrieval strategy across multiplication problems and across individuals was determined only by the physical representation, which therefore is suggested as being responsible for memory storage issues. Interestingly, this impact of physical representation was not observed when predicting performance on subtraction or on general mathematical achievement. In contrast, the impact of the numerical magnitude representation was more general in that it was observed across all arithmetic operations and in general mathematics achievement. Copyright © 2016 Elsevier Inc. All rights reserved.

  8. Dynamic Boolean Mathematics

    ERIC Educational Resources Information Center

    Bossé, Michael J.; Adu-Gyamfi, Kwaku; Chandler, Kayla; Lynch-Davis, Kathleen

    2016-01-01

    Dynamic mathematical environments allow users to reify mathematical concepts through multiple representations, transform mathematical relations and organically explore mathematical properties, investigate integrated mathematics, and develop conceptual understanding. Herein, we integrate Boolean algebra, the functionalities of a dynamic…

  9. Preservice Secondary Mathematics Teachers' Development of Mathematical Knowledge for Teaching and Their Use of Knowledge in Their Instruction

    ERIC Educational Resources Information Center

    Moon, Kyunghee

    2013-01-01

    This study examined how preservice secondary mathematics teachers developed mathematical knowledge for teaching (MKT) around representations and big ideas through mathematics and mathematics education courses. The importance of big ideas and representations in mathematics has been emphasized in national standards as well as in literature. Yet,…

  10. Evidence-Based Practices: Applications of Concrete Representational Abstract Framework across Math Concepts for Students with Mathematics Disabilities

    ERIC Educational Resources Information Center

    Agrawal, Jugnu; Morin, Lisa L.

    2016-01-01

    Students with mathematics disabilities (MD) experience difficulties with both conceptual and procedural knowledge of different math concepts across grade levels. Research shows that concrete representational abstract framework of instruction helps to bridge this gap for students with MD. In this article, we provide an overview of this strategy…

  11. An accurate, compact and computationally efficient representation of orbitals for quantum Monte Carlo calculations

    NASA Astrophysics Data System (ADS)

    Luo, Ye; Esler, Kenneth; Kent, Paul; Shulenburger, Luke

    Quantum Monte Carlo (QMC) calculations of giant molecules, surface and defect properties of solids have been feasible recently due to drastically expanding computational resources. However, with the most computationally efficient basis set, B-splines, these calculations are severely restricted by the memory capacity of compute nodes. The B-spline coefficients are shared on a node but not distributed among nodes, to ensure fast evaluation. A hybrid representation which incorporates atomic orbitals near the ions and B-spline ones in the interstitial regions offers a more accurate and less memory demanding description of the orbitals because they are naturally more atomic like near ions and much smoother in between, thus allowing coarser B-spline grids. We will demonstrate the advantage of hybrid representation over pure B-spline and Gaussian basis sets and also show significant speed-up like computing the non-local pseudopotentials with our new scheme. Moreover, we discuss a new algorithm for atomic orbital initialization which used to require an extra workflow step taking a few days. With this work, the highly efficient hybrid representation paves the way to simulate large size even in-homogeneous systems using QMC. This work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Computational Materials Sciences Program.

  12. The Effect of Dynamic and Interactive Mathematics Learning Environments (DIMLE), Supporting Multiple Representations, on Perceptions of Elementary Mathematics Pre-Service Teachers in Problem Solving Process

    ERIC Educational Resources Information Center

    Ozdemir, S.; Reis, Z. Ayvaz

    2013-01-01

    Mathematics is an important discipline, providing crucial tools, such as problem solving, to improve our cognitive abilities. In order to solve a problem, it is better to envision and represent through multiple means. Multiple representations can help a person to redefine a problem with his/her own words in that envisioning process. Dynamic and…

  13. Simple Mathematical Models Do Not Accurately Predict Early SIV Dynamics

    PubMed Central

    Noecker, Cecilia; Schaefer, Krista; Zaccheo, Kelly; Yang, Yiding; Day, Judy; Ganusov, Vitaly V.

    2015-01-01

    Upon infection of a new host, human immunodeficiency virus (HIV) replicates in the mucosal tissues and is generally undetectable in circulation for 1–2 weeks post-infection. Several interventions against HIV including vaccines and antiretroviral prophylaxis target virus replication at this earliest stage of infection. Mathematical models have been used to understand how HIV spreads from mucosal tissues systemically and what impact vaccination and/or antiretroviral prophylaxis has on viral eradication. Because predictions of such models have been rarely compared to experimental data, it remains unclear which processes included in these models are critical for predicting early HIV dynamics. Here we modified the “standard” mathematical model of HIV infection to include two populations of infected cells: cells that are actively producing the virus and cells that are transitioning into virus production mode. We evaluated the effects of several poorly known parameters on infection outcomes in this model and compared model predictions to experimental data on infection of non-human primates with variable doses of simian immunodifficiency virus (SIV). First, we found that the mode of virus production by infected cells (budding vs. bursting) has a minimal impact on the early virus dynamics for a wide range of model parameters, as long as the parameters are constrained to provide the observed rate of SIV load increase in the blood of infected animals. Interestingly and in contrast with previous results, we found that the bursting mode of virus production generally results in a higher probability of viral extinction than the budding mode of virus production. Second, this mathematical model was not able to accurately describe the change in experimentally determined probability of host infection with increasing viral doses. Third and finally, the model was also unable to accurately explain the decline in the time to virus detection with increasing viral dose. These results

  14. Science modelling in pre-calculus: how to make mathematics problems contextually meaningful

    NASA Astrophysics Data System (ADS)

    Sokolowski, Andrzej; Yalvac, Bugrahan; Loving, Cathleen

    2011-04-01

    'Use of mathematical representations to model and interpret physical phenomena and solve problems is one of the major teaching objectives in high school math curriculum' (National Council of Teachers of Mathematics (NCTM), Principles and Standards for School Mathematics, NCTM, Reston, VA, 2000). Commonly used pre-calculus textbooks provide a wide range of application problems. However, these problems focus students' attention on evaluating or solving pre-arranged formulas for given values. The role of scientific content is reduced to provide a background for these problems instead of being sources of data gathering for inducing mathematical tools. Students are neither required to construct mathematical models based on the contexts nor are they asked to validate or discuss the limitations of applied formulas. Using these contexts, the instructor may think that he/she is teaching problem solving, where in reality he/she is teaching algorithms of the mathematical operations (G. Kulm (ed.), New directions for mathematics assessment, in Assessing Higher Order Thinking in Mathematics, Erlbaum, Hillsdale, NJ, 1994, pp. 221-240). Without a thorough representation of the physical phenomena and the mathematical modelling processes undertaken, problem solving unintentionally appears as simple algorithmic operations. In this article, we deconstruct the representations of mathematics problems from selected pre-calculus textbooks and explicate their limitations. We argue that the structure and content of those problems limits students' coherent understanding of mathematical modelling, and this could result in weak student problem-solving skills. Simultaneously, we explore the ways to enhance representations of those mathematical problems, which we have characterized as lacking a meaningful physical context and limiting coherent student understanding. In light of our discussion, we recommend an alternative to strengthen the process of teaching mathematical modelling - utilization

  15. A Review of the Effects of Visual-Spatial Representations and Heuristics on Word Problem Solving in Middle School Mathematics

    ERIC Educational Resources Information Center

    Kribbs, Elizabeth E.; Rogowsky, Beth A.

    2016-01-01

    Mathematics word-problems continue to be an insurmountable challenge for many middle school students. Educators have used pictorial and schematic illustrations within the classroom to help students visualize these problems. However, the data shows that pictorial representations can be more harmful than helpful in that they only display objects or…

  16. Multi-representation based on scientific investigation for enhancing students’ representation skills

    NASA Astrophysics Data System (ADS)

    Siswanto, J.; Susantini, E.; Jatmiko, B.

    2018-03-01

    This research aims to implementation learning physics with multi-representation based on the scientific investigation for enhancing students’ representation skills, especially on the magnetic field subject. The research design is one group pretest-posttest. This research was conducted in the department of mathematics education, Universitas PGRI Semarang, with the sample is students of class 2F who take basic physics courses. The data were obtained by representation skills test and documentation of multi-representation worksheet. The Results show gain analysis value of .64 which means some medium improvements. The result of t-test (α = .05) is shows p-value = .001. This learning significantly improves students representation skills.

  17. Representing the Electromagnetic Field: How Maxwell's Mathematics Empowered Faraday's Field Theory

    NASA Astrophysics Data System (ADS)

    Tweney, Ryan D.

    2011-07-01

    James Clerk Maxwell `translated' Michael Faraday's experimentally-based field theory into the mathematical representation now known as `Maxwell's Equations.' Working with a variety of mathematical representations and physical models Maxwell extended the reach of Faraday's theory and brought it into consistency with other results in the physics of electricity and magnetism. Examination of Maxwell's procedures opens many issues about the role of mathematical representation in physics and the learning background required for its success. Specifically, Maxwell's training in `Cambridge University' mathematical physics emphasized the use of analogous equations across fields of physics and the repeated solving of extremely difficult problems in physics. Such training develops an array of overlearned mathematical representations supported by highly sophisticated cognitive mechanisms for the retrieval of relevant information from long term memory. For Maxwell, mathematics constituted a new form of representation in physics, enhancing the formal derivational and calculational role of mathematics and opening a cognitive means for the conduct of `experiments in the mind' and for sophisticated representations of theory.

  18. FragBag, an accurate representation of protein structure, retrieves structural neighbors from the entire PDB quickly and accurately.

    PubMed

    Budowski-Tal, Inbal; Nov, Yuval; Kolodny, Rachel

    2010-02-23

    Fast identification of protein structures that are similar to a specified query structure in the entire Protein Data Bank (PDB) is fundamental in structure and function prediction. We present FragBag: An ultrafast and accurate method for comparing protein structures. We describe a protein structure by the collection of its overlapping short contiguous backbone segments, and discretize this set using a library of fragments. Then, we succinctly represent the protein as a "bags-of-fragments"-a vector that counts the number of occurrences of each fragment-and measure the similarity between two structures by the similarity between their vectors. Our representation has two additional benefits: (i) it can be used to construct an inverted index, for implementing a fast structural search engine of the entire PDB, and (ii) one can specify a structure as a collection of substructures, without combining them into a single structure; this is valuable for structure prediction, when there are reliable predictions only of parts of the protein. We use receiver operating characteristic curve analysis to quantify the success of FragBag in identifying neighbor candidate sets in a dataset of over 2,900 structures. The gold standard is the set of neighbors found by six state of the art structural aligners. Our best FragBag library finds more accurate candidate sets than the three other filter methods: The SGM, PRIDE, and a method by Zotenko et al. More interestingly, FragBag performs on a par with the computationally expensive, yet highly trusted structural aligners STRUCTAL and CE.

  19. Helping Children Learn Mathematics through Multiple Intelligences and Standards for School Mathematics.

    ERIC Educational Resources Information Center

    Adams, Thomasenia Lott

    2001-01-01

    Focuses on the National Council of Teachers of Mathematics 2000 process-oriented standards of problem solving, reasoning and proof, communication, connections, and representation as providing a framework for using the multiple intelligences that children bring to mathematics learning. Presents ideas for mathematics lessons and activities to…

  20. Inscriptions Becoming Representations in Representational Practices

    ERIC Educational Resources Information Center

    Medina, Richard; Suthers, Daniel

    2013-01-01

    We analyze the interaction of 3 students working on mathematics problems over several days in a virtual math team. Our analysis traces out how successful collaboration in a later session is contingent upon the work of prior sessions and shows how the development of representational practices is an important aspect of these participants' problem…

  1. What is rate? Does context or representation matter?

    NASA Astrophysics Data System (ADS)

    Herbert, Sandra; Pierce, Robyn

    2011-12-01

    Rate is an important, but difficult, mathematical concept. Despite more than 20 years of research, especially with calculus students, difficulties are reported with this concept. This paper reports the results from analysis of data from 20 Australian Grade 10 students. Interviews targeted students' conceptions of rate, focussing on the influence of representation and context on their expression of their understanding of rate. This analysis shows that different representations of functions provide varying levels of rate-related information for individual students. Understandings of rate in one representation or context are not necessarily transferred to another representation or context. Rate is an important, but commonly misunderstood, mathematical concept with many everyday applications (Swedosh, Dowsey, Caruso, Flynn, & Tynan, 2007). It is a complicated concept comprising many interwoven ideas such as the ratio of two numeric, measurable quantities but in a context where both quantities are changing. In mathematics classes, this is commonly expressed as change in the dependent variable resulting from a unit change in the independent variable, and variously described as constant or variable rate; average or instantaneous rate. In addition, rate may be seen as a purely abstract mathematical notion or embedded in the understanding of real-world applications. This paper explores the research question: Are students' expressions of their conceptions of rate affected by either context or mathematical representation? This question was part of a larger study (Herbert, 2010) conducted with Grade 10 students from the Australian state of Victoria.

  2. Symbolic Representation of Probabilistic Worlds

    ERIC Educational Resources Information Center

    Feldman, Jacob

    2012-01-01

    Symbolic representation of environmental variables is a ubiquitous and often debated component of cognitive science. Yet notwithstanding centuries of philosophical discussion, the efficacy, scope, and validity of such representation has rarely been given direct consideration from a mathematical point of view. This paper introduces a quantitative…

  3. The Binary Representation of Rational Numbers.

    ERIC Educational Resources Information Center

    Schmalz, Rosemary

    1987-01-01

    Presented are the mathematical explanation of the algorithm for representing rational numbers in base two, paper-and-pencil methods for producing the representation, some patterns in these representations, and pseudocode for computer programs to explore these patterns. (MNS)

  4. Mathematical Explorations: Freshwater Scarcity: A Proportional Representation

    ERIC Educational Resources Information Center

    King, Alessandra

    2014-01-01

    Middle school students' mathematical understanding benefits from connecting mathematics to other content areas in the curriculum. This month's activity explores the issue of the scarcity of freshwater, a natural resource (activity sheets are included). This activity concentrates on the critical areas mentioned in the Common Core State…

  5. Student difficulties in translating between mathematical and graphical representations in introductory physics

    NASA Astrophysics Data System (ADS)

    Lin, Shih-Yin; Maries, Alexandru; Singh, Chandralekha

    2013-01-01

    We investigate introductory physics students' difficulties in translating between mathematical and graphical representations and the effect of scaffolding on students' performance. We gave a typical problem that can be solved using Gauss's law involving a spherically symmetric charge distribution (a conducting sphere concentric with a conducting spherical shell) to 95 calculus-based introductory physics students. We asked students to write a mathematical expression for the electric field in various regions and asked them to graph the electric field. We knew from previous experience that students have great difficulty in graphing the electric field. Therefore, we implemented two scaffolding interventions to help them. Students who received the scaffolding support were either (1) asked to plot the electric field in each region first (before having to plot it as a function of distance from the center of the sphere) or (2) asked to plot the electric field in each region after explicitly evaluating the electric field at the beginning, mid and end points of each region. The comparison group was only asked to plot the electric field at the end of the problem. We found that students benefited the most from intervention (1) and that intervention (2), although intended to aid students, had an adverse effect. Also, recorded interviews were conducted with a few students in order to understand how students were impacted by the aforementioned interventions.

  6. Building a Concrete Foundation: A Mixed-Method Study of Teaching Styles and the Use of Concrete, Representational, and Abstract Mathematics Instruction

    ERIC Educational Resources Information Center

    Thigpen, L. Christine

    2012-01-01

    The purpose of this study was to explore teaching styles and how frequently teachers with a variety of teaching styles incorporate multiple representations, such as manipulatives, drawings, counters, etc., in the middle school mathematics classroom. Through this explanatory mixed methods study it was possible to collect the quantitative data in…

  7. Connecting and Using Multiple Representations

    ERIC Educational Resources Information Center

    Nielsen, Maria E.; Bostic, Jonathan D.

    2018-01-01

    "Principles to Actions: Ensuring Mathematical Success for All" (NCTM 2014) emphasizes eight teaching practices for effective mathematics teaching, one of which is to "use and connect multiple representations" (NCTM 2014, p. 24). An action that describes how teachers might promote this practice is to "allocate substantial…

  8. Foundations of children's numerical and mathematical skills: the roles of symbolic and nonsymbolic representations of numerical magnitude.

    PubMed

    Lyons, Ian M; Ansari, Daniel

    2015-01-01

    Numerical and mathematical skills are critical predictors of academic success. The last three decades have seen a substantial growth in our understanding of how the human mind and brain represent and process numbers. In particular, research has shown that we share with animals the ability to represent numerical magnitude (the total number of items in a set) and that preverbal infants can process numerical magnitude. Further research has shown that similar processing signatures characterize numerical magnitude processing across species and developmental time. These findings suggest that an approximate system for nonsymbolic (e.g., dot arrays) numerical magnitude representation serves as the basis for the acquisition of cultural, symbolic (e.g., Arabic numerals) representations of numerical magnitude. This chapter explores this hypothesis by reviewing studies that have examined the relation between individual differences in nonsymbolic numerical magnitude processing and symbolic math abilities (e.g., arithmetic). Furthermore, we examine the extent to which the available literature provides strong evidence for a link between symbolic and nonsymbolic representations of numerical magnitude at the behavioral and neural levels of analysis. We conclude that claims that symbolic number abilities are grounded in the approximate system for the nonsymbolic representation of numerical magnitude are not strongly supported by the available evidence. Alternative models and future research directions are discussed. © 2015 Elsevier Inc. All rights reserved.

  9. Characterization, scaling, and partial representation of diffuse and discrete input junctions to CA3 hippocampus.

    PubMed

    Ascarrunz, F G; Kisley, M A; Flach, K A; Hamilton, R W; MacGregor, R J

    1995-07-01

    This paper applies a general mathematical system for characterizing and scaling functional connectivity and information flow across the diffuse (EC) and discrete (DG) input junctions to the CA3 hippocampus. Both gross connectivity and coordinated multiunit informational firing patterns are quantitatively characterized in terms of 32 defining parameters interrelated by 17 equations, and then scaled down according to rules for uniformly proportional scaling and for partial representation. The diffuse EC-CA3 junction is shown to be uniformly scalable with realistic representation of both essential spatiotemporal cooperativity and coordinated firing patterns down to populations of a few hundred neurons. Scaling of the discrete DG-CA3 junction can be effected with a two-step process, which necessarily deviates from uniform proportionality but nonetheless produces a valuable and readily interpretable reduced model, also utilizing a few hundred neurons in the receiving population. Partial representation produces a reduced model of only a portion of the full network where each model neuron corresponds directly to a biological neuron. The mathematical analysis illustrated here shows that although omissions and distortions are inescapable in such an application, satisfactorily complete and accurate models the size of pattern modules are possible. Finally, the mathematical characterization of these junctions generates a theory which sees the DG as a definer of the fine structure of embedded traces in the hippocampus and entire coordinated patterns of sequences of 14-cell links in CA3 as triggered by the firing of sequences of individual neurons in DG.

  10. On Representations and Situated Tools.

    ERIC Educational Resources Information Center

    Moreno-Armella, Luis

    This paper suggests that the systems of representations that we use in mathematics have a cultural origin and concludes that the knowledge produced with the help of these systems of representation likewise has a cultural origin. This assertion forces a reformulation of the issue of objectivity in terms that differ from those inherited from…

  11. Technology Focus: Multi-Representational Approaches to Equation Solving

    ERIC Educational Resources Information Center

    Garofalo, Joe; Trinter, Christine

    2009-01-01

    Most mathematical functions can be represented in numerous ways. The main representations typically addressed in school, often referred to as "the big three," are graphical, algebraic, and numerical representations, but there are others as well (e.g., diagrams, words, simulations). These different types of representations "often illuminate…

  12. STEM Gives Meaning to Mathematics

    ERIC Educational Resources Information Center

    Hefty, Lukas J.

    2015-01-01

    The National Council of Teachers of Mathematics' (NCTM's) "Principles and Standards for School Mathematics" (2000) outlines fi ve Process Standards that are essential for developing deep understanding of mathematics: (1) Problem Solving; (2) Reasoning and Proof; (3) Communication; (4) Connections; and (5) Representation. The Common Core…

  13. Flawed Mathematical Conceptualizations: Marlon's Dilemma

    ERIC Educational Resources Information Center

    Garrett, Lauretta

    2013-01-01

    Adult developmental mathematics students often work under great pressure to complete the mathematics sequences designed to help them achieve success (Bryk & Treisman, 2010). Results of a teaching experiment demonstrate how the ability to reason can be impeded by flaws in students' mental representations of mathematics. The earnestness of the…

  14. Standard model of knowledge representation

    NASA Astrophysics Data System (ADS)

    Yin, Wensheng

    2016-09-01

    Knowledge representation is the core of artificial intelligence research. Knowledge representation methods include predicate logic, semantic network, computer programming language, database, mathematical model, graphics language, natural language, etc. To establish the intrinsic link between various knowledge representation methods, a unified knowledge representation model is necessary. According to ontology, system theory, and control theory, a standard model of knowledge representation that reflects the change of the objective world is proposed. The model is composed of input, processing, and output. This knowledge representation method is not a contradiction to the traditional knowledge representation method. It can express knowledge in terms of multivariate and multidimensional. It can also express process knowledge, and at the same time, it has a strong ability to solve problems. In addition, the standard model of knowledge representation provides a way to solve problems of non-precision and inconsistent knowledge.

  15. Statistical Representations of Track Geometry : Volume I, Text.

    DOT National Transportation Integrated Search

    1980-03-31

    Mathematical representations of railroad track geometry variations are derived from time series analyses of track measurements. Since the majority of track is free of anomalies (turnouts, crossings, bridges, etc.), representation of anomaly-free trac...

  16. Squeezing, Striking, and Vocalizing: Is Number Representation Fundamentally Spatial?

    ERIC Educational Resources Information Center

    Nunez, Rafael; Doan, D.; Nikoulina, Anastasia

    2011-01-01

    Numbers are fundamental entities in mathematics, but their cognitive bases are unclear. Abundant research points to linear space as a natural grounding for number representation. But, is number representation fundamentally spatial? We disentangle number representation from standard number-to-line reporting methods, and compare numerical…

  17. Mathematical Metaphors: Problem Reformulation and Analysis Strategies

    NASA Technical Reports Server (NTRS)

    Thompson, David E.

    2005-01-01

    This paper addresses the critical need for the development of intelligent or assisting software tools for the scientist who is working in the initial problem formulation and mathematical model representation stage of research. In particular, examples of that representation in fluid dynamics and instability theory are discussed. The creation of a mathematical model that is ready for application of certain solution strategies requires extensive symbolic manipulation of the original mathematical model. These manipulations can be as simple as term reordering or as complicated as discovery of various symmetry groups embodied in the equations, whereby Backlund-type transformations create new determining equations and integrability conditions or create differential Grobner bases that are then solved in place of the original nonlinear PDEs. Several examples are presented of the kinds of problem formulations and transforms that can be frequently encountered in model representation for fluids problems. The capability of intelligently automating these types of transforms, available prior to actual mathematical solution, is advocated. Physical meaning and assumption-understanding can then be propagated through the mathematical transformations, allowing for explicit strategy development.

  18. Visual Representations in Mathematics Teaching: An Experiment with Students

    ERIC Educational Resources Information Center

    Debrenti, Edith

    2015-01-01

    General problem-solving skills are of central importance in school mathematics achievement. Word problems play an important role not just in mathematical education, but in general education as well. Meaningful learning and understanding are basic aspects of all kinds of learning and it is even more important in the case of learning mathematics. In…

  19. The Transition to Formal Thinking in Mathematics

    ERIC Educational Resources Information Center

    Tall, David

    2008-01-01

    This paper focuses on the changes in thinking involved in the transition from school mathematics to formal proof in pure mathematics at university. School mathematics is seen as a combination of visual representations, including geometry and graphs, together with symbolic calculations and manipulations. Pure mathematics in university shifts…

  20. The Number Line as a Representation of Decimal Numbers: A Research with Sixth Grade Students

    ERIC Educational Resources Information Center

    Michaelidou, Niki; Gagatsis, Athanasios; Pitta-Pantazi, Demetra

    2004-01-01

    One of the aims of mathematics instruction is to achieve the understanding of mathematical concepts through the development of rich and well organized cognitive representations (Goldin, 1998; NCTM, 2000; DeWindt-King, & Goldin, 2003). In this study the term representation is interpreted as the tool used for representing mathematical ideas such…

  1. Variational calculation of second-order reduced density matrices by strong N-representability conditions and an accurate semidefinite programming solver.

    PubMed

    Nakata, Maho; Braams, Bastiaan J; Fujisawa, Katsuki; Fukuda, Mituhiro; Percus, Jerome K; Yamashita, Makoto; Zhao, Zhengji

    2008-04-28

    The reduced density matrix (RDM) method, which is a variational calculation based on the second-order reduced density matrix, is applied to the ground state energies and the dipole moments for 57 different states of atoms, molecules, and to the ground state energies and the elements of 2-RDM for the Hubbard model. We explore the well-known N-representability conditions (P, Q, and G) together with the more recent and much stronger T1 and T2(') conditions. T2(') condition was recently rederived and it implies T2 condition. Using these N-representability conditions, we can usually calculate correlation energies in percentage ranging from 100% to 101%, whose accuracy is similar to CCSD(T) and even better for high spin states or anion systems where CCSD(T) fails. Highly accurate calculations are carried out by handling equality constraints and/or developing multiple precision arithmetic in the semidefinite programming (SDP) solver. Results show that handling equality constraints correctly improves the accuracy from 0.1 to 0.6 mhartree. Additionally, improvements by replacing T2 condition with T2(') condition are typically of 0.1-0.5 mhartree. The newly developed multiple precision arithmetic version of SDP solver calculates extraordinary accurate energies for the one dimensional Hubbard model and Be atom. It gives at least 16 significant digits for energies, where double precision calculations gives only two to eight digits. It also provides physically meaningful results for the Hubbard model in the high correlation limit.

  2. Representation control increases task efficiency in complex graphical representations.

    PubMed

    Moritz, Julia; Meyerhoff, Hauke S; Meyer-Dernbecher, Claudia; Schwan, Stephan

    2018-01-01

    In complex graphical representations, the relevant information for a specific task is often distributed across multiple spatial locations. In such situations, understanding the representation requires internal transformation processes in order to extract the relevant information. However, digital technology enables observers to alter the spatial arrangement of depicted information and therefore to offload the transformation processes. The objective of this study was to investigate the use of such a representation control (i.e. the users' option to decide how information should be displayed) in order to accomplish an information extraction task in terms of solution time and accuracy. In the representation control condition, the participants were allowed to reorganize the graphical representation and reduce information density. In the control condition, no interactive features were offered. We observed that participants in the representation control condition solved tasks that required reorganization of the maps faster and more accurate than participants without representation control. The present findings demonstrate how processes of cognitive offloading, spatial contiguity, and information coherence interact in knowledge media intended for broad and diverse groups of recipients.

  3. Representation control increases task efficiency in complex graphical representations

    PubMed Central

    Meyerhoff, Hauke S.; Meyer-Dernbecher, Claudia; Schwan, Stephan

    2018-01-01

    In complex graphical representations, the relevant information for a specific task is often distributed across multiple spatial locations. In such situations, understanding the representation requires internal transformation processes in order to extract the relevant information. However, digital technology enables observers to alter the spatial arrangement of depicted information and therefore to offload the transformation processes. The objective of this study was to investigate the use of such a representation control (i.e. the users' option to decide how information should be displayed) in order to accomplish an information extraction task in terms of solution time and accuracy. In the representation control condition, the participants were allowed to reorganize the graphical representation and reduce information density. In the control condition, no interactive features were offered. We observed that participants in the representation control condition solved tasks that required reorganization of the maps faster and more accurate than participants without representation control. The present findings demonstrate how processes of cognitive offloading, spatial contiguity, and information coherence interact in knowledge media intended for broad and diverse groups of recipients. PMID:29698443

  4. Transformations in the Visual Representation of a Figural Pattern

    ERIC Educational Resources Information Center

    Montenegro, Paula; Costa, Cecília; Lopes, Bernardino

    2018-01-01

    Multiple representations of a given mathematical object/concept are one of the biggest difficulties encountered by students. The aim of this study is to investigate the impact of the use of visual representations in teaching and learning algebra. In this paper, we analyze the transformations from and to visual representations that were performed…

  5. Spatial representation of pitch height: the SMARC effect.

    PubMed

    Rusconi, Elena; Kwan, Bonnie; Giordano, Bruno L; Umiltà, Carlo; Butterworth, Brian

    2006-03-01

    Through the preferential pairing of response positions to pitch, here we show that the internal representation of pitch height is spatial in nature and affects performance, especially in musically trained participants, when response alternatives are either vertically or horizontally aligned. The finding that our cognitive system maps pitch height onto an internal representation of space, which in turn affects motor performance even when this perceptual attribute is irrelevant to the task, extends previous studies on auditory perception and suggests an interesting analogy between music perception and mathematical cognition. Both the basic elements of mathematical cognition (i.e. numbers) and the basic elements of musical cognition (i.e. pitches), appear to be mapped onto a mental spatial representation in a way that affects motor performance.

  6. A Pathway for Mathematical Practices

    ERIC Educational Resources Information Center

    Wenrick, Melanie; Behrend, Jean L.; Mohs, Laura C.

    2013-01-01

    How can teachers engage students in learning essential mathematics? The National Council of Teachers of Mathematics recommends using "contexts that promote problem solving, reasoning, communication, making connections, and designing and analyzing representations" (NCTM 2006, p. 11). Understanding the Process Standards (NCTM 2000) enables teachers…

  7. Representational Translation with Concrete Models in Organic Chemistry

    ERIC Educational Resources Information Center

    Stull, Andrew T.; Hegarty, Mary; Dixon, Bonnie; Stieff, Mike

    2012-01-01

    In representation-rich domains such as organic chemistry, students must be facile and accurate when translating between different 2D representations, such as diagrams. We hypothesized that translating between organic chemistry diagrams would be more accurate when concrete models were used because difficult mental processes could be augmented by…

  8. A Study of Visualization for Mathematics Education

    NASA Technical Reports Server (NTRS)

    Daugherty, Sarah C.

    2008-01-01

    Graphical representations such as figures, illustrations, and diagrams play a critical role in mathematics and they are equally important in mathematics education. However, graphical representations in mathematics textbooks are static, Le. they are used to illustrate only a specific example or a limited set. of examples. By using computer software to visualize mathematical principles, virtually there is no limit to the number of specific cases and examples that can be demonstrated. However, we have not seen widespread adoption of visualization software in mathematics education. There are currently a number of software packages that provide visualization of mathematics for research and also software packages specifically developed for mathematics education. We conducted a survey of mathematics visualization software packages, summarized their features and user bases, and analyzed their limitations. In this survey, we focused on evaluating the software packages for their use with mathematical subjects adopted by institutions of secondary education in the United States (middle schools and high schools), including algebra, geometry, trigonometry, and calculus. We found that cost, complexity, and lack of flexibility are the major factors that hinder the widespread use of mathematics visualization software in education.

  9. Multiple Representations and Connections with the Sierpinski Triangle

    ERIC Educational Resources Information Center

    Kirwan, J. Vince; Tobias, Jennifer M.

    2014-01-01

    To understand multiple representations in algebra, students must be able to describe relationships through a variety of formats, such as graphs, tables, pictures, and equations. NCTM indicates that varied representations are "essential elements in supporting students' understanding of mathematical concepts and relationships" (NCTM…

  10. Integrating Formal and Grounded Representations in Combinatorics Learning

    ERIC Educational Resources Information Center

    Braithwaite, David W.; Goldstone, Robert L.

    2013-01-01

    The terms "concreteness fading" and "progressive formalization" have been used to describe instructional approaches to science and mathematics that use grounded representations to introduce concepts and later transition to more formal representations of the same concepts. There are both theoretical and empirical reasons to…

  11. Mathematics Teachers' Representations of Authority

    ERIC Educational Resources Information Center

    Wagner, David; Herbel-Eisenmann, Beth

    2014-01-01

    Issues of authority abound in education and schooling but have not been interrogated sufficiently. We describe a tool that we have developed to initiate dialogue with teachers about authority in their classrooms--using a diagram to represent authority in their classrooms. Our analysis of the diagrams mathematics teachers created and discussed in…

  12. Effect of the Presence of External Representations on Accuracy and Reaction Time in Solving Mathematical Double-Choice Problems by Students of Different Levels of Instruction

    ERIC Educational Resources Information Center

    Leikin, Roza; Leikin, Mark; Waisman, Ilana; Shaul, Shelley

    2013-01-01

    This study explores the effects of the "presence of external representations of a mathematical object" (ERs) on problem solving performance associated with short double-choice problems. The problems were borrowed from secondary school algebra and geometry, and the ERs were either formulas, graphs of functions, or drawings of geometric…

  13. Representing the Electromagnetic Field: How Maxwell's Mathematics Empowered Faraday's Field Theory

    ERIC Educational Resources Information Center

    Tweney, Ryan D.

    2011-01-01

    James Clerk Maxwell "translated" Michael Faraday's experimentally-based field theory into the mathematical representation now known as "Maxwell's Equations." Working with a variety of mathematical representations and physical models Maxwell extended the reach of Faraday's theory and brought it into consistency with other…

  14. External Representations for Data Distributions: In Search of Cognitive Fit

    ERIC Educational Resources Information Center

    Lem, Stephanie; Onghana, Patrick; Verschaffel, Lieven; Van Dooren, Wim

    2013-01-01

    Data distributions can be represented using different external representations, such as histograms and boxplots. Although the role of external representations has been extensively studied in mathematics, this is less the case in statistics. This study helps to fill this gap by systematically varying the representation that accompanies a task…

  15. Representations in Problem Solving: A Case Study with Optimization Problems

    ERIC Educational Resources Information Center

    Villegas, Jose L.; Castro, Enrique; Gutierrez, Jose

    2009-01-01

    Introduction: Representations play an essential role in mathematical thinking. They favor the understanding of mathematical concepts and stimulate the development of flexible and versatile thinking in problem solving. Here our focus is on their use in optimization problems, a type of problem considered important in mathematics teaching and…

  16. Masculinities in Mathematics. Educating Boys, Learning Gender

    ERIC Educational Resources Information Center

    Mendick, Heather

    2006-01-01

    This book illuminates what studying mathematics means for both students and teachers and offers a broad range of insights into students' views and practices. In addition to the words of young people learning mathematics, the masculinity of mathematics is explored through historical material and cinematic representations. The author discusses the…

  17. Contributions from specific and general factors to unique deficits: two cases of mathematics learning difficulties

    PubMed Central

    Haase, Vitor G.; Júlio-Costa, Annelise; Lopes-Silva, Júlia B.; Starling-Alves, Isabella; Antunes, Andressa M.; Pinheiro-Chagas, Pedro; Wood, Guilherme

    2014-01-01

    Mathematics learning difficulties are a highly comorbid and heterogeneous set of disorders linked to several dissociable mechanisms and endophenotypes. Two of these endophenotypes consist of primary deficits in number sense and verbal numerical representations. However, currently acknowledged endophenotypes are underspecified regarding the role of automatic vs. controlled information processing, and their description should be complemented. Two children with specific deficits in number sense and verbal numerical representations and normal or above-normal intelligence and preserved visuospatial cognition illustrate this point. Child H.V. exhibited deficits in number sense and fact retrieval. Child G.A. presented severe deficits in orally presented problems and transcoding tasks. A partial confirmation of the two endophenotypes that relate to the number sense and verbal processing was obtained, but a much more clear differentiation between the deficits presented by H.V. and G.A. can be reached by looking at differential impairments in modes of processing. H.V. is notably competent in the use of controlled processing but has problems with more automatic processes, such as nonsymbolic magnitude processing, speeded counting and fact retrieval. In contrast, G.A. can retrieve facts and process nonsymbolic magnitudes but exhibits severe impairment in recruiting executive functions and the concentration that is necessary to accomplish transcoding tasks and word problem solving. These results indicate that typical endophenotypes might be insufficient to describe accurately the deficits that are observed in children with mathematics learning abilities. However, by incorporating domain-specificity and modes of processing into the assessment of the endophenotypes, individual deficit profiles can be much more accurately described. This process calls for further specification of the endophenotypes in mathematics learning difficulties. PMID:24592243

  18. Conceptions and Representations: The Circle as an Example.

    ERIC Educational Resources Information Center

    Janvier, Claude

    This paper, which addresses the issue of representation as an internal construct corresponding to an external abstract configuration, attempts to extend A. A. DiSessa's phenomenological primitives to mathematics (particularly to the notion of circle). Various acceptations of the word representation are examined, using the notion of a circle as an…

  19. Using Concrete Manipulatives in Mathematical Instruction

    ERIC Educational Resources Information Center

    Jones, Julie P.; Tiller, Margaret

    2017-01-01

    Concrete, Representational, Abstract (CRA) instruction is a process for teaching and learning mathematical concepts. Starting with manipulation of concrete materials (counters, beans, Unifix cubes), the process moves students to the representational level (tallies, dots, stamps), and peaks at the abstract level, at which numbers and symbols are…

  20. Effects of Finger Counting on Numerical Development – The Opposing Views of Neurocognition and Mathematics Education

    PubMed Central

    Moeller, Korbinian; Martignon, Laura; Wessolowski, Silvia; Engel, Joachim; Nuerk, Hans-Christoph

    2011-01-01

    Children typically learn basic numerical and arithmetic principles using finger-based representations. However, whether or not reliance on finger-based representations is beneficial or detrimental is the subject of an ongoing debate between researchers in neurocognition and mathematics education. From the neurocognitive perspective, finger counting provides multisensory input, which conveys both cardinal and ordinal aspects of numbers. Recent data indicate that children with good finger-based numerical representations show better arithmetic skills and that training finger gnosis, or “finger sense,” enhances mathematical skills. Therefore neurocognitive researchers conclude that elaborate finger-based numerical representations are beneficial for later numerical development. However, research in mathematics education recommends fostering mentally based numerical representations so as to induce children to abandon finger counting. More precisely, mathematics education recommends first using finger counting, then concrete structured representations and, finally, mental representations of numbers to perform numerical operations. Taken together, these results reveal an important debate between neurocognitive and mathematics education research concerning the benefits and detriments of finger-based strategies for numerical development. In the present review, the rationale of both lines of evidence will be discussed. PMID:22144969

  1. Third Graders' Mathematical Thinking of Place Value through the Use of Concrete and Virtual Manipulatives

    ERIC Educational Resources Information Center

    Burris, Justin T.

    2010-01-01

    As one research priority for mathematics education is "to research how mathematical meanings are structured by tools available," the present study examined mathematical representations more closely by investigating instructional modes of representation (Noss, Healy & Hoyles, 1997). The study compared two modes of instruction of place value with…

  2. Making Implicit Multivariable Calculus Representations Explicit: A Clinical Study

    ERIC Educational Resources Information Center

    McGee, Daniel; Moore-Russo, Deborah; Martinez-Planell, Rafael

    2015-01-01

    Reviewing numerous textbooks, we found that in both differential and integral calculus textbooks the authors commonly assume that: (i) students can generalize associations between representations in two dimensions to associations between representations of the same mathematical concept in three dimensions on their own; and (ii) explicit…

  3. Student difficulties regarding symbolic and graphical representations of vector fields

    NASA Astrophysics Data System (ADS)

    Bollen, Laurens; van Kampen, Paul; Baily, Charles; Kelly, Mossy; De Cock, Mieke

    2017-12-01

    The ability to switch between various representations is an invaluable problem-solving skill in physics. In addition, research has shown that using multiple representations can greatly enhance a person's understanding of mathematical and physical concepts. This paper describes a study of student difficulties regarding interpreting, constructing, and switching between representations of vector fields, using both qualitative and quantitative methods. We first identified to what extent students are fluent with the use of field vector plots, field line diagrams, and symbolic expressions of vector fields by conducting individual student interviews and analyzing in-class student activities. Based on those findings, we designed the Vector Field Representations test, a free response assessment tool that has been given to 196 second- and third-year physics, mathematics, and engineering students from four different universities. From the obtained results we gained a comprehensive overview of typical errors that students make when switching between vector field representations. In addition, the study allowed us to determine the relative prevalence of the observed difficulties. Although the results varied greatly between institutions, a general trend revealed that many students struggle with vector addition, fail to recognize the field line density as an indication of the magnitude of the field, confuse characteristics of field lines and equipotential lines, and do not choose the appropriate coordinate system when writing out mathematical expressions of vector fields.

  4. Fraction Representation: The Not-So-Common Denominator among Textbooks

    ERIC Educational Resources Information Center

    Hodges, Thomas E.; Cady, JoAnn; Collins, Lee

    2008-01-01

    Three widely used sixth-grade textbooks were studied to see how fraction concepts were represented. The textbooks selected were "Connected Mathematics," "Middle Grades MathThematics," and Glencoe's "Mathematics: Applications and Concepts Course 1." Three specific areas were examined: representation mode, model, and problem context. Results of…

  5. Nature and origins of mathematics difficulties in very preterm children: a different etiology than developmental dyscalculia.

    PubMed

    Simms, Victoria; Gilmore, Camilla; Cragg, Lucy; Clayton, Sarah; Marlow, Neil; Johnson, Samantha

    2015-02-01

    Children born very preterm (<32 wk) are at high risk for mathematics learning difficulties that are out of proportion to other academic and cognitive deficits. However, the etiology of mathematics difficulties in very preterm children is unknown. We sought to identify the nature and origins of preterm children's mathematics difficulties. One hundred and fifteen very preterm children aged 8-10 y were assessed in school with a control group of 77 term-born classmates. Achievement in mathematics, working memory, visuospatial processing, inhibition, and processing speed were assessed using standardized tests. Numerical representations and specific mathematics skills were assessed using experimental tests. Very preterm children had significantly poorer mathematics achievement, working memory, and visuospatial skills than term-born controls. Although preterm children had poorer performance in specific mathematics skills, there was no evidence of imprecise numerical representations. Difficulties in mathematics were associated with deficits in visuospatial processing and working memory. Mathematics difficulties in very preterm children are associated with deficits in working memory and visuospatial processing not numerical representations. Thus, very preterm children's mathematics difficulties are different in nature from those of children with developmental dyscalculia. Interventions targeting general cognitive problems, rather than numerical representations, may improve very preterm children's mathematics achievement.

  6. Teacher's Representational Fluency in a Context of Technology Use

    ERIC Educational Resources Information Center

    Rocha, Helena

    2016-01-01

    This study focuses on teacher's Knowledge for Teaching Mathematics with Technology (KTMT), paying a special attention to teacher's representational fluency. It intends to characterize how the teacher uses and integrates the different representations provided by the graphing calculator on the process of teaching and learning functions at the high…

  7. The Multicultural Mathematics Classroom: Culturally Aware Teaching through Cooperative Learning & Multiple Representations

    ERIC Educational Resources Information Center

    Jao, Limin

    2012-01-01

    The National Council of Teachers of Mathematics (NCTM, 2000) has created a set of standards to reform mathematics teaching procedures to ensure that all students understand mathematics and learn to think mathematically. The standards also require teachers to use strategies that allow all students to reason and communicate mathematically and…

  8. Reevaluating the two-representation model of numerical magnitude processing.

    PubMed

    Jiang, Ting; Zhang, Wenfeng; Wen, Wen; Zhu, Haiting; Du, Han; Zhu, Xiangru; Gao, Xuefei; Zhang, Hongchuan; Dong, Qi; Chen, Chuansheng

    2016-01-01

    One debate in mathematical cognition centers on the single-representation model versus the two-representation model. Using an improved number Stroop paradigm (i.e., systematically manipulating physical size distance), in the present study we tested the predictions of the two models for number magnitude processing. The results supported the single-representation model and, more importantly, explained how a design problem (failure to manipulate physical size distance) and an analytical problem (failure to consider the interaction between congruity and task-irrelevant numerical distance) might have contributed to the evidence used to support the two-representation model. This study, therefore, can help settle the debate between the single-representation and two-representation models.

  9. Dissociations in mathematical knowledge: case studies in Down's syndrome and Williams syndrome.

    PubMed

    Robinson, Sally J; Temple, Christine M

    2013-02-01

    A study is reported of mathematical vocabulary and factual mathematical knowledge in PQ, a 22 year old with Down's syndrome (DS) who has a verbal mental age (MA) of 9 years 2 months and ST, a 15 year old with Williams syndrome (WS) who has a verbal MA of 9 years 6 months, matched to typically developing controls. The number of mathematical words contained within PQ's lexical stores was significantly reduced as reflected by performance on lexical decision. PQ was also impaired at both naming from descriptions and describing mathematical words. These results contrast with normal lexical decision and item descriptions for concrete words reported recently for PQ (Robinson and Temple, 2010). PQ's recall of mathematical facts was also impaired, whilst his recall of general knowledge facts was normal. This performance in DS indicates a deficit in both lexical representation and semantic knowledge for mathematical words and mathematical facts. In contrast, ST, the teenager with WS had good accuracy on lexical decision, naming and generating definitions for mathematical words. This contrasted with the atypical performance with concrete words recently reported for ST (Robinson and Temple, 2009). Knowledge of addition facts and general knowledge facts was also unimpaired for ST, though knowledge of multiplication facts was weak. Together the cases form a double dissociation and provide support for the distinct representation of mathematical and concrete items within the lexical-semantic system during development. The dissociations between mathematical and general factual knowledge also indicate that different types of factual knowledge may be selectively impaired during development. There is further support for a modular structure within which mathematical vocabulary and mathematical knowledge have distinct representations. This supports the case for the independent representation of factual and language-based knowledge within the semantic system during development. Copyright

  10. Basic and Advanced Numerical Performances Relate to Mathematical Expertise but Are Fully Mediated by Visuospatial Skills

    PubMed Central

    2016-01-01

    Recent studies have highlighted the potential role of basic numerical processing in the acquisition of numerical and mathematical competences. However, it is debated whether high-level numerical skills and mathematics depends specifically on basic numerical representations. In this study mathematicians and nonmathematicians performed a basic number line task, which required mapping positive and negative numbers on a physical horizontal line, and has been shown to correlate with more advanced numerical abilities and mathematical achievement. We found that mathematicians were more accurate compared with nonmathematicians when mapping positive, but not negative numbers, which are considered numerical primitives and cultural artifacts, respectively. Moreover, performance on positive number mapping could predict whether one is a mathematician or not, and was mediated by more advanced mathematical skills. This finding might suggest a link between basic and advanced mathematical skills. However, when we included visuospatial skills, as measured by block design subtest, the mediation analysis revealed that the relation between the performance in the number line task and the group membership was explained by non-numerical visuospatial skills. These results demonstrate that relation between basic, even specific, numerical skills and advanced mathematical achievement can be artifactual and explained by visuospatial processing. PMID:26913930

  11. Handbook for Spoken Mathematics: (Larry's Speakeasy).

    ERIC Educational Resources Information Center

    Chang, Lawrence A.; And Others

    This handbook is directed toward those who have to deal with spoken mathematics, yet have insufficient background to know the correct verbal expression for the written symbolic one. It compiles consistent and well-defined ways of uttering mathematical expressions so listeners will receive clear, unambiguous, and well-pronounced representations.…

  12. Representations of the Extended Poincare Superalgebras in Four Dimensions

    NASA Astrophysics Data System (ADS)

    Griffis, John D.

    Eugene Wigner used the Poincare group to induce representations from the fundamental internal space-time symmetries of (special) relativistic quantum particles. Wigner's students spent considerable amount of time translating passages of this paper into more detailed and accessible papers and books. In 1975, R. Haag et al. investigated the possible extensions of the symmetries of relativistic quantum particles. They showed that the only consistent (super)symmetric extensions to the standard model of physics are obtained by using super charges to generate the odd part of a Lie superalgebra whose even part is generated by the Poincare group; this theory has become known as supersymmetry. In this paper, R. Haag et al. used a notation called supermultiplets to give the dimension of a representation and its multiplicity; this notation is described mathematically in chapter 5 of this thesis. By 1980 S. Ferrara et al. began classifying the representations of these algebras for dimensions greater than four, and in 1986 Strathdee published considerable work listing some representations for the Poincare superalgebra in any finite dimension. This work has been continued to date. We found the work of S. Ferrara et al. to be essential to our understanding extended supersymmetries. However, this paper was written using imprecise language meant for physicists, so it was far from trivial to understand the mathematical interpretation of this work. In this thesis, we provide a "translation" of the previous results (along with some other literature on the Extended Poincare Superalgebras) into a rigorous mathematical setting, which makes the subject more accessible to a larger audience. Having a mathematical model allows us to give explicit results and detailed proofs. Further, this model allows us to see beyond just the physical interpretation and it allows investigation by a purely mathematically adept audience. Our work was motivated by a paper written in 2012 by M. Chaichian et al

  13. Geometric constraints in semiclassical initial value representation calculations in Cartesian coordinates: accurate reduction in zero-point energy.

    PubMed

    Issack, Bilkiss B; Roy, Pierre-Nicholas

    2005-08-22

    An approach for the inclusion of geometric constraints in semiclassical initial value representation calculations is introduced. An important aspect of the approach is that Cartesian coordinates are used throughout. We devised an algorithm for the constrained sampling of initial conditions through the use of multivariate Gaussian distribution based on a projected Hessian. We also propose an approach for the constrained evaluation of the so-called Herman-Kluk prefactor in its exact log-derivative form. Sample calculations are performed for free and constrained rare-gas trimers. The results show that the proposed approach provides an accurate evaluation of the reduction in zero-point energy. Exact basis set calculations are used to assess the accuracy of the semiclassical results. Since Cartesian coordinates are used, the approach is general and applicable to a variety of molecular and atomic systems.

  14. Building Mathematical Models of Simple Harmonic and Damped Motion.

    ERIC Educational Resources Information Center

    Edwards, Thomas

    1995-01-01

    By developing a sequence of mathematical models of harmonic motion, shows that mathematical models are not right or wrong, but instead are better or poorer representations of the problem situation. (MKR)

  15. Mathematical structure of unit systems

    NASA Astrophysics Data System (ADS)

    Kitano, Masao

    2013-05-01

    We investigate the mathematical structure of unit systems and the relations between them. Looking over the entire set of unit systems, we can find a mathematical structure that is called preorder (or quasi-order). For some pair of unit systems, there exists a relation of preorder such that one unit system is transferable to the other unit system. The transfer (or conversion) is possible only when all of the quantities distinguishable in the latter system are always distinguishable in the former system. By utilizing this structure, we can systematically compare the representations in different unit systems. Especially, the equivalence class of unit systems (EUS) plays an important role because the representations of physical quantities and equations are of the same form in unit systems belonging to an EUS. The dimension of quantities is uniquely defined in each EUS. The EUS's form a partially ordered set. Using these mathematical structures, unit systems and EUS's are systematically classified and organized as a hierarchical tree.

  16. Spreadsheets as a Transparent Resource for Learning the Mathematics of Annuities

    ERIC Educational Resources Information Center

    Pournara, Craig

    2009-01-01

    The ability of mathematics teachers to decompress mathematics and to move between representations are two key features of mathematical knowledge that is usable for teaching. This article reports on four pre-service secondary mathematics teachers learning the mathematics of annuities. In working with spreadsheets students began to make sense of…

  17. Student Difficulties Regarding Symbolic and Graphical Representations of Vector Fields

    ERIC Educational Resources Information Center

    Bollen, Laurens; van Kampen, Paul; Baily, Charles; Kelly, Mossy; De Cock, Mieke

    2017-01-01

    The ability to switch between various representations is an invaluable problem-solving skill in physics. In addition, research has shown that using multiple representations can greatly enhance a person's understanding of mathematical and physical concepts. This paper describes a study of student difficulties regarding interpreting, constructing,…

  18. Students’ Mathematical Literacy in Solving PISA Problems Based on Keirsey Personality Theory

    NASA Astrophysics Data System (ADS)

    Masriyah; Firmansyah, M. H.

    2018-01-01

    This research is descriptive-qualitative research. The purpose is to describe students’ mathematical literacy in solving PISA on space and shape content based on Keirsey personality theory. The subjects are four junior high school students grade eight with guardian, artisan, rational or idealist personality. Data collecting methods used test and interview. Data of Keirsey Personality test, PISA test, and interview were analysed. Profile of mathematical literacy of each subject are described as follows. In formulating, guardian subject identified mathematical aspects are formula of rectangle area and sides length; significant variables are terms/conditions in problem and formula of ever encountered question; translated into mathematical language those are measurement and arithmetic operations. In employing, he devised and implemented strategies using ease of calculation on area-subtraction principle; declared truth of result but the reason was less correct; didn’t use and switch between different representations. In interpreting, he declared result as area of house floor; declared reasonableness according measurement estimation. In formulating, artisan subject identified mathematical aspects are plane and sides length; significant variables are solution procedure on both of daily problem and ever encountered question; translated into mathematical language those are measurement, variables, and arithmetic operations as well as symbol representation. In employing, he devised and implemented strategies using two design comparison; declared truth of result without reason; used symbol representation only. In interpreting, he expressed result as floor area of house; declared reasonableness according measurement estimation. In formulating, rational subject identified mathematical aspects are scale and sides length; significant variables are solution strategy on ever encountered question; translated into mathematical language those are measurement, variable, arithmetic

  19. Representations in Calculus: Two Contrasting Cases.

    ERIC Educational Resources Information Center

    Aspinwall, Leslie; Shaw, Kenneth L.

    2002-01-01

    Illustrates the contrasting thinking processes of two beginning calculus students' geometric and analytic schemes for the derivative function. Suggests that teachers can enhance students' understanding by continuing to demonstrate how different representations of the same mathematical concept provide additional information. (KHR)

  20. An Emergent Framework: Views of Mathematical Processes

    ERIC Educational Resources Information Center

    Sanchez, Wendy B.; Lischka, Alyson E.; Edenfield, Kelly W.; Gammill, Rebecca

    2015-01-01

    The findings reported in this paper were generated from a case study of teacher leaders at a state-level mathematics conference. Investigation focused on how participants viewed the mathematical processes of communication, connections, representations, problem solving, and reasoning and proof. Purposeful sampling was employed to select nine…

  1. Spatially variant morphological restoration and skeleton representation.

    PubMed

    Bouaynaya, Nidhal; Charif-Chefchaouni, Mohammed; Schonfeld, Dan

    2006-11-01

    The theory of spatially variant (SV) mathematical morphology is used to extend and analyze two important image processing applications: morphological image restoration and skeleton representation of binary images. For morphological image restoration, we propose the SV alternating sequential filters and SV median filters. We establish the relation of SV median filters to the basic SV morphological operators (i.e., SV erosions and SV dilations). For skeleton representation, we present a general framework for the SV morphological skeleton representation of binary images. We study the properties of the SV morphological skeleton representation and derive conditions for its invertibility. We also develop an algorithm for the implementation of the SV morphological skeleton representation of binary images. The latter algorithm is based on the optimal construction of the SV structuring element mapping designed to minimize the cardinality of the SV morphological skeleton representation. Experimental results show the dramatic improvement in the performance of the SV morphological restoration and SV morphological skeleton representation algorithms in comparison to their translation-invariant counterparts.

  2. Improvement of Word Problem Solving and Basic Mathematics Competencies in Students with Attention Deficit/Hyperactivity Disorder and Mathematical Learning Difficulties

    ERIC Educational Resources Information Center

    González-Castro, Paloma; Cueli, Marisol; Areces, Débora; Rodríguez, Celestino; Sideridis, Georgios

    2016-01-01

    Problem solving represents a salient deficit in students with mathematical learning difficulties (MLD) primarily caused by difficulties with informal and formal mathematical competencies. This study proposes a computerized intervention tool, the integrated dynamic representation (IDR), for enhancing the early learning of basic mathematical…

  3. Characterizing Representational Learning: A Combined Simulation and Tutorial on Perturbation Theory

    ERIC Educational Resources Information Center

    Kohnle, Antje; Passante, Gina

    2017-01-01

    Analyzing, constructing, and translating between graphical, pictorial, and mathematical representations of physics ideas and reasoning flexibly through them ("representational competence") is a key characteristic of expertise in physics but is a challenge for learners to develop. Interactive computer simulations and University of…

  4. A mathematical model of insulin resistance in Parkinson's disease.

    PubMed

    Braatz, Elise M; Coleman, Randolph A

    2015-06-01

    This paper introduces a mathematical model representing the biochemical interactions between insulin signaling and Parkinson's disease. The model can be used to examine the changes that occur over the course of the disease as well as identify which processes would be the most effective targets for treatment. The model is mathematized using biochemical systems theory (BST). It incorporates a treatment strategy that includes several experimental drugs along with current treatments. In the past, BST models of neurodegeneration have used power law analysis and simulation (PLAS) to model the system. This paper recommends the use of MATLAB instead. MATLAB allows for more flexibility in both the model itself and in data analysis. Previous BST analyses of neurodegeneration began treatment at disease onset. As shown in this model, the outcomes of delayed, realistic treatment and full treatment at disease onset are significantly different. The delayed treatment strategy is an important development in BST modeling of neurodegeneration. It emphasizes the importance of early diagnosis, and allows for a more accurate representation of disease and treatment interactions. Copyright © 2015 Elsevier Ltd. All rights reserved.

  5. Good Mathematics Teaching from Mexican High School Students' Perspective

    ERIC Educational Resources Information Center

    Martinez-Sierra, Gustavo

    2014-01-01

    This paper reports a qualitative research that identifies the characteristics of good mathematics teaching from the perspective of Mexican high school students. For this purpose, the social representations of a good mathematics teacher and a good mathematics class were identified in a group of 67 students. In order to obtain information, a…

  6. Using Mental Imagery Processes for Teaching and Research in Mathematics and Computer Science

    ERIC Educational Resources Information Center

    Arnoux, Pierre; Finkel, Alain

    2010-01-01

    The role of mental representations in mathematics and computer science (for teaching or research) is often downplayed or even completely ignored. Using an ongoing work on the subject, we argue for a more systematic study and use of mental representations, to get an intuition of mathematical concepts, and also to understand and build proofs. We…

  7. Motion sensors in mathematics teaching: learning tools for understanding general math concepts?

    NASA Astrophysics Data System (ADS)

    Urban-Woldron, Hildegard

    2015-05-01

    Incorporating technology tools into the mathematics classroom adds a new dimension to the teaching of mathematics concepts and establishes a whole new approach to mathematics learning. In particular, gathering data in a hands-on and real-time method helps classrooms coming alive. The focus of this paper is on bringing forward important mathematics concepts such as functions and rate of change with the motion detector. Findings from the author's studies suggest that the motion detector can be introduced from a very early age and used to enliven classes at any level. Using real-world data to present the main functions invites an experimental approach to mathematics and encourages students to engage actively in their learning. By emphasizing learning experiences with computer-based motion detectors and aiming to involve students in mathematical representations of real-world phenomena, six learning activities, which were developed in previous research studies, will be presented. Students use motion sensors to collect physical data that are graphed in real time and then manipulate and analyse them. Because data are presented in an immediately understandable graphical form, students are allowed to take an active role in their learning by constructing mathematical knowledge from observation of the physical world. By utilizing a predict-observe-explain format, students learn about slope, determining slope and distance vs. time graphs through motion-filled activities. Furthermore, exploring the meaning of slope, viewed as the rate of change, students acquire competencies for reading, understanding and interpreting kinematics graphs involving a multitude of mathematical representations. Consequently, the students are empowered to efficiently move among tabular, graphical and symbolic representation to analyse patterns and discover the relationships between different representations of motion. In fact, there is a need for further research to explore how mathematics teachers

  8. Students' Development and Use of Internal Representations When Solving Algebraic Tasks

    ERIC Educational Resources Information Center

    Cross, Laban J.

    2013-01-01

    The difficulty in observing, recording, and examining internal representations has been well documented (Goldin & Shteingold, 2001). However, the important role that these internal representations play in the learning and understanding of mathematical concepts has been noted (Yackel, 2000). This study sought to develop a framework for…

  9. Students' Use of Mathematical Representations in Problem Solving.

    ERIC Educational Resources Information Center

    Santos-Trigo, Manuel

    2002-01-01

    Documents the experiences of 25 first-year university students with regard to the kinds of tasks calculus instructors should design in order to engage students in mathematical practices that often require the use of a graphing calculator. (MM)

  10. Basic and advanced numerical performances relate to mathematical expertise but are fully mediated by visuospatial skills.

    PubMed

    Sella, Francesco; Sader, Elie; Lolliot, Simon; Cohen Kadosh, Roi

    2016-09-01

    Recent studies have highlighted the potential role of basic numerical processing in the acquisition of numerical and mathematical competences. However, it is debated whether high-level numerical skills and mathematics depends specifically on basic numerical representations. In this study mathematicians and nonmathematicians performed a basic number line task, which required mapping positive and negative numbers on a physical horizontal line, and has been shown to correlate with more advanced numerical abilities and mathematical achievement. We found that mathematicians were more accurate compared with nonmathematicians when mapping positive, but not negative numbers, which are considered numerical primitives and cultural artifacts, respectively. Moreover, performance on positive number mapping could predict whether one is a mathematician or not, and was mediated by more advanced mathematical skills. This finding might suggest a link between basic and advanced mathematical skills. However, when we included visuospatial skills, as measured by block design subtest, the mediation analysis revealed that the relation between the performance in the number line task and the group membership was explained by non-numerical visuospatial skills. These results demonstrate that relation between basic, even specific, numerical skills and advanced mathematical achievement can be artifactual and explained by visuospatial processing. (PsycINFO Database Record (c) 2016 APA, all rights reserved).

  11. Calculus Students' Representation Use in Group-Work and Individual Settings

    ERIC Educational Resources Information Center

    Zazkis, Dov

    2013-01-01

    The study of student representation use and specifically the distinction between analytic and visual representations has fueled a long line of mathematics education literature that began more than 35 years ago. This literature can be partitioned into two bodies of work, one that is primarily cognitive and one that is primarily social. In spite of…

  12. Examining the Task and Knowledge Demands Needed to Teach with Representations

    ERIC Educational Resources Information Center

    Mitchell, Rebecca; Charalambous, Charalambos Y.; Hill, Heather C.

    2014-01-01

    Representations are often used in instruction to highlight key mathematical ideas and support student learning. Despite their centrality in scaffolding teaching and learning, most of our understanding about the tasks involved with using representations in instruction and the knowledge requirements imposed on teachers when using these aids is…

  13. Translations among Mathematical Representations: Teacher Beliefs and Practices

    ERIC Educational Resources Information Center

    Bosse, Michael J.; Adu-Gyamfi, Kwaku; Cheetham, Meredith

    2011-01-01

    Student ability, teacher expectations, respective degrees of difficulty, and curriculum and instructional practices all work together to provide students experiences leading to differing levels of success in respect to mathematical translations. Herein, we discuss teacher beliefs and instructional practices, investigate why some translations seem…

  14. Reformulation of Density Functional Theory for N-Representable Densities and the Resolution of the v-Representability Problem

    DOE PAGES

    Gonis, A.; Zhang, X. G.; Stocks, G. M.; ...

    2015-10-23

    Density functional theory for the case of general, N-representable densities is reformulated in terms of density functional derivatives of expectation values of operators evaluated with wave functions leading to a density, making no reference to the concept of potential. The developments provide a complete solution of the v-representability problem by establishing a mathematical procedure that determines whether a density is v-representable and in the case of an affirmative answer determines the potential (within an additive constant) as a derivative with respect to the density of a constrained search functional. It also establishes the existence of an energy functional of themore » density that, for v-representable densities, assumes its minimum value at the density describing the ground state of an interacting many-particle system. The theorems of Hohenberg and Kohn emerge as special cases of the formalism.« less

  15. iSTEM: Promoting Fifth Graders' Mathematical Modeling

    ERIC Educational Resources Information Center

    Yanik, H. Bahadir; Karabas, Celil

    2014-01-01

    Modeling requires that people develop representations or procedures to address particular problem situations (Lesh et al. 2000). Mathematical modeling is used to describe essential characteristics of a phenomenon or a situation that one intends to study in the real world through building mathematical objects. This article describes how fifth-grade…

  16. Diagnosis and Evaluation in Mathematics Instruction: Making Contact with Students' Mental Representations.

    ERIC Educational Resources Information Center

    Davis, Robert B.

    Research on mathematics instruction is reviewed in order to respond to two questions: (1) Has the influx of talented people who have entered the mathematics instruction field over the last three decades changed anything? and (2) Will any of the work being done actually improve mathematics instruction? The different ways in which parents, students,…

  17. Comparing the Effects of Representational Tools in Collaborative and Individual Inquiry Learning

    ERIC Educational Resources Information Center

    Kolloffel, Bas; Eysink, Tessa H. S.; de Jong, Ton

    2011-01-01

    Constructing a representation in which students express their domain understanding can help them improve their knowledge. Many different representational formats can be used to express one's domain understanding (e.g., concept maps, textual summaries, mathematical equations). The format can direct students' attention to specific aspects of the…

  18. Conditions for the Effectiveness of Multiple Visual Representations in Enhancing STEM Learning

    ERIC Educational Resources Information Center

    Rau, Martina A.

    2017-01-01

    Visual representations play a critical role in enhancing science, technology, engineering, and mathematics (STEM) learning. Educational psychology research shows that adding visual representations to text can enhance students' learning of content knowledge, compared to text-only. But should students learn with a single type of visual…

  19. Assessment of representational competence in kinematics

    NASA Astrophysics Data System (ADS)

    Klein, P.; Müller, A.; Kuhn, J.

    2017-06-01

    A two-tier instrument for representational competence in the field of kinematics (KiRC) is presented, designed for a standard (1st year) calculus-based introductory mechanics course. It comprises 11 multiple choice (MC) and 7 multiple true-false (MTF) questions involving multiple representational formats, such as graphs, pictures, and formal (mathematical) expressions (1st tier). Furthermore, students express their answer confidence for selected items, providing additional information (2nd tier). Measurement characteristics of KiRC were assessed in a validation sample (pre- and post-test, N =83 and N =46 , respectively), including usefulness for measuring learning gain. Validity is checked by interviews and by benchmarking KiRC against related measures. Values for item difficulty, discrimination, and consistency are in the desired ranges; in particular, a good reliability was obtained (KR 20 =0.86 ). Confidence intervals were computed and a replication study yielded values within the latter. For practical and research purposes, KiRC as a diagnostic tool goes beyond related extant instruments both for the representational formats (e.g., mathematical expressions) and for the scope of content covered (e.g., choice of coordinate systems). Together with the satisfactory psychometric properties it appears a versatile and reliable tool for assessing students' representational competency in kinematics (and of its potential change). Confidence judgments add further information to the diagnostic potential of the test, in particular for representational misconceptions. Moreover, we present an analytic result for the question—arising from guessing correction or educational considerations—of how the total effect size (Cohen's d ) varies upon combination of two test components with known individual effect sizes, and then discuss the results in the case of KiRC (MC and MTF combination). The introduced method of test combination analysis can be applied to any test comprising

  20. Phonological Awareness Deficits in Developmental Dyslexia and the Phonological Representations Hypothesis.

    ERIC Educational Resources Information Center

    Swan, Denise; Goswami, Usha

    1997-01-01

    Used picture-naming task to identify accurate/inaccurate phonological representations by dyslexic and control children; compared performance on phonological measures for words with precise/imprecise representations. Found that frequency effects in phonological tasks disappeared after considering representational quality, and that availability of…

  1. Tracing Growth of Teachers' Classroom Interactions with Representations of Functions in the Connected Classroom

    NASA Astrophysics Data System (ADS)

    Morton, Brian Lee

    The purpose of this study is to create an empirically based theoretic model of change of the use and treatment of representations of functions with the use of Connected Classroom Technology (CCT) using data previously collected for the Classroom Connectivity in Promoting Mathematics and Science Achievement (CCMS) project. Qualitative analysis of videotapes of three algebra teachers' instruction focused on different categories thought to influence teaching representations with technology: representations, discourse, technology, and decisions. Models for rating teachers low, medium, or high for each of these categories were created using a priori codes and grounded methodology. A cross case analysis was conducted after the completion of the case studies by comparing and contrasting the three cases. Data revealed that teachers' decisions shifted to incorporate the difference in student ideas/representations made visible by the CCT into their instruction and ultimately altered their orientation to mathematics teaching. The shift in orientation seemed to lead to the teachers' growth with regards to representations, discourse, and technology.

  2. ONTIC: A Knowledge Representation System for Mathematics

    DTIC Science & Technology

    1987-01-01

    rrrPrL rr.t - ~ jr V 1.2. O.NTIC ASA FORMAL LANGUAGE 11 (IS zr) :I. (EXISTS-SOME r) (EXISTS ((xI 71) (U2 T2 ) ... ) ( li . X2, -.) . (FORALL ((xl 71) (x2...language. Jonathan Rees spent j.. about a month defining various mathematical concepts in Ontic. Starting with onlv the fundamental notions described...novelty of the Ontic system lies in the way that the above four inference mechanisms are brought together. Ontic integrates constraint prop- agation

  3. Cognitive, perceptual and action-oriented representations of falling objects.

    PubMed

    Zago, Myrka; Lacquaniti, Francesco

    2005-01-01

    We interact daily with moving objects. How accurate are our predictions about objects' motions? What sources of information do we use? These questions have received wide attention from a variety of different viewpoints. On one end of the spectrum are the ecological approaches assuming that all the information about the visual environment is present in the optic array, with no need to postulate conscious or unconscious representations. On the other end of the spectrum are the constructivist approaches assuming that a more or less accurate representation of the external world is built in the brain using explicit or implicit knowledge or memory besides sensory inputs. Representations can be related to naive physics or to context cue-heuristics or to the construction of internal copies of environmental invariants. We address the issue of prediction of objects' fall at different levels. Cognitive understanding and perceptual judgment of simple Newtonian dynamics can be surprisingly inaccurate. By contrast, motor interactions with falling objects are often very accurate. We argue that the pragmatic action-oriented behaviour and the perception-oriented behaviour may use different modes of operation and different levels of representation.

  4. Automatic Semantic Generation and Arabic Translation of Mathematical Expressions on the Web

    ERIC Educational Resources Information Center

    Doush, Iyad Abu; Al-Bdarneh, Sondos

    2013-01-01

    Automatic processing of mathematical information on the web imposes some difficulties. This paper presents a novel technique for automatic generation of mathematical equations semantic and Arabic translation on the web. The proposed system facilitates unambiguous representation of mathematical equations by correlating equations to their known…

  5. Mathematical Modeling with Middle School Students: The Robot Art Model-Eliciting Activity

    ERIC Educational Resources Information Center

    Stohlmann, Micah S.

    2017-01-01

    Internationally mathematical modeling is garnering more attention for the benefits associated with it. Mathematical modeling can develop students' communication skills and the ability to demonstrate understanding through different representations. With the increased attention on mathematical modeling, there is a need for more curricula to be…

  6. The Object Metaphor and Synecdoche in Mathematics Classroom Discourse

    ERIC Educational Resources Information Center

    Font, Vicenc; Godino, Juan D.; Planas, Nuria; Acevedo, Jorge I.

    2010-01-01

    This article describes aspects of classroom discourse, illustrated through vignettes, that reveal the complex relationship between the forms in which mathematical objects exist and their ostensive representations. We illustrate various aspects of the process through which students come to consider the reality of mathematical objects that are…

  7. Using Mental Computation Training to Improve Complex Mathematical Performance

    ERIC Educational Resources Information Center

    Liu, Allison S.; Kallai, Arava Y.; Schunn, Christian D.; Fiez, Julie A.

    2015-01-01

    Mathematical fluency is important for academic and mathematical success. Fluency training programs have typically focused on fostering retrieval, which leads to math performance that does not reliably transfer to non-trained problems. More recent studies have focused on training number understanding and representational precision, but few have…

  8. On volume-source representations based on the representation theorem

    NASA Astrophysics Data System (ADS)

    Ichihara, Mie; Kusakabe, Tetsuya; Kame, Nobuki; Kumagai, Hiroyuki

    2016-01-01

    We discuss different ways to characterize a moment tensor associated with an actual volume change of ΔV C , which has been represented in terms of either the stress glut or the corresponding stress-free volume change ΔV T . Eshelby's virtual operation provides a conceptual model relating ΔV C to ΔV T and the stress glut, where non-elastic processes such as phase transitions allow ΔV T to be introduced and subsequent elastic deformation of - ΔV T is assumed to produce the stress glut. While it is true that ΔV T correctly represents the moment tensor of an actual volume source with volume change ΔV C , an explanation as to why such an operation relating ΔV C to ΔV T exists has not previously been given. This study presents a comprehensive explanation of the relationship between ΔV C and ΔV T based on the representation theorem. The displacement field is represented using Green's function, which consists of two integrals over the source surface: one for displacement and the other for traction. Both integrals are necessary for representing volumetric sources, whereas the representation of seismic faults includes only the first term, as the second integral over the two adjacent fault surfaces, across which the traction balances, always vanishes. Therefore, in a seismological framework, the contribution from the second term should be included as an additional surface displacement. We show that the seismic moment tensor of a volume source is directly obtained from the actual state of the displacement and stress at the source without considering any virtual non-elastic operations. A purely mathematical procedure based on the representation theorem enables us to specify the additional imaginary displacement necessary for representing a volume source only by the displacement term, which links ΔV C to ΔV T . It also specifies the additional imaginary stress necessary for representing a moment tensor solely by the traction term, which gives the "stress glut." The

  9. Parents' Representations of their Children's Mathematics Learning in Multiethnic Primary Schools

    ERIC Educational Resources Information Center

    De Abreu, Guida; Cline, Tony

    2005-01-01

    There is a growing concern that governmental calls for parental involvement in children's school mathematics learning have not been underpinned by research. In this article the authors aim to offer a contribution to this debate. Links between children's home and school mathematical practices have been researched in sociocultural studies, but the…

  10. An Examination of Connections in Mathematical Processes in Students' Problem Solving: Connections between Representing and Justifying

    ERIC Educational Resources Information Center

    Stylianou, Despina A.

    2013-01-01

    Representation and justification are two central "mathematical practices". In the past, each has been examined to gain insights in the functions that they have in students' mathematical problem solving. Here, we examine the ways that representation and justification interact and influence the development of one another. We focus on the…

  11. Representation of Students in Solving Simultaneous Linear Equation Problems Based on Multiple Intelligence

    NASA Astrophysics Data System (ADS)

    Yanti, Y. R.; Amin, S. M.; Sulaiman, R.

    2018-01-01

    This study described representation of students who have musical, logical-mathematic and naturalist intelligence in solving a problem. Subjects were selected on the basis of multiple intelligence tests (TPM) consists of 108 statements, with 102 statements adopted from Chislet and Chapman and 6 statements equal to eksistensial intelligences. Data were analyzed based on problem-solving tests (TPM) and interviewing. See the validity of the data then problem-solving tests (TPM) and interviewing is given twice with an analyzed using the representation indikator and the problem solving step. The results showed that: the stage of presenting information known, stage of devising a plan, and stage of carrying out the plan those three subjects were using same form of representation. While he stage of presenting information asked and stage of looking back, subject of logical-mathematic was using different forms of representation with subjects of musical and naturalist intelligence. From this research is expected to provide input to the teacher in determining the learning strategy that will be used by considering the representation of students with the basis of multiple intelligences.

  12. Maths Games: A Universal Design Approach to Mathematical Reasoning

    ERIC Educational Resources Information Center

    Buchheister, Kelley; Jackson, Christa; Taylor, Cynthia E.

    2017-01-01

    Providing students with an opportunity to explore mathematical content through games allows teachers to include tasks that: (1) present alternative representations of the content; (2) welcome various expressions of mathematical reasoning; and (3) incorporate variations that empower all students to engage in the problem solving process. Games not…

  13. A Case against Computer Symbolic Manipulation in School Mathematics Today.

    ERIC Educational Resources Information Center

    Waits, Bert K.; Demana, Franklin

    1992-01-01

    Presented are two reasons discouraging computer symbol manipulation systems use in school mathematics at present: cost for computer laboratories or expensive pocket computers; and impracticality of exact solution representations. Although development with this technology in mathematics education advances, graphing calculators are recommended to…

  14. Understanding Linear Functions and Their Representations

    ERIC Educational Resources Information Center

    Wells, Pamela J.

    2015-01-01

    Linear functions are an important part of the middle school mathematics curriculum. Students in the middle grades gain fluency by working with linear functions in a variety of representations (NCTM 2001). Presented in this article is an activity that was used with five eighth-grade classes at three different schools. The activity contains 15 cards…

  15. The Clock Project: Gears as Visual-Tangible Representations for Mathematical Concepts

    ERIC Educational Resources Information Center

    Andrade, Alejandro

    2011-01-01

    As we have noticed from our own classroom experiences, children often find it difficult to identify the adequate operations learned in mathematics class when they are solving mechanical-operators problems in Technology class. We wanted to design a project that exploits the idea of a hands-on relationship between mathematics and technology to teach…

  16. Science Modelling in Pre-Calculus: How to Make Mathematics Problems Contextually Meaningful

    ERIC Educational Resources Information Center

    Sokolowski, Andrzej; Yalvac, Bugrahan; Loving, Cathleen

    2011-01-01

    "Use of mathematical representations to model and interpret physical phenomena and solve problems is one of the major teaching objectives in high school math curriculum" [National Council of Teachers of Mathematics (NCTM), "Principles and Standards for School Mathematics", NCTM, Reston, VA, 2000]. Commonly used pre-calculus textbooks provide a…

  17. Exploring Mental Representations for Literal Symbols Using Priming and Comparison Distance Effects

    ERIC Educational Resources Information Center

    Pollack, Courtney; Leon Guerrero, Sibylla; Star, Jon R.

    2016-01-01

    Higher-level mathematics requires a connection between literal symbols (e.g., "x") and their mental representations. The current study probes the nature of mental representations for literal symbols using both the priming distance effect, in which ease of comparing a target number to a fixed standard is a function of prime-target…

  18. Mathematics Teacher Candidates' Skills of Using Multiple Representations for Division of Fractions

    ERIC Educational Resources Information Center

    Biber, Abdullah Çagri

    2014-01-01

    The aim of this study is to reveal teacher candidates' preference regarding uses of verbal, symbolic, number line, and/or model representations of fraction divisions, and to investigate their skill of transferring from one representation type to the others. Case study was used as the research method in this study. The case that is examined within…

  19. Characterization of mathematics instructional practises for prospective elementary teachers with varying levels of self-efficacy in classroom management and mathematics teaching

    NASA Astrophysics Data System (ADS)

    Lee, Carrie W.; Walkowiak, Temple A.; Nietfeld, John L.

    2017-03-01

    The purpose of this study was to investigate the relationship between prospective teachers' (PTs) instructional practises and their efficacy beliefs in classroom management and mathematics teaching. A sequential, explanatory mixed-methods design was employed. Results from efficacy surveys, implemented with 54 PTs were linked to a sample of teachers' instructional practises during the qualitative phase. In this phase, video-recorded lessons were analysed based on tasks, representations, discourse, and classroom management. Findings indicate that PTs with higher levels of mathematics teaching efficacy taught lessons characterised by tasks of higher cognitive demand, extended student explanations, student-to-student discourse, and explicit connections between representations. Classroom management efficacy seems to bear influence on the utilised grouping structures. These findings support explicit attention to PTs' mathematics teaching and classroom management efficacy throughout teacher preparation and a need for formative feedback to inform development of beliefs about teaching practises.

  20. Intra-mathematical connections made by high school students in performing Calculus tasks

    NASA Astrophysics Data System (ADS)

    García-García, Javier; Dolores-Flores, Crisólogo

    2018-02-01

    In this article, we report the results of research that explores the intra-mathematical connections that high school students make when they solve Calculus tasks, in particular those involving the derivative and the integral. We consider mathematical connections as a cognitive process through which a person relates or associates two or more ideas, concepts, definitions, theorems, procedures, representations and meanings among themselves, with other disciplines or with real life. Task-based interviews were used to collect data and thematic analysis was used to analyze them. Through the analysis of the productions of the 25 participants, we identified 223 intra-mathematical connections. The data allowed us to establish a mathematical connections system which contributes to the understanding of higher concepts, in our case, the Fundamental Theorem of Calculus. We found mathematical connections of the types: different representations, procedural, features, reversibility and meaning as a connection.

  1. A mathematical representation of an advanced helicopter for piloted simulator investigations of control system and display variations

    NASA Technical Reports Server (NTRS)

    Aiken, E. W.

    1980-01-01

    A mathematical model of an advanced helicopter is described. The model is suitable for use in control/display research involving piloted simulation. The general design approach for the six degree of freedom equations of motion is to use the full set of nonlinear gravitational and inertial terms of the equations and to express the aerodynamic forces and moments as the reference values and first order terms of a Taylor series expansion about a reference trajectory defined as a function of longitudinal airspeed. Provisions for several different specific and generic flight control systems are included in the model. The logic required to drive various flight control and weapon delivery symbols on a pilot's electronic display is also provided. Finally, the model includes a simplified representation of low altitude wind and turbulence effects. This model was used in a piloted simulator investigation of the effects of control system and display variations for an attack helicopter mission.

  2. Reading and Mathematics Bound Together: Creating a Home Environment for Preschool Learning

    ERIC Educational Resources Information Center

    Godwin, Amber J.; Rupley, William H.; Capraro, Robert M.; Capraro, Mary Margaret

    2016-01-01

    The combination of mathematics and reading in family reading time can positively impact children's ability to make sense of representations in both mathematics and reading. Four families volunteered to participate in this field based inquiry to learn how to integrate mathematics and reading in parent-supported activities. Four parents and their…

  3. Building Blocks and Cognitive Building Blocks: Playing to Know the World Mathematically

    ERIC Educational Resources Information Center

    Sarama, Julie; Clements, Douglas H.

    2009-01-01

    The authors explore how children's play can support the development of the foundations of mathematics learning and how adults can support children's representation of--and thus the "mathematization" of--their play. The authors review research about the amount and nature of mathematics found in the free play of children. They briefly…

  4. Middle school students' reading comprehension of mathematical texts and algebraic equations

    NASA Astrophysics Data System (ADS)

    Duru, Adem; Koklu, Onder

    2011-06-01

    In this study, middle school students' abilities to translate mathematical texts into algebraic representations and vice versa were investigated. In addition, students' difficulties in making such translations and the potential sources for these difficulties were also explored. Both qualitative and quantitative methods were used to collect data for this study: questionnaire and clinical interviews. The questionnaire consisted of two general types of items: (1) selected-response (multiple-choice) items for which the respondent selects from multiple options and (2) open-ended items for which the respondent constructs a response. In order to further investigate the students' strategies while they were translating the given mathematical texts to algebraic equations and vice versa, five randomly chosen (n = 5) students were interviewed. Data were collected in the 2007-2008 school year from 185 middle-school students in five teachers' classrooms in three different schools in the city of Adıyaman, Turkey. After the analysis of data, it was found that students who participated in this study had difficulties in translating the mathematical texts into algebraic equations by using symbols. It was also observed that these students had difficulties in translating the symbolic representations into mathematical texts because of their weak reading comprehension. In addition, finding of this research revealed that students' difficulties in translating the given mathematical texts into symbolic representations or vice versa come from different sources.

  5. The interaction of representation and reasoning.

    PubMed

    Bundy, Alan

    2013-09-08

    Automated reasoning is an enabling technology for many applications of informatics. These applications include verifying that a computer program meets its specification; enabling a robot to form a plan to achieve a task and answering questions by combining information from diverse sources, e.g. on the Internet, etc. How is automated reasoning possible? Firstly, knowledge of a domain must be stored in a computer, usually in the form of logical formulae. This knowledge might, for instance, have been entered manually, retrieved from the Internet or perceived in the environment via sensors, such as cameras. Secondly, rules of inference are applied to old knowledge to derive new knowledge. Automated reasoning techniques have been adapted from logic, a branch of mathematics that was originally designed to formalize the reasoning of humans, especially mathematicians. My special interest is in the way that representation and reasoning interact. Successful reasoning is dependent on appropriate representation of both knowledge and successful methods of reasoning. Failures of reasoning can suggest changes of representation. This process of representational change can also be automated. We will illustrate the automation of representational change by drawing on recent work in my research group.

  6. The interaction of representation and reasoning

    PubMed Central

    Bundy, Alan

    2013-01-01

    Automated reasoning is an enabling technology for many applications of informatics. These applications include verifying that a computer program meets its specification; enabling a robot to form a plan to achieve a task and answering questions by combining information from diverse sources, e.g. on the Internet, etc. How is automated reasoning possible? Firstly, knowledge of a domain must be stored in a computer, usually in the form of logical formulae. This knowledge might, for instance, have been entered manually, retrieved from the Internet or perceived in the environment via sensors, such as cameras. Secondly, rules of inference are applied to old knowledge to derive new knowledge. Automated reasoning techniques have been adapted from logic, a branch of mathematics that was originally designed to formalize the reasoning of humans, especially mathematicians. My special interest is in the way that representation and reasoning interact. Successful reasoning is dependent on appropriate representation of both knowledge and successful methods of reasoning. Failures of reasoning can suggest changes of representation. This process of representational change can also be automated. We will illustrate the automation of representational change by drawing on recent work in my research group. PMID:24062623

  7. Mathematical Notation in Bibliographic Databases.

    ERIC Educational Resources Information Center

    Pasterczyk, Catherine E.

    1990-01-01

    Discusses ways in which using mathematical symbols to search online bibliographic databases in scientific and technical areas can improve search results. The representations used for Greek letters, relations, binary operators, arrows, and miscellaneous special symbols in the MathSci, Inspec, Compendex, and Chemical Abstracts databases are…

  8. A Study of Pre-Service Teachers Use of Representations in Their Proportional Reasoning

    ERIC Educational Resources Information Center

    Johnson, Kim

    2017-01-01

    Proportional reasoning is important to the field of mathematics education because it lies at the crossroads of additive reasoning in the elementary school and multiplicative reasoning needed for more advanced mathematics. This research reports on the representations used by pre-service teachers (PSTs) as they responded to tasks involving…

  9. Solving Additive Problems at Pre-Elementary School Level with the Support of Graphical Representation

    ERIC Educational Resources Information Center

    Selva, Ana Coelho Vieira; Falcao, Jorge Tarcisio da Rocha; Nunes, Terezinha

    2005-01-01

    This research offers empirical evidence of the importance of supplying diverse symbolic representations in order to support concept development in mathematics. Graphical representation can be a helpful symbolic tool for concept development in the conceptual field of additive structures. Nevertheless, this symbolic tool has specific difficulties…

  10. Quaternionic representation of the genetic code.

    PubMed

    Carlevaro, C Manuel; Irastorza, Ramiro M; Vericat, Fernando

    2016-03-01

    A heuristic diagram of the evolution of the standard genetic code is presented. It incorporates, in a way that resembles the energy levels of an atom, the physical notion of broken symmetry and it is consistent with original ideas by Crick on the origin and evolution of the code as well as with the chronological order of appearance of the amino acids along the evolution as inferred from work that mixtures known experimental results with theoretical speculations. Suggested by the diagram we propose a Hamilton quaternions based mathematical representation of the code as it stands now-a-days. The central object in the description is a codon function that assigns to each amino acid an integer quaternion in such a way that the observed code degeneration is preserved. We emphasize the advantages of a quaternionic representation of amino acids taking as an example the folding of proteins. With this aim we propose an algorithm to go from the quaternions sequence to the protein three dimensional structure which can be compared with the corresponding experimental one stored at the Protein Data Bank. In our criterion the mathematical representation of the genetic code in terms of quaternions merits to be taken into account because it describes not only most of the known properties of the genetic code but also opens new perspectives that are mainly derived from the close relationship between quaternions and rotations. Copyright © 2015 Elsevier Ireland Ltd. All rights reserved.

  11. Theoretical foundations of spatially-variant mathematical morphology part ii: gray-level images.

    PubMed

    Bouaynaya, Nidhal; Schonfeld, Dan

    2008-05-01

    In this paper, we develop a spatially-variant (SV) mathematical morphology theory for gray-level signals and images in the Euclidean space. The proposed theory preserves the geometrical concept of the structuring function, which provides the foundation of classical morphology and is essential in signal and image processing applications. We define the basic SV gray-level morphological operators (i.e., SV gray-level erosion, dilation, opening, and closing) and investigate their properties. We demonstrate the ubiquity of SV gray-level morphological systems by deriving a kernel representation for a large class of systems, called V-systems, in terms of the basic SV graylevel morphological operators. A V-system is defined to be a gray-level operator, which is invariant under gray-level (vertical) translations. Particular attention is focused on the class of SV flat gray-level operators. The kernel representation for increasing V-systems is a generalization of Maragos' kernel representation for increasing and translation-invariant function-processing systems. A representation of V-systems in terms of their kernel elements is established for increasing and upper-semi-continuous V-systems. This representation unifies a large class of spatially-variant linear and non-linear systems under the same mathematical framework. Finally, simulation results show the potential power of the general theory of gray-level spatially-variant mathematical morphology in several image analysis and computer vision applications.

  12. A roadmap for improving the representation of photosynthesis in Earth System Models

    DOE Office of Scientific and Technical Information (OSTI.GOV)

    Rogers, Alistair; Medlyn, Belinda E.; Dukes, Jeffrey S.

    Accurate representation of photosynthesis in terrestrial biosphere models (TBMs) is essential for robust projections of global change. However, current representations vary markedly between TBMs, contributing uncertainty projections of global carbon fluxes.

  13. A roadmap for improving the representation of photosynthesis in Earth System Models

    DOE PAGES

    Rogers, Alistair; Medlyn, Belinda E.; Dukes, Jeffrey S.; ...

    2016-11-28

    Accurate representation of photosynthesis in terrestrial biosphere models (TBMs) is essential for robust projections of global change. However, current representations vary markedly between TBMs, contributing uncertainty projections of global carbon fluxes.

  14. Improving Pupils' Mathematical Communication Abilities through Computer-Supported Reciprocal Peer Tutoring

    ERIC Educational Resources Information Center

    Yang, Euphony F. Y.; Chang, Ben; Cheng, Hercy N. H.; Chan, Tak-Wai

    2016-01-01

    This study examined how to foster pupils' mathematical communication abilities by using tablet PCs. Students were encouraged to generate math creations (including mathematical representation, solution, and solution explanation of word problems) as their teaching materials and reciprocally tutor classmates to increase opportunities for mathematical…

  15. Teaching Mathematics: Computers in the Classroom.

    ERIC Educational Resources Information Center

    Borba, Marcelo C.

    1995-01-01

    Discusses some major changes that computers, calculators, and graphing calculators have brought to the mathematics classroom, including quasi-empirical studies in the classroom, use of multiple representations, emphasis on visualization, emphasis on tables, an altered classroom "ecology," and increasing complexity for students. (SR)

  16. Making Connections among Multiple Graphical Representations of Fractions: Sense-Making Competencies Enhance Perceptual Fluency, but Not Vice Versa

    ERIC Educational Resources Information Center

    Rau, Martina A.; Aleven, Vincent; Rummel, Nikol

    2017-01-01

    Prior research shows that representational competencies that enable students to use graphical representations to reason and solve tasks is key to learning in many science, technology, engineering, and mathematics domains. We focus on two types of representational competencies: (1) "sense making" of connections by verbally explaining how…

  17. Brain activity associated with translation from a visual to a symbolic representation in algebra and geometry.

    PubMed

    Leikin, Mark; Waisman, Ilana; Shaul, Shelley; Leikin, Roza

    2014-03-01

    This paper presents a small part of a larger interdisciplinary study that investigates brain activity (using event related potential methodology) of male adolescents when solving mathematical problems of different types. The study design links mathematics education research with neurocognitive studies. In this paper we performed a comparative analysis of brain activity associated with the translation from visual to symbolic representations of mathematical objects in algebra and geometry. Algebraic tasks require translation from graphical to symbolic representation of a function, whereas tasks in geometry require translation from a drawing of a geometric figure to a symbolic representation of its property. The findings demonstrate that electrical activity associated with the performance of geometrical tasks is stronger than that associated with solving algebraic tasks. Additionally, we found different scalp topography of the brain activity associated with algebraic and geometric tasks. Based on these results, we argue that problem solving in algebra and geometry is associated with different patterns of brain activity.

  18. Modal Representations and Their Role in the Learning Process: A Theoretical and Pragmatic Analysis

    ERIC Educational Resources Information Center

    Gunel, Murat; Yesildag-Hasancebi, Funda

    2016-01-01

    In the construction and sharing of scientific knowledge, modal representations such as text, graphics, pictures, and mathematical expressions are commonly used. Due to the increasing importance of their role in the production and communication of science, modal representations have become a topic of growing interest in science education research…

  19. Computing with scale-invariant neural representations

    NASA Astrophysics Data System (ADS)

    Howard, Marc; Shankar, Karthik

    The Weber-Fechner law is perhaps the oldest quantitative relationship in psychology. Consider the problem of the brain representing a function f (x) . Different neurons have receptive fields that support different parts of the range, such that the ith neuron has a receptive field at xi. Weber-Fechner scaling refers to the finding that the width of the receptive field scales with xi as does the difference between the centers of adjacent receptive fields. Weber-Fechner scaling is exponentially resource-conserving. Neurophysiological evidence suggests that neural representations obey Weber-Fechner scaling in the visual system and perhaps other systems as well. We describe an optimality constraint that is solved by Weber-Fechner scaling, providing an information-theoretic rationale for this principle of neural coding. Weber-Fechner scaling can be generated within a mathematical framework using the Laplace transform. Within this framework, simple computations such as translation, correlation and cross-correlation can be accomplished. This framework can in principle be extended to provide a general computational language for brain-inspired cognitive computation on scale-invariant representations. Supported by NSF PHY 1444389 and the BU Initiative for the Physics and Mathematics of Neural Systems,.

  20. Analytical derivation: An epistemic game for solving mathematically based physics problems

    NASA Astrophysics Data System (ADS)

    Bajracharya, Rabindra R.; Thompson, John R.

    2016-06-01

    Problem solving, which often involves multiple steps, is an integral part of physics learning and teaching. Using the perspective of the epistemic game, we documented a specific game that is commonly pursued by students while solving mathematically based physics problems: the analytical derivation game. This game involves deriving an equation through symbolic manipulations and routine mathematical operations, usually without any physical interpretation of the processes. This game often creates cognitive obstacles in students, preventing them from using alternative resources or better approaches during problem solving. We conducted hour-long, semi-structured, individual interviews with fourteen introductory physics students. Students were asked to solve four "pseudophysics" problems containing algebraic and graphical representations. The problems required the application of the fundamental theorem of calculus (FTC), which is one of the most frequently used mathematical concepts in physics problem solving. We show that the analytical derivation game is necessary, but not sufficient, to solve mathematically based physics problems, specifically those involving graphical representations.

  1. Beauty as Fit: A Metaphor in Mathematics?

    ERIC Educational Resources Information Center

    Raman, Manya; Öhman, Lars-Daniel

    2013-01-01

    Beauty, which plays a central role in the practice of mathematics (Sinclair 2002), is almost absent in discussions of school mathematics (Dreyfus and Eisenberg 1986). This is problematic, because students will decide whether or not to continue their studies in mathematics without having an accurate picture of what the subject is about. In order to…

  2. How Syntactic Reasoners Can Develop Understanding, Evaluate Conjectures, and Generate Counterexamples in Advanced Mathematics

    ERIC Educational Resources Information Center

    Weber, Keith

    2009-01-01

    This paper presents a case study of a highly successful student whose exploration of an advanced mathematical concept relies predominantly on syntactic reasoning, such as developing formal representations of mathematical ideas and making logical deductions. This student is observed as he learns a new mathematical concept and then completes…

  3. Teaching Equivalent Fractions to Secondary Students with Disabilities via the Virtual-Representational-Abstract Instructional Sequence

    ERIC Educational Resources Information Center

    Bouck, Emily C.; Bassette, Laura; Shurr, Jordan; Park, Jiyoon; Kerr, Jackie; Whorley, Abbie

    2017-01-01

    Fractions are an important mathematical concept; however, fractions are also a struggle for many students with disabilities. This study explored a new framework adapted from the evidence-based concrete-representational-abstract framework: the virtual-representational-abstract (VRA) framework. The VRA framework involves teaching students to solve…

  4. Multiple Sparse Representations Classification

    PubMed Central

    Plenge, Esben; Klein, Stefan S.; Niessen, Wiro J.; Meijering, Erik

    2015-01-01

    Sparse representations classification (SRC) is a powerful technique for pixelwise classification of images and it is increasingly being used for a wide variety of image analysis tasks. The method uses sparse representation and learned redundant dictionaries to classify image pixels. In this empirical study we propose to further leverage the redundancy of the learned dictionaries to achieve a more accurate classifier. In conventional SRC, each image pixel is associated with a small patch surrounding it. Using these patches, a dictionary is trained for each class in a supervised fashion. Commonly, redundant/overcomplete dictionaries are trained and image patches are sparsely represented by a linear combination of only a few of the dictionary elements. Given a set of trained dictionaries, a new patch is sparse coded using each of them, and subsequently assigned to the class whose dictionary yields the minimum residual energy. We propose a generalization of this scheme. The method, which we call multiple sparse representations classification (mSRC), is based on the observation that an overcomplete, class specific dictionary is capable of generating multiple accurate and independent estimates of a patch belonging to the class. So instead of finding a single sparse representation of a patch for each dictionary, we find multiple, and the corresponding residual energies provides an enhanced statistic which is used to improve classification. We demonstrate the efficacy of mSRC for three example applications: pixelwise classification of texture images, lumen segmentation in carotid artery magnetic resonance imaging (MRI), and bifurcation point detection in carotid artery MRI. We compare our method with conventional SRC, K-nearest neighbor, and support vector machine classifiers. The results show that mSRC outperforms SRC and the other reference methods. In addition, we present an extensive evaluation of the effect of the main mSRC parameters: patch size, dictionary size, and

  5. On the Prony series representation of stretched exponential relaxation

    NASA Astrophysics Data System (ADS)

    Mauro, John C.; Mauro, Yihong Z.

    2018-09-01

    Stretched exponential relaxation is a ubiquitous feature of homogeneous glasses. The stretched exponential decay function can be derived from the diffusion-trap model, which predicts certain critical values of the fractional stretching exponent, β. In practical implementations of glass relaxation models, it is computationally convenient to represent the stretched exponential function as a Prony series of simple exponentials. Here, we perform a comprehensive mathematical analysis of the Prony series approximation of the stretched exponential relaxation, including optimized coefficients for certain critical values of β. The fitting quality of the Prony series is analyzed as a function of the number of terms in the series. With a sufficient number of terms, the Prony series can accurately capture the time evolution of the stretched exponential function, including its "fat tail" at long times. However, it is unable to capture the divergence of the first-derivative of the stretched exponential function in the limit of zero time. We also present a frequency-domain analysis of the Prony series representation of the stretched exponential function and discuss its physical implications for the modeling of glass relaxation behavior.

  6. Design of multiple representations e-learning resources based on a contextual approach for the basic physics course

    NASA Astrophysics Data System (ADS)

    Bakri, F.; Muliyati, D.

    2018-05-01

    This research aims to design e-learning resources with multiple representations based on a contextual approach for the Basic Physics Course. The research uses the research and development methods accordance Dick & Carey strategy. The development carried out in the digital laboratory of Physics Education Department, Mathematics and Science Faculty, Universitas Negeri Jakarta. The result of the process of product development with Dick & Carey strategy, have produced e-learning design of the Basic Physics Course is presented in multiple representations in contextual learning syntax. The appropriate of representation used in the design of learning basic physics include: concept map, video, figures, data tables of experiment results, charts of data tables, the verbal explanations, mathematical equations, problem and solutions example, and exercise. Multiple representations are presented in the form of contextual learning by stages: relating, experiencing, applying, transferring, and cooperating.

  7. Mathematical Fundamentals of Probabilistic Semantics for High-Level Fusion

    DTIC Science & Technology

    2013-12-02

    understanding of the fundamental aspects of uncertainty representation and reasoning that a theory of hard and soft high-level fusion must encompass...representation and reasoning that a theory of hard and soft high-level fusion must encompass. Successful completion requires an unbiased, in-depth...and soft information is the lack of a fundamental HLIF theory , backed by a consistent mathematical framework and supporting algorithms. Although there

  8. Tensor and Spin Representations of SO(4) and Discrete Quantum Gravity

    NASA Astrophysics Data System (ADS)

    Lorente, M.; Kramer, P.

    Starting from the defining transformations of complex matrices for the SO(4) group, we construct the fundamental representation and the tensor and spinor representations of the group SO(4). Given the commutation relations for the corresponding algebra, the unitary representations of the group in terms of the generalized Euler angles are constructed. These mathematical results help us to a more complete description of the Barret-Crane model in Quantum Gravity. In particular a complete realization of the weight function for the partition function is given and a new geometrical interpretation of the asymptotic limit for the Regge action is presented.

  9. Categorical Working Memory Representations are used in Delayed Estimation of Continuous Colors

    PubMed Central

    Hardman, Kyle O; Vergauwe, Evie; Ricker, Timothy J

    2016-01-01

    In the last decade, major strides have been made in understanding visual working memory through mathematical modeling of color production responses. In the delayed color estimation task (Wilken & Ma, 2004), participants are given a set of colored squares to remember and a few seconds later asked to reproduce those colors by clicking on a color wheel. The degree of error in these responses is characterized with mathematical models that estimate working memory precision and the proportion of items remembered by participants. A standard mathematical model of color memory assumes that items maintained in memory are remembered through memory for precise details about the particular studied shade of color. We contend that this model is incomplete in its present form because no mechanism is provided for remembering the coarse category of a studied color. In the present work we remedy this omission and present a model of visual working memory that includes both continuous and categorical memory representations. In two experiments we show that our new model outperforms this standard modeling approach, which demonstrates that categorical representations should be accounted for by mathematical models of visual working memory. PMID:27797548

  10. Categorical working memory representations are used in delayed estimation of continuous colors.

    PubMed

    Hardman, Kyle O; Vergauwe, Evie; Ricker, Timothy J

    2017-01-01

    In the last decade, major strides have been made in understanding visual working memory through mathematical modeling of color production responses. In the delayed color estimation task (Wilken & Ma, 2004), participants are given a set of colored squares to remember, and a few seconds later asked to reproduce those colors by clicking on a color wheel. The degree of error in these responses is characterized with mathematical models that estimate working memory precision and the proportion of items remembered by participants. A standard mathematical model of color memory assumes that items maintained in memory are remembered through memory for precise details about the particular studied shade of color. We contend that this model is incomplete in its present form because no mechanism is provided for remembering the coarse category of a studied color. In the present work, we remedy this omission and present a model of visual working memory that includes both continuous and categorical memory representations. In 2 experiments, we show that our new model outperforms this standard modeling approach, which demonstrates that categorical representations should be accounted for by mathematical models of visual working memory. (PsycINFO Database Record (c) 2016 APA, all rights reserved).

  11. Neural representations of magnitude for natural and rational numbers.

    PubMed

    DeWolf, Melissa; Chiang, Jeffrey N; Bassok, Miriam; Holyoak, Keith J; Monti, Martin M

    2016-11-01

    Humans have developed multiple symbolic representations for numbers, including natural numbers (positive integers) as well as rational numbers (both fractions and decimals). Despite a considerable body of behavioral and neuroimaging research, it is currently unknown whether different notations map onto a single, fully abstract, magnitude code, or whether separate representations exist for specific number types (e.g., natural versus rational) or number representations (e.g., base-10 versus fractions). We address this question by comparing brain metabolic response during a magnitude comparison task involving (on different trials) integers, decimals, and fractions. Univariate and multivariate analyses revealed that the strength and pattern of activation for fractions differed systematically, within the intraparietal sulcus, from that of both decimals and integers, while the latter two number representations appeared virtually indistinguishable. These results demonstrate that the two major notations formats for rational numbers, fractions and decimals, evoke distinct neural representations of magnitude, with decimals representations being more closely linked to those of integers than to those of magnitude-equivalent fractions. Our findings thus suggest that number representation (base-10 versus fractions) is an important organizational principle for the neural substrate underlying mathematical cognition. Copyright © 2016 Elsevier Inc. All rights reserved.

  12. Action-Based Digital Tools: Mathematics Learning in 6-Year-Old Children

    ERIC Educational Resources Information Center

    Dejonckheere, Peter J. N.; Desoete, Annemie; Fonck, Nathalie; Roderiguez, Dave; Six, Leen; Vermeersch, Tine; Vermeulen, Lies

    2014-01-01

    Introduction: In the present study we used a metaphorical representation in order to stimulate the numerical competences of six-year-olds. It was expected that when properties of physical action are used for mathematical thinking or when abstract mathematical thinking is grounded in sensorimotor processes, learning gains should be more pronounced…

  13. The MATH--Open Source Application for Easier Learning of Numerical Mathematics

    ERIC Educational Resources Information Center

    Glaser-Opitz, Henrich; Budajová, Kristina

    2016-01-01

    The article introduces a software application (MATH) supporting an education of Applied Mathematics, with focus on Numerical Mathematics. The MATH is an easy to use tool supporting various numerical methods calculations with graphical user interface and integrated plotting tool for graphical representation written in Qt with extensive use of Qwt…

  14. Association between basic numerical abilities and mathematics achievement.

    PubMed

    Sasanguie, Delphine; De Smedt, Bert; Defever, Emmy; Reynvoet, Bert

    2012-06-01

    Various measures have been used to investigate number processing in children, including a number comparison or a number line estimation task. The present study aimed to examine whether and to which extent these different measures of number representation are related to performance on a curriculum-based standardized mathematics achievement test in kindergarteners, first, second, and sixth graders. Children completed a number comparison task and a number line estimation task with a balanced set of symbolic (Arabic digits) and non-symbolic (dot patterns) stimuli. Associations with mathematics achievement were observed for the symbolic measures. Although the association with number line estimation was consistent over grades, the association with number comparison was much stronger in kindergarten compared to the other grades. The current data indicate that a good knowledge of the numerical meaning of Arabic digits is important for children's mathematical development and that particularly the access to the numerical meaning of symbolic digits rather than the representation of number per se is important. © 2011 The British Psychological Society.

  15. The language of mathematics: investigating the ways language counts for children's mathematical development.

    PubMed

    Vukovic, Rose K; Lesaux, Nonie K

    2013-06-01

    This longitudinal study examined how language ability relates to mathematical development in a linguistically and ethnically diverse sample of children from 6 to 9 years of age. Study participants were 75 native English speakers and 92 language minority learners followed from first to fourth grades. Autoregression in a structural equation modeling (SEM) framework was used to evaluate the relation between children's language ability and gains in different domains of mathematical cognition (i.e., arithmetic, data analysis/probability, algebra, and geometry). The results showed that language ability predicts gains in data analysis/probability and geometry, but not in arithmetic or algebra, after controlling for visual-spatial working memory, reading ability, and sex. The effect of language on gains in mathematical cognition did not differ between language minority learners and native English speakers. These findings suggest that language influences how children make meaning of mathematics but is not involved in complex arithmetical procedures whether presented with Arabic symbols as in arithmetic or with abstract symbols as in algebraic reasoning. The findings further indicate that early language experiences are important for later mathematical development regardless of language background, denoting the need for intensive and targeted language opportunities for language minority and native English learners to develop mathematical concepts and representations. Copyright © 2013. Published by Elsevier Inc.

  16. Women in Mathematics: Scaling the Heights. MAA Notes Number 46.

    ERIC Educational Resources Information Center

    Nolan, Deborah, Ed.

    Women and mathematics have been thought of as two totally separate subjects for decades. In July, 1994 a group of mathematicians from around the country gathered in Berkeley, CA for three days to discuss ways to increase the representation of women in Ph.D. programs in the mathematical sciences. The primary goal of this conference was to broaden…

  17. Sex Differences in the Spatial Representation of Number

    ERIC Educational Resources Information Center

    Bull, Rebecca; Cleland, Alexandra A.; Mitchell, Thomas

    2013-01-01

    There is a large body of accumulated evidence from behavioral and neuroimaging studies regarding how and where in the brain we represent basic numerical information. A number of these studies have considered how numerical representations may differ between individuals according to their age or level of mathematical ability, but one issue rarely…

  18. Coordinating Multiple Representations in a Reform Calculus Textbook

    ERIC Educational Resources Information Center

    Chang, Briana L.; Cromley, Jennifer G.; Tran, Nhi

    2015-01-01

    Coordination of multiple representations (CMR) is widely recognized as a critical skill in mathematics and is frequently demanded in reform calculus textbooks. However, little is known about the prevalence of coordination tasks in such textbooks. We coded 707 instances of CMR in a widely used reform calculus textbook and analyzed the distributions…

  19. Coordinating Multiple Representations in a Reform Calculus Textbook

    ERIC Educational Resources Information Center

    Chang, Briana L.; Cromley, Jennifer G.; Tran, Nhi

    2016-01-01

    Coordination of multiple representations (CMR) is widely recognized as a critical skill in mathematics and is frequently demanded in reform calculus textbooks. However, little is known about the prevalence of coordination tasks in such textbooks. We coded 707 instances of CMR in a widely used reform calculus textbook and analyzed the distributions…

  20. Teaching Problem Solving to Students Receiving Tiered Interventions Using the Concrete-Representational-Abstract Sequence and Schema-Based Instruction

    ERIC Educational Resources Information Center

    Flores, Margaret M.; Hinton, Vanessa M.; Burton, Megan E.

    2016-01-01

    Mathematical word problems are the most common form of mathematics problem solving implemented in K-12 schools. Identifying key words is a frequent strategy taught in classrooms in which students struggle with problem solving and show low success rates in mathematics. Researchers show that using the concrete-representational-abstract (CRA)…

  1. Role of multiple representations in physics problem solving

    NASA Astrophysics Data System (ADS)

    Maries, Alexandru

    This thesis explores the role of multiple representations in introductory physics students' problem solving performance through several investigations. Representations can help students focus on the conceptual aspects of physics and play a major role in effective problem solving. Diagrammatic representations can play a particularly important role in the initial stages of conceptual analysis and planning of the problem solution. Findings suggest that students who draw productive diagrams are more successful problem solvers even if their approach is primarily mathematical. Furthermore, students provided with a diagram of the physical situation presented in a problem sometimes exhibited deteriorated performance. Think-aloud interviews suggest that this deteriorated performance is in part due to reduced conceptual planning time which caused students to jump to the implementation stage without fully understanding the problem and planning problem solution. Another study investigated two interventions aimed at improving introductory students' representational consistency between mathematical and graphical representations and revealed that excessive scaffolding can have a detrimental effect. The detrimental effect was partly due to increased cognitive load brought on by the additional steps and instructions. Moreover, students who exhibited representational consistency also showed improved problem solving performance. The final investigation is centered on a problem solving task designed to provide information about the pedagogical content knowledge (PCK) of graduate student teaching assistants (TAs). In particular, the TAs identified what they considered to be the most common difficulties of introductory physics students related to graphical representations of kinematics concepts as they occur in the Test of Understanding Graphs in Kinematics (TUG-K). As an extension, the Force Concept Inventory (FCI) was also used to assess this aspect of PCK related to knowledge of

  2. Increasingly complex representations of natural movies across the dorsal stream are shared between subjects.

    PubMed

    Güçlü, Umut; van Gerven, Marcel A J

    2017-01-15

    Recently, deep neural networks (DNNs) have been shown to provide accurate predictions of neural responses across the ventral visual pathway. We here explore whether they also provide accurate predictions of neural responses across the dorsal visual pathway, which is thought to be devoted to motion processing and action recognition. This is achieved by training deep neural networks to recognize actions in videos and subsequently using them to predict neural responses while subjects are watching natural movies. Moreover, we explore whether dorsal stream representations are shared between subjects. In order to address this question, we examine if individual subject predictions can be made in a common representational space estimated via hyperalignment. Results show that a DNN trained for action recognition can be used to accurately predict how dorsal stream responds to natural movies, revealing a correspondence in representations of DNN layers and dorsal stream areas. It is also demonstrated that models operating in a common representational space can generalize to responses of multiple or even unseen individual subjects to novel spatio-temporal stimuli in both encoding and decoding settings, suggesting that a common representational space underlies dorsal stream responses across multiple subjects. Copyright © 2015 Elsevier Inc. All rights reserved.

  3. Evolving Polygons and Spreadsheets: Connecting Mathematics across Grade Levels in Teacher Education

    ERIC Educational Resources Information Center

    Abramovich, Sergei; Brouwer, Peter

    2009-01-01

    This paper was prepared in response to the Conference Board of Mathematical Sciences recommendations for the preparation of secondary teachers. It shows how using trigonometry as a conceptual tool in spreadsheet-based applications enables one to develop mathematical understanding in the context of constructing geometric representations of unit…

  4. Basic mathematical cognition.

    PubMed

    Gaber, David; Schlimm, Dirk

    2015-01-01

    Mathematics is a powerful tool for describing and developing our knowledge of the physical world. It informs our understanding of subjects as diverse as music, games, science, economics, communications protocols, and visual arts. Mathematical thinking has its roots in the adaptive behavior of living creatures: animals must employ judgments about quantities and magnitudes in the assessment of both threats (how many foes) and opportunities (how much food) in order to make effective decisions, and use geometric information in the environment for recognizing landmarks and navigating environments. Correspondingly, cognitive systems that are dedicated to the processing of distinctly mathematical information have developed. In particular, there is evidence that certain core systems for understanding different aspects of arithmetic as well as geometry are employed by humans and many other animals. They become active early in life and, particularly in the case of humans, develop through maturation. Although these core systems individually appear to be quite limited in application, in combination they allow for the recognition of mathematical properties and the formation of appropriate inferences based upon those properties. In this overview, the core systems, their roles, their limitations, and their interaction with external representations are discussed, as well as possibilities for how they can be employed together to allow us to reason about more complex mathematical domains. © 2015 John Wiley & Sons, Ltd.

  5. Detailed 3D representations for object recognition and modeling.

    PubMed

    Zia, M Zeeshan; Stark, Michael; Schiele, Bernt; Schindler, Konrad

    2013-11-01

    Geometric 3D reasoning at the level of objects has received renewed attention recently in the context of visual scene understanding. The level of geometric detail, however, is typically limited to qualitative representations or coarse boxes. This is linked to the fact that today's object class detectors are tuned toward robust 2D matching rather than accurate 3D geometry, encouraged by bounding-box-based benchmarks such as Pascal VOC. In this paper, we revisit ideas from the early days of computer vision, namely, detailed, 3D geometric object class representations for recognition. These representations can recover geometrically far more accurate object hypotheses than just bounding boxes, including continuous estimates of object pose and 3D wireframes with relative 3D positions of object parts. In combination with robust techniques for shape description and inference, we outperform state-of-the-art results in monocular 3D pose estimation. In a series of experiments, we analyze our approach in detail and demonstrate novel applications enabled by such an object class representation, such as fine-grained categorization of cars and bicycles, according to their 3D geometry, and ultrawide baseline matching.

  6. A Plan for Improving Mathematics Instruction in California Elementary Schools. Final Report of the Mathematics Education Task Force.

    ERIC Educational Resources Information Center

    Hoffman, Joseph R.; Tardif, Robert F.

    In a project designed to improve elementary school instruction in mathematics, the California Department of Education collected achievement and profile data from 67 elementary schools. Schools were classified according to size, socioeconomic status, minority representation and mobility of students, city size, and type of community. Profile data…

  7. Accurate finite difference methods for time-harmonic wave propagation

    NASA Technical Reports Server (NTRS)

    Harari, Isaac; Turkel, Eli

    1994-01-01

    Finite difference methods for solving problems of time-harmonic acoustics are developed and analyzed. Multidimensional inhomogeneous problems with variable, possibly discontinuous, coefficients are considered, accounting for the effects of employing nonuniform grids. A weighted-average representation is less sensitive to transition in wave resolution (due to variable wave numbers or nonuniform grids) than the standard pointwise representation. Further enhancement in method performance is obtained by basing the stencils on generalizations of Pade approximation, or generalized definitions of the derivative, reducing spurious dispersion, anisotropy and reflection, and by improving the representation of source terms. The resulting schemes have fourth-order accurate local truncation error on uniform grids and third order in the nonuniform case. Guidelines for discretization pertaining to grid orientation and resolution are presented.

  8. Impact of High Mathematics Education on the Number Sense

    PubMed Central

    Castronovo, Julie; Göbel, Silke M.

    2012-01-01

    In adult number processing two mechanisms are commonly used: approximate estimation of quantity and exact calculation. While the former relies on the approximate number sense (ANS) which we share with animals and preverbal infants, the latter has been proposed to rely on an exact number system (ENS) which develops later in life following the acquisition of symbolic number knowledge. The current study investigated the influence of high level math education on the ANS and the ENS. Our results showed that the precision of non-symbolic quantity representation was not significantly altered by high level math education. However, performance in a symbolic number comparison task as well as the ability to map accurately between symbolic and non-symbolic quantities was significantly better the higher mathematics achievement. Our findings suggest that high level math education in adults shows little influence on their ANS, but it seems to be associated with a better anchored ENS and better mapping abilities between ENS and ANS. PMID:22558077

  9. Impact of high mathematics education on the number sense.

    PubMed

    Castronovo, Julie; Göbel, Silke M

    2012-01-01

    In adult number processing two mechanisms are commonly used: approximate estimation of quantity and exact calculation. While the former relies on the approximate number sense (ANS) which we share with animals and preverbal infants, the latter has been proposed to rely on an exact number system (ENS) which develops later in life following the acquisition of symbolic number knowledge. The current study investigated the influence of high level math education on the ANS and the ENS. Our results showed that the precision of non-symbolic quantity representation was not significantly altered by high level math education. However, performance in a symbolic number comparison task as well as the ability to map accurately between symbolic and non-symbolic quantities was significantly better the higher mathematics achievement. Our findings suggest that high level math education in adults shows little influence on their ANS, but it seems to be associated with a better anchored ENS and better mapping abilities between ENS and ANS.

  10. Computing and STEM in Greece: Gender Representation of Students and Teachers during the Decade 2002/2012

    ERIC Educational Resources Information Center

    Kordaki, Maria; Berdousis, Ioannis

    2017-01-01

    Female student representation in Computing and Science, Technology, Engineering and Mathematics (STEM) Tertiary education is under-researched in a number of countries including Greece, while studies on female secondary level education teacher representation in Computing and STEM have not yet been reported. This study focuses on the investigation…

  11. The field representation language.

    PubMed

    Tsafnat, Guy

    2008-02-01

    The complexity of quantitative biomedical models, and the rate at which they are published, is increasing to a point where managing the information has become all but impossible without automation. International efforts are underway to standardise representation languages for a number of mathematical entities that represent a wide variety of physiological systems. This paper presents the Field Representation Language (FRL), a portable representation of values that change over space and/or time. FRL is an extensible mark-up language (XML) derivative with support for large numeric data sets in Hierarchical Data Format version 5 (HDF5). Components of FRL can be reused through unified resource identifiers (URI) that point to external resources such as custom basis functions, boundary geometries and numerical data. To demonstrate the use of FRL as an interchange we present three models that study hyperthermia cancer treatment: a fractal model of liver tumour microvasculature; a probabilistic model simulating the deposition of magnetic microspheres throughout it; and a finite element model of hyperthermic treatment. The microsphere distribution field was used to compute the heat generation rate field around the tumour. We used FRL to convey results from the microsphere simulation to the treatment model. FRL facilitated the conversion of the coordinate systems and approximated the integral over regions of the microsphere deposition field.

  12. Representational Competence: Towards a Distributed and Embodied Cognition Account

    ERIC Educational Resources Information Center

    Pande, Prajakt; Chandrasekharan, Sanjay

    2017-01-01

    Multiple external representations (MERs) are central to the practice and learning of science, mathematics and engineering, as the phenomena and entities investigated and controlled in these domains are often not available for perception and action. MERs therefore play a twofold constitutive role in reasoning in these domains. Firstly, MERs stand…

  13. Numerical Magnitude Representation in Children With Mathematical Difficulties With or Without Reading Difficulties.

    PubMed

    Tobia, Valentina; Fasola, Anna; Lupieri, Alice; Marzocchi, Gian Marco

    2016-01-01

    This study aimed to explore the spatial numerical association of response codes (SNARC), the flanker, and the numerical distance effects in children with mathematical difficulties. From a sample of 720 third, fourth, and fifth graders, 60 children were selected and divided into the following three groups: typically developing children (TD; n = 29), children with mathematical difficulties only (MD only; n = 21), and children with mathematical and reading difficulties (MD+RD; n = 10). Children were tested with a numerical Eriksen task that was built to assess SNARC, numerical distance, and flanker (first and second order congruency) effects. Children with MD only showed stronger SNARC and second order congruency effects than did TD children, whereas the numerical distance effects were similar across the three groups. Finally, the first order congruency effect was associated with reading difficulties. These results showed that children with mathematical difficulties with or without reading difficulties were globally more impaired when spatial incompatibilities were presented. © Hammill Institute on Disabilities 2014.

  14. On Mathematical Proving

    NASA Astrophysics Data System (ADS)

    Stefaneas, Petros; Vandoulakis, Ioannis M.

    2015-12-01

    This paper outlines a logical representation of certain aspects of the process of mathematical proving that are important from the point of view of Artificial Intelligence. Our starting-point is the concept of proof-event or proving, introduced by Goguen, instead of the traditional concept of mathematical proof. The reason behind this choice is that in contrast to the traditional static concept of mathematical proof, proof-events are understood as processes, which enables their use in Artificial Intelligence in such contexts, in which problem-solving procedures and strategies are studied. We represent proof-events as problem-centered spatio-temporal processes by means of the language of the calculus of events, which captures adequately certain temporal aspects of proof-events (i.e. that they have history and form sequences of proof-events evolving in time). Further, we suggest a "loose" semantics for the proof-events, by means of Kolmogorov's calculus of problems. Finally, we expose the intented interpretations for our logical model from the fields of automated theorem-proving and Web-based collective proving.

  15. The Invisible Link: Using State Space Representations to Investigate the Connection between Variables and Their Referents

    ERIC Educational Resources Information Center

    Pollack, Courtney

    2012-01-01

    The ability to represent numerical quantities in symbolic form is a necessary foundation for mathematical competence. Variables are particularly important symbolic representations for learning algebra and succeeding in higher mathematics, but the mechanisms of how students link a variable to what it represents are not well understood. Research…

  16. Investigating Lebanese Grade Seven Biology Teachers Mathematical Knowledge and Skills: A Case Study

    ERIC Educational Resources Information Center

    Raad, Nawal Abou; Chatila, Hanadi

    2016-01-01

    This paper investigates Lebanese grade 7 biology teachers' mathematical knowledge and skills, by exploring how they explain a visual representation in an activity depending on the mathematical concept "Function". Twenty Lebanese in-service biology teachers participated in the study, and were interviewed about their explanation for the…

  17. Constructions of Mathematicians in Popular Culture and Learners' Narratives: A Study of Mathematical and Non-Mathematical Subjectivities

    ERIC Educational Resources Information Center

    Moreau, Marie-Pierre; Mendick, Heather; Epstein, Debbie

    2010-01-01

    In this paper, based on a project funded by the UK Economic and Social Research Council considering how people position themselves in relation to popular representations of mathematics and mathematicians, we explore constructions of mathematicians in popular culture and the ways learners make meanings from these. Drawing on an analysis of popular…

  18. The effect of mathematics anxiety on the processing of numerical magnitude.

    PubMed

    Maloney, Erin A; Ansari, Daniel; Fugelsang, Jonathan A

    2011-01-01

    In an effort to understand the origins of mathematics anxiety, we investigated the processing of symbolic magnitude by high mathematics-anxious (HMA) and low mathematics-anxious (LMA) individuals by examining their performance on two variants of the symbolic numerical comparison task. In two experiments, a numerical distance by mathematics anxiety (MA) interaction was obtained, demonstrating that the effect of numerical distance on response times was larger for HMA than for LMA individuals. These data support the claim that HMA individuals have less precise representations of numerical magnitude than their LMA peers, suggesting that MA is associated with low-level numerical deficits that compromise the development of higher level mathematical skills.

  19. Approximate numerical abilities and mathematics: Insight from correlational and experimental training studies.

    PubMed

    Hyde, D C; Berteletti, I; Mou, Y

    2016-01-01

    Humans have the ability to nonverbally represent the approximate numerosity of sets of objects. The cognitive system that supports this ability, often referred to as the approximate number system (ANS), is present in early infancy and continues to develop in precision over the life span. It has been proposed that the ANS forms a foundation for uniquely human symbolic number and mathematics learning. Recent work has brought two types of evidence to bear on the relationship between the ANS and human mathematics: correlational studies showing individual differences in approximate numerical abilities correlate with individual differences in mathematics achievement and experimental studies showing enhancing effects of nonsymbolic approximate numerical training on exact, symbolic mathematical abilities. From this work, at least two accounts can be derived from these empirical data. It may be the case that the ANS and mathematics are related because the cognitive and brain processes responsible for representing numerical quantity in each format overlap, the Representational Overlap Hypothesis, or because of commonalities in the cognitive operations involved in mentally manipulating the representations of each format, the Operational Overlap hypothesis. The two hypotheses make distinct predictions for future work to test. © 2016 Elsevier B.V. All rights reserved.

  20. Space-Time Error Representation and Estimation in Navier-Stokes Calculations

    NASA Technical Reports Server (NTRS)

    Barth, Timothy J.

    2006-01-01

    The mathematical framework for a-posteriori error estimation of functionals elucidated by Eriksson et al. [7] and Becker and Rannacher [3] is revisited in a space-time context. Using these theories, a hierarchy of exact and approximate error representation formulas are presented for use in error estimation and mesh adaptivity. Numerical space-time results for simple model problems as well as compressible Navier-Stokes flow at Re = 300 over a 2D circular cylinder are then presented to demonstrate elements of the error representation theory for time-dependent problems.

  1. Characterizing representational learning: A combined simulation and tutorial on perturbation theory

    NASA Astrophysics Data System (ADS)

    Kohnle, Antje; Passante, Gina

    2017-12-01

    Analyzing, constructing, and translating between graphical, pictorial, and mathematical representations of physics ideas and reasoning flexibly through them ("representational competence") is a key characteristic of expertise in physics but is a challenge for learners to develop. Interactive computer simulations and University of Washington style tutorials both have affordances to support representational learning. This article describes work to characterize students' spontaneous use of representations before and after working with a combined simulation and tutorial on first-order energy corrections in the context of quantum-mechanical time-independent perturbation theory. Data were collected from two institutions using pre-, mid-, and post-tests to assess short- and long-term gains. A representational competence level framework was adapted to devise level descriptors for the assessment items. The results indicate an increase in the number of representations used by students and the consistency between them following the combined simulation tutorial. The distributions of representational competence levels suggest a shift from perceptual to semantic use of representations based on their underlying meaning. In terms of activity design, this study illustrates the need to support students in making sense of the representations shown in a simulation and in learning to choose the most appropriate representation for a given task. In terms of characterizing representational abilities, this study illustrates the usefulness of a framework focusing on perceptual, syntactic, and semantic use of representations.

  2. Field Dependency and Performance in Mathematics

    ERIC Educational Resources Information Center

    Onwumere, Onyebuchi; Reid, Norman

    2014-01-01

    Mathematics is an important school subject but one which often poses problems for learners. It has been found that learners do not possess the cognitive capacity to handle understanding procedures, representations, concepts, and applications at the same time. while the extent of field dependency may hold the key to one way by which the working…

  3. Concept Maps Provide a Window onto Preservice Elementary Teachers' Knowledge in the Teaching and Learning of Mathematics

    ERIC Educational Resources Information Center

    Chichekian, Tanya; Shore, Bruce M.

    2013-01-01

    This collaborative concept-mapping exercise was conducted in a second-year mathematics methods course. Teachers' visual representations of their mathematical content and pedagogical knowledge provided insight into their understanding of how students learn mathematics. We collected 28 preservice student teachers' concept maps and analyzed them by…

  4. A scale-invariant internal representation of time.

    PubMed

    Shankar, Karthik H; Howard, Marc W

    2012-01-01

    We propose a principled way to construct an internal representation of the temporal stimulus history leading up to the present moment. A set of leaky integrators performs a Laplace transform on the stimulus function, and a linear operator approximates the inversion of the Laplace transform. The result is a representation of stimulus history that retains information about the temporal sequence of stimuli. This procedure naturally represents more recent stimuli more accurately than less recent stimuli; the decrement in accuracy is precisely scale invariant. This procedure also yields time cells that fire at specific latencies following the stimulus with a scale-invariant temporal spread. Combined with a simple associative memory, this representation gives rise to a moment-to-moment prediction that is also scale invariant in time. We propose that this scale-invariant representation of temporal stimulus history could serve as an underlying representation accessible to higher-level behavioral and cognitive mechanisms. In order to illustrate the potential utility of this scale-invariant representation in a variety of fields, we sketch applications using minimal performance functions to problems in classical conditioning, interval timing, scale-invariant learning in autoshaping, and the persistence of the recency effect in episodic memory across timescales.

  5. Bag of Lines (BoL) for Improved Aerial Scene Representation

    DOE Office of Scientific and Technical Information (OSTI.GOV)

    Sridharan, Harini; Cheriyadat, Anil M.

    2014-09-22

    Feature representation is a key step in automated visual content interpretation. In this letter, we present a robust feature representation technique, referred to as bag of lines (BoL), for high-resolution aerial scenes. The proposed technique involves extracting and compactly representing low-level line primitives from the scene. The compact scene representation is generated by counting the different types of lines representing various linear structures in the scene. Through extensive experiments, we show that the proposed scene representation is invariant to scale changes and scene conditions and can discriminate urban scene categories accurately. We compare the BoL representation with the popular scalemore » invariant feature transform (SIFT) and Gabor wavelets for their classification and clustering performance on an aerial scene database consisting of images acquired by sensors with different spatial resolutions. The proposed BoL representation outperforms the SIFT- and Gabor-based representations.« less

  6. Characterizing and Supporting Change in Algebra Students' Representational Fluency in a CAS/Paper-and-Pencil Environment

    ERIC Educational Resources Information Center

    Fonger, Nicole L.

    2012-01-01

    Representational fluency (RF) includes an ability to interpret, create, move within and among, and connect tool-based representations of mathematical objects. Taken as an indicator of conceptual understanding, there is a need to better support school algebra students' RF in learning environments that utilize both computer algebra systems…

  7. Instructional Supports for Representational Fluency in Solving Linear Equations with Computer Algebra Systems and Paper-and-Pencil

    ERIC Educational Resources Information Center

    Fonger, Nicole L.; Davis, Jon D.; Rohwer, Mary Lou

    2018-01-01

    This research addresses the issue of how to support students' representational fluency--the ability to create, move within, translate across, and derive meaning from external representations of mathematical ideas. The context of solving linear equations in a combined computer algebra system (CAS) and paper-and-pencil classroom environment is…

  8. Creating a Critical Mass Eliminates the Effects of Stereotype Threat on Women's Mathematical Performance

    ERIC Educational Resources Information Center

    Pennington, Charlotte R.; Heim, Derek

    2016-01-01

    Background: Women in mathematical domains may become attuned to situational cues that signal a discredited social identity, contributing to their lower achievement and underrepresentation. Aim: This study examined whether heightened in-group representation alleviates the effects of stereotype threat on women's mathematical performance. It further…

  9. Framing discourse for optimal learning in science and mathematics

    NASA Astrophysics Data System (ADS)

    Megowan, Mary Colleen

    2007-12-01

    This study explored the collaborative thinking and learning that occurred in physics and mathematics classes where teachers practiced Modeling Instruction. Four different classes were videotaped---a middle school mathematics resource class, a 9th grade physical science class, a high school honors physics class and a community college engineering physics course. Videotapes and transcripts were analyzed to discover connections between the conceptual structures and spatial representations that shaped students' conversations about space and time. Along the way, it became apparent that students' and teachers' cultural models of schooling were a significant influence, sometimes positive and sometimes negative, in students' engagement and metaphor selection. A growing number of researchers are exploring the importance of semiotics in physics and mathematics, but typically their unit of analysis is the individual student. To examine the distributed cognition that occurred in this unique learning setting, not just among students but also in connection with their tools, artifacts and representations, I extended the unit of analysis for my research to include small groups and their collaborative work with whiteboarded representations of contextual problems and laboratory exercises. My data revealed a number of interesting insights. Students who constructed spatial representations and used them to assist their reasoning, were more apt to demonstrate a coherent grasp of the elements, operations, relations and rules that govern the model under investigation than those who relied on propositional algebraic representations of the model. In classrooms where teachers permitted and encouraged students to take and hold the floor during whole-group discussions, students learned to probe one another more deeply and conceptually. Shared representations (whether spatial or propositional/algebraic), such as those that naturally occurred when students worked together in small groups to

  10. Enhancing Conceptual Knowledge of Energy in Biology with Incorrect Representations

    ERIC Educational Resources Information Center

    Wernecke, Ulrike; Schütte, Kerstin; Schwanewedel, Julia; Harms, Ute

    2018-01-01

    Energy is an important concept in all natural sciences, and a challenging one for school science education. Students' conceptual knowledge of energy is often low, and they entertain misconceptions. Educational research in science and mathematics suggests that learning through depictive representations and learning from errors, based on the theory…

  11. Nonsymbolic number and cumulative area representations contribute shared and unique variance to symbolic math competence

    PubMed Central

    Lourenco, Stella F.; Bonny, Justin W.; Fernandez, Edmund P.; Rao, Sonia

    2012-01-01

    Humans and nonhuman animals share the capacity to estimate, without counting, the number of objects in a set by relying on an approximate number system (ANS). Only humans, however, learn the concepts and operations of symbolic mathematics. Despite vast differences between these two systems of quantification, neural and behavioral findings suggest functional connections. Another line of research suggests that the ANS is part of a larger, more general system of magnitude representation. Reports of cognitive interactions and common neural coding for number and other magnitudes such as spatial extent led us to ask whether, and how, nonnumerical magnitude interfaces with mathematical competence. On two magnitude comparison tasks, college students estimated (without counting or explicit calculation) which of two arrays was greater in number or cumulative area. They also completed a battery of standardized math tests. Individual differences in both number and cumulative area precision (measured by accuracy on the magnitude comparison tasks) correlated with interindividual variability in math competence, particularly advanced arithmetic and geometry, even after accounting for general aspects of intelligence. Moreover, analyses revealed that whereas number precision contributed unique variance to advanced arithmetic, cumulative area precision contributed unique variance to geometry. Taken together, these results provide evidence for shared and unique contributions of nonsymbolic number and cumulative area representations to formally taught mathematics. More broadly, they suggest that uniquely human branches of mathematics interface with an evolutionarily primitive general magnitude system, which includes partially overlapping representations of numerical and nonnumerical magnitude. PMID:23091023

  12. Humanize Your Classroom with the History of Mathematics.

    ERIC Educational Resources Information Center

    Bidwell, James K.

    1993-01-01

    History humanizes mathematics and explains the "whys" by showing its logical development. Communicating, connecting, and valuing of mathematics can be effected by interjecting historical moments, reading biographies, creating displays for the classroom, and tracing historically accurate developments of topics in class. (Contains 26…

  13. Diagrammatic Representational Constraints of Spatial Scale in Earth-Moon System Astronomy Instruction

    ERIC Educational Resources Information Center

    Taylor, Roger S.; Grundstrom, Erika D.

    2011-01-01

    Given that astronomy heavily relies on visual representations it is especially likely for individuals to assume that instructional materials, such as visual representations of the Earth-Moon system (EMS), would be relatively accurate. However, in our research, we found that images in middle-school textbooks and educational webpages were commonly…

  14. Mathematics for Physics

    NASA Astrophysics Data System (ADS)

    Stone, Michael; Goldbart, Paul

    2009-07-01

    Preface; 1. Calculus of variations; 2. Function spaces; 3. Linear ordinary differential equations; 4. Linear differential operators; 5. Green functions; 6. Partial differential equations; 7. The mathematics of real waves; 8. Special functions; 9. Integral equations; 10. Vectors and tensors; 11. Differential calculus on manifolds; 12. Integration on manifolds; 13. An introduction to differential topology; 14. Group and group representations; 15. Lie groups; 16. The geometry of fibre bundles; 17. Complex analysis I; 18. Applications of complex variables; 19. Special functions and complex variables; Appendixes; Reference; Index.

  15. The link between mental rotation ability and basic numerical representations

    PubMed Central

    Thompson, Jacqueline M.; Nuerk, Hans-Christoph; Moeller, Korbinian; Cohen Kadosh, Roi

    2013-01-01

    Mental rotation and number representation have both been studied widely, but although mental rotation has been linked to higher-level mathematical skills, to date it has not been shown whether mental rotation ability is linked to the most basic mental representation and processing of numbers. To investigate the possible connection between mental rotation abilities and numerical representation, 43 participants completed four tasks: 1) a standard pen-and-paper mental rotation task; 2) a multi-digit number magnitude comparison task assessing the compatibility effect, which indicates separate processing of decade and unit digits; 3) a number-line mapping task, which measures precision of number magnitude representation; and 4) a random number generation task, which yields measures both of executive control and of spatial number representations. Results show that mental rotation ability correlated significantly with both size of the compatibility effect and with number mapping accuracy, but not with any measures from the random number generation task. Together, these results suggest that higher mental rotation abilities are linked to more developed number representation, and also provide further evidence for the connection between spatial and numerical abilities. PMID:23933002

  16. On accurate determination of contact angle

    NASA Technical Reports Server (NTRS)

    Concus, P.; Finn, R.

    1992-01-01

    Methods are proposed that exploit a microgravity environment to obtain highly accurate measurement of contact angle. These methods, which are based on our earlier mathematical results, do not require detailed measurement of a liquid free-surface, as they incorporate discontinuous or nearly-discontinuous behavior of the liquid bulk in certain container geometries. Physical testing is planned in the forthcoming IML-2 space flight and in related preparatory ground-based experiments.

  17. Impaired acuity of the approximate number system underlies mathematical learning disability (dyscalculia)

    PubMed Central

    Mazzocco, Michèle M. M.; Feigenson, Lisa; Halberda, Justin

    2015-01-01

    Many children have significant mathematical learning disabilities (MLD, or dyscalculia) despite adequate schooling. We hypothesize that MLD partly results from a deficiency in the Approximate Number System (ANS) that supports nonverbal numerical representations across species and throughout development. Here we show that ninth grade students with MLD have significantly poorer ANS precision than students in all other mathematics achievement groups (low-, typically-, and high-achieving), as measured by psychophysical assessments of ANS acuity (w) and of the mappings between ANS representations and number words (cv). This relationship persists even when controlling for domain-general abilities. Furthermore, this ANS precision does not differentiate low- from typically-achieving students, suggesting an ANS deficit that is specific to MLD. PMID:21679173

  18. Pre-service teachers' experiences teaching secondary mathematics in English-medium schools in Tanzania

    NASA Astrophysics Data System (ADS)

    Kasmer, Lisa

    2013-09-01

    In order to promote mathematical understanding among English Language Learners (ELLs), it is necessary to modify instructional strategies to effectively communicate mathematical content. This paper discusses the instructional strategies used by four pre-service teachers to teach mathematics to secondary students in English-medium schools in Arusha, Tanzania as a result of the tensions they faced and reflections on their teaching. Strategies such as code switching, attending to sentence structure, non-linguistic representations, and placing the content within a familiar context proved to be beneficial strategies for conveying mathematical ideas.

  19. Tools for Accurate and Efficient Analysis of Complex Evolutionary Mechanisms in Microbial Genomes. Final Report

    DOE Office of Scientific and Technical Information (OSTI.GOV)

    Nakhleh, Luay

    I proposed to develop computationally efficient tools for accurate detection and reconstruction of microbes' complex evolutionary mechanisms, thus enabling rapid and accurate annotation, analysis and understanding of their genomes. To achieve this goal, I proposed to address three aspects. (1) Mathematical modeling. A major challenge facing the accurate detection of HGT is that of distinguishing between these two events on the one hand and other events that have similar "effects." I proposed to develop a novel mathematical approach for distinguishing among these events. Further, I proposed to develop a set of novel optimization criteria for the evolutionary analysis of microbialmore » genomes in the presence of these complex evolutionary events. (2) Algorithm design. In this aspect of the project, I proposed to develop an array of e cient and accurate algorithms for analyzing microbial genomes based on the formulated optimization criteria. Further, I proposed to test the viability of the criteria and the accuracy of the algorithms in an experimental setting using both synthetic as well as biological data. (3) Software development. I proposed the nal outcome to be a suite of software tools which implements the mathematical models as well as the algorithms developed.« less

  20. Life on the Number Line: Routes to Understanding Fraction Magnitude for Students With Difficulties Learning Mathematics.

    PubMed

    Gersten, Russell; Schumacher, Robin F; Jordan, Nancy C

    Magnitude understanding is critical for students to develop a deep understanding of fractions and more advanced mathematics curriculum. The research reports in this special issue underscore magnitude understanding for fractions and emphasize number lines as both an assessment and an instructional tool. In this commentary, we discuss how number lines broaden the concept of fractions for students who are tied to the more general part-whole representations of area models. We also discuss how number lines, compared to other representations, are a superior and more mathematically correct way to explain fraction concepts.

  1. Supporting English Second-Language Learners in Disadvantaged Contexts: Learning Approaches That Promote Success in Mathematics

    ERIC Educational Resources Information Center

    Warren, Elizabeth; Miller, Jodie

    2015-01-01

    In the Australian context, children living in disadvantaged circumstances, whose second language is English, are one of the groups at risk of failing in mathematics. This paper explores the impact purposely developed learning activities (Representations, Oral Language and Engagement in Mathematics Learning activities) have on pupils' mathematics…

  2. Students' Recognition of Function Transformations' Themes Associated with the Algebraic Representation

    ERIC Educational Resources Information Center

    Daher, Wajeeh M.; Anabousi, Anlam A.

    2015-01-01

    The topic of function transformations is a difficult mathematical topic for school and college students. This article examines how students conceive function transformations after working with GeoGebra, when this conceiving relates to the algebraic representation. The research participants were 19 ninth grade high achieving students who learned,…

  3. Space-by-time manifold representation of dynamic facial expressions for emotion categorization

    PubMed Central

    Delis, Ioannis; Chen, Chaona; Jack, Rachael E.; Garrod, Oliver G. B.; Panzeri, Stefano; Schyns, Philippe G.

    2016-01-01

    Visual categorization is the brain computation that reduces high-dimensional information in the visual environment into a smaller set of meaningful categories. An important problem in visual neuroscience is to identify the visual information that the brain must represent and then use to categorize visual inputs. Here we introduce a new mathematical formalism—termed space-by-time manifold decomposition—that describes this information as a low-dimensional manifold separable in space and time. We use this decomposition to characterize the representations used by observers to categorize the six classic facial expressions of emotion (happy, surprise, fear, disgust, anger, and sad). By means of a Generative Face Grammar, we presented random dynamic facial movements on each experimental trial and used subjective human perception to identify the facial movements that correlate with each emotion category. When the random movements projected onto the categorization manifold region corresponding to one of the emotion categories, observers categorized the stimulus accordingly; otherwise they selected “other.” Using this information, we determined both the Action Unit and temporal components whose linear combinations lead to reliable categorization of each emotion. In a validation experiment, we confirmed the psychological validity of the resulting space-by-time manifold representation. Finally, we demonstrated the importance of temporal sequencing for accurate emotion categorization and identified the temporal dynamics of Action Unit components that cause typical confusions between specific emotions (e.g., fear and surprise) as well as those resolving these confusions. PMID:27305521

  4. Changing Mental Representations Using Related Physical Models: The Effects of Analyzing Number Lines on Learner Internal Scale of Numerical Magnitude

    ERIC Educational Resources Information Center

    Bengtson, Barbara J.

    2013-01-01

    Understanding the linear relationship of numbers is essential for doing practical and abstract mathematics throughout education and everyday life. There is evidence that number line activities increase learners' number sense, improving the linearity of mental number line representations (Siegler & Ramani, 2009). Mental representations of…

  5. Strategies That Help Learning-Disabled Students Solve Verbal Mathematical Problems.

    ERIC Educational Resources Information Center

    Giordano, Gerard

    1990-01-01

    Strategies are presented for dealing with factors that can be responsible for failure in mathematical problem solving. The suggestions include personalization of verbal problems, thematic strands based on student interests, visual representation, a laboratory approach, and paraphrasing. (JDD)

  6. Higher-order thinking skill problem on data representation in primary school: A case study

    NASA Astrophysics Data System (ADS)

    Putri, R. I. I.; Zulkardi, Z.

    2018-01-01

    This article aimed at reporting research result on a case study of a lesson using a HOTS problem. The task was about data representation using baby growth context. The study used a design research method consisting of three stages: preparing for an experiment, experiment in the classroom (pilot and teaching), and retrospective analysis. Participants were sixth grade students who were learning data representations in a Primary School in Palembang Indonesia. A set of instructional activities were designed using Indonesian version of Realistic Mathematics Education (PMRI) approach. The result showed that students were able to solve the problem and present their solution in front of the classroom. The conclusion indicated that that HOTS problem using the growth of a child as the context could lead students to use their mathematical thinking. During the learning activities along with teacher orchestra’s guidance, and discussion, students were able to solve the problem using line graph although some of them used a bar graph. In the future, teachers are necessary to focus on the role of real-world figure in mathematics learning.

  7. The Interaction between Semantic Representation and Episodic Memory.

    PubMed

    Fang, Jing; Rüther, Naima; Bellebaum, Christian; Wiskott, Laurenz; Cheng, Sen

    2018-02-01

    The experimental evidence on the interrelation between episodic memory and semantic memory is inconclusive. Are they independent systems, different aspects of a single system, or separate but strongly interacting systems? Here, we propose a computational role for the interaction between the semantic and episodic systems that might help resolve this debate. We hypothesize that episodic memories are represented as sequences of activation patterns. These patterns are the output of a semantic representational network that compresses the high-dimensional sensory input. We show quantitatively that the accuracy of episodic memory crucially depends on the quality of the semantic representation. We compare two types of semantic representations: appropriate representations, which means that the representation is used to store input sequences that are of the same type as those that it was trained on, and inappropriate representations, which means that stored inputs differ from the training data. Retrieval accuracy is higher for appropriate representations because the encoded sequences are less divergent than those encoded with inappropriate representations. Consistent with our model prediction, we found that human subjects remember some aspects of episodes significantly more accurately if they had previously been familiarized with the objects occurring in the episode, as compared to episodes involving unfamiliar objects. We thus conclude that the interaction with the semantic system plays an important role for episodic memory.

  8. Electrophysiological dynamic brain connectivity during symbolic magnitude comparison in children with different mathematics achievement levels.

    PubMed

    Gómez-Velázquez, Fabiola R; Vélez-Pérez, Hugo; Espinoza-Valdez, Aurora; Romo-Vazquez, Rebeca; Salido-Ruiz, Ricardo A; Ruiz-Stovel, Vanessa; Gallardo-Moreno, Geisa B; González-Garrido, Andrés A; Berumen, Gustavo

    2017-02-08

    Children with mathematical difficulties usually have an impaired ability to process symbolic representations. Functional MRI methods have suggested that early frontoparietal connectivity can predict mathematic achievements; however, the study of brain connectivity during numerical processing remains unexplored. With the aim of evaluating this in children with different math proficiencies, we selected a sample of 40 children divided into two groups [high achievement (HA) and low achievement (LA)] according to their arithmetic scores in the Wide Range Achievement Test, 4th ed.. Participants performed a symbolic magnitude comparison task (i.e. determining which of two numbers is numerically larger), with simultaneous electrophysiological recording. Partial directed coherence and graph theory methods were used to estimate and depict frontoparietal connectivity in both groups. The behavioral measures showed that children with LA performed significantly slower and less accurately than their peers in the HA group. Significantly higher frontocentral connectivity was found in LA compared with HA; however, when the connectivity analysis was restricted to parietal locations, no relevant group differences were observed. These findings seem to support the notion that LA children require greater memory and attentional efforts to meet task demands, probably affecting early stages of symbolic comparison.

  9. Basic and Advanced Numerical Performances Relate to Mathematical Expertise but Are Fully Mediated by Visuospatial Skills

    ERIC Educational Resources Information Center

    Sella, Francesco; Sader, Elie; Lolliot, Simon; Cohen Kadosh, Roi

    2016-01-01

    Recent studies have highlighted the potential role of basic numerical processing in the acquisition of numerical and mathematical competences. However, it is debated whether high-level numerical skills and mathematics depends specifically on basic numerical representations. In this study mathematicians and nonmathematicians performed a basic…

  10. From Play to Thoughtful Learning: A Design Strategy to Engage Children with Mathematical Representations

    ERIC Educational Resources Information Center

    Sedig, Kamran

    2008-01-01

    Many children do not like learning mathematics. They do not find mathematics fun, motivating, and engaging, and they think it is difficult to learn. Computer-based games have the potential and possibility of addressing this problem. This paper proposes a strategy for designing game-based learning environments that takes advantage of the…

  11. Real-World Contexts, Multiple Representations, Student-Invented Terminology, and Y-Intercept

    ERIC Educational Resources Information Center

    Davis, Jon D.

    2007-01-01

    One classroom using two units from a "Standards"-based curriculum was the focus of a study designed to examine the effects of real-world contexts, delays in the introduction of formal mathematics terminology, and multiple function representations on student understanding. Students developed their own terminology for y-intercept, which was tightly…

  12. 48 CFR 252.204-7007 - Alternate A, Annual Representations and Certifications.

    Code of Federal Regulations, 2010 CFR

    2010-10-01

    ... Online Representations and Certifications Application (ORCA) Web site at https://orca.bpn.gov/. After... months, are current, accurate, complete, and applicable to this solicitation (including the business size...

  13. Children's Criteria for Representational Adequacy in the Perception of Simple Sonic Stimuli

    ERIC Educational Resources Information Center

    Verschaffel, Lieven; Reybrouck, Mark; Jans, Christine; Van Dooren, Wim

    2010-01-01

    This study investigates children's metarepresentational competence with regard to listening to and making sense of simple sonic stimuli. Using diSessa's (2003) work on metarepresentational competence in mathematics and sciences as theoretical and empirical background, it aims to assess children's criteria for representational adequacy of graphical…

  14. With age comes representational wisdom in social signals.

    PubMed

    van Rijsbergen, Nicola; Jaworska, Katarzyna; Rousselet, Guillaume A; Schyns, Philippe G

    2014-12-01

    In an increasingly aging society, age has become a foundational dimension of social grouping broadly targeted by advertising and governmental policies. However, perception of old age induces mainly strong negative social biases. To characterize their cognitive and perceptual foundations, we modeled the mental representations of faces associated with three age groups (young age, middle age, and old age), in younger and older participants. We then validated the accuracy of each mental representation of age with independent validators. Using statistical image processing, we identified the features of mental representations that predict perceived age. Here, we show that whereas younger people mentally dichotomize aging into two groups, themselves (younger) and others (older), older participants faithfully represent the features of young age, middle age, and old age, with richer representations of all considered ages. Our results demonstrate that, contrary to popular public belief, older minds depict socially relevant information more accurately than their younger counterparts. Copyright © 2014 The Authors. Published by Elsevier Inc. All rights reserved.

  15. How students learn to coordinate knowledge of physical and mathematical models in cellular physiology

    NASA Astrophysics Data System (ADS)

    Lira, Matthew

    This dissertation explores the Knowledge in Pieces (KiP) theory to account for how students learn to coordinate knowledge of mathematical and physical models in biology education. The KiP approach characterizes student knowledge as a fragmented collection of knowledge elements as opposed to stable and theory-like knowledge. This dissertation sought to use this theoretical lens to account for how students understand and learn with mathematical models and representations, such as equations. Cellular physiology provides a quantified discipline that leverages concepts from mathematics, physics, and chemistry to understand cellular functioning. Therefore, this discipline provides an exemplary context for assessing how biology students think and learn with mathematical models. In particular, the resting membrane potential provides an exemplary concept well defined by models of dynamic equilibrium borrowed from physics and chemistry. In brief, membrane potentials, or voltages, "rest" when the electrical and chemical driving forces for permeable ionic species are equal in magnitude but opposite in direction. To assess students' understandings of this concept, this dissertation employed three studies: the first study employed the cognitive clinical interview to assess student thinking in the absence and presence of equations. The second study employed an intervention to assess student learning and the affordances of an innovative assessment. The third student employed a human-computer-interaction paradigm to assess how students learn with a novel multi-representational technology. Study 1 revealed that students saw only one influence--the chemical gradient--and that students coordinated knowledge of only this gradient with the related equations. Study 2 revealed that students benefited from learning with the multi-representational technology and that the assessment detected performance gains across both calculation and explanation tasks. Last, Study 3 revealed how students

  16. Semiclassical propagation: Hilbert space vs. Wigner representation

    NASA Astrophysics Data System (ADS)

    Gottwald, Fabian; Ivanov, Sergei D.

    2018-03-01

    A unified viewpoint on the van Vleck and Herman-Kluk propagators in Hilbert space and their recently developed counterparts in Wigner representation is presented. Based on this viewpoint, the Wigner Herman-Kluk propagator is conceptually the most general one. Nonetheless, the respective semiclassical expressions for expectation values in terms of the density matrix and the Wigner function are mathematically proven here to coincide. The only remaining difference is a mere technical flexibility of the Wigner version in choosing the Gaussians' width for the underlying coherent states beyond minimal uncertainty. This flexibility is investigated numerically on prototypical potentials and it turns out to provide neither qualitative nor quantitative improvements. Given the aforementioned generality, utilizing the Wigner representation for semiclassical propagation thus leads to the same performance as employing the respective most-developed (Hilbert-space) methods for the density matrix.

  17. Mathematical Audio-Podcasts for Teacher Education and School

    ERIC Educational Resources Information Center

    Schreiber, Christof; Klose, Rebecca

    2017-01-01

    Audio-podcasts offer notable opportunities for oral representation of mathematical content through digital media--not only for teacher education but also in primary schools. This article deals with the process of creating such podcasts, as well as their uses in schools, university teaching and research. We allow for various learning groups--which…

  18. Towards a category theory approach to analogy: Analyzing re-representation and acquisition of numerical knowledge.

    PubMed

    Navarrete, Jairo A; Dartnell, Pablo

    2017-08-01

    Category Theory, a branch of mathematics, has shown promise as a modeling framework for higher-level cognition. We introduce an algebraic model for analogy that uses the language of category theory to explore analogy-related cognitive phenomena. To illustrate the potential of this approach, we use this model to explore three objects of study in cognitive literature. First, (a) we use commutative diagrams to analyze an effect of playing particular educational board games on the learning of numbers. Second, (b) we employ a notion called coequalizer as a formal model of re-representation that explains a property of computational models of analogy called "flexibility" whereby non-similar representational elements are considered matches and placed in structural correspondence. Finally, (c) we build a formal learning model which shows that re-representation, language processing and analogy making can explain the acquisition of knowledge of rational numbers. These objects of study provide a picture of acquisition of numerical knowledge that is compatible with empirical evidence and offers insights on possible connections between notions such as relational knowledge, analogy, learning, conceptual knowledge, re-representation and procedural knowledge. This suggests that the approach presented here facilitates mathematical modeling of cognition and provides novel ways to think about analogy-related cognitive phenomena.

  19. Towards a category theory approach to analogy: Analyzing re-representation and acquisition of numerical knowledge

    PubMed Central

    2017-01-01

    Category Theory, a branch of mathematics, has shown promise as a modeling framework for higher-level cognition. We introduce an algebraic model for analogy that uses the language of category theory to explore analogy-related cognitive phenomena. To illustrate the potential of this approach, we use this model to explore three objects of study in cognitive literature. First, (a) we use commutative diagrams to analyze an effect of playing particular educational board games on the learning of numbers. Second, (b) we employ a notion called coequalizer as a formal model of re-representation that explains a property of computational models of analogy called “flexibility” whereby non-similar representational elements are considered matches and placed in structural correspondence. Finally, (c) we build a formal learning model which shows that re-representation, language processing and analogy making can explain the acquisition of knowledge of rational numbers. These objects of study provide a picture of acquisition of numerical knowledge that is compatible with empirical evidence and offers insights on possible connections between notions such as relational knowledge, analogy, learning, conceptual knowledge, re-representation and procedural knowledge. This suggests that the approach presented here facilitates mathematical modeling of cognition and provides novel ways to think about analogy-related cognitive phenomena. PMID:28841643

  20. An accurate and precise representation of drug ingredients.

    PubMed

    Hanna, Josh; Bian, Jiang; Hogan, William R

    2016-01-01

    In previous work, we built the Drug Ontology (DrOn) to support comparative effectiveness research use cases. Here, we have updated our representation of ingredients to include both active ingredients (and their strengths) and excipients. Our update had three primary lines of work: 1) analysing and extracting excipients, 2) analysing and extracting strength information for active ingredients, and 3) representing the binding of active ingredients to cytochrome P450 isoenzymes as substrates and inhibitors of those enzymes. To properly differentiate between excipients and active ingredients, we conducted an ontological analysis of the roles that various ingredients, including excipients, have in drug products. We used the value specification model of the Ontology for Biomedical Investigations to represent strengths of active ingredients and then analyzed RxNorm to extract excipient and strength information and modeled them according to the results of our analysis. We also analyzed and defined dispositions of molecules used in aggregate as active ingredients to bind cytochrome P450 isoenzymes. Our analysis of excipients led to 17 new classes representing the various roles that excipients can bear. We then extracted excipients from RxNorm and added them to DrOn for branded drugs. We found excipients for 5,743 branded drugs, covering ~27% of the 21,191 branded drugs in DrOn. Our analysis of active ingredients resulted in another new class, active ingredient role. We also extracted strengths for all types of tablets, capsules, and caplets, resulting in strengths for 5,782 drug forms, covering ~41% of the 14,035 total drug forms and accounting for ~97 % of the 5,970 tablets, capsules, and caplets in DrOn. We represented binding-as-substrate and binding-as-inhibitor dispositions to two cytochrome P450 (CYP) isoenzymes (CYP2C19 and CYP2D6) and linked these dispositions to 65 compounds. It is now possible to query DrOn automatically for all drug products that contain active

  1. Working Memory in Students with Mathematical Difficulties

    NASA Astrophysics Data System (ADS)

    Nur, I. R. D.; Herman, T.; Ningsih, S.

    2018-04-01

    Learning process is the activities that has important role because this process is one of the all factors that establish students success in learning. oftentimes we find so many students get the difficulties when they study mathematics. This condition is not only because of the outside factor but also it comes from the inside. The purpose of this research is to analyze and give the representation how students working memory happened in physical education students for basic statistics subjects which have mathematical difficulties. The subjects are 4 students which have a mathematical difficulties. The research method is case study and when the describe about students working memory are explanated deeply with naturalistic observation. Based on this research, it was founded that 4 students have a working memory deficit in three components. The components are phonological loop, visuospatial sketchpad, dan episodic buffer.

  2. There aren't Non-Standard Solutions for the Braid Group Representations of the QYBE Associated with 10-D Representations of SU(4)

    NASA Technical Reports Server (NTRS)

    Yijun, Huang; Guochen, Yu; Hong, Sun

    1996-01-01

    It is well known that the quantum Yang-Baxter equations (QYBE) play an important role in various theoretical and mathematical physics, such as completely integrable system in (1 + 1)-dimensions, exactly solvable models in statistical mechanics, the quantum inverse scattering method and the conformal field theories in 2-dimensions. Recently, much remarkable progress has been made in constructing the solutions of the QYBE associated with the representations of lie algebras. It is shown that for some cases except the standard solutions, there also exist new solutions, but the others have not non-standard solutions. In this paper by employing the weight conservation and the diagrammatic techniques we show that the solution associated with the 10-D representations of SU (4) are standard alone.

  3. Factors involved in making post-performance judgments in mathematics problem-solving.

    PubMed

    García Fernández, Trinidad; Kroesbergen, Evelyn; Rodríguez Pérez, Celestino; González-Castro, Paloma; González-Pienda, Julio A

    2015-01-01

    This study examines the impact of executive functions, affective-motivational variables related to mathematics, mathematics achievement and task characteristics on fifth and sixth graders’ calibration accuracy after completing two mathematical problems. A sample of 188 students took part in the study. They were divided into two groups as function of their judgment accuracy after completing the two tasks (accurate= 79, inaccurate= 109). Differences between these groups were examined. The discriminative value of these variables to predict group membership was analyzed, as well as the effect of age, gender, and grade level. The results indicated that accurate students showed better levels of executive functioning, and more positive feelings, beliefs, and motivation related to mathematics. They also spent more time on the tasks. Mathematics achievement, perceived usefulness of mathematics, and time spent on Task 1 significantly predicted group membership, classifying 71.3% of the sample correctly. These results support the relationship between academic achievement and calibration accuracy, suggesting the need to consider a wide range of factors when explaining performance judgments.

  4. Knowledge Representation Of CT Scans Of The Head

    NASA Astrophysics Data System (ADS)

    Ackerman, Laurens V.; Burke, M. W.; Rada, Roy

    1984-06-01

    We have been investigating diagnostic knowledge models which assist in the automatic classification of medical images by combining information extracted from each image with knowledge specific to that class of images. In a more general sense we are trying to integrate verbal and pictorial descriptions of disease via representations of knowledge, study automatic hypothesis generation as related to clinical medicine, evolve new mathematical image measures while integrating them into the total diagnostic process, and investigate ways to augment the knowledge of the physician. Specifically, we have constructed an artificial intelligence knowledge model using the technique of a production system blending pictorial and verbal knowledge about the respective CT scan and patient history. It is an attempt to tie together different sources of knowledge representation, picture feature extraction and hypothesis generation. Our knowledge reasoning and representation system (KRRS) works with data at the conscious reasoning level of the practicing physician while at the visual perceptional level we are building another production system, the picture parameter extractor (PPE). This paper describes KRRS and its relationship to PPE.

  5. Mathematical Thinking Process of Autistic Students in Terms of Representational Gesture

    ERIC Educational Resources Information Center

    Mustafa, Sriyanti; Nusantara, Toto; Subanji; Irawati, Santi

    2016-01-01

    The aim of this study is to describe the mathematical thinking process of autistic students in terms of gesture, using a qualitative approach. Data collecting is conducted by using 3 (three) audio-visual cameras. During the learning process, both teacher and students' activity are recorded using handy cam and digital camera (full HD capacity).…

  6. Connecting Dynamic Representations of Simple Mathematical Objects with the Construction and Exploration of Conic Sections

    ERIC Educational Resources Information Center

    Santos-Trigo, Manuel; Espinosa-Perez, Hugo; Reyes-Rodriguez, Aaron

    2008-01-01

    Different technological artefacts may offer distinct opportunities for students to develop resources and strategies to formulate, comprehend and solve mathematical problems. In particular, the use of dynamic software becomes relevant to assemble geometric configurations that may help students reconstruct and examine mathematical relationships. In…

  7. Development of common neural representations for distinct numerical problems

    PubMed Central

    Chang, Ting-Ting; Rosenberg-Lee, Miriam; Metcalfe, Arron W. S.; Chen, Tianwen; Menon, Vinod

    2015-01-01

    How the brain develops representations for abstract cognitive problems is a major unaddressed question in neuroscience. Here we tackle this fundamental question using arithmetic problem solving, a cognitive domain important for the development of mathematical reasoning. We first examined whether adults demonstrate common neural representations for addition and subtraction problems, two complementary arithmetic operations that manipulate the same quantities. We then examined how the common neural representations for the two problem types change with development. Whole-brain multivoxel representational similarity (MRS) analysis was conducted to examine common coding of addition and subtraction problems in children and adults. We found that adults exhibited significant levels of MRS between the two problem types, not only in the intra-parietal sulcus (IPS) region of the posterior parietal cortex (PPC), but also in ventral temporal-occipital, anterior temporal and dorsolateral prefrontal cortices. Relative to adults, children showed significantly reduced levels of MRS in these same regions. In contrast, no brain areas showed significantly greater MRS between problem types in children. Our findings provide novel evidence that the emergence of arithmetic problem solving skills from childhood to adulthood is characterized by maturation of common neural representations between distinct numerical operations, and involve distributed brain regions important for representing and manipulating numerical quantity. More broadly, our findings demonstrate that representational analysis provides a powerful approach for uncovering fundamental mechanisms by which children develop proficiencies that are a hallmark of human cognition. PMID:26160287

  8. Kendall Demonstration Elementary School: Mathematics Curriculum Guide. Second Edition.

    ERIC Educational Resources Information Center

    Mason, Virgyl

    This mathematics curriculum guide is one of a series developed by the Kendall Demonstration Elementary School (KDES), which serves hearing-impaired students in grades 1-8, to provide a clear representation of the school's programs in various subject areas. Essential classroom practices in the areas of planning, instruction, and evaluation are…

  9. The transition to formal thinking in mathematics

    NASA Astrophysics Data System (ADS)

    Tall, David

    2008-09-01

    This paper focuses on the changes in thinking involved in the transition from school mathematics to formal proof in pure mathematics at university. School mathematics is seen as a combination of visual representations, including geometry and graphs, together with symbolic calculations and manipulations. Pure mathematics in university shifts towards a formal framework of axiomatic systems and mathematical proof. In this paper, the transition in thinking is formulated within a framework of `three worlds of mathematics'- the `conceptual-embodied' world based on perception, action and thought experiment, the `proceptual-symbolic' world of calculation and algebraic manipulation compressing processes such as counting into concepts such as number, and the `axiomatic-formal' world of set-theoretic concept definitions and mathematical proof. Each `world' has its own sequence of development and its own forms of proof that may be blended together to give a rich variety of ways of thinking mathematically. This reveals mathematical thinking as a blend of differing knowledge structures; for instance, the real numbers blend together the embodied number line, symbolic decimal arithmetic and the formal theory of a complete ordered field. Theoretical constructs are introduced to describe how genetic structures set before birth enable the development of mathematical thinking, and how experiences that the individual has met before affect their personal growth. These constructs are used to consider how students negotiate the transition from school to university mathematics as embodiment and symbolism are blended with formalism. At a higher level, structure theorems proved in axiomatic theories link back to more sophisticated forms of embodiment and symbolism, revealing the intimate relationship between the three worlds.

  10. Integrating STEM in Elementary Classrooms Using Model-Eliciting Activities: Responsive Professional Development for Mathematics Coaches and Teachers

    ERIC Educational Resources Information Center

    Baker, Courtney K.; Galanti, Terrie M.

    2017-01-01

    Background: This research highlights a school-university collaboration to pilot a professional development framework for integrating STEM in K-6 mathematics classrooms in a mid-Atlantic suburban school division. Because mathematics within STEM integration is often characterized as the calculations or the data representations in science classrooms,…

  11. Students' Competencies in Working with Functions in Secondary Mathematics Education-Empirical Examination of a Competence Structure Model

    ERIC Educational Resources Information Center

    Nitsch, Renate; Fredebohm, Anneke; Bruder, Regina; Kelava, Augustin; Naccarella, Dominik; Leuders, Timo; Wirtz, Markus

    2015-01-01

    In the subject matter of functional relationships, a student's ability to translate from one form of representation to another is seen as a central competence. In the course of the HEUREKO project (heuristic work with representations of functional relationships and the diagnosis of mathematical competencies of students), a theoretical competence…

  12. Symbolic and Verbal Representation Process of Student in Solving Mathematics Problem Based Polya's Stages

    ERIC Educational Resources Information Center

    Anwar, Rahmad Bustanul; Rahmawati, Dwi

    2017-01-01

    The purpose of this research was to reveal how the construction process of symbolic representation and verbal representation made by students in problem solving. The construction process in this study referred to the problem-solving stage by Polya covering; 1) understanding the problem, 2) devising a plan, 3) carrying out the plan, and 4) looking…

  13. Prototype Images in Mathematics Education: The Case of The Graphical Representation of The Definite Integral

    ERIC Educational Resources Information Center

    Jones, Steven R.

    2018-01-01

    Many mathematical concepts may have prototypical images associated with them. While prototypes can be beneficial for efficient thinking or reasoning, they may also have self-attributes that may impact reasoning about the concept. It is essential that mathematics educators understand these prototype images in order to fully recognize their benefits…

  14. Representational Realism, Closed Theories and the Quantum to Classical Limit

    NASA Astrophysics Data System (ADS)

    de Ronde, Christian

    In this chapter, we discuss the representational realist stance as a pluralistontic approach to inter-theoretic relationships. Our stance stresses the fact that physical theories require the necessary consideration of a conceptual level of discourse which determines and configures the specific field of phenomena discussed by each particular theory. We will criticize the orthodox line of research which has grounded the analysis about QM in two (Bohrian) metaphysical presuppositions - accepted in the present as dogmas that all interpretations must follow. We will also examine how the orthodox project of "bridging the gap" between the quantum and the classical domains has constrained the possibilities of research, producing only a limited set of interpretational problems which only focus in the justification of "classical reality" and exclude the possibility of analyzing the possibilities of non-classical conceptual representations of QM. The representational realist stance introduces two new problems, namely, the superposition problem and the contextuality problem, which consider explicitly the conceptual representation of orthodox QM beyond the mere reference to mathematical structures and measurement outcomes. In the final part of the chapter, we revisit, from representational realist perspective, the quantum to classical limit and the orthodox claim that this inter-theoretic relation can be explained through the principle of decoherence.

  15. Improving basic math skills through integrated dynamic representation strategies.

    PubMed

    González-Castro, Paloma; Cueli, Marisol; Cabeza, Lourdes; Álvarez-García, David; Rodríguez, Celestino

    2014-01-01

    In this paper, we analyze the effectiveness of the Integrated Dynamic Representation strategy (IDR) to develop basic math skills. The study involved 72 students, aged between 6 and 8 years. We compared the development of informal basic skills (numbers, comparison, informal calculation, and informal concepts) and formal (conventionalisms, number facts, formal calculus, and formal concepts) in an experimental group (n = 35) where we applied the IDR strategy and in a Control group (n = 37) in order to identify the impact of the procedure. The experimental group improved significantly in all variables except for number facts and formal calculus. It can therefore be concluded that IDR favors the development of the skills more closely related to applied mathematics than those related to automatic mathematics and mental arithmetic.

  16. On the representation of many-body interactions in water

    DOE PAGES

    Medders, Gregory R.; Gotz, Andreas W.; Morales, Miguel A.; ...

    2015-09-09

    Our recent work has shown that the many-body expansion of the interactionenergy can be used to develop analytical representations of global potential energy surfaces (PESs) for water. In this study, the role of short- and long-range interactions at different orders is investigated by analyzing water potentials that treat the leading terms of the many-body expansion through implicit (i.e., TTM3-F and TTM4-F PESs) and explicit (i.e., WHBB and MB-pol PESs) representations. Moreover, it is found that explicit short-range representations of 2-body and 3-body interactions along with a physically correct incorporation of short- and long-range contributions are necessary for an accurate representationmore » of the waterinteractions from the gas to the condensed phase. Likewise, a complete many-body representation of the dipole moment surface is found to be crucial to reproducing the correct intensities of the infrared spectrum of liquid water.« less

  17. Learning with Multiple Representations: An Example of a Revision Lesson in Mathematics

    ERIC Educational Resources Information Center

    Wong, Darren; Poo, Sng Peng; Hock, Ng Eng; Kang, Wee Loo

    2011-01-01

    We describe an example of learning with multiple representations in an A-level revision lesson on mechanics. The context of the problem involved the motion of a ball thrown vertically upwards in air and studying how the associated physical quantities changed during its flight. Different groups of students were assigned to look at the ball's motion…

  18. Impaired acuity of the approximate number system underlies mathematical learning disability (dyscalculia).

    PubMed

    Mazzocco, Michèle M M; Feigenson, Lisa; Halberda, Justin

    2011-01-01

    Many children have significant mathematical learning disabilities (MLD, or dyscalculia) despite adequate schooling. The current study hypothesizes that MLD partly results from a deficiency in the Approximate Number System (ANS) that supports nonverbal numerical representations across species and throughout development. In this study of 71 ninth graders, it is shown that students with MLD have significantly poorer ANS precision than students in all other mathematics achievement groups (low, typically, and high achieving), as measured by psychophysical assessments of ANS acuity (w) and of the mappings between ANS representations and number words (cv). This relation persists even when controlling for domain-general abilities. Furthermore, this ANS precision does not differentiate low-achieving from typically achieving students, suggesting an ANS deficit that is specific to MLD. © 2011 The Authors. Child Development © 2011 Society for Research in Child Development, Inc.

  19. Stereotypes and Representations of Aging in the Media

    ERIC Educational Resources Information Center

    Mason, Susan E.; Darnell, Emily A.; Prifti, Krisiola

    2010-01-01

    How are older adults presented in print and in the electronic media? Are they underrepresented? Are they accurately portrayed? Based on our examination of several forms of media over a four-month period, we discuss the role of the media in shaping our views on aging. Quantitative and qualitative analyses reveal that media representations often…

  20. Learning semantic histopathological representation for basal cell carcinoma classification

    NASA Astrophysics Data System (ADS)

    Gutiérrez, Ricardo; Rueda, Andrea; Romero, Eduardo

    2013-03-01

    Diagnosis of a histopathology glass slide is a complex process that involves accurate recognition of several structures, their function in the tissue and their relation with other structures. The way in which the pathologist represents the image content and the relations between those objects yields a better and accurate diagnoses. Therefore, an appropriate semantic representation of the image content will be useful in several analysis tasks such as cancer classification, tissue retrieval and histopahological image analysis, among others. Nevertheless, to automatically recognize those structures and extract their inner semantic meaning are still very challenging tasks. In this paper we introduce a new semantic representation that allows to describe histopathological concepts suitable for classification. The approach herein identify local concepts using a dictionary learning approach, i.e., the algorithm learns the most representative atoms from a set of random sampled patches, and then models the spatial relations among them by counting the co-occurrence between atoms, while penalizing the spatial distance. The proposed approach was compared with a bag-of-features representation in a tissue classification task. For this purpose, 240 histological microscopical fields of view, 24 per tissue class, were collected. Those images fed a Support Vector Machine classifier per class, using 120 images as train set and the remaining ones for testing, maintaining the same proportion of each concept in the train and test sets. The obtained classification results, averaged from 100 random partitions of training and test sets, shows that our approach is more sensitive in average than the bag-of-features representation in almost 6%.

  1. Numerical magnitude processing in abacus-trained children with superior mathematical ability: an EEG study*

    PubMed Central

    Huang, Jian; Du, Feng-lei; Yao, Yuan; Wan, Qun; Wang, Xiao-song; Chen, Fei-yan

    2015-01-01

    Distance effect has been regarded as the best established marker of basic numerical magnitude processes and is related to individual mathematical abilities. A larger behavioral distance effect is suggested to be concomitant with lower mathematical achievement in children. However, the relationship between distance effect and superior mathematical abilities is unclear. One could get superior mathematical abilities by acquiring the skill of abacus-based mental calculation (AMC), which can be used to solve calculation problems with exceptional speed and high accuracy. In the current study, we explore the relationship between distance effect and superior mathematical abilities by examining whether and how the AMC training modifies numerical magnitude processing. Thus, mathematical competencies were tested in 18 abacus-trained children (who accepted the AMC training) and 18 non-trained children. Electroencephalography (EEG) waveforms were recorded when these children executed numerical comparison tasks in both Arabic digit and dot array forms. We found that: (a) the abacus-trained group had superior mathematical abilities than their peers; (b) distance effects were found both in behavioral results and on EEG waveforms; (c) the distance effect size of the average amplitude on the late negative-going component was different between groups in the digit task, with a larger effect size for abacus-trained children; (d) both the behavioral and EEG distance effects were modulated by the notation. These results revealed that the neural substrates of magnitude processing were modified by AMC training, and suggested that the mechanism of the representation of numerical magnitude for children with superior mathematical abilities was different from their peers. In addition, the results provide evidence for a view of non-abstract numerical representation. PMID:26238541

  2. Fidelity of the representation of value in decision-making

    PubMed Central

    Dowding, Ben A.

    2017-01-01

    The ability to make optimal decisions depends on evaluating the expected rewards associated with different potential actions. This process is critically dependent on the fidelity with which reward value information can be maintained in the nervous system. Here we directly probe the fidelity of value representation following a standard reinforcement learning task. The results demonstrate a previously-unrecognized bias in the representation of value: extreme reward values, both low and high, are stored significantly more accurately and precisely than intermediate rewards. The symmetry between low and high rewards pertained despite substantially higher frequency of exposure to high rewards, resulting from preferential exploitation of more rewarding options. The observed variation in fidelity of value representation retrospectively predicted performance on the reinforcement learning task, demonstrating that the bias in representation has an impact on decision-making. A second experiment in which one or other extreme-valued option was omitted from the learning sequence showed that representational fidelity is primarily determined by the relative position of an encoded value on the scale of rewards experienced during learning. Both variability and guessing decreased with the reduction in the number of options, consistent with allocation of a limited representational resource. These findings have implications for existing models of reward-based learning, which typically assume defectless representation of reward value. PMID:28248958

  3. War of ontology worlds: mathematics, computer code, or Esperanto?

    PubMed

    Rzhetsky, Andrey; Evans, James A

    2011-09-01

    The use of structured knowledge representations-ontologies and terminologies-has become standard in biomedicine. Definitions of ontologies vary widely, as do the values and philosophies that underlie them. In seeking to make these views explicit, we conducted and summarized interviews with a dozen leading ontologists. Their views clustered into three broad perspectives that we summarize as mathematics, computer code, and Esperanto. Ontology as mathematics puts the ultimate premium on rigor and logic, symmetry and consistency of representation across scientific subfields, and the inclusion of only established, non-contradictory knowledge. Ontology as computer code focuses on utility and cultivates diversity, fitting ontologies to their purpose. Like computer languages C++, Prolog, and HTML, the code perspective holds that diverse applications warrant custom designed ontologies. Ontology as Esperanto focuses on facilitating cross-disciplinary communication, knowledge cross-referencing, and computation across datasets from diverse communities. We show how these views align with classical divides in science and suggest how a synthesis of their concerns could strengthen the next generation of biomedical ontologies.

  4. Negative Numbers in the 18th and 19th Centuries: Phenomenology and Representations

    ERIC Educational Resources Information Center

    Maz-Machado, Alexander; Rico-Romero, Luis

    2009-01-01

    This article presents a categorization of the phenomena and representations used to introduce negative numbers in mathematics books published in Spain during the 18th and 19th centuries. Through a content analysis of fourteen texts which were selected for the study, we distinguished four phenomena typologies: physical, accounting, temporal and…

  5. Sex differences in intrinsic aptitude for mathematics and science?: a critical review.

    PubMed

    Spelke, Elizabeth S

    2005-12-01

    This article considers 3 claims that cognitive sex differences account for the differential representation of men and women in high-level careers in mathematics and science: (a) males are more focused on objects from the beginning of life and therefore are predisposed to better learning about mechanical systems; (b) males have a profile of spatial and numerical abilities producing greater aptitude for mathematics; and (c) males are more variable in their cognitive abilities and therefore predominate at the upper reaches of mathematical talent. Research on cognitive development in human infants, preschool children, and students at all levels fails to support these claims. Instead, it provides evidence that mathematical and scientific reasoning develop from a set of biologically based cognitive capacities that males and females share. These capacities lead men and women to develop equal talent for mathematics and science.

  6. Using Virtual Manipulatives with Pre-Service Mathematics Teachers to Create Representational Models

    ERIC Educational Resources Information Center

    Cooper, Thomas E.

    2012-01-01

    In mathematics education, physical manipulatives such as algebra tiles, pattern blocks, and two-colour counters are commonly used to provide concrete models of abstract concepts. With these traditional manipulatives, people can communicate with the tools only in one another's presence. This limitation poses difficulties concerning assessment and…

  7. Negotiating the Boundaries Between Mathematics and Physics

    NASA Astrophysics Data System (ADS)

    Radtka, Catherine

    2015-07-01

    This paper examines physics and mathematics textbooks published in France at the end of the 1950s and at the beginning of the 1960s for children aged 11-15 years old. It argues that at this "middle school" level, textbooks contributed to shape cultural representations of both disciplines and their mutual boundaries through their contents and their material aspect. Further, this paper argues that far from presenting clearly delimited subjects, late 1950s textbooks offered possible connections between mathematics and physics. It highlights that such connections depended upon the type of schools the textbooks aimed at, at a time when educational organization still differentiated pupils of this age. It thus stresses how the audience and its projected aptitudes and needs, as well as the cultural teaching traditions of the teachers in charge, were inseparable from the diverse conceptions of mathematics and physics and their relationships promoted through textbooks of the time.

  8. Mathematical models used in segmentation and fractal methods of 2-D ultrasound images

    NASA Astrophysics Data System (ADS)

    Moldovanu, Simona; Moraru, Luminita; Bibicu, Dorin

    2012-11-01

    Mathematical models are widely used in biomedical computing. The extracted data from images using the mathematical techniques are the "pillar" achieving scientific progress in experimental, clinical, biomedical, and behavioural researches. This article deals with the representation of 2-D images and highlights the mathematical support for the segmentation operation and fractal analysis in ultrasound images. A large number of mathematical techniques are suitable to be applied during the image processing stage. The addressed topics cover the edge-based segmentation, more precisely the gradient-based edge detection and active contour model, and the region-based segmentation namely Otsu method. Another interesting mathematical approach consists of analyzing the images using the Box Counting Method (BCM) to compute the fractal dimension. The results of the paper provide explicit samples performed by various combination of methods.

  9. Multiscale geometric modeling of macromolecules II: Lagrangian representation

    PubMed Central

    Feng, Xin; Xia, Kelin; Chen, Zhan; Tong, Yiying; Wei, Guo-Wei

    2013-01-01

    Geometric modeling of biomolecules plays an essential role in the conceptualization of biolmolecular structure, function, dynamics and transport. Qualitatively, geometric modeling offers a basis for molecular visualization, which is crucial for the understanding of molecular structure and interactions. Quantitatively, geometric modeling bridges the gap between molecular information, such as that from X-ray, NMR and cryo-EM, and theoretical/mathematical models, such as molecular dynamics, the Poisson-Boltzmann equation and the Nernst-Planck equation. In this work, we present a family of variational multiscale geometric models for macromolecular systems. Our models are able to combine multiresolution geometric modeling with multiscale electrostatic modeling in a unified variational framework. We discuss a suite of techniques for molecular surface generation, molecular surface meshing, molecular volumetric meshing, and the estimation of Hadwiger’s functionals. Emphasis is given to the multiresolution representations of biomolecules and the associated multiscale electrostatic analyses as well as multiresolution curvature characterizations. The resulting fine resolution representations of a biomolecular system enable the detailed analysis of solvent-solute interaction, and ion channel dynamics, while our coarse resolution representations highlight the compatibility of protein-ligand bindings and possibility of protein-protein interactions. PMID:23813599

  10. Creating a YouTube-Like Collaborative Environment in Mathematics: Integrating Animated Geogebra Constructions and Student-Generated Screencast Videos

    ERIC Educational Resources Information Center

    Lazarus, Jill; Roulet, Geoffrey

    2013-01-01

    This article discusses the integration of student-generated GeoGebra applets and Jing screencast videos to create a YouTube-like medium for sharing in mathematics. The value of combining dynamic mathematics software and screencast videos for facilitating communication and representations in a digital era is demonstrated herein. We share our…

  11. Exploring Social Equity Aspects in Integrating Technology in Primary Mathematics Education

    ERIC Educational Resources Information Center

    Stoilescu, Dorian

    2014-01-01

    This research focus on aspects of equity related to the introduction of using technology in classrooms. Technology has the potential to support mathematics pedagogy with visual representations and offer modelling and simulation facilities, increasing the creativity of the learning and teaching processes (Kaput, Ness, & Hoyles, 2008; Stoilescu…

  12. The Effects of Self-Explanation and Metacognitive Instruction on Undergraduate Students' Learning of Statistics Materials Containing Multiple External Representations in a Web-Based Environment

    ERIC Educational Resources Information Center

    Hsu, Yu-Chang

    2009-01-01

    Students in the Science, Technology, Engineering, and Mathematics (STEM) fields are confronted with multiple external representations (MERs) in their learning materials. The ability to learn from and communicate with these MERs requires not only that students comprehend each representation individually but also that students recognize how the…

  13. A Hilbert Space Representation of Generalized Observables and Measurement Processes in the ESR Model

    NASA Astrophysics Data System (ADS)

    Sozzo, Sandro; Garola, Claudio

    2010-12-01

    The extended semantic realism ( ESR) model recently worked out by one of the authors embodies the mathematical formalism of standard (Hilbert space) quantum mechanics in a noncontextual framework, reinterpreting quantum probabilities as conditional instead of absolute. We provide here a Hilbert space representation of the generalized observables introduced by the ESR model that satisfy a simple physical condition, propose a generalization of the projection postulate, and suggest a possible mathematical description of the measurement process in terms of evolution of the compound system made up of the measured system and the measuring apparatus.

  14. Technology and Mathematics Education: A Survey of Recent Developments and Important Problems.

    ERIC Educational Resources Information Center

    Fey, James T.

    1989-01-01

    Provided is an overview and analysis of recent progress in applying electronic information technology to creation of new environments for intellectual work in mathematics. Describes the impact of numerical computation, graphic computation, symbolic computation, multiple representation of information, programing and information, and artificial…

  15. Generating and Analyzing Visual Representations of Conic Sections with the Use of Technological Tools

    ERIC Educational Resources Information Center

    Santos-Trigo, Manuel; Espinosa-Perez, Hugo; Reyes-Rodriguez, Aaron

    2006-01-01

    Technological tools have the potential to offer students the possibility to represent information and relationships embedded in problems and concepts in ways that involve numerical, algebraic, geometric, and visual approaches. In this paper, the authors present and discuss an example in which an initial representation of a mathematical object…

  16. Impact of Context and Representation on Year 10 Students' Expression of Conceptions of Rate

    ERIC Educational Resources Information Center

    Herbert, Sandra

    2010-01-01

    Rate is an important, but difficult mathematical concept. More than twenty years of research, especially with calculus students, report difficulties with this concept. This paper reports on an alternative analysis, from the perspective of multiple representations and context, of interviews probing twenty Victorian Year 10 students' conceptions of…

  17. Machine learning of neural representations of suicide and emotion concepts identifies suicidal youth

    PubMed Central

    Just, Marcel Adam; Pan, Lisa; Cherkassky, Vladimir L.; McMakin, Dana; Cha, Christine; Nock, Matthew K.; Brent, David

    2017-01-01

    The clinical assessment of suicidal risk would be significantly complemented by a biologically-based measure that assesses alterations in the neural representations of concepts related to death and life in people who engage in suicidal ideation. This study used machine-learning algorithms (Gaussian Naïve Bayes) to identify such individuals (17 suicidal ideators vs 17 controls) with high (91%) accuracy, based on their altered fMRI neural signatures of death and life-related concepts. The most discriminating concepts were death, cruelty, trouble, carefree, good, and praise. A similar classification accurately (94%) discriminated 9 suicidal ideators who had made a suicide attempt from 8 who had not. Moreover, a major facet of the concept alterations was the evoked emotion, whose neural signature served as an alternative basis for accurate (85%) group classification. The study establishes a biological, neurocognitive basis for altered concept representations in participants with suicidal ideation, which enables highly accurate group membership classification. PMID:29367952

  18. Student Connections between Algebraic and Graphical Polynomial Representations in the Context of a Polynomial Relation

    ERIC Educational Resources Information Center

    Adu-Gyamfi, Kwaku; Bossé, Michael J.; Chandler, Kayla

    2017-01-01

    When establishing connections among representations of associated mathematical concepts, students encounter different difficulties and successes along the way. The purpose of this study was to uncover information about and gain greater insight into how student processes connections. Pre-calculus students were observed and interviewed while…

  19. Modellus: Learning Physics with Mathematical Modelling

    NASA Astrophysics Data System (ADS)

    Teodoro, Vitor

    Computers are now a major tool in research and development in almost all scientific and technological fields. Despite recent developments, this is far from true for learning environments in schools and most undergraduate studies. This thesis proposes a framework for designing curricula where computers, and computer modelling in particular, are a major tool for learning. The framework, based on research on learning science and mathematics and on computer user interface, assumes that: 1) learning is an active process of creating meaning from representations; 2) learning takes place in a community of practice where students learn both from their own effort and from external guidance; 3) learning is a process of becoming familiar with concepts, with links between concepts, and with representations; 4) direct manipulation user interfaces allow students to explore concrete-abstract objects such as those of physics and can be used by students with minimal computer knowledge. Physics is the science of constructing models and explanations about the physical world. And mathematical models are an important type of models that are difficult for many students. These difficulties can be rooted in the fact that most students do not have an environment where they can explore functions, differential equations and iterations as primary objects that model physical phenomena--as objects-to-think-with, reifying the formal objects of physics. The framework proposes that students should be introduced to modelling in a very early stage of learning physics and mathematics, two scientific areas that must be taught in very closely related way, as they were developed since Galileo and Newton until the beginning of our century, before the rise of overspecialisation in science. At an early stage, functions are the main type of objects used to model real phenomena, such as motions. At a later stage, rates of change and equations with rates of change play an important role. This type of equations

  20. First-Graders' Spatial-Mathematical Reasoning about Plane and Solid Shapes and Their Representations

    ERIC Educational Resources Information Center

    Hallowell, David A.; Okamoto, Yukari; Romo, Laura F.; La Joy, Jonna R.

    2015-01-01

    The primary goal of the study was to explore first-grade children's reasoning about plane and solid shapes across various kinds of geometric representations. Children were individually interviewed while completing a shape-matching task developed for this study. This task required children to compose and decompose geometric figures to identify…

  1. Psycho-Social Determinants of Gender Prejudice in Science, Technology, Engineering and Mathematics

    ERIC Educational Resources Information Center

    Nnachi, N. O.; Okpube, M. N.

    2015-01-01

    This work focused on the "Psycho-social Determinants of Gender Prejudice in Science, Technology, Engineering and Mathematics (STEM)". The females were found to be underrepresented in STEM fields. The under-representation results from gender stereotype, differences in spatial skills, hierarchical and territorial segregations and…

  2. A Multifaceted Mathematical Approach for Complex Systems

    DOE Office of Scientific and Technical Information (OSTI.GOV)

    Alexander, F.; Anitescu, M.; Bell, J.

    2012-03-07

    Applied mathematics has an important role to play in developing the tools needed for the analysis, simulation, and optimization of complex problems. These efforts require the development of the mathematical foundations for scientific discovery, engineering design, and risk analysis based on a sound integrated approach for the understanding of complex systems. However, maximizing the impact of applied mathematics on these challenges requires a novel perspective on approaching the mathematical enterprise. Previous reports that have surveyed the DOE's research needs in applied mathematics have played a key role in defining research directions with the community. Although these reports have had significantmore » impact, accurately assessing current research needs requires an evaluation of today's challenges against the backdrop of recent advances in applied mathematics and computing. To address these needs, the DOE Applied Mathematics Program sponsored a Workshop for Mathematics for the Analysis, Simulation and Optimization of Complex Systems on September 13-14, 2011. The workshop had approximately 50 participants from both the national labs and academia. The goal of the workshop was to identify new research areas in applied mathematics that will complement and enhance the existing DOE ASCR Applied Mathematics Program efforts that are needed to address problems associated with complex systems. This report describes recommendations from the workshop and subsequent analysis of the workshop findings by the organizing committee.« less

  3. Mathematical models of cell motility.

    PubMed

    Flaherty, Brendan; McGarry, J P; McHugh, P E

    2007-01-01

    Cell motility is an essential biological action in the creation, operation and maintenance of our bodies. Developing mathematical models elucidating cell motility will greatly advance our understanding of this fundamental biological process. With accurate models it is possible to explore many permutations of the same event and concisely investigate their outcome. While great advancements have been made in experimental studies of cell motility, it now has somewhat fallen on mathematical models to taking a leading role in future developments. The obvious reason for this is the complexity of cell motility. Employing the processing power of today's computers will give researches the ability to run complex biophysical and biochemical scenarios, without the inherent difficulty and time associated with in vitro investigations. Before any great advancement can be made, the basics of cell motility will have to be well-defined. Without this, complicated mathematical models will be hindered by their inherent conjecture. This review will look at current mathematical investigations of cell motility, explore the reasoning behind such work and conclude with how best to advance this interesting and challenging research area.

  4. Numerical approximation abilities correlate with and predict informal but not formal mathematics abilities

    PubMed Central

    Libertus, Melissa E.; Feigenson, Lisa; Halberda, Justin

    2013-01-01

    Previous research has found a relationship between individual differences in children’s precision when nonverbally approximating quantities and their school mathematics performance. School mathematics performance emerges from both informal (e.g., counting) and formal (e.g., knowledge of mathematics facts) abilities. It remains unknown whether approximation precision relates to both of these types of mathematics abilities. In the present study we assessed the precision of numerical approximation in 85 3- to 7-year-old children four times over a span of two years. Additionally, at the last time point, we tested children’s informal and formal mathematics abilities using the Test of Early Mathematics Ability (TEMA-3; Ginsburg & Baroody, 2003). We found that children’s numerical approximation precision correlated with and predicted their informal, but not formal, mathematics abilities when controlling for age and IQ. These results add to our growing understanding of the relationship between an unlearned, non-symbolic system of quantity representation and the system of mathematical reasoning that children come to master through instruction. PMID:24076381

  5. Numerical approximation abilities correlate with and predict informal but not formal mathematics abilities.

    PubMed

    Libertus, Melissa E; Feigenson, Lisa; Halberda, Justin

    2013-12-01

    Previous research has found a relationship between individual differences in children's precision when nonverbally approximating quantities and their school mathematics performance. School mathematics performance emerges from both informal (e.g., counting) and formal (e.g., knowledge of mathematics facts) abilities. It remains unknown whether approximation precision relates to both of these types of mathematics abilities. In the current study, we assessed the precision of numerical approximation in 85 3- to 7-year-old children four times over a span of 2years. In addition, at the final time point, we tested children's informal and formal mathematics abilities using the Test of Early Mathematics Ability (TEMA-3). We found that children's numerical approximation precision correlated with and predicted their informal, but not formal, mathematics abilities when controlling for age and IQ. These results add to our growing understanding of the relationship between an unlearned nonsymbolic system of quantity representation and the system of mathematics reasoning that children come to master through instruction. Copyright © 2013 Elsevier Inc. All rights reserved.

  6. Ensemble representations: effects of set size and item heterogeneity on average size perception.

    PubMed

    Marchant, Alexander P; Simons, Daniel J; de Fockert, Jan W

    2013-02-01

    Observers can accurately perceive and evaluate the statistical properties of a set of objects, forming what is now known as an ensemble representation. The accuracy and speed with which people can judge the mean size of a set of objects have led to the proposal that ensemble representations of average size can be computed in parallel when attention is distributed across the display. Consistent with this idea, judgments of mean size show little or no decrement in accuracy when the number of objects in the set increases. However, the lack of a set size effect might result from the regularity of the item sizes used in previous studies. Here, we replicate these previous findings, but show that judgments of mean set size become less accurate when set size increases and the heterogeneity of the item sizes increases. This pattern can be explained by assuming that average size judgments are computed using a limited capacity sampling strategy, and it does not necessitate an ensemble representation computed in parallel across all items in a display. Copyright © 2012 Elsevier B.V. All rights reserved.

  7. EliXR-TIME: A Temporal Knowledge Representation for Clinical Research Eligibility Criteria.

    PubMed

    Boland, Mary Regina; Tu, Samson W; Carini, Simona; Sim, Ida; Weng, Chunhua

    2012-01-01

    Effective clinical text processing requires accurate extraction and representation of temporal expressions. Multiple temporal information extraction models were developed but a similar need for extracting temporal expressions in eligibility criteria (e.g., for eligibility determination) remains. We identified the temporal knowledge representation requirements of eligibility criteria by reviewing 100 temporal criteria. We developed EliXR-TIME, a frame-based representation designed to support semantic annotation for temporal expressions in eligibility criteria by reusing applicable classes from well-known clinical temporal knowledge representations. We used EliXR-TIME to analyze a training set of 50 new temporal eligibility criteria. We evaluated EliXR-TIME using an additional random sample of 20 eligibility criteria with temporal expressions that have no overlap with the training data, yielding 92.7% (76 / 82) inter-coder agreement on sentence chunking and 72% (72 / 100) agreement on semantic annotation. We conclude that this knowledge representation can facilitate semantic annotation of the temporal expressions in eligibility criteria.

  8. Know thyself: behavioral evidence for a structural representation of the human body.

    PubMed

    Rusconi, Elena; Gonzaga, Mirandola; Adriani, Michela; Braun, Christoph; Haggard, Patrick

    2009-01-01

    Representing one's own body is often viewed as a basic form of self-awareness. However, little is known about structural representations of the body in the brain. We developed an inter-manual version of the classical "in-between" finger gnosis task: participants judged whether the number of untouched fingers between two touched fingers was the same on both hands, or different. We thereby dissociated structural knowledge about fingers, specifying their order and relative position within a hand, from tactile sensory codes. Judgments following stimulation on homologous fingers were consistently more accurate than trials with no or partial homology. Further experiments showed that structural representations are more enduring than purely sensory codes, are used even when number of fingers is irrelevant to the task, and moreover involve an allocentric representation of finger order, independent of hand posture. Our results suggest the existence of an allocentric representation of body structure at higher stages of the somatosensory processing pathway, in addition to primary sensory representation.

  9. Employing Genetic "Moments" in the History of Mathematics in Classroom Activities

    ERIC Educational Resources Information Center

    Farmaki, Vassiliki; Paschos, Theodorus

    2007-01-01

    The integration of history into educational practice can lead to the development of activities through the use of genetic "moments" in the history of mathematics. In the present paper, we utilize Oresme's genetic ideas--developed during the fourteenth century, including ideas on the velocity-time graphical representation as well as geometric…

  10. Multisensory decisions provide support for probabilistic number representations.

    PubMed

    Kanitscheider, Ingmar; Brown, Amanda; Pouget, Alexandre; Churchland, Anne K

    2015-06-01

    A large body of evidence suggests that an approximate number sense allows humans to estimate numerosity in sensory scenes. This ability is widely observed in humans, including those without formal mathematical training. Despite this, many outstanding questions remain about the nature of the numerosity representation in the brain. Specifically, it is not known whether approximate numbers are represented as scalar estimates of numerosity or, alternatively, as probability distributions over numerosity. In the present study, we used a multisensory decision task to distinguish these possibilities. We trained human subjects to decide whether a test stimulus had a larger or smaller numerosity compared with a fixed reference. Depending on the trial, the numerosity was presented as either a sequence of visual flashes or a sequence of auditory tones, or both. To test for a probabilistic representation, we varied the reliability of the stimulus by adding noise to the visual stimuli. In accordance with a probabilistic representation, we observed a significant improvement in multisensory compared with unisensory trials. Furthermore, a trial-by-trial analysis revealed that although individual subjects showed strategic differences in how they leveraged auditory and visual information, all subjects exploited the reliability of unisensory cues. An alternative, nonprobabilistic model, in which subjects combined cues without regard for reliability, was not able to account for these trial-by-trial choices. These findings provide evidence that the brain relies on a probabilistic representation for numerosity decisions. Copyright © 2015 the American Physiological Society.

  11. Impaired Acuity of the Approximate Number System Underlies Mathematical Learning Disability (Dyscalculia)

    ERIC Educational Resources Information Center

    Mazzocco, Michele M. M.; Feigenson, Lisa; Halberda, Justin

    2011-01-01

    Many children have significant mathematical learning disabilities (MLD, or dyscalculia) despite adequate schooling. The current study hypothesizes that MLD partly results from a deficiency in the Approximate Number System (ANS) that supports nonverbal numerical representations across species and throughout development. In this study of 71 ninth…

  12. Dynamic belief state representations.

    PubMed

    Lee, Daniel D; Ortega, Pedro A; Stocker, Alan A

    2014-04-01

    Perceptual and control systems are tasked with the challenge of accurately and efficiently estimating the dynamic states of objects in the environment. To properly account for uncertainty, it is necessary to maintain a dynamical belief state representation rather than a single state vector. In this review, canonical algorithms for computing and updating belief states in robotic applications are delineated, and connections to biological systems are highlighted. A navigation example is used to illustrate the importance of properly accounting for correlations between belief state components, and to motivate the need for further investigations in psychophysics and neurobiology. Copyright © 2014 Elsevier Ltd. All rights reserved.

  13. Managing Your Mathematics Program: A Total System. A Guide to the U-SAIL Basic Mathematics System.

    ERIC Educational Resources Information Center

    Hales, Carma M.; Jones, Maurine E.

    The Utah System Approach to Individual Learning (U-SAIL) Mathematics System was developed to make it possible for teachers to provide excellence in arithmetic instruction. It is based on the premise that in order to teach arithmetic well, teachers must accurately assess, teach directly, provide students with focused practice, corrective feedback,…

  14. A roadmap for improving the representation of photosynthesis in Earth system models.

    PubMed

    Rogers, Alistair; Medlyn, Belinda E; Dukes, Jeffrey S; Bonan, Gordon; von Caemmerer, Susanne; Dietze, Michael C; Kattge, Jens; Leakey, Andrew D B; Mercado, Lina M; Niinemets, Ülo; Prentice, I Colin; Serbin, Shawn P; Sitch, Stephen; Way, Danielle A; Zaehle, Sönke

    2017-01-01

    Accurate representation of photosynthesis in terrestrial biosphere models (TBMs) is essential for robust projections of global change. However, current representations vary markedly between TBMs, contributing uncertainty to projections of global carbon fluxes. Here we compared the representation of photosynthesis in seven TBMs by examining leaf and canopy level responses of photosynthetic CO 2 assimilation (A) to key environmental variables: light, temperature, CO 2 concentration, vapor pressure deficit and soil water content. We identified research areas where limited process knowledge prevents inclusion of physiological phenomena in current TBMs and research areas where data are urgently needed for model parameterization or evaluation. We provide a roadmap for new science needed to improve the representation of photosynthesis in the next generation of terrestrial biosphere and Earth system models. No claim to original US Government works New Phytologist © 2016 New Phytologist Trust.

  15. Profile of Students’ Mental Model Change on Law Concepts Archimedes as Impact of Multi-Representation Approach

    NASA Astrophysics Data System (ADS)

    Taher, M.; Hamidah, I.; Suwarma, I. R.

    2017-09-01

    This paper outlined the results of an experimental study on the effects of multi-representation approach in learning Archimedes Law on students’ mental model improvement. The multi-representation techniques implemented in the study were verbal, pictorial, mathematical, and graphical representations. Students’ mental model was classified into three levels, i.e. scientific, synthetic, and initial levels, based on the students’ level of understanding. The present study employed the pre-experimental methodology, using one group pretest-posttest design. The subject of the study was 32 eleventh grade students in a Public Senior High School in Riau Province. The research instrument included model mental test on hydrostatic pressure concept, in the form of essay test judged by experts. The findings showed that there was positive change in students’ mental model, indicating that multi-representation approach was effective to improve students’ mental model.

  16. Focus group discussion in mathematical physics learning

    NASA Astrophysics Data System (ADS)

    Ellianawati; Rudiana, D.; Sabandar, J.; Subali, B.

    2018-03-01

    The Focus Group Discussion (FGD) activity in Mathematical Physics learning has helped students perform the stages of problem solving reflectively. The FGD implementation was conducted to explore the problems and find the right strategy to improve the students' ability to solve the problem accurately which is one of reflective thinking component that has been difficult to improve. The research method used is descriptive qualitative by using single subject response in Physics student. During the FGD process, one student was observed of her reflective thinking development in solving the physics problem. The strategy chosen in the discussion activity was the Cognitive Apprenticeship-Instruction (CA-I) syntax. Based on the results of this study, it is obtained the information that after going through a series of stages of discussion, the students' reflective thinking skills is increased significantly. The scaffolding stage in the CA-I model plays an important role in the process of solving physics problems accurately. Students are able to recognize and formulate problems by describing problem sketches, identifying the variables involved, applying mathematical equations that accord to physics concepts, executing accurately, and applying evaluation by explaining the solution to various contexts.

  17. Development of a Mathematical Ability Test: A Validity and Reliability Study

    ERIC Educational Resources Information Center

    Dündar, Sefa; Temel, Hasan; Gündüz, Nazan

    2016-01-01

    The identification of talented students accurately at an early age and the adaptation of the education provided to the students depending on their abilities are of great importance for the future of the countries. In this regard, this study aims to develop a mathematical ability test for the identification of the mathematical abilities of students…

  18. INCREASING ACHIEVEMENT AND HIGHER-EDUCATION REPRESENTATION OF UNDER-REPRESENTED GROUPS IN SCIENCE, TECHNOLOGY, ENGINEERING, AND MATHEMATICS FIELDS: A REVIEW OF CURRENT K-12 INTERVENTION PROGRAMS.

    PubMed

    Valla, Jeffrey M; Williams, Wendy M

    2012-01-01

    The under-representation of women and ethnic minorities in Science, Technology, Engineering, and Mathematics (STEM) education and professions has resulted in a loss of human capital for the US scientific workforce and spurred the development of myriad STEM educational intervention programs. Increased allocation of resources to such programs begs for a critical, prescriptive, evidence-based review that will enable researchers to develop optimal interventions and administrators to maximize investments. We begin by providing a theoretical backdrop for K-12 STEM programs by reviewing current data on under-representation and developmental research describing individual-level social factors undergirding these data. Next, we review prototypical designs of these programs, highlighting specific programs in the literature as examples of program structures and components currently in use. We then evaluate these interventions in terms of overall effectiveness, as a function of how well they address age-, ethnicity-, or gender-specific factors, suggesting improvements in program design based on these critiques. Finally, program evaluation methods are briefly reviewed and discussed in terms of how their empirical soundness can either enable or limit our ability to delineate effective program components. "Now more than ever, the nation's changing demographics demand that we include all of our citizens in science and engineering education and careers. For the U.S. to benefit from the diverse talents of all its citizens, we must grow the pipeline of qualified, underrepresented minority engineers and scientists to fill positions in industry and academia."-Irving P. McPhail..

  19. Beyond Natural Numbers: Negative Number Representation in Parietal Cortex

    PubMed Central

    Blair, Kristen P.; Rosenberg-Lee, Miriam; Tsang, Jessica M.; Schwartz, Daniel L.; Menon, Vinod

    2012-01-01

    Unlike natural numbers, negative numbers do not have natural physical referents. How does the brain represent such abstract mathematical concepts? Two competing hypotheses regarding representational systems for negative numbers are a rule-based model, in which symbolic rules are applied to negative numbers to translate them into positive numbers when assessing magnitudes, and an expanded magnitude model, in which negative numbers have a distinct magnitude representation. Using an event-related functional magnetic resonance imaging design, we examined brain responses in 22 adults while they performed magnitude comparisons of negative and positive numbers that were quantitatively near (difference <4) or far apart (difference >6). Reaction times (RTs) for negative numbers were slower than positive numbers, and both showed a distance effect whereby near pairs took longer to compare. A network of parietal, frontal, and occipital regions were differentially engaged by negative numbers. Specifically, compared to positive numbers, negative number processing resulted in greater activation bilaterally in intraparietal sulcus (IPS), middle frontal gyrus, and inferior lateral occipital cortex. Representational similarity analysis revealed that neural responses in the IPS were more differentiated among positive numbers than among negative numbers, and greater differentiation among negative numbers was associated with faster RTs. Our findings indicate that despite negative numbers engaging the IPS more strongly, the underlying neural representation are less distinct than that of positive numbers. We discuss our findings in the context of the two theoretical models of negative number processing and demonstrate how multivariate approaches can provide novel insights into abstract number representation. PMID:22363276

  20. Beyond natural numbers: negative number representation in parietal cortex.

    PubMed

    Blair, Kristen P; Rosenberg-Lee, Miriam; Tsang, Jessica M; Schwartz, Daniel L; Menon, Vinod

    2012-01-01

    Unlike natural numbers, negative numbers do not have natural physical referents. How does the brain represent such abstract mathematical concepts? Two competing hypotheses regarding representational systems for negative numbers are a rule-based model, in which symbolic rules are applied to negative numbers to translate them into positive numbers when assessing magnitudes, and an expanded magnitude model, in which negative numbers have a distinct magnitude representation. Using an event-related functional magnetic resonance imaging design, we examined brain responses in 22 adults while they performed magnitude comparisons of negative and positive numbers that were quantitatively near (difference <4) or far apart (difference >6). Reaction times (RTs) for negative numbers were slower than positive numbers, and both showed a distance effect whereby near pairs took longer to compare. A network of parietal, frontal, and occipital regions were differentially engaged by negative numbers. Specifically, compared to positive numbers, negative number processing resulted in greater activation bilaterally in intraparietal sulcus (IPS), middle frontal gyrus, and inferior lateral occipital cortex. Representational similarity analysis revealed that neural responses in the IPS were more differentiated among positive numbers than among negative numbers, and greater differentiation among negative numbers was associated with faster RTs. Our findings indicate that despite negative numbers engaging the IPS more strongly, the underlying neural representation are less distinct than that of positive numbers. We discuss our findings in the context of the two theoretical models of negative number processing and demonstrate how multivariate approaches can provide novel insights into abstract number representation.

  1. Accurately tracking single-cell movement trajectories in microfluidic cell sorting devices.

    PubMed

    Jeong, Jenny; Frohberg, Nicholas J; Zhou, Enlu; Sulchek, Todd; Qiu, Peng

    2018-01-01

    Microfluidics are routinely used to study cellular properties, including the efficient quantification of single-cell biomechanics and label-free cell sorting based on the biomechanical properties, such as elasticity, viscosity, stiffness, and adhesion. Both quantification and sorting applications require optimal design of the microfluidic devices and mathematical modeling of the interactions between cells, fluid, and the channel of the device. As a first step toward building such a mathematical model, we collected video recordings of cells moving through a ridged microfluidic channel designed to compress and redirect cells according to cell biomechanics. We developed an efficient algorithm that automatically and accurately tracked the cell trajectories in the recordings. We tested the algorithm on recordings of cells with different stiffness, and showed the correlation between cell stiffness and the tracked trajectories. Moreover, the tracking algorithm successfully picked up subtle differences of cell motion when passing through consecutive ridges. The algorithm for accurately tracking cell trajectories paves the way for future efforts of modeling the flow, forces, and dynamics of cell properties in microfluidics applications.

  2. Students' Development of Representational Competence Through the Sense of Touch

    NASA Astrophysics Data System (ADS)

    Magana, Alejandra J.; Balachandran, Sadhana

    2017-06-01

    Electromagnetism is an umbrella encapsulating several different concepts like electric current, electric fields and forces, and magnetic fields and forces, among other topics. However, a number of studies in the past have highlighted the poor conceptual understanding of electromagnetism concepts by students even after instruction. This study aims to identify novel forms of "hands-on" instruction that can result in representational competence and conceptual gain. Specifically, this study aimed to identify if the use of visuohaptic simulations can have an effect on student representations of electromagnetic-related concepts. The guiding questions is How do visuohaptic simulations influence undergraduate students' representations of electric forces? Participants included nine undergraduate students from science, technology, or engineering backgrounds who participated in a think-aloud procedure while interacting with a visuohaptic simulation. The think-aloud procedure was divided in three stages, a prediction stage, a minimally visual haptic stage, and a visually enhanced haptic stage. The results of this study suggest that students' accurately characterized and represented the forces felt around a particle, line, and ring charges either in the prediction stage, a minimally visual haptic stage or the visually enhanced haptic stage. Also, some students accurately depicted the three-dimensional nature of the field for each configuration in the two stages that included a tactile mode, where the point charge was the most challenging one.

  3. Proceedings of the Annual Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (10th, Dekalb, Illinois, November 2-5, 1988).

    ERIC Educational Resources Information Center

    Behr, Merlyn J., Ed.; Lacampagne, Carole B., Ed.; Wheeler, Margariete Montague, Ed.

    This proceedings of the annual conference of the North American Chapter of the International Group for the Psychology of Mathematics Education includes the following papers: "Affective Representation and Mathematical Problem Solving" (Goldin); "Cognitive Readiness and Process-Oriented Instruction in Secondary School Mathematics" (Harrison); "An…

  4. Pictorial Representations of Simple Arithmetic Problems Are Not Always Helpful: A Cognitive Load Perspective

    ERIC Educational Resources Information Center

    van Lieshout, Ernest C. D. M.; Xenidou-Dervou, Iro

    2018-01-01

    At the start of mathematics education children are often presented with addition and subtraction problems in the form of pictures. They are asked to solve the problems by filling in corresponding number sentences. One type of problem concerns the representation of an increase or a decrease in a depicted amount. A decrease is, however, more…

  5. Advances in visual representation of molecular potentials.

    PubMed

    Du, Qi-Shi; Huang, Ri-Bo; Chou, Kuo-Chen

    2010-06-01

    The recent advances in visual representations of molecular properties in 3D space are summarized, and their applications in molecular modeling study and rational drug design are introduced. The visual representation methods provide us with detailed insights into protein-ligand interactions, and hence can play a major role in elucidating the structure or reactivity of a biomolecular system. Three newly developed computation and visualization methods for studying the physical and chemical properties of molecules are introduced, including their electrostatic potential, lipophilicity potential and excess chemical potential. The newest application examples of visual representations in structure-based rational drug are presented. The 3D electrostatic potentials, calculated using the empirical method (EM-ESP), in which the classical Coulomb equation and traditional atomic partial changes are discarded, are highly consistent with the results by the higher level quantum chemical method. The 3D lipophilicity potentials, computed by the heuristic molecular lipophilicity potential method based on the principles of quantum mechanics and statistical mechanics, are more accurate and reliable than those by using the traditional empirical methods. The 3D excess chemical potentials, derived by the reference interaction site model-hypernetted chain theory, provide a new tool for computational chemistry and molecular modeling. For structure-based drug design, the visual representations of molecular properties will play a significant role in practical applications. It is anticipated that the new advances in computational chemistry will stimulate the development of molecular modeling methods, further enriching the visual representation techniques for rational drug design, as well as other relevant fields in life science.

  6. The relationship between mathematics and language: academic implications for children with specific language impairment and English language learners.

    PubMed

    Alt, Mary; Arizmendi, Genesis D; Beal, Carole R

    2014-07-01

    The present study examined the relationship between mathematics and language to better understand the nature of the deficit and the academic implications associated with specific language impairment (SLI) and academic implications for English language learners (ELLs). School-age children (N = 61; 20 SLI, 20 ELL, 21 native monolingual English [NE]) were assessed using a norm-referenced mathematics instrument and 3 experimental computer-based mathematics games that varied in language demands. Group means were compared with analyses of variance. The ELL group was less accurate than the NE group only when tasks were language heavy. In contrast, the group with SLI was less accurate than the groups with NE and ELLs on language-heavy tasks and some language-light tasks. Specifically, the group with SLI was less accurate on tasks that involved comparing numerical symbols and using visual working memory for patterns. However, there were no group differences between children with SLI and peers without SLI on language-light mathematics tasks that involved visual working memory for numerical symbols. Mathematical difficulties of children who are ELLs appear to be related to the language demands of mathematics tasks. In contrast, children with SLI appear to have difficulty with mathematics tasks because of linguistic as well as nonlinguistic processing constraints.

  7. Metrological traceability in education: A practical online system for measuring and managing middle school mathematics instruction

    NASA Astrophysics Data System (ADS)

    Torres Irribarra, D.; Freund, R.; Fisher, W.; Wilson, M.

    2015-02-01

    Computer-based, online assessments modelled, designed, and evaluated for adaptively administered invariant measurement are uniquely suited to defining and maintaining traceability to standardized units in education. An assessment of this kind is embedded in the Assessing Data Modeling and Statistical Reasoning (ADM) middle school mathematics curriculum. Diagnostic information about middle school students' learning of statistics and modeling is provided via computer-based formative assessments for seven constructs that comprise a learning progression for statistics and modeling from late elementary through the middle school grades. The seven constructs are: Data Display, Meta-Representational Competence, Conceptions of Statistics, Chance, Modeling Variability, Theory of Measurement, and Informal Inference. The end product is a web-delivered system built with Ruby on Rails for use by curriculum development teams working with classroom teachers in designing, developing, and delivering formative assessments. The online accessible system allows teachers to accurately diagnose students' unique comprehension and learning needs in a common language of real-time assessment, logging, analysis, feedback, and reporting.

  8. 3D acquisition and modeling for flint artefacts analysis

    NASA Astrophysics Data System (ADS)

    Loriot, B.; Fougerolle, Y.; Sestier, C.; Seulin, R.

    2007-07-01

    In this paper, we are interested in accurate acquisition and modeling of flint artefacts. Archaeologists needs accurate geometry measurements to refine their understanding of the flint artefacts manufacturing process. Current techniques require several operations. First, a copy of a flint artefact is reproduced. The copy is then sliced. A picture is taken for each slice. Eventually, geometric information is manually determined from the pictures. Such a technique is very time consuming, and the processing applied to the original, as well as the reproduced object, induces several measurement errors (prototyping approximations, slicing, image acquisition, and measurement). By using 3D scanners, we significantly reduce the number of operations related to data acquisition and completely suppress the prototyping step to obtain an accurate 3D model. The 3D models are segmented into sliced parts that are then analyzed. Each slice is then automatically fitted by mathematical representation. Such a representation offers several interesting properties: geometric features can be characterized (e.g. shapes, curvature, sharp edges, etc), and a shape of the original piece of stone can be extrapolated. The contributions of this paper are an acquisition technique using 3D scanners that strongly reduces human intervention, acquisition time and measurement errors, and the representation of flint artefacts as mathematical 2D sections that enable accurate analysis.

  9. Individual Differences in Inhibitory Control, Not Non-Verbal Number Acuity, Correlate with Mathematics Achievement

    PubMed Central

    Gilmore, Camilla; Attridge, Nina; Clayton, Sarah; Cragg, Lucy; Johnson, Samantha; Marlow, Neil; Simms, Victoria; Inglis, Matthew

    2013-01-01

    Given the well-documented failings in mathematics education in many Western societies, there has been an increased interest in understanding the cognitive underpinnings of mathematical achievement. Recent research has proposed the existence of an Approximate Number System (ANS) which allows individuals to represent and manipulate non-verbal numerical information. Evidence has shown that performance on a measure of the ANS (a dot comparison task) is related to mathematics achievement, which has led researchers to suggest that the ANS plays a critical role in mathematics learning. Here we show that, rather than being driven by the nature of underlying numerical representations, this relationship may in fact be an artefact of the inhibitory control demands of some trials of the dot comparison task. This suggests that recent work basing mathematics assessments and interventions around dot comparison tasks may be inappropriate. PMID:23785521

  10. Knowledge representation in metabolic pathway databases.

    PubMed

    Stobbe, Miranda D; Jansen, Gerbert A; Moerland, Perry D; van Kampen, Antoine H C

    2014-05-01

    The accurate representation of all aspects of a metabolic network in a structured format, such that it can be used for a wide variety of computational analyses, is a challenge faced by a growing number of researchers. Analysis of five major metabolic pathway databases reveals that each database has made widely different choices to address this challenge, including how to deal with knowledge that is uncertain or missing. In concise overviews, we show how concepts such as compartments, enzymatic complexes and the direction of reactions are represented in each database. Importantly, also concepts which a database does not represent are described. Which aspects of the metabolic network need to be available in a structured format and to what detail differs per application. For example, for in silico phenotype prediction, a detailed representation of gene-protein-reaction relations and the compartmentalization of the network is essential. Our analysis also shows that current databases are still limited in capturing all details of the biology of the metabolic network, further illustrated with a detailed analysis of three metabolic processes. Finally, we conclude that the conceptual differences between the databases, which make knowledge exchange and integration a challenge, have not been resolved, so far, by the exchange formats in which knowledge representation is standardized.

  11. Using pedagogical discipline representations (PDRs) to enable Astro 101 students to reason about modern astrophysics

    NASA Astrophysics Data System (ADS)

    Wallace, Colin Scott; Prather, Edward E.; Chambers, Timothy G.; Kamenetzky, Julia R.; Hornstein, Seth D.

    2017-01-01

    Instructors of introductory, college-level, general education astronomy (Astro 101) often want to include topics from the cutting-edge of modern astrophysics in their course. Unfortunately, the teaching of these cutting-edge topics is typically confined to advanced undergraduate or graduate classes, using representations (graphical, mathematical, etc.) that are inaccessible to the vast majority of Astro 101 students. Consequently, many Astro 101 instructors feel that they have no choice but to cover these modern topics at a superficial level. Pedagogical discipline representations (PDRs) are one solution to this problem. Pedagogical discipline representations are representations that are explicitly designed to enhance the teaching and learning of a topic, even though these representations may not typically be found in traditional textbooks or used by experts in the discipline who are engaged in topic-specific discourse. In some cases, PDRs are significantly simplified or altered versions of typical discipline representations (graphs, data tables, etc.); in others they may be novel and highly contextualized representations with unique features that purposefully engage novice learners’ pre-existing mental models and reasoning difficulties, facilitating critical discourse. In this talk, I will discuss important lessons that my colleagues and I have learned while developing PDRs and describe how PDRs can enable students to reason about complex modern astrophysical topics.

  12. Accurate representation of organized convection in CFSv2 via a stochastic lattice model

    NASA Astrophysics Data System (ADS)

    Goswami, B. B.; Khouider, B.; Krishna, R. P. M. M.; Mukhopadhyay, P.; Majda, A.

    2016-12-01

    General circulation models (GCM) show limitations of various sorts in their representation of synoptic and intra-seasonal variability associated with tropical convective systems apart from the success of superparameterization and cloud system permitting global models. This systematic deficiency is believed to be due to the inadequate treatment of organized convection by the underlying cumulus parameterizations, which have the quasi-equilibrium assumption as a common denominator. By its nature, this assumption neglects the continuous interactions across scales between convection and the large scale dynamics. By design, the stochastic multicloud model (SMCM) mimics the interactions between the three cloud types, congestus, deep, and stratiform, that are observed to play a central role across multiple scales in the dynamics and physical structure of tropical convective systems. It is based on a stochastic lattice model, overlaid over each GCM grid box, where an order parameter taking the values 0,1,2,3 at each lattice site according to whether the site is clear sky or occupied by a congestus, deep, or stratiform cloud, respectively. As such the SMCM mimics the unresolved variability due to cumulus convection and the interactions across multiple scales of organized convective systems, following the philosophy of superparameterization. Here, we discuss the implementation of the SMCM in NCEP Climate Forecast System model (CFS), version-2, through the use of a simple parametrization of adiabatic heating and moisture sink due to cumulus clouds based on their observed vertical profiles (a.k.a Q1 and Q2). Much like the success of superparameterization but without the burden of high computational cost, a 20 year run showed tremendous improvements in the ability of the CFS-SMCM model to represent synoptic and intraseasonal variability associated with organized convection as well as a few minor improvements in the simulated climatology when compared to the control CFSv2 model

  13. Identifying Representational Competence with Multi-Representational Displays

    ERIC Educational Resources Information Center

    Stieff, Mike; Hegarty, Mary; Deslongchamps, Ghislain

    2011-01-01

    Increasingly, multi-representational educational technologies are being deployed in science classrooms to support science learning and the development of representational competence. Several studies have indicated that students experience significant challenges working with these multi-representational displays and prefer to use only one…

  14. Hierarchical Representation Learning for Kinship Verification.

    PubMed

    Kohli, Naman; Vatsa, Mayank; Singh, Richa; Noore, Afzel; Majumdar, Angshul

    2017-01-01

    Kinship verification has a number of applications such as organizing large collections of images and recognizing resemblances among humans. In this paper, first, a human study is conducted to understand the capabilities of human mind and to identify the discriminatory areas of a face that facilitate kinship-cues. The visual stimuli presented to the participants determine their ability to recognize kin relationship using the whole face as well as specific facial regions. The effect of participant gender and age and kin-relation pair of the stimulus is analyzed using quantitative measures such as accuracy, discriminability index d' , and perceptual information entropy. Utilizing the information obtained from the human study, a hierarchical kinship verification via representation learning (KVRL) framework is utilized to learn the representation of different face regions in an unsupervised manner. We propose a novel approach for feature representation termed as filtered contractive deep belief networks (fcDBN). The proposed feature representation encodes relational information present in images using filters and contractive regularization penalty. A compact representation of facial images of kin is extracted as an output from the learned model and a multi-layer neural network is utilized to verify the kin accurately. A new WVU kinship database is created, which consists of multiple images per subject to facilitate kinship verification. The results show that the proposed deep learning framework (KVRL-fcDBN) yields the state-of-the-art kinship verification accuracy on the WVU kinship database and on four existing benchmark data sets. Furthermore, kinship information is used as a soft biometric modality to boost the performance of face verification via product of likelihood ratio and support vector machine based approaches. Using the proposed KVRL-fcDBN framework, an improvement of over 20% is observed in the performance of face verification.

  15. Representational task formats and problem solving strategies in kinematics and work

    NASA Astrophysics Data System (ADS)

    Ibrahim, Bashirah; Rebello, N. Sanjay

    2012-06-01

    Previous studies have reported that students employed different problem solving approaches when presented with the same task structured with different representations. In this study, we explored and compared students’ strategies as they attempted tasks from two topical areas, kinematics and work. Our participants were 19 engineering students taking a calculus-based physics course. The tasks were presented in linguistic, graphical, and symbolic forms and requested either a qualitative solution or a value. The analysis was both qualitative and quantitative in nature focusing principally on the characteristics of the strategies employed as well as the underlying reasoning for their applications. A comparison was also made for the same student’s approach with the same kind of representation across the two topics. Additionally, the participants’ overall strategies across the different tasks, in each topic, were considered. On the whole, we found that the students prefer manipulating equations irrespective of the representational format of the task. They rarely recognized the applicability of a “qualitative” approach to solve the problem although they were aware of the concepts involved. Even when the students included visual representations in their solutions, they seldom used these representations in conjunction with the mathematical part of the problem. Additionally, the students were not consistent in their approach for interpreting and solving problems with the same kind of representation across the two topical areas. The representational format, level of prior knowledge, and familiarity with a topic appeared to influence their strategies, their written responses, and their ability to recognize qualitative ways to attempt a problem. The nature of the solution does not seem to impact the strategies employed to handle the problem.

  16. Mathematics of Information Processing and the Internet

    ERIC Educational Resources Information Center

    Hart, Eric W.

    2010-01-01

    The mathematics of information processing and the Internet can be organized around four fundamental themes: (1) access (finding information easily); (2) security (keeping information confidential); (3) accuracy (ensuring accurate information); and (4) efficiency (data compression). In this article, the author discusses each theme with reference to…

  17. Getting a Picture that Is Both Accurate and Stable: Situation Models and Epistemic Validation

    ERIC Educational Resources Information Center

    Schroeder, Sascha; Richter, Tobias; Hoever, Inga

    2008-01-01

    Text comprehension entails the construction of a situation model that prepares individuals for situated action. In order to meet this function, situation model representations are required to be both accurate and stable. We propose a framework according to which comprehenders rely on epistemic validation to prevent inaccurate information from…

  18. Teaching Subtraction and Multiplication with Regrouping Using the Concrete-Representational-Abstract Sequence and Strategic Instruction Model

    ERIC Educational Resources Information Center

    Flores, Margaret M.; Hinton, Vanessa; Strozier, Shaunita D.

    2014-01-01

    Based on Common Core Standards (2010), mathematics interventions should emphasize conceptual understanding of numbers and operations as well as fluency. For students at risk for failure, the concrete-representational-abstract (CRA) sequence and the Strategic Instruction Model (SIM) have been shown effective in teaching computation with an emphasis…

  19. Learning with Technology: Video Modeling with Concrete-Representational-Abstract Sequencing for Students with Autism Spectrum Disorder

    ERIC Educational Resources Information Center

    Yakubova, Gulnoza; Hughes, Elizabeth M.; Shinaberry, Megan

    2016-01-01

    The purpose of this study was to determine the effectiveness of a video modeling intervention with concrete-representational-abstract instructional sequence in teaching mathematics concepts to students with autism spectrum disorder (ASD). A multiple baseline across skills design of single-case experimental methodology was used to determine the…

  20. ICT integration in mathematics initial teacher training and its impact on visualization: the case of GeoGebra

    NASA Astrophysics Data System (ADS)

    Dockendorff, Monika; Solar, Horacio

    2018-01-01

    This case study investigates the impact of the integration of information and communications technology (ICT) in mathematics visualization skills and initial teacher education programmes. It reports on the influence GeoGebra dynamic software use has on promoting mathematical learning at secondary school and on its impact on teachers' conceptions about teaching and learning mathematics. This paper describes how GeoGebra-based dynamic applets - designed and used in an exploratory manner - promote mathematical processes such as conjectures. It also refers to the changes prospective teachers experience regarding the relevance visual dynamic representations acquire in teaching mathematics. This study observes a shift in school routines when incorporating technology into the mathematics classroom. Visualization appears as a basic competence associated to key mathematical processes. Implications of an early integration of ICT in mathematics initial teacher training and its impact on developing technological pedagogical content knowledge (TPCK) are drawn.

  1. Women's Leadership in Science, Technology, Engineering and Mathematics: Barriers to Participation

    ERIC Educational Resources Information Center

    McCullough, Laura

    2011-01-01

    Despite gains overall, women are still under-represented in leadership positions in science, technology, engineering, and mathematics (STEM) fields. Data in the US suggest around one-quarter of deans and department heads are women; in science this drops to nearly 1 in 20. Part of this problem of under-representation stems from the population pool:…

  2. Know Thyself: Behavioral Evidence for a Structural Representation of the Human Body

    PubMed Central

    Rusconi, Elena; Gonzaga, Mirandola; Adriani, Michela; Braun, Christoph; Haggard, Patrick

    2009-01-01

    Background Representing one's own body is often viewed as a basic form of self-awareness. However, little is known about structural representations of the body in the brain. Methods and Findings We developed an inter-manual version of the classical “in-between” finger gnosis task: participants judged whether the number of untouched fingers between two touched fingers was the same on both hands, or different. We thereby dissociated structural knowledge about fingers, specifying their order and relative position within a hand, from tactile sensory codes. Judgments following stimulation on homologous fingers were consistently more accurate than trials with no or partial homology. Further experiments showed that structural representations are more enduring than purely sensory codes, are used even when number of fingers is irrelevant to the task, and moreover involve an allocentric representation of finger order, independent of hand posture. Conclusions Our results suggest the existence of an allocentric representation of body structure at higher stages of the somatosensory processing pathway, in addition to primary sensory representation. PMID:19412538

  3. From Rational Numbers to Dirac's Bra and Ket: Symbolic Representation of Physical Laws

    NASA Astrophysics Data System (ADS)

    D'Agostino, Salvo

    2002-05-01

    Beginning at least in the nineteenth century, symbols used by physicists in their equations interacted with their physical concepts. In the 1850s, Wilhelm Eduard Weber introduced a more rational order into symbolization by adopting an absolute system of units, and thus expressing electrodynamic laws in the form of algebraic equations instead of proportionality relationships, the formerly accepted representation of physical laws. In the 1860s, James Clerk Maxwell made a further advance by using dimensional quantities, and more complex symbolic forms such as gradient, convergence, rotor, and the like, in his electromagnetic and kinetic theories. In the twentieth century, Werner Heisenberg, Max Born, Erwin Schrödinger, and others introduced new symbols for complex numbers, operators, and matrices, thus passing from the representation of metrical properties of physical systems to higher-level mathematical objects. This process was enhanced in modern theoretical physics through the introduction of matrices, creation and destruction operators, Paul A. M. Dirac's q and c numbers, and so on. In the 1930s, Dirac radicalized this transformation of symbols, being aware of the profound modification in the method and scope of the mathematical-physical relationship it entailed.

  4. Ontological Representation of Light Wave Camera Data to Support Vision-Based AmI

    PubMed Central

    Serrano, Miguel Ángel; Gómez-Romero, Juan; Patricio, Miguel Ángel; García, Jesús; Molina, José Manuel

    2012-01-01

    Recent advances in technologies for capturing video data have opened a vast amount of new application areas in visual sensor networks. Among them, the incorporation of light wave cameras on Ambient Intelligence (AmI) environments provides more accurate tracking capabilities for activity recognition. Although the performance of tracking algorithms has quickly improved, symbolic models used to represent the resulting knowledge have not yet been adapted to smart environments. This lack of representation does not allow to take advantage of the semantic quality of the information provided by new sensors. This paper advocates for the introduction of a part-based representational level in cognitive-based systems in order to accurately represent the novel sensors' knowledge. The paper also reviews the theoretical and practical issues in part-whole relationships proposing a specific taxonomy for computer vision approaches. General part-based patterns for human body and transitive part-based representation and inference are incorporated to an ontology-based previous framework to enhance scene interpretation in the area of video-based AmI. The advantages and new features of the model are demonstrated in a Social Signal Processing (SSP) application for the elaboration of live market researches.

  5. Using representations in geometry: a model of students' cognitive and affective performance

    NASA Astrophysics Data System (ADS)

    Panaoura, Areti

    2014-05-01

    Self-efficacy beliefs in mathematics, as a dimension of the affective domain, are related with students' performance on solving tasks and mainly on overcoming cognitive obstacles. The present study investigated the interrelations of cognitive performance on geometry and young students' self-efficacy beliefs about using representations for solving geometrical tasks. The emphasis was on confirming a theoretical model for the primary-school and secondary-school students and identifying the differences and similarities for the two ages. A quantitative study was developed and data were collected from 1086 students in Grades 5-8. Confirmatory factor analysis affirmed the existence of a coherent model of affective dimensions about the use of representations for understanding the geometrical concepts, which becomes more stable across the educational levels.

  6. Invariant recognition drives neural representations of action sequences

    PubMed Central

    Poggio, Tomaso

    2017-01-01

    Recognizing the actions of others from visual stimuli is a crucial aspect of human perception that allows individuals to respond to social cues. Humans are able to discriminate between similar actions despite transformations, like changes in viewpoint or actor, that substantially alter the visual appearance of a scene. This ability to generalize across complex transformations is a hallmark of human visual intelligence. Advances in understanding action recognition at the neural level have not always translated into precise accounts of the computational principles underlying what representations of action sequences are constructed by human visual cortex. Here we test the hypothesis that invariant action discrimination might fill this gap. Recently, the study of artificial systems for static object perception has produced models, Convolutional Neural Networks (CNNs), that achieve human level performance in complex discriminative tasks. Within this class, architectures that better support invariant object recognition also produce image representations that better match those implied by human and primate neural data. However, whether these models produce representations of action sequences that support recognition across complex transformations and closely follow neural representations of actions remains unknown. Here we show that spatiotemporal CNNs accurately categorize video stimuli into action classes, and that deliberate model modifications that improve performance on an invariant action recognition task lead to data representations that better match human neural recordings. Our results support our hypothesis that performance on invariant discrimination dictates the neural representations of actions computed in the brain. These results broaden the scope of the invariant recognition framework for understanding visual intelligence from perception of inanimate objects and faces in static images to the study of human perception of action sequences. PMID:29253864

  7. The Effects of Using Drawings in Developing Young Children's Mathematical Word Problem Solving: A Design Experiment with Third-Grade Hungarian Students

    ERIC Educational Resources Information Center

    Csikos, Csaba; Szitanyi, Judit; Kelemen, Rita

    2012-01-01

    The present study aims to investigate the effects of a design experiment developed for third-grade students in the field of mathematics word problems. The main focus of the program was developing students' knowledge about word problem solving strategies with an emphasis on the role of visual representations in mathematical modeling. The experiment…

  8. Examining the Impact of a Framework to Support Prospective Secondary Teachers' Transition from 'Doer' to 'Teacher' of Mathematics

    ERIC Educational Resources Information Center

    Lee, Hea-Jin; Özgün-Koca, S. Asli; Meagher, Michael; Edwards, Michael Todd

    2018-01-01

    A transition from "doer" to "teacher" for prospective teachers requires them to reorient from thinking about how they do mathematics to engaging with students and their work, understanding student representations, and planning instruction accordingly. To scaffold a transition, we developed a five-step mathematics as teacher…

  9. Random Versus Blocked Practice to Enhance Mental Representation in Golf Putting.

    PubMed

    Fazeli, Davoud; Taheri, HamidReza; Saberi Kakhki, Alireza

    2017-06-01

    The purpose of this study was to investigate changes in mental representation from either random or blocked practice when engaged in golf putting. Thirty participants were randomly assigned to random practice, blocked practice, and no-practice groups. First, we measured novice golfers' initial mental representation levels and required them to perform 18 putting trials as a pre-test. We then asked random and blocked groups to practice in accordance with their group assignment for six consecutive days (10 blocks each day, 18 trials each). A week after the last practice session, we re-measured all participants' final mental representation levels and required them to perform 18 putting trials to evaluate learning retention through practice. While those engaged in the random practice method putted more poorly during acquisition (i.e., practice) than those in blocked practice, the random practice group experienced more accurate retention during the final putting trials, and they showed a more structured mental representation than those in blocked practice, one that was more similar to that of skilled golfers. These results support the acquisition of a rich mental representation through random versus blocked practice.

  10. Content Representation in the Human Medial Temporal Lobe

    PubMed Central

    Liang, Jackson C.; Wagner, Anthony D.

    2013-01-01

    Current theories of medial temporal lobe (MTL) function focus on event content as an important organizational principle that differentiates MTL subregions. Perirhinal and parahippocampal cortices may play content-specific roles in memory, whereas hippocampal processing is alternately hypothesized to be content specific or content general. Despite anatomical evidence for content-specific MTL pathways, empirical data for content-based MTL subregional dissociations are mixed. Here, we combined functional magnetic resonance imaging with multiple statistical approaches to characterize MTL subregional responses to different classes of novel event content (faces, scenes, spoken words, sounds, visual words). Univariate analyses revealed that responses to novel faces and scenes were distributed across the anterior–posterior axis of MTL cortex, with face responses distributed more anteriorly than scene responses. Moreover, multivariate pattern analyses of perirhinal and parahippocampal data revealed spatially organized representational codes for multiple content classes, including nonpreferred visual and auditory stimuli. In contrast, anterior hippocampal responses were content general, with less accurate overall pattern classification relative to MTL cortex. Finally, posterior hippocampal activation patterns consistently discriminated scenes more accurately than other forms of content. Collectively, our findings indicate differential contributions of MTL subregions to event representation via a distributed code along the anterior–posterior axis of MTL that depends on the nature of event content. PMID:22275474

  11. Differentiable representations of finite dimensional Lie groups in rigged Hilbert spaces

    NASA Astrophysics Data System (ADS)

    Wickramasekara, Sujeewa

    The inceptive motivation for introducing rigged Hilbert spaces (RHS) in quantum physics in the mid 1960's was to provide the already well established Dirac formalism with a proper mathematical context. It has since become clear, however, that this mathematical framework is lissome enough to accommodate a class of solutions to the dynamical equations of quantum physics that includes some which are not possible in the normative Hilbert space theory. Among the additional solutions, in particular, are those which describe aspects of scattering and decay phenomena that have eluded the orthodox quantum physics. In this light, the RHS formulation seems to provide a mathematical rubric under which various phenomenological observations and calculational techniques, commonly known in the study of resonance scattering and decay as ``effective theories'' (e.g., the Wigner- Weisskopf method), receive a unified theoretical foundation. These observations lead to the inference that a theory founded upon the RHS mathematics may prove to be of better utility and value in understanding quantum physical phenomena. This dissertation primarily aims to contribute to the general formalism of the RHS theory of quantum mechanics by undertaking a study of differentiable representations of finite dimensional Lie groups. In particular, it is shown that a finite dimensional operator Lie algebra G in a rigged Hilbert space can be always integrated, provided one parameter integrability holds true for the elements of any basis for G . This result differs from and extends the well known integration theorem of E. Nelson and the subsequent works of others on unitary representations in that it does not require any assumptions on the existence of analytic vectors. Also presented here is a construction of a particular rigged Hilbert space of Hardy class functions that appears useful in formulating a relativistic version of the RHS theory of resonances and decay. As a contexture for the construction, a

  12. What Matters Most when Students and Teachers Use Interactive Whiteboards in Mathematics Classrooms?

    ERIC Educational Resources Information Center

    McQuillan, Kimberley; Northcote, Maria; Beamish, Peter

    2012-01-01

    Teachers are encouraged to immerse their students in rich and engaging learning environments (NSW Department of Education and Training, 2003). One teaching tool that can facilitate the creation of rich learning environments is the interactive whiteboard (IWB) (Baker, 2009). When teaching mathematics, the varied representational aspects of IWBs can…

  13. Numerical Magnitude Representation in Children with Mathematical Difficulties with or without Reading Difficulties

    ERIC Educational Resources Information Center

    Tobia, Valentina; Fasola, Anna; Lupieri, Alice; Marzocchi, Gian Marco

    2016-01-01

    This study aimed to explore the spatial numerical association of response codes (SNARC), the flanker, and the numerical distance effects in children with mathematical difficulties. From a sample of 720 third, fourth, and fifth graders, 60 children were selected and divided into the following three groups: typically developing children (TD; n =…

  14. Recent Advances in Registration, Integration and Fusion of Remotely Sensed Data: Redundant Representations and Frames

    NASA Technical Reports Server (NTRS)

    Czaja, Wojciech; Le Moigne-Stewart, Jacqueline

    2014-01-01

    In recent years, sophisticated mathematical techniques have been successfully applied to the field of remote sensing to produce significant advances in applications such as registration, integration and fusion of remotely sensed data. Registration, integration and fusion of multiple source imagery are the most important issues when dealing with Earth Science remote sensing data where information from multiple sensors, exhibiting various resolutions, must be integrated. Issues ranging from different sensor geometries, different spectral responses, differing illumination conditions, different seasons, and various amounts of noise need to be dealt with when designing an image registration, integration or fusion method. This tutorial will first define the problems and challenges associated with these applications and then will review some mathematical techniques that have been successfully utilized to solve them. In particular, we will cover topics on geometric multiscale representations, redundant representations and fusion frames, graph operators, diffusion wavelets, as well as spatial-spectral and operator-based data fusion. All the algorithms will be illustrated using remotely sensed data, with an emphasis on current and operational instruments.

  15. Spatiotemporal dynamics of similarity-based neural representations of facial identity.

    PubMed

    Vida, Mark D; Nestor, Adrian; Plaut, David C; Behrmann, Marlene

    2017-01-10

    Humans' remarkable ability to quickly and accurately discriminate among thousands of highly similar complex objects demands rapid and precise neural computations. To elucidate the process by which this is achieved, we used magnetoencephalography to measure spatiotemporal patterns of neural activity with high temporal resolution during visual discrimination among a large and carefully controlled set of faces. We also compared these neural data to lower level "image-based" and higher level "identity-based" model-based representations of our stimuli and to behavioral similarity judgments of our stimuli. Between ∼50 and 400 ms after stimulus onset, face-selective sources in right lateral occipital cortex and right fusiform gyrus and sources in a control region (left V1) yielded successful classification of facial identity. In all regions, early responses were more similar to the image-based representation than to the identity-based representation. In the face-selective regions only, responses were more similar to the identity-based representation at several time points after 200 ms. Behavioral responses were more similar to the identity-based representation than to the image-based representation, and their structure was predicted by responses in the face-selective regions. These results provide a temporally precise description of the transformation from low- to high-level representations of facial identity in human face-selective cortex and demonstrate that face-selective cortical regions represent multiple distinct types of information about face identity at different times over the first 500 ms after stimulus onset. These results have important implications for understanding the rapid emergence of fine-grained, high-level representations of object identity, a computation essential to human visual expertise.

  16. Spatiotemporal dynamics of similarity-based neural representations of facial identity

    PubMed Central

    Vida, Mark D.; Nestor, Adrian; Plaut, David C.; Behrmann, Marlene

    2017-01-01

    Humans’ remarkable ability to quickly and accurately discriminate among thousands of highly similar complex objects demands rapid and precise neural computations. To elucidate the process by which this is achieved, we used magnetoencephalography to measure spatiotemporal patterns of neural activity with high temporal resolution during visual discrimination among a large and carefully controlled set of faces. We also compared these neural data to lower level “image-based” and higher level “identity-based” model-based representations of our stimuli and to behavioral similarity judgments of our stimuli. Between ∼50 and 400 ms after stimulus onset, face-selective sources in right lateral occipital cortex and right fusiform gyrus and sources in a control region (left V1) yielded successful classification of facial identity. In all regions, early responses were more similar to the image-based representation than to the identity-based representation. In the face-selective regions only, responses were more similar to the identity-based representation at several time points after 200 ms. Behavioral responses were more similar to the identity-based representation than to the image-based representation, and their structure was predicted by responses in the face-selective regions. These results provide a temporally precise description of the transformation from low- to high-level representations of facial identity in human face-selective cortex and demonstrate that face-selective cortical regions represent multiple distinct types of information about face identity at different times over the first 500 ms after stimulus onset. These results have important implications for understanding the rapid emergence of fine-grained, high-level representations of object identity, a computation essential to human visual expertise. PMID:28028220

  17. Contacts de langues et representations (Language Contacts and Representations).

    ERIC Educational Resources Information Center

    Matthey, Marinette, Ed.

    1997-01-01

    Essays on language contact and the image of language, entirely in French, include: "Representations 'du' contexte et representations 'en' contexte? Eleves et enseignants face a l'apprentissage de la langue" ("Representations 'of' Context or Representations 'in' Context? Students and Teachers Facing Language Learning" (Laurent…

  18. INCREASING ACHIEVEMENT AND HIGHER-EDUCATION REPRESENTATION OF UNDER-REPRESENTED GROUPS IN SCIENCE, TECHNOLOGY, ENGINEERING, AND MATHEMATICS FIELDS: A REVIEW OF CURRENT K-12 INTERVENTION PROGRAMS

    PubMed Central

    Valla, Jeffrey M.; Williams, Wendy M.

    2012-01-01

    The under-representation of women and ethnic minorities in Science, Technology, Engineering, and Mathematics (STEM) education and professions has resulted in a loss of human capital for the US scientific workforce and spurred the development of myriad STEM educational intervention programs. Increased allocation of resources to such programs begs for a critical, prescriptive, evidence-based review that will enable researchers to develop optimal interventions and administrators to maximize investments. We begin by providing a theoretical backdrop for K-12 STEM programs by reviewing current data on under-representation and developmental research describing individual-level social factors undergirding these data. Next, we review prototypical designs of these programs, highlighting specific programs in the literature as examples of program structures and components currently in use. We then evaluate these interventions in terms of overall effectiveness, as a function of how well they address age-, ethnicity-, or gender-specific factors, suggesting improvements in program design based on these critiques. Finally, program evaluation methods are briefly reviewed and discussed in terms of how their empirical soundness can either enable or limit our ability to delineate effective program components. “Now more than ever, the nation’s changing demographics demand that we include all of our citizens in science and engineering education and careers. For the U.S. to benefit from the diverse talents of all its citizens, we must grow the pipeline of qualified, underrepresented minority engineers and scientists to fill positions in industry and academia.”—Irving P. McPhail.. PMID:22942637

  19. Representations and uses of light distribution functions

    NASA Astrophysics Data System (ADS)

    Lalonde, Paul Albert

    1998-11-01

    small error in the reconstructed signal. The representation can be used to evaluate efficiently some integrals that appear in shading computation which allows fast, accurate computation of local shading. The representation can be used to represent light fields and is used to reconstruct views of environments interactively from a precomputed set of views. The representation of the BRDF also allows the efficient generation of reflected directions for Monte Carlo array tracing applications. The method can be integrated into many different global illumination algorithms, including ray tracers and wavelet radiosity systems.

  20. [Mathematical anatomy: muscles according to Stensen].

    PubMed

    Andrault, Raphaële

    2010-01-01

    In his Elementorum Myologiae Specimen, Steno geometrizes "the new fabric of muscles" and their movement of contraction, so as to refute the main contemporary hypothesis about the functioning of the muscles. This physiological refutation relies on an abstract representation of the muscular fibre as a parallelepiped of flesh transversally linked to the tendons. Those two features have been comprehensively studied. But the method used by Steno, as well as the way he has chosen to present his physiological results, have so far been neglected. Yet, Steno's work follows a true synthetic order, which he conceives as a tool to separate uncertain anatomical "elements" from the certain ones. We will show that the true understanding of this "more geometrico" order is the only way to avoid frequent misconceptions of the scientific aim pursued by Steno, which is neither to give a mathematical explanation of the functioning of the muscles, nor to reduce the muscles to some mathematical shapes.

  1. Reliability in the Location of Hindlimb Motor Representations in Fischer-344 Rats

    PubMed Central

    Frost, Shawn B.; Iliakova, Maria; Dunham, Caleb; Barbay, Scott; Arnold, Paul; Nudo, Randolph J.

    2014-01-01

    Object The purpose of the present study was to determine the feasibility of using a common laboratory rat strain for locating cortical motor representations of the hindlimb reliably. Methods Intracortical Microstimulation (ICMS) techniques were used to derive detailed maps of the hindlimb motor representations in six adult Fischer-344 rats. Results The organization of the hindlimb movement representation, while variable across individuals in topographic detail, displayed several commonalities. The hindlimb representation was positioned posterior to the forelimb motor representation and postero-lateral to the motor trunk representation. The areal extent of the hindlimb representation across the cortical surface averaged 2.00 +/− 0.50 mm2. Superimposing individual maps revealed an overlapping area measuring 0.35 mm2, indicating that the location of the hindlimb representation can be predicted reliably based on stereotactic coordinates. Across the sample of rats, the hindlimb representation was found 1.25–3.75 mm posterior to Bregma, with an average center location ~ 2.6 mm posterior to Bregma. Likewise, the hindlimb representation was found 1–3.25 mm lateral to the midline, with an average center location ~ 2 mm lateral to midline. Conclusions The location of the cortical hindlimb motor representation in Fischer-344 rats can be reliably located based on its stereotactic position posterior to Bregma and lateral to the longitudinal skull suture at midline. The ability to accurately predict the cortical localization of functional hindlimb territories in a rodent model is important, as such animal models are being used increasingly in the development of brain-computer interfaces for restoration of function after spinal cord injury. PMID:23725395

  2. Mental models accurately predict emotion transitions.

    PubMed

    Thornton, Mark A; Tamir, Diana I

    2017-06-06

    Successful social interactions depend on people's ability to predict others' future actions and emotions. People possess many mechanisms for perceiving others' current emotional states, but how might they use this information to predict others' future states? We hypothesized that people might capitalize on an overlooked aspect of affective experience: current emotions predict future emotions. By attending to regularities in emotion transitions, perceivers might develop accurate mental models of others' emotional dynamics. People could then use these mental models of emotion transitions to predict others' future emotions from currently observable emotions. To test this hypothesis, studies 1-3 used data from three extant experience-sampling datasets to establish the actual rates of emotional transitions. We then collected three parallel datasets in which participants rated the transition likelihoods between the same set of emotions. Participants' ratings of emotion transitions predicted others' experienced transitional likelihoods with high accuracy. Study 4 demonstrated that four conceptual dimensions of mental state representation-valence, social impact, rationality, and human mind-inform participants' mental models. Study 5 used 2 million emotion reports on the Experience Project to replicate both of these findings: again people reported accurate models of emotion transitions, and these models were informed by the same four conceptual dimensions. Importantly, neither these conceptual dimensions nor holistic similarity could fully explain participants' accuracy, suggesting that their mental models contain accurate information about emotion dynamics above and beyond what might be predicted by static emotion knowledge alone.

  3. Representation of analysis results involving aleatory and epistemic uncertainty.

    DOE Office of Scientific and Technical Information (OSTI.GOV)

    Johnson, Jay Dean; Helton, Jon Craig; Oberkampf, William Louis

    2008-08-01

    Procedures are described for the representation of results in analyses that involve both aleatory uncertainty and epistemic uncertainty, with aleatory uncertainty deriving from an inherent randomness in the behavior of the system under study and epistemic uncertainty deriving from a lack of knowledge about the appropriate values to use for quantities that are assumed to have fixed but poorly known values in the context of a specific study. Aleatory uncertainty is usually represented with probability and leads to cumulative distribution functions (CDFs) or complementary cumulative distribution functions (CCDFs) for analysis results of interest. Several mathematical structures are available for themore » representation of epistemic uncertainty, including interval analysis, possibility theory, evidence theory and probability theory. In the presence of epistemic uncertainty, there is not a single CDF or CCDF for a given analysis result. Rather, there is a family of CDFs and a corresponding family of CCDFs that derive from epistemic uncertainty and have an uncertainty structure that derives from the particular uncertainty structure (i.e., interval analysis, possibility theory, evidence theory, probability theory) used to represent epistemic uncertainty. Graphical formats for the representation of epistemic uncertainty in families of CDFs and CCDFs are investigated and presented for the indicated characterizations of epistemic uncertainty.« less

  4. Representational Flexibility and Problem-Solving Ability in Fraction and Decimal Number Addition: A Structural Model

    ERIC Educational Resources Information Center

    Deliyianni, Eleni; Gagatsis, Athanasios; Elia, Iliada; Panaoura, Areti

    2016-01-01

    The aim of this study was to propose and validate a structural model in fraction and decimal number addition, which is founded primarily on a synthesis of major theoretical approaches in the field of representations in Mathematics and also on previous research on the learning of fractions and decimals. The study was conducted among 1,701 primary…

  5. A network of spiking neurons for computing sparse representations in an energy-efficient way.

    PubMed

    Hu, Tao; Genkin, Alexander; Chklovskii, Dmitri B

    2012-11-01

    Computing sparse redundant representations is an important problem in both applied mathematics and neuroscience. In many applications, this problem must be solved in an energy-efficient way. Here, we propose a hybrid distributed algorithm (HDA), which solves this problem on a network of simple nodes communicating by low-bandwidth channels. HDA nodes perform both gradient-descent-like steps on analog internal variables and coordinate-descent-like steps via quantized external variables communicated to each other. Interestingly, the operation is equivalent to a network of integrate-and-fire neurons, suggesting that HDA may serve as a model of neural computation. We show that the numerical performance of HDA is on par with existing algorithms. In the asymptotic regime, the representation error of HDA decays with time, t, as 1/t. HDA is stable against time-varying noise; specifically, the representation error decays as 1/√t for gaussian white noise.

  6. Preschoolers' precision of the approximate number system predicts later school mathematics performance.

    PubMed

    Mazzocco, Michèle M M; Feigenson, Lisa; Halberda, Justin

    2011-01-01

    The Approximate Number System (ANS) is a primitive mental system of nonverbal representations that supports an intuitive sense of number in human adults, children, infants, and other animal species. The numerical approximations produced by the ANS are characteristically imprecise and, in humans, this precision gradually improves from infancy to adulthood. Throughout development, wide ranging individual differences in ANS precision are evident within age groups. These individual differences have been linked to formal mathematics outcomes, based on concurrent, retrospective, or short-term longitudinal correlations observed during the school age years. However, it remains unknown whether this approximate number sense actually serves as a foundation for these school mathematics abilities. Here we show that ANS precision measured at preschool, prior to formal instruction in mathematics, selectively predicts performance on school mathematics at 6 years of age. In contrast, ANS precision does not predict non-numerical cognitive abilities. To our knowledge, these results provide the first evidence for early ANS precision, measured before the onset of formal education, predicting later mathematical abilities.

  7. Preschoolers' Precision of the Approximate Number System Predicts Later School Mathematics Performance

    PubMed Central

    Mazzocco, Michèle M. M.; Feigenson, Lisa; Halberda, Justin

    2011-01-01

    The Approximate Number System (ANS) is a primitive mental system of nonverbal representations that supports an intuitive sense of number in human adults, children, infants, and other animal species. The numerical approximations produced by the ANS are characteristically imprecise and, in humans, this precision gradually improves from infancy to adulthood. Throughout development, wide ranging individual differences in ANS precision are evident within age groups. These individual differences have been linked to formal mathematics outcomes, based on concurrent, retrospective, or short-term longitudinal correlations observed during the school age years. However, it remains unknown whether this approximate number sense actually serves as a foundation for these school mathematics abilities. Here we show that ANS precision measured at preschool, prior to formal instruction in mathematics, selectively predicts performance on school mathematics at 6 years of age. In contrast, ANS precision does not predict non-numerical cognitive abilities. To our knowledge, these results provide the first evidence for early ANS precision, measured before the onset of formal education, predicting later mathematical abilities. PMID:21935362

  8. Supporting Students in Learning with Multiple Representation to Improve Student Mental Models on Atomic Structure Concepts

    ERIC Educational Resources Information Center

    Sunyono; Yuanita, L.; Ibrahim, M.

    2015-01-01

    The aim of this research is identify the effectiveness of a multiple representation-based learning model, which builds a mental model within the concept of atomic structure. The research sample of 108 students in 3 classes is obtained randomly from among students of Mathematics and Science Education Studies using a stratified random sampling…

  9. Data handling and representation of freeform surfaces

    NASA Astrophysics Data System (ADS)

    Steinkopf, Ralf; Dick, Lars; Kopf, Tino; Gebhardt, Andreas; Risse, Stefan; Eberhardt, Ramona

    2011-10-01

    Freeform surfaces enable innovative optics. They are not limited by axis symmetry and hence they are almost free in design. They are used to reduce the installation space and enhance the performance of optical elements. State of the art optical design tools are computing with powerful algorithms to simulate freeform surfaces. Even new mathematical approaches are under development /1/. In consequence, new optical designs /2/ are pushing the development of manufacturing processes consequently and novel types of datasets have to proceed through the process chain /3/. The complexity of these data is the huge challenge for the data handling. Because of the asymmetrical and 3-dimensional surfaces of freeforms, large data volumes have to be created, trimmed, extended and fitted. All these processes must be performed without losing the accuracy of the original design data. Additionally, manifold types of geometries results in different kinds of mathematical representations of freeform surfaces and furthermore the used CAD/CAM tools are dealing with a set of spatial transport formats. These are all reasons why manufacture-oriented approaches for the freeform data handling are not yet sufficiently developed. This paper suggests a classification of freeform surfaces based on the manufacturing methods which are offered by diamond machining. The different manufacturing technologies, ranging from servo-turning to shaping, require a differentiated approach for the data handling process. The usage of analytical descriptions in form of splines and polynomials as well as the application of discrete descriptions like point clouds is shown in relation to the previously made classification. Advantages and disadvantages of freeform representations are discussed. Aspects of the data handling in between different process steps are pointed out and suitable exchange formats for freeform data are proposed. The described approach offers the possibility for efficient data handling from optical

  10. UH-60A Black Hawk engineering simulation program. Volume 1: Mathematical model

    NASA Technical Reports Server (NTRS)

    Howlett, J. J.

    1981-01-01

    A nonlinear mathematical model of the UR-60A Black Hawk helicopter was developed. This mathematical model, which was based on the Sikorsky General Helicopter (Gen Hel) Flight Dynamics Simulation, provides NASA with an engineering simulation for performance and handling qualities evaluations. This mathematical model is total systems definition of the Black Hawk helicopter represented at a uniform level of sophistication considered necessary for handling qualities evaluations. The model is a total force, large angle representation in six rigid body degrees of freedom. Rotor blade flapping, lagging, and hub rotational degrees of freedom are also represented. In addition to the basic helicopter modules, supportive modules were defined for the landing interface, power unit, ground effects, and gust penetration. Information defining the cockpit environment relevant to pilot in the loop simulation is presented.

  11. Shaping LGBTQ Identities: Western Media Representations and LGBTQ People's Perceptions in Rural Spain.

    PubMed

    Sanz López, Josep Maria

    2017-10-09

    A growing academic discussion has focused on how, in a globalized world, LGBTQ identities are shaped and influenced by different and international actors, such as the media. This article analyzes how LGBTQ people from a rural region of a Western country-Spain-feel toward their representations on TV series from English-speaking countries. Employing a qualitative approach, this research aims to depict whether the academic conceptualizations to analyze these identity conformation processes are accurate. In addition, it explores how dominating media representations are being adapted in a region that, although within the West, can serve a context of a very different nature. The results found that a major rejection of the TV series representations among participants can suggest both an inaccuracy of the conceptualizations used by some scholars to understand LGBTQ flows and a problematic LGBTQ representation in media products that goes beyond regions and spaces.

  12. RICO: A NEW APPROACH FOR FAST AND ACCURATE REPRESENTATION OF THE COSMOLOGICAL RECOMBINATION HISTORY

    DOE Office of Scientific and Technical Information (OSTI.GOV)

    Fendt, W. A.; Wandelt, B. D.; Chluba, J.

    2009-04-15

    We present RICO, a code designed to compute the ionization fraction of the universe during the epoch of hydrogen and helium recombination with an unprecedented combination of speed and accuracy. This is accomplished by training the machine learning code PICO on the calculations of a multilevel cosmological recombination code which self-consistently includes several physical processes that were neglected previously. After training, RICO is used to fit the free electron fraction as a function of the cosmological parameters. While, for example, at low redshifts (z {approx}< 900), much of the net change in the ionization fraction can be captured by loweringmore » the hydrogen fudge factor in RECFAST by about 3%, RICO provides a means of effectively using the accurate ionization history of the full recombination code in the standard cosmological parameter estimation framework without the need to add new or refined fudge factors or functions to a simple recombination model. Within the new approach presented here, it is easy to update RICO whenever a more accurate full recombination code becomes available. Once trained, RICO computes the cosmological ionization history with negligible fitting error in {approx}10 ms, a speedup of at least 10{sup 6} over the full recombination code that was used here. Also RICO is able to reproduce the ionization history of the full code to a level well below 0.1%, thereby ensuring that the theoretical power spectra of cosmic microwave background (CMB) fluctuations can be computed to sufficient accuracy and speed for analysis from upcoming CMB experiments like Planck. Furthermore, it will enable cross-checking different recombination codes across cosmological parameter space, a comparison that will be very important in order to assure the accurate interpretation of future CMB data.« less

  13. Neuronal foundations of human numerical representations.

    PubMed

    Eger, E

    2016-01-01

    The human species has developed complex mathematical skills which likely emerge from a combination of multiple foundational abilities. One of them seems to be a preverbal capacity to extract and manipulate the numerosity of sets of objects which is shared with other species and in humans is thought to be integrated with symbolic knowledge to result in a more abstract representation of numerical concepts. For what concerns the functional neuroanatomy of this capacity, neuropsychology and functional imaging have localized key substrates of numerical processing in parietal and frontal cortex. However, traditional fMRI mapping relying on a simple subtraction approach to compare numerical and nonnumerical conditions is limited to tackle with sufficient precision and detail the issue of the underlying code for number, a question which more easily lends itself to investigation by methods with higher spatial resolution, such as neurophysiology. In recent years, progress has been made through the introduction of approaches sensitive to within-category discrimination in combination with fMRI (adaptation and multivariate pattern recognition), and the present review summarizes what these have revealed so far about the neural coding of individual numbers in the human brain, the format of these representations and parallels between human and monkey neurophysiology findings. © 2016 Elsevier B.V. All rights reserved.

  14. Prospective Mathematics Teachers' Sense Making of Polynomial Multiplication and Factorization Modeled with Algebra Tiles

    ERIC Educational Resources Information Center

    Caglayan, Günhan

    2013-01-01

    This study is about prospective secondary mathematics teachers' understanding and sense making of representational quantities generated by algebra tiles, the quantitative units (linear vs. areal) inherent in the nature of these quantities, and the quantitative addition and multiplication operations--referent preserving versus referent…

  15. Methods to achieve accurate projection of regional and global raster databases

    USGS Publications Warehouse

    Usery, E. Lynn; Seong, Jeong Chang; Steinwand, Dan

    2002-01-01

    Modeling regional and global activities of climatic and human-induced change requires accurate geographic data from which we can develop mathematical and statistical tabulations of attributes and properties of the environment. Many of these models depend on data formatted as raster cells or matrices of pixel values. Recently, it has been demonstrated that regional and global raster datasets are subject to significant error from mathematical projection and that these errors are of such magnitude that model results may be jeopardized (Steinwand, et al., 1995; Yang, et al., 1996; Usery and Seong, 2001; Seong and Usery, 2001). There is a need to develop methods of projection that maintain the accuracy of these datasets to support regional and global analyses and modeling

  16. Mathematics Intervention for Prevention of Neurocognitive Deficits in Childhood Leukemia

    PubMed Central

    Moore (Ki), Ida M.; Hockenberry, Marilyn J.; Anhalt, Cynthia; McCarthy, Kathy; Krull, Kevin R.

    2011-01-01

    Background Despite evidence that CNS treatment is associated with cognitive and academic impairment, interventions to prevent or mitigate these problems are limited. The purpose was to determine if early intervention can prevent declines in mathematics abilities. Procedures Fifty-seven children with ALL were enrolled and randomized to a Mathematics Intervention or Standard Care. Subjects completed neurocognitive assessments prior to the intervention, post intervention, and one year later. Parents received written results and recommendations for use with their school. The Mathematics Intervention was based on Multiple Representation Theory and delivered individually over one year. Results Thirty-two of 57 subjects completed the study and were included in data analyses. These 32 subjects completed all neurocognitive assessments and, for those in the intervention group, 40–50 hours of the mathematics intervention. There were no group differences on relevant demographic variables; risk stratification; number of intrathecal methotrexate injections or high dose systemic methotrexate. Significant improvements in calculation and applied mathematics from baseline to post-intervention (p = 0.003 and 0.002, respectively) and in visual working memory from baseline to one year follow-up (p = 0.02) were observed in the Intervention but not the Standard Care group. Results from repeated measures ANOVA demonstrated significant between group differences for applied mathematics (F[2, 29] 12.47, p<0.001) and visual working memory (F[2 29]= 5.53, p=0.009). Conclusions The Mathematics Intervention improved mathematics abilities and visual working memory compared to standard care. Future studies are needed to translate the Mathematics Intervention into a “virtual” delivery method more readily available to parents and children. PMID:21938763

  17. Mathematics intervention for prevention of neurocognitive deficits in childhood leukemia.

    PubMed

    Moore, Ida M; Hockenberry, Marilyn J; Anhalt, Cynthia; McCarthy, Kathy; Krull, Kevin R

    2012-08-01

    Despite evidence that CNS treatment is associated with cognitive and academic impairment, interventions to prevent or mitigate these problems are limited. The purpose was to determine if early intervention can prevent declines in mathematics abilities. Fifty-seven children with ALL were enrolled and randomized to a Mathematics Intervention or Standard Care. Subjects completed neurocognitive assessments prior to the intervention, post-intervention, and 1 year later. Parents received written results and recommendations for use with their school. The Mathematics Intervention was based on Multiple Representation Theory and delivered individually over 1 year. Thirty-two of 57 subjects completed the study and were included in data analyses. These 32 subjects completed all neurocognitive assessments and, for those in the Intervention Group, 40-50 hours of the Mathematics Intervention. There were no group differences on relevant demographic variables; risk stratification; number of intrathecal methotrexate injections; or high dose systemic methotrexate. Significant improvements in calculation and applied mathematics from Baseline to Post-Intervention (P = 0.003 and 0.002, respectively) and in visual working memory from Baseline to 1 year Follow-up (P = 0.02) were observed in the Intervention but not the Standard Care Group. Results from repeated measures ANOVA demonstrated significant between group differences for applied mathematics [F(2,29) = 12.47, P < 0.001] and visual working memory [F(2,29) = 5.53, P = 0.009]. The Mathematics Intervention improved mathematics abilities and visual working memory compared to standard care. Future studies are needed to translate the Mathematics Intervention into a "virtual" delivery method more readily available to parents and children. Copyright © 2011 Wiley Periodicals, Inc.

  18. On canonical cylinder sections for accurate determination of contact angle in microgravity

    NASA Technical Reports Server (NTRS)

    Concus, Paul; Finn, Robert; Zabihi, Farhad

    1992-01-01

    Large shifts of liquid arising from small changes in certain container shapes in zero gravity can be used as a basis for accurately determining contact angle. Canonical geometries for this purpose, recently developed mathematically, are investigated here computationally. It is found that the desired nearly-discontinuous behavior can be obtained and that the shifts of liquid have sufficient volume to be readily observed.

  19. A network of spiking neurons for computing sparse representations in an energy efficient way

    PubMed Central

    Hu, Tao; Genkin, Alexander; Chklovskii, Dmitri B.

    2013-01-01

    Computing sparse redundant representations is an important problem both in applied mathematics and neuroscience. In many applications, this problem must be solved in an energy efficient way. Here, we propose a hybrid distributed algorithm (HDA), which solves this problem on a network of simple nodes communicating via low-bandwidth channels. HDA nodes perform both gradient-descent-like steps on analog internal variables and coordinate-descent-like steps via quantized external variables communicated to each other. Interestingly, such operation is equivalent to a network of integrate-and-fire neurons, suggesting that HDA may serve as a model of neural computation. We compare the numerical performance of HDA with existing algorithms and show that in the asymptotic regime the representation error of HDA decays with time, t, as 1/t. We show that HDA is stable against time-varying noise, specifically, the representation error decays as 1/t for Gaussian white noise. PMID:22920853

  20. Tracing the Building of Robert's Connections in Mathematical Problem Solving: A Sixteen-Year Study

    ERIC Educational Resources Information Center

    Ahluwalia, Anoop

    2011-01-01

    This research analyzes how external representations created by a student, Robert, helped him in building mathematical understanding over a sixteen-year period. Robert (also known as Bobby), was an original participant of the Rutgers longitudinal study where students were encouraged to work on problem-solving tasks with minimum intervention (Maher,…

  1. An ellipsoidal representation of human hand anthropometry

    NASA Technical Reports Server (NTRS)

    Buchholz, Bryan; Armstrong, Thomas J.

    1991-01-01

    Anthropometric data concerning the heometry of the hand's surface are presently modeled as a function of gross external hand measurements; an effort is made to evaluate the accuracy with which ellipsoids describe the geometry of the hand segments. Graphical comparisons indicate that differences between the ellipsoidal approximations and the breadth and depth measurements were greatest near the joints. On the bases of the present data, a set of overlapping ellipsoids could furnish a more accurate representation of hand geometry for adaptation to ellipsoid segment-geometry employing biomechanical models.

  2. Cortical circuits for mathematical knowledge: evidence for a major subdivision within the brain's semantic networks.

    PubMed

    Amalric, Marie; Dehaene, Stanislas

    2017-02-19

    Is mathematical language similar to natural language? Are language areas used by mathematicians when they do mathematics? And does the brain comprise a generic semantic system that stores mathematical knowledge alongside knowledge of history, geography or famous people? Here, we refute those views by reviewing three functional MRI studies of the representation and manipulation of high-level mathematical knowledge in professional mathematicians. The results reveal that brain activity during professional mathematical reflection spares perisylvian language-related brain regions as well as temporal lobe areas classically involved in general semantic knowledge. Instead, mathematical reflection recycles bilateral intraparietal and ventral temporal regions involved in elementary number sense. Even simple fact retrieval, such as remembering that 'the sine function is periodical' or that 'London buses are red', activates dissociated areas for math versus non-math knowledge. Together with other fMRI and recent intracranial studies, our results indicated a major separation between two brain networks for mathematical and non-mathematical semantics, which goes a long way to explain a variety of facts in neuroimaging, neuropsychology and developmental disorders.This article is part of a discussion meeting issue 'The origins of numerical abilities'. © 2017 The Author(s).

  3. Using Mathematics, Mathematical Applications, Mathematical Modelling, and Mathematical Literacy: A Theoretical Study

    ERIC Educational Resources Information Center

    Mumcu, Hayal Yavuz

    2016-01-01

    The purpose of this theoretical study is to explore the relationships between the concepts of using mathematics in the daily life, mathematical applications, mathematical modelling, and mathematical literacy. As these concepts are generally taken as independent concepts in the related literature, they are confused with each other and it becomes…

  4. RF tumour ablation: computer simulation and mathematical modelling of the effects of electrical and thermal conductivity.

    PubMed

    Lobo, S M; Liu, Z-J; Yu, N C; Humphries, S; Ahmed, M; Cosman, E R; Lenkinski, R E; Goldberg, W; Goldberg, S N

    2005-05-01

    This study determined the effects of thermal conductivity on RF ablation tissue heating using mathematical modelling and computer simulations of RF heating coupled to thermal transport. Computer simulation of the Bio-Heat equation coupled with temperature-dependent solutions for RF electric fields (ETherm) was used to generate temperature profiles 2 cm away from a 3 cm internally-cooled electrode. Multiple conditions of clinically relevant electrical conductivities (0.07-12 S m-1) and 'tumour' radius (5-30 mm) at a given background electrical conductivity (0.12 S m-1) were studied. Temperature response surfaces were plotted for six thermal conductivities, ranging from 0.3-2 W m-1 degrees C (the range of anticipated clinical and experimental systems). A temperature response surface was obtained for each thermal conductivity at 25 electrical conductivities and 17 radii (n=425 temperature data points). The simulated temperature response was fit to a mathematical model derived from prior phantom data. This mathematical model is of the form (T=a+bRc exp(dR) s(f) exp(g)(s)) for RF generator-energy dependent situations and (T=h+k exp(mR)+n?exp(p)(s)) for RF generator-current limited situations, where T is the temperature (degrees C) 2 cm from the electrode and a, b, c, d, f, g, h, k, m, n and p are fitting parameters. For each of the thermal conductivity temperature profiles generated, the mathematical model fit the response surface to an r2 of 0.97-0.99. Parameters a, b, c, d, f, k and m were highly correlated to thermal conductivity (r2=0.96-0.99). The monotonic progression of fitting parameters permitted their mathematical expression using simple functions. Additionally, the effect of thermal conductivity simplified the above equation to the extent that g, h, n and p were found to be invariant. Thus, representation of the temperature response surface could be accurately expressed as a function of electrical conductivity, radius and thermal conductivity. As a result

  5. Novice Interpretations of Visual Representations of Geosciences Data

    NASA Astrophysics Data System (ADS)

    Burkemper, L. K.; Arthurs, L.

    2013-12-01

    Past cognition research of individual's perception and comprehension of bar and line graphs are substantive enough that they have resulted in the generation of graph design principles and graph comprehension theories; however, gaps remain in our understanding of how people process visual representations of data, especially of geologic and atmospheric data. This pilot project serves to build on others' prior research and begin filling the existing gaps. The primary objectives of this pilot project include: (i) design a novel data collection protocol based on a combination of paper-based surveys, think-aloud interviews, and eye-tracking tasks to investigate student data handling skills of simple to complex visual representations of geologic and atmospheric data, (ii) demonstrate that the protocol yields results that shed light on student data handling skills, and (iii) generate preliminary findings upon which tentative but perhaps helpful recommendations on how to more effectively present these data to the non-scientist community and teach essential data handling skills. An effective protocol for the combined use of paper-based surveys, think-aloud interviews, and computer-based eye-tracking tasks for investigating cognitive processes involved in perceiving, comprehending, and interpreting visual representations of geologic and atmospheric data is instrumental to future research in this area. The outcomes of this pilot study provide the foundation upon which future more in depth and scaled up investigations can build. Furthermore, findings of this pilot project are sufficient for making, at least, tentative recommendations that can help inform (i) the design of physical attributes of visual representations of data, especially more complex representations, that may aid in improving students' data handling skills and (ii) instructional approaches that have the potential to aid students in more effectively handling visual representations of geologic and atmospheric data

  6. A new mathematical formulation of the line-by-line method in case of weak line overlapping

    NASA Technical Reports Server (NTRS)

    Ishov, Alexander G.; Krymova, Natalie V.

    1994-01-01

    A rigorous mathematical proof is presented for multiline representation on the equivalent width of a molecular band which consists in the general case of n overlapping spectral lines. The multiline representation includes a principal term and terms of minor significance. The principal term is the equivalent width of the molecular band consisting of the same n nonoverlapping spectral lines. The terms of minor significance take into consideration the overlapping of two, three and more spectral lines. They are small in case of the weak overlapping of spectral lines in the molecular band. The multiline representation can be easily generalized for optically inhomogeneous gas media and holds true for combinations of molecular bands. If the band lines overlap weakly the standard formulation of line-by-line method becomes too labor-consuming. In this case the multiline representation permits line-by-line calculations to be performed more effectively. Other useful properties of the multiline representation are pointed out.

  7. Multiple Representation Instruction First versus Traditional Algorithmic Instruction First: Impact in Middle School Mathematics Classrooms

    ERIC Educational Resources Information Center

    Flores, Raymond; Koontz, Esther; Inan, Fethi A.; Alagic, Mara

    2015-01-01

    This study examined the impact of the order of two teaching approaches on students' abilities and on-task behaviors while learning how to solve percentage problems. Two treatment groups were compared. MR first received multiple representation instruction followed by traditional algorithmic instruction and TA first received these teaching…

  8. Reliability in the location of hindlimb motor representations in Fischer-344 rats: laboratory investigation.

    PubMed

    Frost, Shawn B; Iliakova, Maria; Dunham, Caleb; Barbay, Scott; Arnold, Paul; Nudo, Randolph J

    2013-08-01

    The purpose of the present study was to determine the feasibility of using a common laboratory rat strain for reliably locating cortical motor representations of the hindlimb. Intracortical microstimulation techniques were used to derive detailed maps of the hindlimb motor representations in 6 adult Fischer-344 rats. The organization of the hindlimb movement representation, while variable across individual rats in topographic detail, displayed several commonalities. The hindlimb representation was positioned posterior to the forelimb motor representation and posterolateral to the motor trunk representation. The areal extent of the hindlimb representation across the cortical surface averaged 2.00 ± 0.50 mm(2). Superimposing individual maps revealed an overlapping area measuring 0.35 mm(2), indicating that the location of the hindlimb representation can be predicted reliably based on stereotactic coordinates. Across the sample of rats, the hindlimb representation was found 1.25-3.75 mm posterior to the bregma, with an average center location approximately 2.6 mm posterior to the bregma. Likewise, the hindlimb representation was found 1-3.25 mm lateral to the midline, with an average center location approximately 2 mm lateral to the midline. The location of the cortical hindlimb motor representation in Fischer-344 rats can be reliably located based on its stereotactic position posterior to the bregma and lateral to the longitudinal skull suture at midline. The ability to accurately predict the cortical localization of functional hindlimb territories in a rodent model is important, as such animal models are being increasingly used in the development of brain-computer interfaces for restoration of function after spinal cord injury.

  9. Relations of Instructional Tasks to Teacher-Student Discourse in Mathematics Classrooms of Chinese Primary Schools

    ERIC Educational Resources Information Center

    Ni, Yujing; Zhou, Dehui; Li, Xiaoqing; Li, Qiong

    2014-01-01

    This study, based on observation of 90 fifth-grade mathematics classes in Chinese elementary schools, examined how the task features, high cognitive demand, multiple representations, and multiple solution methods may relate to classroom discourse. Results indicate that high cognitive demand tasks were associated with teachers' use higher order…

  10. A S[t]imulating Study of Map Projections: An Exploration Integrating Mathematics and Social Studies.

    ERIC Educational Resources Information Center

    Wilkins, Jesse L. M.; Hicks, David

    2001-01-01

    Presents a map-projection activity that combines mathematics and geography through investigating the proportion of land and water that covers the earth. Focuses on helping students become familiar with characteristics of different projections or representations of the world while estimating and graphing and encouraging them to investigate the…

  11. Students' Discourse When Working in Pairs with Etoys in an Eighth-Grade Mathematics Class

    ERIC Educational Resources Information Center

    DeJarnette, Anna F.

    2016-01-01

    I examined students' discourse while working in pairs at the computer in an eighth-grade mathematics class to understand how students kept track of the people and things they discussed. I found that students most often referenced themselves and objects within the environment, through references to shared knowledge and the representations on the…

  12. Progress in fast, accurate multi-scale climate simulations

    DOE PAGES

    Collins, W. D.; Johansen, H.; Evans, K. J.; ...

    2015-06-01

    We present a survey of physical and computational techniques that have the potential to contribute to the next generation of high-fidelity, multi-scale climate simulations. Examples of the climate science problems that can be investigated with more depth with these computational improvements include the capture of remote forcings of localized hydrological extreme events, an accurate representation of cloud features over a range of spatial and temporal scales, and parallel, large ensembles of simulations to more effectively explore model sensitivities and uncertainties. Numerical techniques, such as adaptive mesh refinement, implicit time integration, and separate treatment of fast physical time scales are enablingmore » improved accuracy and fidelity in simulation of dynamics and allowing more complete representations of climate features at the global scale. At the same time, partnerships with computer science teams have focused on taking advantage of evolving computer architectures such as many-core processors and GPUs. As a result, approaches which were previously considered prohibitively costly have become both more efficient and scalable. In combination, progress in these three critical areas is poised to transform climate modeling in the coming decades.« less

  13. Challenges in Designing Appropriate Scaffolding to Improve Students' Representational Consistency: The Case of a Gauss's Law Problem

    ERIC Educational Resources Information Center

    Maries, Alexandru; Lin, Shih-Yin; Singh, Chandralekha

    2017-01-01

    Prior research suggests that introductory physics students have difficulty with graphing and interpreting graphs. Here, we discuss an investigation of student difficulties in translating between mathematical and graphical representations for a problem in electrostatics and the effect of increasing levels of scaffolding on students'…

  14. Towards an accurate representation of electrostatics in classical force fields: Efficient implementation of multipolar interactions in biomolecular simulations

    NASA Astrophysics Data System (ADS)

    Sagui, Celeste; Pedersen, Lee G.; Darden, Thomas A.

    2004-01-01

    The accurate simulation of biologically active macromolecules faces serious limitations that originate in the treatment of electrostatics in the empirical force fields. The current use of "partial charges" is a significant source of errors, since these vary widely with different conformations. By contrast, the molecular electrostatic potential (MEP) obtained through the use of a distributed multipole moment description, has been shown to converge to the quantum MEP outside the van der Waals surface, when higher order multipoles are used. However, in spite of the considerable improvement to the representation of the electronic cloud, higher order multipoles are not part of current classical biomolecular force fields due to the excessive computational cost. In this paper we present an efficient formalism for the treatment of higher order multipoles in Cartesian tensor formalism. The Ewald "direct sum" is evaluated through a McMurchie-Davidson formalism [L. McMurchie and E. Davidson, J. Comput. Phys. 26, 218 (1978)]. The "reciprocal sum" has been implemented in three different ways: using an Ewald scheme, a particle mesh Ewald (PME) method, and a multigrid-based approach. We find that even though the use of the McMurchie-Davidson formalism considerably reduces the cost of the calculation with respect to the standard matrix implementation of multipole interactions, the calculation in direct space remains expensive. When most of the calculation is moved to reciprocal space via the PME method, the cost of a calculation where all multipolar interactions (up to hexadecapole-hexadecapole) are included is only about 8.5 times more expensive than a regular AMBER 7 [D. A. Pearlman et al., Comput. Phys. Commun. 91, 1 (1995)] implementation with only charge-charge interactions. The multigrid implementation is slower but shows very promising results for parallelization. It provides a natural way to interface with continuous, Gaussian-based electrostatics in the future. It is

  15. A Tale of Two Representations: The Misinformation Effect and Children's Developing Theory of Mind.

    ERIC Educational Resources Information Center

    Templeton, Leslie M.; Wilcox, Sharon A.

    2000-01-01

    Investigated children's representational ability as a cognitive factor underlying the suggestibility of their eyewitness memory. Found that the eyewitness memory of children lacking multirepresentational abilities or sufficient general memory abilities (most 3- and 4-year-olds) was less accurate than eyewitness memory of those with…

  16. Angular-spectrum representation of nondiffracting X waves

    NASA Astrophysics Data System (ADS)

    Fagerholm, Juha; Friberg, Ari T.; Huttunen, Juhani; Morgan, David P.; Salomaa, Martti M.

    1996-10-01

    We derive the nondiffracting X waves, first discussed within acoustics by Lu and Greenleaf [IEEE Trans. Ultrason. Ferroelec. Freq. Contr. 39, 19 (1992)], using the general mathematical formalism based on an angular spectrum of plane waves. This serves to provide a unified treatment of not only the fundamental zeroth-order X waves of Lu and Greenleaf, but also of the lesser-known higher-order derivative X waves, first discussed here in terms of a single, universal, angular spectrum. The characteristic crossed (letter-X-like) shape and the special properties of the X waves, as well as of their angular-spectrum representation, are discussed and illustrated in detail. Asymptotically, for increasing order, the appearance of the X waves is found to transform into a triangular wedgelike waveform.

  17. A group matrix representation relevant to scales of measurement of clinical disease states via stratified vectors.

    PubMed

    Sawamura, Jitsuki; Morishita, Shigeru; Ishigooka, Jun

    2016-02-09

    Previously, we applied basic group theory and related concepts to scales of measurement of clinical disease states and clinical findings (including laboratory data). To gain a more concrete comprehension, we here apply the concept of matrix representation, which was not explicitly exploited in our previous work. Starting with a set of orthonormal vectors, called the basis, an operator Rj (an N-tuple patient disease state at the j-th session) was expressed as a set of stratified vectors representing plural operations on individual components, so as to satisfy the group matrix representation. The stratified vectors containing individual unit operations were combined into one-dimensional square matrices [Rj]s. The [Rj]s meet the matrix representation of a group (ring) as a K-algebra. Using the same-sized matrix of stratified vectors, we can also express changes in the plural set of [Rj]s. The method is demonstrated on simple examples. Despite the incompleteness of our model, the group matrix representation of stratified vectors offers a formal mathematical approach to clinical medicine, aligning it with other branches of natural science.

  18. Role of linguistic skills in fifth-grade mathematics.

    PubMed

    Kleemans, Tijs; Segers, Eliane; Verhoeven, Ludo

    2018-03-01

    The current study investigated the direct and indirect relations between basic linguistic skills (i.e., phonological skills and grammatical ability) and advanced linguistic skills (i.e., academic vocabulary and verbal reasoning), on the one hand, and fifth-grade mathematics (i.e., arithmetic, geometry, and fractions), on the other, taking working memory and general intelligence into account and controlling for socioeconomic status, age, and gender. The results showed the basic linguistic representations of 167 fifth graders to be indirectly related to their geometric and fraction skills via arithmetic. Furthermore, advanced linguistic skills were found to be directly related to geometry and fractions after controlling for arithmetic. It can be concluded that linguistic skills directly and indirectly relate to mathematical ability in the upper grades of primary education, which highlights the importance of paying attention to such skills in the school curriculum. Copyright © 2017 Elsevier Inc. All rights reserved.

  19. The Relative Importance of Children's Criteria for Representational Adequacy in the Perception of Simple Sonic Stimuli

    ERIC Educational Resources Information Center

    Verschaffel, Lieven; Reybrouck, Mark; Degraeuwe, Goedele; Van Dooren, Wim

    2013-01-01

    This study investigates children's metarepresentational competence (MRC) with regard to listening to and making sense of simple sonic stimuli. Using diSessa's (2002) seminal work on MRC in mathematics and sciences as background, it aims to assess the relative importance children attribute to several criteria for representational adequacy…

  20. Fractional labelmaps for computing accurate dose volume histograms

    NASA Astrophysics Data System (ADS)

    Sunderland, Kyle; Pinter, Csaba; Lasso, Andras; Fichtinger, Gabor

    2017-03-01

    PURPOSE: In radiation therapy treatment planning systems, structures are represented as parallel 2D contours. For treatment planning algorithms, structures must be converted into labelmap (i.e. 3D image denoting structure inside/outside) representations. This is often done by triangulated a surface from contours, which is converted into a binary labelmap. This surface to binary labelmap conversion can cause large errors in small structures. Binary labelmaps are often represented using one byte per voxel, meaning a large amount of memory is unused. Our goal is to develop a fractional labelmap representation containing non-binary values, allowing more information to be stored in the same amount of memory. METHODS: We implemented an algorithm in 3D Slicer, which converts surfaces to fractional labelmaps by creating 216 binary labelmaps, changing the labelmap origin on each iteration. The binary labelmap values are summed to create the fractional labelmap. In addition, an algorithm is implemented in the SlicerRT toolkit that calculates dose volume histograms (DVH) using fractional labelmaps. RESULTS: We found that with manually segmented RANDO head and neck structures, fractional labelmaps represented structure volume up to 19.07% (average 6.81%) more accurately than binary labelmaps, while occupying the same amount of memory. When compared to baseline DVH from treatment planning software, DVH from fractional labelmaps had agreement acceptance percent (1% ΔD, 1% ΔV) up to 57.46% higher (average 4.33%) than DVH from binary labelmaps. CONCLUSION: Fractional labelmaps promise to be an effective method for structure representation, allowing considerably more information to be stored in the same amount of memory.

  1. Impact of the basic state and MJO representation on MJO Pacific teleconnections in GCMs

    NASA Astrophysics Data System (ADS)

    Henderson, S. A.; Maloney, E. D.; Son, S. W.

    2017-12-01

    Teleconnection patterns induced by the Madden-Julian Oscillation (MJO) are known to significantly alter extratropical weather and climate patterns. However, accurate MJO representation has been difficult for many General Circulation Models (GCMs). Furthermore, many GCMs contain large basic state biases. These issues present challenges to the simulation of MJO teleconnections and, in turn, their associated extratropical impacts. This study examines the impacts of basic state quality and MJO representation on the quality of MJO teleconnection patterns in GCMs from phase 5 of the Coupled Model Intercomparison Project (CMIP5). Results suggest that GCMs assessed to have a good MJO but with large basic state biases have similarly low skill in reproducing MJO teleconnections as GCMs with poor MJO representation. In the good MJO models examined, poor teleconnection quality is associated with large errors in the zonal extent of the Pacific subtropical jet. Whereas the horizontal structure of MJO heating in the Indo-Pacific region is found to have modest impacts on the teleconnection patterns, results suggest that MJO heating east of the dateline can alter the teleconnection pattern characteristics over North America. These findings suggest that in order to accurately simulate the MJO teleconnection patterns and associated extratropical impacts, both the MJO and the basic state must be well represented.

  2. Time-ordered exponential on the complex plane and Gell-Mann—Low formula as a mathematical theorem

    DOE Office of Scientific and Technical Information (OSTI.GOV)

    Futakuchi, Shinichiro; Usui, Kouta

    2016-04-15

    The time-ordered exponential representation of a complex time evolution operator in the interaction picture is studied. Using the complex time evolution, we prove the Gell-Mann—Low formula under certain abstract conditions, in mathematically rigorous manner. We apply the abstract results to quantum electrodynamics with cutoffs.

  3. Integration of Technology, Curriculum, and Professional Development for Advancing Middle School Mathematics: Three Large-Scale Studies

    ERIC Educational Resources Information Center

    Roschelle, Jeremy; Shechtman, Nicole; Tatar, Deborah; Hegedus, Stephen; Hopkins, Bill; Empson, Susan; Knudsen, Jennifer; Gallagher, Lawrence P.

    2010-01-01

    The authors present three studies (two randomized controlled experiments and one embedded quasi-experiment) designed to evaluate the impact of replacement units targeting student learning of advanced middle school mathematics. The studies evaluated the SimCalc approach, which integrates an interactive representational technology, paper curriculum,…

  4. Cognitive Predictors of Achievement Growth in Mathematics: A Five Year Longitudinal Study

    PubMed Central

    Geary, David C.

    2011-01-01

    The study's goal was to identify the beginning of first grade quantitative competencies that predict mathematics achievement start point and growth through fifth grade. Measures of number, counting, and arithmetic competencies were administered in early first grade and used to predict mathematics achievement through fifth (n = 177), while controlling for intelligence, working memory, and processing speed. Multilevel models revealed intelligence, processing speed, and the central executive component of working memory predicted achievement or achievement growth in mathematics and, as a contrast domain, word reading. The phonological loop was uniquely predictive of word reading and the visuospatial sketch pad of mathematics. Early fluency in processing and manipulating numerical set size and Arabic numerals, accurate use of sophisticated counting procedures for solving addition problems, and accuracy in making placements on a mathematical number line were uniquely predictive of mathematics achievement. Use of memory-based processes to solve addition problems predicted mathematics and reading achievement but in different ways. The results identify the early quantitative competencies that uniquely contribute to mathematics learning. PMID:21942667

  5. Use of a Mathematics Word Problem Strategy to Improve Achievement for Students with Mild Disabilities

    ERIC Educational Resources Information Center

    Taber, Mary R.

    2013-01-01

    Mathematics can be a difficult topic both to teach and to learn. Word problems specifically can be difficult for students with disabilities because they have to conceptualize what the problem is asking for, and they must perform the correct operation accurately. Current trends in mathematics instruction stem from the National Council of Teachers…

  6. A Possible Approach to Inclusion of Space and Time in Frame Fields of Quantum Representations of Real and Complex Numbers

    DOE PAGES

    Benioff, Paul

    2009-01-01

    Tmore » his work is based on the field of reference frames based on quantum representations of real and complex numbers described in other work. Here frame domains are expanded to include space and time lattices. Strings of qukits are described as hybrid systems as they are both mathematical and physical systems. As mathematical systems they represent numbers. As physical systems in each frame the strings have a discrete Schrodinger dynamics on the lattices. he frame field has an iterative structure such that the contents of a stage j frame have images in a stage j - 1 (parent) frame. A discussion of parent frame images includes the proposal that points of stage j frame lattices have images as hybrid systems in parent frames. he resulting association of energy with images of lattice point locations, as hybrid systems states, is discussed. Representations and images of other physical systems in the different frames are also described.« less

  7. Computerized measurement and analysis of scoliosis: a more accurate representation of the shape of the curve.

    PubMed

    Jeffries, B F; Tarlton, M; De Smet, A A; Dwyer, S J; Brower, A C

    1980-02-01

    A computer program was created to identify and accept spatial data regarding the location of the thoracic and lumbar vertebral bodies on scoliosis films. With this information, the spine can be mathematically reconstructed and a scoliotic angle calculated. There was a 0.968 positive correlation between the computer and manual methods of measuring scoliosis. The computer method was more reproducible with a standard deviation of only 1.3 degrees. Computerized measurement of scoliosis also provides better evaluation of the true shape of the curve.

  8. From Sensory Signals to Modality-Independent Conceptual Representations: A Probabilistic Language of Thought Approach

    PubMed Central

    Erdogan, Goker; Yildirim, Ilker; Jacobs, Robert A.

    2015-01-01

    People learn modality-independent, conceptual representations from modality-specific sensory signals. Here, we hypothesize that any system that accomplishes this feat will include three components: a representational language for characterizing modality-independent representations, a set of sensory-specific forward models for mapping from modality-independent representations to sensory signals, and an inference algorithm for inverting forward models—that is, an algorithm for using sensory signals to infer modality-independent representations. To evaluate this hypothesis, we instantiate it in the form of a computational model that learns object shape representations from visual and/or haptic signals. The model uses a probabilistic grammar to characterize modality-independent representations of object shape, uses a computer graphics toolkit and a human hand simulator to map from object representations to visual and haptic features, respectively, and uses a Bayesian inference algorithm to infer modality-independent object representations from visual and/or haptic signals. Simulation results show that the model infers identical object representations when an object is viewed, grasped, or both. That is, the model’s percepts are modality invariant. We also report the results of an experiment in which different subjects rated the similarity of pairs of objects in different sensory conditions, and show that the model provides a very accurate account of subjects’ ratings. Conceptually, this research significantly contributes to our understanding of modality invariance, an important type of perceptual constancy, by demonstrating how modality-independent representations can be acquired and used. Methodologically, it provides an important contribution to cognitive modeling, particularly an emerging probabilistic language-of-thought approach, by showing how symbolic and statistical approaches can be combined in order to understand aspects of human perception. PMID

  9. The use of concrete learning objects taken from the history of mathematics in mathematics education

    NASA Astrophysics Data System (ADS)

    Bütüner, Suphi Önder

    2016-11-01

    This study aimed to reveal the effects of teaching with concrete learning objects taken from the history of mathematics on student achievement. Being a quasi-experimental study, it was conducted with two grade 8 classes in a secondary school located in Trabzon. The experimental group consisted of 27 students and the control group consisted of 25. Data were collected by using worksheets, an achievement exam and written opinion forms. The data from the achievement exam were analysed by using the Mann-Whitney U-test while the data from written opinion forms were analysed through content analysis. The Mann-Whitney U-test results showed a significant difference between the mean ranks of the experimental and control groups in favour of the former. Findings from the written opinion forms suggested that the students found the activities to be instructive and fun, enjoyed using concrete models in their classes, and learned from discovering the rules. It was also found that students had previously not engaged in similar activities and had only experienced the history of mathematics through the life stories and works of mathematicians and the representation of ancient numbers at the beginning of each unit.

  10. The Effects of Using Diagramming as a Representational Technique on High School Students' Achievement in Solving Math Word Problems

    ERIC Educational Resources Information Center

    Banerjee, Banmali

    2010-01-01

    Methods and procedures for successfully solving math word problems have been, and continue to be a mystery to many U.S. high school students. Previous studies suggest that the contextual and mathematical understanding of a word problem, along with the development of schemas and their related external representations, positively contribute to…

  11. Mathematics anxiety and mathematics achievement

    NASA Astrophysics Data System (ADS)

    Sherman, Brian F.; Wither (Post.), David P.

    2003-09-01

    This paper is a distillation of the major result from the 1998 Ph.D. thesis of the late David Wither. It details a longitudinal study over five years of the relationship between mathematics anxiety and mathematics achievement. It starts from the already well documented negative correlation between the two, and seeks to establish one of the three hypotheses—that mathematics anxiety causes an impairment of mathematics achievement; that lack of mathematics achievement causes mathematics anxiety; or that there is a third underlying cause of the two.

  12. Prostate segmentation by sparse representation based classification

    PubMed Central

    Gao, Yaozong; Liao, Shu; Shen, Dinggang

    2012-01-01

    Purpose: The segmentation of prostate in CT images is of essential importance to external beam radiotherapy, which is one of the major treatments for prostate cancer nowadays. During the radiotherapy, the prostate is radiated by high-energy x rays from different directions. In order to maximize the dose to the cancer and minimize the dose to the surrounding healthy tissues (e.g., bladder and rectum), the prostate in the new treatment image needs to be accurately localized. Therefore, the effectiveness and efficiency of external beam radiotherapy highly depend on the accurate localization of the prostate. However, due to the low contrast of the prostate with its surrounding tissues (e.g., bladder), the unpredicted prostate motion, and the large appearance variations across different treatment days, it is challenging to segment the prostate in CT images. In this paper, the authors present a novel classification based segmentation method to address these problems. Methods: To segment the prostate, the proposed method first uses sparse representation based classification (SRC) to enhance the prostate in CT images by pixel-wise classification, in order to overcome the limitation of poor contrast of the prostate images. Then, based on the classification results, previous segmented prostates of the same patient are used as patient-specific atlases to align onto the current treatment image and the majority voting strategy is finally adopted to segment the prostate. In order to address the limitations of the traditional SRC in pixel-wise classification, especially for the purpose of segmentation, the authors extend SRC from the following four aspects: (1) A discriminant subdictionary learning method is proposed to learn a discriminant and compact representation of training samples for each class so that the discriminant power of SRC can be increased and also SRC can be applied to the large-scale pixel-wise classification. (2) The L1 regularized sparse coding is replaced by

  13. Prostate segmentation by sparse representation based classification.

    PubMed

    Gao, Yaozong; Liao, Shu; Shen, Dinggang

    2012-10-01

    The segmentation of prostate in CT images is of essential importance to external beam radiotherapy, which is one of the major treatments for prostate cancer nowadays. During the radiotherapy, the prostate is radiated by high-energy x rays from different directions. In order to maximize the dose to the cancer and minimize the dose to the surrounding healthy tissues (e.g., bladder and rectum), the prostate in the new treatment image needs to be accurately localized. Therefore, the effectiveness and efficiency of external beam radiotherapy highly depend on the accurate localization of the prostate. However, due to the low contrast of the prostate with its surrounding tissues (e.g., bladder), the unpredicted prostate motion, and the large appearance variations across different treatment days, it is challenging to segment the prostate in CT images. In this paper, the authors present a novel classification based segmentation method to address these problems. To segment the prostate, the proposed method first uses sparse representation based classification (SRC) to enhance the prostate in CT images by pixel-wise classification, in order to overcome the limitation of poor contrast of the prostate images. Then, based on the classification results, previous segmented prostates of the same patient are used as patient-specific atlases to align onto the current treatment image and the majority voting strategy is finally adopted to segment the prostate. In order to address the limitations of the traditional SRC in pixel-wise classification, especially for the purpose of segmentation, the authors extend SRC from the following four aspects: (1) A discriminant subdictionary learning method is proposed to learn a discriminant and compact representation of training samples for each class so that the discriminant power of SRC can be increased and also SRC can be applied to the large-scale pixel-wise classification. (2) The L1 regularized sparse coding is replaced by the elastic net in

  14. Learning to Read and Write Polysyllabic Words: The Effects of Morphology and Context on the Acquisition of Whole-Word Representations in Fourth and Fifth Grade

    ERIC Educational Resources Information Center

    Al Ghanem, Reem

    2017-01-01

    Accurate and rapid word recognition requires highly-specified phonological, orthographic, and semantic word-specific representations. It has been established that children acquire these representations through phonological decoding in a process known as orthographic learning. Studies examining orthographic learning and its predictors have thus far…

  15. Elementary School Mathematics: What Parents Should Know About... Estimation.

    ERIC Educational Resources Information Center

    Reys, Barbara

    Parents have many opportunities every day to develop, nurture, and refine their children's mathematics skills. This pamphlet was designed to help parents become aware of these opportunities and to encourage them to participate in their children's learning process. Estimation is the skill of making a reasonably accurate guess and is prominently…

  16. Distinguishing Representations as Origin and Representations as Input: Roles for Individual Neurons.

    PubMed

    Edwards, Jonathan C W

    2016-01-01

    It is widely perceived that there is a problem in giving a naturalistic account of mental representation that deals adequately with the issue of meaning, interpretation, or significance (semantic content). It is suggested here that this problem may arise partly from the conflation of two vernacular senses of representation: representation-as-origin and representation-as-input. The flash of a neon sign may in one sense represent a popular drink, but to function as a representation it must provide an input to a 'consumer' in the street. The arguments presented draw on two principles - the neuron doctrine and the need for a venue for 'presentation' or 'reception' of a representation at a specified site, consistent with the locality principle. It is also argued that domains of representation cannot be defined by signal traffic, since they can be expected to include 'null' elements based on non-firing cells. In this analysis, mental representations-as-origin are distributed patterns of cell firing. Each firing cell is given semantic value in its own right - some form of atomic propositional significance - since different axonal branches may contribute to integration with different populations of signals at different downstream sites. Representations-as-input are patterns of local co-arrival of signals in the form of synaptic potentials in dendrites. Meaning then draws on the relationships between active and null inputs, forming 'scenarios' comprising a molecular combination of 'premises' from which a new output with atomic propositional significance is generated. In both types of representation, meaning, interpretation or significance pivots on events in an individual cell. (This analysis only applies to 'occurrent' representations based on current neural activity.) The concept of representations-as-input emphasizes the need for an internal 'consumer' of a representation and the dependence of meaning on the co-relationships involved in an input interaction between

  17. Success in Mathematics within a Challenged Minority: The Case of Students of Ethiopian Origin in Israel (SEO)

    ERIC Educational Resources Information Center

    Mulat, Tiruwork; Arcavi, Abraham

    2009-01-01

    Many studies have reported on the economical, social, and educational difficulties encountered by Ethiopian Jews since their immigration to Israel. Furthermore, the overall academic underachievement and poor representation of students of Ethiopian origin (SEO) in the advanced mathematics and science classes were highlighted and described. Yet,…

  18. Hyperfunction solutions of the zero rest mass equations and representations of LIE groups

    DOE Office of Scientific and Technical Information (OSTI.GOV)

    Dunne, E.G.

    1984-01-01

    Recently, hyperfunctions have arisen in an essential way in separate results in mathematical physics and in representation theory. In the setting of the twistor program, Wells, with others, has extended the Penrose transform to hyperfunction solutions of the zero rest mass equations, showing that the fundamental isomorphisms hold for this larger space. Meanwhile, Schmid has shown the existence of a canonical globalization of a Harish-Chandra module, V, to a representation of the group. This maximal globalization may be realized as the completion of V in a locally convex vector space in the hyperfunction topology. This thesis shows that the formermore » is a particular case of the latter where the globalization can be done by hand. This explicit globalization is then carried out for a more general case of the Radon transform on homogeneous spaces.« less

  19. Dictionary Learning Algorithms for Sparse Representation

    PubMed Central

    Kreutz-Delgado, Kenneth; Murray, Joseph F.; Rao, Bhaskar D.; Engan, Kjersti; Lee, Te-Won; Sejnowski, Terrence J.

    2010-01-01

    Algorithms for data-driven learning of domain-specific overcomplete dictionaries are developed to obtain maximum likelihood and maximum a posteriori dictionary estimates based on the use of Bayesian models with concave/Schur-concave (CSC) negative log priors. Such priors are appropriate for obtaining sparse representations of environmental signals within an appropriately chosen (environmentally matched) dictionary. The elements of the dictionary can be interpreted as concepts, features, or words capable of succinct expression of events encountered in the environment (the source of the measured signals). This is a generalization of vector quantization in that one is interested in a description involving a few dictionary entries (the proverbial “25 words or less”), but not necessarily as succinct as one entry. To learn an environmentally adapted dictionary capable of concise expression of signals generated by the environment, we develop algorithms that iterate between a representative set of sparse representations found by variants of FOCUSS and an update of the dictionary using these sparse representations. Experiments were performed using synthetic data and natural images. For complete dictionaries, we demonstrate that our algorithms have improved performance over other independent component analysis (ICA) methods, measured in terms of signal-to-noise ratios of separated sources. In the overcomplete case, we show that the true underlying dictionary and sparse sources can be accurately recovered. In tests with natural images, learned overcomplete dictionaries are shown to have higher coding efficiency than complete dictionaries; that is, images encoded with an over-complete dictionary have both higher compression (fewer bits per pixel) and higher accuracy (lower mean square error). PMID:12590811

  20. Profile of students’ generated representations and creative thinking skill in problem solving in vocational school

    NASA Astrophysics Data System (ADS)

    Fikri, P. M.; Sinaga, P.; Hasanah, L.; Solehat, D.

    2018-05-01

    This study aims to determine profile of students’ generated representations and creative thinking skill on problem solving in vocational school. This research is a descriptive research to get an idea of comprehend students’ generated representations and creative thinking skill on problem solving of vocational school in Bandung. Technique of collecting data is done by test method, observation, and interview. Representation is something that represents, describes or symbolizes an object or process. To evaluate the multi-representation skill used essay test with rubric of scoring was used to assess multi-depressant student skills. While creative thinking skill on problem solving used essay test which contains the components of skills in finding facts, problem finding skills, idea finding skills and solution finding skills. The results showed generated representations is still relatively low, this is proven by average student answers explanation is mathematically correct but there is no explanation verbally or graphically. While creative thinking skill on problem solving is still relatively low, this is proven by average score for skill indicator in finding the student problem is 1.52 including the non-creative category, average score for the skill indicator in finding the student idea is 1.23 including the non-creative category, and the average score of the students skill in finding this solution is 0.72 belongs to a very uncreative category.

  1. Delaware Longitudinal Study of Fraction Learning: Implications for Helping Children With Mathematics Difficulties.

    PubMed

    Jordan, Nancy C; Resnick, Ilyse; Rodrigues, Jessica; Hansen, Nicole; Dyson, Nancy

    The goal of the present article is to synthesize findings to date from the Delaware Longitudinal Study of Fraction Learning. The study followed a large cohort of children ( N = 536) between Grades 3 and 6. The findings showed that many students, especially those with diagnosed learning disabilities, made minimal growth in fraction knowledge and that some showed only a basic grasp of the meaning of a fraction even after several years of instruction. Children with low growth in fraction knowledge during the intermediate grades were much more likely to fail to meet state standards on a broad mathematics measure at the end of Grade 6. Although a range of general and mathematics-specific competencies predicted fraction outcomes, the ability to estimate numerical magnitudes on a number line was a uniquely important marker of fraction success. Many children with mathematics difficulties have deep-seated problems related to whole number magnitude representations that are complicated by the introduction of fractions into the curriculum. Implications for helping students with mathematics difficulties are discussed.

  2. Accurate color synthesis of three-dimensional objects in an image

    NASA Astrophysics Data System (ADS)

    Xin, John H.; Shen, Hui-Liang

    2004-05-01

    Our study deals with color synthesis of a three-dimensional object in an image; i.e., given a single image, a target color can be accurately mapped onto the object such that the color appearance of the synthesized object closely resembles that of the actual one. As it is almost impossible to acquire the complete geometric description of the surfaces of an object in an image, this study attempted to recover the implicit description of geometry for the color synthesis. The description was obtained from either a series of spectral reflectances or the RGB signals at different surface positions on the basis of the dichromatic reflection model. The experimental results showed that this implicit image-based representation is related to the object geometry and is sufficient for accurate color synthesis of three-dimensional objects in an image. The method established is applicable to the color synthesis of both rigid and deformable objects and should contribute to color fidelity in virtual design, manufacturing, and retailing.

  3. Improving science and mathematics education with computational modelling in interactive engagement environments

    NASA Astrophysics Data System (ADS)

    Neves, Rui Gomes; Teodoro, Vítor Duarte

    2012-09-01

    A teaching approach aiming at an epistemologically balanced integration of computational modelling in science and mathematics education is presented. The approach is based on interactive engagement learning activities built around computational modelling experiments that span the range of different kinds of modelling from explorative to expressive modelling. The activities are designed to make a progressive introduction to scientific computation without requiring prior development of a working knowledge of programming, generate and foster the resolution of cognitive conflicts in the understanding of scientific and mathematical concepts and promote performative competency in the manipulation of different and complementary representations of mathematical models. The activities are supported by interactive PDF documents which explain the fundamental concepts, methods and reasoning processes using text, images and embedded movies, and include free space for multimedia enriched student modelling reports and teacher feedback. To illustrate, an example from physics implemented in the Modellus environment and tested in undergraduate university general physics and biophysics courses is discussed.

  4. Distinguishing Representations as Origin and Representations as Input: Roles for Individual Neurons

    PubMed Central

    Edwards, Jonathan C. W.

    2016-01-01

    It is widely perceived that there is a problem in giving a naturalistic account of mental representation that deals adequately with the issue of meaning, interpretation, or significance (semantic content). It is suggested here that this problem may arise partly from the conflation of two vernacular senses of representation: representation-as-origin and representation-as-input. The flash of a neon sign may in one sense represent a popular drink, but to function as a representation it must provide an input to a ‘consumer’ in the street. The arguments presented draw on two principles – the neuron doctrine and the need for a venue for ‘presentation’ or ‘reception’ of a representation at a specified site, consistent with the locality principle. It is also argued that domains of representation cannot be defined by signal traffic, since they can be expected to include ‘null’ elements based on non-firing cells. In this analysis, mental representations-as-origin are distributed patterns of cell firing. Each firing cell is given semantic value in its own right – some form of atomic propositional significance – since different axonal branches may contribute to integration with different populations of signals at different downstream sites. Representations-as-input are patterns of local co-arrival of signals in the form of synaptic potentials in dendrites. Meaning then draws on the relationships between active and null inputs, forming ‘scenarios’ comprising a molecular combination of ‘premises’ from which a new output with atomic propositional significance is generated. In both types of representation, meaning, interpretation or significance pivots on events in an individual cell. (This analysis only applies to ‘occurrent’ representations based on current neural activity.) The concept of representations-as-input emphasizes the need for an internal ‘consumer’ of a representation and the dependence of meaning on the co-relationships involved

  5. Machine learning predictions of molecular properties: Accurate many-body potentials and nonlocality in chemical space

    DOE PAGES

    Hansen, Katja; Biegler, Franziska; Ramakrishnan, Raghunathan; ...

    2015-06-04

    Simultaneously accurate and efficient prediction of molecular properties throughout chemical compound space is a critical ingredient toward rational compound design in chemical and pharmaceutical industries. Aiming toward this goal, we develop and apply a systematic hierarchy of efficient empirical methods to estimate atomization and total energies of molecules. These methods range from a simple sum over atoms, to addition of bond energies, to pairwise interatomic force fields, reaching to the more sophisticated machine learning approaches that are capable of describing collective interactions between many atoms or bonds. In the case of equilibrium molecular geometries, even simple pairwise force fields demonstratemore » prediction accuracy comparable to benchmark energies calculated using density functional theory with hybrid exchange-correlation functionals; however, accounting for the collective many-body interactions proves to be essential for approaching the “holy grail” of chemical accuracy of 1 kcal/mol for both equilibrium and out-of-equilibrium geometries. This remarkable accuracy is achieved by a vectorized representation of molecules (so-called Bag of Bonds model) that exhibits strong nonlocality in chemical space. The same representation allows us to predict accurate electronic properties of molecules, such as their polarizability and molecular frontier orbital energies.« less

  6. Machine Learning Predictions of Molecular Properties: Accurate Many-Body Potentials and Nonlocality in Chemical Space

    PubMed Central

    2015-01-01

    Simultaneously accurate and efficient prediction of molecular properties throughout chemical compound space is a critical ingredient toward rational compound design in chemical and pharmaceutical industries. Aiming toward this goal, we develop and apply a systematic hierarchy of efficient empirical methods to estimate atomization and total energies of molecules. These methods range from a simple sum over atoms, to addition of bond energies, to pairwise interatomic force fields, reaching to the more sophisticated machine learning approaches that are capable of describing collective interactions between many atoms or bonds. In the case of equilibrium molecular geometries, even simple pairwise force fields demonstrate prediction accuracy comparable to benchmark energies calculated using density functional theory with hybrid exchange-correlation functionals; however, accounting for the collective many-body interactions proves to be essential for approaching the “holy grail” of chemical accuracy of 1 kcal/mol for both equilibrium and out-of-equilibrium geometries. This remarkable accuracy is achieved by a vectorized representation of molecules (so-called Bag of Bonds model) that exhibits strong nonlocality in chemical space. In addition, the same representation allows us to predict accurate electronic properties of molecules, such as their polarizability and molecular frontier orbital energies. PMID:26113956

  7. Embedded Data Representations.

    PubMed

    Willett, Wesley; Jansen, Yvonne; Dragicevic, Pierre

    2017-01-01

    We introduce embedded data representations, the use of visual and physical representations of data that are deeply integrated with the physical spaces, objects, and entities to which the data refers. Technologies like lightweight wireless displays, mixed reality hardware, and autonomous vehicles are making it increasingly easier to display data in-context. While researchers and artists have already begun to create embedded data representations, the benefits, trade-offs, and even the language necessary to describe and compare these approaches remain unexplored. In this paper, we formalize the notion of physical data referents - the real-world entities and spaces to which data corresponds - and examine the relationship between referents and the visual and physical representations of their data. We differentiate situated representations, which display data in proximity to data referents, and embedded representations, which display data so that it spatially coincides with data referents. Drawing on examples from visualization, ubiquitous computing, and art, we explore the role of spatial indirection, scale, and interaction for embedded representations. We also examine the tradeoffs between non-situated, situated, and embedded data displays, including both visualizations and physicalizations. Based on our observations, we identify a variety of design challenges for embedded data representation, and suggest opportunities for future research and applications.

  8. Early identification and interventions for students with mathematics difficulties.

    PubMed

    Gersten, Russell; Jordan, Nancy C; Flojo, Jonathan R

    2005-01-01

    This article highlights key findings from the small body of research on mathematics difficulties (MD) relevant to early identification and early intervention. The research demonstrates that (a) for many children, mathematics difficulties are not stable over time; (b) the presence of reading difficulties seems related to slower progress in many aspects of mathematics; (c) almost all students with MD demonstrate problems with accurate and automatic retrieval of basic arithmetic combinations, such as 6 + 3. The following measures appear to be valid and reliable indicators of potential MD in kindergartners: (a) magnitude comparison (i.e., knowing which digit in a pair is larger), (b) sophistication of counting strategies, (c) fluent identification of numbers, and (d) working memory (as evidenced by reverse digit span). These are discussed in terms of the components of number sense. Implications for early intervention strategies are explored.

  9. The relationship among mathematics anxiety, beliefs about mathematics, mathematics self-efficacy, and mathematics performance in associate degree nursing students.

    PubMed

    Walsh, Kathleen A

    2008-01-01

    This research explored nursing students' mathematics anxiety, beliefs about mathematics, and mathematics self-efficacy in relation to performance on a medication mathematics test. Results revealed that the participants experienced some mathematics anxiety and had positive beliefs about mathematics and mathematics self-efficacy. Qualitative responses indicated that participants worried about the consequences of failing the medication mathematics test and that practice helped reduce this anxiety. In addition, participants acknowledged the importance of correct dosage calculations for nursing practice. Implications for nursing education are discussed.

  10. Changing predictions, stable recognition: Children's representations of downward incline motion.

    PubMed

    Hast, Michael; Howe, Christine

    2017-11-01

    Various studies to-date have demonstrated children hold ill-conceived expressed beliefs about the physical world such as that one ball will fall faster than another because it is heavier. At the same time, they also demonstrate accurate recognition of dynamic events. How these representations relate is still unresolved. This study examined 5- to 11-year-olds' (N = 130) predictions and recognition of motion down inclines. Predictions were typically in error, matching previous work, but children largely recognized correct events as correct and rejected incorrect ones. The results also demonstrate while predictions change with increasing age, recognition shows signs of stability. The findings provide further support for a hybrid model of object representations and argue in favour of stable core cognition existing alongside developmental changes. Statement of contribution What is already known on this subject? Children's predictions of physical events show limitations in accuracy Their recognition of such events suggests children may use different knowledge sources in their reasoning What the present study adds? Predictions fluctuate more strongly than recognition, suggesting stable core cognition But recognition also shows some fluctuation, arguing for a hybrid model of knowledge representation. © 2017 The British Psychological Society.

  11. [Children with developmental coordination disorder have difficulty with action representation].

    PubMed

    Gabbard, Carl; Cacola, Priscila

    The study of children with developmental coordination disorder (DCD) has emerged as a vibrant line of inquiry over the last two decades. The literature indicates quite clearly that children with DCD display deficits with an array of perceptual-motor and daily living skills. The movements of children with DCD are often described as clumsy and uncoordinated and lead to difficulties with performing many of the activities of daily living and sports that typically developing children perform easily. It has been hypothesized, based on limited research, that an underlying problem is a deficit in generating and/or monitoring an action representation termed the internal modeling deficit hypothesis. According to the hypothesis, children with DCD have significant limitations in their ability to accurately generate and utilize internal models of motor planning and control. The focus of this review is on one of the methods used to examine action representation-motor imagery, which theorists argue provides a window into the process of action representation. Included are research methods and possible brain structures involved. An addition, a paradigm unique with this population-estimation of reachability (distance) via motor imagery, will be described.

  12. A Weibull statistics-based lignocellulose saccharification model and a built-in parameter accurately predict lignocellulose hydrolysis performance.

    PubMed

    Wang, Mingyu; Han, Lijuan; Liu, Shasha; Zhao, Xuebing; Yang, Jinghua; Loh, Soh Kheang; Sun, Xiaomin; Zhang, Chenxi; Fang, Xu

    2015-09-01

    Renewable energy from lignocellulosic biomass has been deemed an alternative to depleting fossil fuels. In order to improve this technology, we aim to develop robust mathematical models for the enzymatic lignocellulose degradation process. By analyzing 96 groups of previously published and newly obtained lignocellulose saccharification results and fitting them to Weibull distribution, we discovered Weibull statistics can accurately predict lignocellulose saccharification data, regardless of the type of substrates, enzymes and saccharification conditions. A mathematical model for enzymatic lignocellulose degradation was subsequently constructed based on Weibull statistics. Further analysis of the mathematical structure of the model and experimental saccharification data showed the significance of the two parameters in this model. In particular, the λ value, defined the characteristic time, represents the overall performance of the saccharification system. This suggestion was further supported by statistical analysis of experimental saccharification data and analysis of the glucose production levels when λ and n values change. In conclusion, the constructed Weibull statistics-based model can accurately predict lignocellulose hydrolysis behavior and we can use the λ parameter to assess the overall performance of enzymatic lignocellulose degradation. Advantages and potential applications of the model and the λ value in saccharification performance assessment were discussed. Copyright © 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

  13. Predicting Relationships between Mathematics Anxiety, Mathematics Teaching Anxiety, Self-Efficacy Beliefs towards Mathematics and Mathematics Teaching

    ERIC Educational Resources Information Center

    Unlu, Melihan; Ertekin, Erhan; Dilmac, Bulent

    2017-01-01

    The purpose of the research is to investigate the relationships between self-efficacy beliefs toward mathematics, mathematics anxiety and self-efficacy beliefs toward mathematics teaching, mathematics teaching anxiety variables and testing the relationships between these variables with structural equation model. The sample of the research, which…

  14. The influence of vision, touch, and proprioception on body representation of the lower limbs.

    PubMed

    Stone, Kayla D; Keizer, Anouk; Dijkerman, H Chris

    2018-04-01

    Numerous studies have shown that the representation of the hand is distorted. When participants are asked to localize unseen points on the hand (e.g. the knuckle), it is perceived to be wider and shorter than its physical dimensions. Similar distortions occur when people are asked to judge the distance between two tactile points on the hand; estimates made in the longitudinal direction are perceived as significantly shorter than those made in the transverse direction. Yet, when asked to visually compare the shape and size of one's own hand to a template hand, individuals are accurate at estimating the size of their own hands. Thus, it seems that body representations are, at least in part, a function of the most prominent underlying sensory modality used to perceive the body part. Yet, it remains unknown if the representations of other body parts are similarly distorted. The lower limbs, for example, are structurally and functionally very different from the hands, yet their representation(s) are seldom studied. What does the body representation for the leg look like? And is leg representation dependent on which sense is probed when making judgments about its shape and size? In the current study, we investigated what the representation of the leg looks like in visually-, tactually-, and proprioceptively-guided tasks. Results revealed that the leg, like the hand, is distorted in a highly systematic manner. Distortions seem to rely, at least partly, on sensory input. This is the first study, to our knowledge, to systematically investigate leg representation in healthy individuals. Copyright © 2018 Elsevier B.V. All rights reserved.

  15. Preface of the "Symposium on Mathematical Models and Methods to investigate Heterogeneity in Cell and Cell Population Biology"

    NASA Astrophysics Data System (ADS)

    Clairambault, Jean

    2016-06-01

    This session investigates hot topics related to mathematical representations of cell and cell population dynamics in biology and medicine, in particular, but not only, with applications to cancer. Methods in mathematical modelling and analysis, and in statistical inference using single-cell and cell population data, should contribute to focus this session on heterogeneity in cell populations. Among other methods are proposed: a) Intracellular protein dynamics and gene regulatory networks using ordinary/partial/delay differential equations (ODEs, PDEs, DDEs); b) Representation of cell population dynamics using agent-based models (ABMs) and/or PDEs; c) Hybrid models and multiscale models to integrate single-cell dynamics into cell population behaviour; d) Structured cell population dynamics and asymptotic evolution w.r.t. relevant traits; e) Heterogeneity in cancer cell populations: origin, evolution, phylogeny and methods of reconstruction; f) Drug resistance as an evolutionary phenotype: predicting and overcoming it in therapeutics; g) Theoretical therapeutic optimisation of combined drug treatments in cancer cell populations and in populations of other organisms, such as bacteria.

  16. Explanation, motivation and question posing routines in university mathematics teachers' pedagogical discourse: a commognitive analysis

    NASA Astrophysics Data System (ADS)

    Viirman, Olov

    2015-11-01

    This paper investigates the teaching practices used by university mathematics teachers when lecturing, a topic within university mathematics education research which is gaining an increasing interest. In the study, a view of mathematics teaching as a discursive practice is taken, and Sfard's commognitive framework is used to investigate the teaching practices of seven Swedish university mathematics teachers on the topic of functions. The present paper looks at the discourse of mathematics teaching, presenting a categorization of the didactical routines into three categories - explanation, motivation and question posing routines. All of these are present in the discourses of all seven teachers, but within these general categories, a number of different sub-categories of routines are found, used in different ways and to different extent by the various teachers. The explanation routines include known mathematical facts, summary and repetition, different representations, everyday language, and concretization and metaphor; the motivation routines include reference to utility, the nature of mathematics, humour and result focus; and the question posing routines include control questions, asking for facts, enquiries and rhetorical questions. This categorization of question posing routines, for instance, complements those already found in the literature. In addition to providing a valuable insight into the teaching of functions at the university level, the categorizations presented in the study can also be useful for investigating the teaching of other mathematical topics.

  17. Visual influence on path integration in darkness indicates a multimodal representation of large-scale space

    PubMed Central

    Tcheang, Lili; Bülthoff, Heinrich H.; Burgess, Neil

    2011-01-01

    Our ability to return to the start of a route recently performed in darkness is thought to reflect path integration of motion-related information. Here we provide evidence that motion-related interoceptive representations (proprioceptive, vestibular, and motor efference copy) combine with visual representations to form a single multimodal representation guiding navigation. We used immersive virtual reality to decouple visual input from motion-related interoception by manipulating the rotation or translation gain of the visual projection. First, participants walked an outbound path with both visual and interoceptive input, and returned to the start in darkness, demonstrating the influences of both visual and interoceptive information in a virtual reality environment. Next, participants adapted to visual rotation gains in the virtual environment, and then performed the path integration task entirely in darkness. Our findings were accurately predicted by a quantitative model in which visual and interoceptive inputs combine into a single multimodal representation guiding navigation, and are incompatible with a model of separate visual and interoceptive influences on action (in which path integration in darkness must rely solely on interoceptive representations). Overall, our findings suggest that a combined multimodal representation guides large-scale navigation, consistent with a role for visual imagery or a cognitive map. PMID:21199934

  18. Why Representations?

    ERIC Educational Resources Information Center

    Schultz, James E.; Waters, Michael S.

    2000-01-01

    Discusses representations in the context of solving a system of linear equations. Views representations (concrete, tables, graphs, algebraic, matrices) from perspectives of understanding, technology, generalization, exact versus approximate solution, and learning style. (KHR)

  19. Representation and display of vector field topology in fluid flow data sets

    NASA Technical Reports Server (NTRS)

    Helman, James; Hesselink, Lambertus

    1989-01-01

    The visualization of physical processes in general and of vector fields in particular is discussed. An approach to visualizing flow topology that is based on the physics and mathematics underlying the physical phenomenon is presented. It involves determining critical points in the flow where the velocity vector vanishes. The critical points, connected by principal lines or planes, determine the topology of the flow. The complexity of the data is reduced without sacrificing the quantitative nature of the data set. By reducing the original vector field to a set of critical points and their connections, a representation of the topology of a two-dimensional vector field that is much smaller than the original data set but retains with full precision the information pertinent to the flow topology is obtained. This representation can be displayed as a set of points and tangent curves or as a graph. Analysis (including algorithms), display, interaction, and implementation aspects are discussed.

  20. Representation in Memory.

    ERIC Educational Resources Information Center

    Rumelhart, David E.; Norman, Donald A.

    This paper reviews work on the representation of knowledge from within psychology and artificial intelligence. The work covers the nature of representation, the distinction between the represented world and the representing world, and significant issues concerned with propositional, analogical, and superpositional representations. Specific topics…

  1. Text + Book = Textbook? Development of a Conceptual Framework for Non-Textual Elements in Middle School Mathematics Textbooks

    ERIC Educational Resources Information Center

    Kim, Rae Young

    2009-01-01

    This study is an initial analytic attempt to iteratively develop a conceptual framework informed by both theoretical and practical perspectives that may be used to analyze non-textual elements in mathematics textbooks. Despite the importance of visual representations in teaching and learning, little effort has been made to specify in any…

  2. Impact of representation of hydraulic structures in modelling a Severn barrage

    NASA Astrophysics Data System (ADS)

    Bray, Samuel; Ahmadian, Reza; Falconer, Roger A.

    2016-04-01

    In this study, enhancements to the numerical representation of sluice gates and turbines were made to the hydro-environmental model Environmental Fluid Dynamics Code (EFDC), and applied to the Severn Tidal Power Group Cardiff-Weston Barrage. The extended domain of the EFDC Continental Shelf Model (CSM) allows far-field hydrodynamic impact assessment of the Severn Barrage, pre- and post-enhancement, to demonstrate the importance of accurate hydraulic structure representation. The enhancements were found to significantly affect peak water levels in the Bristol Channel, reducing levels by nearly 1 m in some areas, and even affect predictions as far-field as the West Coast of Scotland, albeit to a far lesser extent. The model was tested for sensitivity to changes in the discharge coefficient, Cd, used in calculating discharge through sluice gates and turbines. It was found that the performance of the Severn Barrage is not sensitive to changes to the Cd value, and is mitigated through the continual, rather than instantaneous, discharge across the structure. The EFDC CSM can now be said to be more accurately predicting the impacts of tidal range proposals, and the investigation of sensitivity to Cd improves the confidence in the modelling results, despite the uncertainty in this coefficient.

  3. The analysis of mathematics literacy on PMRI learning with media schoology of junior high school students

    NASA Astrophysics Data System (ADS)

    Wardono; Mariani, S.

    2018-03-01

    Indonesia as a developing country in the future will have high competitiveness if its students have high mathematics literacy ability. The current reality from year to year rankings of PISA mathematics literacy Indonesian students are still not good. This research is motivated by the importance and low ability of the mathematics literacy. The purpose of this study is to: (1) analyze the effectiveness of PMRI learning with media Schoology, (2) describe the ability of students' mathematics literacy on PMRI learning with media Schoology which is reviewed based on seven components of mathematics literacy, namely communication, mathematizing, representation, reasoning, devising strategies, using symbols, and using mathematics tool. The method used in this research is the method of sequential design method mix. Techniques of data collection using observation, interviews, tests, and documentation. Data analysis techniques use proportion test, appellate test, and use descriptive analysis. Based on the data analysis, it can be concluded; (1) PMRI learning with media Schoology effectively improve the ability of mathematics literacy because of the achievement of classical completeness, students' mathematics literacy ability in PMRI learning with media Schoology is higher than expository learning, and there is increasing ability of mathematics literacy in PMRI learning with media Schoology of 30%. (2) Highly capable students attain excellent mathematics literacy skills, can work using broad thinking with appropriate resolution strategies. Students who are capable of achieving good mathematics literacy skills can summarize information, present problem-solving processes, and interpret solutions. low-ability students have reached the level of ability of mathematics literacy good enough that can solve the problem in a simple way.

  4. Sparse representation-based image restoration via nonlocal supervised coding

    NASA Astrophysics Data System (ADS)

    Li, Ao; Chen, Deyun; Sun, Guanglu; Lin, Kezheng

    2016-10-01

    Sparse representation (SR) and nonlocal technique (NLT) have shown great potential in low-level image processing. However, due to the degradation of the observed image, SR and NLT may not be accurate enough to obtain a faithful restoration results when they are used independently. To improve the performance, in this paper, a nonlocal supervised coding strategy-based NLT for image restoration is proposed. The novel method has three main contributions. First, to exploit the useful nonlocal patches, a nonnegative sparse representation is introduced, whose coefficients can be utilized as the supervised weights among patches. Second, a novel objective function is proposed, which integrated the supervised weights learning and the nonlocal sparse coding to guarantee a more promising solution. Finally, to make the minimization tractable and convergence, a numerical scheme based on iterative shrinkage thresholding is developed to solve the above underdetermined inverse problem. The extensive experiments validate the effectiveness of the proposed method.

  5. Dioptric power: its nature and its representation in three- and four-dimensional space.

    PubMed

    Harris, W F

    1997-06-01

    Dioptric power expressed in the familiar three-component form of sphere, cylinder, and axis is unsuited to mathematical and statistical treatments; there is a particular class of power that cannot be represented in the familiar form; and it is possible that sphere, cylinder, and axis will prove inadequate in future clinical and research applications in optometry and ophthalmology. Dioptric power expressed as the four-component dioptric power matrix, however, overcomes these shortcomings. The intention in this paper is to provide a definitive statement on the nature, function, and mathematical representation of dioptric power in terms of the matrix and within the limitations of paraxial or linear optics. The approach is universal in the sense that its point of departure is not power of the familiar form (that is, of thin systems) but of systems in general (thick or thin). Familiar types of power are then seen within the context of power in general. Dioptric power is defined, for systems that may be thick and astigmatic, in terms of the ray transfer matrix. A functional definition is presented for dioptric power and its components: it defines the additive contribution of incident position to emergent direction of a ray passing through the system. For systems that are thin (or thin-equivalent) it becomes possible to describe an alternative and more familiar function; for such systems dioptric power can be regarded as the increase in reduced surface curvature of a wavefront brought about by the system as the wavefront passes through it. The curvital and torsional components of the power are explored in some detail. Dioptric power, at its most general, defines a four-dimensional inner product space called dioptric power space. The familiar types of power define a three-dimensional subspace called symmetric dioptric power space. For completeness a one-dimensional antisymmetric power space is also defined: it is orthogonal in four dimensions to symmetric dioptric power

  6. The Effect of Mathematics Research on Mathematics Majors' Mathematical Beliefs

    ERIC Educational Resources Information Center

    Goodson, Joshua E.

    2012-01-01

    This is a dissertation about the beliefs that mathematics majors have about mathematics and how their beliefs are affected by the introduction of mathematics research. The mathematics research presented to the students dealt with counting regular orbits of an action. Research has shown that the beliefs that teachers hold about mathematics…

  7. Electrooculography: Connecting Mind, Brain, and Behavior in Mathematics Education Research

    ERIC Educational Resources Information Center

    Shipulina, Olga V.; Campbell, Stephen R.; Cimen, Arda O.

    2009-01-01

    This paper reports on the potential roles and importance of electrooculography (EOG) for mathematics educational neuroscience research. EOG enables accurate measurements of eye-related behavior (i.e., blinks & movements) by recording changes in voltage potentials generated by eye-related behavior. We identify and discuss three main uses of EOG.…

  8. The influence of signal type on the internal auditory representation of a room

    NASA Astrophysics Data System (ADS)

    Teret, Elizabeth

    Currently, architectural acousticians make no real distinction between a room impulse response and the auditory system's internal representation of a room. With this lack of a good model for the auditory representation of a room, it is indirectly assumed that our internal representation of a room is independent of the sound source needed to make the room characteristics audible. The extent to which this assumption holds true is examined with perceptual tests. Listeners are presented with various pairs of signals (music, speech, and noise) convolved with synthesized impulse responses of different reverberation times. They are asked to adjust the reverberation of one of the signals to match the other. Analysis of the data show that the source signal significantly influences perceived reverberance. Listeners are less accurate when matching reverberation times of varied signals than they are with identical signals. Additional testing shows that perception of reverberation can be linked to the existence of transients in the signal.

  9. Threading Mathematics through Symbols, Sketches, Software, Silicon, and Wood: Teachers Produce and Maintain Cohesion to Support STEM Integration

    ERIC Educational Resources Information Center

    Nathan, Mitchell J.; Wolfgram, Matthew; Srisurichan, Rachaya; Walkington, Candace; Alibali, Martha W.

    2017-01-01

    This classroom-based investigation sought to document how, in real time, STEM teachers and students attempt to locate the invariant mathematical relations that are threaded through the range of activities and representations in these classes, and how highlighting this common thread influences student participation and learning. The authors…

  10. STEM Development: A Study of 6th-12th Grade Girls' Interest and Confidence in Mathematics and Science

    ERIC Educational Resources Information Center

    Heaverlo, Carol Ann

    2011-01-01

    Researchers, policymakers, business, and industry have indicated that the United States will experience a shortage of professionals in the Science, Technology, Engineering, and Mathematics (STEM) fields. Several strategies have been suggested to address this shortage, one of which includes increasing the representation of girls and women in the…

  11. Whiteboard Confessionals: Investigating a New Model Using Student Representations in Teaching Astro 101

    NASA Astrophysics Data System (ADS)

    Prather, Edward

    2018-01-01

    Astronomy education researchers in the Department of Astronomy at the University of Arizona have been investigating a new framework for getting students to engage in discussions about fundamental astronomy topics. This framework is intended to also provide students with explicit feedback on the correctness and coherency of their mental models on these topics. This framework builds upon our prior efforts to create productive Pedagogical Discipline Representations (PDR). Students are asked to work collaboratively to generate their own representations (drawings, graphs, data tables, etc.) that reflect important characteristics of astrophysical scenarios presented in class. We have found these representation tasks offer tremendous insight into the broad range of ideas and knowledge students possess after instruction that includes both traditional lecture and actively learning strategies. In particular, we find that some of our students are able to correctly answer challenging multiple-choice questions on topics, however, they struggle to accurately create representations of these same topics themselves. Our work illustrates that some of our students are not developing a robust level of discipline fluency with many core ideas in astronomy, even after engaging with active learning strategies.

  12. Two-tier tissue decomposition for histopathological image representation and classification.

    PubMed

    Gultekin, Tunc; Koyuncu, Can Fahrettin; Sokmensuer, Cenk; Gunduz-Demir, Cigdem

    2015-01-01

    In digital pathology, devising effective image representations is crucial to design robust automated diagnosis systems. To this end, many studies have proposed to develop object-based representations, instead of directly using image pixels, since a histopathological image may contain a considerable amount of noise typically at the pixel-level. These previous studies mostly employ color information to define their objects, which approximately represent histological tissue components in an image, and then use the spatial distribution of these objects for image representation and classification. Thus, object definition has a direct effect on the way of representing the image, which in turn affects classification accuracies. In this paper, our aim is to design a classification system for histopathological images. Towards this end, we present a new model for effective representation of these images that will be used by the classification system. The contributions of this model are twofold. First, it introduces a new two-tier tissue decomposition method for defining a set of multityped objects in an image. Different than the previous studies, these objects are defined combining texture, shape, and size information and they may correspond to individual histological tissue components as well as local tissue subregions of different characteristics. As its second contribution, it defines a new metric, which we call dominant blob scale, to characterize the shape and size of an object with a single scalar value. Our experiments on colon tissue images reveal that this new object definition and characterization provides distinguishing representation of normal and cancerous histopathological images, which is effective to obtain more accurate classification results compared to its counterparts.

  13. A coherent discrete variable representation method on a sphere

    DOE Office of Scientific and Technical Information (OSTI.GOV)

    Yu, Hua -Gen

    Here, the coherent discrete variable representation (ZDVR) has been extended for construct- ing a multidimensional potential-optimized DVR basis on a sphere. In order to deal with the non-constant Jacobian in spherical angles, two direct product primitive basis methods are proposed so that the original ZDVR technique can be properly implemented. The method has been demonstrated by computing the lowest states of a two dimensional (2D) vibrational model. Results show that the extended ZDVR method gives accurate eigenval- ues and exponential convergence with increasing ZDVR basis size.

  14. A coherent discrete variable representation method on a sphere

    DOE PAGES

    Yu, Hua -Gen

    2017-09-05

    Here, the coherent discrete variable representation (ZDVR) has been extended for construct- ing a multidimensional potential-optimized DVR basis on a sphere. In order to deal with the non-constant Jacobian in spherical angles, two direct product primitive basis methods are proposed so that the original ZDVR technique can be properly implemented. The method has been demonstrated by computing the lowest states of a two dimensional (2D) vibrational model. Results show that the extended ZDVR method gives accurate eigenval- ues and exponential convergence with increasing ZDVR basis size.

  15. Teachers' Mathematics as Mathematics-at-Work

    ERIC Educational Resources Information Center

    Bednarz, Nadine; Proulx, Jérôme

    2017-01-01

    Through recognising mathematics teachers as professionals who use mathematics in their workplace, this article traces a parallel between the mathematics enacted by teachers in their practice and the mathematics used in workplaces found in studies of professionals (e.g. nurses, engineers, bankers). This parallel is developed through the five…

  16. A novel knowledge-based system for interpreting complex engineering drawings: theory, representation, and implementation.

    PubMed

    Lu, Tong; Tai, Chiew-Lan; Yang, Huafei; Cai, Shijie

    2009-08-01

    We present a novel knowledge-based system to automatically convert real-life engineering drawings to content-oriented high-level descriptions. The proposed method essentially turns the complex interpretation process into two parts: knowledge representation and knowledge-based interpretation. We propose a new hierarchical descriptor-based knowledge representation method to organize the various types of engineering objects and their complex high-level relations. The descriptors are defined using an Extended Backus Naur Form (EBNF), facilitating modification and maintenance. When interpreting a set of related engineering drawings, the knowledge-based interpretation system first constructs an EBNF-tree from the knowledge representation file, then searches for potential engineering objects guided by a depth-first order of the nodes in the EBNF-tree. Experimental results and comparisons with other interpretation systems demonstrate that our knowledge-based system is accurate and robust for high-level interpretation of complex real-life engineering projects.

  17. A 4-Dimensional Representation of Antennal Lobe Output Based on an Ensemble of Characterized Projection Neurons

    PubMed Central

    Staudacher, Erich M.; Huetteroth, Wolf; Schachtner, Joachim; Daly, Kevin C.

    2009-01-01

    A central problem facing studies of neural encoding in sensory systems is how to accurately quantify the extent of spatial and temporal responses. In this study, we take advantage of the relatively simple and stereotypic neural architecture found in invertebrates. We combine standard electrophysiological techniques, recently developed population analysis techniques, and novel anatomical methods to form an innovative 4-dimensional view of odor output representations in the antennal lobe of the moth Manduca sexta. This novel approach allows quantification of olfactory responses of characterized neurons with spike time resolution. Additionally, arbitrary integration windows can be used for comparisons with other methods such as imaging. By assigning statistical significance to changes in neuronal firing, this method can visualize activity across the entire antennal lobe. The resulting 4-dimensional representation of antennal lobe output complements imaging and multi-unit experiments yet provides a more comprehensive and accurate view of glomerular activation patterns in spike time resolution. PMID:19464513

  18. Tracing the footsteps of Sherlock Holmes: cognitive representations of hypothesis testing.

    PubMed

    Van Wallendael, L R; Hastie, R

    1990-05-01

    A well-documented phenomenon in opinion-revision literature is subjects' failure to revise probability estimates for an exhaustive set of mutually exclusive hypotheses in a complementary manner. However, prior research has not addressed the question of whether such behavior simply represents a misunderstanding of mathematical rules, or whether it is a consequence of a cognitive representation of hypotheses that is at odds with the Bayesian notion of a set relationship. Two alternatives to the Bayesian representation, a belief system (Shafer, 1976) and a system of independent hypotheses, were proposed, and three experiments were conducted to examine cognitive representations of hypothesis sets in the testing of multiple competing hypotheses. Subjects were given brief murder mysteries to solve and allowed to request various types of information about the suspects; after having received each new piece of information, subjects rated each suspect's probability of being the murderer. Presence and timing of suspect eliminations were varied in the first two experiments; the final experiment involved the varying of percentages of clues that referred to more than one suspect (for example, all of the female suspects). The noncomplementarity of opinion revisions remained a strong phenomenon in all conditions. Information-search data refuted the idea that subjects represented hypotheses as a Bayesian set; further study of the independent hypotheses theory and Shaferian belief functions as descriptive models is encouraged.

  19. On Complex Networks Representation and Computation of Hydrologycal Quantities

    NASA Astrophysics Data System (ADS)

    Serafin, F.; Bancheri, M.; David, O.; Rigon, R.

    2017-12-01

    Water is our blue gold. Despite results of discovery-based science keep warning public opinion about the looming worldwide water crisis, water is still treated as a not worth taking resource. Could a different multi-scale perspective affect environmental decision-making more deeply? Can also a further pairing to a new graphical representation of processes interaction sway decision-making more effectively and public opinion consequently?This abstract introduces a complex networks driven way to represent catchments eco-hydrology and related flexible informatics to manage it. The representation is built upon mathematical category. A category is an algebraic structure that comprises "objects" linked by "arrows". It is an evolution of Petri Nets said Time Continuous Petri Nets (TCPN). It aims to display (water) budgets processes and catchment interactions using explicative and self-contained symbolism. The result improves readability of physical processes compared to current descriptions. The IT perspective hinges on the Object Modeling System (OMS) v3. The latter is a non-invasive flexible environmental modeling framework designed to support component-based model development. The implementation of a Directed Acyclic Graph (DAG) data structure, named Net3, has recently enhanced its flexibility. Net3 represents interacting systems as complex networks: vertices match up with any sort of time evolving quantity; edges correspond to their data (fluxes) interchange. It currently hosts JGrass-NewAge components, and those implementing travel time analysis of fluxes. Further bio-physical or management oriented components can be easily added.This talk introduces both graphical representation and related informatics exercising actual applications and examples.

  20. Three-dimensional representation of curved nanowires.

    PubMed

    Huang, Z; Dikin, D A; Ding, W; Qiao, Y; Chen, X; Fridman, Y; Ruoff, R S

    2004-12-01

    Nanostructures, such as nanowires, nanotubes and nanocoils, can be described in many cases as quasi one-dimensional curved objects projecting in three-dimensional space. A parallax method to construct the correct three-dimensional geometry of such one-dimensional nanostructures is presented. A series of scanning electron microscope images was acquired at different view angles, thus providing a set of image pairs that were used to generate three-dimensional representations using a matlab program. An error analysis as a function of the view angle between the two images is presented and discussed. As an example application, the importance of knowing the true three-dimensional shape of boron nanowires is demonstrated; without the nanowire's correct length and diameter, mechanical resonance data cannot provide an accurate estimate of Young's modulus.