Accurate metacognition for visual sensory memory representations.
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. PMID:24549293
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.…
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…
Mathematical representations of turbulent mixing
NASA Technical Reports Server (NTRS)
Farmer, R. C.; Audeh, B.
1973-01-01
A basic description is given of the mathematical tools and models which are presently used to represent turbulent, free shear layers. Recommendations are included for ways in which current modeling techniques can be improved.
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…
Multiple Representations as Sites for Teacher Reflection about Mathematics Learning
ERIC Educational Resources Information Center
Ryken, Amy E.
2009-01-01
This documentary account situates teacher educator, prospective teacher, and elementary students' mathematical thinking in relation to one another, demonstrating shared challenges to learning mathematics. It highlights an important mathematics reasoning skill--creating and analyzing representations. The author examines responses of prospective…
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…
An Accurate Projector Calibration Method Based on Polynomial Distortion Representation
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
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…
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…
A Mathematical Representation of the Genetic Code
NASA Astrophysics Data System (ADS)
Hill, Vanessa J.; Rowlands, Peter
Algebraic and geometric representations of the genetic code are used to show their functions in coding for amino acids. The algebra is a 64-part vector quaternion combination, and the geometry is based on the structure of the regular icosidodecahedron. An almost perfect pattern suggests that this is a biologically significant way of representing the genetic code.
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…
Mathematical representations in science: a cognitive-historical case history.
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. PMID:25163456
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…
Gender Representation on Journal Editorial Boards in the Mathematical Sciences.
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. PMID:27536970
Gender Representation on Journal Editorial Boards in the Mathematical Sciences
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
An accurate analytic representation of the water pair potential.
Cencek, Wojciech; Szalewicz, Krzysztof; Leforestier, Claude; van Harrevelt, Rob; van der Avoird, Ad
2008-08-28
The ab initio water dimer interaction energies obtained from coupled cluster calculations and used in the CC-pol water pair potential (Bukowski et al., Science, 2007, 315, 1249) have been refitted to a site-site form containing eight symmetry-independent sites in each monomer and denoted as CC-pol-8s. Initially, the site-site functions were assumed in a B-spline form, which allowed a precise optimization of the positions of the sites. Next, these functions were assumed in the standard exponential plus inverse powers form. The root mean square error of the CC-pol-8s fit with respect to the 2510 ab initio points is 0.10 kcal mol(-1), compared to 0.42 kcal mol(-1) of the CC-pol fit (0.010 kcal mol(-1) compared to 0.089 kcal mol(-1) for points with negative interaction energies). The energies of the stationary points in the CC-pol-8s potential are considerably more accurate than in the case of CC-pol. The water dimer vibration-rotation-tunneling spectrum predicted by the CC-pol-8s potential agrees substantially and systematically better with experiment than the already very accurate spectrum predicted by CC-pol, while specific features that could not be accurately predicted previously now agree very well with experiment. This shows that the uncertainties of the fit were the largest source of error in the previous predictions and that the present potential sets a new standard of accuracy in investigations of the water dimer. PMID:18688514
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…
Visual-Spatial Representation in Mathematical Problem Solving by Deaf and Hearing Students
ERIC Educational Resources Information Center
Blatto-Vallee, Gary; Kelly, Ronald R.; Gaustad, Martha G.; Porter, Jeffrey; Fonzi, Judith
2007-01-01
This research examined the use of visual-spatial representation by deaf and hearing students while solving mathematical problems. The connection between spatial skills and success in mathematics performance has long been established in the literature. This study examined the distinction between visual-spatial "schematic" representations that…
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…
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…
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…
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…
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…
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…
ERIC Educational Resources Information Center
Sevimli, Eyup; Delice, Ali
2012-01-01
Students' cognitive differences in problem solving have been the focus of much research. One classification of these differences is related to whether visualisation is used. Like mathematical thinking differences, multiple representation preferences are important when considering individual differences. Choosing an appropriate representation is an…
Dynamically Connected Representations: A Powerful Tool for the Teaching and Learning of Mathematics
ERIC Educational Resources Information Center
Lapp, Douglas A.; St. John, Dennis
2009-01-01
This article describes a vision of student use of dynamically connected representations as they investigate mathematical ideas. Although this article is intended to generate discussion about the potential of dynamically connected representations, we situate the discussion within actual descriptions of its use with students in a newly designed…
Students Preference of Non-Algebraic Representations in Mathematical Communications
ERIC Educational Resources Information Center
Neria, Dorit; Amit, Miriam
2004-01-01
This research study deals with the modes of representation that ninth-graders choose in order to communicate their problem solving paths and justifications, and the relation between these modes of representations and achievement level. The findings are based on analysis of 350 answers to problems that demanded communication of reasoning,…
Mathematical Skills in Williams Syndrome: Insight into the Importance of Underlying Representations
ERIC Educational Resources Information Center
O'Hearn, Kirsten; Luna, Beatriz
2009-01-01
Williams syndrome (WS) is a developmental disorder characterized by relatively spared verbal skills and severe visuospatial deficits. Serious impairments in mathematics have also been reported. This article reviews the evidence on mathematical ability in WS, focusing on the integrity and developmental path of two fundamental representations,…
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…
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…
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…
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…
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 course…
From Static to Dynamic Mathematics: Historical and Representational Perspectives
ERIC Educational Resources Information Center
Moreno-Armella, Luis; Hegedus, Stephen J.; Kaput, James J.
2008-01-01
The nature of mathematical reference fields has substantially evolved with the advent of new types of digital technologies enabling students greater access to understanding the use and application of mathematical ideas and procedures. We analyze the evolution of symbolic thinking over time, from static notations to dynamic inscriptions in new…
Functions of Interactive Visual Representations in Interactive Mathematical Textbooks
ERIC Educational Resources Information Center
Yerushalmy, Michal
2005-01-01
The paper explores changes in technology that have implications for the teaching and learning of school mathematics. To this end, it examines aspects of interactive mathematical textbooks; specifically it analyzes functions authors may intend to be carried out by embedded interactive diagrams. The paper analyzes theoretical as well as practical…
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
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). PMID:23587941
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)
A representational approach to developing primary ITT students' confidence in their mathematics
NASA Astrophysics Data System (ADS)
Bolden, D. S.; Barmby, P.; Harries, T.
2013-01-01
Representations of mathematical concepts play an important role in understanding: both in helping learners understand the to-be-learned material and in facilitating teachers' understanding of pedagogical processes which, in turn, are involved in developing learners' understanding. In this paper, we report on work with a cohort of pre-service primary teachers, with the aim of developing their understanding of mathematics, their confidence in their subject knowledge and their confidence in teaching mathematics. This was attempted through the introduction and use of a 'representational approach' to the teaching of the mathematical concepts required of teachers training to teach in primary schools in the UK. We present the results of attitude measures and a follow-up qualitative questionnaire in identifying whether and how the use of this representational approach supported pre-service teachers' understanding and their confidence in teaching mathematics. The results suggest that the representational approach used had a positively significant impact on the attitudes towards studying and teaching mathematics.
Lost in Translation: Examining Translation Errors Associated with Mathematical Representations
ERIC Educational Resources Information Center
Adu-Gyamfi, Kwaku; Stiff, Lee V.; Bosse, Michael J.
2012-01-01
Translation errors and conceptual misunderstandings made by students translating among graphical, tabular, and symbolic representations of linear functions were examined. The study situated student errors in the context of the "Translation-Verification Model" developed specifically for the purpose of explaining student behavior during the process…
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…
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.
Mathematical representation of the incident solar energy as a function of latitude and time
Simmons, P.A.
1988-07-01
A simple mathematical representation of the incoming solar radiation as a function of latitude and time is introduced. The expression approximates the total zonally and daily averaged solar energy incident on the earth's surface before any is absorbed. It includes dependence on both the obliquity and the precession of the equinoxes and, with its accuracy limits, the representation is convenient for use in long-term climate modelling. 7 references.
Comparing Student Use of Mathematical and Physical Vector Representations
NASA Astrophysics Data System (ADS)
Van Deventer, Joel; Wittmann, Michael C.
2007-11-01
Research has shown that students have difficulties with vectors in college introductory physics courses and high school physics courses; furthermore, students have been shown to perform worse on a vector task with a physical context when compared to the same task in a mathematical context. We have used these results to design isomorphic mathematics and physics free-response vector test questions to evaluate student understanding of vectors in both contexts. To validate our test, we carried out task-based interviews with introductory physics students. We used our results to develop a multiple-choice version of the vector test which was then administered to introductory physics students. We report on our test, giving examples of questions and preliminary findings.
ERIC Educational Resources Information Center
Bal, Ayten Pinar
2014-01-01
This study was designed according to the mixed research method in which quantitative and qualitative research methods were used in order to identify the challenges confronted by classroom teacher candidates in solving mathematical problems and the factors affecting how they choose these representations. The population of this study consisted of…
Mathematical Understanding and Representation Ability of Public Junior High School in North Sumatra
ERIC Educational Resources Information Center
Minarni, Ani; Napitupulu, E. Elvis; Husein, Rahmad
2016-01-01
This paper is the result of first phase of the research about the development of students' mathematical understanding and representation ability through Joyful Problem-Based Learning (JPBL) at Public Junior High School in North Sumatra, Indonesia. The population is all of the students of public junior high school (PJHS) in North Sumatra. Samples…
ERIC Educational Resources Information Center
Hegedus, Stephen J.; Penuel, William R.
2008-01-01
Wireless networks are fast becoming ubiquitous in all aspects of society and the world economy. We describe a method for studying the impacts of combining such technology with dynamic, representationally-rich mathematics software, particularly on participation, expression and projection of identity from a local to a public, shared workspace. We…
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)…
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…
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…
Xiao, Suzhi; Tao, Wei; Zhao, Hui
2016-01-01
In order to acquire an accurate three-dimensional (3D) measurement, the traditional fringe projection technique applies complex and laborious procedures to compensate for the errors that exist in the vision system. However, the error sources in the vision system are very complex, such as lens distortion, lens defocus, and fringe pattern nonsinusoidality. Some errors cannot even be explained or rendered with clear expressions and are difficult to compensate directly as a result. In this paper, an approach is proposed that avoids the complex and laborious compensation procedure for error sources but still promises accurate 3D measurement. It is realized by the mathematical model extension technique. The parameters of the extended mathematical model for the ’phase to 3D coordinates transformation’ are derived using the least-squares parameter estimation algorithm. In addition, a phase-coding method based on a frequency analysis is proposed for the absolute phase map retrieval to spatially isolated objects. The results demonstrate the validity and the accuracy of the proposed flexible fringe projection vision system on spatially continuous and discontinuous objects for 3D measurement. PMID:27136553
NASA Astrophysics Data System (ADS)
Hrubý, Jan
2012-04-01
Mathematical modeling of the non-equilibrium condensing transonic steam flow in the complex 3D geometry of a steam turbine is a demanding problem both concerning the physical concepts and the required computational power. Available accurate formulations of steam properties IAPWS-95 and IAPWS-IF97 require much computation time. For this reason, the modelers often accept the unrealistic ideal-gas behavior. Here we present a computation scheme based on a piecewise, thermodynamically consistent representation of the IAPWS-95 formulation. Density and internal energy are chosen as independent variables to avoid variable transformations and iterations. On the contrary to the previous Tabular Taylor Series Expansion Method, the pressure and temperature are continuous functions of the independent variables, which is a desirable property for the solution of the differential equations of the mass, energy, and momentum conservation for both phases.
ERIC Educational Resources Information Center
Bannister, Vanessa R. Pitts
2014-01-01
The concept of multiple representations of functions and the ability to make translations among representations are important topics in secondary school mathematics curricula (Moschkovich, Schoenfeld, & Arcavi, 1993; NCTM, 2000). Research related to students in this domain is fruitful, while research related to teachers is underdeveloped. This…
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…
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…
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…
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…
Elvira, L; Hernandez, F; Cuesta, P; Cano, S; Gonzalez-Martin, J-V; Astiz, S
2013-06-01
Although the intensive production system of Lacaune dairy sheep is the only profitable method for producers outside of the French Roquefort area, little is known about this type of systems. This study evaluated yield records of 3677 Lacaune sheep under intensive management between 2005 and 2010 in order to describe the lactation curve of this breed and to investigate the suitability of different mathematical functions for modeling this curve. A total of 7873 complete lactations during a 40-week lactation period corresponding to 201 281 pieces of weekly yield data were used. First, five mathematical functions were evaluated on the basis of the residual mean square, determination coefficient, Durbin Watson and Runs Test values. The two better models were found to be Pollott Additive and fractional polynomial (FP). In the second part of the study, the milk yield, peak of milk yield, day of peak and persistency of the lactations were calculated with Pollot Additive and FP models and compared with the real data. The results indicate that both models gave an extremely accurate fit to Lacaune lactation curves in order to predict milk yields (P = 0.871), with the FP model being the best choice to provide a good fit to an extensive amount of real data and applicable on farm without specific statistical software. On the other hand, the interpretation of the parameters of the Pollott Additive function helps to understand the biology of the udder of the Lacaune sheep. The characteristics of the Lacaune lactation curve and milk yield are affected by lactation number and length. The lactation curves obtained in the present study allow the early identification of ewes with low milk yield potential, which will help to optimize farm profitability. PMID:23257242
ERIC Educational Resources Information Center
Lin, Yi-Hung; Wilson, Mark; Cheng, Ching-Lin
2013-01-01
In teaching, representations are used as ways to illustrate the concepts underlying a specific topic. For example, use symbols (e.g., 1?+?2?=?3) to express the concept of addition. To compare students' abilities to interpret different representations in mathematics, the symbolic representation (SR) test and the pictorial representation (PR)…
RICO: A New Approach for Fast and Accurate Representation of the Cosmological Recombination History
NASA Astrophysics Data System (ADS)
Fendt, W. A.; Chluba, J.; Rubiño-Martín, J. A.; Wandelt, B. D.
2009-04-01
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 lsim 900), much of the net change in the ionization fraction can be captured by lowering 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 ~10 ms, a speedup of at least 106 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.
Differential-equation-based representation of truncation errors for accurate numerical simulation
NASA Astrophysics Data System (ADS)
MacKinnon, Robert J.; Johnson, Richard W.
1991-09-01
High-order compact finite difference schemes for 2D convection-diffusion-type differential equations with constant and variable convection coefficients are derived. The governing equations are employed to represent leading truncation terms, including cross-derivatives, making the overall O(h super 4) schemes conform to a 3 x 3 stencil. It is shown that the two-dimensional constant coefficient scheme collapses to the optimal scheme for the one-dimensional case wherein the finite difference equation yields nodally exact results. The two-dimensional schemes are tested against standard model problems, including a Navier-Stokes application. Results show that the two schemes are generally more accurate, on comparable grids, than O(h super 2) centered differencing and commonly used O(h) and O(h super 3) upwinding schemes.
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…
Cirelli, Josie; McGregor, Carolyn; Graydon, Brenda; James, Andrew
2013-01-01
Maintaining blood oxygen saturation within the intended target range for preterm infants receiving neonatal intensive care is challenging. Supplemental oxygen is believed to lead to increased risk of retinopathy of prematurity and hence managing the level of oxygen within this population is important within their care. Current quality improvement activities use coarse hourly spot readings to measure supplemental oxygen levels as associated with targeted ranges that vary based on gestational age. In this research we use Artemis, a real-time online healthcare analytics platform to ascertain if the collection of second by second data provides a better representation of retinal exposure to oxygen than an infrequent, intermittent spot reading. We show that Artemis is capable of producing more accurate information from the higher frequency data, as it includes all the episodic events in the activity of the hour, which provides a better understanding of oxygen fluctuation ranges which affect the physiological status of the infant. PMID:23388268
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)…
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…
ERIC Educational Resources Information Center
van Garderen, Delinda; Scheuermann, Amy; Jackson, Christa
2012-01-01
This study was an examination of the extent to which sixth- and seventh-grade mathematics textbooks incorporated recommended instructional practices for students with learning disabilities to help develop representational ability. Results indicated that the textbooks (a) provided very little explicit instructional information about representations…
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…
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.
Mathematical model accurately predicts protein release from an affinity-based delivery system.
Vulic, Katarina; Pakulska, Malgosia M; Sonthalia, Rohit; Ramachandran, Arun; Shoichet, Molly S
2015-01-10
Affinity-based controlled release modulates the delivery of protein or small molecule therapeutics through transient dissociation/association. To understand which parameters can be used to tune release, we used a mathematical model based on simple binding kinetics. A comprehensive asymptotic analysis revealed three characteristic regimes for therapeutic release from affinity-based systems. These regimes can be controlled by diffusion or unbinding kinetics, and can exhibit release over either a single stage or two stages. This analysis fundamentally changes the way we think of controlling release from affinity-based systems and thereby explains some of the discrepancies in the literature on which parameters influence affinity-based release. The rate of protein release from affinity-based systems is determined by the balance of diffusion of the therapeutic agent through the hydrogel and the dissociation kinetics of the affinity pair. Equations for tuning protein release rate by altering the strength (KD) of the affinity interaction, the concentration of binding ligand in the system, the rate of dissociation (koff) of the complex, and the hydrogel size and geometry, are provided. We validated our model by collapsing the model simulations and the experimental data from a recently described affinity release system, to a single master curve. Importantly, this mathematical analysis can be applied to any single species affinity-based system to determine the parameters required for a desired release profile. PMID:25449806
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
2011-01-01
Background Data assimilation refers to methods for updating the state vector (initial condition) of a complex spatiotemporal model (such as a numerical weather model) by combining new observations with one or more prior forecasts. We consider the potential feasibility of this approach for making short-term (60-day) forecasts of the growth and spread of a malignant brain cancer (glioblastoma multiforme) in individual patient cases, where the observations are synthetic magnetic resonance images of a hypothetical tumor. Results We apply a modern state estimation algorithm (the Local Ensemble Transform Kalman Filter), previously developed for numerical weather prediction, to two different mathematical models of glioblastoma, taking into account likely errors in model parameters and measurement uncertainties in magnetic resonance imaging. The filter can accurately shadow the growth of a representative synthetic tumor for 360 days (six 60-day forecast/update cycles) in the presence of a moderate degree of systematic model error and measurement noise. Conclusions The mathematical methodology described here may prove useful for other modeling efforts in biology and oncology. An accurate forecast system for glioblastoma may prove useful in clinical settings for treatment planning and patient counseling. Reviewers This article was reviewed by Anthony Almudevar, Tomas Radivoyevitch, and Kristin Swanson (nominated by Georg Luebeck). PMID:22185645
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)
ERIC Educational Resources Information Center
Capraro, Robert M.; Kulm, Gerald; Capraro, Mary Margaret
This study explored a model for students development of the understandings and skills that are involved in being able to construct graphical representations of data and to interpret these graphs. The study examined four components of prior understanding required for graphic representation that were adapted from a learning map from the Atlas of…
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…
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…
ERIC Educational Resources Information Center
Roche, Anne; Clarke, Doug M.
2013-01-01
This article reports on a study that was conducted with 378 primary teachers from Catholic schools in Victoria who participated in the first year of a 2-year research and professional learning program in mathematics. One aim of the program was to enhance teacher knowledge in mathematics in its many forms. As part of the larger study, the teachers…
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…
Varandas, A J C
2013-04-01
The Combined-Hyperbolic-Inverse-Power-Representation method, which treats evenly both short- and long-range interactions, is used to fit an extensive set of ab initio points for HO2 previously utilized [Xu et al., J. Chem. Phys. 122, 244305 (2005)] to develop a spline interpolant. The novel form is shown to perform accurately when compared with others, while quasiclassical trajectory calculations of the O + OH reaction clearly pinpoint the role of long-range forces at low temperatures. PMID:23574218
Tobia, Valentina; Fasola, Anna; Lupieri, Alice; Marzocchi, Gian Marco
2016-03-01
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. PMID:24737662
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…
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…
ERIC Educational Resources Information Center
Singer, Florence Mihaela
2007-01-01
Effective teaching should focus on representational change, which is fundamental to learning and education, rather than conceptual change, which involves transformation of theories in science rather than the gradual building of knowledge that occurs in students. This article addresses the question about how to develop more efficient strategies for…
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…
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).…
Middle School Students' Understanding of Mathematical Patterns and Their Symbolic Representations.
ERIC Educational Resources Information Center
Bishop, Joyce Wolfer
This study explores seventh- and eighth-grade students' thinking about mathematical patterns. Interviews were conducted in which students solved problems about sequential perimeter and area problems modeled with pattern blocks and tiles, generalized the relationships related to the patterns and represented the relationships symbolically,…
ERIC Educational Resources Information Center
Fan, Lianghuo; Zhu, Yan
2007-01-01
This study examined how selected school mathematics textbooks in China, Singapore, and USA at the lower secondary grade level represent problem-solving procedures. The analysis of problem-solving procedures was carried out in two layers--general strategies, which adopted Polya's four-stage problem-solving model, and specific strategies, which…
Children's Cognitive Representation of the Mathematical Number Line
ERIC Educational Resources Information Center
Rouder, Jeffrey N.; Geary, David C.
2014-01-01
Learning of the mathematical number line has been hypothesized to be dependent on an inherent sense of approximate quantity. Children's number line placements are predicted to conform to the underlying properties of this system; specifically, placements are exaggerated for small numerals and compressed for larger ones. Alternative hypotheses…
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 =…
NASA Astrophysics Data System (ADS)
Kim, Jibeom; Jeon, Joonhyeon
2015-01-01
Recently, related studies on Equation Of State (EOS) have reported that generalized van der Waals (GvdW) shows poor representations in the near critical region for non-polar and non-sphere molecules. Hence, there are still remains a problem of GvdW parameters to minimize loss in describing saturated vapor densities and vice versa. This paper describes a recursive model GvdW (rGvdW) for an accurate representation of pure fluid materials in the near critical region. For the performance evaluation of rGvdW in the near critical region, other EOS models are also applied together with two pure molecule group: alkane and amine. The comparison results show rGvdW provides much more accurate and reliable predictions of pressure than the others. The calculating model of EOS through this approach gives an additional insight into the physical significance of accurate prediction of pressure in the nearcritical region.
NASA Astrophysics Data System (ADS)
Varandas, A. J. C.
2013-04-01
The Combined-Hyperbolic-Inverse-Power-Representation method, which treats evenly both short- and long-range interactions, is used to fit an extensive set of ab initio points for HO2 previously utilized [Xu et al., J. Chem. Phys. 122, 244305 (2005), 10.1063/1.1944290] to develop a spline interpolant. The novel form is shown to perform accurately when compared with others, while quasiclassical trajectory calculations of the O + OH reaction clearly pinpoint the role of long-range forces at low temperatures.
Magombedze, Gesham; Mulder, Nicola
2012-01-01
The majority of individuals infected with Mycobacterium tuberculosis (Mtb) bacilli develop latent infection. Mtb becomes dormant and phenotypically drug resistant when it encounters multiple stresses within the host, and expresses a set of genes, known as the dormancy regulon, in vivo. These genes are expressed in vitro in response to nitric oxide (NO), hypoxia (oxygen deprivation), and nutrient starvation. The occurrence and reactivation of latent tuberculosis (TB) is not clearly understood. The ability of the pathogen to enter and exit from different states is associated with its ability to cause persistent infection. During infection it is not known whether the organism is in a persistent slow replicating state or a dormant non-replicating state, with the latter ultimately causing a latent infection with the potential to reactivate to active disease. We collected gene expression data for Mtb bacilli under different stress conditions that simulate latency or dormancy. Time course experiments were selected and differentially expressed gene profiles were determined at each time point. A mathematical model was then developed to show the dynamics of Mtb latency based on the profile of differentially expressed genes. Analysis of the time course data show the dynamics of latency occurrence in vitro and the mathematical model reveals all possible scenarios of Mtb latency development with respect to the different conditions that may be produced by the immune response in vivo. The mathematical model provides a biological explanation of how Mtb latency occurs based on observed gene expression changes in in vitro latency models. PMID:21968442
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.
NASA Astrophysics Data System (ADS)
Bačić, Z.; Kress, J. D.; Parker, G. A.; Pack, R. T.
1990-02-01
Accurate 3D coupled channel calculations for total angular momentum J=0 for the reaction F+H2→HF+H using a realistic potential energy surface are analyzed. The reactive scattering is formulated using the hyperspherical (APH) coordinates of Pack and Parker. The adiabatic basis functions are generated quite efficiently using the discrete variable representation method. Reaction probabilities for relative collision energies of up to 17.4 kcal/mol are presented. To aid in the interpretation of the resonances and quantum structure observed in the calculated reaction probabilities, we analyze the phases of the S matrix transition elements, Argand diagrams, time delays and eigenlifetimes of the collision lifetime matrix. Collinear (1D) and reduced dimensional 3D bending corrected rotating linear model (BCRLM) calculations are presented and compared with the accurate 3D calculations.
Mui, K W; Wong, L T; Chung, L Y
2009-11-01
Atmospheric visibility impairment has gained increasing concern as it is associated with the existence of a number of aerosols as well as common air pollutants and produces unfavorable conditions for observation, dispersion, and transportation. This study analyzed the atmospheric visibility data measured in urban and suburban Hong Kong (two selected stations) with respect to time-matched mass concentrations of common air pollutants including nitrogen dioxide (NO(2)), nitrogen monoxide (NO), respirable suspended particulates (PM(10)), sulfur dioxide (SO(2)), carbon monoxide (CO), and meteorological parameters including air temperature, relative humidity, and wind speed. No significant difference in atmospheric visibility was reported between the two measurement locations (p > or = 0.6, t test); and good atmospheric visibility was observed more frequently in summer and autumn than in winter and spring (p < 0.01, t test). It was also found that atmospheric visibility increased with temperature but decreased with the concentrations of SO(2), CO, PM(10), NO, and NO(2). The results showed that atmospheric visibility was season dependent and would have significant correlations with temperature, the mass concentrations of PM(10) and NO(2), and the air pollution index API (correlation coefficients mid R: R mid R: > or = 0.7, p < or = 0.0001, t test). Mathematical expressions catering to the seasonal variations of atmospheric visibility were thus proposed. By comparison, the proposed visibility prediction models were more accurate than some existing regional models. In addition to improving visibility prediction accuracy, this study would be useful for understanding the context of low atmospheric visibility, exploring possible remedial measures, and evaluating the impact of air pollution and atmospheric visibility impairment in this region. PMID:18951139
Wang, Guotai; Zhang, Shaoting; Xie, Hongzhi; Metaxas, Dimitris N; Gu, Lixu
2015-01-01
Shape prior plays an important role in accurate and robust liver segmentation. However, liver shapes have complex variations and accurate modeling of liver shapes is challenging. Using large-scale training data can improve the accuracy but it limits the computational efficiency. In order to obtain accurate liver shape priors without sacrificing the efficiency when dealing with large-scale training data, we investigate effective and scalable shape prior modeling method that is more applicable in clinical liver surgical planning system. We employed the Sparse Shape Composition (SSC) to represent liver shapes by an optimized sparse combination of shapes in the repository, without any assumptions on parametric distributions of liver shapes. To leverage large-scale training data and improve the computational efficiency of SSC, we also introduced a homotopy-based method to quickly solve the L1-norm optimization problem in SSC. This method takes advantage of the sparsity of shape modeling, and solves the original optimization problem in SSC by continuously transforming it into a series of simplified problems whose solution is fast to compute. When new training shapes arrive gradually, the homotopy strategy updates the optimal solution on the fly and avoids re-computing it from scratch. Experiments showed that SSC had a high accuracy and efficiency in dealing with complex liver shape variations, excluding gross errors and preserving local details on the input liver shape. The homotopy-based SSC had a high computational efficiency, and its runtime increased very slowly when repository's capacity and vertex number rose to a large degree. When repository's capacity was 10,000, with 2000 vertices on each shape, homotopy method cost merely about 11.29 s to solve the optimization problem in SSC, nearly 2000 times faster than interior point method. The dice similarity coefficient (DSC), average symmetric surface distance (ASD), and maximum symmetric surface distance measurement
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…
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.…
Frolov, Andrey I
2015-05-12
Accurate calculation of solvation free energies (SFEs) is a fundamental problem of theoretical chemistry. In this work we perform a careful validation of the theory of solutions in energy representation (ER method) developed by Matubayasi et al. [J. Chem. Phys. 2000, 113, 6070-6081] for SFE calculations in supercritical solvents. This method can be seen as a bridge between the molecular simulations and the classical (not quantum) density functional theory (DFT) formulated in energy representation. We performed extensive calculations of SFEs of organic molecules of different chemical natures in pure supercritical CO2 (sc-CO2) and in sc-CO2 with addition of 6 mol % of ethanol, acetone, and n-hexane as cosolvents. We show that the ER method reproduces SFE data calculated by a method free of theoretical approximations (the Bennett's acceptance ratio) with the mean absolute error of only 0.05 kcal/mol. However, the ER method requires by an order less computational resources. Also, we show that the quality of ER calculations should be carefully monitored since the lack of sampling can result into a considerable bias in predictions. The present calculations reproduce the trends in the cosolvent-induced solubility enhancement factors observed in experimental data. Thus, we think that molecular simulations coupled with the ER method can be used for quick calculations of the effect of variation of temperature, pressure, and cosolvent concentration on SFE and hence solubility of bioactive compounds in supercritical fluids. This should dramatically reduce the burden of experimental work on optimizing solvency of supercritical solvents. PMID:26574423
Reading Students' Representations
ERIC Educational Resources Information Center
Diezmann, Carmel M.; McCosker, Natalie T.
2011-01-01
Representations play a key role in mathematical thinking: They offer "a medium" to express mathematical knowledge or organize mathematical information and to discern mathematical relationships (e.g., relative household expenditures on a pie chart) using text, symbols, or graphics. They also furnish "tools" for mathematical processes (e.g., use of…
Mavandadi, Sam; Feng, Steve; Yu, Frank; Dimitrov, Stoyan; Nielsen-Saines, Karin; Prescott, William R; Ozcan, Aydogan
2012-01-01
We propose a methodology for digitally fusing diagnostic decisions made by multiple medical experts in order to improve accuracy of diagnosis. Toward this goal, we report an experimental study involving nine experts, where each one was given more than 8,000 digital microscopic images of individual human red blood cells and asked to identify malaria infected cells. The results of this experiment reveal that even highly trained medical experts are not always self-consistent in their diagnostic decisions and that there exists a fair level of disagreement among experts, even for binary decisions (i.e., infected vs. uninfected). To tackle this general medical diagnosis problem, we propose a probabilistic algorithm to fuse the decisions made by trained medical experts to robustly achieve higher levels of accuracy when compared to individual experts making such decisions. By modelling the decisions of experts as a three component mixture model and solving for the underlying parameters using the Expectation Maximisation algorithm, we demonstrate the efficacy of our approach which significantly improves the overall diagnostic accuracy of malaria infected cells. Additionally, we present a mathematical framework for performing 'slide-level' diagnosis by using individual 'cell-level' diagnosis data, shedding more light on the statistical rules that should govern the routine practice in examination of e.g., thin blood smear samples. This framework could be generalized for various other tele-pathology needs, and can be used by trained experts within an efficient tele-medicine platform. PMID:23071544
Mavandadi, Sam; Dimitrov, Stoyan; Nielsen-Saines, Karin; Prescott, William R.; Ozcan, Aydogan
2012-01-01
We propose a methodology for digitally fusing diagnostic decisions made by multiple medical experts in order to improve accuracy of diagnosis. Toward this goal, we report an experimental study involving nine experts, where each one was given more than 8,000 digital microscopic images of individual human red blood cells and asked to identify malaria infected cells. The results of this experiment reveal that even highly trained medical experts are not always self-consistent in their diagnostic decisions and that there exists a fair level of disagreement among experts, even for binary decisions (i.e., infected vs. uninfected). To tackle this general medical diagnosis problem, we propose a probabilistic algorithm to fuse the decisions made by trained medical experts to robustly achieve higher levels of accuracy when compared to individual experts making such decisions. By modelling the decisions of experts as a three component mixture model and solving for the underlying parameters using the Expectation Maximisation algorithm, we demonstrate the efficacy of our approach which significantly improves the overall diagnostic accuracy of malaria infected cells. Additionally, we present a mathematical framework for performing ‘slide-level’ diagnosis by using individual ‘cell-level’ diagnosis data, shedding more light on the statistical rules that should govern the routine practice in examination of e.g., thin blood smear samples. This framework could be generalized for various other tele-pathology needs, and can be used by trained experts within an efficient tele-medicine platform. PMID:23071544
Plotnitsky, Arkady
2016-01-13
The project of this article is twofold. First, it aims to offer a new perspective on, and a new argument concerning, realist and non-realist mathematical models, and differences and affinities between them, using physics as a paradigmatic field of mathematical modelling in science. Most of the article is devoted to this topic. Second, the article aims to explore the implications of this argument for mathematical modelling in other fields, in particular in cognitive psychology and economics. PMID:26621990
ERIC Educational Resources Information Center
Switzer, Matt
2011-01-01
This study reports how 24 grade 4-6 students in one elementary and middle school interpreted formal and informal representations of variables. While interpretations for variables represented as letters (e.g., x and y) have been well established for students in algebra classes and beyond, little research into elementary school students'…
ERIC Educational Resources Information Center
Perkins, D. N.; Simmons, Rebecca
This paper examines the cognitive structures and processes that mediate mathematical and scientific ability. Ability is divided into achieved abilities and precursor abilities. Identified concepts in the area of achieved ability include expertise, understanding, and problem-solving. Other abilities can be seen as precursors to such achieved…
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…
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
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…
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
Connected Representations: From Proportion to Linear Functions
ERIC Educational Resources Information Center
Baltus, Christopher
2010-01-01
Mathematics may be inconceivable without its diagrams and symbols--its representations. Mathematical representations help individuals organize their thinking; they bring a visual component to abstract ideas and serve as templates for computation with understanding. But the inevitability of representations is no guarantee that they are used…
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…
Fernández-Colino, A; Bermudez, J M; Arias, F J; Quinteros, D; Gonzo, E
2016-04-01
Transversality between mathematical modeling, pharmacology, and materials science is essential in order to achieve controlled-release systems with advanced properties. In this regard, the area of biomaterials provides a platform for the development of depots that are able to achieve controlled release of a drug, whereas pharmacology strives to find new therapeutic molecules and mathematical models have a connecting function, providing a rational understanding by modeling the parameters that influence the release observed. Herein we present a mechanism which, based on reasonable assumptions, explains the experimental data obtained very well. In addition, we have developed a simple and accurate “lumped” kinetics model to correctly fit the experimentally observed drug-release behavior. This lumped model allows us to have simple analytic solutions for the mass and rate of drug release as a function of time without limitations of time or mass of drug released, which represents an important step-forward in the area of in vitro drug delivery when compared to the current state of the art in mathematical modeling. As an example, we applied the mechanism and model to the release data for acetazolamide from a recombinant polymer. Both materials were selected because of a need to develop a suitable ophthalmic formulation for the treatment of glaucoma. The in vitro release model proposed herein provides a valuable predictive tool for ensuring product performance and batch-to-batch reproducibility, thus paving the way for the development of further pharmaceutical devices. PMID:26838852
ERIC Educational Resources Information Center
Clements, Peggy; Buffington, Pamela; Tobey, Cheryl
2013-01-01
Rational number concepts underpin many topics in advanced mathematics and understanding these concepts is a prerequisite for students' success in high-school level courses. Students with rational number misconceptions that are not diagnosed and remediated in the middle grades are likely to encounter difficulty in high-school mathematics courses.…
Standard model of knowledge representation
NASA Astrophysics Data System (ADS)
Yin, Wensheng
2016-03-01
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.
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…
ERIC Educational Resources Information Center
Herbst, Patricio; Kosko, Karl W.
2014-01-01
This study compared conversations among groups of teachers of high school geometry that had been elicited by a representation of instruction (either a video or an animation) and facilitated with an open-ended agenda. All artifacts used represented instruction scenarios that departed from what, according to prior work, had been hypothesized as…
ERIC Educational Resources Information Center
Berryman, Sue E.
This paper describes trends in and causes of minority and female representation among holders of advanced science and math degrees. The minority groups studied are Blacks, Hispanic Americans, American Indians, and Asian Americans, all of whom are compared with Whites. The degrees looked at include those in math, the computer sciences, physical…
Mathematics in the Mende Culture: Its General Implication for Mathematics Teaching.
ERIC Educational Resources Information Center
Bockarie, Alex
1993-01-01
Mathematics that exists in the Mende culture, an African tribe in Sierra Leone, includes counting, computation, ratios, fractions, forecasting games, and mathematical applications. Presents The Mende representations of these concepts and discusses implications of their integration into mathematics teaching. (MDH)
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,…
ERIC Educational Resources Information Center
Halat, Erdogan; Peker, Murat
2011-01-01
The purpose of this study was to compare the influence of instruction using WebQuest activities with the influence of an instruction using spreadsheet activities on the motivation of pre-service elementary school teachers in mathematics teaching course. There were a total of 70 pre-service elementary school teachers involved in this study. Thirty…
NASA Technical Reports Server (NTRS)
Kuznetz, L. H.
1976-01-01
Test data and a mathematical model of the human thermoregulatory system were used to investigate control of thermal balance by means of a liquid circulating garment (LCG). The test data were derived from five series of experiments in which environmental and metabolic conditions were varied parametrically as a function of several independent variables, including LCG flowrate, LCG inlet temperature, net environmental heat exchange, surrounding gas ventilation rate, ambient pressure, metabolic rate, and subjective/obligatory cooling control. The resultant data were used to relate skin temperature to LCG water temperature and flowrate, to assess a thermal comfort band, to demonstrate the relationship between metabolic rate and LCG heat dissipation, and so forth. The usefulness of the mathematical model as a tool for data interpretation and for generation of trends and relationships among the various physiological parameters was also investigated and verified.
NASA Technical Reports Server (NTRS)
Peuquet, Donna J.
1987-01-01
A new approach to building geographic data models that is based on the fundamental characteristics of the data is presented. An overall theoretical framework for representing geographic data is proposed. An example of utilizing this framework in a Geographic Information System (GIS) context by combining artificial intelligence techniques with recent developments in spatial data processing techniques is given. Elements of data representation discussed include hierarchical structure, separation of locational and conceptual views, and the ability to store knowledge at variable levels of completeness and precision.
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)
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…
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…
Quantity Cognition: Numbers, Numerosity, Zero and Mathematics.
Harvey, Ben M
2016-05-23
Physical quantities differ from abstract numbers and mathematics, but recent results are revealing the neural representation of both: a new study demonstrates how an absence of quantity is transformed into a representation of zero as a number. PMID:27218850
Mathematical Models and the Experimental Analysis of Behavior
Mazur, James E
2006-01-01
The use of mathematical models in the experimental analysis of behavior has increased over the years, and they offer several advantages. Mathematical models require theorists to be precise and unambiguous, often allowing comparisons of competing theories that sound similar when stated in words. Sometimes different mathematical models may make equally accurate predictions for a large body of data. In such cases, it is important to find and investigate situations for which the competing models make different predictions because, unless two models are actually mathematically equivalent, they are based on different assumptions about the psychological processes that underlie an observed behavior. Mathematical models developed in basic behavioral research have been used to predict and control behavior in applied settings, and they have guided research in other areas of psychology. A good mathematical model can provide a common framework for understanding what might otherwise appear to be diverse and unrelated behavioral phenomena. Because psychologists vary in their quantitative skills and in their tolerance for mathematical equations, it is important for those who develop mathematical models of behavior to find ways (such as verbal analogies, pictorial representations, or concrete examples) to communicate the key premises of their models to nonspecialists. PMID:16673829
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…
Student Nurses and Mathematics.
ERIC Educational Resources Information Center
Hutton, B. Meriel
For the safety of the public, it is essential that nurses are competent at least in the mathematics that enables them to calculate medications accurately. From a survey by G. Hek (1994), it is apparent that mathematics is not universally included in the nursing curricula, nor asked for as a pre-requisite to entry. Changes in the profile of the…
Discipline-Based Remediation: Bridging the Mathematics Gap
NASA Astrophysics Data System (ADS)
Wenner, Jennifer M.; Baer, Eric M.; Burn, Helen E.
2013-10-01
Geoscience relies on numbers, data, equations, graphical representations, and other quantitative skills; therefore, introductory geoscience courses need to accurately portray the science as quantitative [e.g., Wenner et al., 2009]. However, up to 57% of students arrive at college underprepared to perform mathematics at the level necessary to succeed in introductory courses [ACT, 2011]. Although some institutions have turned to prerequisites as a way to ensure appropriate preparation, these extra courses can place undue financial, temporal, and academic burdens on interested students, keeping them from enrolling in science courses that may interest them. As an alternative to mathematics prerequisites, geoscience faculty at the University of Wisconsin Oshkosh and Highline Community College in Des Moines, Wash., funded by the National Science Foundation, developed a model of successful integration of discipline-based mathematics remediation into an introductory geoscience course: The Math You Need, When You Need It (TMYN; http://serc.carleton.edu/mathyouneed/).
Computer aided surface representation
Barnhill, R.E.
1990-02-19
The central research problem of this project is the effective representation, computation, and display of surfaces interpolating to information in three or more dimensions. If the given information is located on another surface, then the problem is to construct a surface defined on a surface''. Sometimes properties of an already defined surface are desired, which is geometry processing''. Visualization of multivariate surfaces is possible by means of contouring higher dimensional surfaces. These problems and more are discussed below. The broad sweep from constructive mathematics through computational algorithms to computer graphics illustrations is utilized in this research. The breadth and depth of this research activity makes this research project unique.
The Development of Numerical Estimation: Evidence against a Representational Shift
ERIC Educational Resources Information Center
Barth, Hilary C.; Paladino, Annie M.
2011-01-01
How do our mental representations of number change over development? The dominant view holds that children (and adults) possess multiple representations of number, and that age and experience lead to a shift from greater reliance upon logarithmically organized number representations to greater reliance upon more accurate, linear representations.…
Representation is representation of similarities.
Edelman, S
1998-08-01
Advanced perceptual systems are faced with the problem of securing a principled (ideally, veridical) relationship between the world and its internal representation. I propose a unified approach to visual representation, addressing the need for superordinate and basic-level categorization and for the identification of specific instances of familiar categories. According to the proposed theory, a shape is represented internally by the responses of a small number of tuned modules, each broadly selective for some reference shape, whose similarity to the stimulus it measures. This amounts to embedding the stimulus in a low-dimensional proximal shape space spanned by the outputs of the active modules. This shape space supports representations of distal shape similarities that are veridical as Shepard's (1968) second-order isomorphisms (i.e., correspondence between distal and proximal similarities among shapes, rather than between distal shapes and their proximal representations). Representation in terms of similarities to reference shapes supports processing (e.g., discrimination) of shapes that are radically different from the reference ones, without the need for the computationally problematic decomposition into parts required by other theories. Furthermore, a general expression for similarity between two stimuli, based on comparisons to reference shapes, can be used to derive models of perceived similarity ranging from continuous, symmetric, and hierarchical ones, as in multidimensional scaling (Shepard 1980), to discrete and nonhierarchical ones, as in the general contrast models (Shepard & Arabie 1979; Tversky 1977). PMID:10097019
ERIC Educational Resources Information Center
Kuntz, Aaron M.
2010-01-01
What can be known and how to render what we know are perpetual quandaries met by qualitative research, complicated further by the understanding that the everyday discourses influencing our representations are often tacit, unspoken or heard so often that they seem to warrant little reflection. In this article, I offer analytic memos as a means for…
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…
Mathematical Approaches to the Composing Process.
ERIC Educational Resources Information Center
Hall, Dennis R.
Rhetoric and mathematics have much in common that can help explain the composing process. Common elements of rhetoric and mathematics important to the teaching of writing are (1) relationships between syntax and semantics, (2) practices of representation, and (3) focus on problem solving. Recent emphasis on "repair processes" in mathematics is…
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…
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…
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…
Computer aided surface representation
Barnhill, R.E.
1991-04-02
Modern computing resources permit the generation of large amounts of numerical data. These large data sets, if left in numerical form, can be overwhelming. Such large data sets are usually discrete points from some underlying physical phenomenon. Because we need to evaluate the phenomenon at places where we don't have data, a continuous representation (a surface'') is required. A simple example is a weather map obtained from a discrete set of weather stations. (For more examples including multi-dimensional ones, see the article by Dr. Rosemary Chang in the enclosed IRIS Universe). In order to create a scientific structure encompassing the data, we construct an interpolating mathematical surface which can evaluate at arbitrary locations. We can also display and analyze the results via interactive computer graphics. In our research we construct a very wide variety of surfaces for applied geometry problems that have sound theoretical foundations. However, our surfaces have the distinguishing feature that they are constructed to solve short or long term practical problems. This DOE-funded project has developed the premiere research team in the subject of constructing surfaces (3D and higher dimensional) that provide smooth representations of real scientific and engineering information, including state of the art computer graphics visualizations. However, our main contribution is in the development of fundamental constructive mathematical methods and visualization techniques which can be incorporated into a wide variety of applications. This project combines constructive mathematics, algorithms, and computer graphics, all applied to real problems. The project is a unique resource, considered by our peers to be a de facto national center for this type of research.
ERIC Educational Resources Information Center
Luther, Kenneth H.
2012-01-01
Mathematical modeling of groundwater flow is a topic at the intersection of mathematics and geohydrology and is rarely encountered in undergraduate mathematics. However, this subject is full of interesting and meaningful examples of truly "applied" mathematics accessible to undergraduates, from the pre-calculus to advanced mathematics levels. This…
Ferrari, Pier Luigi
2003-07-29
Some current interpretations of abstraction in mathematical settings are examined from different perspectives, including history and learning. It is argued that abstraction is a complex concept and that it cannot be reduced to generalization or decontextualization only. In particular, the links between abstraction processes and the emergence of new objects are shown. The role that representations have in abstraction is discussed, taking into account both the historical and the educational perspectives. As languages play a major role in mathematics, some ideas from functional linguistics are applied to explain to what extent mathematical notations are to be considered abstract. Finally, abstraction is examined from the perspective of mathematics education, to show that the teaching ideas resulting from one-dimensional interpretations of abstraction have proved utterly unsuccessful. PMID:12903658
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)
On Blocks, Stairs, and beyond: Learning about the Significance of Representations
ERIC Educational Resources Information Center
Rubel, Laurie H.; Zolkower, Betina A.
2007-01-01
The National Council of Teachers of Mathematics (2000) recommends that students at all grade levels be provided with instructional programs that enable them to "create and use representations to organize, record, and communicate mathematical ideas; select, apply, and translate among mathematical representations to solve problems; and use…
16 CFR 322.3 - Prohibited representations.
Code of Federal Regulations, 2011 CFR
2011-01-01
... and relies upon competent and reliable evidence that substantiates that the representation is true. For the purposes of this paragraph, “competent and reliable evidence” means tests, analyses, research... accepted in the profession to yield accurate and reliable results....
16 CFR 322.3 - Prohibited representations.
Code of Federal Regulations, 2012 CFR
2012-01-01
... and relies upon competent and reliable evidence that substantiates that the representation is true. For the purposes of this paragraph, “competent and reliable evidence” means tests, analyses, research... accepted in the profession to yield accurate and reliable results....
Value and Limitations of Analogs in Teaching Mathematics.
ERIC Educational Resources Information Center
Halford, Graeme S.; Boulton-Lewis, Gillian M.
Analogical reasoning is frequently used in acquisition of mathematical concepts. Concrete representations used to teach mathematics are essentially analogs of mathematical concepts, and it is argued that analogies enter into mathematical concept acquisition in numerous other ways as well. According to Gentner's theory, analogies entail a…
Recchia-Luciani, Angelo N M
2012-04-01
The present paper proposes a definition for the complex polysemic concepts of consciousness and awareness (in humans as well as in other species), and puts forward the idea of a progressive ontological development of consciousness from a state of 'childhood' awareness, in order to explain that humans are not only able to manipulate objects, but also their mental representations. The paper builds on the idea of qualia intended as entities posing regular invariant requests to neural processes, trough the permanence of different properties. The concept of semantic differential introduces the properties of metaphorical qualia as an exclusively human ability. Furthermore this paper proposes a classification of qualia, according to the models-with different levels of abstraction-they are implied in, in a taxonomic perspective. This, in turn, becomes a source of categorization of divergent representations, sign systems, and forms of intentionality, relying always on biological criteria. New emerging image-of-the-world-devices are proposed, whose qualia are likely to be only accessible to humans: emotional qualia, where emotion accounts for the invariant and dominant property; and the qualic self where continuity, combined with the oneness of the self, accounts for the invariant and dominant property. The concept of congruence between different domains in a metaphor introduces the possibility of a general evaluation of truth and falsity of all kinds of metaphorical constructs, while the work of Matte Blanco enables us to classify conscious versus unconscious metaphors, both in individuals and in social organizations. PMID:22347988
Efficient radiometrically accurate synthetic representation of IR scenes
NASA Astrophysics Data System (ADS)
Shaw, Patrick C.; Gover, Robert E.
2003-08-01
A technique is developed for synthesizing a high spectral resolution IR ship signature image, for use in an imaging IR Anti-Ship Cruise Missile (ASCM) model, from an IR scene database provided by the ship signature model NTCS/ShipIR. This synthesized IR ship image is generated for use over ranges representative of an ASCM engagement. The technique presented focuses on the application of in-band averaged transmittance to the source ship signature as a means of reducing the spectral calculations required by the cruise missile model. In order to achieve this reduction in computation, while preserving the fidelity of the apparent ship signature, the idea of sub-banding is introduced. Sub-banding describes the manner in which the IR band is partitioned into smaller bandwidths, such that the error produced in the ship's average contrast radiance due to the use of in-band averaged transmittance is minimized over range. The difference between the average contrast radiance of an IR ship image generated using in-band averaging and the average contrast radiance of a spectrally generated IR ship image is the metric for this minimization. This choice is based on measured data collected from the recent NATO SIMVEX trial, which used high quality IR measurements of the CFAV Quest in an effort to refine the NTCS/ShipIR model. The technique is general and applicable to any band(s) of interest. Results are presented which verify that the use of in-band averaged transmittance over an IR band (3.5-5.0 μm), partitioned using three optimal sub-bands, produces an IR ship image with an average contrast radiance within the desired error bar of a spectrally generated ship image's average contrast radiance.
Formal representation of 3D structural geological models
NASA Astrophysics Data System (ADS)
Wang, Zhangang; Qu, Honggang; Wu, Zixing; Yang, Hongjun; Du, Qunle
2016-05-01
The development and widespread application of geological modeling methods has increased demands for the integration and sharing services of three dimensional (3D) geological data. However, theoretical research in the field of geological information sciences is limited despite the widespread use of Geographic Information Systems (GIS) in geology. In particular, fundamental research on the formal representations and standardized spatial descriptions of 3D structural models is required. This is necessary for accurate understanding and further applications of geological data in 3D space. In this paper, we propose a formal representation method for 3D structural models using the theory of point set topology, which produces a mathematical definition for the major types of geological objects. The spatial relationships between geologic boundaries, structures, and units are explained in detail using the 9-intersection model. Reasonable conditions for describing the topological space of 3D structural models are also provided. The results from this study can be used as potential support for the standardized representation and spatial quality evaluation of 3D structural models, as well as for specific needs related to model-based management, query, and analysis.
ERIC Educational Resources Information Center
Reys, Robert; Reys, Rustin
2011-01-01
In their dual roles as mathematics teachers and tennis coaches, the authors have worked with tennis players who have never thought about how a knowledge of mathematics might help them become "better" tennis players. They have also worked with many mathematics students who have never considered how much mathematics is associated with tennis. This…
Audio representations of multi-channel EEG: a new tool for diagnosis of brain disorders
Vialatte, François B; Dauwels, Justin; Musha, Toshimitsu; Cichocki, Andrzej
2012-01-01
Objective: The objective of this paper is to develop audio representations of electroencephalographic (EEG) multichannel signals, useful for medical practitioners and neuroscientists. The fundamental question explored in this paper is whether clinically valuable information contained in the EEG, not available from the conventional graphical EEG representation, might become apparent through audio representations. Methods and Materials: Music scores are generated from sparse time-frequency maps of EEG signals. Specifically, EEG signals of patients with mild cognitive impairment (MCI) and (healthy) control subjects are considered. Statistical differences in the audio representations of MCI patients and control subjects are assessed through mathematical complexity indexes as well as a perception test; in the latter, participants try to distinguish between audio sequences from MCI patients and control subjects. Results: Several characteristics of the audio sequences, including sample entropy, number of notes, and synchrony, are significantly different in MCI patients and control subjects (Mann-Whitney p < 0.01). Moreover, the participants of the perception test were able to accurately classify the audio sequences (89% correctly classified). Conclusions: The proposed audio representation of multi-channel EEG signals helps to understand the complex structure of EEG. Promising results were obtained on a clinical EEG data set. PMID:23383399
ERIC Educational Resources Information Center
Jones, Thomas A.
1983-01-01
Mathematical techniques used to solve geological problems are briefly discussed (including comments on use of geostatistics). Highlights of conferences/meetings and conference papers in mathematical geology are also provided. (JN)
ERIC Educational Resources Information Center
Hanh, Vu Duc, Ed.
This document gives a listing of mathematical terminology in both the English and Vietnamese languages. Vocabulary used in algebra and geometry is included along with a translation of mathematical symbols. (DT)
... this page: //medlineplus.gov/ency/article/001534.htm Mathematics disorder To use the sharing features on this page, please enable JavaScript. Mathematics disorder is a condition in which a child's ...
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…
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…
ERIC Educational Resources Information Center
Kilpatrick, Jeremy
2014-01-01
This paper addresses the contested way that ethnomathematics has sometimes been received by mathematicians and others and what that disagreement might suggest about issues in mathematics education; namely, (a) the relation of ethnomathematics to academic mathematics; (b) recent efforts to reform secondary school mathematics so that it prepares…
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…
A Reflective Protocol for Mathematics Learning Environments
ERIC Educational Resources Information Center
Kinzer, Cathy Jeanne; Virag, Lisa; Morales, Sara
2011-01-01
How can a teacher use the practice of reflection to create rich mathematical learning environments that are engaging to students? In such environments, one can hear and see a seamless integration of Problem Solving, Reasoning and Proof, Communication, making mathematical Connections, and Representation (the NCTM Process Standards) through Number…
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…
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. PMID:26263425
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…
Structural stability augmentation system design using BODEDIRECT: A quick and accurate approach
NASA Technical Reports Server (NTRS)
Goslin, T. J.; Ho, J. K.
1989-01-01
A methodology is presented for a modal suppression control law design using flight test data instead of mathematical models to obtain the required gain and phase information about the flexible airplane. This approach is referred to as BODEDIRECT. The purpose of the BODEDIRECT program is to provide a method of analyzing the modal phase relationships measured directly from the airplane. These measurements can be achieved with a frequency sweep at the control surface input while measuring the outputs of interest. The measured Bode-models can be used directly for analysis in the frequency domain, and for control law design. Besides providing a more accurate representation for the system inputs and outputs of interest, this method is quick and relatively inexpensive. To date, the BODEDIRECT program has been tested and verified for computational integrity. Its capabilities include calculation of series, parallel and loop closure connections between Bode-model representations. System PSD, together with gain and phase margins of stability may be calculated for successive loop closures of multi-input/multi-output systems. Current plans include extensive flight testing to obtain a Bode-model representation of a commercial aircraft for design of a structural stability augmentation system.
Mathematical Modeling and Pure Mathematics
ERIC Educational Resources Information Center
Usiskin, Zalman
2015-01-01
Common situations, like planning air travel, can become grist for mathematical modeling and can promote the mathematical ideas of variables, formulas, algebraic expressions, functions, and statistics. The purpose of this article is to illustrate how the mathematical modeling that is present in everyday situations can be naturally embedded in…
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…
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…
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…
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…
Developing Students' Representational Fluency Using Virtual and Physical Algebra Balances
ERIC Educational Resources Information Center
Suh, Jennifer; Moyer, Patricia S.
2007-01-01
Both virtual and physical manipulatives are reported as effective learning tools when used with different groups of students in a variety of contexts to learn mathematical content. The use of multiple representations and the flexibility to translate among those representational forms facilitates students' learning and has the potential to deepen…
SNARC Hunting: Examining Number Representation in Deaf Students
ERIC Educational Resources Information Center
Bull, R.; Marschark, M.; Blatto-Vallee, G.
2005-01-01
Many deaf children and adults show lags in mathematical abilities. The current study examines the basic number representations that allow individuals to perform higher-level arithmetical procedures. These representations are normally present in the earliest stages of development, but they may be affected by cultural, developmental, and educational…
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…
NASA Astrophysics Data System (ADS)
Stöltzner, Michael
Answering to the double-faced influence of string theory on mathematical practice and rigour, the mathematical physicists Arthur Jaffe and Frank Quinn have contemplated the idea that there exists a `theoretical' mathematics (alongside `theoretical' physics) whose basic structures and results still require independent corroboration by mathematical proof. In this paper, I shall take the Jaffe-Quinn debate mainly as a problem of mathematical ontology and analyse it against the backdrop of two philosophical views that are appreciative towards informal mathematical development and conjectural results: Lakatos's methodology of proofs and refutations and John von Neumann's opportunistic reading of Hilbert's axiomatic method. The comparison of both approaches shows that mitigating Lakatos's falsificationism makes his insights about mathematical quasi-ontology more relevant to 20th century mathematics in which new structures are introduced by axiomatisation and not necessarily motivated by informal ancestors. The final section discusses the consequences of string theorists' claim to finality for the theory's mathematical make-up. I argue that ontological reductionism as advocated by particle physicists and the quest for mathematically deeper axioms do not necessarily lead to identical results.
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…
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
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.
Experimental Mathematics and Mathematical Physics
Bailey, David H.; Borwein, Jonathan M.; Broadhurst, David; Zudilin, Wadim
2009-06-26
One of the most effective techniques of experimental mathematics is to compute mathematical entities such as integrals, series or limits to high precision, then attempt to recognize the resulting numerical values. Recently these techniques have been applied with great success to problems in mathematical physics. Notable among these applications are the identification of some key multi-dimensional integrals that arise in Ising theory, quantum field theory and in magnetic spin theory.
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.
ERIC Educational Resources Information Center
Langbort, Carol, Ed.; Curtis, Deborah, Ed.
2000-01-01
The focus of this special issue is mathematics education. All articles were written by graduates of the new masters Degree program in which students earn a Master of Arts degree in Education with a concentration in Mathematics Education at San Francisco State University. Articles include: (1) "Developing Teacher-Leaders in a Masters Degree Program…
ERIC Educational Resources Information Center
Flannery, Carol A.
This manuscript provides information and problems for teaching mathematics to vocational education students. Problems reflect applications of mathematical concepts to specific technical areas. The materials are organized into six chapters. Chapter 1 covers basic arithmetic, including fractions, decimals, ratio and proportions, percentages, and…
ERIC Educational Resources Information Center
Siskiyou County Superintendent of Schools, Yreka, CA.
The purpose of this project was to raise the mathematics skills of 100 mathematically retarded students in grades one through eight by one year through the development of an inservice strategy prepared by four teacher specialists. Also used in the study was a control group of 100 students chosen from the median range of stanines on pretest scores…
ERIC Educational Resources Information Center
Prochazka, Helen
2004-01-01
One section of this "scrapbook" section describes Pythagoras' belief in the connections between music and mathematics -- that everything in the universe was a series of harmonies and regulated by music. Another section explains why Phythagoras felt it important for women to be encouraged to learn mathematics. At least 28 women were involved in his…
Influence of TFT-LCD pixel structure on holographic representation
NASA Astrophysics Data System (ADS)
Wang, Hongjun; Wang, Zhao; Tian, Ailing; Liu, Bingcai
2008-09-01
As a new holographic display device, TFT-LCD (Thin Film Transistor Liquid Crystal Displays) is key technical component of holographic representation for easy controlled by computer. With the development of exquisite processing technology, that it instead of the traditional holographic plate become historical necessity and would be the development direction of holographic optics. Based on principles of holography and display character of LCD, the property which the LCD was used as a holographic plate was analyzed. The emphasis on discuss influence of LCD black matrix on holographic representation. First, analyzed on LCD pixel structure, the LCD pixel structure mathematical model was established. LCD was character representation by pixel structure parameters. Then, the influence of LCD pixels structure on holographic representation was analyzed by computer simulation. Meanwhile, the SONY LCX023 was chosen for holographic plate, the He-Ne laser which the wavelength is 0.6328um was holographic representation light source. The holographic representation system was established for test influence of LCD on holographic representation. Final, compared between computer simulations and optical experimental results, the mathematical model of LCD was proved to be true. When aperture ratio is 0.625, the holographic representation wouldn't be distinguished between representation images. At the same time, some useful results was acquired for improve application effects of LCD in holographic representation.
Spatially variant morphological restoration and skeleton representation.
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. PMID:17076415
ERIC Educational Resources Information Center
Rom, Mark Carl
2011-01-01
Grades matter. College grading systems, however, are often ad hoc and prone to mistakes. This essay focuses on one factor that contributes to high-quality grading systems: grading accuracy (or "efficiency"). I proceed in several steps. First, I discuss the elements of "efficient" (i.e., accurate) grading. Next, I present analytical results…
Ephemeris representations for communications satellites
NASA Astrophysics Data System (ADS)
Proulx, R. J.; Cefola, P. J.; McClain, W. D.
1984-08-01
Large orbit determination (OD) centers are the primary source of artificial satellite ephemeris data. The ephemeris message of the OD facility contains implicitly the predicted satellite trajectory. The user can recover ephemeris data on the basis of two conceptual approaches. The current investigation is concerned with an alternative solution to the ephemeris representation problem. According to the procedure employed in this case, the mean equinoctial element time histories corresponding to the predicted satellite trajectory generated by the OD facility are approximated by low degree Legendre polynomials to represent their secular behavior and by trigonometric terms to represent their mean periodic behavior. This approach provides a simple, low cost, and accurate ephemeris representation, which satisfies the potential autonomy requirements for Military Satellite Communications.
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…
ERIC Educational Resources Information Center
McCammon, Richard B.
1979-01-01
The year 1978 marked a continued trend toward practical applications in mathematical geology. Developments included work in interactive computer graphics, factor analysis, the vanishing tons problem, universal kriging, and resource estimating. (BB)
The child may have problems in school, including behavior problems and loss of self-esteem. Some children with mathematics disorder become anxious or afraid when given math problems, making the problem even worse.
ERIC Educational Resources Information Center
Johnson, Jerry
1997-01-01
Presents 12 questions related to a given real-life situation about a man shaving and the number of hairs in his beard in order to help students see the connection between mathematics and the world around them. (ASK)
ERIC Educational Resources Information Center
Gardner, Martin
1978-01-01
Describes the life and work of Charles Peirce, U.S. mathematician and philosopher. His accomplishments include contributions to logic, the foundations of mathematics and scientific method, and decision theory and probability theory. (MA)
Accurate monotone cubic interpolation
NASA Technical Reports Server (NTRS)
Huynh, Hung T.
1991-01-01
Monotone piecewise cubic interpolants are simple and effective. They are generally third-order accurate, except near strict local extrema where accuracy degenerates to second-order due to the monotonicity constraint. Algorithms for piecewise cubic interpolants, which preserve monotonicity as well as uniform third and fourth-order accuracy are presented. The gain of accuracy is obtained by relaxing the monotonicity constraint in a geometric framework in which the median function plays a crucial role.
Accurate Finite Difference Algorithms
NASA Technical Reports Server (NTRS)
Goodrich, John W.
1996-01-01
Two families of finite difference algorithms for computational aeroacoustics are presented and compared. All of the algorithms are single step explicit methods, they have the same order of accuracy in both space and time, with examples up to eleventh order, and they have multidimensional extensions. One of the algorithm families has spectral like high resolution. Propagation with high order and high resolution algorithms can produce accurate results after O(10(exp 6)) periods of propagation with eight grid points per wavelength.
Computer aided surface representation
Barnhill, R.E.
1989-02-09
The central research problem of this project is the effective representation and display of surfaces, interpolating to given information, in three or more dimensions. In a typical problem, we wish to create a surface from some discrete information. If this information is itself on another surface, the problem is to determine a surface defined on a surface,'' which is discussed below. Often, properties of an already constructed surface are desired: such geometry processing'' is described below. The Summary of Proposed Research from our original proposal describes the aims of this research project. This Summary and the Table of Contents from the original proposal are enclosed as an Appendix to this Progress Report. The broad sweep from constructive mathematics through algorithms and computer graphics displays is utilized in the research. The wide range of activity, directed in both theory and applications, makes this project unique. Last month in the first Ardent Titan delivered in the State of Arizona came to our group, funded by the DOE and Arizona State University. Although the Titan is a commercial product, its newness requires our close collaboration with Ardent to maximize results. During the past year, four faculty members and several graduate research assistants have worked on this DOE project. The gaining of new professionals is an important aspect of this project. A listing of the students and their topics is given in the Appendix. The most significant publication during the past year is the book, Curves and Surfaces for Computer Aided Geometric Design, by Dr. Gerald Farin. This 300 page volume helps fill a considerable gap in the subject and includes many new results on Bernstein-Bezier curves and surfaces.
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.
Malkevitch, J. ); McCarthy, D. )
1990-01-01
The papers in this volume represent talks given at the monthly meetings of the Mathematics Section of the New York Academy of Sciences. They reflect the operating philosophy of the Section in its efforts to make a meaningful contribution to the mathematical life of a community that is exceedingly rich in cultural resources and intellectual opportunities. Each week during the academic year a dazzling abundance of mathematical seminars and colloquia is available in the New York metropolitan area. Most of these offer highly technical treatments intended for specialists. At the New York Academy we try to provide a forum of a different sort, where interesting ideas are presented in a congenial atmosphere to a broad mathematical audience. Many of the Section talks contain substantial specialized material, but we ask our speakers to include a strong expository component aimed at working mathematicians presumed to have no expert knowledge of the topic at hand. We urge speakers to try to provide the motivating interest they themselves would like to find in an introduction to a field other than their own. The same advice has been given to the authors of the present papers, with the goal of producing a collection that will be both accessible and stimulating to mathematical minds at large. We have tried to provide variety in the mathematical vistas offered; both pure and applied mathematics are well represented. Since the papers are presented alphabetically by author, some guidance seems appropriate as to what sorts of topics are treated, and where. There are three papers in analysis: those by Engber, Narici and Beckenstein, and Todd. Engber's deals with complex analysis on compact Riemann surfaces; Narici and Beckenstein provide an introduction to analysis on non-Archimendean fields; Todd surveys an area of contemporary functional analysis.
Glimm, J.
2009-10-14
Progress for the past decade or so has been extraordinary. The solution of Fermat's Last Theorem [11] and of the Poincare Conjecture [1] have resolved two of the most outstanding challenges to mathematics. For both cases, deep and advanced theories and whole subfields of mathematics came into play and were developed further as part of the solutions. And still the future is wide open. Six of the original seven problems from the Clay Foundation challenge remain open, the 23 DARPA challenge problems are open. Entire new branches of mathematics have been developed, including financial mathematics and the connection between geometry and string theory, proposed to solve the problems of quantized gravity. New solutions of the Einstein equations, inspired by shock wave theory, suggest a cosmology model which fits accelerating expansion of the universe possibly eliminating assumptions of 'dark matter'. Intellectual challenges and opportunities for mathematics are greater than ever. The role of mathematics in society continues to grow; with this growth comes new opportunities and some growing pains; each will be analyzed here. We see a broadening of the intellectual and professional opportunities and responsibilities for mathematicians. These trends are also occuring across all of science. The response can be at the level of the professional societies, which can work to deepen their interactions, not only within the mathematical sciences, but also with other scientific societies. At a deeper level, the choices to be made will come from individual mathematicians. Here, of course, the individual choices will be varied, and we argue for respect and support for this diversity of responses. In such a manner, we hope to preserve the best of the present while welcoming the best of the new.
Reevaluating the two-representation model of numerical magnitude processing.
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. PMID:26268066
Enhancing Mathematical Communication: "Bag of Tricks" Game
ERIC Educational Resources Information Center
Patahuddin, Sitti Maesuri; Ramful, Ajay; Greenlees, Jane
2015-01-01
An engaging activity which prompts students to listen, talk, reason and write about geometrical properties. The "Bag of Tricks" encourages students to clarify their thoughts and communicate precisely using accurate mathematical language.
Quantum measurement in coherence-vector representation
NASA Astrophysics Data System (ADS)
Zhou, Tao
2016-04-01
We consider the quantum measurements on a finite quantum system in coherence-vector representation. In this representation, all the density operators of an N-level ( N ⩾ 2) quantum system constitute a convex set M (N) embedded in an ( N 2 - 1)-dimensional Euclidean space R^{N^2 - 1}, and we find that an orthogonal measurement is an ( N - 1)-dimensional projector operator on R^{N^2 - 1}. The states unchanged by an orthogonal measurement form an ( N - 1)-dimensional simplex, and in the case when N is prime or power of prime, the space of the density operator is a direct sum of ( N + 1) such simplices. The mathematical description of quantum measurement is plain in this representation, and this may have further applications in quantum information processing.
Compact internal representation as a protocognitive scheme for robots in dynamic environments
NASA Astrophysics Data System (ADS)
Villacorta-Atienza, Jose A.; Salas, Luis; Alba, Luis; Velarde, Manuel G.; Makarov, Valeri A.
2011-05-01
Animals for surviving have developed cognitive abilities allowing them an abstract representation of the environment. This Internal Representation (IR) could contain a huge amount of information concerning the evolution and interactions of the elements in their surroundings. The complexity of this information should be enough to ensure the maximum fidelity in the representation of those aspects of the environment critical for the agent, but not so high to prevent the management of the IR in terms of neural processes, i.e. storing, retrieving, etc. One of the most subtle points is the inclusion of temporal information, necessary in IRs of dynamic environments. This temporal information basically introduces the environmental information for each moment, so the information required to generate the IR would eventually be increased dramatically. The inclusion of this temporal information in biological neural processes remains an open question. In this work we propose a new IR, the Compact Internal Representation (CIR), based on the compaction of spatiotemporal information into only space, leading to a stable structure (with no temporal dimension) suitable to be the base for complex cognitive processes, as memory or learning. The Compact Internal Representation is especially appropriate for be implemented in autonomous robots because it provides global strategies for the interaction with real environments (roving robots, manipulators, etc.). This paper presents the mathematical basis of CIR hardware implementation in the context of navigation in dynamic environments. The aim of such implementation is the obtaining of free-collision trajectories under the requirements of an optimal performance by means of a fast and accurate process.
Children's mapping between symbolic and nonsymbolic representations of number.
Mundy, Eleanor; Gilmore, Camilla K
2009-08-01
When children learn to count and acquire a symbolic system for representing numbers, they map these symbols onto a preexisting system involving approximate nonsymbolic representations of quantity. Little is known about this mapping process, how it develops, and its role in the performance of formal mathematics. Using a novel task to assess children's mapping ability, we show that children can map in both directions between symbolic and nonsymbolic numerical representations and that this ability develops between 6 and 8 years of age. Moreover, we reveal that children's mapping ability is related to their achievement on tests of school mathematics over and above the variance accounted for by standard symbolic and nonsymbolic numerical tasks. These findings support the proposal that underlying nonsymbolic representations play a role in children's mathematical development. PMID:19327782
NASA Astrophysics Data System (ADS)
Itano, Wayne M.; Ramsey, Norman F.
1993-07-01
The paper discusses current methods for accurate measurements of time by conventional atomic clocks, with particular attention given to the principles of operation of atomic-beam frequency standards, atomic hydrogen masers, and atomic fountain and to the potential use of strings of trapped mercury ions as a time device more stable than conventional atomic clocks. The areas of application of the ultraprecise and ultrastable time-measuring devices that tax the capacity of modern atomic clocks include radio astronomy and tests of relativity. The paper also discusses practical applications of ultraprecise clocks, such as navigation of space vehicles and pinpointing the exact position of ships and other objects on earth using the GPS.
Accurate quantum chemical calculations
NASA Technical Reports Server (NTRS)
Bauschlicher, Charles W., Jr.; Langhoff, Stephen R.; Taylor, Peter R.
1989-01-01
An important goal of quantum chemical calculations is to provide an understanding of chemical bonding and molecular electronic structure. A second goal, the prediction of energy differences to chemical accuracy, has been much harder to attain. First, the computational resources required to achieve such accuracy are very large, and second, it is not straightforward to demonstrate that an apparently accurate result, in terms of agreement with experiment, does not result from a cancellation of errors. Recent advances in electronic structure methodology, coupled with the power of vector supercomputers, have made it possible to solve a number of electronic structure problems exactly using the full configuration interaction (FCI) method within a subspace of the complete Hilbert space. These exact results can be used to benchmark approximate techniques that are applicable to a wider range of chemical and physical problems. The methodology of many-electron quantum chemistry is reviewed. Methods are considered in detail for performing FCI calculations. The application of FCI methods to several three-electron problems in molecular physics are discussed. A number of benchmark applications of FCI wave functions are described. Atomic basis sets and the development of improved methods for handling very large basis sets are discussed: these are then applied to a number of chemical and spectroscopic problems; to transition metals; and to problems involving potential energy surfaces. Although the experiences described give considerable grounds for optimism about the general ability to perform accurate calculations, there are several problems that have proved less tractable, at least with current computer resources, and these and possible solutions are discussed.
R. KELSEY
2001-02-01
For focused applications with limited user and use application communities, XML can be the right choice for representation. It is easy to use, maintain, and extend and enjoys wide support in commercial and research sectors. When the knowledge and information to be represented is object-based and use of that knowledge and information is a high priority, then XML-based representation should be considered. This paper discusses some of the issues involved in using XML-based representation and presents an example application that successfully uses an XML-based representation.
ERIC Educational Resources Information Center
Rogness, Jonathan
2011-01-01
Advances in computer graphics have provided mathematicians with the ability to create stunning visualizations, both to gain insight and to help demonstrate the beauty of mathematics to others. As educators these tools can be particularly important as we search for ways to work with students raised with constant visual stimulation, from video games…
ERIC Educational Resources Information Center
Lapointe, Archie E.; And Others
In 1990-91, 20 countries (Brazil, Canada, China, England, France, Hungary, Ireland, Israel, Italy, Jordan, Korea, Mozambique, Portugal, Scotland, Slovenia, Soviet Union, Spain, Switzerland, Taiwan, and the United States) surveyed the mathematics and science performance of 13-year-old students (and 14 countries also assessed 9-year-olds in the same…
ERIC Educational Resources Information Center
Hadlock, Charles R
2013-01-01
The movement of groundwater in underground aquifers is an ideal physical example of many important themes in mathematical modeling, ranging from general principles (like Occam's Razor) to specific techniques (such as geometry, linear equations, and the calculus). This article gives a self-contained introduction to groundwater modeling with…
ERIC Educational Resources Information Center
Catterton, Gene; And Others
This material was developed to be used with the non college-bound student in the senior high school. It provides the student with everyday problems and experiences in which practical mathematical applications are made. The package includes worksheets pertaining to letterhead invoices, sales slips, payroll sheets, inventory sheets, carpentry and…
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 results in the physics of…
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…
A representation of Jacchia's thermospheric models in spherical harmonics
NASA Technical Reports Server (NTRS)
Blum, P.; Harris, I.
1973-01-01
The Jacchia models are represented in terms of spherical harmonic functions. This representation has the advantages of ease of comparison with theoretical and other observational models and data, mathematical analyticity and relative simplicity. The symmetry properties of the models are emphasized by this representation and some physical characteristics like the increase of the amplitude of the diurnal density variation with decreasing solar activity become more apparent.
Graphical representation of the process of solving problems in statics
NASA Astrophysics Data System (ADS)
Lopez, Carlos
2011-03-01
It is presented a method of construction to a graphical representation technique of knowledge called Conceptual Chains. Especially, this tool has been focused to the representation of processes and applied to solving problems in physics, mathematics and engineering. The method is described in ten steps and is illustrated with its development in a particular topic of statics. Various possible didactic applications of this technique are showed.
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
New model accurately predicts reformate composition
Ancheyta-Juarez, J.; Aguilar-Rodriguez, E. )
1994-01-31
Although naphtha reforming is a well-known process, the evolution of catalyst formulation, as well as new trends in gasoline specifications, have led to rapid evolution of the process, including: reactor design, regeneration mode, and operating conditions. Mathematical modeling of the reforming process is an increasingly important tool. It is fundamental to the proper design of new reactors and revamp of existing ones. Modeling can be used to optimize operating conditions, analyze the effects of process variables, and enhance unit performance. Instituto Mexicano del Petroleo has developed a model of the catalytic reforming process that accurately predicts reformate composition at the higher-severity conditions at which new reformers are being designed. The new AA model is more accurate than previous proposals because it takes into account the effects of temperature and pressure on the rate constants of each chemical reaction.
The interaction of representation and reasoning
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
Mathematics Curriculum Guide. Mathematics IV.
ERIC Educational Resources Information Center
Gary City Public School System, IN.
GRADES OR AGES: Grade 12. SUBJECT MATTER: Mathematics. ORGANIZATION AND PHYSICAL APPEARANCE: The subject matter is presented in four columns: major areas, significant outcomes, observations and suggestions, and films and references. The topics include: sets-relations-functions, circular functions, graphs of circular functions, inverses of circular…
For a Learnable Mathematics in the Digital Culture.
ERIC Educational Resources Information Center
Noss, Richard
2001-01-01
Discusses the changed roles of mathematics and novel representations that emerge from the ubiquity of computational models. Considers the implications for learning mathematics. Contends that a central component of knowledge required in modern societies involves the development of a meta-epistemological stance. Maps out implications for the design…
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…
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…
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…
Integrating Concrete and Virtual Manipulatives in Early Childhood Mathematics
ERIC Educational Resources Information Center
Rosen, Dina; Hoffman, Jo
2009-01-01
Early childhood teachers around the country and the world guide children's mathematical learning through the use of manipulatives--pattern blocks, base blocks, geoboards, Unifx cubes, Cuisenaire rods, coins, clocks, and so on. Manipulatives allow concrete, hands-on exploration and representation of mathematical concepts. In the past few years,…
Poverty: Teaching Mathematics and Social Justice
ERIC Educational Resources Information Center
McCoy, Leah P.
2008-01-01
This article presents three mathematics lessons in a social justice setting of learning about poverty. Student activities include budgeting, graphic data representation, and linear regression, all in the context of connecting, communicating, and reasoning about poverty. (Contains 1 table, 5 figures and 6 online resources.)
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…
Negotiating the Boundaries between Mathematics and Physics
ERIC Educational Resources Information Center
Radtka, Catherine
2015-01-01
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…
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.
Mathematical models in biology: from molecules to life
Kaznessis, Yiannis N.
2011-01-01
A vexing question in the biological sciences is the following: can biological phenotypes be explained with mathematical models of molecules that interact according to physical laws? At the crux of the matter lies the doubt that humans can develop physically faithful mathematical representations of living organisms. We discuss advantages that synthetic biological systems confer that may help us describe life’s distinctiveness with tractable mathematics that are grounded on universal laws of thermodynamics and molecular biology. PMID:21472998
Facilitating Students' Problem Solving across Multiple Representations in Introductory Mechanics
NASA Astrophysics Data System (ADS)
Nguyen, Dong-Hai; Gire, Elizabeth; Rebello, N. Sanjay
2010-10-01
Solving problems presented in multiple representations is an important skill for future physicists and engineers. However, such a task is not easy for most students taking introductory physics courses. We conducted teaching/learning interviews with 20 students in a first-semester calculus-based physics course on several topics in introductory mechanics. These interviews helped identify the common difficulties students encountered when solving physics problems posed in multiple representations as well as the hints that help students overcome those difficulties. We found that most representational difficulties arise due to the lack of students' ability to associate physics knowledge with corresponding mathematical knowledge. Based on those findings, we developed, tested and refined a set of problem-solving exercises to help students learn to solve problems in graphical and equational representations. We present our findings on students' common difficulties with graphical and equational representations, the problem-solving exercises and their impact on students' problem solving abilities.
[Mathematical models of hysteresis
Mayergoyz, I.D.
1991-01-01
The research described in this proposal is currently being supported by the US Department of Energy under the contract Mathematical Models of Hysteresis''. Thus, before discussing the proposed research in detail, it is worthwhile to describe and summarize the main results achieved in the course of our work under the above contract. Our ongoing research has largely been focused on the development of mathematical models of hysteretic nonlinearities with nonlocal memories''. The distinct feature of these nonlinearities is that their current states depend on past histories of input variations. It turns out that memories of hysteretic nonlinearities are quite selective. Indeed, experiments show that only some past input extrema leave their marks upon future states of hysteretic nonlinearities. Thus special mathematical tools are needed in order to describe nonlocal selective memories of hysteretic nonlinearities. Our research has been primarily concerned with Preisach-type models of hysteresis. All these models have a common generic feature; they are constructed as superpositions of simplest hysteretic nonlinearities-rectangular loops. Our study has by and large been centered around the following topics: various generalizations and extensions of the classical Preisach model, finding of necessary and sufficient conditions for the representation of actual hysteretic nonlinearities by various Preisach type models, solution of identification problems for these models, numerical implementation and experimental testing of Preisach type models. Although the study of Preisach type models has constituted the main direction of the research, some effort has also been made to establish some interesting connections between these models and such topics as: the critical state model for superconducting hysteresis, the classical Stoner-Wohlfarth model of vector magnetic hysteresis, thermal activation type models for viscosity, magnetostrictive hysteresis and neural networks.
Three essays in mathematical finance
NASA Astrophysics Data System (ADS)
Wang, Ruming
This dissertation uses mathematical techniques to solve three problems in mathematical finance. The first two problems are on model-independent pricing and hedging of financial derivatives. We use asymptotic expansions to express derivative prices and implied volatilities. Then just by using the first few terms in the expansions, we get simple and accurate formulas, which can also help us find connections between different products. The last problem is on optimal trading strategies in a limit order book. Under a very general setup, we solve explicitly for a dynamic decision problem involving choosing between limit order and market order.
Mathematical Models of Gene Regulation
NASA Astrophysics Data System (ADS)
Mackey, Michael C.
2004-03-01
This talk will focus on examples of mathematical models for the regulation of repressible operons (e.g. the tryptophan operon), inducible operons (e.g. the lactose operon), and the lysis/lysogeny switch in phage λ. These ``simple" gene regulatory elements can display characteristics experimentally of rapid response to perturbations and bistability, and biologically accurate mathematical models capture these aspects of the dynamics. The models, if realistic, are always nonlinear and contain significant time delays due to transcriptional and translational delays that pose substantial problems for the analysis of the possible ranges of dynamics.
Quaternionic representation of the genetic code.
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. PMID:26751396
Preschoolers' Nonsymbolic Arithmetic with Large Sets: Is Addition More Accurate than Subtraction?
ERIC Educational Resources Information Center
Shinskey, Jeanne L.; Chan, Cindy Ho-man; Coleman, Rhea; Moxom, Lauren; Yamamoto, Eri
2009-01-01
Adult and developing humans share with other animals analog magnitude representations of number that support nonsymbolic arithmetic with large sets. This experiment tested the hypothesis that such representations may be more accurate for addition than for subtraction in children as young as 3 1/2 years of age. In these tasks, the experimenter hid…
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
Teaching Mathematical Modeling in Mathematics Education
ERIC Educational Resources Information Center
Saxena, Ritu; Shrivastava, Keerty; Bhardwaj, Ramakant
2016-01-01
Mathematics is not only a subject but it is also a language consisting of many different symbols and relations. Taught as a compulsory subject up the 10th class, students are then able to choose whether or not to study mathematics as a main subject. The present paper discusses mathematical modeling in mathematics education. The article provides…
Acuity of mental representations of pitch.
Janata, Petr
2012-04-01
Singing in one's mind or forming expectations about upcoming notes both require that mental images of one or more pitches will be generated. As with other musical abilities, the acuity with which such images are formed might be expected to vary across individuals and may depend on musical training. Results from several behavioral tasks involving intonation judgments indicate that multiple memory systems contribute to the formation of accurate mental images for pitch, and that the functionality of each is affected by musical training. Electrophysiological measures indicate that the ability to form accurate mental images is associated with greater engagement of auditory areas and associated error-detection circuitry when listeners imagine ascending scales and make intonation judgments about target notes. A view of auditory mental images is espoused in which unified mental image representations are distributed across multiple brain areas. Each brain area helps shape the acuity of the unified representation based on current behavioral demands and past experience. PMID:22524362
Edwards, A W F
2011-03-01
Ernst Mayr called the first part of the evolutionary synthesis the 'Fisherian synthesis' on account of the dominant role played by R.A. Fisher in forging a mathematical theory of natural selection together with J.B.S. Haldane and Sewall Wright in the decade 1922-1932. It is here argued that Fisher's contribution relied on a close reading of Darwin's work to a much greater extent than did the contributions of Haldane and Wright, that it was synthetic in contrast to their analytic approach and that it was greatly influenced by his friendship with the Darwin family, particularly with Charles's son Leonard. PMID:21423339
Accurate calculation of diffraction-limited encircled and ensquared energy.
Andersen, Torben B
2015-09-01
Mathematical properties of the encircled and ensquared energy functions for the diffraction-limited point-spread function (PSF) are presented. These include power series and a set of linear differential equations that facilitate the accurate calculation of these functions. Asymptotic expressions are derived that provide very accurate estimates for the relative amount of energy in the diffraction PSF that fall outside a square or rectangular large detector. Tables with accurate values of the encircled and ensquared energy functions are also presented. PMID:26368873
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…
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…
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…
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…
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…
Mental Arithmetic Activates Analogic Representations of Internally Generated Sums
ERIC Educational Resources Information Center
Kallai, Arava Y.; Schunn, Christian D.; Fiez, Julie A.
2012-01-01
The internal representation of numbers generated during calculation has received little attention. Much of the mathematics learning literature focuses on symbolic retrieval of math facts; in contrast, we critically test the hypothesis that internally generated numbers are represented analogically, using an approximate number system. In an fMRI…
Constructing Mental Representations of Complex Three-Dimensional Objects.
ERIC Educational Resources Information Center
Aust, Ronald
This exploratory study investigated whether there are differences between males and females in the strategies used to construct mental representations from three-dimensional objects in a dimensional travel display. A Silicon Graphics IRIS computer was used to create the travel displays and mathematical models were created for each of the objects…
New Representational Infrastructures: Broadening the Focus on Functions
ERIC Educational Resources Information Center
Lagrange, Jean-Baptiste
2014-01-01
For more than 10 years, I had the honour and pleasure to work with Celia Hoyles and Richard Noss. We share a common concern for more learnable mathematics, especially in algebra, and for the need to build new representational infrastructures taking advantage of technology. Beyond this common concern, my choice to work in the French institutional…
Use of Multiple Representations in Technology Rich Environments
ERIC Educational Resources Information Center
Akkoç, Hatice; Ozmantar, Mehmet Fatih
2013-01-01
This study presents part of a research project that aims to develop prospective mathematics teachers' Technological Pedagogical Content Knowledge (TPCK). The project considers various TPCK components. This report focuses on a particular component, namely the "knowledge of using multiple representations (MRs) with technology". A course…
Using Multiple Representations To Improve Conceptions of Average Speed.
ERIC Educational Resources Information Center
Reed, Stephen K.; Jazo, Linda
2002-01-01
Discusses improving mathematical reasoning through the design of computer microworlds and evaluates a computer-based learning environment that uses multiple representations to improve undergraduate students' conception of average speed. Describes improvement of students' estimates of average speed by using visual feedback from a simulation.…
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…
Sense-able Combinatorics: Student's Use of Personal Representations
ERIC Educational Resources Information Center
Tarlow, Lynn D.
2008-01-01
This article describes a mathematics session in which students explored a challenging combinatorics task, a topic shown to be difficult for students when taught traditionally. The students developed a progression of personal representations, increasingly symbolic and abstract, which they used to find a solution and justification. (Contains 8…