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.
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…
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…
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…
Social Representations as Mediators of Mathematics Learning in Multiethnic Classrooms
ERIC Educational Resources Information Center
Gorgorio, Nuria; Planas, Nuria
2005-01-01
Drawing on socio-cultural theory, we understand the norms regulating the practices within the mathematics classroom as resulting from the social representations of the socially dominant groups and of the school culture related to what constitutes learning mathematics. Immigrant students, having their own personal histories as members of particular…
The cortical representation of simple mathematical expressions.
Maruyama, Masaki; Pallier, Christophe; Jobert, Antoinette; Sigman, Mariano; Dehaene, Stanislas
2012-07-16
Written mathematical notation conveys, in a compact visual form, the nested functional relations among abstract concepts such as operators, numbers or sets. Is the comprehension of mathematical expressions derived from the human capacity for processing the recursive structure of language? Or does algebraic processing rely only on a language-independent network, jointly involving the visual system for parsing the string of mathematical symbols and the intraparietal system for representing numbers and operators? We tested these competing hypotheses by scanning mathematically trained adults while they viewed simple strings ranging from randomly arranged characters to mathematical expressions with up to three levels of nested parentheses. Syntactic effects were observed in behavior and in brain activation measured with functional magnetic resonance imaging (fMRI) and magneto-encephalography (MEG). Bilateral occipito-temporal cortices and right parietal and precentral cortices appeared as the primary nodes for mathematical syntax. MEG estimated that a mathematical expression could be parsed by posterior visual regions in less than 180 ms. Nevertheless, a small increase in activation with increasing expression complexity was observed in linguistic regions of interest, including the left inferior frontal gyrus and the posterior superior temporal sulcus. We suggest that mathematical syntax, although arising historically from language competence, becomes "compiled" into visuo-spatial areas in well-trained mathematics students.
[Representation and mathematical analysis of human corneal surface].
Tălu, Stefan; Tălu, Mihai; Giovanzana, Stefano
2011-01-01
In the description and analysis of human corneal surface are used various mathematical models based on parametric representations, used in biomechanical studies and 3D solid modeling of the cornea. Mathematical models are important into the biomechanics of the cornea to model the corneal behavior. Corneal biomechanics also has the potential to improve outcomes in refractive surgery. The objective of this paper is to present the most representative mathematical models currently used for modeling of human corneal in optics and biomechanics fields.
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.
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
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.
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…
[Representation and mathematical analysis of human crystalline lens].
Tălu, Stefan; Giovanzana, Stefano; Tălu, Mihai
2011-01-01
The surface of human crystalline lens can be described and analyzed using mathematical models based on parametric representations, used in biomechanical studies and 3D solid modeling of the lens. The mathematical models used in lens biomechanics allow the study and the behavior of crystalline lens on variables and complex dynamic loads. Also, the lens biomechanics has the potential to improve the results in the development of intraocular lenses and cataract surgery. The paper presents the most representative mathematical models currently used for the modeling of human crystalline lens, both optically and biomechanically.
ERIC Educational Resources Information Center
Jitendra, Asha K.; Nelson, Gena; Pulles, Sandra M.; Kiss, Allyson J.; Houseworth, James
2016-01-01
The purpose of the present review was to evaluate the quality of the research and evidence base for representation of problems as a strategy to enhance the mathematical performance of students with learning disabilities and those at risk for mathematics difficulties. The authors evaluated 25 experimental and quasiexperimental studies according to…
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…
Oral Language, Representations and Mathematical Understanding: Indigenous Australian Students
ERIC Educational Resources Information Center
Warren, Elizabeth; Young, Janelle
2008-01-01
This paper explores the role of oral language and representations in negotiating mathematical understanding. The data were gathered from two Indigenous Australian classrooms in Northern Queensland. The first classroom, a Year 6/7 consisted of 15 students whose ages range from 10 years to 12 years with eight being Aboriginal, six from Torres Strait…
Social Representations of High School Students about Mathematics Assessment
ERIC Educational Resources Information Center
Martínez-Sierra, Gustavo; Valle-Zequeida, María E.; Miranda-Tirado, Marisa; Dolores-Flores, Crisólogo
2016-01-01
The perceptions of students about assessment in mathematics classes have been sparsely investigated. In order to fill this gap, this qualitative study aims to identify the social "representations" (understood as the system of values, ideas, and practices about a social object) of high school students regarding "assessment in…
The Role of Visual Representations for Structuring Classroom Mathematical Activity
ERIC Educational Resources Information Center
David, Maria Manuela; Tomaz, Vanessa Sena
2012-01-01
It is our presupposition that there is still a need for more research about how classroom practices can exploit the use and power of visualization in mathematics education. The aim of this article is to contribute in this direction, investigating how visual representations can structure geometry activity in the classroom and discussing teaching…
Declarative representation of uncertainty in mathematical models.
Miller, Andrew K; Britten, Randall D; Nielsen, Poul M F
2012-01-01
An important aspect of multi-scale modelling is the ability to represent mathematical models in forms that can be exchanged between modellers and tools. While the development of languages like CellML and SBML have provided standardised declarative exchange formats for mathematical models, independent of the algorithm to be applied to the model, to date these standards have not provided a clear mechanism for describing parameter uncertainty. Parameter uncertainty is an inherent feature of many real systems. This uncertainty can result from a number of situations, such as: when measurements include inherent error; when parameters have unknown values and so are replaced by a probability distribution by the modeller; when a model is of an individual from a population, and parameters have unknown values for the individual, but the distribution for the population is known. We present and demonstrate an approach by which uncertainty can be described declaratively in CellML models, by utilising the extension mechanisms provided in CellML. Parameter uncertainty can be described declaratively in terms of either a univariate continuous probability density function or multiple realisations of one variable or several (typically non-independent) variables. We additionally present an extension to SED-ML (the Simulation Experiment Description Markup Language) to describe sampling sensitivity analysis simulation experiments. We demonstrate the usability of the approach by encoding a sample model in the uncertainty markup language, and by developing a software implementation of the uncertainty specification (including the SED-ML extension for sampling sensitivty analyses) in an existing CellML software library, the CellML API implementation. We used the software implementation to run sampling sensitivity analyses over the model to demonstrate that it is possible to run useful simulations on models with uncertainty encoded in this form.
Declarative Representation of Uncertainty in Mathematical Models
Miller, Andrew K.; Britten, Randall D.; Nielsen, Poul M. F.
2012-01-01
An important aspect of multi-scale modelling is the ability to represent mathematical models in forms that can be exchanged between modellers and tools. While the development of languages like CellML and SBML have provided standardised declarative exchange formats for mathematical models, independent of the algorithm to be applied to the model, to date these standards have not provided a clear mechanism for describing parameter uncertainty. Parameter uncertainty is an inherent feature of many real systems. This uncertainty can result from a number of situations, such as: when measurements include inherent error; when parameters have unknown values and so are replaced by a probability distribution by the modeller; when a model is of an individual from a population, and parameters have unknown values for the individual, but the distribution for the population is known. We present and demonstrate an approach by which uncertainty can be described declaratively in CellML models, by utilising the extension mechanisms provided in CellML. Parameter uncertainty can be described declaratively in terms of either a univariate continuous probability density function or multiple realisations of one variable or several (typically non-independent) variables. We additionally present an extension to SED-ML (the Simulation Experiment Description Markup Language) to describe sampling sensitivity analysis simulation experiments. We demonstrate the usability of the approach by encoding a sample model in the uncertainty markup language, and by developing a software implementation of the uncertainty specification (including the SED-ML extension for sampling sensitivty analyses) in an existing CellML software library, the CellML API implementation. We used the software implementation to run sampling sensitivity analyses over the model to demonstrate that it is possible to run useful simulations on models with uncertainty encoded in this form. PMID:22802941
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.
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
Mexican High School Students' Social Representations of Mathematics, Its Teaching and Learning
ERIC Educational Resources Information Center
Martínez-Sierra, Gustavo; Miranda-Tirado, Marisa
2015-01-01
This paper reports a qualitative research that identifies Mexican high school students' social representations of mathematics. For this purpose, the social representations of "mathematics", "learning mathematics" and "teaching mathematics" were identified in a group of 50 students. Focus group interviews were carried…
ERIC Educational Resources Information Center
Hebert, Michael A.; Powell, Sarah R.
2016-01-01
Increasingly, students are expected to write about mathematics. Mathematics writing may be informal (e.g., journals, exit slips) or formal (e.g., writing prompts on high-stakes mathematics assessments). In order to develop an effective mathematics-writing intervention, research needs to be conducted on how students organize mathematics writing and…
A mathematical view on the decoupled sites representation.
Martini, Johannes W R; Ullmann, G Matthias
2013-02-01
The decoupled sites representation (DSR) is a theoretical instrument which allows to regard complex pH titration curves of biomolecules with several interacting proton binding sites as composition of isolated, non-interacting sites, each with a standard Henderson-Hasselbalch titration curve. In this work, we present the mathematical framework in which the DSR is embedded and give mathematical proofs for several statements in the periphery of the DSR. These proofs also identify exceptions. To apply the DSR to any molecule, it is necessary to extend the set of binding energies from R to a stripe within C. An important observation in this context is that even positive interaction energies (repulsion) between the binding sites will not guarantee real binding energies in the decoupled system, at least if the molecule has more than four proton binding sites. Moreover, we show that for a given overall titration curve it is not only possible to find a corresponding system with an interaction energy of zero but with any arbitrary fix interaction energy. This result also effects practical work as it shows that for any given titration curve, there is an infinite number of corresponding hypothetical molecules. Furthermore, this implies that--using a common definition of cooperative binding on the level of interaction energies--a meaningful measure of cooperativity between the binding sites cannot be defined solely on the basis of the overall titration. Consequently, all measures of cooperativity based on the overall binding curve do not measure the type of cooperativity commonly defined on the basis of interaction energies. Understanding the DSR mathematically provides the basis of transferring the DSR to biomolecules with different types of interacting ligands, such as protons and electrons, which play an important role within electron transport chains like in photosynthesis.
ERIC Educational Resources Information Center
Yackel, Erna; Wheatley, Grayson H.
This study investigated the problem representations formed by college students while solving mathematics problems. Problem representation characteristics indicative of understanding were identified by analyzing audio-tapes and written work of sixteen subjects, ages 16 to 24, who solved mathematics problems using the think-aloud technique. These…
ERIC Educational Resources Information Center
Dreher, Anika; Kuntze, Sebastian; Lerman, Stephen
2016-01-01
Dealing with multiple representations and their connections plays a key role for learners to build up conceptual knowledge in the mathematics classroom. Hence, professional knowledge and views of mathematics teachers regarding the use of multiple representations certainly merit attention. In particular, investigating such views of preservice…
ERIC Educational Resources Information Center
Barmby, Patrick; Bolden, David; Raine, Stephanie; Thompson, Lynn
2013-01-01
Background: The research on diagrammatic representations highlights their importance for the teaching and learning of mathematics. However, the empirical evidence to support their use in the classroom is mixed and somewhat lacking. Purpose: The aim of this study was to develop the use of diagrammatic representations of mathematical concepts in…
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…
The Effects of Multiple Linked Representations on Student Learning in Mathematics.
ERIC Educational Resources Information Center
Ozgun-Koca, S. Asli
This study investigated the effects on student understanding of linear relationships using the linked representation software VideoPoint as compared to using semi-linked representation software. It investigated students' attitudes towards and preferences for mathematical representations--equations, tables, or graphs. An Algebra I class was divided…
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
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
Adiguzel, Tufan; Akpinar, Yavuz
2004-01-01
Instructional resources that employ multiple representations have become commonplace in mathematics classrooms. This study will present computer software, LaborScale which was designed to improve seventh grade students' word problem-solving skills through computer-based multiple representations including graphic, symbolic, and audio…
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
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…
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…
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…
The Concrete-Representational-Abstract Sequence of Instruction in Mathematics Classrooms
ERIC Educational Resources Information Center
Mudaly, Vimolan; Naidoo, Jayaluxmi
2015-01-01
The purpose of this paper is to explore how master mathematics teachers use the concrete-representational-abstract (CRA) sequence of instruction in mathematics classrooms. Data was collected from a convenience sample of six master teachers by observations, video recordings of their teaching, and semi-structured interviews. Data collection also…
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…
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…
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…
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,…
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…
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
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).
Fazio, Lisa K; Bailey, Drew H; Thompson, Clarissa A; Siegler, Robert S
2014-07-01
We examined relations between symbolic and non-symbolic numerical magnitude representations, between whole number and fraction representations, and between these representations and overall mathematics achievement in fifth graders. Fraction and whole number symbolic and non-symbolic numerical magnitude understandings were measured using both magnitude comparison and number line estimation tasks. After controlling for non-mathematical cognitive proficiency, both symbolic and non-symbolic numerical magnitude understandings were uniquely related to mathematics achievement, but the relation was much stronger for symbolic numbers. A meta-analysis of 19 published studies indicated that relations between non-symbolic numerical magnitude knowledge and mathematics achievement are present but tend to be weak, especially beyond 6 years of age.
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)
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…
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.
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…
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.
Arellano, Antonio
2011-01-01
Within the social sciences, the standardized notion of representation refers to knowledge expressed through language and sourced from supra-individual thought processes. This notion underscores the social foundation of knowledge while minimizing its inherent material and symbolic foundations, without which it cannot be wholly understood. The article argues that the circulation of heterogeneous inscriptions underpins the elaboration of representations. To demonstrate this, I analyze practices intended to improve a therapeutic treatment for kidney stones and show how scientists place a series of objects, animals, models, texts, and so on in circulation, eventually transformed into the mathematic representation of a treatment.
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
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…
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
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…
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
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
ERIC Educational Resources Information Center
Gierl, Mark J.
1997-01-01
Investigated whether Bloom's taxonomy offers item writers an accurate model for anticipating students' cognitive processes used to solve items on a large-scale mathematics achievement test. Seventh graders thought aloud as they solved problems on the test. Researchers coded their cognitive processes using Bloom's taxonomy. Results suggest that…
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
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
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…
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
Stylianou, Despina A.
2011-01-01
Representation is viewed as central to mathematical problem solving. Yet, it is becoming obvious that students are having difficulty negotiating the various forms and functions of representations. This article examines the functions that representation has in students' mathematical problem solving and how that compares to its function in the…
5D model for accurate representation and visualization of dynamic cardiac structures
NASA Astrophysics Data System (ADS)
Lin, Wei-te; Robb, Richard A.
2000-05-01
Accurate cardiac modeling is challenging due to the intricate structure and complex contraction patterns of myocardial tissues. Fast imaging techniques can provide 4D structural information acquired as a sequence of 3D images throughout the cardiac cycle. To mode. The beating heart, we created a physics-based surface model that deforms between successive time point in the cardiac cycle. 3D images of canine hearts were acquired during one complete cardiac cycle using the DSR and the EBCT. The left ventricle of the first time point is reconstructed as a triangular mesh. A mass-spring physics-based deformable mode,, which can expand and shrink with local contraction and stretching forces distributed in an anatomically accurate simulation of cardiac motion, is applied to the initial mesh and allows the initial mesh to deform to fit the left ventricle in successive time increments of the sequence. The resulting 4D model can be interactively transformed and displayed with associated regional electrical activity mapped onto anatomic surfaces, producing a 5D model, which faithfully exhibits regional cardiac contraction and relaxation patterns over the entire heart. The model faithfully represents structural changes throughout the cardiac cycle. Such models provide the framework for minimizing the number of time points required to usefully depict regional motion of myocardium and allow quantitative assessment of regional myocardial motion. The electrical activation mapping provides spatial and temporal correlation within the cardiac cycle. In procedures which as intra-cardiac catheter ablation, visualization of the dynamic model can be used to accurately localize the foci of myocardial arrhythmias and guide positioning of catheters for optimal ablation.
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)…
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.
Issack, Bilkiss B; Roy, Pierre-Nicholas
2005-08-22
An approach for the inclusion of geometric constraints in semiclassical initial value representation calculations is introduced. An important aspect of the approach is that Cartesian coordinates are used throughout. We devised an algorithm for the constrained sampling of initial conditions through the use of multivariate Gaussian distribution based on a projected Hessian. We also propose an approach for the constrained evaluation of the so-called Herman-Kluk prefactor in its exact log-derivative form. Sample calculations are performed for free and constrained rare-gas trimers. The results show that the proposed approach provides an accurate evaluation of the reduction in zero-point energy. Exact basis set calculations are used to assess the accuracy of the semiclassical results. Since Cartesian coordinates are used, the approach is general and applicable to a variety of molecular and atomic systems.
NASA Astrophysics Data System (ADS)
Lin, Shih-Yin; Maries, Alexandru; Singh, Chandralekha
2013-01-01
We investigate introductory physics students' difficulties in translating between mathematical and graphical representations and the effect of scaffolding on students' performance. We gave a typical problem that can be solved using Gauss's law involving a spherically symmetric charge distribution (a conducting sphere concentric with a conducting spherical shell) to 95 calculus-based introductory physics students. We asked students to write a mathematical expression for the electric field in various regions and asked them to graph the electric field. We knew from previous experience that students have great difficulty in graphing the electric field. Therefore, we implemented two scaffolding interventions to help them. Students who received the scaffolding support were either (1) asked to plot the electric field in each region first (before having to plot it as a function of distance from the center of the sphere) or (2) asked to plot the electric field in each region after explicitly evaluating the electric field at the beginning, mid and end points of each region. The comparison group was only asked to plot the electric field at the end of the problem. We found that students benefited the most from intervention (1) and that intervention (2), although intended to aid students, had an adverse effect. Also, recorded interviews were conducted with a few students in order to understand how students were impacted by the aforementioned interventions.
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
Kribbs, Elizabeth E.; Rogowsky, Beth A.
2016-01-01
Mathematics word-problems continue to be an insurmountable challenge for many middle school students. Educators have used pictorial and schematic illustrations within the classroom to help students visualize these problems. However, the data shows that pictorial representations can be more harmful than helpful in that they only display objects or…
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
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.
ERIC Educational Resources Information Center
Jao, Limin
2012-01-01
The National Council of Teachers of Mathematics (NCTM, 2000) has created a set of standards to reform mathematics teaching procedures to ensure that all students understand mathematics and learn to think mathematically. The standards also require teachers to use strategies that allow all students to reason and communicate mathematically and…
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
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
ERIC Educational Resources Information Center
Bryan, J. A.; Fennell, B. D.
2009-01-01
Because mathematical formulae and problem solving are such prominent components of most introductory physics courses, many students consider these courses to be nothing more than courses in applied mathematics. As a result, students often do not develop an acceptable understanding of the relationship between mathematics and science and of the role…
McDonnell, Mark D; Mohan, Ashutosh; Stricker, Christian
2013-01-01
The release of neurotransmitter vesicles after arrival of a pre-synaptic action potential (AP) at cortical synapses is known to be a stochastic process, as is the availability of vesicles for release. These processes are known to also depend on the recent history of AP arrivals, and this can be described in terms of time-varying probabilities of vesicle release. Mathematical models of such synaptic dynamics frequently are based only on the mean number of vesicles released by each pre-synaptic AP, since if it is assumed there are sufficiently many vesicle sites, then variance is small. However, it has been shown recently that variance across sites can be significant for neuron and network dynamics, and this suggests the potential importance of studying short-term plasticity using simulations that do generate trial-to-trial variability. Therefore, in this paper we study several well-known conceptual models for stochastic availability and release. We state explicitly the random variables that these models describe and propose efficient algorithms for accurately implementing stochastic simulations of these random variables in software or hardware. Our results are complemented by mathematical analysis and statement of pseudo-code algorithms.
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…
ERIC Educational Resources Information Center
Terwel, Jan; van Oers, Bert; van Dijk, Ivanka; van den Eeden, Pieter
2009-01-01
With regard to transfer, is it better to provide pupils with ready-made representations or is it more effective to scaffold pupils' thinking in the process of generating their own representations with the help of peers and under the guidance of a teacher in a process of guided co-construction? The sample comprises 10 classes and 239 Grade 5…
ERIC Educational Resources Information Center
Santos, Manuel
2000-01-01
Documents what high school students showed when asked to work on tasks that involved the use of various representations. Indicates that few students made proper connections between representations and the initial situation or problem on their own and exhibited competence in procedures such as expressing an area or solving quadratic equations, but…
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
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…
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
Sedig, Kamran
2008-01-01
Many children do not like learning mathematics. They do not find mathematics fun, motivating, and engaging, and they think it is difficult to learn. Computer-based games have the potential and possibility of addressing this problem. This paper proposes a strategy for designing game-based learning environments that takes advantage of the…
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
Ward, Robin A.; Anhalt, Cynthia O.; Vinson, Kevin D.
A study was carried out involving K-8 teacher candidates enrolled in an elementary mathematics methods course to investigate and document their thinking as they plan for mathematics instruction. The teacher candidates submitted lesson plans at three intervals during a semester-long methods course, which were coded based on the planned use(s) of…
Tobia, Valentina; Fasola, Anna; Lupieri, Alice; Marzocchi, Gian Marco
2016-01-01
This study aimed to explore the spatial numerical association of response codes (SNARC), the flanker, and the numerical distance effects in children with mathematical difficulties. From a sample of 720 third, fourth, and fifth graders, 60 children were selected and divided into the following three groups: typically developing children (TD; n = 29), children with mathematical difficulties only (MD only; n = 21), and children with mathematical and reading difficulties (MD+RD; n = 10). Children were tested with a numerical Eriksen task that was built to assess SNARC, numerical distance, and flanker (first and second order congruency) effects. Children with MD only showed stronger SNARC and second order congruency effects than did TD children, whereas the numerical distance effects were similar across the three groups. Finally, the first order congruency effect was associated with reading difficulties. These results showed that children with mathematical difficulties with or without reading difficulties were globally more impaired when spatial incompatibilities were presented. PMID:24737662
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
Superfine, Alison Castro; Canty, Reality S.; Marshall, Anne Marie
2009-01-01
The study described herein represents an initial, exploratory attempt to understand what it means to translate between external representation systems. Researchers have traditionally considered translation as an all-or-none activity. We hypothesize that translation is comprised of both knowledge and skill components, and accordingly construe…
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…
ERIC Educational Resources Information Center
Strickland, Tricia K.; Maccini, Paula
2013-01-01
The current study focuses on the effects of incorporating multiple visual representations on students' conceptual understanding of quadratic expressions embedded within area word problems and students' procedural fluency of transforming quadratic expressions in standard form to factored-form and vice versa. The intervention included the…
ERIC Educational Resources Information Center
Flores, Raymond; Koontz, Esther; Inan, Fethi A.; Alagic, Mara
2015-01-01
This study examined the impact of the order of two teaching approaches on students' abilities and on-task behaviors while learning how to solve percentage problems. Two treatment groups were compared. MR first received multiple representation instruction followed by traditional algorithmic instruction and TA first received these teaching…
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.
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 =…
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).…
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.
ERIC Educational Resources Information Center
Bossé, Michael J.; Adu-Gyamfi, Kwaku; Chandler, Kayla; Lynch-Davis, Kathleen
2016-01-01
Dynamic mathematical environments allow users to reify mathematical concepts through multiple representations, transform mathematical relations and organically explore mathematical properties, investigate integrated mathematics, and develop conceptual understanding. Herein, we integrate Boolean algebra, the functionalities of a dynamic…
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…
ERIC Educational Resources Information Center
Thigpen, L. Christine
2012-01-01
The purpose of this study was to explore teaching styles and how frequently teachers with a variety of teaching styles incorporate multiple representations, such as manipulatives, drawings, counters, etc., in the middle school mathematics classroom. Through this explanatory mixed methods study it was possible to collect the quantitative data in…
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…
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..
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
Representations in Primary Mathematics Teaching
ERIC Educational Resources Information Center
Debrenti, Edith
2013-01-01
The OECD PISA (Programme for International Student Assessment) investigates whether students have acquired the applicable knowledge essential for full participation in a modern society (they measure how students can apply their knowledge to novel situations). Meaningful learning and understanding are basic aspects of all kinds of learning and they…
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.
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
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.
Standard model of knowledge representation
NASA Astrophysics Data System (ADS)
Yin, Wensheng
2016-09-01
Knowledge representation is the core of artificial intelligence research. Knowledge representation methods include predicate logic, semantic network, computer programming language, database, mathematical model, graphics language, natural language, etc. To establish the intrinsic link between various knowledge representation methods, a unified knowledge representation model is necessary. According to ontology, system theory, and control theory, a standard model of knowledge representation that reflects the change of the objective world is proposed. The model is composed of input, processing, and output. This knowledge representation method is not a contradiction to the traditional knowledge representation method. It can express knowledge in terms of multivariate and multidimensional. It can also express process knowledge, and at the same time, it has a strong ability to solve problems. In addition, the standard model of knowledge representation provides a way to solve problems of non-precision and inconsistent knowledge.
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
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…
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
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,…
NASA Astrophysics Data System (ADS)
Strocchi, Franco
The general lesson from the GNS theorem is that a state on the algebra of observables, namely a set of expectations, defines a realization of the system in terms of a Hilbert space of states with a reference vector which represents as a cyclic vector (so that all the other vectors of can be obtained by applying the observables to ). In this sense, a state identifies the family of states related to it by observables, equivalently accessible from it by means of physically realizable operations. Thus, one may say that describes a closed world, or phase, to which belongs. An interesting physical and mathematical question is how many closed worlds or phases are associated to a quantum system. In the mathematical language this amounts to investigating how many inequivalent (physically acceptable) representations of the observable algebra which defines the system exist.
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.
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…
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)
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.
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…
Science modelling in pre-calculus: how to make mathematics problems contextually meaningful
NASA Astrophysics Data System (ADS)
Sokolowski, Andrzej; Yalvac, Bugrahan; Loving, Cathleen
2011-04-01
'Use of mathematical representations to model and interpret physical phenomena and solve problems is one of the major teaching objectives in high school math curriculum' (National Council of Teachers of Mathematics (NCTM), Principles and Standards for School Mathematics, NCTM, Reston, VA, 2000). Commonly used pre-calculus textbooks provide a wide range of application problems. However, these problems focus students' attention on evaluating or solving pre-arranged formulas for given values. The role of scientific content is reduced to provide a background for these problems instead of being sources of data gathering for inducing mathematical tools. Students are neither required to construct mathematical models based on the contexts nor are they asked to validate or discuss the limitations of applied formulas. Using these contexts, the instructor may think that he/she is teaching problem solving, where in reality he/she is teaching algorithms of the mathematical operations (G. Kulm (ed.), New directions for mathematics assessment, in Assessing Higher Order Thinking in Mathematics, Erlbaum, Hillsdale, NJ, 1994, pp. 221-240). Without a thorough representation of the physical phenomena and the mathematical modelling processes undertaken, problem solving unintentionally appears as simple algorithmic operations. In this article, we deconstruct the representations of mathematics problems from selected pre-calculus textbooks and explicate their limitations. We argue that the structure and content of those problems limits students' coherent understanding of mathematical modelling, and this could result in weak student problem-solving skills. Simultaneously, we explore the ways to enhance representations of those mathematical problems, which we have characterized as lacking a meaningful physical context and limiting coherent student understanding. In light of our discussion, we recommend an alternative to strengthen the process of teaching mathematical modelling - utilization
Develop Reasoning through Pictorial Representations
ERIC Educational Resources Information Center
Ruchti, Wendy P.; Bennett, Cory A.
2013-01-01
This article describes some of the benefits derived from encouraging math drawing in a class of seventh-and eighth-grade students in line with promoting mathematical proficiency. The authors report teaching pictorial representations as part of the solution process, where both students and teachers gained insight into various areas of…
Symbolic representation of probabilistic worlds.
Feldman, Jacob
2012-04-01
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 measure of the effectiveness of symbolic representation, and develops formal constraints under which such representation is in fact warranted. The effectiveness of symbolic representation hinges on the probabilistic structure of the environment that is to be represented. For arbitrary probability distributions (i.e., environments), symbolic representation is generally not warranted. But in modal environments, defined here as those that consist of mixtures of component distributions that are narrow ("spiky") relative to their spreads, symbolic representation can be shown to represent the environment with a relatively negligible loss of information. Modal environments support propositional forms, logical relations, and other familiar features of symbolic representation. Hence the assumption that our environment is, in fact, modal is a key tacit assumption underlying the use of symbols in cognitive science. PMID:22270145
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.
Encouraging Preservice Mathematics Teachers as Mathematicians
ERIC Educational Resources Information Center
Burroughs, Elizabeth A.
2007-01-01
This article describes an assignment that asks preservice secondary mathematics teachers to make connections between the mathematics they know and the mathematics they will teach. It describes how one preservice teacher's project resulted in a physical representation of the statement and proof that the sum of cubes of the first n natural numbers…
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…
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…
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…
The field representation language.
Tsafnat, Guy
2008-02-01
The complexity of quantitative biomedical models, and the rate at which they are published, is increasing to a point where managing the information has become all but impossible without automation. International efforts are underway to standardise representation languages for a number of mathematical entities that represent a wide variety of physiological systems. This paper presents the Field Representation Language (FRL), a portable representation of values that change over space and/or time. FRL is an extensible mark-up language (XML) derivative with support for large numeric data sets in Hierarchical Data Format version 5 (HDF5). Components of FRL can be reused through unified resource identifiers (URI) that point to external resources such as custom basis functions, boundary geometries and numerical data. To demonstrate the use of FRL as an interchange we present three models that study hyperthermia cancer treatment: a fractal model of liver tumour microvasculature; a probabilistic model simulating the deposition of magnetic microspheres throughout it; and a finite element model of hyperthermic treatment. The microsphere distribution field was used to compute the heat generation rate field around the tumour. We used FRL to convey results from the microsphere simulation to the treatment model. FRL facilitated the conversion of the coordinate systems and approximated the integral over regions of the microsphere deposition field. PMID:17434811
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…
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.
TOWARDS A NEW SPATIAL REPRESENTATION OF BONE REMODELING
Graham, Jason M.; Ayati, Bruce P.; Ramakrishnan, Prem S.; Martin, James A.
2013-01-01
Irregular bone remodeling is associated with a number of bone diseases such as osteoporosis and multiple myeloma. Computational and mathematical modeling can aid in therapy and treatment as well as understanding fundamental biology. Different approaches to modeling give insight into different aspects of a phenomena so it is useful to have an arsenal of various computational and mathematical models. Here we develop a mathematical representation of bone remodeling that can effectively describe many aspects of the complicated geometries and spatial behavior observed. There is a sharp interface between bone and marrow regions. Also the surface of bone moves in and out, i.e. in the normal direction, due to remodeling. Based on these observations we employ the use of a level-set function to represent the spatial behavior of remodeling. We elaborate on a temporal model for osteoclast and osteoblast population dynamics to determine the change in bone mass which influences how the interface between bone and marrow changes. We exhibit simulations based on our computational model that show the motion of the interface between bone and marrow as a consequence of bone remodeling. The simulations show that it is possible to capture spatial behavior of bone remodeling in complicated geometries as they occur in vitro and in vivo. By employing the level set approach it is possible to develop computational and mathematical representations of the spatial behavior of bone remodeling. By including in this formalism further details, such as more complex cytokine interactions and accurate parameter values, it is possible to obtain simulations of phenomena related to bone remodeling with spatial behavior much as in vitro and in vivo. This makes it possible to perform in silica experiments more closely resembling experimental observations. PMID:22901065
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…
An accurate representation of the motion of Pluto
NASA Astrophysics Data System (ADS)
Goffin, E.; Meeus, J.; Steyaert, C.
1986-02-01
Three series of periodic terms are presented which make it possible to calculate the heliocentric coordinates of Pluto (longitude, latitude, radius vector) during a time interval of more than two centuries. The terms and coefficients have been derived indirectly by least-square approximation of a numerical integration of the motion of Pluto. For the years 1885 to 2099, the maximum error is 0.5 arcsec in longitude, 0.1 arcsec in latitude, and 0.00002 AU in radius vector as compared to the numerical integration.
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)
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…
Mathematics anxiety and mathematics achievement
NASA Astrophysics Data System (ADS)
Sherman, Brian F.; Wither (Post.), David P.
2003-09-01
This paper is a distillation of the major result from the 1998 Ph.D. thesis of the late David Wither. It details a longitudinal study over five years of the relationship between mathematics anxiety and mathematics achievement. It starts from the already well documented negative correlation between the two, and seeks to establish one of the three hypotheses—that mathematics anxiety causes an impairment of mathematics achievement; that lack of mathematics achievement causes mathematics anxiety; or that there is a third underlying cause of the two.
Implicit medial representation for vessel segmentation
NASA Astrophysics Data System (ADS)
Pizaine, Guillaume; Angelini, Elsa; Bloch, Isabelle; Makram-Ebeid, Sherif
2011-03-01
In the context of mathematical modeling of complex vessel tree structures with deformable models, we present a novel level set formulation to evolve both the vessel surface and its centerline. The implicit function is computed as the convolution of a geometric primitive, representing the centerline, with localized kernels of continuously-varying scales allowing accurate estimation of the vessel width. The centerline itself is derived as the characteristic function of an underlying signed medialness function, to enforce a tubular shape for the segmented object, and evolves under shape and medialness constraints. Given a set of initial medial loci and radii, this representation first allows for simultaneous recovery of the vessels centerlines and radii, thus enabling surface reconstruction. Secondly, due to the topological adaptivity of the level set segmentation setting, it can handle tree-like structures and bifurcations without additional junction detection schemes nor user inputs. We discuss the shape parameters involved, their tuning and their influence on the control of the segmented shapes, and we present some segmentation results on synthetic images, 2D angiographies, 3D rotational angiographies and 3D-CT scans.
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…
Online Mathematics Instruction: An Analysis of Content.
ERIC Educational Resources Information Center
Snelson, Chareen
This paper presents the results of a pilot study conducted to examine Web-based instructional content for mathematics. Two research questions were posed during the study: (1) how is technology being used to represent mathematical concepts online? and (2) how do the representations work together as a system? A mixed method content analysis design…
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…
Handbook for Spoken Mathematics: (Larry's Speakeasy).
ERIC Educational Resources Information Center
Chang, Lawrence A.; And Others
This handbook is directed toward those who have to deal with spoken mathematics, yet have insufficient background to know the correct verbal expression for the written symbolic one. It compiles consistent and well-defined ways of uttering mathematical expressions so listeners will receive clear, unambiguous, and well-pronounced representations.…
Mathematical Notation in Bibliographic Databases.
ERIC Educational Resources Information Center
Pasterczyk, Catherine E.
1990-01-01
Discusses ways in which using mathematical symbols to search online bibliographic databases in scientific and technical areas can improve search results. The representations used for Greek letters, relations, binary operators, arrows, and miscellaneous special symbols in the MathSci, Inspec, Compendex, and Chemical Abstracts databases are…
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…
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…
Mathematics Placement Tests and Gender Bias.
ERIC Educational Resources Information Center
Dorner, Celine D'Souza; Hutton, Ivan
2002-01-01
Examined whether a mathematics placement system accurately predicted success in university mathematics classes for both genders. The placement system used four variables to predict grades a student would receive if placed in various freshman mathematics classes. Found that the multiple predictors added to the gender bias of the Scholastic…
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…
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…
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…
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…
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…
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…
State-Variable Representations For Moving-Average Sampling
NASA Technical Reports Server (NTRS)
Polites, Michael E.
1991-01-01
Two state-variable representations derived for continuous-time plant driven by control algorithm including zero-order hold and measurements sampled at mutliple rates by multiple-input/multiple-output moving-average processes. New representations enhance observability and controllability of plant. Applications include mathematical modeling of navigation systems including star trackers, gyroscopes, and accelerometers.
Sella, Francesco; Sader, Elie; Lolliot, Simon; Cohen Kadosh, Roi
2016-09-01
Recent studies have highlighted the potential role of basic numerical processing in the acquisition of numerical and mathematical competences. However, it is debated whether high-level numerical skills and mathematics depends specifically on basic numerical representations. In this study mathematicians and nonmathematicians performed a basic number line task, which required mapping positive and negative numbers on a physical horizontal line, and has been shown to correlate with more advanced numerical abilities and mathematical achievement. We found that mathematicians were more accurate compared with nonmathematicians when mapping positive, but not negative numbers, which are considered numerical primitives and cultural artifacts, respectively. Moreover, performance on positive number mapping could predict whether one is a mathematician or not, and was mediated by more advanced mathematical skills. This finding might suggest a link between basic and advanced mathematical skills. However, when we included visuospatial skills, as measured by block design subtest, the mediation analysis revealed that the relation between the performance in the number line task and the group membership was explained by non-numerical visuospatial skills. These results demonstrate that relation between basic, even specific, numerical skills and advanced mathematical achievement can be artifactual and explained by visuospatial processing. (PsycINFO Database Record
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
Sella, Francesco; Sader, Elie; Lolliot, Simon; Cohen Kadosh, Roi
2016-09-01
Recent studies have highlighted the potential role of basic numerical processing in the acquisition of numerical and mathematical competences. However, it is debated whether high-level numerical skills and mathematics depends specifically on basic numerical representations. In this study mathematicians and nonmathematicians performed a basic number line task, which required mapping positive and negative numbers on a physical horizontal line, and has been shown to correlate with more advanced numerical abilities and mathematical achievement. We found that mathematicians were more accurate compared with nonmathematicians when mapping positive, but not negative numbers, which are considered numerical primitives and cultural artifacts, respectively. Moreover, performance on positive number mapping could predict whether one is a mathematician or not, and was mediated by more advanced mathematical skills. This finding might suggest a link between basic and advanced mathematical skills. However, when we included visuospatial skills, as measured by block design subtest, the mediation analysis revealed that the relation between the performance in the number line task and the group membership was explained by non-numerical visuospatial skills. These results demonstrate that relation between basic, even specific, numerical skills and advanced mathematical achievement can be artifactual and explained by visuospatial processing. (PsycINFO Database Record 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.
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
Huckstep, Peter
2002-01-01
Contends teachers must resist the temptation to suggest that, while children can create stories and melodies, they cannot create mathematics. Quotes mathematician G. H. Hardy: "A mathematician, like a painter or poet, is a 'maker' of patterns." Considers mathematics should be able to stand up for itself. (BT)
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
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…
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.
Sex-related Differences in Mathematics Achievement: Myths, Realities and Related Factors.
ERIC Educational Resources Information Center
Fennema, Elizabeth
Many more males than females are involved in post high school mathematics study and in adult occupations that involve mathematics. This paper addresses the issue of whether this unequal representation of females and males is due to females' less adequate learning of mathematics or to deliberate choice of females not to study mathematics. After…
Finite representations of continuum environments
NASA Astrophysics Data System (ADS)
Zwolak, Michael
2008-09-01
Understanding dissipative and decohering processes is fundamental to the study of quantum systems. An accurate and generic method for investigating these processes is to simulate both the system and environment, which, however, is computationally very demanding. We develop a novel approach to constructing finite representations of the environment based on the influence of different frequency scales on the system's dynamics. As an illustration, we analyze a solvable model of an optical mode decaying into a reservoir. The influence of the environment modes is constant for small frequencies, but drops off rapidly for large frequencies, allowing for a very sparse representation at high frequencies that gives a significant computational speedup in simulating the environment. This approach provides a general framework for simulating open quantum systems.
Explorations of Representational Momentum.
ERIC Educational Resources Information Center
Kelly, Michael H.; Freyd, Jennifer J.
1987-01-01
Figures that undergo an implied rotation are remembered as being slightly beyond their final position, a phenomenon called representational momentum. Eight experiments explored the questions of what gets transformed and what types of transformations induce such representational distortions. (GDC)
Representations of fuzzy torus
NASA Astrophysics Data System (ADS)
Aizawa, N.; Chakrabarti, R.
2008-08-01
A classification of Hermitian representations for the recently introduced fuzzy torus algebra is presented. This is carried out by regarding the fuzzy torus algebra as a q-deformation of parafermion. In addition to the known representations, new representations of both finite and infinite dimension are found. Using the infinite dimensional representation, coherent state for the fuzzy torus is constructed. Dirac operator on commutative torus is also discussed.
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)
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)
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…
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.
Mathematical wit and mathematical cognition.
Aberdein, Andrew
2013-04-01
The published works of scientists often conceal the cognitive processes that led to their results. Scholars of mathematical practice must therefore seek out less obvious sources. This article analyzes a widely circulated mathematical joke, comprising a list of spurious proof types. An account is proposed in terms of argumentation schemes: stereotypical patterns of reasoning, which may be accompanied by critical questions itemizing possible lines of defeat. It is argued that humor is associated with risky forms of inference, which are essential to creative mathematics. The components of the joke are explicated by argumentation schemes devised for application to topic-neutral reasoning. These in turn are classified under seven headings: retroduction, citation, intuition, meta-argument, closure, generalization, and definition. Finally, the wider significance of this account for the cognitive science of mathematics is discussed. PMID:23512504
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.
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.
48 CFR 252.204-7007 - Alternate A, Annual Representations and Certifications.
Code of Federal Regulations, 2010 CFR
2010-10-01
... Online Representations and Certifications Application (ORCA) Web site at https://orca.bpn.gov/. After... months, are current, accurate, complete, and applicable to this solicitation (including the business...
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.
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.
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.
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…
Using Mental Imagery Processes for Teaching and Research in Mathematics and Computer Science
ERIC Educational Resources Information Center
Arnoux, Pierre; Finkel, Alain
2010-01-01
The role of mental representations in mathematics and computer science (for teaching or research) is often downplayed or even completely ignored. Using an ongoing work on the subject, we argue for a more systematic study and use of mental representations, to get an intuition of mathematical concepts, and also to understand and build proofs. We…
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…
Accurate Optical Reference Catalogs
NASA Astrophysics Data System (ADS)
Zacharias, N.
2006-08-01
Current and near future all-sky astrometric catalogs on the ICRF are reviewed with the emphasis on reference star data at optical wavelengths for user applications. The standard error of a Hipparcos Catalogue star position is now about 15 mas per coordinate. For the Tycho-2 data it is typically 20 to 100 mas, depending on magnitude. The USNO CCD Astrograph Catalog (UCAC) observing program was completed in 2004 and reductions toward the final UCAC3 release are in progress. This all-sky reference catalogue will have positional errors of 15 to 70 mas for stars in the 10 to 16 mag range, with a high degree of completeness. Proper motions for the about 60 million UCAC stars will be derived by combining UCAC astrometry with available early epoch data, including yet unpublished scans of the complete set of AGK2, Hamburg Zone astrograph and USNO Black Birch programs. Accurate positional and proper motion data are combined in the Naval Observatory Merged Astrometric Dataset (NOMAD) which includes Hipparcos, Tycho-2, UCAC2, USNO-B1, NPM+SPM plate scan data for astrometry, and is supplemented by multi-band optical photometry as well as 2MASS near infrared photometry. The Milli-Arcsecond Pathfinder Survey (MAPS) mission is currently being planned at USNO. This is a micro-satellite to obtain 1 mas positions, parallaxes, and 1 mas/yr proper motions for all bright stars down to about 15th magnitude. This program will be supplemented by a ground-based program to reach 18th magnitude on the 5 mas level.
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
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…
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.
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
The Object Metaphor and Synecdoche in Mathematics Classroom Discourse
ERIC Educational Resources Information Center
Font, Vicenc; Godino, Juan D.; Planas, Nuria; Acevedo, Jorge I.
2010-01-01
This article describes aspects of classroom discourse, illustrated through vignettes, that reveal the complex relationship between the forms in which mathematical objects exist and their ostensive representations. We illustrate various aspects of the process through which students come to consider the reality of mathematical objects that are…
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…
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…
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,…
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…
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…
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
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…
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.)
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…
Constructing Meanings of Mathematical Registers Using Metaphorical Reasoning and Models
ERIC Educational Resources Information Center
Lai, Mun Yee
2013-01-01
Current debates about successful mathematics pedagogy suggest that mathematical learning and problem solving can be enhanced by using metaphors as they provide students with a tool for thinking. But assisting pre-service teachers to understand the importance of careful and accurate explanations for mathematical concepts remains an issue. This…
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.
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…
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
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.
Dilemma in Teaching Mathematics
ERIC Educational Resources Information Center
Md Kamaruddin, Nafisah Kamariah; Md Amin, Zulkarnain
2012-01-01
The challenge in mathematics education is finding the best way to teach mathematics. When students learn the reasoning and proving in mathematics, they will be proficient in mathematics. Students must know mathematics before they can apply it. Symbolism and logic is the key to both the learning of mathematics and its effective application to…
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.
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.
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
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
ERIC Educational Resources Information Center
Flores, Margaret M.; Hinton, Vanessa M.; Burton, Megan E.
2016-01-01
Mathematical word problems are the most common form of mathematics problem solving implemented in K-12 schools. Identifying key words is a frequent strategy taught in classrooms in which students struggle with problem solving and show low success rates in mathematics. Researchers show that using the concrete-representational-abstract (CRA)…
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
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.
Sum over Histories Representation for Chemical Kinetics.
Bai, Shirong; Zhou, Dingyu; Davis, Michael J; Skodje, Rex T
2015-01-01
A new representation for chemical kinetics is introduced that is based on a sum over histories formulation that employs chemical pathways defined at a molecular level. The time evolution of a chemically reactive system is described by enumerating the most important pathways followed by a chemical moiety. An explicit formula for the pathway probabilities is derived and takes the form of an integral over a time-ordered product. When evaluating long pathways, the time-ordered product has a simple Monte Carlo representation that is computationally efficient. A small numerical stochastic simulation was used to identify the most important paths to include in the representation. The method was applied to a realistic H2/O2 combustion problem and is shown to yield accurate results. PMID:26263110
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…
A Technique for Studying the Organization of Mathematics Text Materials. Final Report.
ERIC Educational Resources Information Center
Kane, Robert B.; Holz, Alan W.
The validity and the reliability of a technique for identifying and studying presentation variables in mathematics texts were investigated in this study. A category system for classifying messages in mathematics texts in terms of mathematical content and processes and in terms of mode of representation, procedures for applying this system to…
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…
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…
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…
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…
Building Blocks and Cognitive Building Blocks: Playing to Know the World Mathematically
ERIC Educational Resources Information Center
Sarama, Julie; Clements, Douglas H.
2009-01-01
The authors explore how children's play can support the development of the foundations of mathematics learning and how adults can support children's representation of--and thus the "mathematization" of--their play. The authors review research about the amount and nature of mathematics found in the free play of children. They briefly…
Automatic Semantic Generation and Arabic Translation of Mathematical Expressions on the Web
ERIC Educational Resources Information Center
Doush, Iyad Abu; Al-Bdarneh, Sondos
2013-01-01
Automatic processing of mathematical information on the web imposes some difficulties. This paper presents a novel technique for automatic generation of mathematical equations semantic and Arabic translation on the web. The proposed system facilitates unambiguous representation of mathematical equations by correlating equations to their known…
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
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…
Children's Mapping between Symbolic and Nonsymbolic Representations of Number
ERIC Educational Resources Information Center
Mundy, Eleanor; Gilmore, Camilla K.
2009-01-01
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…
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…
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…
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…
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…
Students' Development and Use of Internal Representations When Solving Algebraic Tasks
ERIC Educational Resources Information Center
Cross, Laban J.
2013-01-01
The difficulty in observing, recording, and examining internal representations has been well documented (Goldin & Shteingold, 2001). However, the important role that these internal representations play in the learning and understanding of mathematical concepts has been noted (Yackel, 2000). This study sought to develop a framework for…
Examining the Task and Knowledge Demands Needed to Teach with Representations
ERIC Educational Resources Information Center
Mitchell, Rebecca; Charalambous, Charalambos Y.; Hill, Heather C.
2014-01-01
Representations are often used in instruction to highlight key mathematical ideas and support student learning. Despite their centrality in scaffolding teaching and learning, most of our understanding about the tasks involved with using representations in instruction and the knowledge requirements imposed on teachers when using these aids is…
Calculus Students' Representation Use in Group-Work and Individual Settings
ERIC Educational Resources Information Center
Zazkis, Dov
2013-01-01
The study of student representation use and specifically the distinction between analytic and visual representations has fueled a long line of mathematics education literature that began more than 35 years ago. This literature can be partitioned into two bodies of work, one that is primarily cognitive and one that is primarily social. In spite of…
Exploring Mental Representations for Literal Symbols Using Priming and Comparison Distance Effects
ERIC Educational Resources Information Center
Pollack, Courtney; Leon Guerrero, Sibylla; Star, Jon R.
2016-01-01
Higher-level mathematics requires a connection between literal symbols (e.g., "x") and their mental representations. The current study probes the nature of mental representations for literal symbols using both the priming distance effect, in which ease of comparing a target number to a fixed standard is a function of prime-target…
Comparing the Effects of Representational Tools in Collaborative and Individual Inquiry Learning
ERIC Educational Resources Information Center
Kolloffel, Bas; Eysink, Tessa H. S.; de Jong, Ton
2011-01-01
Constructing a representation in which students express their domain understanding can help them improve their knowledge. Many different representational formats can be used to express one's domain understanding (e.g., concept maps, textual summaries, mathematical equations). The format can direct students' attention to specific aspects of the…
ERIC Educational Resources Information Center
Selva, Ana Coelho Vieira; Falcao, Jorge Tarcisio da Rocha; Nunes, Terezinha
2005-01-01
This research offers empirical evidence of the importance of supplying diverse symbolic representations in order to support concept development in mathematics. Graphical representation can be a helpful symbolic tool for concept development in the conceptual field of additive structures. Nevertheless, this symbolic tool has specific difficulties…
A Multifaceted Mathematical Approach for Complex Systems
Alexander, F.; Anitescu, M.; Bell, J.; Brown, D.; Ferris, M.; Luskin, M.; Mehrotra, S.; Moser, B.; Pinar, A.; Tartakovsky, A.; Willcox, K.; Wright, S.; Zavala, V.
2012-03-07
Applied mathematics has an important role to play in developing the tools needed for the analysis, simulation, and optimization of complex problems. These efforts require the development of the mathematical foundations for scientific discovery, engineering design, and risk analysis based on a sound integrated approach for the understanding of complex systems. However, maximizing the impact of applied mathematics on these challenges requires a novel perspective on approaching the mathematical enterprise. Previous reports that have surveyed the DOE's research needs in applied mathematics have played a key role in defining research directions with the community. Although these reports have had significant impact, accurately assessing current research needs requires an evaluation of today's challenges against the backdrop of recent advances in applied mathematics and computing. To address these needs, the DOE Applied Mathematics Program sponsored a Workshop for Mathematics for the Analysis, Simulation and Optimization of Complex Systems on September 13-14, 2011. The workshop had approximately 50 participants from both the national labs and academia. The goal of the workshop was to identify new research areas in applied mathematics that will complement and enhance the existing DOE ASCR Applied Mathematics Program efforts that are needed to address problems associated with complex systems. This report describes recommendations from the workshop and subsequent analysis of the workshop findings by the organizing committee.
Accurate finite difference methods for time-harmonic wave propagation
NASA Technical Reports Server (NTRS)
Harari, Isaac; Turkel, Eli
1994-01-01
Finite difference methods for solving problems of time-harmonic acoustics are developed and analyzed. Multidimensional inhomogeneous problems with variable, possibly discontinuous, coefficients are considered, accounting for the effects of employing nonuniform grids. A weighted-average representation is less sensitive to transition in wave resolution (due to variable wave numbers or nonuniform grids) than the standard pointwise representation. Further enhancement in method performance is obtained by basing the stencils on generalizations of Pade approximation, or generalized definitions of the derivative, reducing spurious dispersion, anisotropy and reflection, and by improving the representation of source terms. The resulting schemes have fourth-order accurate local truncation error on uniform grids and third order in the nonuniform case. Guidelines for discretization pertaining to grid orientation and resolution are presented.
Contacts de langues et representations (Language Contacts and Representations).
ERIC Educational Resources Information Center
Matthey, Marinette, Ed.
1997-01-01
Essays on language contact and the image of language, entirely in French, include: "Representations 'du' contexte et representations 'en' contexte? Eleves et enseignants face a l'apprentissage de la langue" ("Representations 'of' Context or Representations 'in' Context? Students and Teachers Facing Language Learning" (Laurent Gajo); "Le crepuscule…
Three representations of the Ising model
NASA Astrophysics Data System (ADS)
Kruis, Joost; Maris, Gunter
2016-10-01
Statistical models that analyse (pairwise) relations between variables encompass assumptions about the underlying mechanism that generated the associations in the observed data. In the present paper we demonstrate that three Ising model representations exist that, although each proposes a distinct theoretical explanation for the observed associations, are mathematically equivalent. This equivalence allows the researcher to interpret the results of one model in three different ways. We illustrate the ramifications of this by discussing concepts that are conceived as problematic in their traditional explanation, yet when interpreted in the context of another explanation make immediate sense.
Three representations of the Ising model
Kruis, Joost; Maris, Gunter
2016-01-01
Statistical models that analyse (pairwise) relations between variables encompass assumptions about the underlying mechanism that generated the associations in the observed data. In the present paper we demonstrate that three Ising model representations exist that, although each proposes a distinct theoretical explanation for the observed associations, are mathematically equivalent. This equivalence allows the researcher to interpret the results of one model in three different ways. We illustrate the ramifications of this by discussing concepts that are conceived as problematic in their traditional explanation, yet when interpreted in the context of another explanation make immediate sense. PMID:27698356
ERIC Educational Resources Information Center
Ding, Meixia; Li, Xiaobao
2014-01-01
Through examining a representative Chinese textbook series' presentation of the distributive property, this study explores how mathematics curriculum may structure representations in ways that facilitate the transition from concrete to abstract so as to support students' learning of mathematical principles. A total of 319 instances of…
ERIC Educational Resources Information Center
Pollack, Courtney
2012-01-01
The ability to represent numerical quantities in symbolic form is a necessary foundation for mathematical competence. Variables are particularly important symbolic representations for learning algebra and succeeding in higher mathematics, but the mechanisms of how students link a variable to what it represents are not well understood. Research…
Numerical Magnitude Representations and Individual Differences in Children's Arithmetic Strategy Use
ERIC Educational Resources Information Center
Vanbinst, Kiran; Ghesquiere, Pol; De Smedt, Bert
2012-01-01
Against the background of neuroimaging studies on how the brain processes numbers, there is now converging evidence that numerical magnitude representations are crucial for successful mathematics achievement. One major drawback of this research is that it mainly investigated mathematics performance as measured through general standardized…
Mathematical Modelling with Technology: The Role of Dynamic Representations
ERIC Educational Resources Information Center
Arzarello, Ferdinando; Ferrara, Francesca; Robutti, Ornella
2012-01-01
In this research we present the use of some technologies in problem solving activities (at different secondary school grades), aimed at finding a model for a geometric configuration, and representing this model in various ways: through a construction, through a Cartesian graph, etc. The task is part of a teaching experiment, in which students used…
Authenticity of Mathematical Modeling
ERIC Educational Resources Information Center
Tran, Dung; Dougherty, Barbara J.
2014-01-01
Some students leave high school never quite sure of the relevancy of the mathematics they have learned. They fail to see links between school mathematics and the mathematics of everyday life that requires thoughtful decision making and often complex problem solving. Is it possible to bridge the gap between school mathematics and the mathematics in…
Absolute value equations - what can we learn from their graphical representation?
NASA Astrophysics Data System (ADS)
Stupel, Moshe; Ben-Chaim, David
2014-08-01
Understanding graphical representations of algebraic equations, particularly graphical representations of absolute value equations, significantly improves students' mathematical comprehension and ignites within them an appreciation of the beauty and aesthetics of mathematics. In this paper, we focus on absolute value equations of linear and quadratic expressions, by examining various cases, presenting different methods of solving them by graphical representation, exhibiting the advantage of using dynamic software such as GeoGebra in solving them, and illustrating some examples of interesting graphical solutions. We recommend that teachers take advantage of the rapid development in technology to help learners tangibly visualize the solutions of absolute value equations before proceeding to the analytical solutions.
NNLOPS accurate associated HW production
NASA Astrophysics Data System (ADS)
Astill, William; Bizon, Wojciech; Re, Emanuele; Zanderighi, Giulia
2016-06-01
We present a next-to-next-to-leading order accurate description of associated HW production consistently matched to a parton shower. The method is based on reweighting events obtained with the HW plus one jet NLO accurate calculation implemented in POWHEG, extended with the MiNLO procedure, to reproduce NNLO accurate Born distributions. Since the Born kinematics is more complex than the cases treated before, we use a parametrization of the Collins-Soper angles to reduce the number of variables required for the reweighting. We present phenomenological results at 13 TeV, with cuts suggested by the Higgs Cross section Working Group.
Space-by-time manifold representation of dynamic facial expressions for emotion categorization
Delis, Ioannis; Chen, Chaona; Jack, Rachael E.; Garrod, Oliver G. B.; Panzeri, Stefano; Schyns, Philippe G.
2016-01-01
Visual categorization is the brain computation that reduces high-dimensional information in the visual environment into a smaller set of meaningful categories. An important problem in visual neuroscience is to identify the visual information that the brain must represent and then use to categorize visual inputs. Here we introduce a new mathematical formalism—termed space-by-time manifold decomposition—that describes this information as a low-dimensional manifold separable in space and time. We use this decomposition to characterize the representations used by observers to categorize the six classic facial expressions of emotion (happy, surprise, fear, disgust, anger, and sad). By means of a Generative Face Grammar, we presented random dynamic facial movements on each experimental trial and used subjective human perception to identify the facial movements that correlate with each emotion category. When the random movements projected onto the categorization manifold region corresponding to one of the emotion categories, observers categorized the stimulus accordingly; otherwise they selected “other.” Using this information, we determined both the Action Unit and temporal components whose linear combinations lead to reliable categorization of each emotion. In a validation experiment, we confirmed the psychological validity of the resulting space-by-time manifold representation. Finally, we demonstrated the importance of temporal sequencing for accurate emotion categorization and identified the temporal dynamics of Action Unit components that cause typical confusions between specific emotions (e.g., fear and surprise) as well as those resolving these confusions. PMID:27305521
Sparse Representation of Deformable 3D Organs with Spherical Harmonics and Structured Dictionary
Wang, Dan; Tewfik, Ahmed H.; Zhang, Yingchun; Shen, Yunhe
2011-01-01
This paper proposed a novel algorithm to sparsely represent a deformable surface (SRDS) with low dimensionality based on spherical harmonic decomposition (SHD) and orthogonal subspace pursuit (OSP). The key idea in SRDS method is to identify the subspaces from a training data set in the transformed spherical harmonic domain and then cluster each deformation into the best-fit subspace for fast and accurate representation. This algorithm is also generalized into applications of organs with both interior and exterior surfaces. To test the feasibility, we first use the computer models to demonstrate that the proposed approach matches the accuracy of complex mathematical modeling techniques and then both ex vivo and in vivo experiments are conducted using 3D magnetic resonance imaging (MRI) scans for verification in practical settings. All results demonstrated that the proposed algorithm features sparse representation of deformable surfaces with low dimensionality and high accuracy. Specifically, the precision evaluated as maximum error distance between the reconstructed surface and the MRI ground truth is better than 3 mm in real MRI experiments. PMID:21941524
Spacecraft Attitude Representations
NASA Technical Reports Server (NTRS)
Markley, F. Landis
1999-01-01
The direction cosine matrix or attitude matrix is the most fundamental representation of the attitude, but it is very inefficient: It has six redundant parameters, it is difficult to enforce the six (orthogonality) constraints. the four-component quaternion representation is very convenient: it has only one redundant parameter, it is easy to enforce the normalization constraint, the attitude matrix is a homogeneous quadratic function of q, quaternion kinematics are bilinear in q and m. Euler angles are extensively used: they often have a physical interpretation, they provide a natural description of some spacecraft motions (COBE, MAP), but kinematics and attitude matrix involve trigonometric functions, "gimbal lock" for certain values of the angles. Other minimum (three-parameter) representations: Gibbs vector is infinite for 180 deg rotations, but useful for analysis, Modified Rodrigues Parameters are nonsingular, no trig functions, Rotation vector phi is nonsingular, but requires trig functions.
On the representation of many-body interactions in water
Medders, Gregory; Gotz, Andreas; Morales, Miguel A.; Bajaj, Pushp; Paesani, Francesco
2015-09-09
Recent work has shown that the many-body expansion of the interaction energy can be used to develop analytical representations of global potential energy surfaces (PESs) for water. In this study, the role of short- and long-range interactions at different orders is investigated by analyzing water potentials that treat the leading terms of the many-body expansion through implicit (i.e., TTM3-F and TTM4-F PESs) and explicit (i.e., WHBB and MB-pol PESs) representations. It is found that explicit short-range representations of 2-body and 3-body interactions along with a physically correct incorporation of short- and long-range contributions are necessary for an accurate representation of the water interactions from the gas to the condensed phase. Similarly, a complete many-body representation of the dipole moment surface is found to be crucial to reproducing the correct intensities of the infrared spectrum of liquid water.
On the representation of many-body interactions in water
Medders, Gregory R.; Gotz, Andreas W.; Morales, Miguel A.; Bajaj, Pushp; Paesani, Francesco
2015-09-09
Our recent work has shown that the many-body expansion of the interactionenergy can be used to develop analytical representations of global potential energy surfaces (PESs) for water. In this study, the role of short- and long-range interactions at different orders is investigated by analyzing water potentials that treat the leading terms of the many-body expansion through implicit (i.e., TTM3-F and TTM4-F PESs) and explicit (i.e., WHBB and MB-pol PESs) representations. Moreover, it is found that explicit short-range representations of 2-body and 3-body interactions along with a physically correct incorporation of short- and long-range contributions are necessary for an accurate representation of the waterinteractions from the gas to the condensed phase. Likewise, a complete many-body representation of the dipole moment surface is found to be crucial to reproducing the correct intensities of the infrared spectrum of liquid water.
The representation of knowledge within model-based control systems
Weygand, D.P.; Koul, R.
1987-01-01
Representation of knowledge in artificially intelligent systems is discussed. Types of knowledge that might need to be represented in AI systems are listed, and include knowledge about objects, events, knowledge about how to do things, and knowledge about what human beings know (meta-knowledge). The use of knowledge in AI systems is discussed in terms of acquiring and retrieving knowledge and reasoning about known facts. Different kinds of reasonings or representations are ghen described with some examples given. These include formal reasoning or logical representation, which is related to mathematical logic, production systems, which are based on the idea of condition-action pairs (production), procedural reasoning, which uses pre-formed plans to solve problems, frames, which provide a structure for representing knowledge in an organized manner, direct analogical representations, which represent knowledge in such a manner that permits some observation without deduction. (LEW)
Umbra's system representation.
McDonald, Michael James
2005-07-01
This document describes the Umbra System representation. Umbra System representation, initially developed in the spring of 2003, is implemented in Incr/Tcl using concepts borrowed from Carnegie Mellon University's Architecture Description Language (ADL) called Acme. In the spring of 2004 through January 2005, System was converted to Umbra 4, extended slightly, and adopted as the underlying software system for a variety of Umbra applications that support Complex Systems Engineering (CSE) and Complex Adaptive Systems Engineering (CASE). System is now a standard part Of Umbra 4. While Umbra 4 also includes an XML parser for System, the XML parser and Schema are not described in this document.
Mathematical Language and Advanced Mathematics Learning
ERIC Educational Resources Information Center
Ferrari, Pier Luigi
2004-01-01
This paper is concerned with the role of language in mathematics learning at college level. Its main aim is to provide a perspective on mathematical language appropriate to effectively interpret students' linguistic behaviors in mathematics and to suggest new teaching ideas. Examples are given to show that the explanation of students' behaviors…
Mathematical Story: A Metaphor for Mathematics Curriculum
ERIC Educational Resources Information Center
Dietiker, Leslie
2015-01-01
This paper proposes a theoretical framework for interpreting the content found in mathematics curriculum in order to offer teachers and other mathematics educators comprehensive conceptual tools with which to make curricular decisions. More specifically, it describes a metaphor of "mathematics curriculum as story" and defines and…
Mathematics for Life: Sustainable Mathematics Education
ERIC Educational Resources Information Center
Renert, Moshe
2011-01-01
Ecological sustainability has not been a major focus of mathematics education research, even though it has attracted considerable attention in other areas of educational research in the past decade. The connections between mathematics education and ecological sustainability are not readily apparent. This paper explores how mathematics educators…
Mathematical Modelling Approach in Mathematics Education
ERIC Educational Resources Information Center
Arseven, Ayla
2015-01-01
The topic of models and modeling has come to be important for science and mathematics education in recent years. The topic of "Modeling" topic is especially important for examinations such as PISA which is conducted at an international level and measures a student's success in mathematics. Mathematical modeling can be defined as using…
Mathematics Anxiety and Attitudes toward Mathematics.
ERIC Educational Resources Information Center
Rounds, James B., Jr.; Hendel, Darwin D.
1980-01-01
Results indicate that the Mathematics Anxiety Rating Scale and the Math Anxiety Scale measure similar components of mathematics-anxiety domain. Mathematics-anxiety scales and Fennema-Sherman scales of Confidence and Effectance Motivation measure similar affective domains and are approximately equal predictors of arithmetic performance. (Author)
Gonis, A.; Zhang, X. G.; Stocks, G. M.; Nicholson, D. M.
2015-10-23
Density functional theory for the case of general, N-representable densities is reformulated in terms of density functional derivatives of expectation values of operators evaluated with wave functions leading to a density, making no reference to the concept of potential. The developments provide a complete solution of the v-representability problem by establishing a mathematical procedure that determines whether a density is v-representable and in the case of an affirmative answer determines the potential (within an additive constant) as a derivative with respect to the density of a constrained search functional. It also establishes the existence of an energy functional of themore » density that, for v-representable densities, assumes its minimum value at the density describing the ground state of an interacting many-particle system. The theorems of Hohenberg and Kohn emerge as special cases of the formalism.« less
Gonis, A.; Zhang, X. G.; Stocks, G. M.; Nicholson, D. M.
2015-10-23
Density functional theory for the case of general, N-representable densities is reformulated in terms of density functional derivatives of expectation values of operators evaluated with wave functions leading to a density, making no reference to the concept of potential. The developments provide a complete solution of the v-representability problem by establishing a mathematical procedure that determines whether a density is v-representable and in the case of an affirmative answer determines the potential (within an additive constant) as a derivative with respect to the density of a constrained search functional. It also establishes the existence of an energy functional of the density that, for v-representable densities, assumes its minimum value at the density describing the ground state of an interacting many-particle system. The theorems of Hohenberg and Kohn emerge as special cases of the formalism.
ERIC Educational Resources Information Center
Tervo, Juuso
2012-01-01
In "Postphysical Vision: Art Education's Challenge in an Age of Globalized Aesthetics (AMondofesto)" (2008) and "Beyond Aesthetics: Returning Force and Truth to Art and Its Education" (2009), jan jagodzinski argued for politics that go "beyond" representation--a project that radically questions visual culture…
Women and political representation.
Rathod, P B
1999-01-01
A remarkable progress in women's participation in politics throughout the world was witnessed in the final decade of the 20th century. According to the Inter-Parliamentary Union report, there were only eight countries with no women in their legislatures in 1998. The number of women ministers at the cabinet level worldwide doubled in a decade, and the number of countries without any women ministers dropped from 93 to 48 during 1987-96. However, this progress is far from satisfactory. Political representation of women, minorities, and other social groups is still inadequate. This may be due to a complex combination of socioeconomic, cultural, and institutional factors. The view that women's political participation increases with social and economic development is supported by data from the Nordic countries, where there are higher proportions of women legislators than in less developed countries. While better levels of socioeconomic development, having a women-friendly political culture, and higher literacy are considered favorable factors for women's increased political representation, adopting one of the proportional representation systems (such as a party-list system, a single transferable vote system, or a mixed proportional system with multi-member constituencies) is the single factor most responsible for the higher representation of women.
Computing with scale-invariant neural representations
NASA Astrophysics Data System (ADS)
Howard, Marc; Shankar, Karthik
The Weber-Fechner law is perhaps the oldest quantitative relationship in psychology. Consider the problem of the brain representing a function f (x) . Different neurons have receptive fields that support different parts of the range, such that the ith neuron has a receptive field at xi. Weber-Fechner scaling refers to the finding that the width of the receptive field scales with xi as does the difference between the centers of adjacent receptive fields. Weber-Fechner scaling is exponentially resource-conserving. Neurophysiological evidence suggests that neural representations obey Weber-Fechner scaling in the visual system and perhaps other systems as well. We describe an optimality constraint that is solved by Weber-Fechner scaling, providing an information-theoretic rationale for this principle of neural coding. Weber-Fechner scaling can be generated within a mathematical framework using the Laplace transform. Within this framework, simple computations such as translation, correlation and cross-correlation can be accomplished. This framework can in principle be extended to provide a general computational language for brain-inspired cognitive computation on scale-invariant representations. Supported by NSF PHY 1444389 and the BU Initiative for the Physics and Mathematics of Neural Systems,.
[Time perceptions and representations].
Tordjman, S
2015-09-01
Representations of time and time measurements depend on subjective constructs that vary according to changes in our concepts, beliefs, societal needs and technical advances. Similarly, the past, the future and the present are subjective representations that depend on each individual's psychic time and biological time. Therefore, there is no single, one-size-fits-all time for everyone, but rather a different, subjective time for each individual. We need to acknowledge the existence of different inter-individual times but also intra-individual times, to which different functions and different rhythms are attached, depending on the system of reference. However, the construction of these time perceptions and representations is influenced by objective factors (physiological, physical and cognitive) related to neuroscience which will be presented and discussed in this article. Thus, studying representation and perception of time lies at the crossroads between neuroscience, human sciences and philosophy. Furthermore, it is possible to identify several constants among the many and various representations of time and their corresponding measures, regardless of the system of time reference. These include the notion of movements repeated in a stable rhythmic pattern involving the recurrence of the same interval of time, which enables us to define units of time of equal and invariable duration. This rhythmicity is also found at a physiological level and contributes through circadian rhythms, in particular the melatonin rhythm, to the existence of a biological time. Alterations of temporality in mental disorders will be also discussed in this article illustrated by certain developmental disorders such as autism spectrum disorders. In particular, the hypothesis will be developed that children with autism would need to create discontinuity out of continuity through stereotyped behaviors and/or interests. This discontinuity repeated at regular intervals could have been
Updating representations of temporal intervals.
Danckert, James; Anderson, Britt
2015-12-01
Effectively engaging with the world depends on accurate representations of the regularities that make up that world-what we call mental models. The success of any mental model depends on the ability to adapt to changes-to 'update' the model. In prior work, we have shown that damage to the right hemisphere of the brain impairs the ability to update mental models across a range of tasks. Given the disparate nature of the tasks we have employed in this prior work (i.e. statistical learning, language acquisition, position priming, perceptual ambiguity, strategic game play), we propose that a cognitive module important for updating mental representations should be generic, in the sense that it is invoked across multiple cognitive and perceptual domains. To date, the majority of our tasks have been visual in nature. Given the ubiquity and import of temporal information in sensory experience, we examined the ability to build and update mental models of time. We had healthy individuals complete a temporal prediction task in which intervals were initially drawn from one temporal range before an unannounced switch to a different range of intervals. Separate groups had the second range of intervals switch to one that contained either longer or shorter intervals than the first range. Both groups showed significant positive correlations between perceptual and prediction accuracy. While each group updated mental models of temporal intervals, those exposed to shorter intervals did so more efficiently. Our results support the notion of generic capacity to update regularities in the environment-in this instance based on temporal information. The task developed here is well suited to investigations in neurological patients and in neuroimaging settings.
Updating representations of temporal intervals.
Danckert, James; Anderson, Britt
2015-12-01
Effectively engaging with the world depends on accurate representations of the regularities that make up that world-what we call mental models. The success of any mental model depends on the ability to adapt to changes-to 'update' the model. In prior work, we have shown that damage to the right hemisphere of the brain impairs the ability to update mental models across a range of tasks. Given the disparate nature of the tasks we have employed in this prior work (i.e. statistical learning, language acquisition, position priming, perceptual ambiguity, strategic game play), we propose that a cognitive module important for updating mental representations should be generic, in the sense that it is invoked across multiple cognitive and perceptual domains. To date, the majority of our tasks have been visual in nature. Given the ubiquity and import of temporal information in sensory experience, we examined the ability to build and update mental models of time. We had healthy individuals complete a temporal prediction task in which intervals were initially drawn from one temporal range before an unannounced switch to a different range of intervals. Separate groups had the second range of intervals switch to one that contained either longer or shorter intervals than the first range. Both groups showed significant positive correlations between perceptual and prediction accuracy. While each group updated mental models of temporal intervals, those exposed to shorter intervals did so more efficiently. Our results support the notion of generic capacity to update regularities in the environment-in this instance based on temporal information. The task developed here is well suited to investigations in neurological patients and in neuroimaging settings. PMID:26303026
Representing the Electromagnetic Field: How Maxwell's Mathematics Empowered Faraday's Field Theory
NASA Astrophysics Data System (ADS)
Tweney, Ryan D.
2011-07-01
James Clerk Maxwell `translated' Michael Faraday's experimentally-based field theory into the mathematical representation now known as `Maxwell's Equations.' Working with a variety of mathematical representations and physical models Maxwell extended the reach of Faraday's theory and brought it into consistency with other results in the physics of electricity and magnetism. Examination of Maxwell's procedures opens many issues about the role of mathematical representation in physics and the learning background required for its success. Specifically, Maxwell's training in `Cambridge University' mathematical physics emphasized the use of analogous equations across fields of physics and the repeated solving of extremely difficult problems in physics. Such training develops an array of overlearned mathematical representations supported by highly sophisticated cognitive mechanisms for the retrieval of relevant information from long term memory. For Maxwell, mathematics constituted a new form of representation in physics, enhancing the formal derivational and calculational role of mathematics and opening a cognitive means for the conduct of `experiments in the mind' and for sophisticated representations of theory.
Neural representations of magnitude for natural and rational numbers.
DeWolf, Melissa; Chiang, Jeffrey N; Bassok, Miriam; Holyoak, Keith J; Monti, Martin M
2016-11-01
Humans have developed multiple symbolic representations for numbers, including natural numbers (positive integers) as well as rational numbers (both fractions and decimals). Despite a considerable body of behavioral and neuroimaging research, it is currently unknown whether different notations map onto a single, fully abstract, magnitude code, or whether separate representations exist for specific number types (e.g., natural versus rational) or number representations (e.g., base-10 versus fractions). We address this question by comparing brain metabolic response during a magnitude comparison task involving (on different trials) integers, decimals, and fractions. Univariate and multivariate analyses revealed that the strength and pattern of activation for fractions differed systematically, within the intraparietal sulcus, from that of both decimals and integers, while the latter two number representations appeared virtually indistinguishable. These results demonstrate that the two major notations formats for rational numbers, fractions and decimals, evoke distinct neural representations of magnitude, with decimals representations being more closely linked to those of integers than to those of magnitude-equivalent fractions. Our findings thus suggest that number representation (base-10 versus fractions) is an important organizational principle for the neural substrate underlying mathematical cognition.
Students as Mathematics Consultants
ERIC Educational Resources Information Center
Jensen, Jennifer L.
2013-01-01
If students are going to develop reasoning and thinking skills, use their mathematical knowledge, and recognize the relevance of mathematics in their lives, they need to experience mathematics in meaningful ways. Only then will their mathematical skills be transferrable to all other parts of their lives. To promote such flexible mathematical…
Transforming Primary Mathematics
ERIC Educational Resources Information Center
Askew, Mike
2011-01-01
What is good mathematics teaching? What is mathematics teaching good for? Who is mathematics teaching for? These are just some of the questions addressed in "Transforming Primary Mathematics", a highly timely new resource for teachers which accessibly sets out the key theories and latest research in primary maths today. Under-pinned by findings…
ERIC Educational Resources Information Center
Morsy, Zaghloul, Ed.; Draxler, Alexandra, Ed.
1979-01-01
A set of articles with the theme "Mathematics for Real Life" is presented. These article titles are: (1) Teaching Mathematics as a Tool for Problem Solving; (2) New Math or New Education; (3) Hand Calculators and Math in Primary School; (4) Mass Media in the Mathematical Training of Polish Primary Teachers; (5) The Goals of Mathematics Teaching in…
Mathematical Epistemologies at Work.
ERIC Educational Resources Information Center
Noss, Richard
2002-01-01
Investigates young people's expression of mathematical ideas with a computer, the nature of mathematical practices, and the problem of mathematical meaning from cognitive and socio-cultural perspectives. Describes a mathematical activity system designed for learning and the role of digital technologies in helping to understand and reshape the…
ERIC Educational Resources Information Center
Cain, David
2007-01-01
This article presents the first part of the closing address given by the author to the 2007 Association of Teachers of Mathematics (ATM) Easter conference at Loughborough. In his closing address, the author focuses on functioning mathematically as opposed to functional mathematics. His view of functional mathematics is that the focus is on someone…
Mathematics Lessons without ...
ERIC Educational Resources Information Center
Cross, Kath; Hibbs, John
2006-01-01
In the Association of Teachers of Mathematics (ATM) Easter conference, 2006, the authors presented a list of important aspects of mathematics lessons, recommended for students to have a positive attitude to mathematics and for teachers to acquire effective teaching. The following are discussed in detail: (1) Mathematics lessons without good…
Identifying Representational Competence with Multi-Representational Displays
ERIC Educational Resources Information Center
Stieff, Mike; Hegarty, Mary; Deslongchamps, Ghislain
2011-01-01
Increasingly, multi-representational educational technologies are being deployed in science classrooms to support science learning and the development of representational competence. Several studies have indicated that students experience significant challenges working with these multi-representational displays and prefer to use only one…
Representations in Simulated Workplaces
ERIC Educational Resources Information Center
van Schaik, Martijn; Terwel, Jan; van Oers, Bert
2014-01-01
In vocational education students are to be prepared to participate in communities of practice. Hence they need technical skills as well as content knowledge e.g. science and mathematics. Research has shown that the instructional strategy of guided co-construction may lead to deeper understandings within a practice. The research questions in this…
ERIC Educational Resources Information Center
International Group for the Psychology of Mathematics Education.
The papers presented in these proceedings are organized into seven categories: (1) number and number operations (13 papers); (2) spatial representation and geometrical understanding (10 papers); (3) developing and/or using models of mathematical learning (15 papers); (4) mathematical concept formation (17 papers); (5) the mathematical learning…
ERIC Educational Resources Information Center
Kosko, Karl Wesley; Norton, Anderson
2012-01-01
The current body of literature suggests an interactive relationship between several of the process standards advocated by National Council of Teachers of Mathematics. Verbal and written mathematical communication has often been described as an alternative to typical mathematical representations (e.g., charts and graphs). Therefore, the…
ERIC Educational Resources Information Center
Chichekian, Tanya; Shore, Bruce M.
2013-01-01
This collaborative concept-mapping exercise was conducted in a second-year mathematics methods course. Teachers' visual representations of their mathematical content and pedagogical knowledge provided insight into their understanding of how students learn mathematics. We collected 28 preservice student teachers' concept maps and analyzed them by…
The Functions of Multiple Representations.
ERIC Educational Resources Information Center
Ainsworth, Shaaron
1999-01-01
Discusses multiple representations and multimedia learning environments; describes a functional taxonomy of MERs (multiple external representations); and considers how MERs are used to support cognitive processes in learning and problem solving with computers. (Contains 41 references.) (Author/LRW)
Naturalising Representational Content
Shea, Nicholas
2014-01-01
This paper sets out a view about the explanatory role of representational content and advocates one approach to naturalising content – to giving a naturalistic account of what makes an entity a representation and in virtue of what it has the content it does. It argues for pluralism about the metaphysics of content and suggests that a good strategy is to ask the content question with respect to a variety of predictively successful information processing models in experimental psychology and cognitive neuroscience; and hence that data from psychology and cognitive neuroscience should play a greater role in theorising about the nature of content. Finally, the contours of the view are illustrated by drawing out and defending a surprising consequence: that individuation of vehicles of content is partly externalist. PMID:24563661
Representation in incremental learning
NASA Technical Reports Server (NTRS)
1993-01-01
Work focused on two areas in machine learning: representation for inductive learning and how to apply concept learning techniques to learning state preferences, which can represent search control knowledge for problem solving. Specifically, in the first area the issues of the effect of representation on learning, on how learning formalisms are biased, and how concept learning can benefit from the use of a hybrid formalism are addressed. In the second area, the issues of developing an agent to learn search control knowledge from the relative values of states, of the source of that qualitative information, and of the ability to use both quantitative and qualitative information in order to develop an effective problem-solving policy are examined.
ERIC Educational Resources Information Center
González-Castro, Paloma; Cueli, Marisol; Areces, Débora; Rodríguez, Celestino; Sideridis, Georgios
2016-01-01
Problem solving represents a salient deficit in students with mathematical learning difficulties (MLD) primarily caused by difficulties with informal and formal mathematical competencies. This study proposes a computerized intervention tool, the integrated dynamic representation (IDR), for enhancing the early learning of basic mathematical…
ERIC Educational Resources Information Center
Clarkson, Philip C., Ed.
This document contains papers presented at the 19th annual conference of the Mathematics Education Research Group of Australasia. Topics of the presentations include learning research, mathematical representations, problem solving, strategic learning behaviors, algebraic thinking and learning environments, teaching and learning of algebra,…
Multiple Forms of Dynamic Representation
ERIC Educational Resources Information Center
Ainsworth, Shaaron; VanLabeke, Nicolas
2004-01-01
The terms dynamic representation and animation are often used as if they are synonymous, but in this paper we argue that there are multiple ways to represent phenomena that change over time. Time-persistent representations show a range of values over time. Time-implicit representations also show a range of values but not the specific times when…
[Time perceptions and representations].
Tordjman, S
2015-09-01
Representations of time and time measurements depend on subjective constructs that vary according to changes in our concepts, beliefs, societal needs and technical advances. Similarly, the past, the future and the present are subjective representations that depend on each individual's psychic time and biological time. Therefore, there is no single, one-size-fits-all time for everyone, but rather a different, subjective time for each individual. We need to acknowledge the existence of different inter-individual times but also intra-individual times, to which different functions and different rhythms are attached, depending on the system of reference. However, the construction of these time perceptions and representations is influenced by objective factors (physiological, physical and cognitive) related to neuroscience which will be presented and discussed in this article. Thus, studying representation and perception of time lies at the crossroads between neuroscience, human sciences and philosophy. Furthermore, it is possible to identify several constants among the many and various representations of time and their corresponding measures, regardless of the system of time reference. These include the notion of movements repeated in a stable rhythmic pattern involving the recurrence of the same interval of time, which enables us to define units of time of equal and invariable duration. This rhythmicity is also found at a physiological level and contributes through circadian rhythms, in particular the melatonin rhythm, to the existence of a biological time. Alterations of temporality in mental disorders will be also discussed in this article illustrated by certain developmental disorders such as autism spectrum disorders. In particular, the hypothesis will be developed that children with autism would need to create discontinuity out of continuity through stereotyped behaviors and/or interests. This discontinuity repeated at regular intervals could have been
Translation between representation languages
NASA Technical Reports Server (NTRS)
Vanbaalen, Jeffrey
1994-01-01
A capability for translating between representation languages is critical for effective knowledge base reuse. A translation technology for knowledge representation languages based on the use of an interlingua for communicating knowledge is described. The interlingua-based translation process consists of three major steps: translation from the source language into a subset of the interlingua, translation between subsets of the interlingua, and translation from a subset of the interlingua into the target language. The first translation step into the interlingua can typically be specified in the form of a grammar that describes how each top-level form in the source language translates into the interlingua. In cases where the source language does not have a declarative semantics, such a grammar is also a specification of a declarative semantics for the language. A methodology for building translators that is currently under development is described. A 'translator shell' based on this methodology is also under development. The shell has been used to build translators for multiple representation languages and those translators have successfully translated nontrivial knowledge bases.
Mapping Pedagogical Opportunities Provided by Mathematics Analysis Software
ERIC Educational Resources Information Center
Pierce, Robyn; Stacey, Kaye
2010-01-01
This paper proposes a taxonomy of the pedagogical opportunities that are offered by mathematics analysis software such as computer algebra systems, graphics calculators, dynamic geometry or statistical packages. Mathematics analysis software is software for purposes such as calculating, drawing graphs and making accurate diagrams. However, its…
Negotiating the Boundaries Between Mathematics and Physics
NASA Astrophysics Data System (ADS)
Radtka, Catherine
2015-07-01
This paper examines physics and mathematics textbooks published in France at the end of the 1950s and at the beginning of the 1960s for children aged 11-15 years old. It argues that at this "middle school" level, textbooks contributed to shape cultural representations of both disciplines and their mutual boundaries through their contents and their material aspect. Further, this paper argues that far from presenting clearly delimited subjects, late 1950s textbooks offered possible connections between mathematics and physics. It highlights that such connections depended upon the type of schools the textbooks aimed at, at a time when educational organization still differentiated pupils of this age. It thus stresses how the audience and its projected aptitudes and needs, as well as the cultural teaching traditions of the teachers in charge, were inseparable from the diverse conceptions of mathematics and physics and their relationships promoted through textbooks of the time.
Representation of probabilistic scientific knowledge.
Soldatova, Larisa N; Rzhetsky, Andrey; De Grave, Kurt; King, Ross D
2013-04-15
The theory of probability is widely used in biomedical research for data analysis and modelling. In previous work the probabilities of the research hypotheses have been recorded as experimental metadata. The ontology HELO is designed to support probabilistic reasoning, and provides semantic descriptors for reporting on research that involves operations with probabilities. HELO explicitly links research statements such as hypotheses, models, laws, conclusions, etc. to the associated probabilities of these statements being true. HELO enables the explicit semantic representation and accurate recording of probabilities in hypotheses, as well as the inference methods used to generate and update those hypotheses. We demonstrate the utility of HELO on three worked examples: changes in the probability of the hypothesis that sirtuins regulate human life span; changes in the probability of hypotheses about gene functions in the S. cerevisiae aromatic amino acid pathway; and the use of active learning in drug design (quantitative structure activity relation learning), where a strategy for the selection of compounds with the highest probability of improving on the best known compound was used. HELO is open source and available at https://github.com/larisa-soldatova/HELO. PMID:23734675
Representation of probabilistic scientific knowledge
2013-01-01
The theory of probability is widely used in biomedical research for data analysis and modelling. In previous work the probabilities of the research hypotheses have been recorded as experimental metadata. The ontology HELO is designed to support probabilistic reasoning, and provides semantic descriptors for reporting on research that involves operations with probabilities. HELO explicitly links research statements such as hypotheses, models, laws, conclusions, etc. to the associated probabilities of these statements being true. HELO enables the explicit semantic representation and accurate recording of probabilities in hypotheses, as well as the inference methods used to generate and update those hypotheses. We demonstrate the utility of HELO on three worked examples: changes in the probability of the hypothesis that sirtuins regulate human life span; changes in the probability of hypotheses about gene functions in the S. cerevisiae aromatic amino acid pathway; and the use of active learning in drug design (quantitative structure activity relation learning), where a strategy for the selection of compounds with the highest probability of improving on the best known compound was used. HELO is open source and available at https://github.com/larisa-soldatova/HELO PMID:23734675
Classroom Aids for Mathematics, Volume 1: Polynomials.
ERIC Educational Resources Information Center
Holden, Herbert L.
The goal of this pamphlet is to provide instructors of various scientific disciplines with mathematically accurate graphs of elementary polynomial functions. The figures in this pamphlet are intended to provide suitable material for the preparation of classroom handouts and overhead transparencies. In addition, sample sets of exercises are…
Mathematics of Information Processing and the Internet
ERIC Educational Resources Information Center
Hart, Eric W.
2010-01-01
The mathematics of information processing and the Internet can be organized around four fundamental themes: (1) access (finding information easily); (2) security (keeping information confidential); (3) accuracy (ensuring accurate information); and (4) efficiency (data compression). In this article, the author discusses each theme with reference to…
An ellipsoidal representation of human hand anthropometry
NASA Technical Reports Server (NTRS)
Buchholz, Bryan; Armstrong, Thomas J.
1991-01-01
Anthropometric data concerning the heometry of the hand's surface are presently modeled as a function of gross external hand measurements; an effort is made to evaluate the accuracy with which ellipsoids describe the geometry of the hand segments. Graphical comparisons indicate that differences between the ellipsoidal approximations and the breadth and depth measurements were greatest near the joints. On the bases of the present data, a set of overlapping ellipsoids could furnish a more accurate representation of hand geometry for adaptation to ellipsoid segment-geometry employing biomechanical models.
Profitable capitation requires accurate costing.
West, D A; Hicks, L L; Balas, E A; West, T D
1996-01-01
In the name of costing accuracy, nurses are asked to track inventory use on per treatment basis when more significant costs, such as general overhead and nursing salaries, are usually allocated to patients or treatments on an average cost basis. Accurate treatment costing and financial viability require analysis of all resources actually consumed in treatment delivery, including nursing services and inventory. More precise costing information enables more profitable decisions as is demonstrated by comparing the ratio-of-cost-to-treatment method (aggregate costing) with alternative activity-based costing methods (ABC). Nurses must participate in this costing process to assure that capitation bids are based upon accurate costs rather than simple averages. PMID:8788799
Social Work Scholars' Representation of Rawls: A Critique
ERIC Educational Resources Information Center
Banerjee, Mahasweta M.
2011-01-01
Although Rawls is the most cited social justice theorist in social work, he is not always accurately represented in the literature. To clarify this claim, the author reviews social work scholars' views about social justice, shows social work scholars' representation of Rawls, and highlights aspects of Rawls' theory of social justice. The author's…
48 CFR 52.219-1 - Small Business Program Representations.
Code of Federal Regulations, 2011 CFR
2011-10-01
... as defined in 13 CFR 124.1002. (3) (Complete only if the offeror represented itself as a small...; and (ii) It * is, * is not a joint venture that complies with the requirements of 13 CFR part 127, and... 13 CFR part 127, and the representation in paragraph (b)(5)(i) of this provision is accurate...
48 CFR 52.219-1 - Small Business Program Representations.
Code of Federal Regulations, 2010 CFR
2010-10-01
... as defined in 13 CFR 124.1002. (3) (Complete only if the offeror represented itself as a small... accordance with 13 CFR part 126; and (ii) It is, is not a joint venture that complies with the requirements of 13 CFR part 126, and the representation in paragraph (b)(6)(i) of this provision is accurate...
Moeller, Korbinian; Martignon, Laura; Wessolowski, Silvia; Engel, Joachim; Nuerk, Hans-Christoph
2011-01-01
Children typically learn basic numerical and arithmetic principles using finger-based representations. However, whether or not reliance on finger-based representations is beneficial or detrimental is the subject of an ongoing debate between researchers in neurocognition and mathematics education. From the neurocognitive perspective, finger counting provides multisensory input, which conveys both cardinal and ordinal aspects of numbers. Recent data indicate that children with good finger-based numerical representations show better arithmetic skills and that training finger gnosis, or "finger sense," enhances mathematical skills. Therefore neurocognitive researchers conclude that elaborate finger-based numerical representations are beneficial for later numerical development. However, research in mathematics education recommends fostering mentally based numerical representations so as to induce children to abandon finger counting. More precisely, mathematics education recommends first using finger counting, then concrete structured representations and, finally, mental representations of numbers to perform numerical operations. Taken together, these results reveal an important debate between neurocognitive and mathematics education research concerning the benefits and detriments of finger-based strategies for numerical development. In the present review, the rationale of both lines of evidence will be discussed. PMID:22144969
ERIC Educational Resources Information Center
Bengtson, Barbara J.
2013-01-01
Understanding the linear relationship of numbers is essential for doing practical and abstract mathematics throughout education and everyday life. There is evidence that number line activities increase learners' number sense, improving the linearity of mental number line representations (Siegler & Ramani, 2009). Mental representations of…
NASA Astrophysics Data System (ADS)
Gates, D. E. A.; Gates, S. James; Stiffler, Kory
2016-08-01
We present an expanded and detailed discussion of the mathematical tools required to cull and filter representations of the Coxeter Group BC 4 into providing bases for the construction of minimal off-shell representations of the 4D, {N} = 1 spacetime supersymmetry algebra.
ERIC Educational Resources Information Center
Fonger, Nicole L.
2012-01-01
Representational fluency (RF) includes an ability to interpret, create, move within and among, and connect tool-based representations of mathematical objects. Taken as an indicator of conceptual understanding, there is a need to better support school algebra students' RF in learning environments that utilize both computer algebra systems…
ERIC Educational Resources Information Center
Muzheve, Michael T.; Capraro, Robert M.
2012-01-01
Using qualitative data collection and analyses techniques, we examined mathematical representations used by sixteen (N=16) teachers while teaching the concepts of converting among fractions, decimals, and percents. We also studied representational choices by their students (N=581). In addition to using geometric figures and manipulatives, teachers…
Mathematical Modelling: A New Approach to Teaching Applied Mathematics.
ERIC Educational Resources Information Center
Burghes, D. N.; Borrie, M. S.
1979-01-01
Describes the advantages of mathematical modeling approach in teaching applied mathematics and gives many suggestions for suitable material which illustrates the links between real problems and mathematics. (GA)
Development of common neural representations for distinct numerical problems
Chang, Ting-Ting; Rosenberg-Lee, Miriam; Metcalfe, Arron W. S.; Chen, Tianwen; Menon, Vinod
2015-01-01
How the brain develops representations for abstract cognitive problems is a major unaddressed question in neuroscience. Here we tackle this fundamental question using arithmetic problem solving, a cognitive domain important for the development of mathematical reasoning. We first examined whether adults demonstrate common neural representations for addition and subtraction problems, two complementary arithmetic operations that manipulate the same quantities. We then examined how the common neural representations for the two problem types change with development. Whole-brain multivoxel representational similarity (MRS) analysis was conducted to examine common coding of addition and subtraction problems in children and adults. We found that adults exhibited significant levels of MRS between the two problem types, not only in the intra-parietal sulcus (IPS) region of the posterior parietal cortex (PPC), but also in ventral temporal-occipital, anterior temporal and dorsolateral prefrontal cortices. Relative to adults, children showed significantly reduced levels of MRS in these same regions. In contrast, no brain areas showed significantly greater MRS between problem types in children. Our findings provide novel evidence that the emergence of arithmetic problem solving skills from childhood to adulthood is characterized by maturation of common neural representations between distinct numerical operations, and involve distributed brain regions important for representing and manipulating numerical quantity. More broadly, our findings demonstrate that representational analysis provides a powerful approach for uncovering fundamental mechanisms by which children develop proficiencies that are a hallmark of human cognition. PMID:26160287
With age comes representational wisdom in social signals.
van Rijsbergen, Nicola; Jaworska, Katarzyna; Rousselet, Guillaume A; Schyns, Philippe G
2014-12-01
In an increasingly aging society, age has become a foundational dimension of social grouping broadly targeted by advertising and governmental policies. However, perception of old age induces mainly strong negative social biases. To characterize their cognitive and perceptual foundations, we modeled the mental representations of faces associated with three age groups (young age, middle age, and old age), in younger and older participants. We then validated the accuracy of each mental representation of age with independent validators. Using statistical image processing, we identified the features of mental representations that predict perceived age. Here, we show that whereas younger people mentally dichotomize aging into two groups, themselves (younger) and others (older), older participants faithfully represent the features of young age, middle age, and old age, with richer representations of all considered ages. Our results demonstrate that, contrary to popular public belief, older minds depict socially relevant information more accurately than their younger counterparts. PMID:25455036
With age comes representational wisdom in social signals.
van Rijsbergen, Nicola; Jaworska, Katarzyna; Rousselet, Guillaume A; Schyns, Philippe G
2014-12-01
In an increasingly aging society, age has become a foundational dimension of social grouping broadly targeted by advertising and governmental policies. However, perception of old age induces mainly strong negative social biases. To characterize their cognitive and perceptual foundations, we modeled the mental representations of faces associated with three age groups (young age, middle age, and old age), in younger and older participants. We then validated the accuracy of each mental representation of age with independent validators. Using statistical image processing, we identified the features of mental representations that predict perceived age. Here, we show that whereas younger people mentally dichotomize aging into two groups, themselves (younger) and others (older), older participants faithfully represent the features of young age, middle age, and old age, with richer representations of all considered ages. Our results demonstrate that, contrary to popular public belief, older minds depict socially relevant information more accurately than their younger counterparts.
ERIC Educational Resources Information Center
Taylor, Roger S.; Grundstrom, Erika D.
2011-01-01
Given that astronomy heavily relies on visual representations it is especially likely for individuals to assume that instructional materials, such as visual representations of the Earth-Moon system (EMS), would be relatively accurate. However, in our research, we found that images in middle-school textbooks and educational webpages were commonly…
Impact of High Mathematics Education on the Number Sense
Castronovo, Julie; Göbel, Silke M.
2012-01-01
In adult number processing two mechanisms are commonly used: approximate estimation of quantity and exact calculation. While the former relies on the approximate number sense (ANS) which we share with animals and preverbal infants, the latter has been proposed to rely on an exact number system (ENS) which develops later in life following the acquisition of symbolic number knowledge. The current study investigated the influence of high level math education on the ANS and the ENS. Our results showed that the precision of non-symbolic quantity representation was not significantly altered by high level math education. However, performance in a symbolic number comparison task as well as the ability to map accurately between symbolic and non-symbolic quantities was significantly better the higher mathematics achievement. Our findings suggest that high level math education in adults shows little influence on their ANS, but it seems to be associated with a better anchored ENS and better mapping abilities between ENS and ANS. PMID:22558077
On the Spontaneous Discovery of a Mathematical Relation during Problem Solving
ERIC Educational Resources Information Center
Dixon, James A.; Bangert, Ashley S.
2004-01-01
People spontaneously discover new representations during problem solving. Discovery of a mathematical representation is of special interest, because it shows that the underlying structure of the problem has been extracted. In the current study, participants solved gear-system problems as part of a game. Although none of the participants initially…
Machine learning of user profiles: Representational issues
Bloedorn, E.; Mani, I.; MacMillan, T.R.
1996-12-31
As more information becomes available electronically, tools for finding information of interest to users becomes increasingly important. The goal of the research described here is to build a system for generating comprehensible user profiles that accurately capture user interest with minimum user interaction. The research described here focuses on the importance of a suitable generalization hierarchy and representation for learning profiles which are predictively accurate and comprehensible. In our experiments we evaluated both traditional features based on weighted term vectors as well as subject features corresponding to categories which could be drawn from a thesaurus. Our experiments, conducted in the context of a content-based profiling system for on-line newspapers on the World Wide Web (the IDD News Browser), demonstrate the importance of a generalization hierarchy and the promise of combining natural language processing techniques with machine learning (ML) to address an information retrieval (ER) problem.
Mathematics and Sports. Mathematical World. Volume 3.
ERIC Educational Resources Information Center
Sadovskii, L. E.; Sadovskii, A. L.
This volume contains some examples of mathematical applications in sports. Sports discussed include tennis, figure skating, gymnastics, track and field, soccer, skiing, hockey, and swimming. Problems and situations are posed and answers with thorough explanations are provided. Chapters include: (1) Mathematics and Sports; (2) What Is Applied…
Mathematics Coursework Regulates Growth in Mathematics Achievement
ERIC Educational Resources Information Center
Ma, Xin; Wilkins, Jesse L. M.
2007-01-01
Using data from the Longitudinal Study of American Youth (LSAY), we examined the extent to which students' mathematics coursework regulates (influences) the rate of growth in mathematics achievement during middle and high school. Graphical analysis showed that students who started middle school with higher achievement took individual mathematics…
Mathematics for Language, Language for Mathematics
ERIC Educational Resources Information Center
Prochazkova, Lenka Tejkalova
2013-01-01
The author discusses the balance and mutual influence of the language of instruction and mathematics in the context of CLIL, Content and Language Integrated Learning. Different aspects of the relationship of language and Mathematics teaching and learning are discussed: the benefits of using a foreign language of instruction, as well as the…
Negotiation of Mathematical Meaning and Learning Mathematics.
ERIC Educational Resources Information Center
Voigt, Jorg
1994-01-01
Presents a case study of a first-grade class and their teacher who were observed as they ascribed mathematical meanings of numbers and of numerical operations to empirical phenomena. Differences in ascriptions led to negotiation of meanings. Discusses some indirect relations between social interaction and mathematics learning. (Contains 60…
Combination of direct matching and collaborative representation for face recognition
NASA Astrophysics Data System (ADS)
Zhang, Chongyang
2013-06-01
It has been proved that representation-based classification (RBC) can achieve high accuracy in face recognition. However, conventional RBC has a very high computational cost. Collaborative representation proposed in [1] not only has the advantages of RBC but also is computationally very efficient. In this paper, a combination of direct matching of images and collaborative representation is proposed for face recognition. Experimental results show that the proposed method can always classify more accurately than collaborative representation! The underlying reason is that direct matching of images and collaborative representation use different ways to calculate the dissimilarity between the test sample and training sample. As a result, the score obtained using direct matching of images is very complementary to the score obtained using collaborative representation. Actually, the analysis shows that the matching scores generated from direct matching of images and collaborative representation always have a low correlation. This allows the proposed method to exploit more information for face recognition and to produce a better result.
[Suffering: representations and challenges].
Dany, Lionel; Dormieux, Anne; Futo, Francette; Favre, Roger
2006-03-01
Research that we carried out aimed to analyzing the suffering as an object of the relation of care and common experiment of this relation. In this way, we have conducted 55 interviews with patients and nurses in an oncology unit. The results underline the central role of the relation for giving meaning to the suffering. The knowledge of the suffering representations appears as a tool for understanding the complex realities connected to the experiences of suffering, and allows to imagine more adapted evaluation methods which could be anchored on "practical knowledge". PMID:16711083
Spatial representations are specific to different domains of knowledge.
Beecham, Rowena; Reeve, Robert A; Wilson, Sarah J
2009-05-20
There is evidence that many abstract concepts are represented cognitively in a spatial format. However, it is unknown whether similar spatial processes are employed in different knowledge domains, or whether individuals exhibit similar spatial profiles within and across domains. This research investigated similarities in spatial representation in two knowledge domains--mathematics and music. Sixty-one adults completed analogous number magnitude and pitch discrimination tasks: the Spatial-Numerical Association of Response Codes and Spatial-Musical Association of Response Codes tasks. Subgroups of individuals with different response patterns were identified through cluster analyses. For both the mathematical and musical tasks, approximately half of the participants showed the expected spatial judgment effect when explicitly cued to focus on the spatial properties of the stimuli. Despite this, performances on the two tasks were largely independent. Consistent with previous research, the study provides evidence for the spatial representation of number and pitch in the majority of individuals. However, there was little evidence to support the claim that the same spatial representation processes underpin mathematical and musical judgments.
Spatial Representations Are Specific to Different Domains of Knowledge
Beecham, Rowena; Reeve, Robert A.; Wilson, Sarah J.
2009-01-01
There is evidence that many abstract concepts are represented cognitively in a spatial format. However, it is unknown whether similar spatial processes are employed in different knowledge domains, or whether individuals exhibit similar spatial profiles within and across domains. This research investigated similarities in spatial representation in two knowledge domains – mathematics and music. Sixty-one adults completed analogous number magnitude and pitch discrimination tasks: the Spatial-Numerical Association of Response Codes and Spatial-Musical Association of Response Codes tasks. Subgroups of individuals with different response patterns were identified through cluster analyses. For both the mathematical and musical tasks, approximately half of the participants showed the expected spatial judgment effect when explicitly cued to focus on the spatial properties of the stimuli. Despite this, performances on the two tasks were largely independent. Consistent with previous research, the study provides evidence for the spatial representation of number and pitch in the majority of individuals. However, there was little evidence to support the claim that the same spatial representation processes underpin mathematical and musical judgments. PMID:19461994
Mental arithmetic activates analogic representations of internally generated sums.
Kallai, Arava Y; Schunn, Christian D; Fiez, Julie A
2012-08-01
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 study, the spontaneous processing of arithmetical expressions was tested. Participants passively viewed a sequence of double-digit addition expressions that summed to the same number. Adaptation was found in number-related regions in a fronto-parietal network. Following adaptation, arrays of dots were introduced, differing in their numerical distance from the sum of the addition expressions. Activation in voxels that showed adaptation to a repeated sum was also sensitive to the distance of the dot quantity from the sum. We conclude that participants exhibited adaptation to an internally generated number, that adapted representations were analogic in nature, and that these analogic representations may undergird arithmetic calculation. PMID:22732492
Bag of Lines (BoL) for Improved Aerial Scene Representation
Sridharan, Harini; Cheriyadat, Anil M.
2014-09-22
Feature representation is a key step in automated visual content interpretation. In this letter, we present a robust feature representation technique, referred to as bag of lines (BoL), for high-resolution aerial scenes. The proposed technique involves extracting and compactly representing low-level line primitives from the scene. The compact scene representation is generated by counting the different types of lines representing various linear structures in the scene. Through extensive experiments, we show that the proposed scene representation is invariant to scale changes and scene conditions and can discriminate urban scene categories accurately. We compare the BoL representation with the popular scale invariant feature transform (SIFT) and Gabor wavelets for their classification and clustering performance on an aerial scene database consisting of images acquired by sensors with different spatial resolutions. The proposed BoL representation outperforms the SIFT- and Gabor-based representations.
Bag of Lines (BoL) for Improved Aerial Scene Representation
Sridharan, Harini; Cheriyadat, Anil M.
2014-09-22
Feature representation is a key step in automated visual content interpretation. In this letter, we present a robust feature representation technique, referred to as bag of lines (BoL), for high-resolution aerial scenes. The proposed technique involves extracting and compactly representing low-level line primitives from the scene. The compact scene representation is generated by counting the different types of lines representing various linear structures in the scene. Through extensive experiments, we show that the proposed scene representation is invariant to scale changes and scene conditions and can discriminate urban scene categories accurately. We compare the BoL representation with the popular scalemore » invariant feature transform (SIFT) and Gabor wavelets for their classification and clustering performance on an aerial scene database consisting of images acquired by sensors with different spatial resolutions. The proposed BoL representation outperforms the SIFT- and Gabor-based representations.« less
Accurate documentation and wound measurement.
Hampton, Sylvie
This article, part 4 in a series on wound management, addresses the sometimes routine yet crucial task of documentation. Clear and accurate records of a wound enable its progress to be determined so the appropriate treatment can be applied. Thorough records mean any practitioner picking up a patient's notes will know when the wound was last checked, how it looked and what dressing and/or treatment was applied, ensuring continuity of care. Documenting every assessment also has legal implications, demonstrating due consideration and care of the patient and the rationale for any treatment carried out. Part 5 in the series discusses wound dressing characteristics and selection.
Perspectives from Mathematics Education.
ERIC Educational Resources Information Center
Sowder, Judith Threadgill
1998-01-01
Explores possible reasons for the gender differences in mathematics problem-solving strategies found in primary-grade students in the study by E. Fennema and others and considers implications of the findings for mathematics instruction. (SLD)
Mathematical and statistical analysis
NASA Technical Reports Server (NTRS)
Houston, A. Glen
1988-01-01
The goal of the mathematical and statistical analysis component of RICIS is to research, develop, and evaluate mathematical and statistical techniques for aerospace technology applications. Specific research areas of interest include modeling, simulation, experiment design, reliability assessment, and numerical analysis.
Developing My Mathematics Identity
ERIC Educational Resources Information Center
Gonzalez, Lidia
2016-01-01
Assuming the role of storyteller, the author uses her experiences as a graduate student and beginning teacher to reflect critically on issues related to mathematics, mathematics education, gender, and diversity.
Mental Mathematics Moves Ahead.
ERIC Educational Resources Information Center
Jones, Pamela
1988-01-01
The author suggests that the efficient use of mathematics in everyday life means translating situations into mathematical contexts, using a calculator and mental methods of calculation. Suggestions for teaching these concepts are included. (PK)
Leikin, Mark; Waisman, Ilana; Shaul, Shelley; Leikin, Roza
2014-03-01
This paper presents a small part of a larger interdisciplinary study that investigates brain activity (using event related potential methodology) of male adolescents when solving mathematical problems of different types. The study design links mathematics education research with neurocognitive studies. In this paper we performed a comparative analysis of brain activity associated with the translation from visual to symbolic representations of mathematical objects in algebra and geometry. Algebraic tasks require translation from graphical to symbolic representation of a function, whereas tasks in geometry require translation from a drawing of a geometric figure to a symbolic representation of its property. The findings demonstrate that electrical activity associated with the performance of geometrical tasks is stronger than that associated with solving algebraic tasks. Additionally, we found different scalp topography of the brain activity associated with algebraic and geometric tasks. Based on these results, we argue that problem solving in algebra and geometry is associated with different patterns of brain activity.
Aerial Scene Recognition using Efficient Sparse Representation
Cheriyadat, Anil M
2012-01-01
Advanced scene recognition systems for processing large volumes of high-resolution aerial image data are in great demand today. However, automated scene recognition remains a challenging problem. Efficient encoding and representation of spatial and structural patterns in the imagery are key in developing automated scene recognition algorithms. We describe an image representation approach that uses simple and computationally efficient sparse code computation to generate accurate features capable of producing excellent classification performance using linear SVM kernels. Our method exploits unlabeled low-level image feature measurements to learn a set of basis vectors. We project the low-level features onto the basis vectors and use simple soft threshold activation function to derive the sparse features. The proposed technique generates sparse features at a significantly lower computational cost than other methods~\\cite{Yang10, newsam11}, yet it produces comparable or better classification accuracy. We apply our technique to high-resolution aerial image datasets to quantify the aerial scene classification performance. We demonstrate that the dense feature extraction and representation methods are highly effective for automatic large-facility detection on wide area high-resolution aerial imagery.
Incidental statistical summary representation over time.
Oriet, Chris; Hozempa, Kadie
2016-01-01
Information taken in by the human visual system allows individuals to form statistical representations of sets of items. One's knowledge of natural categories includes statistical information, such as average size of category members and the upper and lower boundaries of the set. Previous research suggests that when subjects attend to a particular dimension of a set of items presented over an extended duration, they quickly learn about the central tendency of the set. However, it is unclear whether such learning can occur incidentally, when subjects are not attending to the relevant dimension of the set. The present study explored whether subjects could reproduce global statistical properties of a set presented over an extended duration when oriented to task-irrelevant properties of the set. Subjects were tested for their memory of its mean, its smallest and largest exemplars, the direction of its skew, and the relative distribution of the items. Subjects were able to accurately recall the average size circle, as well as the upper and lower boundaries of a set of 4,200 circles displayed over an extended period. This suggests that even without intending to do so, they were encoding and updating a statistical summary representation of a task-irrelevant attribute of the circles over time. Such incidental encoding of statistical properties of sets is thus a plausible mechanism for establishing a representation of typicality in category membership. PMID:26830709
The MATH--Open Source Application for Easier Learning of Numerical Mathematics
ERIC Educational Resources Information Center
Glaser-Opitz, Henrich; Budajová, Kristina
2016-01-01
The article introduces a software application (MATH) supporting an education of Applied Mathematics, with focus on Numerical Mathematics. The MATH is an easy to use tool supporting various numerical methods calculations with graphical user interface and integrated plotting tool for graphical representation written in Qt with extensive use of Qwt…
ERIC Educational Resources Information Center
Pennington, Charlotte R.; Heim, Derek
2016-01-01
Background: Women in mathematical domains may become attuned to situational cues that signal a discredited social identity, contributing to their lower achievement and underrepresentation. Aim: This study examined whether heightened in-group representation alleviates the effects of stereotype threat on women's mathematical performance. It further…
ERIC Educational Resources Information Center
Hawkins, William A.
This report examines issues in the low achievement of American students in mathematics, with emphasis on the low representation of minority students in this field. American myths about mathematics which emphasize the importance of innate ability rather than hard work are seen as reinforcing racial and gender stereotypes about who can do…
Investigating Lebanese Grade Seven Biology Teachers Mathematical Knowledge and Skills: A Case Study
ERIC Educational Resources Information Center
Raad, Nawal Abou; Chatila, Hanadi
2016-01-01
This paper investigates Lebanese grade 7 biology teachers' mathematical knowledge and skills, by exploring how they explain a visual representation in an activity depending on the mathematical concept "Function". Twenty Lebanese in-service biology teachers participated in the study, and were interviewed about their explanation for the…
ERIC Educational Resources Information Center
Yang, Euphony F. Y.; Chang, Ben; Cheng, Hercy N. H.; Chan, Tak-Wai
2016-01-01
This study examined how to foster pupils' mathematical communication abilities by using tablet PCs. Students were encouraged to generate math creations (including mathematical representation, solution, and solution explanation of word problems) as their teaching materials and reciprocally tutor classmates to increase opportunities for mathematical…
Action-Based Digital Tools: Mathematics Learning in 6-Year-Old Children
ERIC Educational Resources Information Center
Dejonckheere, Peter J. N.; Desoete, Annemie; Fonck, Nathalie; Roderiguez, Dave; Six, Leen; Vermeersch, Tine; Vermeulen, Lies
2014-01-01
Introduction: In the present study we used a metaphorical representation in order to stimulate the numerical competences of six-year-olds. It was expected that when properties of physical action are used for mathematical thinking or when abstract mathematical thinking is grounded in sensorimotor processes, learning gains should be more pronounced…
The Curious--and Crucial--Case of Mathematical Knowledge for Teaching
ERIC Educational Resources Information Center
Hill, Heather; Ball, Deborah Loewenberg
2009-01-01
Mathematics teachers need specialized math knowledge that is different from the knowledge needed by mathematicians, including the mathematical understanding involved in posing questions, interpreting students' answers, providing explanations, and using representations. Based on the authors' work involving over 300 teachers, they have found that…
ERIC Educational Resources Information Center
Warren, Elizabeth; Miller, Jodie
2015-01-01
In the Australian context, children living in disadvantaged circumstances, whose second language is English, are one of the groups at risk of failing in mathematics. This paper explores the impact purposely developed learning activities (Representations, Oral Language and Engagement in Mathematics Learning activities) have on pupils' mathematics…
ERIC Educational Resources Information Center
Sella, Francesco; Sader, Elie; Lolliot, Simon; Cohen Kadosh, Roi
2016-01-01
Recent studies have highlighted the potential role of basic numerical processing in the acquisition of numerical and mathematical competences. However, it is debated whether high-level numerical skills and mathematics depends specifically on basic numerical representations. In this study mathematicians and nonmathematicians performed a basic…
Evolving Polygons and Spreadsheets: Connecting Mathematics across Grade Levels in Teacher Education
ERIC Educational Resources Information Center
Abramovich, Sergei; Brouwer, Peter
2009-01-01
This paper was prepared in response to the Conference Board of Mathematical Sciences recommendations for the preparation of secondary teachers. It shows how using trigonometry as a conceptual tool in spreadsheet-based applications enables one to develop mathematical understanding in the context of constructing geometric representations of unit…
Metaphorical Reasoning in Mathematics: Experts and Novices Solving Negative Number Problems.
ERIC Educational Resources Information Center
Chiu, Ming Ming
Metaphorical reasoning explains how people can interpret abstract representations through a complex activity and then apply them to new problems. In particular, metaphors can facilitate both conceptual understanding and problem solving by: (1) intuitively justifying mathematical operations, (2) integrating mathematical knowledge, (3) enhancing the…
Implementing CRA with Secondary Students with Learning Disabilities in Mathematics
ERIC Educational Resources Information Center
Witzel, Bradley S.; Riccomini, Paul J.; Schneider, Elke
2008-01-01
Students with learning disabilities struggle to acquire essential mathematical concepts and skills, especially at the secondary level. One effective approach to improving secondary math performance supported by research is the concrete-to-representational-to-abstract (CRA) sequence of instruction. Although CRA is an evidenced-based instructional…
Affect in Mathematics Education--Exploring Theoretical Frameworks. Research Forum
ERIC Educational Resources Information Center
Hannula, Markku; Evans, Jeff; Philippou, George; Zan, Rosetta
2004-01-01
This document brings into a dialogue some of the theoretical frameworks used to study affect in mathematics education. It presents affect as a representational system, affect as one regulator of the dynamic self, affect in a socio-constructivist framework, and affect as embodied. It also evaluates these frameworks from different perspectives:…
Expanding Notions of "Learning Trajectories" in Mathematics Education
ERIC Educational Resources Information Center
Weber, Eric; Walkington, Candace; McGalliard, William
2015-01-01
Over the past 20 years learning trajectories and learning progressions have gained prominence in mathematics and science education research. However, use of these representations ranges widely in breadth and depth, often depending on from what discipline they emerge and the type of learning they intend to characterize. Learning trajectories…
ERIC Educational Resources Information Center
Clary, Joseph R.; Nery, Karen P.
This set of 20 modules was designed for use primarily to help teach and reinforce the basic mathematics skills in electronics classes. The modules are based on electronics competencies that require mathematics skills, as determined by a panel of high school electronics and mathematics teachers. Each module consists of one or two pages of basic…
ERIC Educational Resources Information Center
Pratt, Dave
2012-01-01
Mathematics is often portrayed as an "abstract" cerebral subject, beyond the reach of many. In response, research with digital technology has led to innovative design in which mathematics can be experienced much like everyday phenomena. This lecture examines how careful design can "phenomenalise" mathematics and support not only engagement but…
Translations toward Connected Mathematics
ERIC Educational Resources Information Center
Applebaum, Mark; Leikin, Roza
2010-01-01
The translation principle allows students to solve problems in different branches of mathematics and thus to develop connectedness in their mathematical knowledge. Successful application of the translation principle depends on the classroom mathematical norms for the development of discussions and the comparison of different solutions to one…
Mathematics and Comprehensive Ideals
ERIC Educational Resources Information Center
Watson, Anne
2011-01-01
This article revisits methods and debates about teaching mathematics that were common in the 1980s and early 1990s, and then moves up to date with the findings from three mathematics departments that set out to make a difference for their lowest attaining students. The methods they used were distinctly focused on core mathematical ideas, and how…
Winning Women into Mathematics.
ERIC Educational Resources Information Center
Kenschaft, Patricia Clark, Ed.; Keith, Sandra Zaroodny, Ed.
Mathematics is an auspicious discipline for young people of both sexes and all ethnic groups. This booklet aims to help members of the Mathematical Association of America (MAA) and others to increase future participation of women in mathematics, to better understand their present roles, and to develop a vision of a world with greater equal…
ERIC Educational Resources Information Center
Martin, Tami S.; Speer, William R.
2009-01-01
This article describes features, consistent messages, and new components of "Mathematics Teaching Today: Improving Practice, Improving Student Learning" (NCTM 2007), an updated edition of "Professional Standards for Teaching Mathematics" (NCTM 1991). The new book describes aspects of high-quality mathematics teaching; offers a model for observing,…
Revisiting Mathematics Manipulative Materials
ERIC Educational Resources Information Center
Swan, Paul; Marshall, Linda
2010-01-01
It is over 12 years since "APMC" published Bob Perry and Peter Howard's research on the use of mathematics manipulative materials in primary mathematics classrooms. Since then the availability of virtual manipulatives and associated access to computers and interactive whiteboards have caused educators to rethink the use of mathematics manipulative…
Rural Mathematics Educator, 2002.
ERIC Educational Resources Information Center
Rural Mathematics Educator, 2002
2002-01-01
This document contains the two issues of "Rural Mathematics Educator" published in 2002. This newsletter of the Appalachian Collaborative Center for Learning, Assessment, and Instruction in Mathematics (ACCLAIM) includes articles on rural mathematics education, as well as information and descriptions of professional development opportunities for…
Mathematics and Global Survival.
ERIC Educational Resources Information Center
Schwartz, Richard H.
This resource was written to provide students with an awareness of critical issues facing the world today. In courses for college students, it can motivate their study of mathematics, teach them how to solve mathematical problems related to current global issues, provide coherence to mathematical studies through a focus on issues of human…
ERIC Educational Resources Information Center
Walshaw, Margaret
2014-01-01
This paper explores contemporary thinking about learning mathematics, and within that, social justice within mathematics education. The discussion first looks at mechanisms offered by conventional explanations on the emancipatory project and then moves towards more recent insights developed within mathematics education. Synergies are drawn between…
Making Mathematics Culturally Relevant.
ERIC Educational Resources Information Center
Moyer, Patricia
2001-01-01
Examines three strands of elementary mathematics--numerals and counting, recording and calculating, and mathematics exploration and play--and provides ways to integrate culture and mathematics experiences in each area. Specific topics include Egyptian methods for multiplication, the abacus, and the words for the numbers 1-10 in seven different…
Mathematical Epistemologies at Work.
ERIC Educational Resources Information Center
Noss, Richard
In this paper, I draw together a corpus of findings derived from two sources: studies of students using computers to learn mathematics, and research into the use of mathematics in professional practice. Using this as a basis, I map some elements of a theoretical framework for understanding the nature of mathematical knowledge in use, and how it is…
ERIC Educational Resources Information Center
Darlington, Ellie
2014-01-01
This article describes part of a study which investigated the role of questions in students' approaches to learning mathematics at the secondary-tertiary interface, focussing on the enculturation of students at the University of Oxford. Use of the Mathematical Assessment Task Hierarchy taxonomy revealed A-level Mathematics and Further Mathematics…
ERIC Educational Resources Information Center
Gallian, Joseph A., Ed.
2010-01-01
"Mathematics and Sports", edited by Joseph A. Gallian, gathers 25 articles that illuminate the power and role of mathematics in the worlds of professional and recreational play. Divided into sections by the kind of sports, the book offers source materials for classroom use and student projects. Readers will encounter mathematical ideas from an…
A "Mathematics Background Check"
ERIC Educational Resources Information Center
Hubisz, John
2009-01-01
Early in my career someone else reported that the best indicator of success in calculus-based physics (CBP) at our school was whether students had taken mathematics in a certain region of New Brunswick. I sat down with a very longtime mathematics teacher and asked him what he thought students should know in mathematics after high school to succeed…
ERIC Educational Resources Information Center
Posey, Johnsie Jo, Ed.; And Others
This manual is a collection of materials and teaching strategies to motivate the development of mathematical ideas in secondary school mathematics programs or in beginning college mathematics programs. The unit is written for the instructor with step-by-step procedures including lists of needed materials. The exercises in this unit also appear in…
Mathematics and Mobile Learning
ERIC Educational Resources Information Center
Sayed, Fayez
2015-01-01
The wide range of Mathematical Apps targeting different mathematical concepts and the various types of mobile devices available present a demanding and challenging problem to the teaching and learning in the field of mathematics. In an attempt to address this issue, a few Apps were selected, implemented and tested in this work. [For complete…
ERIC Educational Resources Information Center
Ortiz-Franco, Luis
An historical perspective reveals that sophisticated mathematical activity has been going on in the Latino culture for thousands of years. This paper provides a general definition of the area of mathematics education that deals with issues of culture and mathematics (ethnomathematics) and defines what is meant by the term Latino in this essay.…
Mathenger Hunt: Mathematics Matters.
ERIC Educational Resources Information Center
Falba, Christy J.; Weiss, Maria J.
1991-01-01
Presented is an activity which shows how mathematics is used in real life and helps to establish a need for mathematics in students' futures. Adapted from a scavenger-hunt idea, this activity helps students to discover that almost every career makes use of mathematics. (KR)
Mathematics. [SITE 2001 Section].
ERIC Educational Resources Information Center
Connell, Michael L., Ed.; Lowery, Norene Vail, Ed.; Harnisch, Delwyn L., Ed.
This document contains the following papers on mathematics from the SITE (Society for Information Technology & Teacher Education) 2001 conference: "Secondary Mathematics Methods Course with Technology Units: Encouraging Pre-Service Teachers To Use Technology" (Rajee Amarasinghe); "Competency Exams in College Mathematics" (Kathy R. Autrey and Leigh…
Applying Mathematical Processes (AMP)
ERIC Educational Resources Information Center
Kathotia, Vinay
2011-01-01
This article provides insights into the "Applying Mathematical Processes" resources, developed by the Nuffield Foundation. It features Nuffield AMP activities--and related ones from Bowland Maths--that were designed to support the teaching and assessment of key processes in mathematics--representing a situation mathematically, analysing,…
Abraham, Ralph
2015-12-01
Is there a world of mathematics above and beyond ordinary reality, as Plato proposed? Or is mathematics a cultural construct? In this short article we speculate on the place of mathematical reality from the perspective of the mystical cosmologies of the ancient traditions of meditation, psychedelics, and divination.
ERIC Educational Resources Information Center
Galligan, Linda
2016-01-01
A "National Numeracy Report" and the Australian Curriculum (2014) have recognised the importance of language in mathematics. The general capabilities contained within the "Australian Curriculum: Mathematics" (2014) highlight literacy as an important tool in the teaching and learning of mathematics, from the interpretation of…
ERIC Educational Resources Information Center
Moreau, Marie-Pierre; Mendick, Heather; Epstein, Debbie
2010-01-01
In this paper, based on a project funded by the UK Economic and Social Research Council considering how people position themselves in relation to popular representations of mathematics and mathematicians, we explore constructions of mathematicians in popular culture and the ways learners make meanings from these. Drawing on an analysis of popular…
Mathematical models of diabetes progression.
De Gaetano, Andrea; Hardy, Thomas; Beck, Benoit; Abu-Raddad, Eyas; Palumbo, Pasquale; Bue-Valleskey, Juliana; Pørksen, Niels
2008-12-01
Few attempts have been made to model mathematically the progression of type 2 diabetes. A realistic representation of the long-term physiological adaptation to developing insulin resistance is necessary for effectively designing clinical trials and evaluating diabetes prevention or disease modification therapies. Writing a good model for diabetes progression is difficult because the long time span of the disease makes experimental verification of modeling hypotheses extremely awkward. In this context, it is of primary importance that the assumptions underlying the model equations properly reflect established physiology and that the mathematical formulation of the model give rise only to physically plausible behavior of the solutions. In the present work, a model of the pancreatic islet compensation is formulated, its physiological assumptions are presented, some fundamental qualitative characteristics of its solutions are established, the numerical values assigned to its parameters are extensively discussed (also with reference to available cross-sectional epidemiologic data), and its performance over the span of a lifetime is simulated under various conditions, including worsening insulin resistance and primary replication defects. The differences with respect to two previously proposed models of diabetes progression are highlighted, and therefore, the model is proposed as a realistic, robust description of the evolution of the compensation of the glucose-insulin system in healthy and diabetic individuals. Model simulations can be run from the authors' web page.
SPLASH: Accurate OH maser positions
NASA Astrophysics Data System (ADS)
Walsh, Andrew; Gomez, Jose F.; Jones, Paul; Cunningham, Maria; Green, James; Dawson, Joanne; Ellingsen, Simon; Breen, Shari; Imai, Hiroshi; Lowe, Vicki; Jones, Courtney
2013-10-01
The hydroxyl (OH) 18 cm lines are powerful and versatile probes of diffuse molecular gas, that may trace a largely unstudied component of the Galactic ISM. SPLASH (the Southern Parkes Large Area Survey in Hydroxyl) is a large, unbiased and fully-sampled survey of OH emission, absorption and masers in the Galactic Plane that will achieve sensitivities an order of magnitude better than previous work. In this proposal, we request ATCA time to follow up OH maser candidates. This will give us accurate (~10") positions of the masers, which can be compared to other maser positions from HOPS, MMB and MALT-45 and will provide full polarisation measurements towards a sample of OH masers that have not been observed in MAGMO.
Accurate thickness measurement of graphene
NASA Astrophysics Data System (ADS)
Shearer, Cameron J.; Slattery, Ashley D.; Stapleton, Andrew J.; Shapter, Joseph G.; Gibson, Christopher T.
2016-03-01
Graphene has emerged as a material with a vast variety of applications. The electronic, optical and mechanical properties of graphene are strongly influenced by the number of layers present in a sample. As a result, the dimensional characterization of graphene films is crucial, especially with the continued development of new synthesis methods and applications. A number of techniques exist to determine the thickness of graphene films including optical contrast, Raman scattering and scanning probe microscopy techniques. Atomic force microscopy (AFM), in particular, is used extensively since it provides three-dimensional images that enable the measurement of the lateral dimensions of graphene films as well as the thickness, and by extension the number of layers present. However, in the literature AFM has proven to be inaccurate with a wide range of measured values for single layer graphene thickness reported (between 0.4 and 1.7 nm). This discrepancy has been attributed to tip-surface interactions, image feedback settings and surface chemistry. In this work, we use standard and carbon nanotube modified AFM probes and a relatively new AFM imaging mode known as PeakForce tapping mode to establish a protocol that will allow users to accurately determine the thickness of graphene films. In particular, the error in measuring the first layer is reduced from 0.1-1.3 nm to 0.1-0.3 nm. Furthermore, in the process we establish that the graphene-substrate adsorbate layer and imaging force, in particular the pressure the tip exerts on the surface, are crucial components in the accurate measurement of graphene using AFM. These findings can be applied to other 2D materials.
Accurate thickness measurement of graphene.
Shearer, Cameron J; Slattery, Ashley D; Stapleton, Andrew J; Shapter, Joseph G; Gibson, Christopher T
2016-03-29
Graphene has emerged as a material with a vast variety of applications. The electronic, optical and mechanical properties of graphene are strongly influenced by the number of layers present in a sample. As a result, the dimensional characterization of graphene films is crucial, especially with the continued development of new synthesis methods and applications. A number of techniques exist to determine the thickness of graphene films including optical contrast, Raman scattering and scanning probe microscopy techniques. Atomic force microscopy (AFM), in particular, is used extensively since it provides three-dimensional images that enable the measurement of the lateral dimensions of graphene films as well as the thickness, and by extension the number of layers present. However, in the literature AFM has proven to be inaccurate with a wide range of measured values for single layer graphene thickness reported (between 0.4 and 1.7 nm). This discrepancy has been attributed to tip-surface interactions, image feedback settings and surface chemistry. In this work, we use standard and carbon nanotube modified AFM probes and a relatively new AFM imaging mode known as PeakForce tapping mode to establish a protocol that will allow users to accurately determine the thickness of graphene films. In particular, the error in measuring the first layer is reduced from 0.1-1.3 nm to 0.1-0.3 nm. Furthermore, in the process we establish that the graphene-substrate adsorbate layer and imaging force, in particular the pressure the tip exerts on the surface, are crucial components in the accurate measurement of graphene using AFM. These findings can be applied to other 2D materials.
Resource representation in COMPASS
NASA Technical Reports Server (NTRS)
Fox, Barry R.
1991-01-01
A set of viewgraphs on resource representation in COMPASS is given. COMPASS is an incremental, interactive, non-chronological scheduler written in Ada with an X-windows user interface. Beginning with an empty schedule, activities are added to the schedule one at a time, taking into consideration the placement of the activities already on the timeline and the resources that have been reserved for them. The order that the activities are added to the timeline and their location on the timeline are controlled by selection and placement commands invoked by the user. The order that activities are added to the timeline and their location are independent. The COMPASS code library is a cost effective platform for the development of new scheduling applications. It can be effectively used off the shelf for compatible scheduling applications or it can be used as a parts library for the development of custom scheduling systems.
Intentionality, Representation, and Anticipation
NASA Astrophysics Data System (ADS)
De Preester, Helena
2002-09-01
Both Brentano and Merleau-Ponty have developed an account of intentionality, which nevertheless differ profoundly in the following respect. According to Brentano, intentionality mainly is a matter of mental presentations. This marks the beginning of phenomenology's difficult relation with the nature of the intentional reference. Merleau-Ponty, on the other hand, has situated intentionality on the level of the body, a turn which has important implications for the nature of intentionality. Intentionality no longer is primarily based on having (re)presentations, but is rooted in the dynamics of the living body. To contrast those approaches enables us to make clear in what way intentionality is studied nowadays. On the one hand, intentionality is conceived of as a matter of formal-syntactical causality in cognitive science, and in particular in classical-computational theory. On the other hand, a interactivist approach offers a more Merleau-Ponty-like point of view, in which autonomy, embodiment and interaction are stressed.
Parental representations of transsexuals.
Parker, G; Barr, R
1982-06-01
The parental representations of 30 male-to-female transsexuals were rated using a measure of fundamental parental dimensions and shown to be of acceptable validity as a measure both of perceived and actual parental characteristics. Scores on that measure were compared separately against scores returned by matched male and female controls. The transsexuals did not differ from the male controls in their scoring of their mothers but did score their fathers as less caring and more overprotective. These differences were weaker for the comparisons made against the female controls. Item analyses suggested that the greater paternal "overprotection" experienced by transsexuals was due to their fathers being perceived as offering less encouragement to their sons' independence and autonomy. Several interpretations of the findings are considered. PMID:7138296
Age Differences in Symbolic Representation: Fluidity in Representational Construction.
ERIC Educational Resources Information Center
Reifel, Stuart
This paper reports a cross-sectional, developmental study of the fluidity of children's mental functioning (representational skills) in contexts involving the representational use of blocks. Data were collected from a sample of 40 children from a laboratory school: 20 four-year-olds and 20 seven-year-olds, with an equal number of boys and girls in…
Neuronal foundations of human numerical representations.
Eger, E
2016-01-01
The human species has developed complex mathematical skills which likely emerge from a combination of multiple foundational abilities. One of them seems to be a preverbal capacity to extract and manipulate the numerosity of sets of objects which is shared with other species and in humans is thought to be integrated with symbolic knowledge to result in a more abstract representation of numerical concepts. For what concerns the functional neuroanatomy of this capacity, neuropsychology and functional imaging have localized key substrates of numerical processing in parietal and frontal cortex. However, traditional fMRI mapping relying on a simple subtraction approach to compare numerical and nonnumerical conditions is limited to tackle with sufficient precision and detail the issue of the underlying code for number, a question which more easily lends itself to investigation by methods with higher spatial resolution, such as neurophysiology. In recent years, progress has been made through the introduction of approaches sensitive to within-category discrimination in combination with fMRI (adaptation and multivariate pattern recognition), and the present review summarizes what these have revealed so far about the neural coding of individual numbers in the human brain, the format of these representations and parallels between human and monkey neurophysiology findings. PMID:27339006
ERIC Educational Resources Information Center
Chen, Minjie
2009-01-01
The sheer amount of American children's and young adult literature, boasting an outpouring of 5,000 titles every year, often amazes a person who is new to this field. Not only is a large proportion of these books of high printing and binding quality, but, at a quick glance, among them is also a pleasant diversity of genre, format, targeted age…
The History of Mathematics and Mathematical Education
ERIC Educational Resources Information Center
Grattan-Guinness, I.
1977-01-01
Answers to questions which were asked after the author's various lectures in Australia are gathered here. Topics touched upon include "new" mathematics, unknown constants and free variables, propositional functions, linear algebra, arithmetic and geometry, and student assessment. (MN)
Core number representations are shaped by language.
Salillas, Elena; Carreiras, Manuel
2014-03-01
Language and math have been predominantly related through exact calculation. In the present study we investigated a more fundamental link between language and math: whether the most basic quantity representation used for the contrast of numerosities could be shaped by language. We selected two groups of balanced, equally proficient Basque-Spanish bilinguals. Crucially, the two groups differed with respect to the language in which math had been learned at the point of earliest formal instruction in mathematics (Language of learning Math - LL(math)). They performed a simple comparison task between pairs of Arabic digits related through the decimal system or through the vigesimal system. The vigesimal system is retained in Basque for the naming of certain numerals, while for other numerals the decimal system is used, just as for all Spanish number words. Event-related potential (ERP) distance effects were taken as the dependent variable, indexing the activation of quantity. Results showed an N1-P2 distance effect during the comparison of digit pairs related through the base-10 system in both groups. Importantly, this N1-P2 effect appeared only for the group whose LL(math) was Basque when base-20 related digits were compared, even if both groups were perfectly fluent in Basque. Thus the early N1-P2 component appears to be sensitive to verbal components contained in quantity representation. Since the task did not contain any verbal input, the present data suggest that quantity representation may have verbal traces inherited from early learning. In turn, LL(math) should be the optimal medium for numerical communication.
Vukovic, Rose K; Lesaux, Nonie K
2013-06-01
This longitudinal study examined how language ability relates to mathematical development in a linguistically and ethnically diverse sample of children from 6 to 9 years of age. Study participants were 75 native English speakers and 92 language minority learners followed from first to fourth grades. Autoregression in a structural equation modeling (SEM) framework was used to evaluate the relation between children's language ability and gains in different domains of mathematical cognition (i.e., arithmetic, data analysis/probability, algebra, and geometry). The results showed that language ability predicts gains in data analysis/probability and geometry, but not in arithmetic or algebra, after controlling for visual-spatial working memory, reading ability, and sex. The effect of language on gains in mathematical cognition did not differ between language minority learners and native English speakers. These findings suggest that language influences how children make meaning of mathematics but is not involved in complex arithmetical procedures whether presented with Arabic symbols as in arithmetic or with abstract symbols as in algebraic reasoning. The findings further indicate that early language experiences are important for later mathematical development regardless of language background, denoting the need for intensive and targeted language opportunities for language minority and native English learners to develop mathematical concepts and representations.
Computer aided surface representation
Barnhill, R E
1987-11-01
The aims of this research are the creation of new surface forms and the determination of geometric and physical properties of surfaces. The full sweep from constructive mathematics through the implementation of algorithms and the interactive computer graphics display of surfaces is utilized. Both three-dimensional and multi- dimensional surfaces are considered. Particular emphasis is given to the scientific computing solution of Department of Energy problems. The methods that we have developed and that we are proposing to develop allow applications such as: Producing smooth contour maps from measured data, such as weather maps. Modeling the heat distribution inside a furnace from sample measurements. Terrain modeling based on satellite pictures. The investigation of new surface forms includes the topics of triangular interpolants, multivariate interpolation, surfaces defined on surfaces and monotone and/or convex surfaces. The geometric and physical properties considered include contours, the intersection of surfaces, curvatures as a interrogation tool, and numerical integration.
Beyond natural numbers: negative number representation in parietal cortex.
Blair, Kristen P; Rosenberg-Lee, Miriam; Tsang, Jessica M; Schwartz, Daniel L; Menon, Vinod
2012-01-01
Unlike natural numbers, negative numbers do not have natural physical referents. How does the brain represent such abstract mathematical concepts? Two competing hypotheses regarding representational systems for negative numbers are a rule-based model, in which symbolic rules are applied to negative numbers to translate them into positive numbers when assessing magnitudes, and an expanded magnitude model, in which negative numbers have a distinct magnitude representation. Using an event-related functional magnetic resonance imaging design, we examined brain responses in 22 adults while they performed magnitude comparisons of negative and positive numbers that were quantitatively near (difference <4) or far apart (difference >6). Reaction times (RTs) for negative numbers were slower than positive numbers, and both showed a distance effect whereby near pairs took longer to compare. A network of parietal, frontal, and occipital regions were differentially engaged by negative numbers. Specifically, compared to positive numbers, negative number processing resulted in greater activation bilaterally in intraparietal sulcus (IPS), middle frontal gyrus, and inferior lateral occipital cortex. Representational similarity analysis revealed that neural responses in the IPS were more differentiated among positive numbers than among negative numbers, and greater differentiation among negative numbers was associated with faster RTs. Our findings indicate that despite negative numbers engaging the IPS more strongly, the underlying neural representation are less distinct than that of positive numbers. We discuss our findings in the context of the two theoretical models of negative number processing and demonstrate how multivariate approaches can provide novel insights into abstract number representation.
"Ladettes," Social Representations, and Aggression.
ERIC Educational Resources Information Center
Muncer, Steven; Campbell, Anne; Jervis, Victoria; Lewis, Rachel
2001-01-01
Examined the relationship among "laddishness" (traditionally working-class, youthful, male social behavior by young women), social representations, and self-reported aggression among English college students. Measures of aggression correlated with holding more instrumental representations of aggression. Females indicated no relationship between…
Knowledge Representation: A Brief Review.
ERIC Educational Resources Information Center
Vickery, B. C.
1986-01-01
Reviews different structures and techniques of knowledge representation: structure of database records and files, data structures in computer programming, syntatic and semantic structure of natural language, knowledge representation in artificial intelligence, and models of human memory. A prototype expert system that makes use of some of these…
Getting a Picture that Is Both Accurate and Stable: Situation Models and Epistemic Validation
ERIC Educational Resources Information Center
Schroeder, Sascha; Richter, Tobias; Hoever, Inga
2008-01-01
Text comprehension entails the construction of a situation model that prepares individuals for situated action. In order to meet this function, situation model representations are required to be both accurate and stable. We propose a framework according to which comprehenders rely on epistemic validation to prevent inaccurate information from…
Revealing children's implicit spelling representations.
Critten, Sarah; Pine, Karen J; Messer, David J
2013-06-01
Conceptualizing the underlying representations and cognitive mechanisms of children's spelling development is a key challenge for literacy researchers. Using the Representational Redescription model (Karmiloff-Smith), Critten, Pine and Steffler (2007) demonstrated that the acquisition of phonological and morphological knowledge may be underpinned by increasingly explicit levels of spelling representation. However, their proposal that implicit representations may underlie early 'visually based' spelling remains unresolved. Children (N = 101, aged 4-6 years) were given a recognition task (Critten et al., 2007) and a novel production task, both involving verbal justifications of why spellings are correct/incorrect, strategy use and word pattern similarity. Results for both tasks supported an implicit level of spelling characterized by the ability to correctly recognize/produce words but the inability to explain operational strategies or generalize knowledge. Explicit levels and multiple representations were also in evidence across the two tasks. Implications for cognitive mechanisms underlying spelling development are discussed. PMID:23659891
Revealing children's implicit spelling representations.
Critten, Sarah; Pine, Karen J; Messer, David J
2013-06-01
Conceptualizing the underlying representations and cognitive mechanisms of children's spelling development is a key challenge for literacy researchers. Using the Representational Redescription model (Karmiloff-Smith), Critten, Pine and Steffler (2007) demonstrated that the acquisition of phonological and morphological knowledge may be underpinned by increasingly explicit levels of spelling representation. However, their proposal that implicit representations may underlie early 'visually based' spelling remains unresolved. Children (N = 101, aged 4-6 years) were given a recognition task (Critten et al., 2007) and a novel production task, both involving verbal justifications of why spellings are correct/incorrect, strategy use and word pattern similarity. Results for both tasks supported an implicit level of spelling characterized by the ability to correctly recognize/produce words but the inability to explain operational strategies or generalize knowledge. Explicit levels and multiple representations were also in evidence across the two tasks. Implications for cognitive mechanisms underlying spelling development are discussed.
A generalized wavelet extrema representation
Lu, Jian; Lades, M.
1995-10-01
The wavelet extrema representation originated by Stephane Mallat is a unique framework for low-level and intermediate-level (feature) processing. In this paper, we present a new form of wavelet extrema representation generalizing Mallat`s original work. The generalized wavelet extrema representation is a feature-based multiscale representation. For a particular choice of wavelet, our scheme can be interpreted as representing a signal or image by its edges, and peaks and valleys at multiple scales. Such a representation is shown to be stable -- the original signal or image can be reconstructed with very good quality. It is further shown that a signal or image can be modeled as piecewise monotonic, with all turning points between monotonic segments given by the wavelet extrema. A new projection operator is introduced to enforce piecewise inonotonicity of a signal in its reconstruction. This leads to an enhancement to previously developed algorithms in preventing artifacts in reconstructed signal.
Conceptions of mathematics and student identity: implications for engineering education
NASA Astrophysics Data System (ADS)
Craig, Tracy S.
2013-10-01
Lecturers of first-year mathematics often have reason to believe that students enter university studies with naïve conceptions of mathematics and that more mature conceptions need to be developed in the classroom. Students' conceptions of the nature and role of mathematics in current and future studies as well as future career are pedagogically important as they can impact on student learning and have the potential to influence how and what we teach. As part of ongoing longitudinal research into the experience of a cohort of students registered at the author's institution, students' conceptions of mathematics were determined using a coding scheme developed elsewhere. In this article, I discuss how the cohort of students choosing to study engineering exhibits a view of mathematics as conceptual skill and as problem-solving, coherent with an accurate understanding of the role of mathematics in engineering. Parallel investigation shows, however, that the students do not embody designated identities as engineers.
NASA Astrophysics Data System (ADS)
Srivastava, H. M.; Lin, Shy-Der; Liu, Shuoh-Jung; Lu, Han-Chun
2012-03-01
Motivated essentially by their potential for applications in the mathematical, physical, and statistical sciences, the object of this paper is to investigate several general families of hypergeometric polynomials and their associated multiple integral representations. By suitably specializing the main results presented here, the corresponding integral representations are derived for familiar simpler classes of hypergeometric polynomials such as (for example) the Lagrange polynomials, Shively's pseudo-Laguerre polynomials, and generalized Bessel polynomials. Each of the integral representations derived in this paper may be also viewed as a linearization relationship for the product of two different members of the associated family of hypergeometric polynomials.
Archival Representation in the Digital Age
ERIC Educational Resources Information Center
Zhang, Jane
2012-01-01
This study analyzes the representation systems of three digitized archival collections using the traditional archival representation framework of provenance, order, and content. The results of the study reveal a prominent role of provenance representation, a compromised role of order representation, and an active role of content representation in…
Representation of research hypotheses
2011-01-01
Background Hypotheses are now being automatically produced on an industrial scale by computers in biology, e.g. the annotation of a genome is essentially a large set of hypotheses generated by sequence similarity programs; and robot scientists enable the full automation of a scientific investigation, including generation and testing of research hypotheses. Results This paper proposes a logically defined way for recording automatically generated hypotheses in machine amenable way. The proposed formalism allows the description of complete hypotheses sets as specified input and output for scientific investigations. The formalism supports the decomposition of research hypotheses into more specialised hypotheses if that is required by an application. Hypotheses are represented in an operational way – it is possible to design an experiment to test them. The explicit formal description of research hypotheses promotes the explicit formal description of the results and conclusions of an investigation. The paper also proposes a framework for automated hypotheses generation. We demonstrate how the key components of the proposed framework are implemented in the Robot Scientist “Adam”. Conclusions A formal representation of automatically generated research hypotheses can help to improve the way humans produce, record, and validate research hypotheses. Availability http://www.aber.ac.uk/en/cs/research/cb/projects/robotscientist/results/ PMID:21624164
Jenkins, Rob; Burton, A. Mike
2011-01-01
Photographs are often used to establish the identity of an individual or to verify that they are who they claim to be. Yet, recent research shows that it is surprisingly difficult to match a photo to a face. Neither humans nor machines can perform this task reliably. Although human perceivers are good at matching familiar faces, performance with unfamiliar faces is strikingly poor. The situation is no better for automatic face recognition systems. In practical settings, automatic systems have been consistently disappointing. In this review, we suggest that failure to distinguish between familiar and unfamiliar face processing has led to unrealistic expectations about face identification in applied settings. We also argue that a photograph is not necessarily a reliable indicator of facial appearance, and develop our proposal that summary statistics can provide more stable face representations. In particular, we show that image averaging stabilizes facial appearance by diluting aspects of the image that vary between snapshots of the same person. We review evidence that the resulting images can outperform photographs in both behavioural experiments and computer simulations, and outline promising directions for future research. PMID:21536553
Early childhood mathematics intervention.
Clements, Douglas H; Sarama, Julie
2011-08-19
Preschool and primary grade children have the capacity to learn substantial mathematics, but many children lack opportunities to do so. Too many children not only start behind their more advantaged peers, but also begin a negative trajectory in mathematics. Interventions designed to facilitate their mathematical learning during ages 3 to 5 years have a strong positive effect on these children's lives for many years thereafter.
Amorphous computing: examples, mathematics and theory.
Stark, W Richard
2013-01-01
The cellular automata model was described by John von Neumann and his friends in the 1950s as a representation of information processing in multicellular tissue. With crystalline arrays of cells and synchronous activity, it missed the mark (Stark and Hughes, BioSystems 55:107-117, 2000). Recently, amorphous computing, a valid model for morphogenesis in multicellular information processing, has begun to fill the void. Through simple examples and elementary mathematics, this paper begins a computation theory for this important new direction. PMID:23946719
ERIC Educational Resources Information Center
House, Peggy A.
1994-01-01
Describes some mathematical investigations of the necktie which includes applications of geometry, statistics, data analysis, sampling, probability, symmetry, proportion, problem solving, and business. (MKR)
[Mathematical anatomy: muscles according to Stensen].
Andrault, Raphaële
2010-01-01
In his Elementorum Myologiae Specimen, Steno geometrizes "the new fabric of muscles" and their movement of contraction, so as to refute the main contemporary hypothesis about the functioning of the muscles. This physiological refutation relies on an abstract representation of the muscular fibre as a parallelepiped of flesh transversally linked to the tendons. Those two features have been comprehensively studied. But the method used by Steno, as well as the way he has chosen to present his physiological results, have so far been neglected. Yet, Steno's work follows a true synthetic order, which he conceives as a tool to separate uncertain anatomical "elements" from the certain ones. We will show that the true understanding of this "more geometrico" order is the only way to avoid frequent misconceptions of the scientific aim pursued by Steno, which is neither to give a mathematical explanation of the functioning of the muscles, nor to reduce the muscles to some mathematical shapes.
Mathematics for Teaching: A Form of Applied Mathematics
ERIC Educational Resources Information Center
Stylianides, Gabriel J.; Stylianides, Andreas J.
2010-01-01
In this article we elaborate a conceptualisation of "mathematics for teaching" as a form of applied mathematics (using Bass's idea of characterising mathematics education as a form of applied mathematics) and we examine implications of this conceptualisation for the mathematical preparation of teachers. Specifically, we focus on issues of design…
Using Mathematics Literature with Prospective Secondary Mathematics Teachers
ERIC Educational Resources Information Center
Jett, Christopher C.
2014-01-01
Literature in mathematics has been found to foster positive improvements in mathematics learning. This manuscript reports on a mathematics teacher educator's use of literature via literature circles with 11 prospective secondary mathematics teachers in a mathematics content course. Using survey and reflection data, the author found that…
Finite Mathematics and Discrete Mathematics: Is There a Difference?
ERIC Educational Resources Information Center
Johnson, Marvin L.
Discrete mathematics and finite mathematics differ in a number of ways. First, finite mathematics has a longer history and is therefore more stable in terms of course content. Finite mathematics courses emphasize certain particular mathematical tools which are useful in solving the problems of business and the social sciences. Discrete mathematics…
Exploring Differential Effects of Mathematics Courses on Mathematics Achievement
ERIC Educational Resources Information Center
Ma, Xin; McIntyre, Laureen J.
2005-01-01
Using data from the Longitudinal Study of Mathematics Participation (N = 1,518 students from 34 schools), we investigated the effects of pure and applied mathematics courses on mathematics achievement, controlling for prior mathematics achievement. Results of multilevel modelling showed that the effects of pure mathematics were significant after…
Hands-On Mathematics: Two Cases from Ancient Chinese Mathematics
ERIC Educational Resources Information Center
Wang, Youjun
2009-01-01
In modern mathematical teaching, it has become increasingly emphasized that mathematical knowledge should be taught by problem-solving, hands-on activities, and interactive learning experiences. Comparing the ideas of modern mathematical education with the development of ancient Chinese mathematics, we find that the history of mathematics in…
A Capstone Mathematics Course for Prospective Secondary Mathematics Teachers
ERIC Educational Resources Information Center
Artzt, Alice F.; Sultan, Alan; Curcio, Frances R.; Gurl, Theresa
2012-01-01
This article describes an innovative capstone mathematics course that links college mathematics with school mathematics and pedagogy. It describes how college juniors in a secondary mathematics teacher preparation program engage in leadership experiences that enable them to learn mathematics for teaching while developing student-centered…
Ensemble polarimetric SAR image classification based on contextual sparse representation
NASA Astrophysics Data System (ADS)
Zhang, Lamei; Wang, Xiao; Zou, Bin; Qiao, Zhijun
2016-05-01
Polarimetric SAR image interpretation has become one of the most interesting topics, in which the construction of the reasonable and effective technique of image classification is of key importance. Sparse representation represents the data using the most succinct sparse atoms of the over-complete dictionary and the advantages of sparse representation also have been confirmed in the field of PolSAR classification. However, it is not perfect, like the ordinary classifier, at different aspects. So ensemble learning is introduced to improve the issue, which makes a plurality of different learners training and obtained the integrated results by combining the individual learner to get more accurate and ideal learning results. Therefore, this paper presents a polarimetric SAR image classification method based on the ensemble learning of sparse representation to achieve the optimal classification.
Learning in Lectures: Multiple Representations
ERIC Educational Resources Information Center
Wood, Leigh N.; Joyce, Sadhbh; Petocz, Peter; Rodd, Melissa
2007-01-01
Lectures remain the lynchpin of mathematics teaching at university even with advances in information technology and access to the internet. This paper examines the requirements for learning mathematics and shows how important it is for lecturers to be aware of the different modes of presentation they are using. Ways to assist students to make the…
Representations of filtered solvable Lie algebras
Panov, Alexander N
2012-01-31
The representation theory of filtered solvable Lie algebras is constructed. In this framework a classification of irreducible representations is obtained and spectra of some reducible representations are found. Bibliography: 9 titles.
Attitude Representations for Kalman Filtering
NASA Technical Reports Server (NTRS)
Markley, F. Landis; Bauer, Frank H. (Technical Monitor)
2001-01-01
The four-component quaternion has the lowest dimensionality possible for a globally nonsingular attitude representation, it represents the attitude matrix as a homogeneous quadratic function, and its dynamic propagation equation is bilinear in the quaternion and the angular velocity. The quaternion is required to obey a unit norm constraint, though, so Kalman filters often employ a quaternion for the global attitude estimate and a three-component representation for small errors about the estimate. We consider these mixed attitude representations for both a first-order Extended Kalman filter and a second-order filter, as well for quaternion-norm-preserving attitude propagation.
Remedial Mathematics for Quantum Chemistry
ERIC Educational Resources Information Center
Koopman, Lodewijk; Brouwer, Natasa; Heck, Andre; Buma, Wybren Jan
2008-01-01
Proper mathematical skills are important for every science course and mathematics-intensive chemistry courses rely on a sound mathematical pre-knowledge. In the first-year quantum chemistry course at this university, it was noticed that many students lack basic mathematical knowledge. To tackle the mathematics problem, a remedial mathematics…
The Real Story Behind Story Problems: Effects of Representations on Quantitative Reasoning
ERIC Educational Resources Information Center
Koedinger, Kenneth R.; Nathan, Mitchell J.
2004-01-01
This article explores how differences in problem representations change both the performance and underlying cognitive processes of beginning algebra students engaged in quantitative reasoning. Contrary to beliefs held by practitioners and researchers in mathematics education, students were more successful solving simple algebra story problems than…
ERIC Educational Resources Information Center
Santos-Trigo, Manuel; Espinosa-Perez, Hugo; Reyes-Rodriguez, Aaron
2006-01-01
Technological tools have the potential to offer students the possibility to represent information and relationships embedded in problems and concepts in ways that involve numerical, algebraic, geometric, and visual approaches. In this paper, the authors present and discuss an example in which an initial representation of a mathematical object…
ERIC Educational Resources Information Center
Daher, Wajeeh M.; Anabousi, Anlam A.
2015-01-01
The topic of function transformations is a difficult mathematical topic for school and college students. This article examines how students conceive function transformations after working with GeoGebra, when this conceiving relates to the algebraic representation. The research participants were 19 ninth grade high achieving students who learned,…
Wickramasekara, S.
2009-07-15
We construct two rigged Hilbert spaces that furnish differentiable representations of the causal Poincare semigroup. These rigged Hilbert spaces provide the mathematical foundation for a theory of relativistic quasistable states that synthesizes the S-matrix description of resonance scattering with the Bakamjian-Thomas construction for interacting relativistic quantum systems.
Beneath the Tip of the Iceberg: Using Representations to Support Student Understanding
ERIC Educational Resources Information Center
Webb, David C.; Boswinkel, Nina; Dekker, Truus
2008-01-01
In the Netherlands, the "iceberg model," developed by the Freudenthal Institute, has been used to support teacher identification of informal and preformal representations that build students' understanding of formal mathematics. This article offers suggestions on how this model can be used to support professional development, collaborative…
Multi-Modal Representations in Primary Science: What's Offered by Interactive Whiteboard Technology
ERIC Educational Resources Information Center
Murcia, Karen
2010-01-01
This paper reports on exploratory research that examined how students learn Science with an interactive whiteboard. In this study, the IWB was found to support a range of multimodal representation types including verbal, graphic, tabular, mathematical, pictorial and kinaesthetic. The affordances offered by the technology are discussed in this…
ERIC Educational Resources Information Center
Presmeg, Norma; Nenduradu, Rajeev
2005-01-01
As part of a larger investigation of preservice teachers' use of, and movement amongst, various modes of representing exponential relationships, this report focuses on one case study, that of Mike, whose facility in moving amongst representational registers was not matched by conceptual understanding of the underlying mathematical ideas as he…
ERIC Educational Resources Information Center
Sunyono; Yuanita, L.; Ibrahim, M.
2015-01-01
The aim of this research is identify the effectiveness of a multiple representation-based learning model, which builds a mental model within the concept of atomic structure. The research sample of 108 students in 3 classes is obtained randomly from among students of Mathematics and Science Education Studies using a stratified random sampling…
Negative Numbers in the 18th and 19th Centuries: Phenomenology and Representations
ERIC Educational Resources Information Center
Maz-Machado, Alexander; Rico-Romero, Luis
2009-01-01
This article presents a categorization of the phenomena and representations used to introduce negative numbers in mathematics books published in Spain during the 18th and 19th centuries. Through a content analysis of fourteen texts which were selected for the study, we distinguished four phenomena typologies: physical, accounting, temporal and…
Subitizing, Magnitude Representation, and Magnitude Retrieval in Deaf and Hearing Adults
ERIC Educational Resources Information Center
Bull, Rebecca; Blatto-Vallee, Gary; Fabich, Megan
2006-01-01
This study examines basic number processing (subitizing, automaticity, and magnitude representation) as the possible underpinning of mathematical difficulties often evidenced in deaf adults. Hearing and deaf participants completed tasks to assess the automaticity with which magnitude information was activated and retrieved from long-term memory…
ERIC Educational Resources Information Center
Huang, Rongjin; Cai, Jinfa
2011-01-01
This study investigates Chinese and U.S. teachers' construction and use of pedagogical representations surrounding implementation of mathematical tasks. It does this by analyzing video-taped lessons from the Learner's Perspective Study, involving 15 Chinese and 10 U.S. consecutive lessons on the topic of linear equations/linear relations. We…
Solving Common Mathematical Problems
NASA Technical Reports Server (NTRS)
Luz, Paul L.
2005-01-01
Mathematical Solutions Toolset is a collection of five software programs that rapidly solve some common mathematical problems. The programs consist of a set of Microsoft Excel worksheets. The programs provide for entry of input data and display of output data in a user-friendly, menu-driven format, and for automatic execution once the input data has been entered.
Issues in Teaching Mathematics
ERIC Educational Resources Information Center
Ediger, Marlow
2013-01-01
In this article, the author states that there are selected issues in mathematics instruction that educators should be well aware of when planning lessons and units of study. These issues provide a basis for thought and discussion when assisting pupils to attain more optimally. Purposeful studying of issues guides mathematics teachers in…
Mathematics: The Universal Language?
ERIC Educational Resources Information Center
Hoffert, Sharon B.
2009-01-01
Mathematics is considered the universal language, but students who speak languages other than English have difficulty doing mathematics in English. For instance, because of a lack of familiarity with the problem's context, many have trouble understanding exactly what operations to perform. In the United States, approximately one in seven students…
Mathematical techniques: A compilation
NASA Technical Reports Server (NTRS)
1975-01-01
Articles on theoretical and applied mathematics are introduced. The articles cover information that might be of interest to workers in statistics and information theory, computational aids that could be used by scientists and engineers, and mathematical techniques for design and control.
Modularizing Remedial Mathematics
ERIC Educational Resources Information Center
Wong, Aaron
2013-01-01
As remedial mathematics education has become an increasingly important topic of conversation in higher education. Mathematics departments have been put under increased pressure to change their programs to increase the student success rate. A number of models have been introduced over the last decade that represent a wide range of new ideas and…
Dyslexia, Dyspraxia and Mathematics.
ERIC Educational Resources Information Center
Yeo, Dorian
This book explores how primary school children with dyslexia or dyspraxia and difficulty in math can learn math and provides practical support and detailed teaching suggestions. It considers cognitive features that underlie difficulty with mathematics generally or with specific aspects of mathematics. It outlines the ways in which children usually…
Developing Mathematically Promising Students.
ERIC Educational Resources Information Center
Sheffield, Linda Jensen, Ed.
This book, written on the recommendation of the Task Force on Mathematically Promising Students, investigates issues involving the development of promising mathematics students. Recommendations are made concerning topics such as the definition of promising students; the identification of such students; appropriate curriculum, instruction, and…
Mathematics in Power Technology.
ERIC Educational Resources Information Center
Trombley, Carl; And Others
This mathematics curriculum is designed to be taught by the technology education teacher during the power technology class over a period of 2 years. It is intended to be elective in nature; upon successful completion of both years, one-half credit in mathematics is to be awarded. A list of the academic competencies contained in the curriculum…
Experimenting with Mathematical Biology
ERIC Educational Resources Information Center
Sanft, Rebecca; Walter, Anne
2016-01-01
St. Olaf College recently added a Mathematical Biology concentration to its curriculum. The core course, Mathematics of Biology, was redesigned to include a wet laboratory. The lab classes required students to collect data and implement the essential modeling techniques of formulation, implementation, validation, and analysis. The four labs…
ERIC Educational Resources Information Center
Sharp, Karen Tobey
This paper cites information received from a number of sources, e.g., mathematics teachers in two-year colleges, publishers, and convention speakers, about the nature of discrete mathematics and about what topics a course in this subject should contain. Note is taken of the book edited by Ralston and Young which discusses the future of college…
Consumer Mathematics Curriculum Guide.
ERIC Educational Resources Information Center
Louisiana State Dept. of Education, Baton Rouge.
This guide for high school consumer mathematics (one in a set of curriculum guides developed by Louisiana statewide mathematics curriculum committees) contains a course outline, performance objectives, and coordinated activities designed to teach skills that students will need as citizens and consumers. Background on the development,…
Business Mathematics Curriculum.
ERIC Educational Resources Information Center
EASTCONN Regional Educational Services Center, North Windham, CT.
This curriculum guide for teaching business mathematics in the Connecticut Vocational-Technical School System is based on the latest thinking of instructors in the field, suggestions from mathematics authorities, and current instructional approaches in education. The curriculum guide consists of six sections: (1) career relationships and…
Astronomy and Mathematics Education
NASA Astrophysics Data System (ADS)
Ros, Rosa M.
There are many European countries where Astronomy does not appear as a specific course on the secondary school. In these cases Astronomy content can be introduced by means of other subjects. There are some astronomical topics within the subject of Physics but this talk concerns introducing Astronomy in Mathematics classes. Teaching Astronomy through Mathematics would result in more exposure than through Physics as Mathematics is more prevalent in the curriculum. Generally it is not easy to motivate students in Mathematics but they are motivated to find out more about the universe and Astronomy current events than appears in the media. This situation can be an excellent introduction to several mathematics topics. The teachers in secondary and high school can use this idea in order to present more attractive mathematics courses. In particular some different examples will be offered regarding * Angles and spherical coordinates considering star traces * Logarithms and visual magnitudes * Plane trigonometry related orbital movements * Spherical trigonometry in connection with ecliptic obliquity * Conic curves related to sundial at several latitudes Some students do not enjoy studying Mathematics but they can be attracted by practical situations using Applied Mathematics: Astronomy is always very attractive to teenagers.
Quality Teaching in Mathematics
ERIC Educational Resources Information Center
Ediger, Marlow
2012-01-01
The best teaching possible needs to accrue in the mathematics curriculum. Pupils also need to become proficient in using mathematics in every day situations in life. Individuals buy goods and services. They pay for these in different ways, including cash. Here, persons need to be able to compute the total cost of items purchased and then pay for…
Learning Mathematics while Black
ERIC Educational Resources Information Center
Martin, Danny Bernard
2012-01-01
While research by scholars has contributed greatly to an emerging knowledge base on Black children and mathematics, there continues to be a dire need for insightful research that de-centers longstanding accounts that have contributed to the construction of Black children as mathematically illiterate and as less than ideal learners relative to…
Intermediate Mathematics Study Guide.
ERIC Educational Resources Information Center
Stanford Univ., CA. School Mathematics Study Group.
This SMSG study guide is intended to provide teachers who use "Intermediate Mathematics," as a textbook with references to materials which will help them to gain a better understanding of the mathematics contained in the text. For each chapter of the text a brief resume of its content is followed by a list of annotated references which are…
ERIC Educational Resources Information Center
Hallenberg, Harvey
1995-01-01
Presents ideas for creating mathematical classroom activities associated with the history of mathematics: calculating sums and products the way ancient Greeks did it, using an abacus or moving stones on a sanded floor, and engaging elementary students through role playing specific mathematicians. Suggests that through such techniques, mathematics…
Designing Assessment for Mathematics
ERIC Educational Resources Information Center
Depka, Eileen
2007-01-01
Teaching mathematics in today's world requires practices and procedures integrated with performance tasks that actively involve students. In this second edition of Designing Rubrics for Mathematics, Eileen Depka clarifies the purpose of rubrics in math instruction and illustrates the relationship between assessment, rubrics, and the National…
Mathematical Graphic Organizers
ERIC Educational Resources Information Center
Zollman, Alan
2009-01-01
As part of a math-science partnership, a university mathematics educator and ten elementary school teachers developed a novel approach to mathematical problem solving derived from research on reading and writing pedagogy. Specifically, research indicates that students who use graphic organizers to arrange their ideas improve their comprehension…
Mathematics. [SITE 2002 Section].
ERIC Educational Resources Information Center
Connell, Michael L., Ed.; Lowery, Norene Vail, Ed.; Harnisch, Delwyn L., Ed.
This document contains the following papers on mathematics from the SITE (Society for Information Technology & Teacher Education) 2002 conference: (1) "Teachers' Learning of Mathematics in the Presence of Technology: Participatory Cognitive Apprenticeship" (Mara Alagic); (2) "A Fractal Is a Pattern in Your Neighborhood" (Craig N. Bach); (3)…
ERIC Educational Resources Information Center
Stallings, L. Lynn
2007-01-01
This article proposes four strategies for posing mathematics problems that raise the cognitive demands of the tasks given to students. Each strategy is illustrated with three common middle school mathematics examples: finding the greatest common factor, finding area or perimeter, and finding the equation of a line. Posing these types of problems…
The Impossible in Mathematics.
ERIC Educational Resources Information Center
Adler, Irving
The material in this reprint, with minor editorial changes, is from the chapter "Doing the Impossible" in MONKEY BUSINESS by Irving Adler. This 25-page booklet contains brief accounts of historical attempts to prove impossible problems in mathematics. The mathematical recreations in this booklet of geometric constructions include the trisection…
Strengthen Your Mathematical Muscles
ERIC Educational Resources Information Center
Wohlhuter, Kay A.; Breyfogle, M. Lynn; McDuffie, Amy Roth
2010-01-01
Developing deep knowledge and understanding of mathematics is a lifelong process, and building the foundation for teachers' development must begin in preservice preparation and continue throughout one's professional life. While teaching mathematics content courses and methods courses, the authors have found that preservice elementary school…
Mathematics: Content and Pedagogy
ERIC Educational Resources Information Center
Ediger, Marlow
2009-01-01
The debate has gone on for some time in terms of which is more salient for the teacher to be well versed in, mathematical content versus methods and approaches in teaching. Both are salient. They cannot be separated from each other. The mathematics teacher must indeed have broad, in-depth knowledge of subject matter as well as in teaching and…
Mathematics Education in Argentina
ERIC Educational Resources Information Center
Varsavsky, Cristina; Anaya, Marta
2009-01-01
This article gives an overview of the state of mathematics education in Argentina across all levels, in the regional and world contexts. Statistics are drawn from Mercosur and UNESCO data bases, World Education Indicators and various national time-series government reports. Mathematics results in national testing programmes, Programme for…
[Collected Papers on Mathematics.
ERIC Educational Resources Information Center
Connell, Michael L., Ed.
This document contains the following papers on issues related to mathematics in technology and teacher education: "A Case for Strong Conceptualization in Technology Enhanced Mathematics Instruction" (Michael L. Connell and Delwyn L. Harnisch); "Faculty/Student Collaboration in Education and Math--Using the Web To Improve Student Learning and…
Discrete Mathematics Re "Tooled."
ERIC Educational Resources Information Center
Grassl, Richard M.; Mingus, Tabitha T. Y.
1999-01-01
Indicates the importance of teaching discrete mathematics. Describes how the use of technology can enhance the teaching and learning of discrete mathematics. Explorations using Excel, Derive, and the TI-92 proved how preservice and inservice teachers experienced a new dimension in problem solving and discovery. (ASK)
ERIC Educational Resources Information Center
Fennema, Elizabeth, Ed.; Leder, Gilah C., Ed.
This book reports on various studies that have increased our understanding of why females and males learn different kinds and amounts of mathematics. In particular, this book explicates the Autonomous Learning Behavior model, proposed by Fennema and Peterson, which is a possible explanation of the development of gender differences in mathematics.…
Mathematics Knowledge for Teaching
ERIC Educational Resources Information Center
Mohr, Margaret
2006-01-01
Research indicates that U. S. teachers' mathematical knowledge continues to be weak, and there is an inherent difference between the mathematical knowledge needed to be an effective teacher and that needed by a mathematician. In this article, the author discusses pedagogical content knowledge. According to Schulman (1987), pedagogical content…
ERIC Educational Resources Information Center
Iben, Miriam F.
1991-01-01
Examines seventh and eighth grade students in Australia, Japan, and the United States for attitudes related to mathematics, and the relationship these attitudes have to students' development of abstract mathematical thought and spatial relations. Study uses the Iowa Algebra Aptitude Test, Differential Aptitude Test-Spatial Relations, and the…
Developing Mathematical Vocabulary.
ERIC Educational Resources Information Center
Monroe, Eula Ewing; Orme, Michelle P.
2002-01-01
This article discusses the importance of mathematical vocabulary, difficulties students encounter in learning this vocabulary, and some instructional strategies. Two general methods for teaching vocabulary are discussed: context and explicit vocabulary instruction. The methods are summarized as they apply to mathematical vocabulary instruction and…
ERIC Educational Resources Information Center
Sharp, John
2012-01-01
This relationship is omnipresent to those who appreciate the shared attributes of these two areas of creativity. The dynamic nature of media, and further study, enable mathematicians and artists to present new and exciting manifestations of the "mathematics in art", and the "art in mathematics". The illustrative images of the relationship--that…
NASA Astrophysics Data System (ADS)
Rohrlich, Fritz
2011-12-01
Classical and the quantum mechanical sciences are in essential need of mathematics. Only thus can the laws of nature be formulated quantitatively permitting quantitative predictions. Mathematics also facilitates extrapolations. But classical and quantum sciences differ in essential ways: they follow different laws of logic, Aristotelian and non-Aristotelian logics, respectively. These are explicated.
On canonical cylinder sections for accurate determination of contact angle in microgravity
Concus, P.; Zabihi, F. California Univ., Berkeley, CA . Dept. of Mathematics); Finn, R. . Dept. of Mathematics)
1992-07-01
Large shifts of liquid arising from small changes in certain container shapes in zero gravity can be used as a basis for accurately determining contact angle. Canonical'' geometries for this purpose, recently developed mathematically, are investigated here computationally. It is found that the desired nearly- discontinuous'' behavior can be obtained and that the shifts of liquid have sufficient volume to be readily observed.
On canonical cylinder sections for accurate determination of contact angle in microgravity
Concus, P.; Zabihi, F. |; Finn, R.
1992-07-01
Large shifts of liquid arising from small changes in certain container shapes in zero gravity can be used as a basis for accurately determining contact angle. ``Canonical`` geometries for this purpose, recently developed mathematically, are investigated here computationally. It is found that the desired ``nearly- discontinuous`` behavior can be obtained and that the shifts of liquid have sufficient volume to be readily observed.
ERIC Educational Resources Information Center
Rosales, Javier; Vicente, Santiago; Chamoso, Jose M.; Munez, David; Orrantia, Josetxu
2012-01-01
Word problem solving involves the construction of two different mental representations, namely, mathematical and situational. Although educational research in word problem solving has documented different kinds of instruction at these levels, less is known about how both representational levels are evoked during word problem solving in day-to-day…
ERIC Educational Resources Information Center
Nitsch, Renate; Fredebohm, Anneke; Bruder, Regina; Kelava, Augustin; Naccarella, Dominik; Leuders, Timo; Wirtz, Markus
2015-01-01
In the subject matter of functional relationships, a student's ability to translate from one form of representation to another is seen as a central competence. In the course of the HEUREKO project (heuristic work with representations of functional relationships and the diagnosis of mathematical competencies of students), a theoretical competence…
Computer representation of molecular surfaces
Max, N.L.
1981-07-06
This review article surveys recent work on computer representation of molecular surfaces. Several different algorithms are discussed for producing vector or raster drawings of space-filling models formed as the union of spheres. Other smoother surfaces are also considered.
Vietnamese Document Representation and Classification
NASA Astrophysics Data System (ADS)
Nguyen, Giang-Son; Gao, Xiaoying; Andreae, Peter
Vietnamese is very different from English and little research has been done on Vietnamese document classification, or indeed, on any kind of Vietnamese language processing, and only a few small corpora are available for research. We created a large Vietnamese text corpus with about 18000 documents, and manually classified them based on different criteria such as topics and styles, giving several classification tasks of different difficulty levels. This paper introduces a new syllable-based document representation at the morphological level of the language for efficient classification. We tested the representation on our corpus with different classification tasks using six classification algorithms and two feature selection techniques. Our experiments show that the new representation is effective for Vietnamese categorization, and suggest that best performance can be achieved using syllable-pair document representation, an SVM with a polynomial kernel as the learning algorithm, and using Information gain and an external dictionary for feature selection.
38 CFR 4.46 - Accurate measurement.
Code of Federal Regulations, 2010 CFR
2010-07-01
... 38 Pensions, Bonuses, and Veterans' Relief 1 2010-07-01 2010-07-01 false Accurate measurement. 4... RATING DISABILITIES Disability Ratings The Musculoskeletal System § 4.46 Accurate measurement. Accurate measurement of the length of stumps, excursion of joints, dimensions and location of scars with respect...
38 CFR 4.46 - Accurate measurement.
Code of Federal Regulations, 2013 CFR
2013-07-01
... 38 Pensions, Bonuses, and Veterans' Relief 1 2013-07-01 2013-07-01 false Accurate measurement. 4... RATING DISABILITIES Disability Ratings The Musculoskeletal System § 4.46 Accurate measurement. Accurate measurement of the length of stumps, excursion of joints, dimensions and location of scars with respect...
38 CFR 4.46 - Accurate measurement.
Code of Federal Regulations, 2011 CFR
2011-07-01
... 38 Pensions, Bonuses, and Veterans' Relief 1 2011-07-01 2011-07-01 false Accurate measurement. 4... RATING DISABILITIES Disability Ratings The Musculoskeletal System § 4.46 Accurate measurement. Accurate measurement of the length of stumps, excursion of joints, dimensions and location of scars with respect...
38 CFR 4.46 - Accurate measurement.
Code of Federal Regulations, 2014 CFR
2014-07-01
... 38 Pensions, Bonuses, and Veterans' Relief 1 2014-07-01 2014-07-01 false Accurate measurement. 4... RATING DISABILITIES Disability Ratings The Musculoskeletal System § 4.46 Accurate measurement. Accurate measurement of the length of stumps, excursion of joints, dimensions and location of scars with respect...
38 CFR 4.46 - Accurate measurement.
Code of Federal Regulations, 2012 CFR
2012-07-01
... 38 Pensions, Bonuses, and Veterans' Relief 1 2012-07-01 2012-07-01 false Accurate measurement. 4... RATING DISABILITIES Disability Ratings The Musculoskeletal System § 4.46 Accurate measurement. Accurate measurement of the length of stumps, excursion of joints, dimensions and location of scars with respect...
Competition and Cooperation among Relational Memory Representations
Schwarb, Hillary; Watson, Patrick D.; Campbell, Kelsey; Shander, Christopher L.; Monti, Jim M.; Cooke, Gillian E.; Wang, Jane X.; Kramer, Arthur F.; Cohen, Neal J.
2015-01-01
Mnemonic processing engages multiple systems that cooperate and compete to support task performance. Exploring these systems’ interaction requires memory tasks that produce rich data with multiple patterns of performance sensitive to different processing sub-components. Here we present a novel context-dependent relational memory paradigm designed to engage multiple learning and memory systems. In this task, participants learned unique face-room associations in two distinct contexts (i.e., different colored buildings). Faces occupied rooms as determined by an implicit gender-by-side rule structure (e.g., male faces on the left and female faces on the right) and all faces were seen in both contexts. In two experiments, we use behavioral and eye-tracking measures to investigate interactions among different memory representations in both younger and older adult populations; furthermore we link these representations to volumetric variations in hippocampus and ventromedial PFC among older adults. Overall, performance was very accurate. Successful face placement into a studied room systematically varied with hippocampal volume. Selecting the studied room in the wrong context was the most typical error. The proportion of these errors to correct responses positively correlated with ventromedial prefrontal volume. This novel task provides a powerful tool for investigating both the unique and interacting contributions of these systems in support of relational memory. PMID:26619203
Competition and Cooperation among Relational Memory Representations.
Schwarb, Hillary; Watson, Patrick D; Campbell, Kelsey; Shander, Christopher L; Monti, Jim M; Cooke, Gillian E; Wang, Jane X; Kramer, Arthur F; Cohen, Neal J
2015-01-01
Mnemonic processing engages multiple systems that cooperate and compete to support task performance. Exploring these systems' interaction requires memory tasks that produce rich data with multiple patterns of performance sensitive to different processing sub-components. Here we present a novel context-dependent relational memory paradigm designed to engage multiple learning and memory systems. In this task, participants learned unique face-room associations in two distinct contexts (i.e., different colored buildings). Faces occupied rooms as determined by an implicit gender-by-side rule structure (e.g., male faces on the left and female faces on the right) and all faces were seen in both contexts. In two experiments, we use behavioral and eye-tracking measures to investigate interactions among different memory representations in both younger and older adult populations; furthermore we link these representations to volumetric variations in hippocampus and ventromedial PFC among older adults. Overall, performance was very accurate. Successful face placement into a studied room systematically varied with hippocampal volume. Selecting the studied room in the wrong context was the most typical error. The proportion of these errors to correct responses positively correlated with ventromedial prefrontal volume. This novel task provides a powerful tool for investigating both the unique and interacting contributions of these systems in support of relational memory. PMID:26619203
A numerical study of thin flame representations
Rotman, D.A.; Pindera, M.Z.
1989-08-11
In studies of reacting flows, the flame may be viewed as a moving discontinuity endowed with certain properties; notably, it acts as a source of velocity and vorticity. Asymptotic analysis shows this to be justified provided that the flame curvature is small compared to the flame thickness. Such an approach is useful when one is interested in the hydrodynamic effects of the flame on the surrounding flowfield. In numerical models of this kind it is customary to treat the discontinuity as a collection of discrete velocity blobs. In this study, we show that the velocities associated with such a representation can be very non-smooth, particularly very near to the flame surface. As an alternative, we propose the use of a finite line source as the basic flame element. Comparisons of the two flame representations are made for several simple test cases as well as for a flame propagating through an enclosure forming the tulip shape. The results show that the use of line sources eliminates spurious fluctuations in nearfield velocities thus allowing for a more accurate calculation of flame propagation and flame-flowfield interactions. 7 refs., 15 figs.
Role of multiple representations in physics problem solving
NASA Astrophysics Data System (ADS)
Maries, Alexandru
This thesis explores the role of multiple representations in introductory physics students' problem solving performance through several investigations. Representations can help students focus on the conceptual aspects of physics and play a major role in effective problem solving. Diagrammatic representations can play a particularly important role in the initial stages of conceptual analysis and planning of the problem solution. Findings suggest that students who draw productive diagrams are more successful problem solvers even if their approach is primarily mathematical. Furthermore, students provided with a diagram of the physical situation presented in a problem sometimes exhibited deteriorated performance. Think-aloud interviews suggest that this deteriorated performance is in part due to reduced conceptual planning time which caused students to jump to the implementation stage without fully understanding the problem and planning problem solution. Another study investigated two interventions aimed at improving introductory students' representational consistency between mathematical and graphical representations and revealed that excessive scaffolding can have a detrimental effect. The detrimental effect was partly due to increased cognitive load brought on by the additional steps and instructions. Moreover, students who exhibited representational consistency also showed improved problem solving performance. The final investigation is centered on a problem solving task designed to provide information about the pedagogical content knowledge (PCK) of graduate student teaching assistants (TAs). In particular, the TAs identified what they considered to be the most common difficulties of introductory physics students related to graphical representations of kinematics concepts as they occur in the Test of Understanding Graphs in Kinematics (TUG-K). As an extension, the Force Concept Inventory (FCI) was also used to assess this aspect of PCK related to knowledge of
Business Mathematics. Mathematics Curriculum Guide (Career Oriented).
ERIC Educational Resources Information Center
Nuschler, Alexandra; And Others
The curriculum guide correlates concepts in business mathematics with career-oriented concepts and activities. The curriculum outline format gives the concepts to be taught, matched with related career-oriented performance objectives, concepts, and suggested instructional activities in facing page layouts. The outline is divided into the major…
Technical Mathematics: Restructure of Technical Mathematics.
ERIC Educational Resources Information Center
Flannery, Carol A.
Designed to accompany a series of videotapes, this textbook provides information, examples, problems, and solutions relating to mathematics and its applications in technical fields. Chapter I deals with basic arithmetic, providing information on fractions, decimals, ratios, proportions, percentages, and order of operations. Chapter II focuses on…
NASA Astrophysics Data System (ADS)
Kasmer, Lisa
2013-09-01
In order to promote mathematical understanding among English Language Learners (ELLs), it is necessary to modify instructional strategies to effectively communicate mathematical content. This paper discusses the instructional strategies used by four pre-service teachers to teach mathematics to secondary students in English-medium schools in Arusha, Tanzania as a result of the tensions they faced and reflections on their teaching. Strategies such as code switching, attending to sentence structure, non-linguistic representations, and placing the content within a familiar context proved to be beneficial strategies for conveying mathematical ideas.
ERIC Educational Resources Information Center
Hsu, Yu-Chang
2009-01-01
Students in the Science, Technology, Engineering, and Mathematics (STEM) fields are confronted with multiple external representations (MERs) in their learning materials. The ability to learn from and communicate with these MERs requires not only that students comprehend each representation individually but also that students recognize how the…
Mathematical model for gyroscope effects
NASA Astrophysics Data System (ADS)
Usubamatov, Ryspek
2015-05-01
Gyroscope effects are used in many engineering calculations of rotating parts, and a gyroscope is the basic unit of numerous devices and instruments used in aviation, space, marine and other industries. The primary attribute of a gyroscope is a spinning rotor that persists in maintaining its plane of rotation, creating gyroscope effects. Numerous publications represent the gyroscope theory using mathematical models based on the law of kinetic energy conservation and the rate of change in angular momentum of a spinning rotor. Gyroscope theory still attracts many researchers who continue to discover new properties of gyroscopic devices. In reality, gyroscope effects are more complex and known mathematical models do not accurately reflect the actual motions. Analysis of forces acting on a gyroscope shows that four dynamic components act simultaneously: the centrifugal, inertial and Coriolis forces and the rate of change in angular momentum of the spinning rotor. The spinning rotor generates a rotating plane of centrifugal and Coriols forces that resist the twisting of the spinning rotor with external torque applied. The forced inclination of the spinning rotor generates inertial forces, resulting in precession torque of a gyroscope. The rate of change of the angular momentum creates resisting and precession torques which are not primary one in gyroscope effects. The new mathematical model for the gyroscope motions under the action of the external torque applied can be as base for new gyroscope theory. At the request of the author of the paper, this corrigendum was issued on 24 May 2016 to correct an incomplete Table 1 and errors in Eq. (47) and Eq. (48).