Technical Reports

Technical Reports 2009

König-Egerváry graphs are non-Edmonds

P. Mark Kayll
Department of Mathematical Sciences
University of Montana
Missoula MT 59812-0864, USA
mark.kayll@umontana.edu

Abstract

König-Egerváry graphs are those whose maximum matchings are equicardinal to their minimum- order coverings by vertices. Edmonds [J. Res. Nat. Bur. Standards Sect. B 69B (1965), 125–130] characterized the perfect matching polytope of a graph G = (V,E) as the set of nonnegative vectors x ∈ RE satisfying two families of constraints: ‘vertex saturation’ and ‘blossom’. Graphs for which the latter constraints are implied by the former are termed non-Edmonds. This note presents two proofs—one combinatorial, one algorithmic—of its title’s assertion. Neither proof relies on the characterization of non-Edmonds graphs due to de Carvalho et al. [J. Combin. Theory Ser. B 92 (2004), 319–324].

Keywords: matching, perfect matching polytope, covering

AMS Subject Classification: Primary 05C70; Secondary 05C35, 90C35, 68R10, 52B99

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Another short proof of the Joni-Rota-Godsil integral formula for counting bipartite matchings

Erin E. Emerson*
Ann Arbor MI 48104, USA
erinbeth@umich.edu

P. Mark Kayll
Department of Mathematical Sciences
University of Montana
Missoula MT 59812-0864, USA
mark.kayll@umontana.edu


Abstract

How many perfect matchings are contained in a given bipartite graph? An exercise in Godsil’s 1993 Algebraic Combinatorics solicits proof that this question’s answer is an integral involving a certain rook polynomial. Though not widely known, this result appears implicitly in Riordan’s 1958 An Introduction to Combinatorial Analysis. It was stated more explicitly and proved independently by S.A. Joni and G.-C. Rota [JCTA 29(1980), 59–73] and C.D. Godsil [Combinatorica 1 (1981), 257–262]. Another generation later, perhaps it’s time both to revisit the theorem and to broaden the formula’s reach.

Keywords: perfect matching, rook polynomial, bipartite complement, inclusion-exclusion

AMS Subject Classification: Primary 05A15 Secondary 05C70 33B15

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_________________________________________________________
*based on work done while attending University of Montana
†contact author

Reduction and identification of dynamic models. Simple example: generic receptor model.

Heikki Haario*, Leonid Kalachev**, and Marko Laine***
*Lappeenranta University of Technology, Lappeenranta, Finland
**University of Montana, Missoula, MT, USA
***Finnish Meteorological institute, Helsinki, Finland

Abstract

We consider a general scheme for reduction and identification of dynamic models using available experimental data. Analysis of reliability regions for estimated parameter values is performed using Markov Chain Monte Carlo simulation methods. In cases where some of the model parameters are not reliably defined, and when the values of certain model parameters turn out to be small (or large), asymptotic reduction techniques are used to reduce the models (i.e., to reduce the number of equations, number of reliably identifiable parameter, etc.). Consecutive application of parameters estimation (together with their reliability regions) and asymptotic reduction procedures will produce the new simpler model with the smallest number of parameters reliably identifiable by the available data (i.e., the model that is optimal with respect to available data). The ideas are illustrated using a simple example related to biomedical applications: a model of a generic receptor.

Keywords: Model identification, asymptotic methods, Boundary Function Method, model reduction, Markov chain Monte Carlo (MCMC)

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A Computational Framework for Total Variation-Regularized Positron Emission Tomography

Johnathan M. Bardsley and John Goldes

Abstract

In positron emission tomography, image data corresponds to measurements of emitted photons from a radioactive tracer in the subject. Such count data is typically modeled using a Poisson random variable, leading to the use of the negative-log Poisson likelihood fit-to-data function. Regularization is needed, however, in order to guarantee reconstructions with minimal artifacts. Given that tracer densities are primarily smoothly varying, but also contain sharp jumps (or edges), total variation regularization is a natural choice. However, the resulting computational problem is quite challenging. In this paper, we present a efficient computational method for this problem. We also introduce three regularization parameter choice methods for use on total variation-regularized negative-log Poisson likelihood problems, which to our knowledge, have not been presented elsewhere. We test the computational and regularization parameter selection methods on synthetic data.

Keywords: total variation, positron emission tomography, inverse problems, and statistical imaging

MSC numbers: 65J22, 65K10, and 65F22

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Mathematics Achievement and Gender: A Case of "No Difference" from Turkey

Safure Bulut
Middle East Technical University

Bekir Gur
Yuzuncuyil University

Bharath Sriraman

The University of Montana


Abstract

The canon of literature in gender studies points to the fact that culture and socioeconomic circumstances affect the mathematics education of individuals. Accordingly, girls’ mathematics achievement shows some variations across different countries. Unlike popular misconceptions about the roles of males and females in Turkish society, in this commentary we will show that the majority of studies conducted in Turkey have found no significant mean difference between mathematics achievement of boys and girls especially in primary and high schools. Hence, we do not think we need to offer an alternate pedagogy for girls. We briefly point out some points of convergences and divergences with the Jacobs’ article. And then, we give some background information on Turkish education. Subsequently, we discuss the literature related to mathematics achievement and gender in Turkey. The discussion includes the results of TIMMS, national exams in Turkey in addition to articles, theses and dissertations.

Keywords: education in Turkey; meta-analysis of gender studies in Turkey; mathematics achievement; mathematics achievement and gender

AMS Subject Classification: 97

Pdf: Preprint of chapter to appear in B. Sriraman & L. English (Eds). Theories of Mathematics Education- Seeking New Frontiers. Springer Science, Berlin/Heidelberg

Politicizing Mathematics Education: Has Politics gone too far? Or not far enough?

Bharath Sriraman, Matt Roscoe
The University of Montana

Lyn English
Queensland University of Technology, Australia

Abstract

In this chapter we tackle increasingly sensitive questions in mathematics and mathematics education, particularly those that have polarized the community into distinct schools of thought as well as impacted reform efforts. We attempt to address the following questions:

  • What are the origins of politics in mathematics education, with the progressive educational movement of Dewey as a starting point?
  • How can critical mathematics education improve the democratization of society?
  • What role, if any, does politics play in mathematics education, in relation to assessment, research and curricular reform?
  • How is the politicization of mathematics education linked to policy on equity, equal access and social justice?
  • Is the politicization of research beneficial or damaging to the field?
  • Does the philosophy of mathematics (education) influence the political orientation of policy makers, researchers, teachers and other stake holders?
  • What role does technology play in pushing society into adopting particular views on teaching and learning and mathematics education in general?
  • What does the future bear for mathematics as a field, when viewed through the lens of equity and culture?

Keywords: critical mathematics education; equity; sociopolitical theories; technology and mathematics; politics of mathematics education

AMS Subject Classification: 97

Pdf: Preprint of chapter to appear in B. Sriraman & L. English (Eds). Theories of Mathematics Education- Seeking New Frontiers. Springer Science, Berlin/Heidelberg

Lakatos-Hersh-Ernest: Triangulating Philosophy-Mathematics-Mathematics Education

Bharath Sriraman
The University of Montana

Nick Haverhals
The University of Montana

Abstract

A preface is made to Paul Ernest’s Reflections on theories of learning with a particular emphasis on exploring whether social constructivism as a philosophy of mathematics is applicable to mathematics education for the purposes of theory generation. The works of Imre Lakatos and Reuben Hersh are referred to in relation to their fit with the contributions of Ernest.

Keywords: Lakatos; Learning theories; Social Constructivism; Philosophy of mathematics

AMS Subject Classification: 97

Pdf: Preprint of chapter to appear in B. Sriraman & L. English (Eds). Theories of Mathematics Education- Seeking New Frontiers. Springer Science, Berlin/Heidelberg

On Proof and Certainty- Some Educational Implications

Bharath Sriraman
The University of Montana

Hillary VanSpronsen
Michigan Technological University

Nick Haverhals
The University of Montana

Abstract

In this commentary to Guershon Harel’s chapter DNR-Based Instruction in Mathematics as a Conceptual Framework, we contrast his theory within the larger scheme of deductivist and heuristic approaches to proof, and explore some of its instructional implications.

Keywords: DNR-Based instruction; Proof theory; Teaching and learning Proof

AMS Subject Classification: 97

Pdf: Preprint of chapter to appear in B. Sriraman & L. English (Eds). Theories of Mathematics Education- Seeking New Frontiers. Springer Science, Berlin/Heidelberg

The existential void in learning: Juxtaposing mathematics and literature

Bharath Sriraman
The University of Montana

Harry Adrian
Ottawa Township High School, IL

Abstract

In this chapter unusual relations between mathematics and literature are examined in the context of a number theory problem and contemporary literary work. Lakatos’ proofs and refutations is employed as a methodology for the purposes of classroom implementation.

Keywords: interdisciplinarity; proofs and refutations; philosophy of learning

AMS Subject Classification: 97

Pdf: Preprint of chapter to appear in B. Sriraman, V. Freiman & N. Lirette-Pitre (Eds). Interdisciplinarity, Creativity and Learning. Information Age Publishing, Charlotte, NC.

Social justice and mathematics education: Issues, Dilemmas, Excellence and Equity

Bharath Sriraman
The University of Montana

Olof Steinthorsdottir
University of North Carolina, Chapel Hill

Abstract

This article explores reasons for educational research and practice in social justice from evolutionary, ideological and philosophical viewpoints. The tension between nihilistic and empathetic tendencies within our history is used to reflexively examine the origins and causes of inequity with emphasis on the works of giants such as Paolo Freire, John Dewey, Karl Marx, and Vivekananda. Finally we address one particular issue in depth, namely the tension between excellence and equity in talent development in schools, east and west.

Keywords: East versus West; political issues in mathematics education; philosophy of education; social justice issues; talent development gifted and talented learners

AMS Subject Classification: 97

Pdf: Preprint of chapter to appear in P. Ernest, B. Greer & B. Sriraman (Eds). Critical Issues in mathematics education. Information Age Publishing, Charlotte, NC.

Problem solving for the 21st century


Lyn English
Queensland University of Technology

Bharath Sriraman
The University of Montana

Abstract

Mathematical problem solving has been the subject of substantial and often controversial research for several decades. We use the term, problem solving, here in a broad sense to cover a range of activities that challenge and extend one’s thinking. In this chapter, we initially present a sketch of past decades of research on mathematical problem solving and its impact on the mathematics curriculum. We then consider some of the factors that have limited previous research on problem solving. In the remainder of the chapter we address some ways in which we might advance the fields of problem-solving research and curriculum development.

Keywords: advances in problem solving; literature review; mathematical problem solving; modeling

AMS Subject Classification: 97

Pdf: Preprint of chapter to appear in B. Sriraman & L. English (Eds). Theories of Mathematics Education: Seeking new frontiers. Springer Science, Berlin/Heidelberg.

Agency in Mathematics Education

Paul Ernest
University of Exeter

Brian Greer
Portland State University

Bharath Sriraman
The University of Montana

Abstract

The authors use the definition of critical mathematics education in order to propose re-examining the nature and purpose of mathematics, suggest a new sense of agency on the meaning and purpose of mathematics education as well as explore the ethical responsibilities of mathematics educators.

Keywords: Agency; Critical mathematics education

AMS Subject Classification: 97

Pdf: Preprint of chapter to appear P.Ernest, B.Greer, B. Sriraman (Eds) . Critical Issues in mathematics education. Information Age Publishing, Charlotte, NC.

Regularization Parameter Selection Methods for Ill-Posed Poisson Maximum Likelihood Estimation

Johnathan M. Bardsley and John Goldes

Abstract

In image processing applications, image intensity is often measured via the counting of incident photons emitted by the object of interest. In such cases, image data-noise is accurately modeled by a Poisson distribution. This motivates the use of Poisson maximum likelihood estimation for image reconstruction. However, when the underlying model equation is ill-posed, regularization is needed. Regularized Poisson likelihood estimation has been studied extensively by the authors, though a problem of high importance remains: the choice of the regularization parameter. We will present three statistically motivated methods for choosing the regularization parameter, and numerical examples will be presented to illustrate their effectiveness.

Keywords: regularization parameter selection methods, Poisson maximum likelihood estimation, ill-posed problems, image reconstruction.

MSC numbers: 15A29, 65F22, 94A08

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Interdisciplinarity, Creativity and Learning: Mathematics with Literature, Paradoxes, History, Technology & Modeling

Bharath Sriraman
The University of Montana

Viktor Freiman & Nicole Lirette-Pitre

University of Moncton

Abstract

Interdisciplinarity is increasingly viewed as a necessary ingredient in the training of future oriented 21st century disciplines that rely on both analytic and synthetic abilities across disciplines. Nearly every curricular document or vision statement of schools and universities include a call for promoting creativity in students. Yet the construct of creativity and giftedness across disciplines remains elusive in the sense that the prototypical examples of such work come from eminent scientists, artists and mathematicians, and little if any work has been conducted with non-eminent individuals. This monograph is an attempt to fill this gap by putting forth the view that interdisciplinarity and creativity are related constructs, and that the cultivation of domain general creativity is possible. Mathematics has historically been anchored to numerous disciplines like theology, natural philosophy, culture and art, allowing for a flexibility of thought that is difficult to cultivate in other disciplines. In this monograph, the numerous chapters from Australia, U.S.A., Canada, Cyprus, Denmark and Japan provide a compelling illustration of the intricate connection of mathematics with literature,paradoxes, history, technology and modeling, thus serving as a conduit for interdisciplinarity, creativity and learning to occur.

Keywords: Mathematics, Psychology, Paradoxes, Literature, Modeling, Teaching & Learning

AMS Subject Classification: 97

Pdf: Preprint of table of contents of book to be published by Information Age Publishing, Charlotte, NC.

PMENA-Atlanta 2009 Working Group on Research Advances in Theories of Mathematics Education

Bharath Sriraman
The University of Montana

Gabriele Kaiser

University of Hamburg

Abstract

This working group revolves around the launch of a new book series entitled Advances in Mathematics Education by Springer Science, Heidelberg, and in particular on the first book in the series which focuses on Theories of Mathematics Education. This edited book in turn is based on a research forum on Theories of Mathematics Education at PME 29 in Melbourne, 2005, which resulted in two ZDM special issues on theories of mathematics education(issue 6/2005 and issue 1/2006). Since the research forum in Melbourne, numerous advances have taken place in the area of theory development in mathematics education in Europe and in North America. The purpose of this working group on research advances in theories of mathematics education is to integrate, synthesize and present a coherent picture on the state of the art. The working group will attempt to be both summative as well as forward looking by highlighting theories from psychology, philosophy and social sciences that continue to influence theory building, as well as provide participants insights into new developments in feminist, critical and political theories of mathematics education.

Keywords: Mathematics Education, Theories, Research working group

AMS Subject Classification: 97

Pdf: Pre-print of PMENA 2009 Conference Proceedings

Advances in Mathematics Education

Gabriele Kaiser
University of Hamburg

Bharath Sriraman
The University of Montana

Abstract

"Advances in Mathematics Education" is a brand new forward looking monograph series which continues the tradition of the international journal "ZDM - The International Journal on Mathematics Education" of producing themed issues with invited guest editors and peer-review process. The new monograph series aims to integrate, synthesize and extend papers from already published themed issues of relevance today, by orientating issues of relevance towards the future state of the art. The planned monographs will enrich the ZDM issues with newly invited chapters and commentaries from renowned international experts in which ideas are pushed to new frontiers. Submissions undergo two rounds of critical peer review, externally from experts in the field and internally from the international editorial board. Although the monograph series aims mainly to produce monographs based on important ZDM issues from the past, the series is open to proposals from the community on other topics of interest to the field.

Keywords: Mathematics Education, Research Advances, Theories

AMS Subject Classification: 97

Pdf: Flyer of new research series in mathematics education

An Nonnegatively Constrained Iterative Method for Positron Emission Tomography

See #23 2010 above for an expanded and corrected version of this paper.


Johnathan M. Bardsley

Abstract

In positron emission tomography (PET), data is collected via the detection of photons emitted by a radioactive tracer within the subject. The noise in PET data is Poisson, and hence, it is typical to reconstruct the tracer density distribution via the computation of an approximate minimizer of the negative-log Poisson likelihood function. In this paper, we present an iterative method for the solution of the PET inverse problem. A weighted least squares approximation of the negative-log Poisson likelihood is used. The method is nonnegatively constrained and, due to the ill-conditioned nature of the PET inverse problem, requires a stopping rule for its iterations. We present a statistically motivated stopping rule based on the χ2-test. The approach is implemented on a synthetically generated example, and is shown to be effective.

Keywords: positron emission tomography, iterative methods, regularization, statistical methods.

AMS Subject Classification: 65J22, 65K10, 65F22

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A Multi-Stage Model for Quantitative PCR

Emily Stone
The University of Montana

John Goldes
The University of Montana

Martha Garlick
Utah State University

Abstract

PCR (Polymerase Chain Reaction), a method which replicates a selected sequence of DNA, has revolutionized the study of genomic material, but mathematical study of the process has been limited to simple deterministic models or descriptions relying on stochastic processes. In this paper we develop a deterministic model for the reactions of quantitative PCR (Polymerase Chain Reaction) based on the law of mass action. Maps are created from DNA copy number in one cycle to the next, with ordinary differential equations describing the evolution of different molecular species during each cycle. The advantage of this type of model is the ability to vary the time spent in each stage of the reaction, which is critical to predicting optimal protocols. Qualitative analysis of the models are performed and parameters are estimated by fitting each model to data from Roche LightCycler (TM) runs.

Keywords: polymerase chain reaction, mathematical model

AMS Subject Classification: 92C45, 92C50, 92D20

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Theories of Mathematics Education: Seeking new frontiers


Bharath Sriraman
The University of Montana

Lyn English
Queensland University of Technology, Australia

Abstract

The inaugural monograph of Advances in Mathematics Education is based on the two highly acclaimed ZDM special issues on theories of mathematics education (issue 6/2005 and issue 1/2006), which in turn stemmed from the revival of the Theories of Mathematics Education (TME group) of Hans-Georg Steiner, at PME 29 in Melbourne, 2005, organized by the monograph editors (Sriraman & English). This monograph consists of articles with updated prefaces and commentaries from leading thinkers who have worked on theory building in mathematics education. The monograph attempts to be both summative as well as forward looking by highlighting theories from psychology, philosophy and social sciences that continue to influence theory building, as well as provide new developments in feminist, complexity and critical theories of mathematics education. To this end the monograph includes new chapters focused on networking theories, neuroscience research, complexity theory for mathematics education, political and feminist theories The cast of authors of this book include Paul Ernest, Stephen Lerman, Frank Lester, David Tall, John Pegg, Richard Lesh, Gerald Goldin, Luis Moreno Armella, Bharath Sriraman, Lyn English, Angelika Bikner-Ahsbahs, Guenter Toerner, Gabriele Kaiser, Gilah Leder, Norma Presmeg, Tommy Dreyfus, Guershon Harel, Stephen Campbell, Simon Goodchild and others.

Keywords: theories; theory development; mathematics education

AMS Subject Classification: 97

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Pre-print of Sriraman, B., & English, L. (Eds) (2009). Theories of Mathematics Education: Seeking new frontiers. Monograph 1 of Advances in Mathematics Education. Springer Science.

Critical Issues in Mathematics Education


Paul Ernest, University of Exeter, UK
Brian Greer, Portland State University
Bharath Sriraman, The University of Montana

Abstract

The word "critical" in the title of this collection has three meanings, all of which are relevant. One meaning, as applied to a situation or problem, is "at a point of crisis". A second meaning is "expressing adverse or disapproving comments or judgments". A third is related to the verb "to critique", meaning "to analyze the merits and faults of". The authors contributing to this book pose challenging questions, from multiple perspectives, about the roles of mathematics in society and the implications for education. Traditional reasons for teaching mathematics include: preparing a new generation of mathematics researchers and a cadre of technically competent users of mathematics; training students to think logically; and because mathematics is as much part of cultural heritage as literature or music. These reasons remain valid, though open to critique, but a deeper analysis is required that recognizes the roles of mathematics in framing many aspects of contemporary society, that will connect mathematics education to the lived experiences of students, their communities, and society in general, and that acknowledges the global ethical responsibilities of mathematicians and mathematics educators.

The book is organized in four sections (1) Mathematics education: For what and why? (2) Globalization and cultural diversity, (3) Mathematics, education, and society and (4) Social justice in, and through, mathematics education

Keywords: critical theory; equity; mathematics education; social justice; mathematics in society

AMS Subject Classification: 97

Pre-print of Monograph 6 of The Montana Mathematics Enthusiast: Monograph Series in Mathematics Education published by Information Age Publishing, Charlotte, NC.

The characteristics of mathematical creativity


Bharath Sriraman
The University of Montana

Abstract

Mathematical creativity ensures the growth of mathematics as a whole. However the source of this growth, the creativity of the mathematician is a relatively unexplored area in mathematics and mathematics education. In order to investigate how mathematicians create mathematics; a qualitative study involving five creative mathematicians was conducted. The mathematicians in this study verbally reflected on the thought processes involved in creating mathematics. Analytic induction was used to analyze the qualitative data in the interview transcripts and to verify the theory driven hypotheses. The results indicate that in general, the mathematicians’ creative process followed the four-stage Gestalt model of preparation– incubation–illumination–verification. It was found that social interaction, imagery, heuristics, intuition, and proof were the common characteristics of mathematical creativity. In addition contemporary models of creativity from psychology were reviewed and used to interpret the characteristics of mathematical creativity.

Keywords: Domain specific creativity ; Gestalt psychology ; Jacques Hadamard; Systems views of creativity; Theories of creativity; Mathematical creativity

AMS Subject Classification: 97

Download Technical Report: pdf (226 KB)

Pre-print of Sriraman, B. (2009). The characteristics of mathematical creativity. ZDM- The International Journal on Mathematics education, vol41, nos 1&2, pp. 13-27.

Interdisciplinarity in Mathematics Education: Psychology, Philosophy, Aesthetics, Modelling and Curriculum


Bharath Sriraman
The University of Montana

Abstract

Human beings are by definition “interdisciplinary”- we are complex neurobiological organisms capable of juggling a wide array of tasks that intertwine the physical, psychological, inter-personal, intuitional, intellectual, cultural and spiritual dimensions of being. Yet schooling and institutional practices utilize a fraction of our capabilities with the goal of reducing us to functional specialists who propagate the existing status quos. This double issue of ZDM-The International Journal on Mathematics Education has been organized into 5 sections, namely psychology, philosophy, aesthetics, modelling and curriculum. This double issue includes extended papers from the MACAS1, MACAS2 and IHPST9 symposia after two, sometimes three rounds of critical peer review. This publication provides the current state of the art on interdisciplinarity in mathematics education based on the last 5 years of ongoing research initiatives.

Keywords: Interdisciplinary education; Mathematics Education; Creativity; Curriculum

AMS Subject Classification: 97

Download Technical Report: pdf (375 KB)

Pre-print of ZDM- The International Journal on Mathematics Education, vol 41. Nos 1& 2, pp. 1-256 . January 2009 [in press]

URL: http://www.springerlink.com/content/vg6772846pgv8136

Relationship of creativity to intelligence

Bharath Sriraman
The University of Montana

Yasemin Kýymaz
Ahi Evran Üniversitesi Egitim Fakültesi, Turkey

Abstract

In this chapter traditional and contemporary theories of the relationship of creativity to intelligence are examined. The traditional approach to creativity can be characterized as the four P’s approach, namely studying the person, the process, the product and productive conditions (the environment). A number of confluence theories of creativity, such as investment theory, systems theory are considered in which the general intelligence (g) of a person is a necessary component but not sufficient for creativity (C) to manifest. In these confluence theories Creativity (big C) is that which is domain specific and a creative product is one that causes a significant shift within a specialized domain of knowledge. Ordinary creativity (little c) is also considered as well as the overlap between the constructs of creativity and intelligence.

Keywords: Creativity testing; Creativity theories; Flow; Fluid and Crystallized Intelligence; General creativity; Intelligence theories

AMS Subject Classification: 97

Contact author for details on obtaining the Encyclopedia (pre-print unavailable due to Publisher policy)

Preprint of chapter to appear in B. Kerr (Ed). Encyclopedia of Giftedness, Creativity and Talent. Sage Publications

Mathematics Curriculum

Bharath Sriraman
Dept. of Mathematical Sciences
The University of Montana


Abstract

This chapter examines mathematics curriculum in general from the point of view of the programming needs for gifted and talented learners . NSF reform based curricula, AP courses , the international baccalaureate program and the calls of the National Research Council (NRC) are critiqued.

Keywords: mathematics curriculum; gifted and talented learners; programming options

AMS Subject Classification: 97

Contact author for details on obtaining the Encyclopedia (pre-print unavailable due to Publisher policy)

Preprint of chapter to appear in B. Kerr (Ed). Encyclopedia of Giftedness, Creativity and Talent. Sage Publications

Elementary School, Mathematics Curriculum

Bharath Sriraman
Dept. of Mathematical Sciences
The University of Montana


Abstract

The elementary school mathematics curriculum is a crucial component in the general education of students and provides opportunities for giftedness and creativity to manifest, as well as serves as a foundation for initiating a deeper interest in mathematically oriented fields. Elementary mathematics also plays a key role in identification procedures used in gifted education. In this chapter the recommendations of the National Council of Teachers of Mathematics for the elementary curriculum are summarized, followed by an exposition of the major theorists that have impacted the elementary school mathematics curriculum. The entry concludes with a discussion on identification and a brief synopsis of programming options available at the elementary level.

Keywords: Cognition; Early Identification; Identification; Mathematics curriculum

AMS Subject Classification: 97

Contact author for details on obtaining the Encyclopedia (pre-print unavailable due to Publisher policy)

Preprint of chapter to appear in B. Kerr (Ed). Encyclopedia of Giftedness, Creativity and Talent. Sage Publications

Collaborative Learning: Mathematics and Science

Bharath Sriraman
Dept. of Mathematical Sciences
The University of Montana

Abstract

The pedagogical notion of collaborative learning can be traced back to the writings of John Dewey and Lev Vygotsky, among others. The image of an isolated and solitary learner is a thing of the past since most 21st century professions in industry require the ability to work collaboratively on projects in groups/teams where communicating, sharing and synthesizing ideas are paramount to success and accountability is accorded to the group. This chapter examines theory, research and practice of collaborative learning in mathematics and science education, particularly social constructivism, and from the point of view of the literature in psychology and organizational sciences on group dynamics.

Keywords: Collaboration, Cluster grouping, Creative communities, Group dynamics

AMS Subject Classification: 97

Contact author for details on obtaining the Encyclopedia (pre-print unavailable due to Publisher policy)

Preprint of chapter to appear in B. Kerr (Ed). Encyclopedia of Giftedness, Creativity and Talent. Sage Publications

Hierarchical regularization for edge-preserving reconstruction of PET images

Johnathan M. Bardsley , Daniela Calvetti, and Erkki Somersalo

Abstract

The data in PET emission and transmission tomography and in low dose X-ray tomography, consists of counts of photons originating from random events. The need to model the data as a Poisson process poses a challenge for traditional integral geometry-based reconstruction algorithms. Although qualitative a priori information of the target may be available, it may be difficult to encode it as a regularization functional in a minimization algorithm. This is the case, for example, when the target is known to consist of well defined structures, but how many, and their location, form and size are not specified. Following the Bayesian paradigm, we model the data and the target as random variables, and we account for the qualitative nature of the a priori information by introducing a hierarchical model in which the a priori variance is unknown and therefore part of the estimation problem. We present a numerically effective algorithm for estimating both the target and its prior variance. Computed examples with simulated and real data demonstrate that the algorithm gives good quality reconstruction for both emission and transmission PET problems at very low computational cost.

Keywords: edge-preserving regularization, positron emission tomography, Bayesian statistical methods, nonnegatively constrained optimization

AMS Subject Classification:

Download Technical Report: Pdf (260 KB)