# Technical Reports

## Technical Reports 2008

### Combinatorial Proof of an Abel-type Identity

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

David Perkins
Department of Mathematics and Computer Science
Houghton College, Houghton NY 14744, USA
david.perkins@houghton.edu

AMS Subject Classification: Primary 05A19; Secondary 05C30, 60C05

Preprint to appear in J. Combin. Math. Combin. Comput. Pdf (90 KB)

### A Fixed Point Formulation of the k-Means Algorithm for Image Segmentation and a Connection to Mumford-Shaw

J. M. Bardsley

Department of Mathematical Sciences
University of Montana
Missoula, Montana 59812, USA

Aaron Luttman
Division of Mathematics and Computer Science
Clarkson University, Science Center
Potsdam, New York, 13699

Abstract

In this note, we present a fixed point formulation of the k-means segmentation algorithm and show that the iteration's fixed points are solutions of the Euler-Lagrange equation for the k-phase Mumford-Shah energy functional.

Keywords:Mumford-Shah segmentation, fixed point methods, k-means

AMS Subject Classification:

### Aesthetics and Creativity: An exploration of the relationships between the constructs

Astrid Brinkmann
University of Muenster, Germany

Bharath Sriraman
The University of Montana, USA

Abstract

In this contribution, we report on an ongoing study that examines the relationship between aesthetics and creativity among working mathematicians. Writings of eminent individuals indicate that aesthetics is an important component of mathematical creativity, however we were interested in researching this relationship among the normal working mathematician. Anecdotally speaking, many working mathematicians often convey a reciprocal relationship between aesthetics and creativity, particularly when mathematical results and proofs are arrived at with considerable strain and stamina. We report on the findings of our ongoing study among working mathematicians in the U.S.A and Germany, and make a case for emphasizing the aesthetic dimension in mathematics education.

Keywords: affect; beliefs; aesthetics; creativity; mathematicians; aesthetics in mathematics education

AMS Subject Classification: 97

### Does interdisciplinary instruction raise students' interest in mathematics and the subjects of the natural sciences

Claus Michelsen
Centre for Science and Mathematics Education
University of Southern Denmark

Bharath Sriraman
Dept. of Mathematical Sciences
The University of Montana

Abstract

This paper presents the research project IFUN1 - Interest and Interdisciplinary Instruction in Science2 and Mathematics. The aim of the project was to investigate on how upper secondary students’ interest in the subjects of mathematics, physics, chemistry and biology might be improved by increased instructional interplay and integration between the subjects. The individual student’s interests in interdisciplinary domains of mathematics and science are studied within a three-dimensional framework: (i) The student’s interest in a particular interdisciplinary domain of mathematics and science. (ii) The characteristics of a specific learning setting that causes a situational interest in the topic and promotes and supports a shift from catching interest to holding interest (iii) The student’s affiliation with and valuation of mathematics and science. We present the main results from an interest study conducted with a 147 item Likert questionnaire administered to 255 grade 11 students. The results of the study show that students have a high interest in mathematics and are positive towards interdisciplinary instruction. When it comes to the individual student’s affiliation with and valuation of mathematics and science, the study shows that future studies and careers play an important role. We conclude that the results indicate it is possible to expand interest in one subject to another subject through interdisciplinary instruction.

Keywords: affect; IFUN; interdisciplinarity; mathematics and science instruction; models of interest; Southern Denmark

AMS Subject Classification: 97

Pre-print of paper accepted in ZDM- The International Journal on Mathematics Education, vol.41, nos.1&2, xx-xx
_____________________________________________

1 The acronym IFUN refers to Interesse og Fagoverskrindende Undervisning i Naturvidenskab and Interesse und Fächerübergreifender Unterricht in den Naturwisseschaften which is Danish and German respectively for Interest and Interdisciplinary Instruction in Science and Mathematics.
2We use the term science as a common denominator for the subjects of physics, chemistry and biology.

### The variational Kalman filter and an efficient implementation using limited memory BFGS

H. Auvinen, H. Haario, and T. Kauranne
Department of Mathematics and Physics
Lappeenranta University of Technology
Lappeenranta, Finland

J. M. Bardsley

Department of Mathematical Sciences
University of Montana
Missoula, Montana 59812, USA

Abstract

The standard formulations of the Kalman filter (KF) and extended Kalman filter (EKF) require storing and multiplication of matrices of size n x n, where n is the size of the state space, and the inversion of matrices of size m x m, where m is the size of the observation space. For large dimensions implementation issues arise. In this paper we introduce a Variational Kalman Filter (VKF) method to provide a low storage approximation of KF/EKF methods. In stead of using the KF formulae, we solve the underlying maximum a posteriori optimization problem using the limited memory BFGS (LBFGS) method. Moreover, the LBFGS optimization method is used to obtain a low storage approximation of state estimate covariances and prediction error covariances. A detailed description of the VKF method with LBFGS is given. The methodology is tested on linear and nonlinear test examples. Our simulations indicate that the approach yields results that are comparable with those obtained using KF and EKF, respectively, and can be used on much larger scale problems.

Keywords:Kalman filter, Bayesian inversion, large-scale optimization

AMS Subject Classification:

### A Matrix Theoretic Derivation of the Kalman Filter

John Bardsley
University of Montana

Abstract

We present a matrix theoretic derivation of the Kalman filter—motivated by the statistical technique of minimum variance estimation—in order to make its theoretical underpinnings accessible to a broader audience. Standard derivations of the filter utilize probabilistic arguments that are less familiar to the matrix analyst and computational mathematician.

Keywords:

AMS Subject Classification: 15A99, 65C60

### Aha! Experiences

Bharath Sriraman
Dept. of Mathematical Sciences
The University of Montana

Abstract

This chapter examines the role of aha! experiences in the discovery process from the point of view of historical literature on the topic, as well as from the literature in Gestalt and modern psychology.

Keywords: Problem solving; Gestalt psychology and creativity; General creativity; Mathematical creativity

AMS Subject Classification: 97

Contact author for details on obtaining the Encyclopedia

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

### Let Lakatos Be! - A Commentary to "Would the real Lakatos please stand up"

Bharath Sriraman
Dept. of Mathematical Sciences
The University of Montana

Abstract

In this commentary, some remarks are offered on David Pimm, Mary Beisiegel and Irene Meglis’ article “Would the real Lakatos please stand up”. The commentary focuses on relatively recent developments in the philosophy of mathematics based on the work of Lakatos; on theory development in mathematics education; and offers critique on whether Lakatos’ Proofs and Refutations can be directly implicated in mathematics education.

Keywords: mathematical methodology, philosophy of mathematics, mathematics education, philosophy of mathematics education, classroom discussion, school mathematics, Imre Lakatos, Proofs and Refutations

AMS Subject Classification: 97

Preprint of paper to appear in Interchange: A Quarterly Review of Education

### Astronomy

Bharath Sriraman
The University of Montana

Abstract

This chapter summarizes the origins and development of astronomy in numerous world cultures with a focus on the contributions in the post-Renaissance period.

Keywords: Eminence; History of science; History of mathematics; Polymaths; Visualization

AMS Subject Classification: 97

Contact author for details on obtaining the Encyclopedia

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

### Cognition

Bharath Sriraman
The University of Montana

Lyn English
Queensland University of Technology, Australia

Abstract

The term cognition encompasses a vast, diverse array of terms, concepts, processes and meanings. In this chapter we restrict ourselves to the meanings and descriptions of this term found in the domains of linguistics, psychology, philosophy, and phenomenology. We focus on some features of cognition such as analogical or metaphorical thinking and generalizing since they have been reported as characteristics of creative thinkers.

Keywords: Consciousness; Gestalt psychology and creativity; Habits of mind; Intuition; Learning sciences; Problem solving

AMS Subject Classification: 97

Contact author for details on obtaining the Encyclopedia

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

### Creativity, Giftedness and Talent Development in Mathematics

Bharath Sriraman
Dept. of Mathematical Sciences
The University of Montana

Abstract

Our innovative spirit and creativity lies beneath the comforts and security of today’s technologically evolved society. Scientists, inventors, investors, artists and leaders play a vital role in the advancement and transmission of knowledge. Mathematics, in particular, plays a central role in numerous professions and has historically served as the gatekeeper to numerous other areas of study, particularly the hard sciences, engineering and business. Mathematics is also a major component in standardized tests in the United States, and in university entrance exams in numerous parts of world. Creativity and imagination is often evident when young children begin to develop numeric and spatial concepts, and explore mathematical tasks that capture their interest. Creativity is also an essential ingredient in the work of professional mathematicians. Yet, the bulk of mathematical thinking encouraged in the institutionalized setting of schools is focused on rote learning, memorization, and the mastery of numerous skills to solve specific problems prescribed by the curricula or aimed at standardized testing. Given the lack of research based perspectives on talent development in mathematics education, this monograph is specifically focused on contributions towards the constructs of creativity and giftedness in mathematics. This monograph presents new perspectives for talent development in the mathematics classroom and gives insights into the psychology of creativity and giftedness. The book is aimed at classroom teachers, coordinators of gifted programs, math contest coaches, graduate students and researchers interested in creativity, giftedness, and talent development in mathematics.

AMS Subject Classification: 97

Pre-print of Monograph 4 of The Montana Mathematics Enthusiast: Monograph Series in Mathematics Education published by Information Age Publishing, Charlotte, NC.

### An Iterative Method for Edge-Preserving MAP Estimation when Data-Noise is Poisson

John Bardsley
University of Montana
&

John Goldes
University of Montana

Abstract

In numerous applications of image processing, e.g. astronomical and medical imaging, data-noise is well-modeled by a Poisson distribution. This motivates the use of the negative-log Poisson likelihood function for data fitting. (The fact that application scientists in both astronomical and medical imaging regularly choose this function for data fitting provides further motivation.) However difficulties arise when the negative-log Poisson likelihood is used. Chief among them are the facts that it is non-quadratic and is defined only for vectors with nonnegative values. The nonnegatively constrained, convex optimization problems that arise when the negative-log Poisson likelihood is used are therefore more challenging than when least squares is the fit-to-data function.

Edge preserving deblurring and denoising has long been a problem of keen interest in the image processing community. While total variation regularization is the gold standard for such problems, its use yields computationally intensive optimization problems. This motivates the desire to develop regularization functions that are edge preserving, but are less difficult to use. We present one such regularization function here. This function is quadratic, and can be viewed as the discretization of a diffusion operator with a diffusion function that is approximately 1 in smooth regions of the true image and is less than 1 (but still positive) at or near an edge.

Combining the negative-log Poisson likelihood function with this quadratic, edge preserving regularization function yields a strictly convex, nonnegatively constrained optimization problem. A large portion of this paper is dedicated to the presentation of and convergence proof for an algorithm designed for this problem. Finally, we apply the algorithm to synthetically generated data in order to test the methodology.

Keywords: edge-preserving regularization, inverse problems, nonnegatively constrained optimization, Bayesian statistical methods.

AMS Subject Classification: 15A29, 65K10, 65F22.

### A Theoretical Framework for the Regularization of Poisson Likelihood Estimation Problems

John Bardsley
University of Montana

Abstract

Let $$z = Au+\gamma$$, where $$\gamma>0$$ is constant, be an ill-posed, linear operator equation. Such a model arises, for example, in both astronomical and medical imaging, in which case \gamma corresponds to background light intensity. Regularized solutions of this equation can be obtained by solving

$$R_{\alpha }\left ( A,z \right )=\arg \min_{u\geq 0}\left \{ T_{0}\left ( Au;z \right )+\alpha J\left ( u \right ) \right \},$$

where $$T_{0}\left ( Au;z \right )$$ is the negative-log of the Poisson likelihood functional, and $$\alpha>0$$ and $$J$$ are the regularization parameter and functiosnal, respectively. This variational problem can be motivated from the fact that typical image data contains Poisson noise, and it has been analyzed, for three different choices of J, in previous work of the author. Our goal in this paper is to prove that these previous results imply that the approach defines a regularization scheme—which we rigorously define here—for each choice of J. Determining the appropriate definition for regularization scheme in this context is important: not only will it serve to unify the previously mentioned theoretical arguments, it will provide a framework for future theoretical analysis. In addition, we modify our presentation somewhat in order to improve understandability and provide missing arguments from our previous analysis.

Keywords: regularization, Poisson likelihood, statistical estimation, mathematical imaging

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

### The Handbook of International Research in Mathematics Education (2nd edition)

Lyn English
Queensland University of Technology, Australia

Richard Lesh
Indiana University

Graham Jones
Griffith University, Australia

Bharath Sriraman
The University of Montana

Dina Tirosh
School of Education, Tel-Aviv University, Israel

Mariolina Bartolini Bussi
Dipartimento di Matematica Pura ed Applicata - Università di Modena e Reggio Emilia – Italy

Abstract

The Second Edition of The Handbook of International Research in Mathematics Educationcontinues the mission of bringing together important new mathematics education research that makes a difference in both theory and practice. It updates and extends the Handbook’s original key themes and issues for international research in mathematics education for the 21st century, namely:

• priorities in international mathematics education research

Each of these themes is examined in terms of learners, teachers, and learning contexts, with theory development being an important component of all these aspects. This edition also examines other catalysts that have gained increased import in recent years including a stronger focus on the teacher and teacher practice, a renewed interest in theory development, an increased focus on the mathematics needed in work place settings, and a proliferation of research designs and methodologies that have provided unprecedented opportunities for investigating (and ultimately improving) mathematical teaching and learning. This edition includes ten totally new chapters; all other chapters are thoroughly revised and updated.

Keywords: research; policy; practice; learning contexts; methodologies; technology; theory development; international mathematics education; international comparisons

AMS Subject Classification: 97

### An Analysis of Regularization by Diffusion for Ill-Posed Poisson Likelihood Estimation

Johnathan M. Bardsley
The University of Montana

&

N'djekornom Laobeul
The University of Montana

Abstract

The noise contained in images collected by a charge coupled device (CCD) camera is predominantly of Poisson type. This motivates the use of the negative-log of the Poisson likelihood function in place of the ubiquitous least squares fit-to-data. However if the underlying mathematical model is assumed to have the form z = Au, where z is the data and A is a linear, compact operator, minimizing the negative-log of the Poisson likelihood function is an ill-posed problem, and hence some form of regularization is required. In previous work, the authors have performed theoretical analyses of two approaches for regularization in this setting: standard Tikhonov regularization in Bardsley and Laobeul 2008, and total variation regularization in Bardsley and Luttman 2008. In this paper, we consider a class of regularization functionals defined by differential operators of diffusion type, and our main results constitute a theoretical justification of this approach. However, in order to demonstrate that the approach is effective in practice, we follow our theoretical analysis with a numerical experiment.

Keywords:

AMS Subject Classification:

### A Morphology of Teacher Discourse in the Mathematics Classroom

Libby Knott
University of Montana

Bharath Sriraman
The University of Montana

&

Irv Jacob

Abstract

Discourse in mathematics classrooms is surprisingly complex and both student and teacher mathematical discourse contain distinct, identifiable elements. Student discourse is necessarily focused on understanding concepts and solving mathematical problems. Teacher discourse contains some of these same elements, but when examined critically it gives rise to major distinctions. Teacher discourse is directed at improving student understanding and also the logistics of the classroom, and thus is often meta-mathematical in nature. We shine a light on the tactics teachers use which are part of meta-mathematical discourse such as re-voicing, redirecting, questioning, clarifying. Contrasts are explored between student and teacher discourse.

Keywords: discourse morphology; discourse tactics; discourse tools; meta-mathematical discourse; student discourse; teacher discourse

AMS Subject Classification: 97

Pre-print of paper in press in International Journal of Mathematics Education Policy and Practice

### Exact bootstrap k-nearest neighbor learners

Brian M. Steele
Dept. of Mathematical Sciences
The University of Montana
Missoula MT 59812

Abstract

Bootstrap aggregation, or bagging, is a method of reducing the prediction error of a statistical learner. The goal of bagging is to construct a new learner which is the expectation of the original learner with respect to the empirical distribution function. In nearly all cases, the expectation cannot be computed analytically, and bootstrap sampling is used to produce an approximation. The k-nearest neighbor learners are exceptions to this generalization, and exact bagging of many k-nearest neighbor learners is straightforward. This article presents computationally simple and fast formulae for exact bagging of k-nearest neighbor learners and extends exact bagging methods from the conventional bootstrap sampling (sampling n observations with replacement from a set of n observations) to bootstrap sub-sampling schemes (with and without replacement). In addition, a partially exact k-nearest neighbor regression learner is developed. The article also compares the prediction error associated with elementary and exact bagging k-nearest neighbor learners, and several other ensemble methods using a suite of publicly available data sets.

Keywords: bagging, k-nearest neighbor, classication, regression, ensemble methods

AMS Subject Classification:

### On the internal structure of goals and beliefs

Günter Törner
University of Duisburg-Essen (Germany)

Katrin Rolka
University of Dortmund (Germany)

Bettina Rösken
University of Duisburg-Essen (Germany)

Bharath Sriraman
The University of Montana (USA)

Abstract

The theory ‚Teaching-In-Context’ (abbreviated here as KGB), first introduced by Schoenfeld in 1998, has the objective of making the actions of a teacher in mathematics lessons rationally understandable. According to this theory, it suffices to locate the behavior as a function of the following three parameters: goals, beliefs and available teacher’s knowledge. Schoenfeld´s hypothesis implies that spontaneous alterations in the teaching trajectories can be explained through shifts in the structure of goals and beliefs. In the following we discuss a particular videoed classroom lesson with a remarkable turning point on the background of this approach.

Keywords: Beliefs; Goals; Knowledge structures; Theory of teaching

AMS Subject Classification: 97

### Shaping the Future of Mathematics Education

Lyn D. English
Queensland University of Technology, Australia

Bharath Sriraman
The University of Montana

Abstract

This book will convey the state of the art on current theoretical perspectives in mathematics education and directions in which the future will/might shape up based on what is happening currently. The audience is the field of mathematics education [researchers, graduate and undergraduate students, as well as teachers]. Suggested chapters for inclusion in this book are as follows (titles are tentative). The chapters will be comprehensive to the extent possible to make the book an accessible introduction to the field of mathematics education and thus serve as both a resource book for researchers as well as a textbook for upper division and graduate courses.

Chapter 1. Why a book on shaping the future of mathematics education?
Chapter 2 Crossroads in current theories of mathematical thinking and learning.
Chapter 3. The critical role of theory and philosophy in conceptualizing and conducting research.
Chapter 4. The impact of cultural, social, and political forces on theory development in mathematical thinking and learning.
Chapter 5. Neuroscience and mathematical thinking and learning.
Chapter 6. Design science paradigms for learning and teaching mathematics from a European perspective.
Chapter 7. Design science paradigms for learning and teaching mathematics (from a U.S perspective).
Chapter 8 Comparative perspectives on mathematical modeling. This chapter will address the different perspectives on math modeling from Europe and the US.
Chapter 9. Mathematical modeling and its application to the solution of cross-disciplinary problems.
Chapter 10. Concluding chapter that summarizes the key ideas presented in the book and recommends directions for future development.

Keywords: cognitive science; design based research; learning theories; neuroscience and mathematics education; mathematics education; philosophy of mathematics education; politics of mathematics education; mathematical modeling

AMS Subject Classification: 97

Pre-print of accepted book proposal with Information Age Publishing, Charlotte, NC.

### Mathematics education and the legacy of Zoltan Paul Dienes

Bharath Sriraman
Dept. of Mathematical Sciences
The University of Montana

Abstract

The name of Zoltan P. Dienes (1916-) stands with those of Jean Piaget and Jerome Bruner as a legendary figure whose theories of learning have left a lasting impression on the field of mathematics education. Dienes’ name is synonymous with the Multi-base blocks (also known as Dienes blocks) which he invented for the teaching of place value. He also is the inventor of Algebraic materials and logic blocks, which sowed the seeds of contemporary uses of manipulative materials in mathematics instruction. Dienes’ place is unique in the field of mathematics education because of his theories on how mathematical structures can be taught from the early grades onwards using multiple embodiments through manipulatives, games, stories and dance. Dienes’ notion of embodied knowledge presaged other cognitive scientists who eventually came to recognize the importance of embodied knowledge and situated cognition – where knowledge and abilities are organized around experience as much as they are organized around abstractions. Dienes was an early pioneer in what was later to be called sociocultural perspectives and democratization of learning.

This monograph honors the seminal contributions of Dienes to mathematics education and includes several recent unpublished articles written by Dienes himself. These articles exemplify his principles of guided discovery learning and reveal the non-trivial mathematical structures that can be made accessible to any student. The monograph also includes a rare interview with Dienes in which he reflects on his life, his work, the role of context, language and technology in mathematics teaching and learning today. The book finds an important place in any mathematics education library and is vital reading for mathematics education researchers, cognitive scientists, prospective teachers, graduate students and teachers of mathematics.

Keywords: cognitive psychology; discovery learning; mathematical structures; multiple embodiments; Zoltan Dienes

AMS Subject Classification: 97

Pre-print of Monograph 2 of The Montana Mathematics Enthusiast: Monograph Series in Mathematics Education published by Information Age Publishing, Charlotte, NC.

### Models-modeling perspectives and the philosophy of pragmatism

Richard Lesh
Indiana University

Bharath Sriraman
The University of Montana

Abstract

Long ago, the American pragmatist John Dewey emphasized the fact that making science practical involves significantly different educational goals than making practice scientific. Recent studies based on models & modeling perspectives of learning and problem solving (Lesh & Doerr, 2003; Lesh & Sriraman, 2005) demonstrate that the knowledge and abilities that students develop tends to be significantly different depending on whether learning activities focus on: (a) realizing mathematics - by first teaching what is to be learned and then applying these concepts or abilities in realistic situations, or (b) mathematizing reality – that is by first putting students in sense-making situations where the conceptual systems that they develop on their own are later de-contextualized and formalized. Throughout this paper, the examples and observations suggest that, regardless whether we focus on conceptual development or skill development, a large share of what it means to “understand” is likely to be neglected unless adequate attention is given to learning activities where students are encouraged to mathematize reality – by expressing>testing>revising your own ways of thinking as opposed to being guided along artificially narrow paths toward idealized versions of the teacher’s or textbook’s ways of thinking.

Keywords: conceptual systems; model eliciting activities; models and modeling; Pragmatism

AMS Subject Classification: 97

Pre-print of paper accepted in ZDM- The International Journal on Mathematics Education, vol.41, no.1, xx-xx

### General Creativity

Bharath Sriraman
Dept. of Mathematical Sciences
The University of Montana

Abstract

This chapter examines the notion of general creativity taking into consideration Gestalt psychology, domain specific theories versus domain general theories, and distinguishing between extra-ordinary creativity versus day-to-day creativity. Five principles for promoting creativity in the classroom are synthesized from the literature.

Keywords: “Aha!” experience; Cognition; Intelligence Testing; Gestalt psychology and creativity; Mathematical creativity

AMS Subject Classification: 97

Contact author for details on obtaining the Encyclopedia

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

### Iceland and rural/urban girls- PISA 2003 examined from emancipatory viewpoint.

Olof Steinthorsdottir
Faculty of Education
University of North Carolina-Chapel Hill

Bharath Sriraman
Dept. of Mathematical Sciences
The University of Montana

Abstract

Scholarly research related to gender and mathematics is not as frequently published as it was in the 1980’s and the 1990’s. In Lubienski’s (2000) survey of Mathematical Education Research from 1982 to 1998 there are 367 publications in Journal for Research in Mathematics Education (JRME) and 385 publications in Educational Studies in Mathematics (ESM) about gender. This gives us approximately 21 publications a year in JRME and 22 publications a year in ESM. We did a search of publications about gender in JRME and ESM from 1999 to 2005 (or today) and saw a very different picture. Over this period 14 publications were in JRME and 17 publications in ESM, which gives approximately 4 publications a year in JRME and 3 publications a year in ESM. So what do these numbers tell us about the status of research about gender and mathematics? Does this mean that the gender gap has been closed? If so, for whom is that true? Does it mean that we don’t have to worry about gender differences in mathematics any more? And if it is true, is it certain that it will sustain itself without any follow up? Finally, why are there still differences in women entering fields such as mathematics and physics?

Keywords: Achievement; Beliefs; Gender gap; Gender and Mathematics; Iceland; PISA 2003

AMS Subject Classification: 97

Pre-print of chapter in B. Sriraman (Ed). International Perspectives on Social Justice in Mathematics Education (pp. 231-244). Monograph1 of The Montana Mathematics Enthusiast: Monograph Series in Mathematics Education published by Information Age Publishing, Charlotte, NC.

### Beliefs and Mathematics

Bharath Sriraman
Dept. of Mathematical Sciences
The University of Montana

Abstract

Beliefs and Mathematics is a Festschrift honoring the contributions of Günter Törner to mathematics education and mathematics. Mathematics Education as a legitimate area of research emerged from the initiatives of well known mathematicians of the last century such as Felix Klein and Hans Freudenthal. Today there is an increasing schism between researchers in mathematics education and those in mathematics as evidenced in the Math wars in the U.S and other parts of the world. Günter Törner represents an international voice of reason, well respected and known in both groups, one who has successfully bridged and worked in both domains for three decades. His contributions in the domain of beliefs theory are well known and acknowledged. The articles in this book are written by many prominent researchers in the area of mathematics education, several of whom are editors of leading journals in the field and have been at the helm of cutting edge advances in research and practice. The contents cover a wide spectrum of research, teaching and learning issues that are relevant for anyone interested in mathematics education and its multifaceted nature of research. The book as a whole also conveys the beauty and relevance of mathematics in societies around the world. It is a must read for anyone interested in mathematics education.

Keywords: attitudes; beliefs; mathematicians and mathematics educators; “math wars”; modeling; teaching and learning

AMS Subject Classification: 97

Pre-print of Monograph 3 of The Montana Mathematics Enthusiast: Monograph Series in Mathematics Education published by Information Age Publishing, Charlotte, NC.

### International Perspectives on Social Justice in Mathematics Education

Bharath Sriraman
Dept. of Mathematical Sciences
The University of Montana

Abstract

International Perspectives and Research on Social Justice in Mathematics Education is the highly acclaimed inaugural monograph of The Montana Mathematics Enthusiast now available through IAP. The book covers prescient social, political and ethical issues for the domain of education in general and mathematics education in particular from the perspectives of critical theory, feminist theory and social justice research. The major themes in the book are (1) relevant mathematics, teaching and learning practices for minority and marginalized students in Australia, Brazil, South Africa, Israel, Palestine, and the United States., (2) closing the achievement gap in the U.K, U.S and Iceland across classes, ethnicities and gender, and (3) the political dimensions of mathematics. The fourteen chapters are written by leading researchers in the international community interested and active in research issues of equity and social justice.

Keywords: education; educational equalization; equity; minorities; political dimensions of mathematics education; social justice

AMS Subject Classification: 97

Pre-print of Monograph1 of The Montana Mathematics Enthusiast: Monograph Series in Mathematics Education published by Information Age Publishing, Charlotte, NC.

### The Modeling Perspective in the teaching and learning of mathematical Problem Solving at the elementary and secondary school level

Nicholas Mousoulides
The University of Cyprus

Bharath Sriraman
The University of Montana

Constantinos Christou
The University of Cyprus

Abstract

The purpose of this study was to examine in depth the modeling processes used by students in working with modeling activities and to examine how students’ modeling abilities are changed over time. Two student populations, one experimental and one control group, were involved in the study. To examine modeling processes in students’ work, experimental group students participated in an intervention program consisting of a sequence of six modeling activities. To examine students’ modeling abilities, experimental and control group students completed a modeling abilities test three times. Results showed that students’ models improved as students worked through the sequence of the modeling activities. Results also revealed that a number of factors, such as students’ grade, students’ experiences with modeling activities, and students’ modeling abilities influence the modeling processes students used in their work. Results related to students’ modeling abilities showed that participation in the intervention program had a significant impact on the students’ modeling abilities. Finally, the study proposes a three- layer theoretical model for examining students’ modeling behavior, which may have some implications on the teaching and learning of mathematical problem solving.

Keywords: Cyprus; 6th and 8th grade achievement differences; local developmental trajectories; intervention program; modeling abilities; modeling activities; mathematical modeling

AMS Subject Classification: 97

Pre-print of paper to appear in Mathematical Thinking & Learning: An International Journal

### On the Identit(ies) of Mathematics Education

Bharath Sriraman
Dept. of Mathematical Sciences
The University of Montana

Abstract

This critical notice reviews sections of three recently published Handbooks of relevance to the community of mathematics education researchers. In particular it examines (1) the overlap between the historic foundations of mathematics education and educational psychology; (2) the shared problem of identity; (3) areas of consonance and dissonance, particularly the need to bring in socio-political issues; and (4) the complexity of understanding and modeling human cognition.

Keywords: Education Psychology; Foundations; Handbooks; Mathematical Cognition; Mathematics Education

AMS Subject Classification: 97

Pre-print of paper to appear in Mathematical Thinking & Learning: An International Journal, vol.10, no.3, pp.xx-xx

### Tracing Student's modeling processes in school

Nicholas Mousoulides
The University of Cyprus

Bharath Sriraman
The University of Montana

Marios Pittalis & Constantinos Christou
The University of Cyprus

Abstract

In this study we report on an analysis of the mathematization processes of one 6th and one 8th grade group, with emphasis on the similarities and differences between the two groups in solving a modeling problem. Results provide evidence that all students developed the necessary mathematical constructs and processes to actively solve the problem through meaningful problem solving. Eighth graders were involved in a higher level of understanding the problem presented in the activity, employed more sophisticated mathematical concepts and operations, better validated and communicated their results and reached more efficient models. Finally, a reflection on the differences in the diversity and sophistication of the constructed models and mathematization processes between the two groups raises issues regarding the design and implementation of modeling activities in elementary and lower secondary school level.

Keywords: Achievement differences; Cyprus; mathematizaton; Mathematical modeling

AMS Subject Classification: 97

Pre-print of chapter to appear in R. Lesh., P. Galbraith., W. Blum., and A. Hurford (Eds). Education and the Design Sciences. Springer Science and Business

### A historic overview of the interplay of theology and philosophy in the arts, mathematics and sciences

Bharath Sriraman
Dept. of Mathematical Sciences
The University of Montana

Abstract

The etymology of the word "mathematics" can be traced to Greek and Latin roots with meanings such as "to think or have one's mind aroused" or "the art of knowing". The natural philosophers of the Renaissance did not draw an explicit distinction between mathematics, the sciences and to an extent the arts. In this paper we first explore connections forged by the thinkers of the Renaissance between mathematics, the arts and the sciences, with attention to the nature of the underlying questions that call for a particular mode of inquiry. Second, we will examine both the relationship and individual differences between innovative behaviors across domains. Recently Robert Root-Bernstein (2003) introduced the construct of polymathy to suggest that innovative individuals are equally likely to contribute both to the arts and the sciences and either consciously or unconsciously forge links between the two. Several contemporary examples are presented of individuals who pursued multiple fields of research and were able to combine the aesthetic with the scientific. Finally, we will also discuss the possibilities for re-introducing university courses on natural philosophy as a means to integrate mathematics, the arts and the sciences.

Keywords: history of mathematics; philosophy of mathematics; polymathy; theology

AMS Subject Classification: 97

Pre-print of paper accepted in ZDM- The International Journal on Mathematics Education, vol.41, no.1, xx-xx

### Classroom Practice: Challenging mathematics classroom practices.

Bharath Sriraman
Dept. of Mathematical Sciences
The University of Montana

(with G.Stillman, C.Kwok-cheung, R.Mason, L.Sheffield & K. Ueno)

Abstract

In this chapter we examine classroom practice issues related to teachers providing mathematical challenges in their everyday classrooms. We examine how challenging mathematics can become the essence of mathematics classrooms, how challenging mathematics can be designed for the everyday classroom and how classroom artefacts and practices can be designed for mathematical challenges. Finally, the question of suitable research designs for research into classroom practices associated with the use of challenging mathematics in everyday classrooms is addressed and illustrated.

Keywords: Classroom practice; Discovery learning; Mathematical challenges; Mathematical Structures; International initiatives in mathematical challenges

AMS Subject Classification: 97

Preprint of chapter to appear in E. Barbeau & P. Taylor (Eds), ICMI Study 16 Study, Volume on Mathematical Challenges, Springer Science & Business

### Mathematical precociousness

Bharath Sriraman
Dept. of Mathematical Sciences
The University of Montana

Abstract

This chapter describes mathematical precociousness from historical and contemporary viewpoints taking into consideration psychometric methods, history of mathematics and extant models of talent development.

Keywords: Eminence; Genius; Mathematical giftedness; Prodigies; Study for Mathematically Precocious Youth; Very young gifted

AMS Subject Classification: 97

Contact author for details on obtaining the Encyclopedia

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

### Mathematical Intelligence

Bharath Sriraman
Dept. of Mathematical Sciences
The University of Montana

Abstract

This chapter takes a critical stance on the construct of mathematical intelligence and synthesizes and summarizes the research from psychometric, socio-cultural, mathematics education and educational psychology viewpoints.

Keywords: Cognition; Intelligence Testing; Mathematical giftedness; Mathematical creativity; Mathematics curriculum

AMS Subject Classification: 97

Contact author for details on obtaining the Encyclopedia

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

### Future directions for mathematics education

Lyn English
Queensland University of Technology, Australia

Richard Lesh
Indiana University

Graham Jones
Griffith University, Australia

Bharath Sriraman
The University of Montana

Dina Tirosh
School of Education, Tel-Aviv University, Israel

Mariolina Bartolini Bussi
Dipartimento di Matematica Pura ed Applicata - Università di Modena e Reggio Emilia – Italy

Abstract

This chapter summarizes promising trends within international research in mathematics education as well as outlines and discusses promising directions for future research, policy and practice in the 21st century.

Keywords:Directions for research in mathematics education; mathematics education; Research in Mathematics Education; Policy in mathematics education

AMS Subject Classification: 97

Pre-print of chapter to appear in L. English (Editor) Handbook of International Research in Mathematics Education (2nd edition), Taylor and Francis.

### Explaining the Icelandic Gender "Anomaly" in PISA 2003

Olof Steinthorsdottir
Faculty of Education
University of North Carolina-Chapel Hill

Bharath Sriraman
Dept. of Mathematical Sciences
The University of Montana

Abstract

PISA 2003 presented interesting results about students' mathematical achievement in Iceland, where Iceland was the only country that showed significant gender differences in mathematics in favor of girls. These unique results when statistically analyzed, it became evident that the gender differences were only measurable in the rural areas of Iceland. This poses a very interesting question about differences in rural and urban educational communities. The authors conducted a qualitative study in Iceland in 2007, in which 19 students from rural and urban Iceland who participated in PISA 2003 were interviewed in order to investigate these differences and determine factors that contributed to gender differences. The purpose of these interviews was to get students to elicit their thoughts on their mathematical experiences, their beliefs about mathematical learning, their thoughts about the PISA results, and their ideas on the reasons behind the unusual PISA 03 results. The data was transcribed, coded and analyzed using techniques from analytic induction in order to build themes and to present feminine and masculine student perspectives on the Icelandic anomaly.

Keywords:Achievement; Beliefs; Gendered discourse; Iceland; PISA 2003

AMS Subject Classification: 97

Pre-print of paper to appear in ZDM- The International Journal on Mathematics Education, vol.40, no.5, xx-xx

### Proceedings of the Second International Symposium on Mathematics and its Connections to the Arts and Sciences

Bharath Sriraman
Dept. of Mathematical Sciences
The University of Montana

Claus Michelsen
Dept. of Mathematics and Computer Sciences
University of Southern Denmark

Astrid Beckmann
University of Education, Scwhaebisch Gmuend,Germany

Viktor Freiman
University of Moncton, New Brunswick, Canada

Abstract

The Proceedings are based on the recently held Symposium on mathematics and its connections to the arts and sciences, namely the MACAS2 Symposium in Odense, Denmark (May 29-31, 2007). Simply put the proceedings are an eclectic collection of interdisciplinary research initiatives undertaken by mathematics educators with implications for practitioners concerned with teaching and learning processes. The papers cover a wide genre of research domains within mathematics education (cognition, modelling, problem solving, teacher education, ethnomathematics, mathematical/statistical literacy, curricular and technological initiatives and research related to science education). One of the papers presents an argument for the interdisciplinary nature of the field of mathematics education. The major interdisciplinary themes of the papers in this book are:

1. How can modelling activities be used to foster interdisciplinary projects in the school and university setting?
2. How can the intricate connections between mathematics and physics be used to design and research interdisciplinary activities in schools and the university?
3. How can research within the ethnomathematics domain of mathematics education be linked to critical mathematics education and interdisciplinary projects involving mathematics, art and culture?
4. How can the push for mathematical and statistical literacy be connected to other subjects in the school curricula and emphasized via interdisciplinary activities?
5. What are concrete examples of classroom experiments with empirical data that demonstrate new and unusual connections/relations between mathematics, arts and the sciences with implications for pedagogy?
6. What is the role of technology and new ICT interfaces in linking communities of learners in interdisciplinary activities involving problem solving?

Keywords:

AMS Subject Classification: 97

Pre-print of Book to be published by University of Southern Denmark Press, Odense.

### Response to Multicultural visions of Globalization

Bharath Sriraman
Dept. of Mathematical Sciences
The University of Montana

Ottawa, Illinois

Abstract

The paper by White in this issue of Interchange contains an interesting model for a global educational perspective based on the writings of Aurobindo and Pierre Teilhard de Chardin. White proposes a foundation for this new perspective based on the synthesis of Aurobindo’s and de Chardin’s theories of global, social, and conscious evolution. In our response we critique the author’s proposal from the perspective of the current challenges to social justice within the educational community. In particular the writings of Charles Darwin, Paolo Freire, Karl Marx, and Vivekananda (Aurobindo’s peer) are examined to present evolutionary and philosophical viewpoints on the origins and causes of inequity. We contend that no global multicultural perspective on education is possible unless we first address the fundamental inequities present within the socio-economic and educational structures that characterize the world today. Thus any attempt to introduce a “global educational agenda” within an educational institutional structure is bound to be problematic.

Keywords: Darwin; Education theories; Equity; Freire; Globalization; Karl Marx; Social Justice; Vivekananda

AMS Subject Classification: 97

Pre-print of paper accepted in Interchange: A Quarterly Review of Education, vol. 39, no.1, pp. xx-xx

### Mathematical paradoxes as pathways into beliefs and polymathy

Bharath Sriraman
Dept. of Mathematical Sciences
The University of Montana

Abstract

This paper addresses the role of mathematical paradoxes in fostering polymathy among pre-service elementary teachers. The results of a 3-year study with 120 students are reported with implications for mathematics pre-service education as well as interdisciplinary education. A hermeneutic-phenomenological approach is used to recreate the emotions, voices and struggles of students as they tried to unravel Russell’s paradox presented in its linguistic form. Based on the gathered evidence some arguments are made for the benefits and dangers in the use of paradoxes in mathematics pre-service education to foster polymathy, change beliefs, discover structures and open new avenues for interdisciplinary pedagogy.