Technical Reports

2013 Technical Reports

Watersheds for Solutions of Parabolic Systems

J. A. Cima
Department of Mathematics
University of North Carolina, Chapel Hill, NC 27599
cima@email.unc.edu

W. R. Derrick
Department of Mathematical Sciences
University of Montana, Missoula, MT 59802
derrick@mso.umt.edu

L. V. Kalachev
Department of Mathematical Sciences
University of Montana, Missoula, MT 59802
kalachev@mso.umt.edu

Abstract: In this paper we describe a technique that we have used in a number of publications to find the “watershed” under which the initial condition of a positive solution of a nonlinear reaction-diffusion equation must lie, so that this solution does not develop into a traveling wave, but decays into a trivial solution. The watershed consists of the positive solution of the steady-state problem together with positive pieces of nodal solutions (with identical boundary conditions). We prove in this paper that our method for finding watersheds works in Rk, k ≥ 1, for increasing functions ƒ(z) ⁄ z. In addition, we weaken the condition that ƒ(z) ⁄ z be increasing, and show that the method also works in R1 when f(z) ⁄ z is bounded. The decay rate is exponential.

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A Natural Experiment in Reservoir Levels and Recreational Use: Modeling Visitation on Lake Mead and Lake Powell

Chris J. NeherA,C, John W. DuffieldA, David A. PattersonA
ADepartment of Mathematical Sciences, University of Montana, Missoula, MT 59812,USA.

C Corresponding author. Email: bioecon@montana.com

Abstract: The Colorado River is one of the most highly developed watersheds in the United States and has relatively unique long term data sets for both recreational visitation and water levels on the two largest reservoirs in the United States—Lake Mead and Lake Powell. Previous efforts to model the relationship of recreation and reservoir water levels have primarily relied on survey-based estimates of visitor response to past actual or future hypothetical water levels. We provide the first major reservoir-recreation study based entirely on long-term observed data that also includes observed economic impacts.

Keywords: Lake Powell, Recreational use, Reservoir levels, Visitor spending, National Park Service

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A Framework for Quality Assurance in Globalization of Higher Education- a view toward the future

Jennifer Job
University of North Carolina at Chapel Hill

Bharath Sriraman
The University of Montana

Abstract: In the paper we theorize a framework for discussing quality assurance of globalized models of education used internationally. The philosophical assumption that homogeneity of perspectives achieves objectivity in practice is argued against using the examples of (1) Brain drain, and (2) Profit over quality. We present a coherent real world scenario for the abstract ideas on inequities in Sriraman & Adrian (2008) with a meta-analytical support of the literature.

Keywords: borderology; quality assurance; higher education; brain drain; meta-analysis of globalization of “higher education”; philosophy of education

AMS Subject Classification: 97

Pre-print of Revision to #19/2011 submitted to Interchange: A Quarterly Review of Education PDF (147 KB)

Abstracts of the First Sourcebook on Asian Research in Mathematics Education

Bharath Sriraman
The University of Montana

Jinfa Cai
University of Delaware

Kyeonghwa Lee
Seoul National University

Fan Lianghuo
University of Southampton

Yoshinori Shimuzu
University of Tsukuba

Lim Chap Sam
Universiti Sains Malaysia

K. Subramaniam
Tata Institute of Fundamental Research, India

Abstract: In this edited book, extended abstracts cover existing mathematics education research in China, Korea, Singapore, Japan, Malaysia and India. The 76 abstracts cover historical, philosophical and cultural origins of mathematics education in these countries, the role of policy in the implementation of new curricula, advances in assessment, cross national studies, and last but not least basic research and development in mathematical thinking and learning. The book represents the state of the art of advances in mathematics education research in these Asian countries.

Keywords: mathematics education; Confucianism; history of mathematics education; history of mathematics reform; China; Korea; Singapore; Japan; Malaysia; India

AMS Subject Classification: 97

Table of contents of Sriraman, B et al (2013) Abstracts of the First Sourcebook on Asian Research in Mathematics Education. Information Age Publishing, Charlotte, NC. PDF (3253 KB)

Commentary: Mathematical Creativity and Giftedness- A review of theory, new operational views and ways forward

Bharath Sriraman
The University of Montana

Per Haavold
University of Tromso, Norway

Kyeonghwa Lee
Seoul National University, South Korea

Abstract: In this commentary we synthesize and critique three articles in this special issue of ZDM (Leikin, & Lev; Kattou, Kontoyianni, Pitta-Pantazi, & Christou ; Pitta-Pantazi, Sophocleous, & Christou). In particular we address the theory that bridges the constructs of “mathematical creativity” and “mathematical giftedness” by reviewing the related literature. Finally, we discuss the need for a reliable metric to assess problem difficulty and problem sequencing in instruments that purport to measure mathematical creativity, as well as the need to situate mathematics education research within an existing canon of work in mainstream psychology.

Keywords: Creativity; Mathematical Creativity; Gestaltism; Problem sequencing and metrics; Psychology, mathematics education; Qualitative psychometrics

AMS Subject Classification: 97

Pre-print of commentary paper submitted to ZDM-The International Journal on Mathematics Education, Springer Science & Business. Pdf (959 KB)

Creatively Gifted Students are not like other gifted students

Kyung Hee Kim
College of William and Mary, Virginia, USA

James C. Kaufman
California State University, San Bernardino, USA

John Baer
Rider University, New Jersey, USA

Bharath Sriraman
The University of Montana

Abstract: This book focuses on the needs of creatively gifted students and how schools can meet those needs. Creatively gifted students show exceptional levels of creativity. These students may or may not have developed other talents and abilities, yet. Even when their abilities and talents are apparent, the needs of creatively gifted students may not be recognized by current gifted education programs. Regardless of whether a creatively gifted student is included in these programs, schools often inadvertently ignore their special needs. The goal of this book is to share the newest research about the attributes and needs of creatively gifted students and the kinds of programs that best address those special needs.

Keywords: creativity; giftedness; exceptional children; gifted education programs

AMS Subject Classification: 97

Table of Contents and Chapter 1 to appear in Creatively Gifted Students are not like other gifted students, Advances in Creativity and Giftedness, Monograph 4. Sense Publishers, The Netherlands. Pdf (527 KB)

Probabilistic Thinking: Presenting Plural Perspectives

Egan Chernoff
University of Saskatchewan

Bharath Sriraman
The University of Montana

Abstract: In this edited collection the state of the art on advances in research on probabilistic thinking is presented via perspectives in philosophy, psychology, stochastics and mathematics education.

Keywords: decision making; teaching and learning probability; teaching experiments; philosophy of probability; psychology of probability; mathematics education

AMS Subject Classification: 97

Table of Contents with Abstracts to appear in E. Chernoff & B. Sriraman (Eds), Probabilistic Thinking: Presenting Plural Perspectives, Springer Science and Business Pdf (229 KB)

Theories of Learning Mathematics

Richard Lesh
Indiana University

Bharath Sriraman
The University of Montana

Lyn English
Queensland University of Technology, Australia

Abstract: According to Karl Popper, widely regarded as one of the greatest philosophers of science in the 20th century, falsifiability is the primary characteristic that distinguishes scientific theories from ideologies – or dogma. For example, for people who argue that schools should treat creationism as a scientific theory, comparable to modern theories of evolution, advocates of creationism would need to become engaged in the generation of falsifiable hypothesis, and would need to abandon the practice of discouraging questioning and inquiry. Ironically, scientific theories themselves are accepted or rejected based on a principle that might be called survival of the fittest. In this article we examine developments of learning theories in mathematics education through the lens of four essential Darwinian functions.

Keywords: Complexity; Learning Theories; Models and Modeling; Models versus Theories; Theories of mathematics education.

AMS Subject Classification: 97

Excerpt of article to appear in S. Lerman (Ed), The Encyclopedia of Mathematics Education, Springer Reference Works. Full article unavailable due to publisher embargo. Pdf (176 KB)

Quasi-empirical reasoning (Lakatos)

Bharath Sriraman
The University of Montana

Nicholas Mousoulides
University of Cyprus

Abstract: This entry examines Lakatos’ assertion that the nature of mathematical knowledge is quasi-empirical, in attempting to describe the growth of mathematical knowledge and its implications for mathematics education.

Keywords: Lakatos; Maverick traditions in the philosophy of mathematics; Fallibilism; Proofs and Refutations; Proof; Philosophy of Mathematics

AMS Subject Classification: 97

Excerpt of article to appear in S. Lerman (Ed), The Encyclopedia of Mathematics Education, Springer Reference Works. Full article unavailable due to publisher embargo. Pdf (174 KB)

Mathematical games in learning and teaching

Nicholas Mousoulides
University of Cyprus

Bharath Sriraman
The University of Montana

Abstract: Piaget, Bruner and Dienes suggest that games have a very important part to play in the learning of mathematics. In the last four decades, games have been proposed by a number of researchers as a potential learning tool in the mathematics classroom. Papert (1980) was among the first who suggested that students could learn mathematics effectively not only by playing (video) games, but also by designing their own computer games, using for instance authoring programming tools like Scratch and ToonTalk. In this article we discuss the findings in the literature on the use of mathematical games in learning and teaching.

Keywords: Games, videogames, computer games, engagement, motivation, programming

AMS Subject Classification: 97

Excerpt of article to appear in S. Lerman (Ed), The Encyclopedia of Mathematics Education, Springer Reference Works. Full article unavailable due to publisher embargo. Pdf (170 KB)

Creativity in Mathematics Education

Bharath Sriraman
The University of Montana

Per Haavold
University of Tromso, Norway

Kyeonghwa Lee
Seoul National University

Abstract: In this encyclopedia entry the relationship between mathematical creativity and giftedness and ability is looked at through a synthesis of contemporary research. In particular we analyze the findings in relation to a larger canon of literature in psychology.

Keywords: mathematical creativity; mathematical ability; psychology of creativity

AMS Subject Classification: 97

Excerpt of article to appear in S. Lerman (Ed), The Encyclopedia of Mathematics Education, Springer Reference Works. Full article unavailable due to publisher embargo. Pdf (168 KB)

Argumentation in Mathematics

Bharath Sriraman
Dept. of Mathematical Sciences
The University of Montana

Kristin Umland
Dept of Mathematics
University of New Mexico

Abstract: The origins of argumentation in mathematics are traced from Aristotelian logic onto the work of medieval theologians in modal logic, culminating in work in didactics on the pedagogy of argumentation. The characteristics of mathematical argumentation are unpacked.

Keywords: argumentation; history of logic; didactics of argumentation; history of mathematics

AMS Subject Classification: 01, 97

Excerpt of article to appear in S. Lerman (Ed), The Encyclopedia of Mathematics Education, Springer Reference Works. Full article unavailable due to publisher embargo. Pdf (168 KB)

Argumentation in Mathematics Education

Kristin Umland
Dept of Mathematics
University of New Mexico

Bharath Sriraman
Dept. of Mathematical Sciences
The University of Montana

Abstract: In this article argumentation in mathematics education is defined as the mathematical arguments that students and teachers produce in mathematics classrooms. The characteristics of argumentation are described by examining the canonical literature, particularly the works of Harold Fawcett, George Polya and Imre Lakatos.

Keywords: argumentation; deductive reasoning; heuristic reasoning

AMS Subject Classification: 97

Excerpt of article to appear in S. Lerman (Ed), The Encyclopedia of Mathematics Education, Springer Reference Works. Full article unavailable due to publisher embargo. Pdf (161 KB)