Technical Reports

2010 Technical Reports

Polynomial Identity Rings as Rings of Functions, II

Abstract: In characteristic zero, Zinovy Reichstein and the author generalized the usual relationship between irreducible Zariski closed subsets of the affine space, their defining ideals, coordinate rings, and function fields, to a non-commutative setting, where "varieties" carry a PGLn-action, regular and rational "functions" on them are matrix-valued, "coordinate rings" are prime polynomial identity algebras, and "function fields" are central simple algebras of degree n. In the present paper, much of this is extended to prime characteristic. In addition, a mistake in the earlier paper is corrected. One of the results is that the finitely generated prime PI-algebras of degree n are precisely the rings that arise as "coordinate rings" of "n-varieties" in this setting. For n = 1 the definitions and results reduce to those of classical affine algebraic geometry.

Keywords: Polynomial identity ring, central simple algebra, trace ring, coordinate ring, the Nullstellensatz, n-variety.

Mathematics Subject Classification. Primary: 16R30, 16R20; Secondary 14L30, 14A10.

Download Technical Report: Pdf (222 KB) (Revised November 12, 2011)

Krylov space approximate Kalman filtering

Johnathan M. Bardsley†, Marylesa Howard†, and Albert Parker*

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

* Center for Biolm Engineering, Montana State University, Bozeman, Montana, 59717, USA.

Abstract: The Kalman filter is a technique for estimating a time-varying state given a dynamical model for, and indirect measurements of, the state. It is used, for example, on the control problems associated with a variety of navigation systems. Even in the case of nonlinear state and/or measurement models, standard implementations require only linear algebra. However, for sufficiently large-scale problems, such as arise in weather forecasting and oceanography, the matrix inversion and storage requirements of the Kalman filter are prohibitive, and hence, approximations must be made. In this paper, we describe how the conjugate gradient iteration can be used within the Kalman filter for quadratic minimization, as well as for obtaining low-rank, low-storage approximations of the covariance and inverse-covariance matrices required for its implementation. The approach requires that we exploit the connection between the conjugate gradient and Lanczos iterations. Copyright © 2000 John Wiley & Sons, Ltd.

Keywords: Krylov subspace methods, Kalman filter, data assimilation, inverse problems.

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An Efficient MCMC Method for Uncertainty Quantification in Inverse Problems

Johnatha M. Bardsley, Univ. of Montana
Department of Mathematical Sciences
University of Montana
Missoula, Montana 59812
USA

E-mail: bardsleyj@mso.umt.edu

Abstract

The connection between Bayesian statistics and the technique of regularization for inverse problems has been given significant attention in recent years. For example, Bayes’ law is frequently used as motivation for variational regularization methods of Tikhonov type. In this setting, the regularization function corresponds to the negative-log of the prior probability density; the fit-to-data function corresponds to the negative-log of the likelihood; and the regularized solution corresponds to the maximizer of the posterior density, known as the maximum a posteriori (MAP) estimator. While a great deal of attention has been focused on the development of techniques for efficient computation of MAP estimators (regularized solutions), less explored is the problem of uncertainty quantification, which corresponds to the problem of determining the shape, at least to some degree, of the posterior density in high probability regions. One way to do this is to sample from the posterior density using a Markov chain Monte Carlo (MCMC) method. In this paper, we present an MCMC method for use on linear inverse problems with independent and identically distributed Gaussian noise and Gaussian priors (quadratic regularization functions). From the MCMC samples, an estimator (regularized solution), and measures of variability in the estimator, are computed. Additionally, samples of the noise and prior precision parameters are computed, making regularization parameter selection unnecessary.

Keywords: inverse problems, regularization, image reconstruction, Bayesian inference, Markov chain Monte Carlo, uncertainty quantification.

MSC numbers: 15A29, 62F15, 65F22, 94A08

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Integrals don't have anything to do with discrete math, do they?

Kal M. Karply*
*correspondence to mark.kayll@umontana.edu


Abstract

Most people think of integration as belonging to the realm of continuous mathematics, far removed from its ‘polar opposite’, discrete mathematics. We present a few examples illustrating the ubiquity of integration, even in the discrete world. Our goal is to convince students to keep up on their analysis, even if they lean more to the mathematically discrete. One never knows when an integral might rise seemingly out of nowhere and play an interesting role in a discrete problem.

Keywords: gamma function, perfect matching, graph, bipartite graph, rook polynomial, derangement

AMS Subject Classification: Primary 05C70 Secondary 05A15 33B15 05A05

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The Elements of Creativity and Giftedness in Mathematics

Bharath Sriraman, The University of Montana

Kyeonghwa Lee, Seoul National University

Abstract

The Elements of Creativity and Giftedness in Mathematics covers recent advances in mathematics education pertaining to the development of creativity and giftedness. The book is international in scope in the “sense” that it includes numerous studies on mathematical creativity and giftedness conducted in the U.S.A, China, Korea, Turkey, Israel, Sweden, and Norway in addition to cross-national perspectives from Canada and Russia. The topics include problem -posing, problem-solving and mathematical creativity; the development of mathematical creativity with students, pre and in-service teachers; cross-cultural views of creativity and giftedness; the unpacking of notions and labels such as high achieving, inclusion, and potential; as well as the theoretical state of the art on the constructs of mathematical creativity and giftedness. The book also includes some contributions from the first joint meeting of the American Mathematical Society and the Korean Mathematical Society in Seoul, 2009.Topics covered in the book are essential reading for graduate students and researchers interested in researching issues and topics within the domain of mathematical creativity and mathematical giftedness. It is also accessible to pre-service and practicing teachers interested in developing creativity in their classrooms, in addition to professional development specialists, mathematics educators, gifted educators, and psychologists.

Keywords: creativity; cross-cultural views of creativity; giftedness; mathematical creativity; mathematical giftedness; First KMS-AMS joint meeting.

AMS Subject Classification: 97

Preprint of Volume 1, Advances in Creativity and Giftedness, Sense Publishers, Rotterdam Pdf (169 KB)

Gifted girls and non-mathematical aspirations: A longitudinal case study of two gifted Korean girls

Kyeonghwa Lee
Seoul National University

Bharath Sriraman
The University of Montana


Abstract

In this longitudinal study of two gifted Korean girls, the experiences with early admittance into a gifted program are charted alongside their family and societal experiences that ultimately influenced their career choices in non-mathematical fields. Over the course of 8 years, interviews were conducted with the two gifted girls and their parents to determine factors that led to their choosing a non mathematical area of specialization in spite of early identification and support of their mathematical talent. Qualitative analysis of the longitudinal data led to the development of three main factors as contributing to the career choices. One of the startling finding of this study was contrary to the findings of other gifted education research, the two girls' early experiences with gifted education kept them from choosing careers related to math.

Keywords: career aspirations; early identification; enculturation; gender inequalities; gifted education, Korea, mathematics, self-concept.

AMS Subject Classification: 97

Preprint of paper submitted to Gifted Child Quarterly Pdf (432 KB)

Teachers as stakeholders of mathematics education research: Projects from Southern Norway

Simon Goodchild
University of Agder, Norway

Bharath Sriraman
The University of Montana


Abstract

Reports from the mathematics teaching developmental research projects that have been conducted in the southern part of Norway over the past six years. The projects have been established on principles of community and inquiry, in which didacticians from the university and teachers from schools and kindergartens have collaborated to improve the quality of teaching and learning mathematics, and to research the developmental processes. Developmental activity has been based on reports and experience of action research, design research, Japanese lesson study and learning study. Theoretically the projects have been developed, and data analysed from sociocultural perspectives, particularly communities of practice theory and cultural historical activity theory. The projects have pursued developmental research in which teachers and didacticians collaborate in a co-learning agreement with the intended outcome to develop communities of inquiry within schools and the project.

The projects have resulted in the development of school teams of mathematics teachers collaborating more closely in their practice. Teachers claim their practice has been transformed, and that the approaches to teaching and learning promoted through the projects have helped them to address fundamental tensions in their practice. However, outcomes from the projects are also consistent with reports from other curriculum and development activity: development takes place slowly: it is evolutionary rather than transformative; it marks an extrapolation of practice rather than a radical expansion of action possibilities; new tasks are adapted to existing practice rather than taken as an opportunity for practice to be adapted given the new possibilities opened by the tasks. The pace of development is understandable given the forces favouring alignment to the reproduction of practice: a national curriculum and textbook schemes, high-stakes examinations, expectations of students, parents, school leaders and authorities, teachers’ deep and decisive personal experience in a culture of practice, teachers’ workloads. Inquiry, it appears, is still seen mostly as an approach to and within mathematics, rather than, additionally, an approach to developing individual and collective agency in the practice of teaching and learning mathematics. In future projects it is, perhaps, necessary to emphasise more that inquiry is about experimenting and creativity in teaching as well as curiosity in learning.

Keywords: teacher development; school development; communities of practice; Norway mathematics education

AMS Subject Classification: 97

Abstract of paper to be presented at Banff International Research Station, December 5-10, 2010.

A Nonnegatively Constrained Iterative Method with Statistically Based Stopping Rules for CT, PET, and SPECT Imaging

Johnatha M. Bardsley, Univ. of Montana

Abstract

In this paper, we extend a nonnegatively constrained iterative method and three stopping rules for its iterations to the medical imaging inverse problems of computed tomography (CT), positron emission tomography (PET), and single photon emission computed tomography (SPECT); the iterative method and stopping rules were introduced for use in astronomical imaging. The paper begins with a brief introduction to the CT, PET, and SPECT mathematical and statistical models. The iterative method and stopping rules are then presented. And finally, numerical experiments are presented to validate the methods.

Keywords: nonnegatively constrained iterative methods, inverse problems, statistical methods, medical imaging

MSC numbers: 65J22, 65K10, 65F22

Download Technical Report: Pdf (167 KB) 

An Exploratory Study of Relationships between Students' Creativity and Mathematical Problem-Posing abilities

Xianwei Yuan
Dept of Mathematics
Illinois State University
USA

Bharath Sriraman
Dept of Mathematical Sciences
The University of Montana
USA

Abstract

In this paper we report on a cross cultural study involving U.S and Chinese high school students exploring the construct of mathematical creativity from a problem posing point of view. Numerous quantitative batteries such as the Torrance Tests of Creative Thinking [TTCT] and problem posing tasks are used to discern similarities and differences in creative abilities of comparable groups of U.S and Chinese students. Among the findings reported in this paper are a significant relationship between creativity and mathematical problem-posing abilities of high school students taking advanced courses in the Chinese group but not in the U.S. group.

Keywords: creativity; divergent thinking; generalizing; mathematical creativity; mathematical problem posing; problem solving; Torrance Tests of Creative thinking;

AMS Subject Classification: 97

Preprint of chapter to appear in B.Sriraman, K. Lee (eds.), The Elements of Creativity and Giftedness in Mathematics, xx–xy. © 2010 Sense Publishers (239 KB)

Interdisciplinarity for the 21st Century: Proceedings of the 3rd International Symposium on Mathematics and its connections to the Arts and Sciences

Bharath Sriraman, The University of Montana, USA
Viktor Freiman, University of Moncton, Canada

Abstract

Interdisciplinarity has become increasingly important for emergent professions of the 21st century yet there is a dearth of systematic studies aimed at implementing it in the school and university curricula. The Mathematics and its Connections to the Arts and Sciences (MACAS ) group places Mathematics as a vehicle through which deep and meaningful connections can be forged with the Arts and the Sciences and as a means of promoting interdisciplinary and transdisciplinary thinking traits amongst students. The Third International Symposium held by the MACAS group in Moncton, Canada in 2009 included numerous initiaves and ideas for interdisciplinarity that are implementable in both the school and university setting. The chapters in this book cover interdisciplinary links with mathematics found in the domains of culture, art, aesthetics, music, cognition, history, philosophy, engineering, technology and science with contributors from Canada, U.S, Denmark, Germany, Mexico, Iran and Poland amongst others.

Keywords: interdisciplinary mathematics; art and math; science and math; mathematics pedagogy

AMS Subject Classification: 97

Preprint of B.Sriraman & V.Freiman (Eds). Interdisciplinarity for the 21st Century: Proceedings of the 3rd International Symposium on Mathematics and its connections the Arts and Sciences, Information Age Publishing, Charlotte, NC. Pdf (239 KB)

Link to book on Information Age Publishing

Commentary on the Cognitive aspects of Early Algebraization

Bharath Sriraman, The University of Montana, USA
Kyeong-Hwa Lee, Seoul National University, Korea

Abstract

In this commentary to the nine chapters in the cognitive section of early algebraization, we synthesize and critically discuss common themes found in them such as components of non-formal algebraic thinking, the purported dichotomy between arithmetic and algebra; meaningful arithmetic, and generalizing ability, among others using the frameworks of William Brownell, Ernst Haeckl and Jean-Baptiste Lamarck.

Keywords: abstraction; early algebraization; generalization; generalized arithmetic; recapitulation; cultural evolution; longitudinal algebra projects; history of mathematics

AMS Subject Classification: 97

Pre-print of Chapter to appear in J. Cai & E. Knuth (Eds). EARLY ALGEBRAIZATION-CURRICULAR, COGNITIVE AND INSTRUCTIONAL PERSPECTIVES. ADVANCES IN MATHEMATICS EDUCATION, VOLUME 2, SPRINGER SCIENCE, BERLIN/HEIDELBERG Pdf (58 KB)

Advances in Creativity and Giftedness

Bharath Sriraman, The University of Montana

Abstract

Advances in Creativity and Giftedness is the first internationally established book series that focuses exclusively on the constructs of creativity and giftedness as pertaining to the psychology, philosophy, pedagogy and ecology of talent development across the milieus of family, school, institutions, organizations and society. ACG strives to synthesize both domain specific and domain general efforts at developing creativity, giftedness and talent. The books in the series are international in scope and include the efforts of researchers, clinicians and practitioners across the globe.

Keywords: Creativity; Giftedness; Talent development; Human development; Domain specific creativity (mathematics)

AMS Subject Classification: 97

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Conjecturing via reconceived classical analogy

Kyeonghwa Lee, Seoul National University, Korea
Bharath Sriraman, The University of Montana

Abstract

Reasoning and conjecturing by analogy is a fundamental human trait. One encounters excellent examples of this propensity to “analogize” in ancient Greek philosophy. If an ancient Greek philosopher were asked: why do we create analogies? The answer would simply be to create a framework by which we could better understand the dimensions of human experience (Sriraman, 2005). An important finding of English (2004) was that teachers must understand analogies themselves and know how to use them effectively (and also know which analogies are appropriate and which aren’t when it come to their use). They sometimes have to make the relationships explicit for the child. The OCA framework we have developed through reflective discourse practices by the teacher Seo, illustrate that analogies arising in mathematics are quite different from those arising in a discipline such as the life sciences where spontaneous analogies work well because children have a much larger a priori linguistic base, whereas in mathematics children’s pre-existing knowledge base is limited. This necessitates that both practitioners and researchers are sensitive to the major role that the knowledge base plays in the use of analogies for mathematics learning. The present study makes an important contribution for following this line of mathematical thinking initiated by the likes Newton, Euler and Polya. Further research is needed on a more typical classroom-type group of students. Another limitation relates to the limited content area. The focus of this paper was a triangle. Further studies involving a variety of OCA problems in different content areas are encouraged to verify the possibility of including OCA in mathematics learning. Finally, it will be necessary to not only identify but also clarify the kinds of norms or teaching interventions essential for effective integration of OCA into mathematics lessons.

Keywords: Analogical reasoning; Lakatos; Conjecturing; Classical Analogy; OCA framework

AMS Subject Classification: 97

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Mathematics Education in Turkey- At the crossroads of cultural, political and economic currents

Bharath Sriraman
Dept of Mathematical Sciences & The Department of Central/SW Asian Studies
The University of Montana
Missoula, MT 59812
USA

Abstract

In this opening introductory article to the issue of ZDM focused on mathematics education in Turkey, a short survey of the deep historical and cultural foundations of modern day Turkey is offered. This is followed by an outline of the development of researchers, as well as doctoral programs in mathematics education at universities in Turkey. The article also summarizes recently completed influential dissertations in Turkey and explains the aims and scope of this particular issue in relation to its conception and development.

Keywords: curriculum development; doctoral programs in Turkey; history of Turkey; mathematics education research in Turkey

AMS Subject Classification: 97

Revised version of pdf (295 KB)

A Brief History of Mathematics Education in Turkey: K-12 Mathematics Curricula

Ziya Argün, Ahmet Arikan, Safure Bulut, Bharath Sriraman

1Gazi University, Faculty of Education, Ankara, TURKEY
E-mail: ziya@gazi.edu.tr
2Gazi University, Faculty of Education, Ankara, TURKEY
E-mail: arikan@gazi.edu.tr
3Middle East Technical University, Ankara, TURKEY
E-mail: sbulut@metu.edu.tr
4The University of Montana, Dept. Of Mathematical Sciences, USA
E-mail: SriramanB@mso.umt.edu

Abstract

In this study, we survey the history of mathematics education in Turkey starting with its historical roots in the foundation of the republic. The changes in mathematics education in Turkey over the last century are investigated through an analysis of changes in curricular documents for K-12 schools. We consider the factors and reasons affecting curriculum developments, changes in philosophy and structure in terms of standards, objective and instructions. This article utilizes archival research techniques by examining original sources and illustrates the nature of the changes benefiting from a historical perspective. As a result of such analysis of the aforesaid sources, we have seen that the main reasons for changing mathematics curricula are: to build up a modern civilization in Turkey; the reports of John Dewey and the recommendations of Kate Wofford, William C. Varaceus and Watson Dickerman; the desire to become a member of the European Union; international factors and political situations.

Keywords: Turkey - Mathematics education - Curriculum - History

AMS Subject Classification: 97

Pre-print of article in press in – ZDM-The International Journal on Mathematics Education, vol. 42 Pdf (275 KB)

Tracing Students' Modeling Processes in School

N. Mousoulides, M. Pittalis, C. Christou
University of Cyprus, Nicosia, Cyprus
&
B. Sriraman
The University of Montana, Missoula, MT, USA

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 who 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: mathematical modeling; mathematization; models and modeling

AMS Subject Classification: 97

Prepress version of Chapter in R. Lesh et al. (eds.) (2010), Modeling Students’ Mathematical Modeling Competencies, (pp.119-129).DOI 10.1007/978-1-4419-0561-1_10, C _ Springer Science+Business Media, LLC 2010 Pdf (649 KB)

Mathematics Education in Iceland: Explaining the Non-homogeneity in a Homogenous System

Guðný Helga Gunnarsdóttir, Guðbjörg Pálsdóttir
School of Education
University of Iceland

&

Bharath Sriraman
The University of Montana


Abstract

In this opening salvo to the Icelandic section, we synthesize some of the themes present in the different chapters and address aspects that the reader may not realize about Icelandic mathematics education. We briefly outline distinct features and anomalies within mathematics education in Iceland, which in spite of the homogeneity of the system reveals many threads of thought in terms of reform and implementation of new ideas.

Keywords: mathematics education in Iceland; The First Sourcebook on Nordic Research in mathematics education

AMS Subject Classification: 97

Pre-print from Monograph 10 of The Montana Mathematics Enthusiast: Monograph Series in Mathematics Education published by Information Age Publishing, Charlotte, NC. Pdf (44 KB)

The First Sourcebook on Nordic Research in Mathematics Education

Mathematics Education Research in Denmark: A Brief Introduction

Bettina Dahl, Aarhus University, Denmark
&
Bharath Sriraman, The University of Montana

Abstract

This section of The First Sourcebook on Nordic Research in Mathematics Education presents nine chapters describing and discussing recent mathematics education research at primary, secondary, and tertiary level as well as adult mathematics education in Denmark. The papers represent a wide range of research methodologies such as qualitative, quantitative, historical, sociological, and philosophical accounts, and as a whole constitute a representative state of the art of research in mathematics education in Denmark.

Keywords: mathematics education; Denmark; The First Sourcebook of Nordic Research in Mathematics Education

AMS Subject Classification: 97

Pre-print from Monograph 10 of The Montana Mathematics Enthusiast: Monograph Series in Mathematics Education published by Information Age Publishing, Charlotte, NC. Pdf (44 KB)

The First Sourcebook on Nordic Research in Mathematics Education

Review of Creativity in Mathematics and the Education of Gifted Students- Edited by Roza Leikin, Abraham Berman and Boris Koichu

Bharath Sriraman
Department of Mathematical Sciences
The University of Montana
&
Kyeong-Hwa Lee
Seoul National University, Korea

Abstract

A critical review of Leikin, Berman & Koichu’s (2009) book Creativity in Mathematics and the Education of Gifted Students is offered in relation to the existing research and development in the domain of mathematical giftedness, creativity and talent.

Keywords: Giftedness; Theories of creativity; Talent development; Mathematical creativity

AMS Subject Classification: 97

Pre-print of Sriraman, B & Lee, K. (2010). Book Review, Educational Studies in Mathematics Pdf (426 KB)

Commentary on Theorizing in mathematics education research: differences in modes and quality

Bharath Sriraman
Department of Mathematical Sciences
The University of Montana

Abstract

Jameson’s critique of post-modernism is used as a general framework to discuss the analysis and recommendations of Jablonka & Bergsten on the different modes of theorizing in mathematics education, as well as their limitations and scope. In particular their arguments for transparency, inter-textuality and development of its inherently discursive nature are closely scrutinized.

Keywords:discursive approaches; post modernism; theories of mathematics education; inter-textuality

AMS Subject Classification: 97

Pre-print of article to appear in – Nordic Studies in Mathematics Education, vol 15, [in press] Pdf (50 KB)

Understanding a Teacher's Actions in the Classroom by Applying Schoenfeld's Theory Teaching-In-Context: Reflecting on Goals and Beliefs

G. Törner , B. Rösken
Department of Mathematics, University of Duisburg-Essen, Duisburg, Germany

K. Rolka
Department of Mathematics, University of Cologne, Cologne, Germany

B. Sriraman
Department of Mathematical Sciences, The University of Montana, Missoula, USA

Abstract

The theory teaching-in-context, 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: available teacher knowledge, goals, and beliefs. In the following, we discuss a particular videoed classroom lesson with a remarkable turning point on the background of this approach. A teacher, who recently attained an in-service training course about the use of open tasks, tried to adapt the imparted issues to the topic of linear functions. While the lesson did not develop as desired, she shifted back to her hitherto established traditional teaching repertoire. Schoenfeld’s hypothesis implies that such spontaneous alterations in the teaching trajectories can be explained through shifts in the interplay of knowledge, goals and beliefs.

Keywords: Beliefs theory; teaching-in-context; teaching trajectories; knowledge development

AMS Subject Classification: 97

Prepress version of Toerner, Rolka, K., Roesken, B., & Sriraman, B.(2010 ).. In° B. Sriraman & L. D. English (Eds.), Theories of mathematics education: Seeking new frontiers (pp. 401-420). Advances in Mathematics Education, Series: Springer. Pdf (326 KB)

Symbols and mediation in mathematics education (Book chapter of developed version of journal article)

Luis Moreno-Armella
Cinvestav-IPN
Mexico
&
Bharath Sriraman
Dept. of Mathematical Sciences
The University of Montana

Abstract

In this paper we discuss topics that are relevant for designing a theory of mathematics education. More precisely, they are elements of a pre-theory of mathematics education and consist of a set of interdisciplinary ideas which may lead to understand what occurs in the central nervous system—our metaphor for the classroom, and eventually, in larger educational settings. In particular, we highlight the crucial role of representations, the mediation role of artifacts, symbols viewed from an evolutionary perspective, and mathematics as symbolic technology.

Keywords: representation theory in philosophy; symbolic technology; digital media; philosophy of mathematics education

AMS Subject Classification: 97

Prepress version of Moreno, L., & Sriraman, B.(2010 ). Symbols and mediation in mathematics education. In° B. Sriraman & L. D. English (Eds.), Theories of mathematics education: Seeking new frontiers (pp. 213-232). Advances in Mathematics Education, Series: Springer. Pdf (485 KB)

Reconceptualizing mathematics education as a design science

Richard Lesh
School of Education
Indiana University

Bharath Sriraman
Dept. of Mathematical Sciences
The University of Montana

Abstract

In this chapter we propose re-conceptualizing the field of mathematics education research as that of a design science akin to engineering and other emerging interdisciplinary fields which involve the interaction of “subjects”, conceptual systems and technology influenced by social constraints and affordances. Numerous examples from the history and philosophy of science and mathematics and ongoing findings of M&M research are drawn to illustrate our notion of mathematics education research as a design science. Our ideas are intended as a framework and do not constitute a “grand” theory. That is, we provide a framework (a system of thinking together with accompanying concepts, language, methodologies, tools, and so on) that provides structure to help mathematics education researchers develop both models and theories, which encourage diversity and emphasize Darwinian processes such as: (a) selection (rigorous testing), (b) communication (so that productive ways of thinking spread throughout relevant communities), and (c) accumulation (so that productive ways of thinking are not lost and get integrated into future developments).

Keywords: design sciences; engineering education; engineering sciences; models and modeling; theories of mathematics education

AMS Subject Classification: 97

Prepress version of Lesh, R., & Sriraman, B.(2010). Reconceptualizing mathematics education as a design science. In° B. Sriraman & L. D. English (Eds.), Theories of mathematics education: Seeking new frontiers (pp. 123-146). Advances in Mathematics Education, Series: Springer. Pdf (659 KB)

Surveying theories and philosophies of mathematics education

Bharath Sriraman
Dept. of Mathematical Sciences
The University of Montana

Lyn English
Queensland University of Technology
Brisbane, Australia

Abstract

Any theory of thinking or teaching or learning rests on an underlying philosophy of knowledge. Mathematics education is situated at the nexus of two fields of inquiry, namely mathematics and education. However, numerous other disciplines interact with these two fields, which compound the complexity of developing theories that define mathematics education. We first address the issue of clarifying a philosophy of mathematics education before attempting to answer whether theories of mathematics education are constructible.

Keywords: philosophy of mathematics; philosophy of mathematics education; theory of knowledge; theory of mathematics education

AMS Subject Classification: 97

Prepress version of Sriraman, B., & English, L.(2010). Surveying theories and philosophies of mathematics education. In° B. Sriraman & L. D. English (Eds.), Theories of mathematics education: Seeking new frontiers (pp. 7-29). Advances in Mathematics Education, Series: Springer. Pdf (322 KB)

Prediction of river hydraulic conditions via satellite multi-spectral imagery and statistical learners

Brian M. Steele, Univ. of Montana
Department of Mathematical Sciences, University of Montana, Missoula MT, 59812, USA

Mark S. Lorang
Flathead Lake Biological Station, University of Montana, 32125 Bio Station Lane Polson,
MT 59860-9659 MT, USA

Abstract

The hydrologic regimes and ecology of many rivers have been altered through human action. Understanding the natural processes and the extent of alteration is a crucial component of management and restoration of these systems. River and floodplain systems are often highly dynamic, requiring data sets that cover a large spatial extent to fully assess ecological structure and function. However, in situ data collection is extremely challenging in these environments, particularly in localities of high river velocity. This article discusses an approach that exploits high resolution remotely sensed images for expanding data coverage, and thereby enhancing the quantification of important hydraulic parameters, specifically water depth, flow velocity and energetic state through Froude number. The primary elements of this approach are (i) acoustic Doppler profilers coupled with GPS for in situ data collection of water depth and velocity (ii) satellite multispectral imagery, (iii) modern distribution-free statistical learners, and (iv) error analysis. Herein, we provide motivation and background for the study of riverine landscapes via remote sensing, though the article concentrates on the third and fourth elements of the approach. An analysis of a 12 km reach of the Tagliamento River located in Northern Italy provides an example of our approach.

Keywords: random forest, Quickbird

Download Technical Report: Pdf (1413 KB)

MATLAB Software for Supervised Classification in Remote Sensing and Image Processing

J. M. Bardsley, Univ. of Montana
Marylesa Wilde, Univ. of Montana
Chris Gotschalk, U.C. Santa Barbara
M. S. Lorang, Univ. of Montana

Abstract

We present a software package for the supervised classification of images. By supervised, we mean that the user has in hand a representative subset of the pixels in the image of interest. A statistical model is then built from this subset to assign every pixel in the image to a best fit group based on reflectance or spectral similarity. In remote sensing, this approach is typical, and the subset of known pixels is called the ground-truth data.

Ideally, a classifier incorporates both spectral and spatial information. In our software, we implement quadratic discriminant analysis (QDA) for spectral classification and a choice of three spatial methods – mode filtering, probability label relaxation, and Markov random fields – for the incorporation of spatial context after the spectral classification has been computed. Each of these techniques is discussed with some detail in the text.

Finally, we introduce a graphical-user-interface (GUI) that facilitates the creation of ground-truth data subsets – based on individual pixels, lines of pixels, or polygons of pixels – that appear to the user to have spectral similarity. Once a ground-truth subset is created, histogram plots for each band are outputted in order to aid the user in determining whether to accept or reject it. Therefore the GUI makes the software quantitatively robust, broadly applicable and easily usable. We test our classification software on several examples.

Keywords: Markov random fields, probability label relaxation, quadratic discriminant analysis, remote sensing, supervised classification.

Download Technical Report: Pdf (2599 KB)

Regularization parameter selection for penalized-maximum likelihood methods in PET

Johnathan M. Bardsley, and John Goldes

Abstract

Penalized maximum likelihood methods are commonly used in positron emission tomography (PET). Due to the fact that a Poisson data-noise model is typically assumed, standard regularization parameter choice methods, such as the discrepancy principle or generalized cross validation, can not be directly applied. In recent work of the authors, regularization parameter choice methods for penalized negative-log Poisson likelihood problems are introduced, and the application is image deconvolution. In this paper, we extend those methods to the application of PET, introducing a minor modification that seems to improve the performance of the methods. Moreover, we show how these methods can be used to choose the hyper-parameters in a Bayesian hierarchical regularization approach, also of the authors' previous work.

Keywords: positron emission tomography, inverse problems, regularization parameter selection, Bayesian statistical methods, Poisson noise

MSC numbers: 65J22, 65K10, 65F22

Download Technical Report: Pdf (241 KB)

Distinguishing division algebras by finite splitting fields

Daniel Krashen and Kelly McKinnie

Abstract

This paper is concerned with the problem of determining the number of division algebras which share the same collection of finite splitting fields. As a corollary we are able to determine when two central division algebras may be distinguished by their finite splitting fields over certain fields.

Keywords: Division algebras, Brauer group, splitting fields

AMS Subject Classification: 16K50, 14F22

Download Technical Report: Pdf (374 KB) 

Some Virtually Special Hyperbolic 3-Manifold Groups

ERIC CHESEBRO, JASON DEBLOIS, AND HENRY WILTON

Abstract

Let M be a complete hyperbolic 3-manifold of finite volume that admits a decomposition into right-angled ideal polyhedra. We show that M has a deformation retraction that is a virtually special square complex, in the sense of Haglund and Wise. A variety of attractive properties follow: such manifolds are virtually fibered; their fundamental groups are LERF; and their geometrically finite subgroups are virtual retracts. Examples of 3-manifolds admitting such a decomposition include augmented link complements. We classify the low-complexity augmented links and describe an infinite family with complements not commensurable to any 3-dimensional reflection orbifold.

Keywords: hyperbolic 3-manifold, residual properties of groups, virtually fibered.

AMS Subject Classification: 20E26, 57M10

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Singularly perturbed periodic parabolic equations with alternating boundary layer type solutions in spatially two-dimensional domains

Adelaida B. Vasil'eva
Department of Physics,
Moscow State University,
Moscow, 119899 Russia

and

Leonid V. Kalachev
Department of Mathematical Sciences,
University of Montana,
Missoula, MT 59812, USA

Abstract

In this article we continue the analysis of a class of singularly perturbed parabolic equations with alternating boundary layer type solutions. For such problems the degenerate (reduced) equations obtained by setting small parameter equal to zero correspond to algebraic equations that have several isolated roots. As time increases, solutions of these equations periodically go through two comparatively long lasting stages with fast transitions between these stages. During one of these stages the solution outside the boundary layer (i.e., the regular part of the asymptotic solution) is close to one of the roots of the degenerate equation. During the other stage the regular part of the asymptotic solution is close to the other root. Here we discuss some specific features of solutions' behavior for such problems in certain two-dimensional spatial domains.

Keywords: singular perturbations, parabolic equations, boundary function method, two-dimensional spatial domains

AMS Subject Classification:34E10, 35B05, 35B25

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