Technical Reports

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2016

Convergence in Creativity Development for Mathematical Capacity

Ai-Girl Tan, Nanyang Technological University Singapore
Bharath Sriraman, University of Montana

Abstract: In this chapter, we highlight the role of convergence in developing creativity and mathematical capacity. We renew our understanding of creativity from the relations of three creativity mechanisms: Convergence in divergence for emergence, and three principles of experience: Continuity, interaction and complementarity. Convergence in the context of creativity development is an incidence of learning for capacity building and knowledge construction. Examples of convergent processes in learning are: setting a plan, having a structure, and possessing coordinated capacity to complete a task. To elaborate, we refer to theories of development and creativity on how people develop their capacity in convergence (e.g., collaboration), through mathematical learning (e.g., with coherence, congruence), and for creativity (e.g., imagination). We make reference to convergent creativity of an eminent mathematician Srinivasa Ramanujan (1887-1920) for a reflection on developing creativity.

Keywords: Convergence; mathematics; collaboration;  convergent thinking; creativity.

AMS Subject Classification: 97

Pre-print of chapter submitted to edited volume: Creativity and giftedness- Interdisciplinary perspectives and beyond; Springer Science and Business, 2016

Sámi teachers' development of culturally based examinations with a focus on self-determination

Anne Birgitte Fyhn, University of Tromsø, Norway
Ylva Jannok Nutti, Kristine Nystad, Sami University College
Bharath Sriraman, University of Montana
Ellen J. Sara Eira, Ole Einar Hætta , Guovdageainnu Nuoraidskuvla- Kautokeino Ungdomsskole, Norway

Abstract: In this article we examine the issues relevant to developing culturally congruent examination tasks for Sami students. The study focusses on teachers’ autonomy and teachers’ self-determination with respect to a framework consisting of the following four categories: i) Deci and Ryan’s (2000) four types of regulation: external, introjected, identified and integrated; ii) the four levels in Smith’s (1999/2006) tide metaphor: survival, recovery, development and self-determination; iii) creativity as the presence of a) everyday creativity (Feldhusen 2006) and b) flexibility (Torrance (1988); and iv) teachers’ beliefs about culturally based teaching and learning mathematics. Student tasks are categorized as belonging either to the exercise paradigm or to landscapes of investigation, and as to whether they are culturally congruent or not.

Keywords: mathematics teacher education; culturally congruent examination items; Sami; Norway examinations

AMS Subject Classification: 97

Pre-print unavailable due to journal policy

Pre-service teacher’s creative mathematical reasoning: Development of a theoretical framework for a research study.

Alv Birkeland, Anne Birgitte Fyhn, University of Tromsø, Norway
Bharath Sriraman, University of Montana

Abstract: To investigate if pre-service teachers’ mathematical reasoning could be creative, the first author conducted a study on his own students work with mathematical tasks based on a loose framework. The aim of this chapter is to illuminate the development of a new theoretical framework for ensuing studies on the creative mathematical reasoning. In order to do so the authors analyze four texts written over a period of four years and use inductive analysis to describe the development of the theoretical framework. We also give some possible explanations of how support from various researchers has influenced the development of the study. The analysis showed that the initial framework was not reliable for reasons elicited in the chapter. We present the new framework towards the end of the chapter.

Keywords: theoretical frameworks; mathematical creativity; creative mathematical reasoning

AMS Subject Classification: 97

Pre-print of Chapter (in Norwegian) to appear in: Festschrift for Marit Johnsen-Høines, Caspar Vorlag, Bergen

Creativity and Giftedness in Mathematics Education: A Pragmatic view

Bharath Sriraman, University of Montana
Per Haavold, University of Tromsø, Norway

Abstract: One purpose of this chapter is to unpack the confusion between the constructs of giftedness (often synonymous with highly able, high potential, high achieving) and creativity (often synonymous with deviance and divergent thinking) and give the reader a clear picture of the two constructs within the context of mathematics education. The second purpose of this chapter is to provide a synthesis of international perspectives in the area of gifted education and suggest implications for mathematics education. The third and last objective is to explore the state of the art within mathematics education, and explore futuristic issues.

Keywords: Mathematical creativity; Giftedness; Mathematical giftedness; High ability

AMS Subject Classification: 97

Revision of chapter (Tech report #2,2015) submitted to J. Cai (Ed). Third Handbook on Mathematics Teaching and Learning, NCTM.

Creative Contradictions in Education: Cross Disciplinary Paradoxes and Perspectives

Ronald Beghetto, University of Connecticut
Bharath Sriraman, University of Montana

Abstract: Creativity is a paradoxical construct. One reason it’s paradoxical is because its definitions tend to be elusive for many people, yet everyone knows creativity when they see it. Numerous other contradictions are present in characterizations of creativity. For instance, most people tend to equate creativity with originality and ‘thinking outside of the box,’ however creativity researchers note that it often requires constraints (Sternberg & Kaufman, 2010). Some people view creativity as being associated with more clear-cut and legendary contributions, yet creativity researchers have long recognized more everyday and subjective forms of creativity (Craft, 2001; Stein, 1953). People also tend to associate creativity with artistic endeavors (Runco & Pagnani, 2011), yet scientific insights and innovation are some of the clearest examples of creative expression. Although there is general consensus amongst creativity researchers on the defining criteria of creativity, minority views persist from the artistic domain, which view any definition as being too constrictive. At present, the field of creativity studies is perhaps best thought of as a transdiscipline. This means that the study of creativity does not belong to any one discipline and that the study of creativity can inform and be informed by multiple disciplines. The transciplinary nature of creativity presents an opportunity to examine the paradoxes facing creativity in education with fresh, multidisciplinary eyes. This is the purpose of the proposed volume. More specifically, the purpose of this volume is to bring together leading cross-disciplinary experts to weigh-in on the creative contradictions in education. Not only will these experts identify and describe key creative contradictions in education, they will provide fresh cross-disciplinary into how these paradoxes might be resolved or better addressed.

Keywords: Creativity; paradoxes; paradoxes in education; contradictions in education

AMS Subject Classification: n/a

Pre-print of TOC and Abstracts of R. Beghetto and B. Sriraman (Eds) "Creative Contradictions in Education: Cross Disciplinary Paradoxes and Perspectives". Springer Science and Business, Switzerland.

Interdisciplinary perspectives to the development of high ability in the 21st century Commentary to Don Ambrose’s “Borrowing Insights from Other Disciplines to Strengthen the Conceptual Foundations for Gifted Education

Bharath Sriraman, University of Montana
Matt Roscoe, University of Montana

Abstract: Ambrose posits that gifted education is mired in the conceptual folds of psychology with dogmatic trends spilling into its application in educational settings. In particular he calls into question issues of socio-economic fairness, epistemological entrenchments within the discipline, and the need to adopt an interdisciplinary approach that can make it relevant for the 21st century. Arguments are proposed for interdisciplinary frameworks to help Gifted Education move beyond its existing theoretical status quo, and to make it relevant for the needs of 21st century societies. Other disciplines such as philosophy, economics, and sociology, which became encumbered in dogmatism were able to develop as a result of being open to conceptual frameworks from other disciplines that helped scholars rise above dogmatic quagmires (Ambrose, Sternberg, Sriraman, 2012). We discuss an interdisciplinary framework for talent development within the macro context of the changing needs of societies. More specifically we give examples of interdisciplinary work arising within mathematics and mathematics education that have freed these disciplines from their foundations in logic and psychology respectively.

Keywords: interdisciplinary education; experimental mathematics; model eliciting activities; high ability; talent development; mathematics education; societal needs

AMS Subject Classification: 97

Pre-print of article to appear in The International Journal for Talent Development and Creativity, vol. 3, no. 2

2015

Generalising separating families of fixed size

Fabrício S. Benevides
Departamento de Matemática|
Universidade Federal do Cear
á
Fortaleza, CE 60455-760, Brazil
fabricio@mat.ufc.br
Research supported by CNPq.

Dániel Gerbner
Hungarian Academy of Sciences
Alfréd Rényi Institute of Mathematics
P.O.B. 127, Budapest H-1364, Hungary
gerbner.daniel@renyi.mta.hu
Research supported by Hungarian National Science Fund (OTKA), grant PD 109537.

Cory Palmer
Department of Mathematical Sciences
University of Montana, Missoula, MT 59812, USA.
cory.palmer@umontana.edu
Research supported by Hungarian National Science Fund (OTKA), grant NK 78439.

Dominik K. Vu
Department of Mathematical Sciences
University of Memphis
Memphis, Tennessee 38152, USA
dominik.vu@memphis.edu
Research supported in part by the National Science Foundation, grants  DMS-0906634 and CNS-0721983, and by the Heilbronn Fund.

Abstract: We examine the following version of a classic combinatorial search problem introduced by Rényi:  Given a finite set \(X\) of \(n\) elements we want to identify an unknown subset \(Y \subset X\) of exactly \(d\) elements by testing, by as few as possible subsets \(A\) of \(X\), whether \(A\) contains an element of \(Y\) or not. We are primarily concerned with the model where the family of test sets is specified in advance (non-adaptive) and each test set is of size at most a given \(k\). Our main results are asymptotically sharp bounds on the minimum number of tests necessary for fixed \(d\) and \(k\) and for \(n\) tending to infinity.

Keywords:

AMS Subject Classification:

Download Technical Report: PDF(261KB)

Extremal results for Berge-hypergraphs


Dániel Gerbner

Hungarian Academy of Sciences
Alfréd Rényi Institute of Mathematics
P.O.B. 127, Budapest H-1364, Hungary.
Research supported by OTKA grant PD-109537.

Cory Palmer
Department of Mathematical Sciences
University of Montana, Missoula, MT 59812, USA.
Research supported by University of Montana UGP grant 325341.

Abstract: Let \(G\) be a graph and \(\mathcal{H}\) be a hypergraph both on the same vertex set. We say that a hypergraph \(\mathcal{H}\) is a Berge-\(G\) if there is a bijection \(f : E(G) \rightarrow  E(\mathcal{H})\) such that for \(e \in E(G)\) we have \(e \subset f(e)\). This generalizes the established definitions of "Berge path"' and "Berge cycle'' to general graphs. For a fixed graph \(G\) we examine the maximum possible size (i.e. the sum of the cardinality of each edge) of a hypergraph with no Berge-\(G\) as a subhypergraph.  In the present paper we prove general bounds for this maximum when \(G\) is an arbitrary graph. We also consider the specific case when \(G\) is a complete bipartite graph and prove an analogue of the Kővári-Sós-Turán theorem.

AMS Subject Classification: 05C65 and 05C35

Download Technical Report: PDF (253KB)

The Maximum of Δu + f (u) = 0 on an Isosceles Triangle

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: We use the moving planes method to prove that if u is a positive solution to the equation Δu + f(u) = 0, on an isosceles triangle T in R2, with u = 0 on ∂T and f Lipschitz continuous and restoring, then u has a unique maximum value on the axis of symmetry of T. We conjecture that the location of the maximum is independent of f, and extend the result to a set in R3.

Keywords: Maximum Principle, Moving Planes, Symmetric-Convex Domains.

AMS Subject Classification: 35B50

Download Technical Report: PDF (72KB)

Indigenous Universalities and Peculiarities of Innovation

Elizabeth Sumida Huaman, Arizona State University
Bharath Sriraman, University of Montana

Abstract: Rooted in diverse cultures and in distinct regions of the world, Indigenous communities and people have for centuries negotiated unique relationships with their surroundings, including new forms of conquest that continue to emerge today. The intent of this book is to showcase instances of Indigenous innovation in sustainability, ecological stewardship, and oral knowledge that demonstrate cosmological conceptions of time and place and critical practices that can be shared in order to shape new conceptualizations of relationships within the world that are transformative and just. The book is based on empirical and/or other research that has been collaboratively conducted with Indigenous communities, narratives and counter-narratives, qualitative fieldwork reflections, and theoretical perspectives. Authors should center their work around major definitions and ideas of innovation within Indigenous contexts that are respectful of and exemplify local knowledge. At the same time, authors are asked to engage in conscientious discussion regarding dominant discourse that posits Western modes of progress that impact those Indigenous communities and any tensions that result between communities and province, state, and/or country.

Keywords: Indigeneous Education; Traditional Knowledge; Innovation

AMS Subject Classification: n/a

TOC and Abstracts of vol.2 in Advances in Innovation Education, Sense Publishers, Rotterdam

The First Sourcebook on Asian Research in Mathematics Education: China, Korea, Singapore, Japan, Malaysia and India (1782 pages)

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: The First Sourcebook on Asian Research in Mathematics Education: China, Korea, Singapore, Japan, Malaysia and India provides the first synthesized treatment of mathematics education that has both developed and is now prominently emerging in the Asian and South Asian world. The book is organized in sections coordinated by leaders in mathematics education in these countries and editorial teams for each country affiliated with them. The purpose of unique sourcebook is to both consolidate and survey the established body of research in these countries with findings that have influenced ongoing research agendas and informed practices in Europe, North America (and other countries) in addition to serving as a platform to showcase existing research that has shaped teacher education, curricula and policy in these Asian countries. The book will serve as a standard reference for mathematics education researchers, policy makers, practitioners and students both in and outside Asia, and complement the Nordic and NCTM perspectives.

Keywords: Asian Mathematics Education; Mathematics Education; Mathematics Education Research; China; Korea; Singapore;Japan;Malaysia; India

AMS Subject Classification: 97

Preprint of TOC available here

The Hobbesian trap in contemporary India and Korea: Implications for education in the 21st century

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

Abstract: In this commentary to Ambrose’s focus chapter on “21st Century Contextual Influences on the Life Trajectories of the Gifted, Talented and Creative”, we examine the socio-economic, cultural and ideological constraints to development in education and society in India and Korea, with a particular focus on issues that fall through the cracks and segments of society that get left behind. In spite of the phenomenal economic growth in these countries and advances at the frontiers of technology (e.g the success of the 2014 India's Mars mission; the information technology sector in Korea), educational opportunities are still mired within a socio-economic and cultural context that hinders opportunities for young people. Ideology and social Darwinism in the 21st century play a role in both countries, to a lesser extent in Korea where a homogeneous society and a smaller albeit dense population has been able to reap some of the benefits of socio economic and technological advances. In this commentary to Ambrose’s chapter, the Darwinian nature and constraints of educational opportunities in these countries is examined framed within the macro-context of historical forces that shaped the structure of society in these countries, particularly cultural ideology that creates a Hobbesian trap.

Keywords: Globalization; Hobbes' trap; India; Korea; Socio-economic class struggle; Ideology in India; Confucianism

AMS Subject Classification: n/a

Pre-print of Chapter to appear in Ambrose, D., and Sternberg, B. (Eds), Giftedness and Talent in the 21st Century: Adapting to the Turbulence of Globalization

Refractions of Mathematics Education

Christer Bergsten, Linkoepings University, Sweden
Bharath Sriraman, University of Montana

Abstract: An ancient Sanskrit adage states that Knowledge is something that grows when shared, but shrinks when hoarded. Academics engaged in the generation of new Knowledge are blessed with both the time and the freedom to engage in pursuits that allow for intellectual pleasure. As a phenomenon of the Zeitgeist many have succumbed to the increased corporatization of academic work, engaging in activities for monetary and self advancement purposes. Are there any real intellectuals left in academia, alà Adorno, Bourdieu, Chomsky, Foucault, among others? This Festschrift is dedicated to academics that don't bother with self promotion or aggrandizement of themselves or their ideas in simplistic terms.

Keywords: Mathematics Education; Essays on Mathematics Education; Sociology of Education

AMS Subject Classification: 97

Pre-print of Book (TOC) in Cognition, Equity & Society: International Perspectives, Critical Mathematics Education: Theory, Praxis and Reality on Infromation Age Publiching.

Critical Mathematics Education: Theory, Praxis and Reality

Paul Ernest, University of Exeter, UK
Bharath Sriraman, University of Montana
Nuala Ernest

Abstract: Mathematics is traditionally seen as the most neutral of disciplines, the furthest removed from the arguments and controversy of politics and social life. However, critical mathematics challenges these assumptions of neutrality and actively attacks the idea that mathematics is pure, objective, and value neutral. It argues that history, society and politics have shaped mathematics – not only through its applications and uses, but through moulding its concepts, methods, and even mathematical truth and proof, the very means of establishing truth. Critical mathematics education also attacks the neutrality of the teaching and learning of mathematics, showing how these are value laden activities indissolubly linked to social and political life. Instead it argues that the values of openness, dialogicality, criticality towards received opinion, empowerment of the learner and social/political engagement and citizenship are necessary dimensions of the teaching and learning of mathematics, if it is to contribute towards democracy and social justice.

The books draws together critical theoretic contributions on mathematics and mathematics education from leading researchers in the field. It explores many facets of the practical implications of these critical views for the varying stages and phases of mathematics education around the globe.

Recurring themes in the book include:

  • The natures of mathematics and critical mathematics education, issues of epistemology and ethics;
  • Ideology, the hegemony of mathematics, ethnomathematics, and real-life education;
  • Capitalism, globalization, politics, social class, habitus, citizenship and equity.

The book demonstrates the links between these themes and the discipline of mathematics and its critical teaching and learning. The outcome is a groundbreaking collection unified by a shared concern with critical perspectives of mathematics and education, and of the ways they impact on practice and the realities of society and schooling.

Keywords: Critical Mathematics Education; Real life education; Globalization; Social Class; Equity

AMS Subject Classification: 97

Pre-print of Book (TOC)  in Cognition, Equity & Society: International Perspectives, Critical Mathematics Education: Theory, Praxis and Reality on Infromation Age Publiching.

The Concept of Teacher-Student/Student-Teacher in Higher Education Trends

Jennifer Job, Oklahoma State University
Bharath Sriraman, University of Montana

Abstract: In this article we examine the use and misuse of the term “Freirean” with respect to student-centered classrooms, flipped classrooms and problem-based learning. In doing so, we unpack teaching methods which are contrary to dialogism and yet subsumed and reported as Freirean.

Keywords: Paolo Freire; constructivism; problem based learning; flipped classrooms; Khan academy

AMS Subject Classification: 97

Download Technical Report: Preprint of paper submitted to Interchange: A Quarterly Review of Education

Creativity and Giftedness in Mathematics Education: A Pragmatic view

Bharath Sriraman, University of Montana
Per Haavold, University of Tromsø, Norway

Abstract: One purpose of this chapter is to unpack the confusion between the constructs of giftedness (often synonymous with highly able, high potential, high achieving) and creativity (often synonymous with deviance and divergent thinking) and give the reader a clear picture of the two constructs within the context of mathematics education. The second purpose of this chapter is to provide a synthesis of international perspectives in the area of gifted education and suggest implications for mathematics education. The third and last objective is to explore the state of the art within mathematics education, and explore futuristic issues.

Keywords: Creativity; Mathematical creativity; Giftedness; Mathematical giftedness; High ability

AMS Subject Classification: 97

Download Technical Report: Preprint of Chapter submitted to  J. Cai (Ed). Third Handbook on Mathematics Teaching and Learning, NCTM.

The Teaching and Learning of Probabilistic Thinking: Heuristic, Informal and Fallacious Reasoning

Egan Chernoff, The University of Saskatchewan
Bharath Sriraman, University of Montana

Abstract: In this chapter we examine the use of heuristic principles in mathematics by paying particular attention to the counterintuitive nature of problems in probability and ways in which heuristics can be used to overcome fallacious reasoning.

Keywords: Heuristics; Probabilistic heuristics; Fallacious reasoning; Counter-intuitive probability

AMS Subject Classification: 97

Download Technical Report: Pre-print of chapter submitted to Routledge Handbook on Research on Teaching Thinking, Routledge, Taylor & Francis