# Technical Reports

## 2015

### Generalising separating families of fixed size

#### Dominik K. Vu Department of Mathematical Sciences University of Memphis Memphis, Tennessee 38152, USAdominik.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:

### Extremal results for Berge-hypergraphs

#### Cory PalmerDepartment of Mathematical SciencesUniversity 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

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

#### L. V. KalachevDepartment of Mathematical SciencesUniversity of Montana, Missoula, MT 59802kalachev@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

### Indigenous Universalities and Peculiarities of Innovation

#### Elizabeth Sumida Huaman, Arizona State UniversityBharath 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)

#### K. SubramaniamTata 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, USAKyeonghwa 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, SwedenBharath 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

### Critical Mathematics Education: Theory, Praxis and Reality

#### Paul Ernest, University of Exeter, UKBharath Sriraman, University of MontanaNuala 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

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

#### Jennifer Job, Oklahoma State UniversityBharath 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.

AMS Subject Classification: 97

### 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