# Technical Reports

## 2017

### Turán numbers for Berge-hypergraphs and related extremal problems

#### Cory Palmer, Department of Mathematical Sciences, University of Montana Michael Tait, Department of Mathematical Sciences, Carnegie Mellon University Craig Timmons, Department of Mathematics and Statistics, California State University Sacramento Adam Zsolt Wagner, Department of Mathematics, University of Illinois at Urbana-Champaign

Abstract: Let $$F$$ be a graph. We say that a hypergraph $$H$$ is a Berge-$$F$$ if there is a bijection $$f : E(F) \rightarrow E(H )$$ such that $$e \subseteq f(e)$$ for every $$e \in E(F)$$. Note that Berge-$$F$$ actually denotes a class of hypergraphs. The maximum number of edges in an $$n$$-vertex $$r$$-graph with no subhypergraph isomorphic to any Berge-$$F$$ is denoted ex$$_r(n,\textrm{Berge-}F)$$. In this paper we establish new upper and lower bounds on ex$$_r(n,\textrm{Berge-}F)$$ for general graphs $$F$$, and investigate connections between ex$$_r(n,\textrm{Berge-}F)$$ and other recently studied extremal functions for graphs and hypergraphs. One case of specific interest will be when $$F = K_{s,t}$$. Additionally, we prove a counting result for $$r$$-graphs of girth five that complements the asymptotic formula ex$$_3 (n , \textrm{Berge-}\{ C_2 , C_3 , C_4 \} ) = \frac{1}{6} n^{3/2} + o( n^{3/2} )$$ of Lazebnik and Verstraëte [Electron. J. of Combin. 10, (2003)].

### Dimensions of Mathematical Thinking and Learning in ACCEL

#### Bharath Sriraman Dept of Mathematical Sciences University of Montana – Missoula

Abstract: Sternberg summarizes the history of identification of giftedness in the 20th century and presents a case for the shortcomings of measures such as IQ for problem solving skills required in the 21st century. The Active Concerned Citizenship and Ethical Leadership (ACCEL) model is proposed to replace the outdated construct of IQ, particularly for the field of gifted education. In this commentary, the mathematical dimensions of ACCEL are teased out in contrast to its presence in psychometric testing. Further, what is considered as relevant in mathematics for learners today is addressed in relation to the skills outlined in the ACCEL model..

Keywords: IQ; ACCEL model; Interdisciplinary mathematics; Mathematical Modeling; Standardized Testing; Mathematics Curricula

AMS Subject Classification: 97

#### Ron Beghetto, University of Connecticut Bharath Sriraman, University of Montana- Missoula

Abstract: Creative Contradictions in Education is a provocative collection of essays by international experts who tackle difficult questions about creativity in education from a cross-disciplinary perspective. The contributors to this volume examine and provide fresh insights into the tensions and contradictions that researchers and educators face when attempting to understand and apply creativity in educational contexts.    Creativity in education is surrounded by many contradictions. Teachers generally value creativity, but question the role it can and should play in their classroom. Many educators find themselves feeling caught between the push to promote students’ creative thinking skills and the pull to meet external curricular mandates, increased performance monitoring, and various other curricular constraints. This book brings together leading experts who provide fresh, cross-disciplinary insights into how creative contradictions in education might be addressed. Contributors will draw from existing empirical and theoretical work, but push beyond “what currently is” and comment on future possibilities. This includes challenging the orthodoxy of traditional conceptions of creativity in education or making a case for maintaining particular orthodoxies.

Keywords: creativity; creativity in education; paradoxes in creativity; cross-disciplinary perspectives on creativity

AMS Subject Classification: n/a

Pdf of Book cover and TOC (Springer Switzerland). Preprint unavailable due to publisher embargo

### Creativity and Giftedness: Interdisciplinary Perspectives from Mathematics and Beyond

#### Roza Leikin, University of Haifa, Israel Bharath Sriraman, University of Montana- Missoula

Abstract: This volume in Advances in Mathematics Education provides readers with a broad view on the variety of issues related to the educational research and practices in the field of Creativity in Mathematics and Mathematical Giftedness. The book explores (a) the relationship between creativity and giftedness;  (b) empirical work with high ability (or gifted) students in the classroom and its implications for teaching mathematics; (c) interdisciplinary work which views creativity as a complex phenomena that cannot be understood from within the borders of disciplines, i.e., to present research and theorists from disciplines such as neuroscience and complexity theory; and (d) findings from psychology that pertain the creatively gifted students. As a whole, this volume brings together perspectives from mathematics educators, psychologists, neuroscientists, and teachers to present a collection of empirical, theoretical and philosophical works that address the complexity of mathematical creativity and giftedness, its origins, nature, nurture and ways forward. In keeping with the spirit of the series, the anthology substantially builds on previous ZDM volumes on interdisciplinarity (2009), creativity and giftedness (2013).

Keywords: mathematical creativity; mathematical giftedness; high ability; interdisciplinary perspectives; psychology of giftedness; psychology of creativity; philosophy of creativity

AMS Subject Classification: 97

Pdf of Book cover and TOC (Springer Netherlands). Preprint unavailable due to publisher embargo

### Mathematics Education as a matter of cognition

#### Bharath Sriraman, University of Montana- Missoula Kyeonghwa Lee, Seoul National University, South Korea

Abstract: The word cognition is defined in most dictionaries as (1) process of knowing, (2) something that is known, (3) thinking, (4) perception, (5) study of the mind. There are numerous other meanings that can be found in the domains of psychology, biology, philosophy, sociology, linguistics and phenomenology. However for mathematics education we focus primarily on psychology and secondarily on biology, philosophy, and sociology. More specifically we describe and analyze the domain of mathematics education as evolving in its notion of cognition from its roots in psychology and moving onto domains that broaden the notion of “cognition” for mathematics education researchers. This encyclopedia entry has three objectives:

a.   To determine a “starting point” (if any) for research on cognition in mathematics education.

b.   To unfold the development of mathematics education as a field of research based on its interaction with research in cognition.

Keywords: mathematical cognition; mathematics education; psychology of mathematical cognition; history of mathematical cognition; philosophy of cognition

AMS Subject Classification: 97

Preprint of encyclopedia entry in press in M.A. Peters (Ed), Encyclopedia of Educational Philosophy and Theory, Springer Singapore.

### “Integrating” Creativity and Technology through Interpolation

#### Bharath Sriraman, University of Montana - Missoula Daniel Lande, Sentinel High School, Missoula, Montana

Abstract: The digital age of the 21st century is ubiquitous with easy access to information. Students of mathematics find at their fingertips (literally) immense resources such as Wolfram Math and other digital repositories where anything can be looked up in a few clicks. The purpose of this chapter is to convey to the reader that Mathematics as a discipline offers examples of how hand calculations using first principles can result in deep insights that present students with the opportunities of learning and understanding. By first principles we are referring to fundamental definitions of mathematical concepts that enable one to derive results (e.g., definition of a derivative; definition of a Taylor series etc.). We also highlight the value of integrating (pun intended) technology to understand functions that were obtained via mathematical interpolation by the likes of John Wallis (1616-1703), Lord Brouncker (1620- 1684), Johann Lambert (1728-1777) and Edward Wright (1558-1615). The interpolation techniques used by these eminent mathematicians reveals their creativity in deriving representations for functions without the aid of modern technology. Their techniques are contrasted with modern graphing techniques for the same functions.

Keywords: circular functions; pi; John Wallis; quadratures (areas); conformal maps; secant function; integration; history of Calculus; history of infinite series; interpolation; mathematical creativity

AMS Subject Classification: 97

Preprint of chapter in press in V. Freiman, V. & J. Tassell, J., (Eds) Creativity and Technology in Mathematics Education. A 2017 volume in Mathematics Education in the Digital Era, New York, NY: Springer

### Mathematical Pathologies as Pathways into Creativity

#### Bharath Sriraman University of Montana Benjamin Dickman The Hewitt School

Abstract: In this paper, the role of mathematical pathologies as a means of fostering creativity in the classroom is discussed. In particular, it delves into what constitutes a mathematical pathology, examines historical mathematical pathologies as well as pathologies in contemporary classrooms, and indicates how the Lakatosian heuristic can be used to formulate problems that illustrate mathematical pathologies. The paper concludes with remarks on mathematical pathologies from the perspective of creativity studies at large.

Keywords: creativity; convergent thinking; divergent thinking; heuristics; Lakatosian heuristics; pathologies; mathematical pathologies; problem-posing

AMS Subject Classification: 97

Preprint of paper in press, ZDM Mathematics Education, vol.49, no.2, 2017, Springer Nature.