# Technical Reports

## 2016

### On the number of cycles in a graph with restricted cycle lengths

#### Dániel Gerbner, Hungarian Academy of Sciences, Alfréd Rényi Institute of Mathematics Bálazs Keszegh, Hungarian Academy of Sciences, Alfréd Rényi Institute of Mathematics Cory Palmer, University of Montana Balázs Patkós, Hungarian Academy of Sciences, Alfréd Rényi Institute of Mathematics

Abstract: Let $$L$$ be a set of positive integers. We call a (directed) graph $$G$$ an $$L\mathit{-cycle graph}$$ if all cycle lengths in $$G$$ belong to $$L$$. Let $$c(L,n)$$ be the maximum number of cycles possible in an $$n$$-vertex $$L$$-cycle graph (we use $$\vec{c}(L,n)$$ for the number of cycles indirected graphs). In the undirected case we show that for any fixed set $$L$$, we have $$c(L,n)=\Theta_L(n^{\lfloor k/\ell \rfloor})$$ where $$k$$ is the largest element of $$L$$ and $$2\ell$$ is the smallest even element of $$L$$ (if $$L$$ contains only odd elements, then $$c(L,n)=\Theta_L(n)$$ holds.) We also give a characterization of $$L$$-cycle graphs when $$L$$ is a single element.

In the directed case we prove that for any fixed set $$L$$ we have $$\vec{c}(L,n)=(1+o(1))(\frac{n-1}{k-1})^{k-1}$$, where $$k$$ is the largest element of $$L$$. We determine the exact value of $$\vec{c}(\{k\},n)$$ for every $$k$$ and characterize all graphs attaining this maximum.

### Rainbow Turán problems for paths and forests of stars

#### Daniel Johnston, University of Montana Cory Palmer, University of Montana Amites Sarkarz, Western Washington University

Abstract: For a fixed graph $$F$$, we would like to determine the maximum number of edges in a properly edge-colored graph on $$n$$ vertices which does not contain a rainbow copy of $$F$$, that is, a copy of $$F$$ all of whose edges receive a different color. This maximum, denoted by ex$$^*(n,F)$$ is the rainbow Turán number of $$F$$, and its systematic study was initiated by Keevash, Mubayi, Sudakov and Verstraëte in 2007. We determine ex$$^*(n,F)$$ exactly when $$F$$ is a forest of stars, and give bounds on ex$$^*(n,F)$$ when $$F$$ is a path with $$k$$ edges.

### 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

### 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

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

#### 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.

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