# Technical Reports

## 2012 Technical Reports

### #12/2012: W. R. Derrick, L. V. Kalachev, J. A. Cima

** Heat Waves in a Supercritical Fluid**

**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

**J. A. Cima**

Department of Mathematics

University of North Carolina, Chapel Hill, NC 27599

cima@email.unc.edu

**Abstract:** Experiments reveal that thermal energy transfers as an acoustic wave near the critical point of a classical fluid under microgravity. In this paper we use asymptotic methods and Fourier analysis to provide a complete mathematical proof of this piston effect. Our two-step method provides a “fast-time” solution describing the piston effect in detail, and a “slow-time” solution leading to solutions that agree in nature with those obtained by numerical methods applied to the original model system on “slow-time” scale.

**Download Technical Report:** Pdf (126 KB)

### #11/2012: Eric Chesebro

**Formulas for Character Varieties of 2-Bridge Knots**

**Abstract:** We introduce the first closed-form formula for the defining polynomials for the character varieties of an infinite collection of hyperbolic knot manifolds. In particular, these knots belong to a subfamily of 2-bridge knots.

**Keywords:** 3-manifold, knot, character variety

**AMS Subject Classification:** 57M27

**Download Technical Report:** Pdf (250 KB)

### #10/2012: Kyeonghwa Lee, Bharath Sriraman

**An Eastern Learning Paradox: Paradoxes in Two Korean Mathematics Teachers’ Pedagogy of Silence in the Classroom**

**Kyeonghwa Lee**

Seoul National University

**Bharath Sriraman**

University of Montana

**Abstract:**Eastern philosophies of education such as Confucianism and Taosim advocate the use of silence in the teacher-pupil tradition of pedagogy. We investigate contemporary classrooms in Korea, and study whether teachers in Korea today incorporate this method implicitly or explicitly in their classrooms. Empirical data in the form of video-taped classroom lessons and the analysis of the videos reveal the learning paradoxes that arise from a rote adherence to constructivism by teachers that are trained within a larger Confucian society.

**Keywords:** Confucianism; Constructivism; Culture; Learning paradox; Mathematics teaching; Silence; Korea

**AMS Subject Classification:** 97

**Preprint of paper submitted to Interchange:** A Quarterly Review of Education, Springer. Pdf (188 KB)

### #9/2012: George McRae, Demitri Plessas, Liam Rafferty

**On the Concrete Categories of Graphs**

**George McRae**

University of Montana

**Demitri Plessas**

University of Montana

**Liam Rafferty**

University of Montana

**Abstract:**In the standard Category of Graphs, the graphs allow only one edge to be incident to any two vertices, not necessarily distinct, and the graph morphisms must map edges to edges and vertices to vertices while preserving incidence. We refer to these graph morphisms as Strict Morphisms. We relax the condition on the graphs allowing any number of edges to be incident to any two vertices, as well as relaxing the condition on graph morphisms by allowing edges to be mapped to vertices, provided that incidence is still preserved. We call this broader graph category The Category of Conceptual Graphs, and define four other graph categories created by combinations of restrictions of the graph morphisms as well as restrictions on the allowed graphs.

We investigate which Lawvere axioms for the category of Sets and Functions apply to each of these Categories of Graphs, as well as the other categorial constructions of free objects, projective objects, generators, and their categorial duals.

**Keywords:** graph morphisms, graph homomorphisms, concrete categories, cartesian closed category, topos, Categories of Graphs

**AMS Subject Classification:** Primary: 18B99, Secondary: 05C25, 18A40, 18D15

**Download Technical Report:** Pdf (535 KB)

### #8/2012: P. Mark Kayll, Dave Perkins

**A Chip-Firing Variation and A New Proof of Cayley's Formula**

**P. Mark Kayll**

Department of Mathematical Sciences

University of Montana

Missoula MT 59812-0864, USA

mark.kayll@umontana.edu

**Dave Perkins**

Department of Mathematics

Luzerne County Community College

Nanticoke PA 18634, USA

dperkins@luzerne.edu

**Abstract:** We introduce a variation of chip-firing games on connected graphs. These 'burn-off' games incorporate the loss of energy that may occur in the physical processes that classical chip-firing games have been used to model. For a graph G = (V,E), a configuration of 'chips' on its nodes is a mapping C: V → ℕ. We study the configurations that can arise in the course of iterating a burn-off game. After characterizing the 'relaxed legal' configurations for general graphs, we enumerate the 'legal' ones for complete graphs Kn. The number of relaxed legal configurations on Kn coincides with the number tn+1 of spanning trees of Kn+1. Since our algorithmic, bijective proof of this fact does not invoke Cayley's Formula for tn, our main results yield secondarily a new proof of this formula.

**Keywords:** chip-firing, burn-off games, relaxed legal configurations, Cayley's Formula

**AMS Subject Classification:** Primary 05C57; Secondary 90D42, 05C30, 05A19, 05C85, 68R10

**Download Technical Report:** Pdf (309 KB)

### #7/2012: Kyeonghwa Lee, Bharath Sriraman

**Gifted Girls and Non-mathematical Aspirations: A Longitudinal Case Study of Two Gifted Korean Girls**

**Kyeonghwa Lee**

Seoul National University

**Bharath Sriraman**

University of Montana

**Abstract:** In this longitudinal study of two gifted Korean girls, experiences with early admittance into a gifted program are charted alongside their family and societal experiences which ultimately influenced their career choices in non-mathematical fields. The 8-year long qualitative study involved extensive interviews with the two gifted girls and their parents to determine factors that led to their choice of a non mathematical area of specialization in spite of early identification and support of their mathematical talent. Using tenets of qualitative inquiry to code the longitudinal data, we identified three main factors that contributed to these career choices, which are presented in the form of narratives. One of the startling findings of this study, contrary to the literature in gifted education research, is that the two girls' early experiences with gifted education kept them from choosing careers related to mathematics. The article also narrates the enculturation of mathematically girls in Korea which leads to non-mathematical career aspirations.

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

**AMS Subject Classification:** 97

SAGE Pre-print of second revision of Technical Report Pdf (338 KB)

### #6/2012: Xianwei Y. Van Harpen, Bharath Sriraman

**Creativity and Mathematical Problem Posing: An Analysis of High School Students' Mathematical Problem Posing in China and the United States**

**Xianwei Y. Van Harpen**

Illinois State University

**Bharath Sriraman**

University of Montana

**Abstract:** In the literature, problem posing abilities are reported to be an important aspect/indicator of creativity in mathematics. The importance of problem posing activities in mathematics is emphasized in educational documents in many countries, including the United States and China. This study was aimed at exploring high school students' creativity in mathematics by analyzing their abilities in posing problems in geometric scenarios. The participants in this study were from one location in the United States and two locations in China. All participants were enrolled in advanced mathematical courses in the local high school. Differences in the problems posed by the three groups are discussed in terms of quality as well as quantity. The analysis of the data indicated that even mathematically advanced high school students had trouble posing good quality and/or novel mathematical problems. We discuss our findings in terms of the culture and curricula of the respective school systems and suggest implications for directions in problem posing research within mathematics education.

**Keywords:**advanced high school students; cross cultural thinking; geometry; mathematical creativity; novelty; problem posing; problem solving; U.S and Chinese students; rural and urban Chinese students

**AMS Subject Classification:**97

Revised version Pdf (7900 KB)

### #5/2012: Claire Berg, A.B, Fuglestad, S. Goodchild, B. Sriraman

**Extrapolation and Expansion: Characteristics of Change Occurring in Mathematics Teaching Development Projects.**

**Claire V. Berg, Anne Berit Fuglestad and Simon Goodchild**

University of Agder, Norway

**Bharath Sriraman**

University of Montana

**Abstract:** This paper commences by critically examining how mathematics teaching development projects based on the creation of communities of inquiry (CoI) are theorised from communities of practice theory (CPT) and cultural historical activity theory (CHAT). Two types of change, which can be developed from these sociocultural theories are articulated. Change as ‘extrapolation’ derived from CPT, and change as ‘expansion’ developed within CHAT. The differences between these types of change and their underlying principles are examined by contrasting conceptualisations of CPT and CHAT; attention focuses on mediation, goals, and agency. It is argued that the introduction of inquiry to CPT transforms the practice and entails a paradigm shift, with CoI as category within the critical paradigm. Expansion and extrapolation are illustrated using narrative evidence from a longitudinal case study of one school team that worked within a series of mathematics teaching developmental research projects over a period of six years. The paper emerges from the analysis and synthesis of a large volume of qualitative data accrued in teaching development projects.

**Keywords:** Communities of practice; Communities of inquiry; Cultural historical activity theory; extrapolation; expansion; Large scale qualitative data analysis

**AMS Subject Classification:** 97

Revised version (202 KB) of #2/2011

### #4/2012: Gabriel Johnson, Bharath Sriraman, Rachel Saltzstein

**Where are the Plans?- A socio-critical and architectural survey of early Egyptian Mathematics**

**Gabriel Johnson & Bharath Sriraman**

The University of Montana

**Rachel Saltzstein**

Volcano Vista High School,

Albuquerque, New Mexico

**Abstract:** The majority of the mathematics taught in secondary schools was invented in the ancient world, and this fact opens the curriculum up to organic opportunities to teach mathematics from a socio critical historical perspective. Teachers and education theorists have observed a fascination on the part of students of all ages with the culture of ancient Egypt, which makes a study of the mathematics of ancient Egypt a particularly attractive topic for the public school mathematics classroom. How is one to approach this topic given the controversy surrounding the Egyptians and their place in the history of mathematics? This controversy is discussed by Joseph (1991) but most explicitly examined in Bernal’s (1987) still controversial, multivolume work Black Athena, in which he defends his thesis that there are two competing models of the genesis of Ancient Greek language and mathematics, one that is Eurocentric and the other, Bernal’s (1987) “revised ancient model,” that is essentially Egyptian and Semitic. Bernal (1987) presents evidence that the Eurocentric model emerged during the late 17th through 19th centuries and was motivated by skin color racism that developed during the era of African slavery and European and American imperialism. In this chapter we examine early Egyptian mathematics from a socio-critical as well as an mathematical viewpoint by calculating quantities and relating it to feasibility construction problems in engineering and architecture in modern times.

**Keywords:** Black Athena; Egyptian mathematics; History of mathematics; Architecture and Engineering problems

AMS Subject Classification: 01, 97

Full preprint unavailable due to publisher embargo. Excerpt from Chapter to appear in: B. Sriraman (Ed). (2012) Crossroads in the History of Mathematics and Mathematics Education. Information Age Publishing, Charlotte, NC. Pdf (467 KB)

### #3/2012: Kajsa BrĂ¥ting, Nicholas Kallem, Bharath Sriraman

**The Didactical Nature of Some Lesser Known Historical Examples in Mathematics**

**Kajsa Bråting**

Uppsala University, Sweden

**Nicholas Kallem**

The University of Montana

**Bharath Sriraman**

The University of Montana

**Abstract:**In the field of history of mathematics the work of famous mathematicians, such as Cauchy, Newton and Leibniz, have been carefully studied which certainly have provided valuable knowledge regarding the development of mathematics. The results are well documented in the literature. However, it may also be important to take into account the efforts of the less known (or sometimes unknown) mathematicians, contemporary to the famous mathematicians. For instance, by studying the work of the less known mathematicians we get more information of the struggle behind famous mathematical results, which often include valuable didactical knowledge. Moreover, the general view of certain mathematical concepts at particular time periods can be better understood on the basis of not only the work of the leading mathematicians, but also on the basis of contemporary mathematicians working in the shadow of the famous mathematicians. Furthermore, in the work of the less known mathematicians, one can sometimes find alternative solutions to a difficult mathematical problem, even before it was posed as a problem. However, these alternative solutions may not be as straightforward as the solutions documented in the modern textbooks. But they can provide valuable knowledge regarding mathematical thinking as well as the development in mathematics. In this paper we try to highlight the didactical nature of some historical mathematical examples. We use examples from some less-known mathematicians in order to show how didactics of mathematics is naturally interweaved in the methods devised in the history of mathematics to solve difficult problems.

**Keywords:**Didactics of mathematics; Björling; Hobbes and Wallis; History of Calculus; Basel problem; Indian mathematics; Islamic mathematics

**AMS Subject Classification:**01, 97

Full Preprint unavailable due to publisher embargo. Excerpt from chapter to appear in: B. Sriraman (Ed). (2012) Crossroads in the History of Mathematics and Mathematics Education. Information Age Publishing, Charlotte, NC. Pdf (463 KB)

### #2/2012: Bharath Sriraman

**Crossroads in the History of Mathematics and Mathematics Education**

**Bharath Sriraman**

Dept. of Mathematical Sciences

The University of Montana

**Abstract:** The interaction of the history of mathematics and mathematics education has long been construed as an esoteric area of inquiry. Much of the research done in this realm has been under the auspices of the history and pedagogy of mathematics group. However there is little systematization or consolidation of the existing literature aimed at undergraduate mathematics education, particularly in the teaching and learning of the history of mathematics and other undergraduate topics. In this monograph, the chapters cover topics such as the development of Calculus through the actuarial sciences and map making, logarithms, the people and practices behind real world mathematics, and fruitful ways in which the history of mathematics informs mathematics education. The book is meant to serve as a source of enrichment for undergraduate mathematics majors and for mathematics education courses aimed at teachers.

**Keywords:** History of Mathematics; Mathematics Education; Undergraduate Mathematics Education; History of Analysis; History of Geometry

**AMS Subject Classification:** 01, 97

Preprint of TOC in Book to appear in 2012, Information Age Publishing Pdf (485 KB)

### #1/2012: Bharath Sriraman, Elena Nardi

**Theories in Mathematics Education: Some Developments and Ways Forward**

**Bharath Sriraman**

The University of Montana

**Elena Nardi**

University of East Anglia

**Abstract:** In this survey of the state of the art, roots of mathematics education are traced from Piaget onto the current work on theorizing which utilize sociological and commognitive frameworks. Attention is given to the critiques of “Theories of Mathematics Education” (Sriraman & English, 2010), and productive discussions from the reviews are unpacked. The notions of “operational” versus “functional”, and “models” versus “theories” are also tackled by focusing on conceptual frameworks that harmonize the terms as opposed to exemplify their polarities.

**Keywords:** Models of Mathematics Education; Theories; Stage theories; Discursive frameworks; Philosophy of Knowledge; Structuralism; Conceptual Frameworks in Mathematics Education; Local theories versus Global theories; Operational definitions

**AMS Subject Classification:** 97