Technical Reports

2011 Technical Reports

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
The 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

Download Technical Report: Pdf (620 KB)

Revised version of #2/2011

Closed Surfaces and Character Varieties

Eric Chesebro

Abstract: We give some algebraic characterizations for when the character variety techniques of Culler and Shalen can be used to construct a closed essential surface in a knot manifold.

Keywords: 3-manifold, knot, character variety, essential surface

AMS Subject Classification: 57M27

Download Technical Report: Pdf (456 KB) 

Algebraic Invariants, Mutation, and Commensursbility of Link Complements

Eric Chesebro and Jason Deblois

Abstract: We construct a family of hyperbolic link complements, all with trace field Q(i,Sqrt(2)), by gluing tangles along totally geodesic four-punctured spheres, and investigate the commensurability relation among its members. Those with different volume are incommensurable, distinguished by their scissors congruence classes. Mutation produces arbitrarily large finite subfamilies of non-isometric manifolds with the same volume and scissors commensurability class. Depending on the choice of mutation, these manifolds may be commensurable or incommensurable, distinguished in the latter case by cusp parameters. Examples with integral and nonintegral traces are also produced.

Keywords: hyperbolic 3-manifold, commensurable, trace field, mutant

AMS Subject Classification: 57M50

Download Technical Report: Pdf (620 KB)

The Interfaces of Innovation in Mathematics and the Arts

Bharath Sriraman
The University of Montana

Kristina Juter
Kristianstad University, Sweden

Abstract: The chapter outlines human innovation in architecture and art with an emphasis on mathematical creativity, e.g. the work of Buckminster Fuller who was inspired by popular psychology and human consciousness in his creations. Architectural creation in society is tightly connected to geometry, topology and other parts of mathematics. Buildings and art are results of human minds linking abstract mathematical representations and concrete physical structures. For such links to occur, inventors need to be able to work in interdisciplinary settings. We present some findings in mathematics education in the light of fostering creative innovators in mathematics related to the arts.

Keywords: Buckminster Fuller; Innovation education; Geometry; Creativity; Architectiure; Interdisciplinarity

AMS Subject Classification: 97

Chapter to appear in L.V. Shavinina (Ed), The International handbook of Innovation Education, Taylor and Francis

Pre-print unavailable due to publisher embargo

Polarities in (Nordic) Mathematics Education: Scaling the field

Bharath Sriraman
Department of Mathematical Sciences
The University of Montana

Abstract: Mathematics education research in the Nordic world is compared to that occurring elsewhere by using the metaphor of scaling. Mathematics education as a research entity is scaled with respect to the knowledge industry, namely economics, its place within professional societies, and disciplines such as mathematics, social sciences, psychology and the humanities. Polarities, consistencies and contradictions are discussed in the role of an (external) observer and a collaborator with researchers in the Nordic mathematics education milieu. Some implications for the future are elaborated.

Keywords: Mathematics education; Nordic mathematics education; International comparisions; Bibliometrics; Scaling

AMS Subject Classification: 97

Download Technical Report: Paper to appear in Proceedings of NORMA11-The 6th Nordic Congress of Mathematics Education,Reykavik, Iceland (176 KB pdf)

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
The 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 of #11/2011

Download Technical Report: Pdf (229 KB)

A Framework for Quality Assurance in Globalization of Higher Education

Jennifer Job and Dwight Irvin
The University of North Carolina at Chapel Hill

Bharath Sriraman
The University of Montana

Abstract: In the paper we theorize a framework for discussing quality assurance of globalized models of education used internationally. The philosophical assumption that homogeneity of perspectives achieves objectivity in practice is argued against using the examples of (1) Brain drain, and (2) Profit over quality. We present a coherent real world scenario for the abstract ideas presented in Sriraman & Adrian (2008) with a meta-review support of the literature.

Keywords: borderology; quality assurance; higher education; brain drain; meta-review of globalization of “higher education”; philosophy of education

AMS Subject Classification: 97

Preprint of paper submitted to Interchange: A Quarterly Review of Education. Springer Science and Business Pdf (135 KB)

An MCMC Method for Uncertainty Quantification in Nonnegativity Constrained Inverse Problems

Johnathan M. Bardsley
Department of Mathematical Sciences
University of Montana
Missoula, Montana 59812
USA
E-mail: bardsleyj@mso.umt.edu

Colin Fox
Department of Physics
University of Otago
Dunedin 9054
New Zealand
E-mail: fox@physics.otago.ac.nz

Abstract: The development of computational algorithms for solving inverse problems is, and has been, a primary focus of the inverse problems community. Less studied, but of increased interest, is uncertainty quantification for solutions of inverse problems obtained using computational methods. In this paper, we present a method of uncertainty quantification for linear inverse problems with nonnegativity constraints. Our approach utilizes a Bayesian statistical framework, and we present a simple Markov chain Monte Carlo (MCMC) method for sampling from a particular posterior distribution. From the posterior samples, estimation and uncertainty quantification for both the unknown (image in our case) and regularization parameter are performed. The primary challenge of the approach is that for each sample a large-scale nonnegativity constrained quadratic minimization problem must be solved. We perform numerical tests on both one- and two-dimensional image deconvolution problems, as well as on a computed tomography test case. Our results show that our nonnegativity constrained sampler is effective and computationally feasible.

Keywords: inverse problems, image reconstruction, nonnegativity constrained optimization, Bayesian inference, Markov chain Monte Carlo, uncertainty quantification.

MSC numbers: 15A29, 62F15, 65F22, 94A08

Download Technical Report: Pdf (300 KB) 

High School Mathematics Curricula, University Mathematics Placement Recommendations, and Student University Mathematics Performance

Ke Wu Norman
University of Montana

Amanuel Medhanie, Michael Harwell, Edward Anderson, Thomas Post
University of Minnesota

Abstract: Recent “math wars” have drawn attention to how well various high school mathematics curricula prepare students for college level mathematics. The purpose of this study was to investigate the relationship between the high school mathematics curriculum and students’ post-secondary mathematics placement recommendation. Specifically how students responded to the mathematics placement recommendations and their performance in the first college mathematics class.

The results showed no relationship between student participation in a particular high school mathematics curriculum and mathematics placement recommendation, nor between student high school mathematics curriculum and student’s response to a university mathematics placement recommendation. However, students who took a more (less) difficult class than what was recommended achieved significantly lower (higher) grades than those who followed the recommendation. The findings have implications for high school mathematics curriculum selection, post-secondary student placement, and for future research in this area.

Keywords: : High school mathematics curriculum, mathematics placement, post-secondary mathematics achievement

AMS Subject Classification: 97

Paper was accepted and will appear in PRIMUS

A Call for Integrating Engineering Through Cooperative Learning in the Mathematics and Science Teacher Education Program

Ke Wu Norman
University of Montana

Anne L. Kern
University of Idaho- Coeur d'Alene

Tamara J. Moore
University of Montana

Abstract: The challenge for many teacher education programs is to provide balanced coursework that help teachers be successful. The purpose of this paper is to discuss offering courses that provide cooperative learning experiences in interdisciplinary (STEM-science, technology, engineering, and mathematics) contexts, for mathematics and science teachers. The following paragraphs cover four main topics: (1) Shulman’s theoretical framework of teacher knowledge, (2) the interconnections among engineering, mathematics, and science disciplines, (3) collaboration among faculties in engineering, mathematics, and science education at universities, and (4) improving the communication and collaboration between mathematics and science teachers using cooperative learning strategies.

Keywords: Mathematics Education, Engineering, Science Education

AMS Subject Classification: 97

Paper appears in The Proceedings of the MACAS3 Symposium which are now available at Information Age Publishing

The Preparation of Students for Intense College Mathematics Coursework

Michael Harwell, Amanuel Medhanie, Thomas R. Post
University of Minnesota

Ke Wu Norman
University of Montana

Daniele Dupuis
University of Montana

Abstract: Longitudinal data were used to explore the criticism that National Science Foundation-funded (Core-Plus) mathematics curricula do not adequately prepare college bound high school students for intense college mathematics coursework. The college mathematics achievement and coursetaking of students who completed a commercially developed or Core-Plus curriculum, and who completed a minimum of two college mathematics courses of difficulty level at or beyond precalculus mathematics, was examined. The results suggested that students (including STEM majors) were equally prepared for intense college mathematics coursework regardless of the high school mathematics curriculum they completed. These findings inform high school mathematics curriculum adoption decisions for college bound students, and college policies and practices for advising students enrolling in mathematics courses.

Keywords: Mathematics Education, Curriculum, Achievement, Secondary Education, Post Secondary Education

AMS Subject Classification: 97

Paper was accepted and will appear in The Journal of Experimental Education

Mathematics Teacher Education in the Public Interest

Laura Jacobsen
Radford University

Jean Mistele
Radford University

Bharath Sriraman
The University of Montana

Abstract: Mathematics teacher education has a critical role to play in preparing teachers to put at center stage goals to support equity in mathematics education and to diversify student interest and participation in mathematics. These goals must also resonate with broader public interest goals to improve educational and social conditions both in the U.S. and abroad. The Mathematics Teacher Education in the Public Interest book aims to support mathematics teacher educators to prepare teachers with new knowledge and skills to support all students to learn mathematics and to become informed, engaged, and critical citizens within their community, nation, and world. While internationally there is considerable interest among mathematics educators in issues of equity and social justice, the literature on mathematics teacher education for equity and social justice thus far has been very limited. The book provides theoretical discussions on the need for equity and social justice emphases in mathematics teacher education, as well as practical examples from mathematics teacher educators, documenting their own professional efforts to center practices on equity and social justice. Section emphases include critical perspectives on mathematics teacher education, the use of equity and social justice-themed activities in mathematics teacher preparation courses, and issues of identity and community and cultural contexts in mathematics teacher education. In addition syntheses of major ideas of the book are offered by experienced researchers.

Keywords: critical theory; equity; mathematics teacher education; social justice; mathematics in society

AMS Subject Classification: 97

Pre-print of Monograph 9 in Cognition Equity and Society Series published by Information Age Publishing, Charlotte, NC.

Dogmatism and the Knowledge Industry: More Accurately Assessing the Work of Gifted Scholars

Bharath Sriraman
The University of Montana

Abstract: In this chapter I address the tension between the objectivity of knowledge embodied in academic citation indexes, impact factors and other bibliometric measures in vogue today and used by the Knowledge industry, with the subjectivity inherent in the review, selection, publication and eventual propagation of ideas in scholarly articles in journals, books and other peer reviewed outlets. Kant's anti-thesis of dogmatism and criticism is used to explain the fallibility of the Knowledge industry, and the need to move towards a more critical stance on the objectivity of bibliometric measures.

Keywords: Bibliometrics; Dogmatism; Kant; Latour; Philosophy of Knowledge

AMS Subject Classification: n/a

Abstract of chapter to appear in D. Ambrose, R. Sternberg, B. Sriraman (Eds). Confronting Dogmatism in Gifted Education, Routledge, Taylor & Francis. Pre-print not made available due to publisher policy.

Considering the Effects of Dogmatism on Giftedness and Talent Development

Don Ambrose
Rider University

Robert J. Sternberg
Oklahoma State University

Bharath Sriraman
The University of Montana

Abstract: In the field of gifted education there is lack of consensus about definitions (who are the gifted) and identification (how should the gifted be “selected” for gifted programs). The unsettled, contested nature of the conceptual foundations for gifted education causes problems for educators who do the practical work of program development, curricular refinement, and instruction. Moreover, lack of conceptual clarity hampers the efforts of professionals in the field when it comes to advocacy for the needs of the gifted and securing the necessary resources for program implementation. This chapter explores ways in which narrow-minded, shortsighted thinking might be exacerbating the conceptual turbulence in the field. After a discussion of these conceptual problems we discuss the contributions in this book Confronting Dogmatism in Gifted Education (Routledge, Taylor and Francis) and the ways in which they can establish more clarity about the nature and nuances of giftedness and talent.

Keywords: dogmatism; philosophy of giftedness; philosophy of education; philosophy

AMS Subject Classification: n/a

Preliminary version of chapter to appear in D. Ambrose, R. Sternberg, B. Sriraman (Eds). Confronting Dogmatism in Gifted Education, Routledge, Taylor & Francis Pdf (59 KB)

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

Xianwei Yuan Van Harpen
Illinois State University

Bharath Sriraman
The 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

Preprint of paper submitted to Educational Studies in Mathematics Pdf (660 KB)

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

Kyeonghwa Lee
Seoul National University

Bharath Sriraman
The 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 gifted 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

Preprint of revision submitted Gifted Child Quarterly. Pdf (172 KB)

Original Technical Report available at #25/2010

Circumpolar Indigenous Issues, Knowledge, Relations to Education, Science and Mathematics

Bharath Sriraman
The University of Montana

Anne Birgitte Fyhn
University of Tromsø, Norway

Abstract: The articles in the special issue of vol.42, no. 2, 2011 of Interchange: A Quarterly Review of Education (Springer, in press) capture an essence of educational initiatives in the circumpolar regions in question and bring Inuit, Yup’ik, Athabaskan, and Sámi voices into the fray. The concluding article is a synthesis that tackles the notions of “indigenous” as stipulated by the U.N versus the reality inherent in how the term is interpreted in a world carved by nation-states and vested interests. The journal issue as whole covers topics that include indigenous knowledge, autonomy, educational policy, cultural preservation and developmental issues. The only voices absent in this issue are from arctic Greenland and Russia.

Keywords: Circumpolar first peoples; Indigenous Issues; Traditional knowledge versus Institutional knowledge; Geopolitics; Mathematics; Science

AMS Subject Classification: 97

Preprint of articles to appear in Interchange: A Quarterly Review of Education, vol. 42 (Springer) Pdf (37 KB)

Creatively Gifted Students Are Not Like Other Gifted Students: Research, Theory, and Practice

Kyung Hee Kim
College of William and Mary

James C. Kaufman
Learning Research Institute
California State University at San Bernardino

John Baer
Rider University

Bharath Sriraman
The University of Montana

Lauren Skidmore
California State University at San Bernardino

Abstract: Despite this attention to the need to promote and nurture creativity of students in gifted education programs, there is an almost invisible lacuna in the way gifted education treats creatively gifted students. Exhibiting creativity may help a student in the selection process and creative-thinking activities may be part of the program itself. The special and important needs of creatively gifted students, however, are often overlooked. In contrast, a student in a gifted education program with extreme math or science or language abilities will likely be given opportunities to accelerate her math or science or language arts studies, work with a mentor in that area, or be given other opportunities related to her special area of ability and interest. Similarly, a student with outstanding music or art abilities will often be given opportunities to develop the domain-specific skills and acquire the domain-specific knowledge important in her area of special talent. But there is a rarely any program, or provision within a broader gifted/talented program, for a student who is extremely creative, but not necessarily (at least yet) highly accomplished in one particular area. This book will address the following topics:

  •     cultural influences on the kinds of constraints that influence creative performance, both positively and negatively
  •     social needs of creatively gifted students
  •     assessment for student selection - aligning program goals with selection procedures
  •     developing teachers' skills and comfort in teaching creatively gifted students
  •     applying a dual process (conscious/unconscious vs. explicit/implicit) model to understanding creative giftedness
  •     career development for creatively gifted students
  •     making gifted programs work for creatively gifted minority students
  •     engaging creatively gifted but underachieving students
  •     the interplay of nature and nurture in the development of creatively gifted students' thinking
  •     the complex relationship between intelligence and creativity
  •     techniques that increase and utilize creativity in play
  •     how to improve the critical and evaluative thinking skills of creatively gifted students in ways that enhance both idea generation and selection in the writing process

Keywords: creativity; giftedness; creativity education; gifted education; talent development; mathematics education; intelligence; pedagogy

AMS Subject Classification: None

Pre-print of Accepted Book Proposal under contract with Sense Publishers (Rotterdam) Pdf (170 KB)

Immigrant and “Alien” Reactions to Obama’s Educational Policy: Disposition of Authenticity or the Politics of The Emperor’s New Clothes

Bharath Sriraman
The University of Montana

Per Haavold Øystein & Gunnar Kristiansen
University of Tromsø, Norway

Abstract: The article “Obama’s Educational Policy: Disposition of Authenticity” [see pdf attachment] brings into sharp relief issues confronting education in general in the U.S.A through the narrative of an immigrant inside the system. In particular Author addresses current policy level rhetoric and subsequent decision making that has led to increased polarization of views about the nature and purpose of public schools, the role of teachers, and measured accountability through high stakes testing as the shibboleth for society. The reaction in the form of voices is composed by another immigrant to the United States, and two Norwegian “aliens” to the system.

Keywords: accountability; equity; mathematics education policy; Obama’s educational policy; public school systems

AMS Subject Classification: 97

Pre-print of Commentary submitted to Journal of Educational Thought Pdf (191 KB)

Probabilistic Thinking: Presenting Plural Perspectives

Egan J Chernoff
University of Sasketchwan, Canada

Bharath Sriraman
The University of Montana

Abstract: Probabilistic Thinking: Presenting Plural Perspectives (PT: PPP) will serve four main purposes. First, in 2010 research will have been in the Contemporary Period for nearly 20 years and, as such, we contend it is no longer premature to evaluate historically the significance of probability research in this period. Second, to address the volume and diversity (and the historical evaluation) of the research in the Contemporary Period, the structure (and the title) of this book are derived from cubism. Whereas a cubist artwork presents multiple perspectives, which represents the subject in a greater context, the different sections of this book will present a variety of perspectives, which will represent the subject (i.e., probabilistic thinking) in a greater context. Further, each of the sections of the book will be comprised of multiple chapters, which will also represent a variety of subjects (i.e., the different sections of the book) in greater context. Third, and within the forward looking spirit of the monograph series "Advances in Mathematics Education," the structure of PT: PPP will allow for an international group of scholars to begin to define the next Period of research in probabilistic thinking. Fourth, we plan, with this book, to contribute to the major pieces of literature on probabilistic thinking, as detailed above. To achieve the four main purposes, the content of the ten sections, consisting of anywhere between 5 to 10 chapters, has been strategically chosen and will now been commented on and justified in turn. The book is planned in two volumes given its substantial scope. A sound synthesis of the existing research on probabilistic thinking is presented in this prospectus in addition to a sneak preview of the developing Table of Contents.

Keywords: Probability; Interpretations of Probability; Probabilistic Thinking

AMS Subject Classification: 60, 97

Pre-print of Accepted Book Proposal, Advances in Mathematics Education, vols.6&7, 2013 [Springer Science] Pdf (101 KB)

New Perspectives on the Didactic Triangle: Teacher-Student-Content

Simon Goodchild
University of Agder, Norway

Bharath Sriraman
The University of Montana

Abstract: What constitutes development when applied to the teaching of mathematics? Responses to this question might focus on the nature of the tasks and activities in which teachers engage their students. The introduction of ’inquiry’ tasks, problem solving activities, and open, rather than closed tasks can all be taken as evidence of teaching development, indeed there has, over the past couple of decades, and recently been considerable attention on the nature of tasks in the literature of mathematics teaching development. In classrooms tasks are a ‘mediating artefact’ used by the teacher with the intention of leading students to develop new understanding or knowing, that is tasks are used in a wider context of teaching. This special issue of ZDM brings together leading researchers and thinkers in the field of mathematics education to address, from the perspective of their own research the relationships between mathematics, student and teacher.

  •     How does/can the introduction of inquiry or investigational tasks impact upon the relationships within the didactical triangle?
  •     How does/can the development of a problem oriented approach to mathematics teaching and learning affect the relationships within the didactical triangle?
  •     How does/can the introduction of digital technologies to teaching and learning mathematics affect the relationships within the didactical triangle? Does the technology introduce another ‘vertex’ such that it is necessary to refer to a didactical quadrilateral?
  •     How do/can teachers transform the relationships between mathematics, students and themselves?
  •     How can those working in teaching development projects influence teaching so that teaching, and the didactical relationships are accommodated to new artefacts (inquiry tasks, problems, ICT, etc?).

Keywords: Didactic Triangle; Didactical Transpositions; Mediation between Teacher-Student-Content; Mathematics Education Theory

AMS Subject Classification: 97

Pre-print of ZDM- The International Journal on Mathematics Education, vol 43. No.7, 2011 [Accepted proposal on the Didactic Triangle Pdf (107 KB)

Perspectives on Sámi Mathematics Education

Anne Birgitte Fyhn
University of Tromsø, Norway

Ellen J. Sara Eira
Guovdageainnu Nuoraidskuvla- Kautokeino Ungdomsskole, Norway

Bharath Sriraman
The University of Montana

Abstract: The Sámi are an indigenous people of the Arctic, and through a resolution of the United Nations, Norway is bound to take care of the Sámi culture and language. Since 1987 the Sámi have had their curriculum, but they have no mathematics syllabus. In this paper we summarize the legal acts that take care of the Sámi culture within the Norwegian educational system, and then discuss three examples of Sámi mathematics, which can be part of a possible Sámi mathematics syllabus. First, a unit of measurement is presented, second, a unique way of treating the ratios 2 : 1 and 1 : 2 is described, and finally the use of some Sámi versus Norwegian geometry terms are exposed. The three examples are situated in relation to the Yupik Eskimo Mathematics in a Cultural Context (MCC), as described by Lipka, Webster and Yanez (2005) and their contribution in this special issue of Interchange.

Keywords: beali unnit; beali eanet; indigenous issues; Mathematics in Cultural Context; multiplicative structures; Sámi measurement; Sámi; Sámi mathematics education

AMS Subject Classification: 97

Preprint of article to appear in Interchange: A Quarterly Review of Education, vol. 42 (Springer) Pdf (762 KB)

Syntheses of Circumpolar Indigenous Issues, Knowledge, Relations to Education, Science and Mathematics

Bharath Sriraman
The University of Montana

Abstract: In this concluding article of the special issue of Interchange focused on circumpolar indigenous issues, relations to education, science and mathematics, major themes are explored in the context of the 2007 U.N declaration of indigenous rights, as well as the political connotations of the notion of “indigenous” in current geo-economic flux. Commonalties from the five articles exploring Inuit, Yup’ik, Inuktitut, Sami and Athabaskan perspectives are also summarized.

Keywords: Circumpolar first peoples; Indigenous Issues; Traditional knowledge versus Institutional knowledge; Geopolitics; Mathematics; Science

AMS Subject Classification: 97

Preprint of article to appear in Interchange: A Quarterly Review of Education, vol. 42 (Springer) Pdf (137 KB)

Extrapolation or Expansion?: Characteristics of impact exposed in a longitudinal study of one school’s participation in successive mathematics teaching development projects

Claire V. Berg, Anne Berit Fuglestad and Simon Goodchild
University of Agder, Norway

Bharath Sriraman
The University of Montana

Abstract: Communities of practice theory models development as extrapolation, whereas cultural historical activity theory models development as expansion. This paper explores the differences between these models and their underlying principles. 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 is analysed to expose evidence of development, which is examined for indications of extrapolation and expansion. The projects were designed on principles of communities of inquiry, which it is claimed radically transform community of practice theory, entailing a shift into the critical paradigm. The paper engages with the analysis and synthesis of a large volume of qualitative data that accrues 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

Preprint of Paper submitted to Journal for Research in Mathematics Education Pdf (204 KB)

A Quantitative Study of the Effects of Informal Mathematics Activities on the Beliefs of Pre-service Elementary School Teachers

Matt Roscoe
The University of Montana

Bharath Sriraman
The University of Montana

Abstract: This study sought to determine the relationship between participation in informal mathematics activities and the formal/informal beliefs of pre-service elementary teachers. Three classes of preservice teachers participated in the study through their enrollment in a content mathematics course for elementary education majors. Four informal mathematics activities were employed as part of the course requirements. Pre and post formal/informal beliefs about mathematics and mathematics instruction were measured using a Likert-scale beliefs assessment instrument used by Collier (1972) and Seaman et al. (2005). Changes in beliefs about mathematics and mathematics instruction were compared to a control group. Student reflections derived from participation in the activities was analyzed for formal and informal belief statements.

Keywords: Beliefs theory; Mathematics Beliefs instrumentation; Informal learning; Pre-service mathematics teachers; Quantitative analysis; Reflection; Reflective mathematical activities

AMS Subject Classification: 97

Preprint of paper submitted to ZDM- The International Journal on Mathematics Education [Special issue on beliefs] Pdf (1440 KB)