Mathematics Commons^™

Visualizations And Intuitive Reasoning In Mathematics, Kajsa Bråting

The Mathematics Enthusiast

On the basis of historical and didactical examples we consider the role of visualizations and intuitive thinking in mathematics. Examples from the 17th and 19th century have been used as well as smaller empirical studies at upper secondary school level and university level. We emphasize that a mathematical visualization does not reveal its intended meaning. With experience we can learn to interpret the visualization in different ways, depending on what is asked for.

If Not, Then What If As Well- Unexpected Trigonometric Insights, Stanley Barkan

The Mathematics Enthusiast

In performing an exercise of “What if not”, one can end up with a paucity of structure. Adding alternative structure can be a rich source of discovery, as we present here. The framework of this presentation is the original voyage of discovery, from a trivial geometric problem to the derivation of some unexpected trigonometric formulae based on regular polygons. The original “voyage” has been changed only sufficiently to make the text readable.

Reflections On Teaching With A Standards-Based Curriculum: A Conversation Among Mathematics Educators, Jill Newton, Ricki Geller, Lindsay Umbeck, Lisa Kasmer

The Mathematics Enthusiast

Many teachers and researchers have written about the challenges inherent in adopting new teaching practices in mathematics classrooms (e.g., Chazan, 2000; Clarke, 1997; Heaton, 2000). The authors of this article, all with secondary mathematics teaching experience, are convinced by research suggesting that Standards-based mathematics curricula are beneficial for student learning.1 However, the first three authors had not used such curriculum materials in their own classrooms, and we desired experience using a Standards-based mathematics curriculum with secondary students. To this end, we taught a week-long summer course with a focus on linear functions to high school students who had previously struggled …

Increasing The Use Of Practical Activities Through Changed Practice, Frode Olav Haara, Kari Smith

The Mathematics Enthusiast

This study sets out to examine the influence of a value-based intervention on two elementary school teachers’ use of practical activities in mathematics teaching. The intervention was a “Values and Knowledge Education” (VaKE)-based in-service course that introduced the two teachers to a value-based approach to mathematics teaching. The introduction included examples that were supported by use of practical activities. Interviews prior to the intervention made the teachers aware of an inconsistency between the desired and actual practice of their own teaching. The intervention provided them with a possibility of narrowing the gap between vision and practice by changing practice. Qualitative …

Number Theory And The Queen Of Mathematics, Megan Wagner

The Mathematics Enthusiast

Geometry is an integral component of secondary mathematics curriculum. From my experience in a mathematics teacher preparation program, I have seen a real push to connect geometry to other areas of mathematics. Secondary geometry can often be presented without clear or any connections to other areas of mathematics. One main purpose of this paper is to explore geometry and its rightful connection to other areas of mathematics, specifically number theory. Such strong emphasis is placed on drawing connections to number theory because of its intrinsic value in enhancing understanding of mathematical concepts. Learning number theory has positive ramifications for students, …

Translating And Adapting The Mathematical Knowledge For Teaching (Mkt) Measures: The Cases Of Indonesia And Norway, Dicky Ng, Reidar Mosvold, Janne Fauskanger

The Mathematics Enthusiast

This paper examines issues of translating and adapting an instrument that aims at measuring mathematical knowledge for teaching in Indonesia and Norway. The instrument was created for use in the U.S., and we discuss problematic and challenging issues of translation and adaptation. Two items from the released items pool were translated using a common framework modified from a previous study to exemplify critical issues that need to be resolved prior to using such instrument in another country. Themes identified in this study include a) minor challenges due to cultural differences; b) the use of technical language in schools; c) incommensurable …

Was Pythagoras Chinese- Revisiting An Old Debate, Ross Gustafson

The Mathematics Enthusiast

Two of the great ancient civilizations were those of the Greeks and the Chinese. Many great works of art, architecture, philosophy and literature have been produced by both of these civilizations. When it comes to mathematics in the Western World, the Greeks have also been credited with many contributions to the field, especially geometry. Anyone who has completed a standard high school mathematical curriculum has been introduced to the names of Pythagoras, Euclid and Archimedes and their methods. Not as widely credited in the Western world are the mathematical contributions of the ancient Chinese civilization. An examination into Chinese mathematics …

Tme Volume 9, Numbers 1 And 2

The Mathematics Enthusiast

No abstract provided.

Research On Practical Rationality: Studying The Justification Of Actions In Mathematics Teaching, Patricio Herbst, Daniel Chazan

The Mathematics Enthusiast

Building on our earlier work conceptualizing teaching as the management of instructional exchanges, we lay out a theory of the practical rationality of mathematics teaching—that is, a theory of the grounds upon which instructional actions specific to mathematics can be justified or rebuffed. We do that from a perspective informed by what experienced practitioners consider viable but also in ways that suggest operational avenues for the study of instructional improvement, in particular for improvements that enable students to do more authentic mathematical work. We show how different kinds of experiments can be used to engage in theory building and provide …

Complex Learning Through Cognitively Demanding Tasks, Lyn D. English

The Mathematics Enthusiast

The world’s increasing complexity, competitiveness, interconnectivity, and dependence on technology generate new challenges for nations and individuals that cannot be met by “continuing education as usual” (The National Academies, 2009). With the proliferation of complex systems have come new technologies for communication, collaboration, and conceptualization. These technologies have led to significant changes in the forms of mathematical thinking that are required beyond the classroom. This paper argues for the need to incorporate future-oriented understandings and competencies within the mathematics curriculum, through intellectually stimulating activities that draw upon multidisciplinary content and contexts. The paper also argues for greater recognition of children’s …

Conceptualizations And Issues Related To Learning Progressions, Learning Trajectories, And Levels Of Sophistication, Michael T. Battista

The Mathematics Enthusiast

In this paper the nature of learning progressions and related concepts are discussed. The notions of learning progressions and learning trajectories are conceptualized and their usage is illustrated with the help of examples. In particular the nuances of instructional interventions utilizing these concepts are also discussed with implications for the teaching and learning of mathematics.

On The Idea Of Learning Trajectories: Promises And Pitfalls, Susan B. Empson

The Mathematics Enthusiast

Learning mathematics is a complex and multidimensional if not an inherently indeterminate process. A necessary goal of research on learning is to simplify this complexity without sacrificing the ability of research to inform teaching. This goal has been addressed in part by researchers focusing on how to represent research on learning for teachers and on how to support teachers to use and generate models of students’ learning (e.g., Franke, Carpenter, Levi, & Fennema, et al., 2001; Hammer & Schifter, 2001; Simon & Tzur, 2004; Steffe, 2004). Recently, the idea of learning trajectories has gained attention as a way to focus …

Can Dual Processing Theories Of Thinking Inform Conceptual Learning In Mathematics?, Ron Tzur

The Mathematics Enthusiast

Concurring with Uri Leron’s (2010) cross-disciplinary approach to two distinct modes of mathematical thinking, intuitive and analytic, I discuss his elaboration and adaptation to mathematics education of the cognitive psychology dual-processing theory (DPT) in terms of (a) the problem significance and (b) features of the theory he adapts. Then, I discuss DPT in light of a constructivist stance on the inseparability between thinking and learning. In particular, I propose a brain-based account of conceptual learning—the Reflection on Activity-Effect Relationship (Ref*AER) framework—as a plausible alternative to DPT. I discuss advantages of the Ref*AER framework over DPT for mathematics education.

Tme Volume 8, Number 3

The Mathematics Enthusiast

No abstract provided.

Practical Rationality, The Disciplinary Obligation, And Authentic Mathematical Work: A Look At Geometry, Deborah Moore-Russo, Michael Weiss

The Mathematics Enthusiast

Grossman and McDonald (2008) recently argued that the research community needs to move its “attention beyond the cognitive demands of teaching … to an expanded view of teaching that focuses on teaching as a practice (p. 185).” Building on the work of Bourdieu (Bourdieu and Wacquent, 1992; Bourdieu, 1985, 1998), Herbst and Chazan (2003, 2006) have written about mathematics teaching as a practice, just as law and medicine are considered practices, in an attempt to better understand the rationality that produces, regulates, and sustains mathematics instruction. This practical rationality is the commonly held system of dispositions or the “feel for …

Editorial: “Glocal”, “Glocavores”: Good Gadgetry?, Bharath Sriraman

The Mathematics Enthusiast

No abstract provided.

Location And Scale Estimation With Correlation Coefficients, Rudy Gideon, Adele Marie Rothan

Mathematical Sciences Faculty Publications

This article shows how to use any correlation coefficient to produce an estimate of location and scale. It is part of a broader system, called a correlation estimation system (CES), that uses correlation coefficients as the starting point for estimations. The method is illustrated using the well-known normal distribution. This article shows that any correlation coefficient can be used to fit a simple linear regression line to bivariate data and then the slope and intercept are estimates of standard deviation and location. Because a robust correlation will produce robust estimates, this CES can be recommended as a tool for everyday …

Designing And Teaching An Elementary School Enrichment Program: What The Students Were Taught And What I Learned, Angela M. Smart

The Mathematics Enthusiast

This article is a reflection on the experiences I had designing and teaching an elementary school enrichment program to gifted students in mathematics. In particular, I consider not just what I taught the students in the program but what I learned throughout the entire process. This article first focuses on a description of the program and my role within the program. I then describe in detail four of the lessons I designed and taught for the program. Central to the description of the lessons are my observations of the students’ reactions to the lessons and my own growth as the …

An Overview Of The Gifted Education Portfolio For The John Templeton Foundation, Mark Saul

The Mathematics Enthusiast

The John Templeton Foundation supported a philanthropic portfolio concerning the development of human genius. The work was contoured to some of the big questions of human activity: the nature/nurture question, the question of how cultures value and institutionalize support of exceptional students, and the ‘continuum hypothesis’ for gifted education. The first strikes at the heart of what makes us human while the second relates questions about high intelligence to the great social issues.

Mathematical Intuition (Poincaré, Polya, Dewey), Reuben Hersh

The Mathematics Enthusiast

Practical calculation of the limit of a sequence often violates the definition of convergence to a limit as taught in calculus. Together with examples from Euler, Polya and Poincare, this fact shows that in mathematics, as in science and in everyday life, we are often obligated to use knowledge that is derived, not rigorously or deductively, but simply by making the best use of available information— plausible reasoning. The “philosophy of mathematical practice” fits into the general framework of “warranted assertibility,” the pragmatist view of the logic of inquiry developed by John Dewey.

Seeking More Than Nothing: Two Elementary Teachers Conceptions Of Zero, Gale Russell, Egan J. Chernoff

The Mathematics Enthusiast

Zero is a complex and important concept within mathematics, yet prior research has demonstrated that students, pre-service teachers, and teachers all have misconceptions about and/or lack of knowledge of zero. Using a hermeneutic approach based upon Gadamer’s philosophy, this study examined how two elementary mathematics teachers understand zero and how and when zero enters into their teaching of mathematics. The results of this study add new insights into the understandings of teachers and students related to zero and the origins, relationships between, and consequences of those understandings. Significant gaps and misconceptions within both teachers’ understandings of zero suggest the need …

The Education Of Mathematically Gifted Students: Some Complexities And Questions, Roza Leikin

The Mathematics Enthusiast

In this paper I analyze some complexities in the education of mathematically gifted students. The list of issues presented in this paper is not inclusive; however, all of them seem to be typical on the international scope. Among these issues are: (1) the gap between research in mathematics education and the research in gifted education; (2) the role of creativity in the education of the gifted and the theoretical perspective on the relationship between creativity and giftedness, and (3) teaching the gifted and the teachers of gifted, including relationships between the equity principle in mathematics education and views on the …

Historical Perspectives On A Program For Mathematically Talented Students, Harvey B. Keynes, Jonathan Rogness

The Mathematics Enthusiast

The University of Minnesota Talented Youth Mathematics Program (UMTYMP) is a highly accelerated program for students who are extremely talented in mathematics. This paper describes our experiences running UMTYMP since its inception thirty years ago, the challenges in implementing such a program, and how changes in the student body have necessitated changes in the program over three decades.

Prospective Teachers’ Conceptions About Teaching Mathematically Talented Students: Comparative Examples From Canada And Israel, Mark Applebaum, Viktor Freiman, Roza Leikin

The Mathematics Enthusiast

In this paper we analyze prospective mathematics teachers' conceptions about teaching mathematically talented students. Forty-two Israeli participants learning at mathematics education courses for getting their teaching certificates, and fifty-four Canadian pre-service (K-8) teachers participating in mathematics didactics course were asked to solve a challenging mathematical task. We performed comparative analysis of problem-solving strategies, solution results and participants' success. Based on the discussion with 25 Israeli participants we composed an attitude questionnaire, in which prospective teachers were asked to express their degree of agreement with statements expressing different beliefs about education of mathematically talented students. The questionnaire was presented to 56 …

Mathematical And Didactical Enrichment For Pre-Service Teachers: Mentoring Online Problem Solving In The Casmi Project, Manon Leblanc, Viktor Freiman

The Mathematics Enthusiast

In order to teach successfully, future teachers should not only be educated about students’ conceptions, but also about different forms of knowledge and classroom culture. In our research, we examined whether the participation in the Internet-based challenging problem solving community CASMI contributes to the development of the aforementioned awareness and understanding in order to meet the needs of all students including the gifted ones. The results obtained enabled us to note that the pre-service teachers’ perceptions of the project as a source of enrichment are mainly positive. However, analyzing schoolchildren’s strategies, the participants preferred to use pre-determined criteria instead of …

Disrupting Gifted Teenager’S Mathematical Identity With Epistemological Messiness, Paul Betts, Laura Mcmaster

The Mathematics Enthusiast

Mathematics is widely perceived as a universal and uncontested discipline, contrary to the philosophy of mathematics literature. Other researchers have considered the potential role of philosophy in school, but there is little work with gifted students engaged with issues concerning the nature of mathematics. We developed a philosophy of mathematics unit intended to enlarge gifted students’ perceptions of the nature of mathematics by exposing the uncritical and tidy rendering of mathematics within school math. Using a narrative methodology, we attended to gifted student’s students’ stories of relationship with mathematics, based on the premise that a person’s relationship with mathematics is …

Gifted Students And Advanced Mathematics, Edward J. Barbeau

The Mathematics Enthusiast

The extension to a wide population of secondary education in many countries seems to have led to a weakening of the mathematics curriculum. In response, many students have been classified as “gifted” so that they can access a stronger program. Apart from the difficulties that might arise in actually determining which students are gifted (is it always clear what the term means?), there are dangers inherent in programs that might be devised even for those that are truly talented.

Sometimes students are moved ahead to more advanced mathematics. Elementary students might be taught algebra or even subjects like trigonometry and …

The Promise Of Interconnecting Problems For Enriching Students’ Experiences In Mathematics, Margo Kondratieva

The Mathematics Enthusiast

The interconnecting problem approach suggests that often one and the same mathematical problem can be used to teach various mathematical topics at different grade levels. How is this approach useful for the development of mathematical ability and the enrichment of mathematical experiences of all students including the gifted ones? What are the benefits for teachers’ and what would teachers need to implement this approach? What directions would further research on these issues take? The paper discusses these and closely related questions.

I propose that a long-term study of a progression of mathematical ideas revolved around one interconnecting problem is useful …

Creativity Assessment In School Settings Through Problem Posing Tasks, Ildikó Pelczer, Fernando Gamboa Rodríguez

The Mathematics Enthusiast

Research in math education on mathematical creativity relies on the idea that creativity is potentially within all students and it can be fostered by properly structured activities. The tasks most commonly used for its assessment are problem solving and problem posing. In our approach we use problem posing tasks to get insight into students’ creativity. Based on a qualitative analysis of the participants’ answers to the questionnaire that followed the task, we define algorithmic, combined and innovative creativity as constructs that can be put in correspondence with the types and level of knowledge involved in the problem posing task. We …

Forthcoming Tmme Vol8,No3 [August 2011: Special Section On The North Calotte Conference In Mathematics Education : Tromsø-2010], Bharath Sriraman

The Mathematics Enthusiast

No abstract provided.