Mathematics Commons^™

Associative Operations On A Three-Element Set, Friðrik Diego, Kristín Halla Jónsdóttir

The Mathematics Enthusiast

A set with a binary operation is a fundamental concept in algebra and one of the most fundamental properties of a binary operation is associativity. In this paper the authors discuss binary operations on a three-element set and show, by an inclusion-exclusion argument, that exactly 113 operations out of the 19,683 existing operations on the set are associative. Moreover these 113 associative operations are accounted for by means of their operation tables.

Magic Math Cards, Steve Humble

The Mathematics Enthusiast

Starting lessons with a good story, captures students imagination. The story then leads nicely into a magic card experiment which can be used to help teach the topic. I have used magic with a variety of age groups and found that they ask different questions of the task, and search for different levels of answers.

Mathematical Cognition And The Final Report Of The National Mathematics Advisory Panel: A Critical, Cultural-Historical Activity Theoretic Analysis, Wolff-Michael Roth

The Mathematics Enthusiast

In its Final Report, the National Mathematics Advisory Panel has depicted a stark image of mathematical competencies and achievement among U.S. students. The Advisory Panel notes the lack of research with “truly scientific” rigor and, mentioning Vygotsky’s cultural-historical activity theory in passing, suggests its utility to be untested. In reading the report, I noted the limited understanding of the mathematics education literature it articulates and a complete failure to draw on established, “tried and proven,” theory and practice in mathematics education founded upon an encompassing culturalhistorical activity theory. This theory is comprehensive and encompassing, because it retains activity in its …

E(Race)Ing Race From A National Conversation On Mathematics Teaching And Learning: The National Mathematics Advisory Panel As White Institutional Space, Danny Bernard Martin

The Mathematics Enthusiast

In this paper, I wish to argue that several factors support a characterization of the National Mathematics Advisory Panel as an instantiation of the white institutional space (Moore, 2008) that characterizes mathematics education research and policy contexts more generally. In particular, my analysis highlights that mathematics education research and policy contexts such as the National Mathematics Advisory Panel are not immune to the structural and institutional racism that characterize many other areas of U.S. society.

Guest Editorial: Reaction To The Final Report Of The National Mathematics Advisory Panel, Brian Greer

The Mathematics Enthusiast

In 2006, President Bush appointed the National Mathematics Advisory Panel (NMAP henceforth) which issued its Final Report on March 13, 2008. The Final Report (summary), plus reports of three subcommittees and five Task Groups, together with two appendices containing the Presidential Executive Order setting up the panel, and details of the panel membership and other personnel, can be downloaded from www.ed.gov/about/bdscomm/list/mathpanel/index.html

Algebra For All?, Brian Greer

The Mathematics Enthusiast

In 1957, the director of one of the major math curriculum projects in the UK was quoted in a newspaper as saying: "Up went Sputnik and down came all the pure mathematicians saying we must do sets and be saved". The corresponding message from the National Math Panel Final Report is "Up went the economic performance of China and India and down came all the pure mathematicians saying we must do algebra and be saved".

The Political Context Of The National Mathematics Advisory Panel, Eric Rico Gutstein

The Mathematics Enthusiast

The National Mathematics Advisory Panel needs to be situated in its broader political context to more fully understand it. Who created it, for what purpose, and who will (and will not) benefit from it are key questions I address in this article. My argument is that the NMAP, as part of a larger initiative undertaken by the Bush Administration and US financial/corporate elites, serves capital’s efforts to shore up the US’s weakening economic global position and does not benefit the majority of the US people—particularly marginalized and excluded students of color and low-income students.

Meet The Authors

The Mathematics Enthusiast

No abstract provided.

New Titles In Mathematics Education

The Mathematics Enthusiast

No abstract provided.

Editorial: Expanding Spheres Of Influence- The Zenith, The Nadir And Everything In-Between, Bharath Sriraman

The Mathematics Enthusiast

No abstract provided.

Historical Tidbits, The Shoah, And The Teaching Of Mathematics, Wolff-Michael Roth

The Mathematics Enthusiast

In this extended essay, I use cultural-historical activity theory to look at the questions Theodore Eisenberg raises about the inclusion of historical facts, both historical tidbits and ethically questionable tendencies and horrific actions (the Shoah), in the teaching of mathematics. I conclude by suggesting that the ultimate answer has to be one that involves a decision, which means that an answer cannot be provided a priori or be determined by any antecedent. Deciding to include this or that in a mathematical curriculum is an ethical act.

Mathematically Gifted Elementary Students' Revisiting Of Euler's Polyhedron Theorem, Jaehoon Yim, Sanghun Song, Jiwon Kim

The Mathematics Enthusiast

This paper explores how the constructions of mathematically gifted fifth and sixth grade students using Euler’s polyhedron theorem compare to those of mathematicians as discussed by Lakatos (1976). Eleven mathematically gifted elementary school students were asked to justify the theorem, find counterexamples, and resolve conflicts between the theorem and counterexamples. The students provided two types of justification of the theorem. The solid figures suggested as counterexamples were categorized as 1) solids with curved surfaces, 2) solids made of multiple polyhedra sharing points, lines, or faces, 3) polyhedra with holes, and 4) polyhedra containing polyhedra. In addition to using the monster-barring …

Final Comments On Eisenberg's Paper, Brian Greer

The Mathematics Enthusiast

I must begin by expressing appreciation for the spirit in which Ted accepted my comments as intended to be intellectually and not personally provocative. Let me also assure him that I was not at all offended by the paper – it would be more accurate to say that I was stimulated by it to voice some strong disagreements. Beyond the particular aspects on which we have different perspectives, I applaud his decision to raise ethical issues, too often ignored in writing about mathematics education.

Flaws And Idiosyncrasies In Mathematicians: Food For The Classroom, Ted Eisenberg

The Mathematics Enthusiast

This paper raises an ethical question: should aspects of a mathematician’s personality, political beliefs, physical handicaps, and the ironies surrounding their life be mentioned parenthetically or otherwise in our lessons? What about the political and social norms of the times in the countries in which they lived? There are no hard and fast guidelines on this other than to use good taste; but what is in good taste to one is often in bad taste to another. At the very least this paper presents tidbits of information and innuendo about mathematicians the reader might not know. But hopefully this paper …

Reflections Upon Teaching A Poorly-Conceived Lesson, Yuichi Handa

The Mathematics Enthusiast

Using a “failed” mathematics lesson as a mini-case study, I explore some of the challenges inherent in teaching an inquiry-style mathematics lesson. The ensuing discussion centers around two issues: the preparation and subsequent scaffolding involved in guiding students to desired understandings, along with the tension inherent in “telling” and “not telling” in an inquiry-style pedagogy. I resort to personal reflection, as well as to Wenger’s (1998) theoretical construct of participation and reification in revealing the underlying link between the manner of lesson preparation and consequent engagement of students during the enactment of the lesson.

Tme Volume 5, Number 1

The Mathematics Enthusiast

No abstract provided.

Reaction To The Reactors, Ted Eisenberg

The Mathematics Enthusiast

1. A comment. I discussed the contents of my paper with colleagues on many different occasions. By their reactions they seem to divide themselves into two camps. One camp takes the stance of: Are you crazy to mention such things in the classroom? We are charged to teach mathematics, and that’s it. Who cares if Einstein never learned to swim? The other camp however takes an opposite stance: Such items are really important for students to know because they help students to think about the lives and times of the men who created the material they are studying; our lessons …

Comments Provoked By "Flaws And Idiosyncrasies In Mathematicians: Food For The Classroom?", Brian Greer

The Mathematics Enthusiast

In mathematics classes, it is appropriate for many reasons to discuss mathematicians as people with lives, personal problems, both influenced by and influencing cultural movements and societal issues. Mathematics is a human activity, and mathematicians are human. Eisenberg's paper raises important and fascinating issues, such as the extent to which intellectual achievements can be kept separate from the personality and actions of their creator (such as Wagner). However, in my reactions, I suggest ways in which I believe the discussion needs to be broadened and refocused.

Values In Mathematics And Science Education: Similarities And Differences, Alan Bishop

The Mathematics Enthusiast

literacy and expertise from their citizens than ever before. At the heart of such demands is the need for greater engagement by students with school mathematics and science. As the OECD/PISA definition of numeracy puts it:

“Mathematical literacy is an individual’s capacity to identify and understand the role that mathematics plays in the world, to make well-founded judgements and to use and engage with mathematics in ways that meet the needs of that individual’s life as a constructive, concerned and reflective citizen”(OECD, 2003)

Values are an inherent part of the educational process at all levels, from the systemic, institutional macro-level, …

Critique On Eisenberg's Article, Renuka Vithal

The Mathematics Enthusiast

I enjoyed reading the article and learned some interesting (and disturbing) information about well known mathematicians that is not so well known. However in its current form it is more appropriate for the popular media rather than as an academic or scholarly article.

Having said that it does raise some serious questions of ethics and values that all mathematics educators should be engaging. The increasingly popular view that mathematics teaching be socially contextualised means that this kind of historical information may be communicated in lecture rooms and classroom with little understanding or awareness of the "hidden curriculum" being enacted.

Doin' The Math: On Meaningful Mathematics-Ethics Connections, Kurt Stemhagen

The Mathematics Enthusiast

In this essay mathematics is conceived of as intentional human activity. Since intention implies choice, there are ethical dimensions to making mathematical choices. Embracing these dimensions requires acknowledging the contextual nature of mathematics. First John Dewey’s philosophy of mathematics and a reconsideration of mathematical empiricism are posited as ways to foster a context sensitive understanding of mathematics. Next, I address the ways in which existent conceptions of mathematics—even those which support reform in mathematics education—are insufficient with regard to their ability to recognize its human dimensions. The essay concludes with a distinction between mathematics education that ethically applies existing versions …

The Action Map As A Tool For Assessing Situated Mathematical Problem Solving Performance, Murad Jurdak

The Mathematics Enthusiast

The aim of this paper is to investigate the appropriateness, concurrent and construct validity of action map as a tool for assessing situated problem solving performance. Action map is rooted in activity theory whose stipulations are compatible with situated problem solving. Thirty-one last year secondary students were given three tasks with real –world context. Based on the analysis of the written solutions and interviews, evidence is presented on the appropriateness and validity of action map as an instrument to assess situated problem solving performance.

A Reflection On Mathematical Cognition: How Far Have We Come And Where Are We Going?, Kristin Umland

The Mathematics Enthusiast

Any theory of mathematical cognition and learning must ultimately articulate with our current understanding of how the mind works and with current theories of how knowledge is acquired from both an individual and social perspective. Certainly the human mind is very complex; understanding how it works in general and identifying the components that contribute to “doing mathematics” in particular is no easy task. To understand how the mind does mathematics, we must identify what mathematicians actually do in the context of everyday cognition. Here we discuss some of what is known and about this and point out directions for future …

Relationship Of Some Psychological Variables In Predicting Problem Solving Ability Of In-Service Mathematics Teachers, M. K. Akinsola

The Mathematics Enthusiast

This paper examines some psychological variables in predicting problem solving ability of inservice mathematics teachers. The sample consists of 122 in-service teachers enrolled in degree programme. Five standardized instruments were used to collect the data on teachers’ mathematics anxiety, mathematics teaching efficacy belief, locus of control, study habits and problem solving ability. Multiple regression, Chi-square analysis,and Pearson moment correlation coefficient were used to analyze the data. The results show that mathematics anxiety, mathematics teaching efficacy belief, locus of control and study habits all have significant relationships with problem solving ability with mathematics anxiety having the highest and study habits the …

How Is Mathematics Education Philosophy Reflected In The Math Wars?, David M. Davison, Johanna E. Mitchell

The Mathematics Enthusiast

Throughout the duration of what has been termed the “math wars”, many overlygeneralized statements by both sides have detracted from the quest for a solution to the conflict between conceptual and procedural approaches to mathematics study. In terms of philosophies of mathematics education, the absolutist view posits that mathematical knowledge is certain and unchallengeable while the fallibilist view is that mathematical knowledge is never beyond revision and correction. We suggest that the major mathematics education reforms have been absolutist in focus and have not reflected the changing nature of the discipline. Thus we believe that true reform will reflect changing …

Teaching Proof At Ks4, Sue Waring

The Mathematics Enthusiast

Proof is a fundamental component of mathematics and so, in my opinion, it should be part of mathematical education in schools. It is an important link between topics and so can help pupils achieve a deeper understanding of the “wholeness” of mathematics and, moreover, it is vital for able pupils if they are to be inspired by mathematics and given a firm basis for future mathematical studies.

Consecutive Numbers, Steve Humble

The Mathematics Enthusiast

The hidden secrets of our number system can often reveal the magical quality of mathematics. Through the process of discovery and discussion with fellow classmates, the hidden depths of maths takes on new appeal. Consecutive numbers is one such area that gives this excitement.

New And Noteworthy Books From Sense Publishers

The Mathematics Enthusiast

No abstract provided.

Tme Worldwide Daily Circulation [Dec 6-11, 2007]

The Mathematics Enthusiast

No abstract provided.

A Hexagon Result And Its Generalization Via Proof, Michael De Villiers

The Mathematics Enthusiast

This paper presents the discovery of a hexagon result on Geometer’s Sketchpad and its generalization via proof for any 2n-gon. The result is : If ABCDEF is a hexagon with opposite sides parallel (not necessarily equal), then the respective centroids G, H, I, J, K and L of triangles ABC, BCD, CDE, DEF, EFA and FAB, form a hexagon with opposite sides both equal and parallel.