Uri Leron

On Learning Fundamental Concepts of Group Theory.

The research reported in this paper explores the nature of student knowledge about group theory, and how an individual may develop an understanding of certain topics in this domain. As part of a long-term research and development project in learning and teaching undergraduate mathematics, this report is one of a series of papers on the abstract algebra component of that project.The observations discussed here were collected during a six-week summer workshop where 24 high school teachers took a course in Abstract Algebra as part of their work. By comparing written samples, and student interviews with our own theoretical analysis, we attempt to describe ways in which these individuals seemed to be approaching the concepts of group, subgroup, coset, normality, and quotient group. The general pattern of learning that we infer here illustrates an action-process-object-schema framework for addressing these specific group theory issues. We make here only some quite general observations about learning these specific topics, the complex nature of “understanding”, and the role of errors and misconceptions in light of an action-process-schema framework. Seen as research questions for further exploration, we expect these observations to inform our continuing investigations and those of other researchers.We end the paper with a brief discussion of some pedagogical suggestions arising out of our considerations. We defer, however, a full consideration of instructional strategies and their effects on learning these topics to some future time when more extensive research can provide a more solid foundation for the design of specific pedagogies.

Being sloppy about slope: The effect of changing the scale

What is the slope of a(linear) function? Due to the ubiquitous use of mathematical software, this seemingly simple question is shown to lead to some subtle issues that are not usually addressed in the school curriculum. In particular, we present evidence that there exists much confusion regarding the connection between the algebraic and geometric aspects of slope, scale and angle. The confusion arises when some common but undeclared default assumptions, concerning the isomorphism between the algebraic and geometric systems, are undermined. The participants in the study were 11th-grade students, prospective and in-service secondary mathematics teachers, mathematics educators and mathematicians — a total of 124 people. All participants responded to a simple but non-standard task, concerning the behavior of slope under a non-homogeneous change of scale. Analysis of the responses reveals two main approaches, which we have termed ‘analytic’ and ‘visual’, as well as some combinations of the two.

Learning Abstract Algebra with ISETL

From the Publisher: Most students in abstract algebra classes have great difficulty making sense of what the instructor is saying. Moreover, this seems to remain true almost independently of the quality of the lecture. This book is based on the constructivist belief that, before students can make sense of any presentation of abstract mathematics, they need to be engaged in mental activities which will establish an experiential base for any future verbal explanations. No less, they need to have the opportunity to reflect on their activities. This approach is based on extensive theoretical and empirical studies as well as on the substantial experience of the authors in teaching abstract algebra. The main source of activities in this course is computer constructions, specifically, small programs written in the mathlike programming language ISETL; the main tool for reflection is work in teams of 2-4 students, where the activities are discussed and debated. Because of the similarity of ISETL expressions to standard written mathematics, there is very little programming overhead: learning to program is inseparable from learning the mathematics. Each topic is first introduced through computer activities, which are then followed by a text section and exercises. This text section is written in an informal, discursive style, closely relating definitions and proofs to the constructions in the activities. Notions such as cosets and quotient groups become much more meaningful to the students than when they are presented in a lecture.

How intuitive is object-oriented design?

Intuition is a powerful tool that helps us navigate through life, but it can get in the way of more formal processes.

Trace identities and polynomial identities of n × n matrices

Abstract A simple criterion for trace identities of n × n matrices over a commutative ring is given, from which the Procesi-Razmyslov criterion in terms of Young symmetrizers is deduced. A new criterion for polynomial identities of n × n matrices is then obtained.

Computers and applied constructivism

In this paper we discuss and demonstrate the nature of computational learning environments which support and encourage learners’ constructions in a way which is compatible with constructivist learning theory. We highlight the process of learning by successive refinement as a way for the human mind to cope with complexity, and the essential role the computer can play in this process. Three examples of constructivist environments (ISETL, Dynamic Geometry and Logo), are used to describe the dynamics of how the computer can facilitate the process of successive refinement, as well as to delineate issues of classroom culture, assessment and time.

Matrix methods in decompositions of modules

Abstract Matrix methods of elementary linear algebra are extended to general direct-sum decompositions of modules. These methods are then shown to yield simple proofs of some well-known theorems, notably the Beck-Warfield “mutual exchange property” and the Krull-Schmidt theorem.

Turtle goes to college: intrinsic representations and graphical integration

A general method is presented of obtaining intrinsic representations of functions, using the Logo turtle. These approximate representations consist of repetitions of the basic ‘action step’ RIGHT FORWARD , with varying values for the angle and length. The method is based on the idea of orienting the turtle at each point in the direction determined by the derivative of the function at that point. As a corollary, a ‘graphical integration’ algorithm is obtained: given a function, the algorithm produces a graph of its indefinite integral. The algorithm still works in cases where the integral cannot be expressed in terms of the elementary functions. Turtle representations are discrete, local, intrinsic and procedural, therefore are largely complementary to standard representations, and can add fresh insights to familiar topics. Also, since they are expressed as procedures in a programming language, running them on a computer offers many opportunities for exploration and discovery.

Controversy corner: Disciplined and free-spirited: 'Time-out behaviour' at the Agile conference

In this article we observe and try to understand a peculiar duality in the agile community, whereby on the one hand, we see a serious professional community working hard to improve the quality of software products and submitting to the strictest discipline of high professional standards, while on the other hand, in its conferences, we see the same community adopting a playful free-spirited stance. Invoking an anthropological perspective, we propose that both the serious professional aspects and the playful free-spirited atmosphere at the conference, as well as the connection between the two, can all be seen to emerge from the fundamental principles of the agile community as expressed by its Manifesto.

The false‐coin problem ‐‐ a colorful approach

The author discusses the relationship between puzzles and mathematics teaching in this paper.