Every educator knows the sinking feeling of a lesson gone wrong. As teachers look around the room and realize that many of their students are just not getting it, they often feel like failures. However, the struggle students experience as they persevere through high-quality challenging tasks is not a sign of failure, but rather a key aspect of…

This paper analyses how pre-service teachers approach the notion of the infinite limit of a sequence from two perspectives: Specialized Content Knowledge and Advanced Mathematical Thinking. The aim of this study is to identify the difficulties associated with this notion and to classify them. In order to achieve this, an exploratory qualitative…

Reports on two types of unsolvability in mathematics. One is an instance of impossibility and the second is a judgment of undecidability. Presents a number of examples of undecidable propositions. (GS)

A traditional question is whether or not computers shall ever think like humans. This question is redirected to a discussion of whether computers shall ever be truly creative. Creativity is defined and a program is described that is designed to complete creatively a series problem in mathematics. (MP)

Develops a division algorithm in terms of familiar manipulations of concrete objects and presents it with a series of questions for diagnosis of students' understanding of the algorithm in terms of the concrete model utilized. Also offers general guidelines for using concrete illustrations to explain algorithms and other mathematical principles.…

Abstracts of 28 research reports are provided. The reports were prepared by investigators for presentation at the 54th annual meeting of the National Council of Teachers of Mathematics. A broad range of topics related to mathematics education are covered. Three reports concern the effects of differing presentations of mathematics, four are related…

An approach to how to expand explorations of determinants is detailed that allows evaluation of the fourth order. The method is built from a close examination of the product terms found in the expansions of second- and third-order determinants. Students are provided with an experience in basic mathematical investigation. (MP)

This document contains two units that examine integral transforms and series expansions. In the first module, the user is expected to learn how to use the unified method presented to obtain Laplace transforms, Fourier transforms, complex Fourier series, real Fourier series, and half-range sine series for given piecewise continuous functions. In…

An attempt is made to show that algebra is rarely obvious, and merely expecting children to learn rules is an oversimplification. Sections cover: (1) The Non-visual Nature of Algebra; (2) The Apparently Arbitrary Nature of Algebra; (3) The Relationship Between Symbolism, System and Question; (4) The Complex Nature of Algebra; and (5) Some…

The need for incorporating algorithmics into mathematics instruction is presented. The proliferation of computers is seen to have made the designing of algorithms an essential skill. Examples are given, and the view that mathematics will lose much prestige and importance if algorithmics is not integrated into it is presented. (MP)

A tool to aid elementary calculus students in the computation of unusual limits is reviewed. (MP)

Describes an activity whose objectives are to encode and decode messages using linear functions and their inverses; to use modular arithmetic, including use of the reciprocal for simple equation solving; to analyze patterns and make and test conjectures; to communicate procedures and algorithms; and to use problem-solving strategies. (ASK)

The author promotes an earlier introduction of decimals than is currently standard practice in most elementary mathematics programs. Suggested activities and related skills are reviewed. (MP)

Presented is a method where a quadratic equation is solved and from its roots the eigenvalues and corresponding eigenvectors are determined immediately. Included are the proposition, the procedure, and comments. (KR)