A teaching experiment-using Mathematica to investigate the convergence of sequence of functions visually as a sequence of objects (graphs) converging onto a fixed object (the graph of the limit function)-is here used to analyze how the approach can support the dynamic blending of visual and symbolic representations that has the potential to lead…

Jim Kaput lived a full life in mathematics education and we have many reasons to be grateful to him, not only for his vision of the use of technology in mathematics, but also for his fundamental humanity. This paper considers the origins of his "big ideas" as he lived through the most amazing innovations in technology that have changed our lives…

How do students think about algebra? Here we consider a theoretical framework which builds from natural human functioning in terms of embodiment--perceiving the world, acting on it and reflecting on the effect of the actions--to shift to the use of symbolism to solve linear equations. In the main, the students involved in this study do not…

A number of general ideas intended to be helpful in analyzing differences in concept images among individuals are formulated. These ideas are applied to the specific concepts of continuity and limits, as taught in the secondary school and university. (MP)

A concept of infinity is described which extrapolates the measuring rather than counting aspects of number. Various theorems are proved in detail to show that "false" properties of infinity in a cardinal sense are "true" in a measuring sense. (MP)

Discusses the place of infinity in the history and epistemology of mathematics. (Author/MM)

The construction of both natural and formal infinities are products of human thought and may be considered in terms of embodied cognition. Forwards the viewpoint that formal deduction focuses as far as possible on formal logic in preference to perceptual imagery, developing a network of formal properties that do not depend on specific embodiments.…

Presents computer approach formulated within a framework of versatile thinking designed to overcome obstacles to understanding algebra. Results from related studies comparing the computer and traditional approaches taught to 11, 12, and 13 year olds showed that the computer approach significantly improved the understanding of higher order concepts…

Studies the effects of a course encouraging cooperative problem solving and reflection on thinking activities on students' attitudes. Reports that a majority declared negative attitudes but during the problem-solving course, the changes were almost all in the desired direction. Contains 18 references. (Author/ASK)

Considers the cognitive mechanisms available to individuals which enable them to operate successfully in different parts of the mathematics curriculum, such as children's arithmetic shows divergence in performance. Explains how students cope with the transition to advanced mathematical thinking in different ways, leading once more to a diverging…