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

This paper focuses on the changes in thinking involved in the transition from school mathematics to formal proof in pure mathematics at university. School mathematics is seen as a combination of visual representations, including geometry and graphs, together with symbolic calculations and manipulations. Pure mathematics in university shifts…

This paper considers mathematical abstraction as arising through a natural mechanism of the biological brain in which complicated phenomena are compressed into thinkable concepts. The neurons in the brain continually fire in parallel and the brain copes with the saturation of information by the simple expedient of suppressing irrelevant data and…

The concept of function is considered as foundational in mathematics. Yet, it proves to be elusive and subtle for students. A generic image that can act as a cognitive root for the concept is the function box. This is not a simple pattern-spotting device, but a concept that embodies the salient features of the idea of function, including process…

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.…

This paper examines the question of whether the introduction and use of the function machine representation as a scaffolding device helps undergraduates enrolled in a developmental algebra course to form a rich, foundational concept of function. It describes students' developing understanding of function as an input/output process and as an…

The major focus of this study is to trace the cognitive development of students throughout a mathematics course and to seek the qualitative differences between those of different levels of achievement. The aspect of the project described here concerns the use of concept maps constructed by the students at intervals during the course. From these…

The terms generalization and abstraction are used with various shades of meaning by mathematicians and mathematics educators. Introduced is the idea of "generic abstraction" that gives the student an operative sense of a mathematical concept and provides a passage point in the process toward formal abstraction. (MDH)

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)

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…