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…

This paper introduces the notion of "crystalline concept" as a focal idea in long-term mathematical thinking, bringing together the geometric development of Van Hiele, process-object encapsulation, and formal axiomatic systems. Each of these is a strand in the framework of "three worlds of mathematics" with its own special characteristics, but all…

This study combines the Japanese lesson study approach and mathematics teachers' professional development. The first year of a 4-year project in which 3 Dutch secondary school teachers worked cooperatively on introducing making sense of the calculus is reported. The analysis focusses on instrumental and relational student understanding of…

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…

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…

In this commentary on Matthew Inglis' "Three Worlds and the Imaginary Sphere" (see EJ1106688), David Tall develops the theme that the building of theories is not an easy process. A theory in progress is a particularly delicate creation. Theories do not appear fully formed. There is a period of exploration and incubation that precedes the…

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.…

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)

Cardinal infinities, the superrational numbers, and intuitions of infinity in limiting processes are discussed. (MP)

The major idea in this paper is the formulation of a theory of three distinct but interrelated worlds of mathematical thinking each with its own sequence of development of sophistication, and its own sequence of developing warrants for truth, that in total spans the range of growth from the mathematics of new-born babies to the mathematics of…

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)

Discussed are the: (1) Notion of Mathematical Objects and the Realist Approach; (2) Psychological Need for Mathematical Realities, A Psychological Point of View; (3) Formalist Approach; (4) Meaning of Mathematical Existence; (5) Relative versus Absolute Existence; (6) Psychological Need for Mathematical Realities and the Offer of Mathematics; and…

In this paper, we discuss the (potentially positive) pedagogical role of intrinsic limitations of computational descriptions for mathematical concepts, with special focus on the concept of derivative. Our claim is that, in a suitable approach, those limitations can act for the enrichment of learners' concept images. We report a case study with a…

Discusses using the computer to promote versatile learning of higher order concepts in algebra and calculus. Generic organizers, generic difficulties, and differences between mathematical and cognitive approaches are considered. (YP)