The view of infinity as a metaphor, a basic premise of modern cognitive theory of embodied knowledge, suggests in particular that there may be alternative ways in which one could formalize mathematical ideas about infinity. We discuss the key ideas about infinitesimals via a proceptual analysis of the meaning of the ellipsis "..." in the real…

As part of individual interviews incorporating whole number and rational number tasks, 323 grade 6 children in Victoria, Australia were asked to nominate the larger of two fractions for eight pairs, giving reasons for their choice. All tasks were expected to be undertaken mentally. The relative difficulty of the pairs was found to be close to that…

Although proof comprehension is fundamental in advanced undergraduate mathematics courses, there has been limited research on what it means to understand a mathematical proof at this level and how such understanding can be assessed. In this paper, we address these issues by presenting a multidimensional model for assessing proof comprehension in…

This paper focuses on teaching and learning the set of real numbers "R" and its completeness property at the university level. It studies, in particular, the opportunities for understanding this property that students are offered in four undergraduate correlative courses in Calculus and Analysis. The theoretical framework used in the study draws…

This paper discusses variation in reasoning strategies among expert mathematicians, with a particular focus on the degree to which they use examples to reason about general conjectures. We first discuss literature on the use of examples in understanding and reasoning about abstract mathematics, relating this to a conceptualisation of syntactic and…

A semiotic perspective on mathematical activity provides a way of conceptualizing the teaching and learning of mathematics that transcends and encompasses both psychological perspectives focussing exclusively on mental structures and functions, and performance-focussed perspectives concerned only with student's behaviours. Instead it considers the…

This paper proposes a genetic development of the concept of limit of a sequence leading to a definition, through a succession of proofs rather than through a succession of sequences or a succession of epsilons. The major ideas on which it is based are historical and depend on Euclid, Archimedes, Fermat, Wallis and Newton. Proofs of equality by…

Describes the constructs of referents, relationships, and modes and illustrates how these constructs may be reflected in students' written responses to a decimal task that requests an explanation. Examines sets of responses from two classrooms using the proposed framework to illustrate the type of information that teachers may acquire through the…