In mathematics education, students are repeatedly confronted with the tasks of interpreting and relating different representations. In particular, switching between equations and diagrams plays a major role in learning mathematical procedures and solving mathematical problems. In this article, we investigate a rather unexplored topic with…

Secondary-tertiary transition issues are explored from the perspective of ways of doing mathematics that are constituted in the implicit aspects of teachers' action. Theories of culture (Hall, 1959) and ethnomethodology (Garfinkel, 1967) provide us with a basis for describing and explicating the ways of doing mathematics specific to each teaching…

In mathematical activities and in the analysis of mathematics teaching-learning processes, "objects," "signs", and "representations" are often mentioned, where the meaning assigned to those words is sometimes very broad, sometimes limited, other times intuitive, allusive, or not completely clear. On the other hand, as…

In many mathematics curricula, the notion of limit is introduced three times: the limit of a sequence, the limit of a function at a point and the limit of a function at infinity. Despite the use of very similar symbols, few connections between these notions are made explicitly and few papers in the large literature on student understanding of…

The hypothesis that mathematically superfluous brackets can be useful when teaching the rules for the order of operations is challenged. The idea of the hypothesis is that with brackets it is possible to emphasize the order priority of one operation over another. An experiment was conducted where expressions with mixed operations were studied,…

This paper investigates how three widely used calculus textbooks in the U.S. realize the derivative as a point-specific object and as a function using Sfard's communicational approach. For this purpose, the study analyzed word-use and visual mediators for the "limit process" through which the derivative at a point was objectified, and…

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…

We report on our analysis of data from a dataset of 26 videotapes of university students working in groups of 2 and 3 on different proving problems. Our aim is to understand the role of example generation in the proving process, focusing on deliberate changes in representation and symbol manipulation. We suggest and illustrate four aspects of…

The objective of this paper is to study students' difficulties when they have to ascribe the same meaning to different representations of the same mathematical object. We address two theoretical tools that are at the core of Radford's cultural semiotic and Godino's onto-semiotic approaches: objectification and the semiotic function. The analysis…

Everyday Mathematics has contributed in important ways to long-standing debates about mathematical concepts, symbolic representation, and the role of contexts in thinking--the latter topic reaching back at least as far as Kant's notion of scheme. The descriptive work plays a role, of course. But it is only by making sense of the observations that…

Research in the didactics of mathematics has shown the importance of the problem of the particular and its relation to the general in teaching and learning mathematics as well as the complexity of factors related to them. In particular, one of the central open questions is the nature and diversity of objects that carry out the role of particular…

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…

The main focus of this paper is on the study of students' conceptual understanding of two major concepts of Set Theory--the concepts of inclusion and belonging. To do so, we analyze two experimental classroom episodes. Our analysis rests on the theoretical idea that, from an ontogenetic viewpoint, the cognitive activity of representation of…

In this paper, we distinguish between two ways that an individual can construct a formal proof. We define a syntactic proof production to occur when the prover draws inferences by manipulating symbolic formulae in a logically permissible way. We define a semantic proof production to occur when the prover uses instantiations of mathematical…

The interplay among and connections between objects (structured or unstructured), images, language and symbols that lead to mathematical reasoning and the stating of mathematical propositions of very wide generality is well worth closer study. I believe that the subtle distinction between the way mathematical ideas are constructed from objects and…