How to influence and assess whether students engage in conceptual thinking are longstanding methodological problems in mathematics education. Recently, eye-tracking technology has fueled a discussion on whether eye movement analysis can support valid inferences about mathematical thinking. This study investigates whether eye movement analysis can…

According to the design principle of progressive schematization, learning trajectories towards procedural rules can be organized as independent discoveries when the learning arrangement invites the students first to develop models for mathematical concepts and model-based informal strategies; then to explore the strategies and to discover pattern…

This paper describes five growth points in linking representations of function developed from a study of secondary school learners. Framed within the cognitivist perspective and process-object conception of function, the growth points were identified and described based on linear and quadratic function tasks learners can do and their strategies…

The links between the mathematical and cognitive models that interact during problem solving are explored with the purpose of developing a reference framework for designing problem-posing tasks. When the process of solving is a successful one, a solver successively changes his/her cognitive stances related to the problem via transformations that…

This paper reports a classroom-based study involving investigation activities in a university numerical analysis course. The study aims to analyse students' mathematical processes and to understand how these activities provide opportunities for problem posing. The investigations were intended to stimulate students in asking questions, to trigger…

In this paper we describe learners being asked to generate examples of new mathematical concepts, thus developing and exploring example spaces. First we elaborate the theoretical background for learner generated examples (LGEs) in learning new concepts. The data we then present provides evidence of the possibility of learning new concepts through…

Meaning is one of the recent terms which have gained great currency in mathematics education. It is generally used as a correlate of individuals' intentions and considered a central element in contemporary accounts of knowledge formation. One important question that arises in this context is the following: if, in one way or another, knowledge…

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

Considers young peoples' views of infinity prior to instruction in the methods mathematicians use in addressing the subject of infinity. Presents a partially historical account of studies examining young peoples' ideas of infinity. Four sections address potential pitfalls for research in this area and the work of Piaget, issues concerning the…

Demonstrates how research-based knowledge about students' incompatible answers to different representations of the same task could be used in mathematics instruction. Describes the 'It's the Same Task' research-based activity which encourages students to reflect upon their own thinking about infinite quantities and avoid contradictions by using…

Analyses several examples of tacit influences exerted by mental models on the interpretation of various mathematical concepts in the domain of actual infinity. Specifically addresses the unconscious effect of the figural-pictorial models of statements related to the infinite sets of geometrical points related to the concepts of function and…

Focuses on children creating representations on paper for situations that change over time. Articulates the distinction between homogeneous and heterogeneous spaces and reflects on children's tendency to create hybrids between them. (Author/MM)

Considers examples of aspects of the infinitely small and large as they unfolded in the history of calculus from the 17th through the 20th centuries. Presents didactic observations at relevant places in the historical account. (Author/MM)

Discusses the problems of conceptualization of the function limit in technological environments (principally graphing calculators today and symbolic calculators tomorrow) that are gradually being adopted in precalculus teaching. Explains how the instrumentation process and the conceptualization process are dependent on each other. Sets forth a…

Proposes a framework for discussing the nature of the relationships between contexts and the formation of mathematical knowledge through model and field-of-experience concepts. Supports and illustrates this framework with references to curricular innovation and educational research. (Contains 31 references.) (Author/ASK)