Building on the papers in this special issue as well as on our own experience and research, we try to shed light on the construct of "example spaces" and on how it can inform research and practice in the teaching and learning of mathematical concepts. Consistent with our way of working, we delay definition until after appropriate reader experience…

With our conceptualization of Harre and van Langenhove's (1999) positioning theory, we draw attention to immanent experience and read transcendent discursive practices through the moment of interaction. We use a series of spatial images as metaphors to analyze the way positioning is conceptualized in current mathematics education literature and…

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…

The definition of "definition" cannot be taken for granted. The problem has been treated from various angles in different journals. Among other questions raised on the subject we find: the notions of "concept definition" and "concept image", conceptions of mathematical definitions, redefinitions, and from a more axiomatic point of view, how to…

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…

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…

Suggests that trivialized versions of constructivism are promulgated due to misunderstandings of vocabulary adopted by its advocates. Discusses specifics of frustrations in an effort to communicate the insights and implications of radical and social constructivist theories to preservice and practicing teachers. Presents a classroom episode to show…

Students in a mathematics classroom are motivated to do many things, not only the ones we expect them to do. In order to understand student behaviour in classrooms we need to increase our understanding of what motivation is and how it is regulated. Two issues relevant to a critique of mainstream motivation research need consideration: (a) the…

The framework for this paper is a recently developed theory of abstraction in context. The paper reports on data collected from one student working on tasks concerned with absolute value functions. It examines the relationship between mathematical constructions and abstractions. It argues that an abstraction is a consolidated construction that can…

In recent years, semiotics has become an innovative theoretical framework in mathematics education. The purpose of this article is to show that semiotics can be used to explain learning as a process of experimenting with and communicating about one's own representations (in particular "diagrams") of mathematical problems. As a paradigmatic…

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…

Investigates proficiencies when students work on mathematics problems and the extent captured by knowledge categories. Explores competency that goes beyond knowledge to include mathematical 'dispositions' and 'practices' in which students engage. Considers ways in which research knowledge is conceived and developed, reflecting upon the role of…

Used are test and interview data to extract evidence as to what level tenth graders have coordinated the worlds of arithmetic and algebra and can move freely between them. More dissociation is shown than was expected. (Author/MVL)

This paper is concerned with pupils' learning of the concept of variable in the LOGO programing environment and how this relates to teaching. Results from three research projects are described and compared in a discussion of how improvements in learning have been influenced by a refinement in teaching method. (Contains 21 references.) (Author/MDH)

Described is the Educational Leaders in Mathematics (ELM) Project, a four-stage inservice teacher program designed to stimulate teachers' development of a constructivist view of learning. The impact of the intervention was studied, and results indicated that teachers' beliefs were affected, producing changes in teaching strategies. (MDH)