This study describes the types of explanations one student, Sharon, gives and prefers at different ages. Sharon is interviewed in the second grade regarding multiplication of one-digit numbers, in the fifth grade regarding even and odd numbers, and in the sixth grade regarding equivalent fractions. In the tenth grade, she revisits each of these…

Not only are the problems teachers pose throughout their teaching of great importance but also the ways in which they use those problems make this a critical component of teaching. A problem-posing episode includes the problem setup, the statement of the problem, and the follow-up questions. Analysis of problem-posing episodes of precalculus…

Much of the evidence provided in support of the argument that mathematical knowing is embodied/enacted is based on the analysis of gestures and bodily configurations, and, to a lesser extent, on certain vocal features (e.g., prosody). However, there are dimensions involved in the emergence of mathematical knowing and the production of mathematical…

There is broad acceptance that mathematics teachers' beliefs about the nature of mathematics influence the ways in which they teach the subject. It is also recognised that mathematics as practised in typical school classrooms is different from the mathematical activity of mathematicians. This paper presents case studies of two secondary…

"From Another Perspective" is a year-long course for teachers of mathematics that is designed to enhance teachers' awareness of the way that their students think when they are experiencing difficulties in geometry. It also aims at equipping teachers with tools needed to analyze and cope with Problematic Learning Situations in geometry (Gal &…

Working with Navajo Indian informants in Arizona, USA we became aware of the capabilities of children and adults to find their way in vast and clearly "chaotic" canyons. One thing we did was describe what people actually did and said about their ways to find the way back home in such contexts. A second one was to use these data in order to build a…

This article presents a description of a South African mathematics teacher's struggles to mediate a new national assessment tool called the Common Tasks for Assessment with her grade 9 learners, who are unprepared for the assessment. The purpose of the study is to describe and to identify the constraints on the teacher's mediation attempts. The…

The aim of this study is to describe and analyse the structure of 3D geometry thinking by identifying different types of reasoning and to examine their relation with spatial ability. To achieve this goal, two tests were administered to students in grades 5 to 9. The results of the study showed that 3D geometry thinking could be described by four…

The notion of equivalence relation is arguably one of the most fundamental ideas of mathematics. Accordingly, it plays an important role in teaching mathematics at all levels, whether explicitly or implicitly. Our success in introducing this notion for its own sake or as a means to teach other mathematical concepts, however, depends largely on our…

We study in this article mathematics teachers' "documentation work": looking for resources, selecting/designing mathematical tasks, planning their succession, managing available artifacts, etc. We consider that this documentation work is at the core of teachers' professional activity and professional development. We introduce a distinction between…

In the mathematics education literature, there is currently a debate about the mechanisms by which group discussion can contribute to mathematical learning and under what conditions this learning is likely to occur. In this paper, we contribute to this debate by illustrating three learning opportunities that group discussions can create. In…

In this paper we describe two episodes of instructional interaction, in which examples are used in order to help students face their misconceptions. We introduce the notions of pivotal example and bridging example and highlight their role in creating and resolving a cognitive conflict. We suggest that the convincing power of counterexamples…

In this response we address some of the significant issues that Tony Brown raised in his analysis and critique of the Special Issue of "Educational Studies in Mathematics" on "Semiotic perspectives in mathematics education" (Saenz-Ludlow & Presmeg, Educational Studies in Mathematics 61(1-2), 2006). Among these issues are conceptualizations of…

This paper examines a Special Issue of "Educational Studies in Mathematics" comprising research reports centred on Peircian semiotics in mathematics education, written by some of the major authors in the area. The paper is targeted at inspecting how subjectivity is understood, or implied, in those reports. It seeks to delineate how the conceptions…

One of the great strengths of Jim Kaput's research program was his relentless drive towards scaling up his innovative approach to teaching the mathematics of change and variation. The SimCalc mission, "democratizing access to the mathematics of change," was enacted by deliberate efforts to reach an increasing number of teachers and students each…