This paper describes and analyses two mathematics lessons, one with very young pupils, about subtraction, the other for lower secondary school pupils, about gradients [slopes] and graphs. The focus of the analysis is on teacher knowledge, and on the fundamental mathematical and mathematics-pedagogical prerequisites that underpin teaching these…

In the last two decades some researchers have been using elements of Critical Theory to rethink the aims and the possibilities of school mathematics. The field known as critical mathematics education has become an important research area, and has been informing teachers who felt the need to change their practices. In this article, we explore the…

In this article, we explore the influence of mathematics graduate students' experiences with the teaching assistantships and texts that they encounter in their programs on their development and identity as teachers. We look to von Foerster's notion of trivial and non-trivial machines to interpret the graduate students' experiences and explain how…

The Math Wars have been raging since the 1990's in the United States, where the world of mathematics education has become polarised into two camps: the reformers and the traditionalists. In this article I explore the background to the Math Wars, with specific reference to conflicting ideologies of mathematics education. I draw parallels with…

This article uses Ranciere's notion of the ignorant schoolmaster and McElheny's differentiation of artist's models from those of the architect and scientist to propose the reconceptualization of mathematics education as the support of emancipated spectators and sculptors of utopia.

Story and audience form the cornerstones of the mathematics classroom work discussed in this article. Put in the simplest terms, our starting point for collaborative lesson planning is to imagine a good mathematics story to be shared by our students with family and friends and with the wider community: a story that will offer a new and wonderful…

This paper explores the role of narrative in decoding diagrams. I focus on two fundamental facets of narrative: (1) the recounting of causally related sequences of events, and (2) the positioning of the narrator through point-of-view and voice. In the first two sections of the paper I discuss philosophical and semiotic frameworks for making sense…

One of the challenges of learning ratio concepts is that it involves intensive quantities, a type of quantity that is more conceptually demanding than those that are evaluated by counting or measuring (extensive quantities). In this paper, we engage in an exploration of the possibility of developing reasoning about intensive quantities during the…

In this paper I argue my opposition to the consensus which has dominated the literature that young children view shapes as a whole and pay no attention to shape structure and that geometrical thinking can be described through a hierarchical model formed by levels. This consensus is linked to van Hiele's weok by van Hiele-based research. In the…

What are the theoretical foundations of mathematics education? Recently disciplines other than mathematics and psychology have grown in importance, including philosophy. But which branch of philosophy is the most fundamental for mathematics education? In this article, I consider the claims of five branches of philosophy to be our "first…

To learn means coming to know something new at the end of, or subsequent to, a (learning) process. Because students do not yet know at the beginning of the process what they will know subsequent to the process, they cannot actively orient towards the object of learning. In this article, I propose a phenomenological perspective that theorizes…

This article describes the development of the authors' understanding of the contradictions in their mathematics teacher education practice. This understanding emerged from contrasting analyses of the impact of the authors' practices in mathematics content courses versus mathematics methods courses. Examples of the authors' work with two students,…

In this article, I build on the work of mathematics education researchers (Hackenberg, 2005; Sztajn, 2008) who have used Noddings's (1984) notion of caring to lend insights to the teaching of mathematics. In particular, I retrospectively analyze my actions when I was serving as a mathematics teacher educator seeking to provide professional…

The purpose of this paper is to reclaim the place of specific examples. It is argued that a plausible step in experiencing the general is to start with a certain general process that can be applied to an already known and specific-for-the-learner object; the specific object is supposed to be there to be treated as a non-specific, particular…

Weber and Alcock's (2004, 2009) syntactic/semantic framework provides a useful means of delineating two basic categories of proof-oriented activity. They define their dichotomy using Goldin's (1998) theory of representation systems. In this paper, I intend to clarify the framework by providing criteria for classifying student reasoning into…