Many universities throughout the world apply student-centered approaches. Common to these is that projects, problems, challenges etc. stem from "real life". Mathematics is an effective tool to solve problems, but in this paper, we discuss how we reconcile the fact that mathematics is both a pure and an applied discipline with…

Analogical reasoning has played an important role in the development of modern mathematics. However, there has been critique of analogies for the purpose of learning new mathematics. In this article, I counter that students can productively reason by analogy to learn new mathematics and even develop new mathematics themselves. I display examples…

This paper is about the role and the potential of the history of mathematics in mathematics education. We report a discussion that we had about theoretical issues in this context, particularly about the notion of Otherness as developed by the French philosopher Emmanuel Levinas. The focus is on the reading of historical texts, both in the context…

This article is a celebration of the pleasure inherent in the teaching of mathematics, with a particular focus on a university undergraduate context, and a taught course in the Theory of Numbers. The paper describes a conscious effort by one lecturer to respect the 'genetic sequence', whereby students are encouraged and enabled to investigate the…

The paper argues that there is a need for new approaches to teaching proof with newly-available technology. It contributes to filling this need by opening a discussion on digital proof assistants, programs that allow one to do mathematics with the aid of a computer, construct proofs, and check their correctness. The paper starts by exploring such…

In this paper, we discuss the relationship between rhetoric and mathematics, focusing on mathematical proofs. We offer a theoretical framework based on Perelman's New Rhetoric for analyzing the teaching of proof, taking into account rhetorical aspects. We illustrate the practicality and applicability of the proposed framework and methodology by…

Critical to constructing and interpreting graphs is an individual's understanding of the underlying coordinate systems, yet coordinate systems are often overlooked or taken-for-granted in both mathematics education research and curricula. In this paper, we foreground coordinate systems and present a distinction between two uses of coordinate…

This paper sets forth a construct that describes how many undergraduate students understand mathematical terms to refer to mathematical objects, namely that they only refer to those objects that satisfy the term. I call this students' pronominal sense of reference (PSR) because it means they treat terms as pronouns that point to objects, like…

How do students cope with and make sense of polysemy in mathematics? Zazkis (1998) tackled these questions in the case of 'divisor' and 'quotient'. When requested to determine the quotient in the division of 12 by 5, some of her pre-service teachers operated in the domain of integers and argued for 2, while others adhered to rational numbers and…

Prior learning provides the basis for new learning. Mathematics educators employ formative assessment to "elicit" and "use" student thinking as the foundation of their instruction. Yet, information can be elicited and used in a variety of ways, so not all formative assessment is equally "formative." This means that…

Two mathematics teacher educators describe their responses during a year disrupted by a serious earthquake and the ongoing effects on students and staff. We discuss our pedagogical responses to unforeseen changes in courses and in students' lives, including the re-calibrating of social relationships. We describe our work in a postgraduate course…

This paper demonstrates how questions of "provability" can help students engaged in reinvention of mathematical theory to understand the axiomatic game. While proof demonstrates how conclusions follow from assumptions, "provability" characterizes the dual relation that assumptions are "justified" when they afford…

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,…

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…

In this dialog, the notion of mathematical understanding as might be needed by classroom teachers is critically examined by mathematics educators, mathematicians, and a classroom teacher, based on the outcomes of recent work with expert classroom teachers. Terminology, assumptions and examples are discussed and analysed from a number of points of…