In this paper we explore the growth of understanding of a group of preservice teachers, and their instructor, as they work collectively together on what we describe as a "mathematics for teaching" task. Through drawing on elements of improvisational theory, in particular the notion of the "better idea," we analyse how the preservice teachers'…

The ability of mathematics teachers to decompress mathematics and to move between representations are two key features of mathematical knowledge that is usable for teaching. This article reports on four pre-service secondary mathematics teachers learning the mathematics of annuities. In working with spreadsheets students began to make sense of…

Mathematical knowledge used in teaching has attracted the interest of many researchers, but was mainly explored considering teaching at the elementary school level. This paper attends to mathematical knowledge used in teaching at the University level. We present a story about a student suggesting reconsideration of Cantor's diagonal method and the…

The first goal of this article to show the profound difference between how equality and similarity are understood in Greek geometry and how they are presented in modern mathematics classes. It highlights that the formula "equal-and-similar" reflects the distinct character of "equal" and "similar" as signs in Greek mathematical discourse. The…

This article discusses the integration of history of modern applied mathematics in mathematics education as well as the possible teaching and learning benefits of introducing a newer history of mathematics over an old(er) one--something that seems to be done most often when integrating history. Three cases of the history of modern applied…

I adopt a poststructural perspective to argue that the current focus in mathematics education on measuring, reducing, and explaining the achievement gap is dangerous and can be counter-productive to the equity-oriented arguments that those mobilizing "gap" language want to make. The reiteration of the achievement gap in scholarly arguments can…

Classroom mathematics in Malta is generally communicated through code-switching between the national language Maltese and the country's second language English. This is because not all mathematical words have Maltese equivalents and furthermore, mathematics textbooks are UK publications. I consider the language used for spoken communication and…

This article explores the factors that may cause teachers to defer to the textbook as the ultimate authority on curriculum and pedagogy, drawing on Bakhtin's notion of authoritative and personally persuasive discourse. We found that although teachers deferred to the textbook as the "authority" of what should be taught, they did not fully implement…

Few studies in mathematics education explicitly address ethical issues arising from student interactions. The ethical concerns held by students are expressed in their words, actions, and interactions. The purpose of this article is to explore the ethical nature of copying as it arises in a mathematics classroom. We investigate the basis for…

We introduce Lesson Play as an imaginary interaction between teacher and students presented in a form of a dialogue or play. We suggest that lesson plays are a valuable professional development tool in preparing for teaching that can be juxtaposed with, or used as a replacement for, traditional lesson planning. The article begins with an…

This article addresses the learning paradox, which obliges researchers to explain how cognition can advance from a lower level of reasoning to a higher one. Although the question is at least as old as Plato, two major flaws have inhibited progress in developing solutions: the assumption that learning is an inductive process, and the conflation of…

It is argued that the discourse currently regulating early years mathematical development in the UK and elsewhere is "producing its own truths" in an unhelpfully narrowing way, restricting the objects, imagery, and symbols involved to those arising within "everyday" or "realistic" situations, and producing an apparently "natural" development to…

We offer a theoretical contribution to the discussion of transfer from a situated point of view. We reconceptualise what, within the school context, might be thought of in Bernstein's terms as the transfer of knowledge between two insulated vertical discourses. We describe an interdisciplinary project carried out by secondary mathematics and…

The primary focus for this article involves aspects of professional mathematical writing and examines the possibility of a form of literary criticism in relation to it. By means of examples from contemporary style guides for academic articles in mathematics (AMS, MAA), as well as the writing of mathematicians (Hamilton, Dedekind) from earlier…

While Socrates and Meno are discussing the nature of knowledge through considering the square-root of two with one of Meno's slaves, their wives are discussing the nature of knowledge through considering the square-root of three with one of the slave girls, using a diagram based on Raphael's "The School of Athens" being contemplated by Euclid.