The main aim of this paper is to propose a critical view of the application of Toulmin's model of argument to mathematics education, focusing on aspects that have strong teaching implications. It is claimed that Toulmin's description of arguments in mathematics, in particular as regards the relations between form and meaning, is misleading and out…

What is the paradigmatic direction of teaching and learning mathematics? According to Yves Chevallard, father of the anthropological theory of the didactic, the current paradigm is characterized by an obsolescent form of monumentalism. But is a new paradigm possibly on the rise? And what is the role of powerful organizations such as the OECD? We…

The author has been influenced throughout his time in mathematics education by the work of Caleb Gattegno. Gattegno made extensive use of the word awareness whereas much educational literature from a psychological perspective talks about memory (for example, Justicia-Galiano, MartÌn-Puga, Linares & Pelegrina, 2017). This has, amongst other…

Research in mathematics education has long identified the ideology of certainty as central to mathematical instruction and to the way in which mathematics is (mis)used in social and political discourses. As we will see, this ideology is not merely a matter of individual beliefs, but is deeply embedded in the material practices and power structures…

Drawing on research in the context of Swedish school-age educare and adopting a post-humanist theoretical--methodological approach, we put forward the notion of mathemat-ing to conceptualise mathematical events that emerge in out-of-school configurations of practice. In them, ethical sensibilities as affects of engagement and rejection may be…

Earlier approaches to sense-making in mathematics have looked at the ways students comprehend a mathematical concept. Recent research suggests that some students make sense not only of mathematical objects that have a being, but also of objects that have yet to become. In such cases, learning mathematics is not just an act of comprehending a given…

Mathematical writing in school is difficult to teach. Reasons include that it is a domain in which there are rules for some parts of the writing like for example mathematical notation while others must be negotiated locally in the classroom. Writing is also seldom separated from the process of solving mathematical problems, creating a situation…

Covariational reasoning, essential for understanding functions and modeling dynamic situations, is traditionally introduced in secondary education. This paper proposes introducing covariational reasoning in primary school settings through an artifact, the Tracer. Previous studies investigate a covariational approach to functions with younger…

Years of failed attempts to engage my mathematics teacher students in exploration and co-creation led me to radically change my teaching approach. Inspired by literature that uses the jazz improv metaphor for teaching, I derived three tenets based on principles from jazz improv. First, I would bring no more than a single problem-solving task to…

It is widely accepted that the classroom teaching practices of in-service secondary mathematics teachers can benefit from the advanced mathematics they learned during their university education. However, pinpointing explicit instances of how this advanced knowledge informs teaching practices has proven challenging for both teachers and…

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…

Valeria and Emilia are two experienced high-school teachers, who position themselves as enthusiasts of using instructional mathematical videos in their teaching. They both see the videos as adjuvant to students' learning, but their respective lessons turn out to be very different, in particular with respect to students' participation. Teachers'…

In this article, I argue that the common practice across many school mathematics curricula of using a variety of different representations of number may diminish the coherence of mathematics for students. Instead, I advocate prioritising a single representation of number (the number line) and applying this repeatedly across diverse content areas.…

Backward transfer refers to the influence on prior knowledge of the acquisition and generalisation of new knowledge. Studies of backward transfer of mathematical knowledge have focused on content that is closely related in time and in curricular sequencing. Employing the notion of thickening understanding, we describe instances of transfer that…

We introduce, motivate, and exemplify a proposed theoretical construct guiding the design and facilitation of collaborative geometry activities, conceptually generative perspectival complementarity (CGPC). Participants in CGPC activities learn content by negotiating their respective perceptions of situated features they manipulate to accomplish…