This article examines what constitutes a 'good' mathematical problem and proposes to conceptualize the notion of "collective mathematical problems" as phenomena that arise through and by way of the inter-actions enabled by the collective itself. The author examines how these collective mathematical problems can emerge in the classroom…

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…

This essay extends beyond the characteristics and discourse of word problems to, more generally, school mathematical problem-solving and the implications entailed by a predominant paper-and-pencil mode of learning and instruction since the modern era of education. Contrasting what I call "one-handed" (with paper-and-pencil) with…

The research on mathematical argumentation has mainly adopted a dialectic lens which focuses on understanding the abstract and logical development of reasoning in argumentation. However, this approach may have overlooked other key aspects of mathematical argumentation, including the unfolding of the meaning-making experience and process during…

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…

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…

The aim of this study is to explore the possibility of introducing the general notions of function and inverse function through a mathematical activity on linear functions, focusing on the quantitative meaning associated to the connection between a relation and its inverse. We present a genetic decomposition, that is, a viable cognitive path for…

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…

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…

Research in mathematics education suggests that learning to solve a problem should involve modelling and visual representation (e.g., Lesh & Zawojewski, 2007). According to researchers, transforming a mental representation of a situation into a visual representation of mathematical relationships between quantities enhances students'…

This essay aims to advance the debate on the conceptualization of interdisciplinarity in mathematics education by presenting the idea of the border as a dynamic semiotic cross-cultural space. We refer to three theories, that originate in different research fields and use specific languages: De Luca Picione and Valsiner's idea of the border as a…

In this paper, we propose a potential interactional explanation of tertiary-to-secondary (dis)continuity: that of authority relations. Using secondary mathematics teachers' proof validations across two contexts, we suggest that secondary teachers' conceptions of authority shape their capacity to reconcile their positions as former mathematics…

Tentativeness is often framed as a deficit, synonymous with timidity or a lack of confidence. In this article, I situate the notion of tentativeness in an enactivist framework and describe its role as both a strategy and affordance in a spatial visualization exercise. Drawing on insights from mathematics education and ecological psychology, I…

This theoretical article explores the affordances and challenges of Euler diagrams as tools for supporting undergraduate introduction-to-proof students to make sense of, and reason about, logical implications. To theoretically frame students' meaning making with Euler diagrams, we introduce the notion of logico-spatial linked structuring (or…

We describe how a group of secondary mathematics teachers posed and solved a real-world problem. The problem was posed by teachers on a sandbar off the coast of an island in the Bahamas, where panoramic views of water meeting sky spurred the teachers to wonder how far away they could see. We analyze how the teachers translated the natural…