Transitioning from material to theoretical geometry is often considered a significant challenge in compulsory education. Over the last twenty years, research rooted in the Theory of Didactical Situations (TDS) has explored how material geometry can support the teaching and learning of theoretical geometry, promoting continuity in teaching. These…

In this paper, we are interested in a dynamic vision of diagrams, focusing on their evocative power, their force in mathematical activity and their potential for interaction with mathematical objects. We are inspired by the work of C. S. Pierce to see diagrams as "inscriptions that organise space and articulate relations" and that of the…

The production and use of explicit models of mathematics are important elements of research in Didactics of Mathematics. In this paper, we present and compare two different European approaches to didactical modelling. The first, centred around the notion of basic idea (Grundvorstellung), arose in German-speaking countries as a development within…

In this paper, we draw on our recent research to inspect again some of the theoretical perspectives we have been using to analyze data and to characterize teaching-learning in university settings. We focus particularly, within a sociocultural perspective on Activity Theory and the construct the "Teaching Triad," seeking to embed the…

In this study, we present a mathematical analysis distinguishing two conceptions of equivalence: "proportional equivalence" and "unit equivalence." These two conceptions have distinct meanings in relation to equivalent fractions: one is grounded in proportionality, while the other is grounded in equal wholes. We argue that (a)…

In mathematics education research, proofs are often conceptualized as sequences of mathematical assertions. We argue that this ignores proofs that contain instructions to perform mathematical actions, often in the form of imperatives, which are common both in mathematical practice and in undergraduate mathematics textbooks. We consider in detail a…

In this paper, learning is conceptualized as a process of routinization of learners' actions. We begin by operationalizing the keyword "routine." Two categorizations of routines are then offered. First, we distinguish between "practical" and "discursive" routines, discuss the process of their historical…

Mathematics content courses for prospective elementary teachers (PTs) should provide them with opportunities to deepen their understanding of and develop productive dispositions toward mathematics. Supporting mathematics teacher educators (MTEs) to learn how to develop practices that create rich learning opportunities for novice teachers comprises…

International comparative studies on mathematics teaching and learning often provide unitary and harmonious images of classroom practices. This paper aims to complement those studies by describing the complex aspects of those practices. Adopting activity theory as a framework, this paper considers classroom teaching and learning of mathematics as…

Recent scholarship around teaching elementary mathematics supports the learning of early algebra with 5- to 12-year olds. However, in spite of the recognition of the affordances of early algebra, issues about how to introduce it remain open. Within this context, Davydov's work is often cited as a source of impressive demonstration of young…

This study explores relationships among prospective secondary teachers' skills of attending to relevant mathematics elements in students' answers, interpreting students' mathematical understanding, and proposing instructional actions. Thirty prospective secondary mathematics teachers analyzed three high school students' answers to three problems…

Mathematicians commonly distinguish two modes of work in the discipline: "Problem solving," and "theory building." Mathematics education offers many opportunities to learn problem solving. This paper explores the possibility, and value, of designing instructional activities that provide supported opportunities for students to…

This article is concerned with cognitive aspects of students' struggles in situations in which familiar concepts are reconsidered in a new mathematical domain. Examples of such cross-curricular concepts are divisibility in the domain of integers and in the domain of polynomials, multiplication in the domain of numbers and in the domain of vectors,…

We offer criteria that an observer can use to determine whether an argument that uses an example to argue for a general claim appeals to that example generically. We review existing literature on generic example and note the strengths of each contribution, as well as inconsistencies among uses of the term. We offer several examples from the…

This paper provides an analysis of a teaching episode of the multidigit algorithm for multiplication, with a focus on the influence of the teacher's mathematical knowledge on their teaching. The theoretical framework uses Mathematical Knowledge for Teaching, mathematical pertinence of the teacher and structuration of the milieu in a descending and…