According to our approach to mathematics education, the optimal aim of the teaching of mathematics is to assist students in achieving efficient mathematical work. But, what does efficient exactly mean in that case? And how can teachers reach this objective? The model of Mathematical Working Spaces with its three dimensions--semiotic, instrumental,…

Word problems are among the most difficult kinds of problems that mathematics learners encounter. Perhaps as a result, they have been the object of a tremendous amount research over the past 50 years. This opening article gives an overview of the research literature on word problem solving, by pointing to a number of major topics, questions, and…

Pictorial representations are often used to help children understand the situation described in a given number-sentence scheme. These static pictorial problems essentially attempt to depict a dynamic situation (e.g., one bird flies away while there are three birds still sitting on the fence). Previous research suggested that such pictorial…

The cognitive relationship between intuition and proof is complex and often students struggle when they need to find mathematical justifications to explain what appears as self-evident. In this paper, we address this complexity in the specific case of open geometrical problems that ask for a conjecture and its proof. We analyze four meaningful…

The aim of this paper is on the one hand to discuss from an APOS (Action-Process-Object-Schema) theory perspective the mental constructions involved in the learning of linear algebra, through examples concerning the linear transformation concept and related notions. On the other hand, methodological issues related to the design of research…

Stimulating thinking related to mathematical content is the focus of many tasks in the mathematics classroom. Beyond such content-related thinking, promoting forms of higher order thinking is among the goals of mathematics instruction as well. So-called hybrid tasks focus on combining both goals: they aim at fostering mathematical thinking and…

One of the main challenges for teachers during teaching in class is the diagnosis of students' learning and thinking processes. For this purpose, teachers must perceive relevant information, they need to interpret this information and finally, they need to respond and select suitable opportunities to learn. In this paper, diagnostic processes in…

This paper comments on the theoretical formulations and usage of the construct of teacher noticing in a selection of the papers in this special issue of "ZDM Mathematics Education." The analysis of how the notion of teacher noticing is used in the papers suggests that it draws attention to several interdependencies involved that have not…

Lesson study (LS) has been practiced in China as an effective way to advance teachers' professional development for decades. This study explores how LS improves teaching that promotes students' understanding. A LS group including didacticians (practice-based teaching research specialist and University-based mathematics educators) and mathematics…

This issue of "ZDM" collects research works sharing a common reference to the theoretical framework of Mathematical Working Spaces (MWS), a construction which emerged about one decade ago, and has progressively found its way in the mathematics education community, thanks to the collaborative work of an international group of researchers.…

This article is a commentary on the mathematical working space (MWS) approach and draws on the articles contained in this "ZDM" issue. The article is divided into three parts. In the first part I discuss the place of the MWS approach among the French theories of didactique des mathématiques. In the second part I outline what I think are…

The aim of this study is to analyze and document the extent to which high school teachers rely on a set of technology affordances to articulate epistemological and cognitive actions in problem solving approaches. Participants were encouraged to construct dynamic representations of tasks and always to look for different ways to identify and support…

Scaffolding is an important tool for meeting the challenging needs of heterogeneous groups of students in inclusive classrooms. It is especially useful when supporting low achievers. A video study (funded by the Swiss National Science Foundation, Project Nr. 134652) of 36 inclusive classes, (3rd grade, aged 9 years), was conducted to examine how…

In this paper, we explore the implications of adopting (and developing the capacities necessary to adopt) an empathic second-person research perspective. Such a perspective aims to mediate participants' access to their own experience, thereby providing a rich source of first-person data as well as a powerful pedagogical tool. Working within the…

My purpose in this paper is to illustrate the way in which an enactivist methodological approach guided me as I conducted a two-case longitudinal study where the learning of algebra was explored in different contexts throughout time. Three groups of students in two different schools in the city of Puebla, Mexico, were followed from the last year…