Mathematical induction is a powerful method of proof, taught in most undergraduate programs involving mathematics and in secondary schools in some countries. It is also commonly known to be complex and difficult to comprehend. During the last five decades, mathematics education research has produced numerous studies on the learning and teaching of…

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 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…

Before the pandemic (2019), we asked: "On what themes should research in mathematics education focus in the coming decade?" The 229 responses from 44 countries led to eight themes plus considerations about mathematics education research itself. The themes can be summarized as teaching approaches, goals, relations to practices outside…

The concept of "proof" has attracted considerable research attention over the past decades in part due to its indisputable importance to the discipline of mathematics and to students' learning of mathematics. Yet, the teaching and learning of proof is an instructionally arduous territory, with proof being recognized as a hard-to-teach…

Storytelling is an engaging way through which lived experience can be shared and reflected upon, and a tool through which difference, diversity--and even conflict--can be acknowledged and elaborated upon. Narrative approaches to research bring the richness and vibrancy of storytelling into how data is collected and interpretations of it shared. In…

This paper describes five growth points in linking representations of function developed from a study of secondary school learners. Framed within the cognitivist perspective and process-object conception of function, the growth points were identified and described based on linear and quadratic function tasks learners can do and their strategies…

Inversion is a fundamental relational building block both within mathematics as the study of structures and within people's physical and social experience, linked to many other key elements such as equilibrium, invariance, reversal, compensation, symmetry, and balance. Within purely formal arithmetic, the inverse relationships between addition and…

It is our presupposition that there is still a need for more research about how classroom practices can exploit the use and power of visualization in mathematics education. The aim of this article is to contribute in this direction, investigating how visual representations can structure geometry activity in the classroom and discussing teaching…

Representation is viewed as central to mathematical problem solving. Yet, it is becoming obvious that students are having difficulty negotiating the various forms and functions of representations. This article examines the functions that representation has in students' mathematical problem solving and how that compares to its function in the…

The article discusses the understanding of infinity in children, teachers and primary teacher students. It focuses on a number of difficulties that people cope with when dealing with problems related to infinity such as its abstract nature, understanding of infinity as an ongoing process which never ends, understanding of infinity as a set of an…

In the multilingual mathematics classroom, the assignment for teachers to scaffold students by means of instruction and guidance in order to facilitate language progress and learning for all is often emphasized. In Sweden, where mathematics education is characterized by a low level of teacher responsibility for students' performance, this…

The secondary-tertiary transition has been studied in a great amount of research in mathematics education, adopting different focuses and theoretical approaches. I present here how these focuses led the authors to identify and study different students' difficulties and to develop different means of didactical action. Individual, social, but also…

What is the relationship between research into multilingualism and research concerned more generally with language and communication in mathematics education? Diversity in linguistic practices is universal in modern society and poses problems for teaching and learning even in apparently monolingual contexts. Research in multilingualism and…

One of the great strengths of Jim Kaput's research program was his relentless drive towards scaling up his innovative approach to teaching the mathematics of change and variation. The SimCalc mission, "democratizing access to the mathematics of change," was enacted by deliberate efforts to reach an increasing number of teachers and students each…