This article discusses the concepts of "key ideas" and "memorability" and how they relate to the metric "width of a proof" put forward by the Fields medalist Timothy Gowers (2007) in a recent essay entitled "Mathematics, memory and mental arithmetic". The paper looks at the meaning of these concepts and…

This paper demonstrates how questions of "provability" can help students engaged in reinvention of mathematical theory to understand the axiomatic game. While proof demonstrates how conclusions follow from assumptions, "provability" characterizes the dual relation that assumptions are "justified" when they afford…

This paper explores contemporary thinking about learning mathematics, and within that, social justice within mathematics education. The discussion first looks at mechanisms offered by conventional explanations on the emancipatory project and then moves towards more recent insights developed within mathematics education. Synergies are drawn between…

An argument is presented for including exponentiation as a "basic" of school mathematics. Key elements of the argument include the topic's increasing relevance in a rapidly changing world and its utility for supporting understandings of role of analogy in mathematical thought. The discussion is illustrated with an account of a…

Findings from international assessments present an opportunity to reconsider mathematics education across the grades. If concepts taught in elementary grades lay the foundation for continued study, then children's introduction to school mathematics deserves particular attention. We consider Davydov's theory (1966), which sequences…

To encourage students to do geometry, the art of Islamic geometric ornamentation was chosen as the central theme of a lesson strand which was developed using the newly presented didactical tool called "Learning by Acting". The Dutch students who took these lessons in 2010 to 2013 were challenged to act as if they themselves were Persian…

It has become gradually accepted that proof and proving are essential at all grades of mathematical learning. Among the various aspects of proof and proving, this study addresses proofs and refutations described by Lakatos, in particular a part of increasing content by deductive guessing, to introduce an authentic process into mathematics…

In Sweden often immigrant students' failure in mathematics is explained by referring to deficit discourses. To critique that our aim have been to highlight the complexity of the situation in which immigrant students are positioned, by interrogating their perspectives on mathematics homework and the importance of parental support, as well as how…

Based on a study about using original sources with Danish upper secondary students, the paper addresses the potential outcome of such an approach in regard to the so-called recruitment problem to the mathematical sciences. 24 students were exposed to questionnaire questions and 16 of these to follow-up interviews, which form the basis for both a…

The observation that the human mind operates in two distinct thinking modes--intuitive and analytical- have occupied psychological and educational researchers for several decades now. Much of this research has focused on the explanatory power of intuitive thinking as source of errors and misconceptions, but in this article, in contrast, we view…

This article examines the shifts in attention and focus as one teacher introduces and explains an image that represents the processes involved in a numeric problem that his students have been working on. This paper takes a micro-analytic approach to examine how the focus of attention shifts through what the teacher and students do and say in the…

In this article I present examples of young children's interaction in collaborative group work in mathematics and consider how the children shared intentions, that is, how they influenced the thinking of another. By analysing the children's use of deixis as an aspect of indexicality, I examined how the students pointed out mathematical…

It might be trite to observe that every research study is framed within a cultural background. In this paper we argue that the description of the cultural background is important for discussing, evaluating and exploiting internationally the findings of local educational studies. This issue is fundamental in every study in mathematics education…

Abstraction is a key adaptive mechanism of human cognition and an essential process in the personal construction of mathematical knowledge. Based on the notion of abstraction, this paper aims to conceptualise a framework for analysing textbooks. First, I search for the meaning of abstraction from a constructive-empirical and a dialectic…

The most common distinction within research in metacognition, is between metacognitive knowledge and metacognitive skill. This distinction leads to a teaching dilemma: which aspect to prioritise? As part of an enactivist study, I analyse the development of student metacognition in one teacher's mathematics classroom. While it is possible…