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…

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…

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…

In this article, I present the notion of a set-oriented perspective for solving counting problems that emerged during task-based interviews with postsecondary students. Framing the findings within Harel's "ways of thinking", I argue that students may benefit from this perspective, in which they view attending to sets of outcomes as…

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…

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…

Many of the central ideas in an introductory undergraduate linear algebra course are closely tied to a set of interpretations of the matrix equation Ax = b (A is a matrix, x and b are vectors): linear combination interpretations, systems interpretations, and transformation interpretations. We consider graphic and symbolic representations for each,…

We analyze the role of generic proofs in helping students access difficult proofs more easily and naturally. We present three examples of generic proving--an elementary one on numbers, a more advanced one on permutations, and yet more advanced one on groups--and consider the affordances and pitfalls of the method by reflecting on these examples. A…

Research in ethnomathematics has become predominantly focused on "local cultures" and non-scholarized forms of mathematics, thus becoming less a critical reflection on the sociopolitical roots of academic mathematics and the place it occupies in the popular imagination and in schooling, and more of a learning device. Such a development…

Findings from a 1.5 year study of black adolescent mathematics students attending an African-centered school in the US are used to highlight the benefits of separate schooling for this population of students. Critical race theory is used to frame a dialogue surrounding the ways in which this type of school environment and embedded racialized…

Characterizing how quantities change (or vary) in tandem has been an important historical focus in mathematics that extends into the current teaching of mathematics. Thus, how students conceptualize quantities that change in tandem becomes critical to their mathematical development. In this paper, we propose two images of change: chunky and…

The limit concept is a fundamental mathematical notion both for its practical applications and its importance as a prerequisite for later calculus topics. Past research suggests that limit conceptualizations promoted in introductory calculus are far removed from the formal epsilon-delta definition of limit. In this article, I provide an overview…