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…

Drawing on the work of Gattegno, it is suggested that a powerful way of teaching mathematics is to introduce symbols as relationships between visible or tangible resources. The symbols are abstract (formal) from the beginning and yet there are concrete resources to support their use. Drawing on data from a research project in primary schools in…

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…

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…

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…

The notion of temporal range is introduced and discussed in this paper. Two dimensions of temporal range are identified: mathematical temporality relating to mathematical ideas, their precursors and horizons; and a mathematical learning temporality where what students say/do provides the ground on which future learning can be built. These…

This paper proposes a framework for reading mathematics texts as narratives. Building from a narrative framework of Meike Bal, a reader's experience with the mathematical content as it unfolds in the text (the "mathematical story") is distinguished from his or her logical reconstruction of the content beyond the text (the…

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…