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…

Starting from an interest in the teachers' use of diagrams and gestures during a traditional front lesson at tertiary level, this research takes a narratologic perspective to see a mathematical lesson as a story, and hence the students' notes as re-tellings of a mathematical story. The first minutes of a traditional mathematics lecture…

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…

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…

Becoming a teacher is not developing an identity, but is developing identity as a continuous process of constructing and deconstructing understandings within the complexities of social practice, beliefs, experiences, and social norms. I take up this stance on identity as articulated in Judith Butler's (1999) work with gender identity and…

In his "Proofs and Refutations," Lakatos identifies the "Primitive Conjecture" as the first stage in the pattern of mathematical discovery. In this article, I am interested in ways of reaching the "Primitive Conjecture" stage in an undergraduate classroom. I adapted Realistic Mathematics Education methods in an…

Identifying mathematical and didactical conditions under which mathematics learners can encounter an intellectual need for defining and proving is recognized as a challenging research enterprise. This paper presents a particular configuration of conditions under which a group of pre-service mathematics teachers successfully constructed a…

Motivated by the question, "What exactly about a mathematical concept should students discover, when they study it via discovery learning?", I present and demonstrate an interpretation of discovery pedagogy that attempts to address its criticism. My approach hinges on decoupling the solution process from its resultant product. Whereas theories of…