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…

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…

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…

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…

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…

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…

This article uses Ranciere's notion of the ignorant schoolmaster and McElheny's differentiation of artist's models from those of the architect and scientist to propose the reconceptualization of mathematics education as the support of emancipated spectators and sculptors of utopia.

Story and audience form the cornerstones of the mathematics classroom work discussed in this article. Put in the simplest terms, our starting point for collaborative lesson planning is to imagine a good mathematics story to be shared by our students with family and friends and with the wider community: a story that will offer a new and wonderful…

To learn means coming to know something new at the end of, or subsequent to, a (learning) process. Because students do not yet know at the beginning of the process what they will know subsequent to the process, they cannot actively orient towards the object of learning. In this article, I propose a phenomenological perspective that theorizes…

This article describes the development of the authors' understanding of the contradictions in their mathematics teacher education practice. This understanding emerged from contrasting analyses of the impact of the authors' practices in mathematics content courses versus mathematics methods courses. Examples of the authors' work with two students,…

The purpose of this paper is to reclaim the place of specific examples. It is argued that a plausible step in experiencing the general is to start with a certain general process that can be applied to an already known and specific-for-the-learner object; the specific object is supposed to be there to be treated as a non-specific, particular…