David John Pimm

Simon Fraser University · Faculty of Education

Gestures, whether derived from teachers/lecturers or students in relation to mathematics education, are becoming more and more present in our field. Our focus here is on concept gesture, which can also be seen as a body metaphor, where (part of) the body attempts to enact or depict the mathematical concept dynamically, both somewhat idiosyncratical...

In this article, we present a narrative review of mathematics education research on language and on communication over 2019–2022, but also look ahead by addressing challenges posed by the lack of distinction between language and communication. The persistence and significance of the problem of the distinction between language and communication are...

This short chapter is in response to my reading of all the other chapters in section 2 of this book, including Rina Zazkis’s introduction. It is certainly not an evaluation (as in a review, though I did cite a couple of reviews), nor a description (as to which Rina’s introduction relates) of these chapters (though I did mention something from each...

Not often cited as one of the countries making major contributions to new/modern math movements in school mathematics, the case(s) in Canada may seem of lesser interest to the international community. However, in this chapter, we show how the multiple forces acting on school mathematics curricula across the country led to a slow brew of quiet chang...

The development of touchscreen technology is providing alternative ways for learners to conceptualise, visualise, experiment with and communicate about mathematical ideas and relationships. While the multi-touch affordances of touchscreens enable children to produce and transform ‘screen objects’ with their fingers (by means of varied forms of pres...

This chapter primarily offers a commentary on Chap. 3, before moving off at the end into some wider issues. I have organized my comments under four broad headings – links among what is said, written and gestured vis-à-vis number; place value; the (dissolving) distinction and its pertinence between count nouns and mass nouns (and the place of Englis...

While the section title focuses on inclusion, what caught my attention most when reading the chapters that constitute this section was the variety of foci and voices employed in them, not least in regard to making claims or suggestions about practices, about teachers and teaching, and about students and learning. And about various aspects of risk,...

Responding to widespread interest within cultural studies and social inquiry, this book addresses the question 'what is a mathematical concept?' using a variety of vanguard theories in the humanities and posthumanities. Tapping historical, philosophical, sociological and psychological perspectives, each chapter explores the question of how mathemat...

In this companion piece to the article ?A tale of two metaphors: Storylines about mathematics education in Canadian national media? (see this issue), we further explore constructed meanings through the use of positioning theory. In our examination of 71 articles in the two Canadian national newspapers (The Globe and Mail and The National Post), we...

This chapter comprises a conversation between the two authors focused on half-a-dozen core words concerned with teaching and their mutual interaction: dilemmas, tensions, commitments, obligations, ambivalence and, indirectly, technique. Additionally, we engage Jill Adler in an indirect conversation through her writing on the dilemmas teachers face...

Throughout much of her work, Jill Adler’s abiding interest has lain in the political implications of language in practice in mathematics classrooms, not solely because of the cultural importance ascribed to success in mathematics, but also because of there being some specific interactions of significance to be found within mathematics–language, our...

TouchCounts is a novel iPad application, one which makes full use of its multi-touch affordance to engage young children in aspects of the cultures of counting and adding/subtracting, by means of engagement with the combined sensory modalities of the visible, the audible and the tangible. Drawing on various excerpts with children aged three to six...

This paper takes the other papers in this issue as ‘data’—in other words, as a source of observation and comment—and then proceeds to further discussion, analysis and surmise. By focusing in substantial measure on the notions of authority, explanation, contention and register, ideas which occur both explicitly and tacitly in different articles here...

Traducción al español y edición: Cecilia Agudelo Valderrama agudelo.cecilia@gmail.com El desarrollo del pensamiento algebraico – particularmente la identificación y expresión de regularidades – está al alcance de todo estudiante (todo ser humano), y es vital para su activa participación como ser social y como ciudadano productivo. Esta obra constit...

Discourse practices warrant the attention of mathematics educators because discourse is the primary medium of education. Evidence about one’s hopes or expectations can be found in discourse practice whether the goal is performance in mathematical procedures, creativity in problem solving, or a classroom environment that uses the diversity of voices...

In The Tacit Dimension, Michael Polanyi declares, “I shall reconsider human knowledge by starting from the fact that we can know more than we can tell.” (1966, p.4; italics in original). I am struck by the directness with which he unobtrusively asserts the existence of the arena which Sinclair’s paper addresses and expands upon, with respect to a m...

What is the role of mathematics in the secondary classroom?What is expected of a would-be maths teacher? How is mathematics best taught and learnt? Learning to Teach Mathematics in the Secondary School combines theory and practice to present a broad introduction to the opportunities and challenges of teaching mathematics in modern secondary school...

This paper starts from some observations about Presmeg’s paper ‘Mathematics education research embracing arts and sciences’ also published in this issue. The main topics discussed here are disciplinary boundaries, method and, briefly, certainty and trust. Specific interdisciplinary examples of work come from the history of mathematics (Diophantus’s...

This article discusses issues of geometrical awareness by means of an exploration of the concepts of area and volume drawing on a historical theorem (now-named Cavalieri's principle) due to the seventeenth-century mathematician Bonaventura Cavalieri. We draw ideas from and work through this principle to illustrate insightful geometrical awareness t...

Although Imre Lakatos described the work published in his book Proofs and Refutations as a study of mathematical methodology, work which has been responded to and criticized by philosophers and historians of mathematics more on its own terms, a significant body of writing in the 30years since its appearance has used it as a pertinent cognate text a...

Italo Calvino (1992), con l’intento di offrire sei proposte sull’arte della scrittura, espone in questo paragrafo un argomento a favore della visibilità. Egli mette in contrasto il visibile con lo scritto, l’immagine concepita con l’espressione verbale. Il suo argomento non riguarda la scelta di uno piuttosto che dell’altro, ma riguarda invece il f...

This opening article of the Special Issue makes an argument for parallel definitions of scientific literacy and mathematical literacy that have shared features: importance of general cognitive and metacognitive abilities and reasoning/thinking and discipline-specific language, habits-of-mind/emotional dispositions, and information communication tec...

Listen, reader - the clock is ticking. Have you shored up your days against pleasure and the strange? (Mark Cochrane, in Cran, 2002, p. 82)

Studies on the foundations of mathematics and mathematical method should make substantial room for psychology, indeed even for the aesthetic. (Henri Lebesgue, 1941, p. 122)

No matter how correct a mathematical theorem may appear to be, one ought never to be satisfied that there was not something imperfect about it until it gives the impression of also being beautiful. (George Boole, in MacHale, 1993, p. 107)

The essays in this book explore the ancient affinity between the mathematical and the aesthetic, focusing on the fundamental connections between these two modes of reasoning and communicating. From historical, philosophical and psychological perspectives, with particular attention to certain mathematical areas such as geometry and analysis, the aut...

This is a book all mathematics teachers and teacher educators should read! It brings together a wealth of insights from a range of authors! The major issues confronting teachers of mathematics who wish to use ICT in different domains of mathematics are addressed in a clear and accessible way." Professor Celia Hoyles OBE, Dean of Research and Consul...

The viability of a genre like the viability of a family is based on survival, and the indispensable property of a surviving family is a continuing ability to take in new members who bring fresh genetic material into the old reservoir. So the viability of a genre may depend fairly heavily on an avant-garde activity that has often been seen as threat...

We began this book with a claim that induction represents a phase and not simply a program. In arguing for recognizing induction as a unique period, we are reminded of the lessons of Aries (1996) and others about childhood. That research demonstrated how ‘childhood’ came to be constructed as a category during the nineteenth century, created in part...

For a cultural outsider to shadow a French secondary mathematics teacher for a week during the first year of teaching is perhaps repeatedly to be surprised. The work required is varied and not even predominantly based in a single institution, a school. Indeed, at first sight, it may seem an error even to consider stagiaires (as all first-year teach...

Beginning to teach involves both starting a new job and entering a new way of life. There may be more or less ceremony (of welcome or initiation) at the school where one starts; there may be formal or informal procedures intended to aid or ease the challenges of this beginning. There may be nothing in evidence at all. Whether officially inducted or...

This book provides a detailed examination of how systems located within five countries shape the early career learning of beginning teachers. It describes, discusses and analyzes comprehensive teacher induction found within France, Japan, New Zealand, Shanghai and Switzerland. We refer to the phenomena we observed as induction ‘systems’ because the...

Concrete materials have a long history in the mathematics classroom, although they have not always been readily accepted or used appropriately. They disappeared when written computational methods arose and little premium was placed on understanding the algorithms being learned. Comenius and Pestalozzi began the process of reintroduction, with Monte...

This article offers an extended essay review of a major published book whose title specifies an area:Advanced Mathematical Thinking. Each chapter is discussed, connections and reactions detailed, and the response set in a context both of research in mathematics education and of writingabout mathematics.

Symbols and Meanings in School Mathematics explores the various uses and aspects of symbols in school mathematics and also examines the notion of mathematical meaning. It is concerned with the power of language which enables us to do mathematics, giving us the ability to name and rename, to transform names and to use names and descriptions to conju...

One major component of an educational researcher’s task is to document and understand the world of schools, teaching and learning: in the case of mathematics education researchers, that of mathematics classrooms. Using metaphors is one way of undertaking this complex task. For instance, at the most general level, when looking at discourse generated...

This paper is a rejoinder to John Wilson's piece “Power, Paranoia, and Education” which appeared inInterchange in 1991. Following a brief summary of Wilson's piece, I criticize his article under two major headings: some reactions to his psychological claims and terminology, and Wilson's style and form of argument. I conclude by offering some altern...

This paper explores some of the ambiguities inherent in the notions of generality and genericity, drawing parallels between natural language and mathematics, and thereby obliquely attacking the entrenched view that mathematics is unambiguous. Alternative ways of construing 2N, for example, suggest approaches to some of the difficulties which studen...

In order to examine some of the problems and benefits of using a microcomputer as a classroom teaching aid, we observed 174 school lessons during which 17 teachers employed a microcomputer regularly with a chosen class for a whole term. It is argued that the ergonomic factors here differ considerably from those in other man/machine interactions. Pr...

The following is intended as an analysis of a specific conceptual change in mathematics during the early 19th century. From this some pedagogical conclusions are drawn. Analyzing a special example seems to be a more appropriate way of handling the relevance of historical and philosophical considerations for mathematical education than dealing with...

One of the more taxing questions implicated in the complex interrelationship between language and mathematics has to do with the shaping of form by content and of content by form. One of the less considered aspects of this mutual influence has to do with the nature and influence of the audience for the language, especially written mathematical lang...

Throughout recorded history, devices have been created in order to assist with the doing of mathematics, especially for the carrying out of algorithmic computations and the coming up with (and holding by means of tables) the values of particular functions. Such devices have ranged from ancient Babylonian table texts (e.g. of multiples, squares or s...

This working group on mathematics classroom discourse will focus attention on the specifically mathematical characteristics of discourse in mathematics classrooms. Participants will work together in small groups to respond to various artifacts from mathematics classroom discourse. In large-group discussion, we will hear from the small groups and wo...