David Pimm

Investigation, mathematics education and genre: An essay review of Candia Morgan's Writing Mathematically: the Discourse of Investigation

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 threatening its very existence, but is more accurately seen as opening its present to its past and to its future. (Antin, 1987, p. 479) The noun le genre is an everyday sort of word in French, meaning “kind”, “sort”, “type” or “category”. Ce n’est pas mon genre simply means “It’s not my sort of thing”. The allied English adjective ‘generic’ has some specifically mathematical overtones, while the cognate noun ‘kind’ has links to those ancient words ‘kindred’ and ‘kin’, evoking Wittgenstein’s notion of ‘family resemblance’. What kinds of mathematics are there? And what are possible bases for distinction or grouping, what are some salient features that could be stressed or ignored? One way, important both to libraries and to Mathematical Reviews, is by means of the traditional yet still-evolving categories such as ‘geometry’, ‘algebra’, ‘calculus’, ‘analysis’, and ‘number theory’ – though these can generate turbulence at the boundaries, as well as increasingly requiring hybrids: algebraic geometry, topological algebra, analytic number theory, geometric topology, and so on.

A Historical Gaze at the Mathematical Aesthetic

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)

A Tale of Two More Metaphors: Storylines About Mathematics Education in Canadian National Media

ABSTRACTIn this companion piece to the article “A Tale of Two Metaphors: Storylines About Mathematics Education in Canadian National Media” (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 focus primarily on communication acts and further our investigation into storylines and positionings concerned with mathematics education. Elaborating on aspects of two more metaphors, those of war and competition, we highlight the production of dichotomies and the shared aspect of taking sides. An investigation of the ramifications and effects that are brought to the fore in the articles we examined involves two general themes: the purpose of education and the point of mathematics education itself. We draw on David Labarees framework identifying social efficiency and social mobility as different patterns of educational goals and highlight its parallels wit...

Bakhtin and Some Dilemmas of Mathematics–Language

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 equally weighted hybrid term coined to signal their unseparateness. In the closing chapter of her 2001 book, she offers a number of questions that remain relevant fifteen years later: “If the costs of obtaining meaningful mathematical communication are so high, can they possibly be made widely available? Or does meaningful mathematics conversation as a route to mathematical learning, become, however unintentionally, the preserve of the privileged few? Expressed in more political terms: in whose interests is the dominant construction of mathematically rich and meaningful communication?” In this chapter, we explore these questions. To do so, we critically review some key ideas in Adler’s work, notably the concepts of dilemma and resource in relation to language. Our review of these ideas is informed by and elaborated through a Bakhtinian theoretical perspective.

Making Sense of Induction: Looking across International Cases

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 to go against a then-prevailing view of a child as simply a ‘little adult’. This was done in order to argue for a protected period in the life of the young human, where particular things both should and should not happen. Yet, nowadays, childhood has become a ‘natural’ category.

Being and Becoming a Mathematics Teacher: Ambiguities in Teacher Formation in France

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 teachers are commonly known: broadly, ‘probationers’ or even ‘inductees’) as beginning teachers at all.

Commentary on Knowing More Than We Can Tell

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 mathematics education opened far wider than is customary. The fact that Polanyi opts for the word ‘fact’ also puts his claim into the realm of knowledge, though Caleb Gattegno’s expression ‘a fact of my awareness’ might also be one to bear in mind. These oxymoronic or catachretic ripples—although the instance in the title of Polanyi’s (1958) more famous book Personal Knowledge may have provoked a tsunami—can make us aware even today of what is often taken-for-granted with regard to the epistemic.

Aesthetics and the ‘Mathematical Mind’

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

Sixty Years (or so) of Language Data in Mathematics Education

This chapter, based both on pre-ICME-13 conference documents as well as on the author’s actual panel presentation made at TSG 31, covers a range of themes concerned with the issues of ‘language data’ in mathematics education. It also addresses several instances from its history, including word problems, classroom language and transcription, in addition to the mathematics register, its syntax, semantics and pragmatics.

On Number Language: A Commentary on Chapter 3

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-a-vis number; place value; the (dissolving) distinction and its pertinence between count nouns and mass nouns (and the place of English-language number words in relation to this distinction); and some similarities and differences among the systems of cardinal, ordinal and fractional number words – before concluding with a (very) few summary remarks.