1 Introduction

[Art is the] making well of whatever needs making. (Coomaraswamy, 1956, p. 91)

In this piece, I offer a partial response to Presmeg’s article in this issue of ZDM, and do so as someone who has worked interdisciplinarily in mathematics education for most of his career, whether drawing on cognate mathematical disciplines (its history and philosophy, as well as mathematics itself) or on ideas and occasionally on techniques from linguistics writ large. Of late, I have become increasingly interested in contemporary poetry and the way poetry deals with issues of considerable pertinence to both mathematics and mathematics education (such as its extensive use of particulars and the varied ways of alluding to the general through the particular, a core engagement with metaphor, and a determined, aesthetic focus on questions of truth and Truth)Footnote 1. These are my three main ‘others’ that I bring to mathematics education and I provide instances of each (at varying levels of specificity) in the discussion below.

In light of this experience, I pick up on certain themes that caught me in Presmeg’s paper, specifically her strong focus on theories and method as well as certainty (and also ‘rigor’), before going on to consider ‘trust’ which was only present tacitly, hovering beneath the surface. But before starting this more systematic exploration, I offer a brief discussion of the nature of disciplines and their boundaries, also one of Presmeg’s concerns.

2 On boundary disputes

Mathematics education is a particular second-order enterprise, as all forms of education are, though this need not mean it is “twice removed from the centre of events” (Stoppard, 1972, p. 36). But this fact does speak to its struggle (felt more strongly by some than others) to be both seen and taken as a separate, free-standing discipline. In addition, there is a specific inheritance from the fact that it is mathematics, a subject where rightness of method has become the customary core guarantor of truth, that is its closest intellectual kin. Geography educators, I surmise, do not experience a need for certainty in their work in the same way.

But I was struck by the fact that it was actually the methods of science rather than mathematics to which Presmeg referred: “At the same time, I acknowledge and celebrate the rigor and certainty (albeit contingent) of the methods of the sciences.” I have two things to say here. The first is that it is not the methods themselves which are deemed certain rather the results of those methods, and few if any scientists (and no historians or philosophers of science) I feel would attribute certainty to the results of scientific methods. The second is to delight in the oxymoronic quality of contingent certainty.

In relation to certainty in general, psychoanalyst Adam Phillips (1993) has asked about its origin:

The psychoanalytic question becomes not, Is that true? But what in your personal history disposes you to believe that? […] always an interesting question to ask someone in a state of conviction, What kind of person would you be if you no longer believed that? (p. 112)

It is for this reason (among others, naturally), namely the presumed conferring of certainty through method, that I feel that issues of method have risen to the top (in North America at least), far beyond their rightful place if I may so venture in the consideration of things. The preoccupation with issues of method has occurred to the detriment of consideration of the actual questions themselves and that the former should be firmly subordinated to the latter. In passing, discussion of ‘mixed methods’ immediately brings to my mind the less-comfortable notion of ‘mixed metaphors’.

In Presmeg’s early anecdote about Dena, she draws attention to questions of subject boundaries. Frequently, disputes over boundaries speak to significant issues in discipline perception. One such instance has arisen over the past thirty or so years with ethnomathematics and its relation to mathematics, not least as explored for over two decades in the journal For the Learning of Mathematics (and despite the then-editor’s occasional private reservations). It provides, I feel, a good example of how new interdisciplinary work needs to find a sufficiently accommodating home in order to flourish.

Wheeler’s (1974, 1978) own thinking around the notion of mathematisation (a word whose roots as a verb date back at least to the early eighteenth centuryFootnote 2) is significant in this regard. In his 1978 presentation to an ICMI symposium, he called for a “serious, subtle study of mathematization” (p. 155) and offered the following as a possible task mathematicians could undertake for mathematics education:

[They could] describe the mental processes that produce mathematics—i.e., help us understand how mathematics comes into being. […] In common with many others I adopt the word “mathematization” to refer to the mental processes which produce mathematics […] it can most easily be detected in situations where something not obviously mathematical is being converted into something which obviously is. (pp. 147, 149–150)

And we should not forget the most splendid methodological monster-barring retort of Paul Gordan to David Hilbert, “Das ist nicht Mathematik. Das ist Theologie.” (“That is not mathematics. That is theology”, quoted in Reid, 1970, p. 34.) Or even that Lakatos’s (1976) Proofs and Refutations can be seen in terms of a methods handbook for boundary disputes in mathematics (and elsewhere). One thing to remember is that turbulence occurs near boundary surfaces, so turbulent discussion around definitions, methods or claims may well be indicative of a boundary dispute.

3 Some notes on method and ‘methodologies’

As I mentioned above, Presmeg’s paper presumes that what is transferred between disciplines are theories (and ‘theoretical frameworks’) and methods (and ‘methodologies’Footnote 3), not least when in relation to anthropology she emphasizes “particularly with regard to methodology and construction of theory”). This has not been my experience. While I am certainly interested in the awarenesses, attentions and practices of a discipline and how they are theorized, what I have transported across various frontiers has usually been individual notions, both identifying and conceptualizing phenomena (I later give a more detailed example concerned with the linguistic notion of ‘hedging’).

I have become increasingly interested in the origins of the notion of method and, in particular, what a method is in mathematics itself. In addition, there is a small cluster of related words, whose more careful delineation and distinction might lead to some clarity in wider realms, such as mathematics education. These words (in arguably decreasing order of specificity of application) are ‘algorithm’, ‘method’, ‘technique’, ‘strategy’, ‘heuristic’ and ‘sutra’ (for a discussion of this final term, see Joseph, 1991). All of the other words have Greek etymology, and ‘strategy’ has a nice inadvertent joke sitting inside it, as its (military) gloss is ‘art of the general’. And it is worth recalling that ‘art’ has a significant sense of its own, separate from ‘the arts’, just as skill has a disappearing sense separate from objectified ‘skills’.

An etymology for the Greek word methodos as meta+hodos seems plausible, where hodos (οδος) can refer prosaically to ‘street’ or ‘road’, but also to ‘way’.Footnote 4 This is a place where research in the history of mathematics can prove more broadly informative. For example, Christianidis (2007) details the results of his careful linguistic-mathematical study of Diophantus’s Arithmetica from late Antiquity, paying close attention to Diophantus’s assertion that to solve arithmetical problems one should “follow the way [οδος] I will show” (p. 289). Christianidis sets out to examine a persistent issue in the literature on Diophantus, which concerns:

the question of whether or not Diophantus elaborated and employed a single general strategy for the treatment of arithmetical problems. To my mind, this issue cannot be adequately treated without prior clarification of another issue: the exact description and characterization of the mathematical practice of Diophantus. (p. 291)

In addition, Christianidis seeks to weigh in on a much-discussed boundary dispute about whether and when this work of Diophantus’s is algebra or arithmetic, as well as offering a cogent view on why Diophantus frequently posed a problem type in general and then only proceeded to solve a single particular instance of each type. And as this article makes clear, to speak of Diophantus’s method is to do a considerable disservice to the subtlety of precisely what this mathematician was and was not offering his readers.

Problems of particular and general beset discussions of method, not least as for some generalisation is presumed to be the only game in town (as compared with appropriability, for example, or enlightenment). In passing I will make an allusion to poems, where a relatively current aesthetic eschews generalities while cramming in particulars, allowing the one to stand for the other. Diophantus’s solutions can profitably be seen in this light.

Technique (with its resonant sense of techne, know-how or craft, a term Presmeg mentions near the end of her piece, when discussing Habermas) requires a greater art on the part of its user. Sinclair (2008) makes fine use of a distinction that could be seen as one between method and technique in the teaching around different obstetric devices and procedures (here, the Apgar scale and forceps vs Caesarian section, with forceps involving far greater technique and artistry—‘craft’ is Sinclair’s word—than Caesarian section which ‘only’ involves a method), in order to draw attention to different forms of (non-gold-standard) research in mathematics education.Footnote 5

As a fleeting final instance, I find I import notions more than ‘methods’ or ‘techniques’ into my work (though on occasion these are not as clearly distinguished as this remark might suggest). One recent exception, one with a poetic overtone, has come with my learning more about some techniques of ethnopoetics in relation to speech transcription. One instance related to this can be seen in Staats (2008) recent account of use of poetic attention (in the sense of Roman Jakobson’s poetic function of language) to the structures inherent in mathematical speech and bringing it out in the way a transcript is presented on the page. Staats claims that mathematical language is particularly susceptible to such rendering due to the significance of mathematical speech structures in carrying mathematical meaning.

4 An example from linguistics: hedging

The notion of ‘hedging’, as an explicitly identified linguistic phenomenon, dates at least back to Lakoff in the early 1970s. Lakoff (1972, p. 195) specified a hedge as a word or phrase “whose job it is to make things fuzzier”, and examples in English include maybe, perhaps, possibly, about, more or less, and many more. The applied linguist Ken Hyland (1998) has written an extensive account of hedging among professional scientists in published research articles, but to my knowledge there has been no comparable study in mathematics (I later offer a tentative reason for why this might be so).

I first came across this idea in the late 1980s in conversation with applied linguist Joanna Channell, while she was working on her doctoral dissertation on the linguistics of approximation and vague utterances (for a later version of this work, see Channell, 1994). One seminal study in the history of hedging is Prince, Frader and Bosk (1982), whose authors offer a taxonomy of hedging types based initially either on their effect on the proposition itself (approximators) or on the effect on the degree of commitment expressed by the speaker in relation to the proposition (shields). Tim Rowland, my first doctoral student, has done quite a lot of work on hedging within mathematics education (see, e.g., Rowland, 1995, 2000).

Hedging is clearly not a mathematics-specific phenomenon. It has been identified, named and explored by linguists. However, as with many linguistic phenomena, I have found there are specific features or wrinkles that occur when hedging comes into contact with mathematics and mathematics teaching and learning settings. And once again it is linguistic notions far more than methods that I have found useful to import into my work. As Halliday (1978) has claimed about languages as a whole in relation to the world-views of members of different cultures, I find in particular within English in relation to certain linguistic notions, increasingly ones from pragmatics:

languages have different patterns of meaning—different ‘semantic structures’, in the terminology of linguistics. These are significant for the ways their speakers interact with one another; not in the sense that they determine the ways in which the members of the community perceive the world around them, but in the sense that they determine what the members of the community attend to. (p. 198; italics in original)

I find this idea very resonant in terms of different academic disciplines. Each discipline, among other things, encodes and structures particular patterns of meaning, orients towards certain things in certain ways and away from others: it influences both what is attended to and how, and among other means does so through language. By importing ideas from linguistics,Footnote 6 I offer myself things to look for, to attend to in their particular mathematical formulation or embodiment. Even absences can be informative: the absence of hedging in formal published mathematics, for example, is well worth thinking about (given its extensive presence in formal published science as documented by Hyland). Without the orientation that this notion provided, I am sure I would not have noticed.

But the example goes on. It is a commonplace to remark that potentially the same phenomenon can be interpreted in different ways. But when this occurs in a significant manner, we may learn more both about the phenomenon and about the nature and process of trans-disciplinary importation.

Inglis et al. (2007) neatly indicate a way in which one needs to pay close attention to the full context of a model when importing it into mathematics education. They open their article by convincingly documenting how past authors using Toulmin’s model of argumentation within mathematics education have only included some of its components, missing out a central element which has to do, in Toulmin’s terms, with ‘modal qualification’ of the claim under consideration as well as the identified possibility of a rebuttal. They illustrate the significance of this absence by reporting conversations with individual mathematics graduate students working on a range of conjectures the researchers provided them with. The transcripts include a significant amount of hedging (primarily shields), which for me connects to Toulmin’s ‘modal qualification’ category. Inglis et al. argue convincingly to my mind that by neglecting to attend to such modal qualifiers, accounts of even these mathematically sophisticated individuals’ mathematical argumentation would be seriously deficient.

One possible reason for neglecting this aspect of mathematical language is that, unlike in science, there is almost no hedging in published academic mathematics, even though there is in spoken contexts. In his book Disciplinary Discourses: Social Interactions in Academic Writing, Hyland (2000) talks about a range of ways in which academics attend to the interpersonal, one of Halliday’s (1973) three metafunctions of language (the others being the ideational and experiential function and the textual function: for more on these in a mathematics education setting, see Morgan, 1996). Halliday glosses the interpersonal as the range of ways the author (speaker) relates to the reader (listener), “including all forms of the speaker’s intrusion into the speech situation and the speech act” (p. 41). While Hyland again documents the considerable diversity of ways in which professional scientists attend to the interpersonal in their academic writing, unfortunately for us he did not work with mathematicians and their writing.

There is a further connection that Sinclair and Pimm (2008) are starting to explore, namely among Peirce’s notion of abduction (see also Sinclair, Lee and Strickland, 2008), the modal qualifier in Toulmin’s argumentation scheme and hedging. Toulmin (even in the revised 2003 edition of his classic The Uses of Argument) makes no reference to Peirce or abduction. As there is no room here to go further into these links, I offer it simply as a way in which mathematics education as a site can offer something back to these various borrowings from philosophy and linguistics, by allowing the bringing together of what are seen as quite different things and, for me at least, offer hedging (specifically modal qualification) as a phenomenon to help identify abductions.

5 Rhetoric and poetry

We make out of the quarrel with others, rhetoric, but of the quarrel with ourselves, poetry. (Yeats, 1918, p. 21)

Mathematics education rhetoric, again arguably arising in part from its proximity to mathematics with its restricted and cramped ‘elegant’ rhetorical style (see, e.g., Csiszar, 2003), has a tendency towards the declarative, to what Bruner (1986) terms paradigmatic. Mathematics says, endlessly: This, therefore this, therefore this, therefore this. While some significant moves have occurred in relation to developing a more narrative style (as well as exploring its appropriateness for mathematics as well), one of the tensions between the arts and the sciences that Presmeg’s paper alluded to is in rhetorical style and the trust that comes with one or other form. In mathematics, our students need to learn to trust to pattern while at the same time being open to new possibilities that also might become patterns of the future. Likewise, our field is exploring new patterns of writing about phenomena that may later become established (ethnopoetic transcription might provide one very small instance).

In an extraordinary chapter ‘Lyric, narrative, memory’, Canadian poet and philosopher Jan Zwicky (2006) characterises two different ‘styles’ of poetry, narrative and lyric, in terms two different connective tissues, the latter of which I do not think we have yet seen in mathematics education. But I end with the gentlest of suggestions that we might.

Narrative addresses us as members of a tribe, it binds us together in a common mythos. Narrative makes things hang together causally; we use it to tame experience so it does not overwhelm us.

Narrative is the genre of choice for the historical treatment of memory. And then, it says. And then, and then, and then. (pp. 95–96)

Lyric attempts to listen—to remember—without constructing, without imposing a logical or temporal order on experience. This, it says. This. And this. And this. (p. 98)