Researchers’ descriptions and the construction of mathematical thinking
Barwell R. (2009) Researchers’ descriptions and the construction of mathematical thinking. Educational Studies in Mathematics 72(2): 255–269. Available at http://cepa.info/3731
Table of Contents
1 Discursive psychology: a theoretical and methodological perspective
1.1 Descriptions and facts
2 Example 1: Mathematical thinking in Sfard’s (2001) paper
2.1 The construction of ‘cognitive conflict’
2.2 The construction of ‘discursive conflict’
2.3 A higher order contrast structure
3 Example 2: Mathematical thinking in Gray and Tall’s (1994) paper
5 Implications and concluding remarks
Research in mathematics education is a discursive process: It entails the analysis and production of texts, whether in the analysis of what learners say, the use of transcripts, or the publication of research reports. Much research in mathematics education is concerned with various aspects of mathematical thinking, including mathematical knowing, understanding and learning. In this paper, using ideas from discursive psychology, I examine the discursive construction of mathematical thinking in the research process. I focus, in particular, on the role of researchers’ descriptions. Specifically, I examine discursive features of two well-known research papers on mathematical thinking. These features include the use of contrast structures, categorisation and the construction of facts. Based on this analysis, I argue that researchers’ descriptions of learners’ or researchers’ behaviour and interaction make possible subsequent accounts of mathematical thinking.
Key words: Mathematical thinking, research methodology, discursive psychology, contrast structures, descriptions, epistemology
One of the primary concerns of research in mathematics education is to understand various aspects of mathematical thinking, including mathematical learning, knowing, understanding, meaning, remembering, feeling, and so on. Research, however, is a discursive process: Research in mathematics education, for example, involves the production and interpretation of various kinds of spoken and written texts, such as interviews, classroom observations, transcripts, conference presentations or journal articles. Theories, analyses or discussion of mathematical thinking are part of this discursive process. In this paper, I highlight some of the discursive through which mathematical thinking is discursively constructed by examining examples from published research reports.
Discursive accounts of mathematical thinking are being increasingly well established (e.g. Kieran, Forman & Sfard 2001; Moschkovich 2003; Sfard 2008; see Barwell 2008, for a more comprehensive review). Much of this work shares the influence of Vygotsky, including the idea of language as a cultural tool, what Wertsch (1991) calls ‘mediational means’, and the idea of a dialectic discourse-based relationship between individual and social. More recently, however, some, including Sfard (2008), have gone beyond this position to argue that cognition is a form of communication, that “interpersonal communication and individual thinking are two faces of the same phenomenon” (Sfard 2008, p. 262). This position suggests an increasing focus on discourse itself, rather than on discourse as a way to understand specific examples of cognition.
Moving towards an at least partially discursive account of mathematical thinking leads, however, to some tricky methodological issues. In particular, since all data arises from some kind of discursive activity (interviews, observations of classroom interaction, written tests, etc.), it is not possible to simply read off what learners are thinking from what they say or do. As Lerman (2001) argues:
…classroom discourse does not offer the observer a window on the mind precisely because ‘mind’ is not static, or decontextualised, but responds to the context, the activity, and power/knowledge, and is oriented to communicate and to act. The tools for analysis, then, must view the data as student-in-mathematics-classroom-in-student in a holistic approach, taking into account both structure and agency. (p. 108)
Whilst most researchers would agree with the point that classroom discourse, or any data for that matter, do not provide a simple ‘window on the mind’, in most research, such data are nevertheless treated as relatively transparent; interpretation of discourse is dealt with as a technical issue (e.g. through triangulation). In the above quote, for example, the researcher appears as ‘the observer’ or, implicitly, as the analyst who ‘views’ the data. For me, there is a danger that ‘observing’ and ‘viewing’ have an air of ‘apartness’. The student-in-mathematics-classroom-in-student is doing the thinking whilst the researcher is merely observing. But what does it entail, this observing? Surely, if learners’ mathematical thinking is seen as a discursive process, then so too must researchers’ thinking. Observing or viewing are cognitive processes and, as such, subject to the same general theoretical ideas as the cognitive processes involved in mathematical thinking. Doing research in mathematics education, then, is as much a discursive process as doing mathematics. As Walkerdine (1988) has argued:
The central concepts in […] mathematics education may themselves be regarded as signifiers, that is, aspects of discourse. […] what is crucial about this analysis is that language and cognition become similarly amenable to the analysis. (p. 202, original emphasis)
And whilst Walkerdine made her remarks over 30 years ago, there has been little detailed analysis of how the discourse we use as researchers constructs mathematical thinking. In this paper, therefore, my aim is to examine some of the specific discursive practices through which mathematical thinking is constructed. I will focus on the published research report, and in particular, journal articles, since such reports are the public face of research and are, in some senses, definitive versions of our work. My conclusions, however, are generally applicable to other parts of the research process, although the nature of some of the discursive practices may vary.
As with classroom discourse, journal articles are not ‘windows’ on the minds of researchers or indeed on the minds of the participants in the research. They are carefully constructed texts, one of the purposes of which is to propose a plausible account of a piece of research, including various accounts of what happened and of participants’ mathematical thinking. In what follows, I present an analysis of two research papers, selected in part because they draw on slightly different perspectives on mathematical thinking. My analysis is designed to be illustrative, rather than exhaustive, allowing me to expose some of the ways in which mathematical thinking is constructed. This analysis is based on ideas developed in discursive psychology, which are summarised in the next section.
1 Discursive psychology: a theoretical and methodological perspective
At the heart of discursive psychology (e.g. Edwards 1997; Edwards & Potter 1992) is a simple observation. Whilst academic psychologists are busy trying to understand how people think, the people about whom they are theorising are busy doing more or less the same thing. Making sense of what people mean or think or feel is one of the most basic aspects of human social life.1 Most people, however, do not base this sense making on formal theories. Since it is impossible to directly experience other minds, moreover, the sense-making process is necessarily discursive. Discursive psychologists, therefore, set out to investigate how ‘most people’ go about interpreting each other’s mental states. This anti-cognitivist position entails a shift from a focus on ‘what happens in the mind’ (as an individual mental process) to how ‘what happens in the mind’ is constructed through discursive practice:
The emphasis in discursive psychology is on the publicly available social practices which constitute the psychological. Much work (following Wittgenstein) has focused on the discursive practices or language games which determine how mental predicates […] gain meaning as recognisable activities […] Through looking at how people talk about mental states, researchers can therefore study […] the criteria and practices a community develops and through which it recognises and constitutes its psychological life. (Wetherell 2007: 664; see also Edwards 1997; Potter 2003)
From this perspective, the nature of mathematical thinking is jointly produced and developed through interaction, including through written text, within a community. It follows that discursive psychology entails an anti-realist perspective:
Discursive psychology took, for the most part, an anti-realist position, in line with its constructionist allegiances, rejecting simple, naïve realist or correspondence models of language. Language, it was maintained, does not act like a mirror faithfully reflecting the world, and, most importantly for psychology, there is no easy route through self-description to the true nature of worlds and minds beyond. (Wetherell 2007: 663)
From this perspective, language is not seen a means through which psychological activities like thinking or knowing or experiencing the world are reported. Rather, discourse is intimately involved in these activities. Accounts of reality, for example, are shaped and interpreted to suit particular, on-going circumstances. Since these accounts are discursively produced, their meaning is produced through interaction as described above. Reality is reflexively (and so relativistically) constituted through interaction. Rather than mathematical meaning, for example, being pre-determined by words, symbols or diagrams, participants read such meanings into these things through their interaction (Edwards & Potter 1992; Edwards 1997).
As Edwards (1997: 48) points out, this perspective is to some degree related to socio-cultural approaches to psychology. Both approaches recognise the central role played by social processes, culture and language in the development of the human mind. For much research influenced by sociocultural theory, however, the aim is to understand how the mind works, even if mind is constructed through participation in society (see, for example, Lerman 2001). Discursive psychology, by contrast, is more interested in how ideas like ‘mind’ are discursively constructed in particular situations (Edwards 1997 2006; Potter 2003). The difference, Edwards argues, is broadly between ontological and epistemological concerns:
In discursive psychology, the major sense of ‘social construction’ is epistemic: it is about the constructive nature of descriptions, rather than of the entities that (according to descriptions) exist beyond them. (Edwards 1997: 47-48)
Discursive psychologists are not interested in developing theories of what mathematical thinking is really like (i.e. its ontology); they are interested in how mathematical thinking is discursively constructed in situated interaction – in simple terms, in how people talk or write about mathematical thinking. The researcher’s focus is then on the discursive practices that are used to do this.
There is a substantial literature in discursive psychology devoted to the identification and investigation of the various discursive practices through which mental states are constructed in interaction.2 Rather than attempt to summarise all of this work, I will focus on two related features of discourse: descriptions and facts. These two features are two of the most basic elements of human interaction, wherever it arises.
1.1 Descriptions and facts
One of the functions of description in human interaction is to construct thinking. The idea that thinking is constructed by participants through descriptions does not mean that participants are continually describing what they are thinking or what other people are thinking (though they may often do this). Rather, it is to say that descriptions (of people, events, ideas, places etc.) necessarily also involve constructing thinking. Edwards (2006), for example, discusses a police interview, in which a police officer is interrogating someone who had admitted punching and smashing a car window. In the extract discussed by Edwards, the police officer and the accused work to establish ‘what happened’, i.e. to come up with a description of what the accused did. The extract begins:
P: So a- as you punched the window, have y’wanted t’put the window through?A: I dunno. I didn’t think of that. I jus’ punched it.(Edwards 2006: 44, transcription simplified)
The police officer proposes a description that is partially rejected by the accused. The difference concerns what A was thinking at the time of the incident (‘have y’wanted’ vs. ‘I didn’t think’). The significance of this difference in this situation, Edwards argues, is that A’s intention (a mental state) has legal relevance. Intentional criminal damage is a more serious offence than reckless or careless behaviour. This example highlights how descriptions are highly variable; in this case, two versions are proposed in two turns of talk, but many others could be imagined. The choice of ‘punched’, for example, as opposed to ‘smashed’ or ‘broke’, signals a concern with the action (of A’s fist) rather than the outcome (a broken car window). This is a discussion about punching, with what is eventually constructed as an unintended outcome, rather than a discussion about smashing, with intentionality already implicit in the choice of verb. Hence, descriptions are not neutral accounts of behaviour or incident; they serve to construct participants’ mental states, both explicitly (‘I didn’t think’) and implicitly (the use of ‘punched’ rather than ‘smashed’). It is apparent from this example, moreover, that descriptions are closely bound up with the construction of facts.
From the perspective of discursive psychology, at issue in the case of facts is not whether they are true or not but how they are constructed to seem that they must be true. Consideration of a newspaper or television news reports, for example, quickly suggests several ways in which facts may be constructed and supported. Such methods include the construction of eye-witness reports, independent corroboration, telling details and the use of an objective style of description (see Edwards & Potter 1992). Smith (1978), in an analysis of how a young woman, K, comes to be identified by her friends as mentally ill, goes further:
The actual events are not facts. It is the proper procedure for categorizing events which transforms them into facts. A fact is something which is already categorized, which is already worked up so that it conforms to the model of what that fact would be like. (p. 35)
Smith shows how her informant’s descriptions of K’s circumstances are carefully designed (unconsciously, presumably) to bring about the apparently uncontroversial fact that she is mentally ill. Therefore, descriptions of K’s behaviour, for example, are designed to conform to a model of what mental illness is like. These descriptions include what Smith calls contrast structures, through which K’s behaviour is constructed as abnormal through first establishing her friends’ behaviour as normal. Smith supports her argument by proposing how alternative forms of description would lead to a rather different ‘fact’, namely that K has been ostracised by her ‘friends’ and is upset. Her point is not necessarily to dispute her informant’s version of events but to highlight how descriptions and facts, and in particular their design, are crucial in constituting what is the case, why people act in the way that they do, and what they are thinking when they act. These last three points are, of course, central concerns in research into mathematical thinking. In the next sections of this paper, I will discuss aspects of two journal articles concerning mathematical thinking. I will argue that researchers’ descriptions and reporting of facts are key elements in the construction of children’s mathematical thinking.
2 Example 1: Mathematical thinking in Sfard’s (2001) paper
The first paper I have selected, by Sfard (2001), argues that a ‘traditional’ cognitivist, learning-as-acquisition approach to research into mathematical thinking has “left many important problems unresolved” (p. 13) and then proposes an alternative “metaphor of thinking-as-communicating” (p. 13). Sfard sees these two approaches as complementary (p. 49). I have selected this paper in part because it has an interesting rhetorical structure, involving the explicit comparison of two different perspectives, drawing on empirical data to illustrate the comparison. This approach makes more salient the way descriptions and facts are used to construct mathematical thinking in different ways. One strand within the paper involves the presentation and discussion of an exchange between a pre-service teacher and a 7-year-old girl (p. 19):
Teacher: What is the biggest number you can think of?
Teacher: What happens when we add one to million?
Noa: Million and one.
Teacher: Is it bigger than million?
Teacher: So what is the biggest number?
Noa: Two millions.
Teacher: And if we add one to two millions?
Noa: It’s more than two millions.
Teacher: So can one arrive at the biggest number?
Teacher: Let’s assume that googol is the biggest number. Can we add one to googol?
Noa: Yes. There are numbers bigger than googol.
Teacher: So what is the biggest number?
Noa: There is no such number!
Teacher: Why there is no biggest number?
Noa: Because there is always a number which is bigger than that?
As the paper proceeds, Sfard first discusses this exchange from what she describes as a cognitivist perspective, before later proposing an alternative account presented from a participationist perspective. I will first discuss Sfard’s cognitivist account:
Clearly, for Noa, this very brief conversation becomes an opportunity for learning. The girl begins the dialogue convinced that there is a number that can be called ‘the biggest’ and she ends by emphatically stating the opposite: “There is no such number!” The question is whether this learning may be regarded as learning-with-understanding, and whether it is therefore the desirable kind of learning.
To answer this question, one has to look at the way in which the learning occurs. The seemingly most natural thing to say if one approaches the task from the traditional perspective […] is that the teacher leads the girl to realize the contradiction in her conception of number: Noa views the number set as finite, but she also seems aware of the fact that adding one to any number leads to an even bigger number. These two facts, put together, lead to what is called in the literature ‘a cognitive conflict’ […] and thus call for revision and modification of her number schema. This is what the girl eventually does. On the face of it, the change occurs as a result of rational considerations, and may thus count as an instance of learning with understanding.
And yet, something seems to be missing in this explanation. Why is it that Noa stays quite unimpressed by the contradiction the first time she is asked about the number obtained by adding one? Why doesn’t she modify her answer when exposed to it for the second time? Why is it that when she eventually puts together the two contradicting claims – the claim that adding one leads to a bigger number and the claim that there is such a thing as the biggest number – her conclusion ends with a question mark rather than with a firm assertion? Isn’t the girl aware of the logical necessity of this conclusion? (Sfard 2001: 19)
These paragraphs give a description of what happened in the conversation. The description constructs various aspects of mathematical thinking on the part of Noa. The key claim, however, is that Noa is experiencing ‘cognitive conflict’. Several discursive forms contribute to the establishment of the claim.
2.1 The construction of ‘cognitive conflict’
First, a number of contrast structures are apparent. In her paper about K, Smith (1978) defines contrast structures as “those where a description of K’s behaviour is preceded by a statement which supplies the instructions for how to see that behaviour anomalous” (p. 39). A similar kind of structure can be seen in Sfard’s description, in which descriptions of Noa’s behaviour are preceded by a statement that supplies instructions for how to see that behaviour as cognitively anomalous. A key example in the above extract consists of the following two parts:
The girl begins the dialogue, convinced that there is a number that can be called ‘the biggest’
and she ends by emphatically stating the opposite: ‘There is no such number!’
The first part of this structure is a statement that Noa is convinced, a description of her cognitive state. This description is presented as fact, although a number of alternative formulations can be imagined. The statement that Noa is convinced tells us how to interpret the next part of the description; it sets up the contrast that arises in the second part of the structure, in which she is described as ‘emphatically stating the opposite’ (my emphasis). Again, the choice of ‘emphatic’ implies a cognitive state, in the form of conviction or certainty. The structure also involves a related contrast, between ‘a number that can be called the biggest’ and ‘there is no such number’. It is the organisation of Sfard’s description in the form of a contrast structure that constructs Noa’s responses as (a) contradictory and (b) illogical, both readings that are taken up later in the text. Indeed, Sfard needs to establish a contradiction, in order to be able to apply the description ‘cognitive conflict’. Finally, the structure involves a temporal contrast, between ‘begins’ and ‘ends’.
Second, Sfard’s description constructs a variety of facts. Notably, learning is constructed as a fact. The contrast structure mentioned above is presented as evidence for learning, since, by implication, a change in Noa’s thinking between beginning and ending means that learning has taken place. The choice of words like ‘begins’ and ‘ends’ are part of the description, however. We are not party to what happened before or after the reported exchange, whether immediately or over a longer period. The use of ‘begins’ and ‘ends’ to enclose the episode contributes to the establishment of learning as a fact, subsequently referred to unproblematically as ‘this learning’, ‘the learning’ or ‘the change’. The construction of facts also arises in relation to mathematics. Sfard describes, in another contrast structure, how:
Noa views the number set as finite,
but she also seems aware of the fact that adding one to any number leads to an even bigger number.
She then refers to ‘these two facts’, a description of her own description. There is some subtle construction of mathematical thinking involved in these constructed facts, however, since to refer to them as facts is to imply something about Noa’s understanding of these ideas.
Third, the description relies on the preceding transcription of the discussion. As Ochs (1979) long ago observed, transcriptions are not neutral descriptions; they embed various theoretical assumptions about the nature of interaction. Sfard’s subsequent prose description of the exchange relies on the transcript to support its facticity. The use of the word ‘emphatic’, for example, is linked to the presence of an exclamation mark in the transcript. Similarly, Noa is described (in a curious kind of synaesthesia) as ending her conclusion ‘with a question mark rather than with a firm assertion’, a description that relies (literally) on the presence of a question mark in the transcript. More subtly, Sfard must rely on the transcript to support her claim that Noa is convinced that there is a number that can be called ‘the biggest’. Given that Noa appears to contribute short, often single-word responses to the opening questions, Sfard is relying on more than Noa’s utterances. Her description also relies on the sequence of questions attributed to the teacher. This point is apparent when Sfard refers to Noa staying “quite unimpressed,” the sense of ‘staying’ being linked to the repetition of the questions, as well as constructing a sense of stability in Noa’s thinking.
Fourth, Sfard’s description constructs Noa’s behaviour as that which needs accounting for; the teacher’s behaviour is barely referred to, is hence unmarked and is, as such, treated as normative (see Smith’s similar argument in relation to the construction of K as mentally ill). It would not be unreasonable to describe the same exchange in such a way as to treat the teacher’s behaviour as needing to be accounted for; Noa’s responses could be described, for example, as reasonable in the face of repetitive, somewhat obsessive questioning on the part of the teacher. This last point is not intended to imply a more correct description but to show how other, sometimes opposing descriptions may plausibly have been made.
2.2 The construction of ‘discursive conflict’
Later in the same article, Sfard gives a second description of the same exchange between Noa and her teacher, which she characterises in terms of a participationist perspective. This later description uses the same kinds of discursive forms as those I have discussed above, as I will illustrate, though more briefly than before. Here is an extract from the later description:
[…] much of what is happening between Noa and Rada may be explained by the fact that unlike the teacher, the girl uses the number-related words in an unobjectified way. The term ‘number’ functions in Noa’s discourse as an equivalent of the term ‘number-word’, and such words as hundred or million are things in themselves rather than mere pointers to some intangible entities. […] both interlocutors seem interested in aligning their positions. The teacher keeps repeating her question about the existence of ‘the biggest number’, thus issuing meta-level cue signaling that the girl’s response failed to meet expectations. In order to go on, Noa tries to adjust her answers to these expectations, and she does it in spite of the fact that what she is supposed to say evidently does not fit with her use of the words the biggest number. (Sfard 2001: 46, original emphasis)
Again, the description makes use of contrast structures. Consider, for example, the three-part contrast structure:
Much of what is happening between Noa and Rada may be explained by the fact that
unlike the teacher,
the girl uses the number-related words in an unobjectified way.
The first part of the structure sets up the rest as an explanation, so (again) constructing Noa’s behaviour as needing to be explained. It also constructs the subsequent parts as being a ‘fact’. The second part introduces the ‘standard’ behaviour, setting up Noa’s behaviour in contrast to the teacher, whose behaviour is again treated as normative. The third part describes what Noa was doing, to be read in the light of the previous two parts. Hence, Noa’s use of “number-related words in an unobjectified way” should be seen as explaining something or at least as preparatory to some such explanation, as well as being non-standard in contrast to the teacher’s use of the same kind of words. This description makes possible Sfard’s subsequent explanation of Noa’s behaviour in terms of “the clash of habitual uses of words” (p. 48).
Similarly, the use of facts and the basis for the description in the transcript are apparent. The second part of the above extract, for example, gives an account of the conversation in largely discursive terms, that is, in terms of cues, repeats and alignments. Even in this description, however, a degree of intention is read into Noa’s behaviour: She “tries to adjust her answers,” for example (my emphasis). This discursive reading is then overlaid, with a more cognitively oriented account of ignorance, presumptions, motivations and thinking:
The notion of discursive conflict stresses the clash of habitual uses of words, which is an inherently discursive phenomenon […] While aware of the fact that the teacher was applying these terms in a way quite different from her own, Noa was ignorant of the reasons for this incompatibility. In this case, therefore, the girl had to presume the superiority of her teacher’s use in order to have any motivation at all to start thinking of rational justification for a change in her own discursive habits. (Sfard 2001: 48, original emphasis)
Again, then, it is the nature of the earlier description that makes possible the inferences about Noa’s mathematical thinking.
2.3 A higher order contrast structure
In order to understand what, discursively, is going on in Sfard’s account, however, the relation between the first and second versions needs to be understood. In effect, the two versions form another contrast structure, though broader in scope than the sentence level examples referred to so far. This contrast structure can be summarised as follows:
Description of the exchange in ‘cognitive’ terms, making possible the inference of“Something seems to be missing in this [cognitive] explanation” (p. 19)
Description of the exchange in ‘participationist’ terms making possible the inference of ‘discursive conflict’.
In this structure, the initial description already discussed above is problematised through a series of questions:
And yet, something seems to be missing in this explanation. Why is it that Noa stays quite unimpressed by the contradiction the first time she is asked about the number obtained by adding one? Why doesn’t she modify her answer when exposed to it for the second time? Why is it that when she eventually puts together the two contradicting claims – the claim that adding one leads to a bigger number and the claim that there is such a thing as the biggest number – her conclusion ends with a question mark rather than with a firm assertion? Isn’t the girl aware of the logical necessity of this conclusion? (Sfard 2001: 19, original emphasis)
It is notable that all these questions are about Noa’s mathematical thinking, which is broadly characterised as somewhat baffling. The second, later, participationist description must be read through the filter of the first two parts of the broad contrast structure. This later description is not a neutrally different one, an alternative that happened to be available; it must be read as dealing with the ‘something’ that is missing. In particular, Sfard seeks to propose the notion of discursive conflict as an explanation, a rationalisation, of Noa’s behaviour. A key feature of the description is the idea that Noa is interpreting the word ‘number’ as “an equivalent of the term ‘number word’” (p. 46). Based on this description, Sfard is able to provide a rational account of Noa’s utterances. Indeed, the later part of the above paragraph is devoted to setting out the linguistic and mathematical basis for that rationality, which amounts to a reading of Noa’s mathematical understanding in the earlier stages of the exchange. Thus, the nature of the description is intimately related with the argument that Sfard is pursuing. By setting out a particular version of what is happening in the conversation, Sfard makes available particular inferences about Noa’s mathematical thinking, which in turn fit in with Sfard’s larger argument that the conversation also represents an example of discursive conflict.
3 Example 2: Mathematical thinking in Gray and Tall’s (1994) paper
The two examples discussed in this paper are designed to illustrate my argument that researchers’ descriptions contribute to the construction of the mathematical thinking they report. The preceding section on Sfard’s paper has, to a large extent, already provided such illustration. My argument is, however, a general one; I am not highlighting something that is peculiar to Sfard. It is therefore useful to consider a second example briefly, in order to offer some sense of how these discursive forms might appear in a different context. I have selected the paper by Gray and Tall (1994) for this purpose. First, Gray and Tall draw on a slightly different perspective on mathematical thinking from Sfard (2001). Second, they also include some basic quantitative data, thus illustrating how the role of description is not restricted to qualitative research presented in prose form.
In their paper, Gray and Tall (1994) use the notion of procept to show how successful school students make use of “qualitatively different approaches” to arithmetic (p. 126), drawing on data collected by Gray (1991). I will highlight two aspects of their paper: their description of part of their research design and an example of a contrast structure. At the start of the discussion of their data, Gray and Tall include a description of how their sample of students was constructed:
[Gray] interviewed a cross-section of children aged 7 to 12 from two mixed-ability English schools […] Toward the end of the school year, when the teachers had developed intimate knowledge of the children for over 6 months, he asked the teachers of each class to divide their children into three groups – “above average,” “average,” and “below average” according to their performance of arithmetic – and to select two children from each group who were “representative” of each group. (Gray & Tall 1994: 126)
This description is apparently fairly neutral in tone and, as such, typical of research discourse. Nevertheless, it is not neutral in its relation to Gray and Tall’s broader analysis of children’s mathematical thinking. The description establishes three categories of ability, implicitly constructing the idea of ability itself. These categories are crucial for Gray and Tall’s later argument that “the more able tend to display more flexible techniques […] whereas the less able rely on less flexible procedural methods of counting” (p. 130). This last point, the concluding sentence of this section of the paper, is a contrast structure concerning aspects of children’s mathematical thinking. It depends on some way of distinguishing ‘the more able’ from ‘the less able’. The description from p. 126 provides the basis for this contrast. The categories are not simply stated, however: An account is given of how the categories were operationalised. This account includes several features designed to establish the facticity of the description. First, they give an account of how well the teacher knew the students. This point provides a basis for the second, an account of how the teachers divided up their students, and the third, an account of how the teachers selected representative members of each group. Hence, the categorisation is constructed as reasonable because (1) it was conducted by teachers, not by Gray, (2) the teachers had ‘intimate knowledge’ of their students, and (3) they followed a divide and select procedure. The demonstrable reasonableness of the categorisation is crucial, since their analysis of the children’s mathematical thinking depends entirely on the categorisations. Having established their categories, Gray and Tall use them to describe differences in children’s mathematical thinking.
As with Sfard’s (2001) paper, contrast structures prove to be a key discursive form. One example occurs after Gray and Tall have introduced two graphs, which “consider the types of response made by the above-average and the below-average children to a range of addition and subtraction problems subdivided into three levels” (p. 126). An account is then given of the categorisation of arithmetic problems into three levels, about which similar observations can be made to those concerning the categorisation of the children. The contrast structure then takes the following form:
The striking difference between the two groups is seen by a comparison of
the use of procedural methods (counting)
and the use of derived facts. (modified from Gray & Tall 1994: 126)
The first part of the structure instructs the reader that the subsequent parts involve a ‘striking difference’. The next two parts set out the nature of this difference. As such, this contrast is based on: the categorisation of children into three types, the categorisation of children into age groups, the categorisation of arithmetic problems into three types and their visual representation in the two figures. This last aspect is particularly relevant to the description of the difference as ‘striking’. This contrast structure in turn makes possible Gray and Tall’s later introduction of the idea of a proceptual divide (pp. 132–135). Hence, as in the case of Sfard’s (2001) paper, descriptions, in this case of both research procedures and of children’s behaviour, make possible subsequent inferences about children’s mathematical thinking.
My purpose in examining papers by Sfard and by Gray and Tall in such detail is not to challenge or undermine their work. Both have made significant contributions to our field. Rather, I am interested in how mathematical thinking is discursively constructed through the process of doing and communicating research in mathematics education. The preceding examples illustrate the idea that researchers’ descriptions of learners’ mathematical behaviour, or of their own research activity, serve to construct the learners’ mathematical thinking. This point is significant, since a major part of the research endeavour consists in working up and presenting such descriptions, in order to develop analyses and theories about mathematical thinking. As Edwards (1997: 37–43) has argued in the case of cognitive psychology, such descriptions play an important role: They make available particular interpretations of cognitive processes whilst shutting out alternatives. Sfard’s descriptions, for example, build in cognitive or discursive conflict, which can then be made explicit. More generally, then, in written research reports, descriptions of mathematical behaviour are shaped to suit authors’ wider arguments about mathematical thinking and learning. The above examples show some of the specific discursive practices that are implicated. These practices include:
The construction of factsThe construction of some behaviours or actors as normativeContrast structuresThe use of transcriptsThe use of graphsCategorisation.
Contrast structures, in particular, seem to play a significant role in the construction of mathematical thinking. The establishment of many research findings is based on identifying differences, such as between groups of learners or at different times. Contrasts are compelling. However, what is presented in a research paper is an account of various events, not the events themselves. The challenge for the author of a research paper is to give an account that allows certain distinctions to be drawn on in order to demonstrate some outcome – that, for example, more able students use different strategies from less able ones. The use of contrast structures is one discursive mechanism through which such distinctions are made available. And since alternative descriptions are always possible, the particular contrasts built into a description of mathematical activity are, in some sense, not arbitrary. Sfard’s paper illustrates this point well – one set of contrasts makes possible the idea of cognitive conflict; a different set of contrasts makes possible the idea of discursive conflict. Therefore, the distinctions are not ‘there’ in the world, waiting to be described; if they are anywhere, they are in the descriptions that researchers provide.
Most researchers accept that no description can be perfectly accurate. Indeed, most accept that all descriptions include some degree of bias or subjectivity on the part of the author. Both these points, however, assume that the problem is, simplistically put, one of accuracy. My point is slightly different: It is that the descriptions themselves constitute mathematical thinking as it is theorised and conjectured about in research reports. Therefore, Sfard’s descriptions of Noa’s conversation with the teacher make possible the ideas about cognitive and discursive conflict that Sfard wishes to discuss. Similarly, Gray and Tall’s descriptions of their research procedures make possible their ideas about a proceptual divide. I emphasise again that I am not suggesting that researchers present misleading information or select their data to suit their argument. My point is more subtle: Any description is constructed, and the way it is constructed makes available the inferences the researcher subsequently draws. Discursive psychologists, moreover, argue that this feature of the use of descriptions is related to a concern with accountability.
Edwards and Potter (1992: 154) argue that descriptions routinely attend to the accountability of the actors in the described events, that is, to why these actors did the things they are reported to have done. They also argue that these same descriptions routinely attend to the accountability of the author of the report, that is, for the reliability of their claims and the consequences that may result from them. Finally, Edwards and Potter argue that these two levels of accountability are closely related. These ideas help to understand how researchers’ descriptions are organised in their construction of mathematical thinking. The excerpts from Gray and Tall’s paper, for example, include descriptions of differences in students’ mathematical behaviour that are explained in terms of differences in ability. Gray and Tall also deal with their own accountability in their descriptions of the various procedures that were followed. These procedures are described as systematic and rigorous, implying that the research was well conducted. These two levels of accountability are inter-related; it is necessary to persuade the reader that robust procedures were followed by the researchers to identify students of different abilities in order for the differences in these students’ mathematical behaviour to be significant.
The role of descriptions in research reports is complex. They serve a number of purposes and are shaped by these purposes. In particular, they must deal with the accountability of their author with respect to, for example, their veracity, and they must also deal with the accountability of the actors they describe who need to be seen, for example, as rational in their behaviour. The list of discursive practices shown above highlights some of the ways in which researchers’ descriptions manage this accountability.
5 Implications and concluding remarks
The preceding discussion and examples are designed to demonstrate that researchers’ descriptions of mathematical behaviour construct the mathematical thinking under investigation. Contrast structures build in distinctions; transcripts build in relevant features like exclamation marks; facts build in intentions. Researchers cannot, however, get away from doing descriptions: “There is no non-discursive discourse for doing proper, accurate, non-action oriented description” (Edwards & Potter 1992: 173). Our descriptions are always action-oriented: designed to suit our purpose; designed to deal with accountability. What, then, is my analysis good for? First, it underlines the non-transparency of data. More than that, it underlines the non-transparency of research reports; it draws attention back to the surface of the glass that, habitually, we tend to look through in search of a coherent image. All research is description of some sort, and alternative accounts are always possible. Second, my analysis provides a more detailed way to look at how descriptions are constructed and deployed to construct mathematical thinking. Since descriptions are situated, however, it would not make sense to make general claims about the value of or problem with any particular discursive practice. Nevertheless, the discursive practices mentioned above can be looked for and examined in any particular research report to see what they make possible and what alternative versions of mathematical thinking they shut out. In reading research (including this paper), readers might ask themselves:
How have the facts in this report been constructed? What procedures are involved? What intentions are implicit in these constructions? Could different facts have been constructed?Are any of the behaviours or actors in this description constructed as normative? What if other participants were constructed as normative instead?What contrast structures are used? What inferences about mathematical thinking do they make possible?How are transcripts presented? What inferences about mathematical thinking does this presentation make possible? How would alternative versions of the transcript affect the inferences that could be made? How are graphs used? How could they be presented differently? What difference would that make?What kinds of categorisation appear in the description? Where do they come from? What alternative forms of categorisation could be used? What difference would they make?How do these and other discursive practices combine to construct a particular version of mathematical thinking? How do they manage the accountability of their authors?
Providing descriptions of the interaction or behaviour of people engaged in doing or learning mathematics is an integral part of much published research in the field. There are, however, particular ways of writing such descriptions that are specific to the kind of writing found in research papers, as opposed to, say, newspaper reports. Moreover, whilst I have highlighted the role of description in the construction of mathematical thinking in written research reports, other parts of the research process are likely to draw on other practices. Research interviews, discussions between researchers or informal conversations at conferences, for example, can analogously be seen to construct mathematical thinking in particular ways. An interesting question then arises on how these different discursive practices work together to produce accepted accounts of mathematical thinking within a wider community. It is also interesting to consider how descriptions work in different ways at different points in the process. The transcript of Noa and her teacher, for example, can be seen as another form of description, perhaps produced prior to analysis and the writing of a research paper. The structuring of such a transcript is also implicated in the construction of mathematical thinking that arises, as with, for example, the role of the exclamation mark I pointed out earlier. These are complex issues that merit further research.
The practices I have highlighted are likely to be familiar to mathematics education researchers. We are not, however, the only people interested in mathematical thinking; it is also of interest to mathematics teachers, government advisors, students or textbook writers among others. Mathematics curricula, for example, often include descriptive lists of the mathematics that students should learn, thereby constructing mathematical thinking as something that can be listed and broken down into small components to which teachers can be held accountable. In this case, descriptions are used to construct mathematical thinking in rather different ways to those found in research papers. This observation leads to more general questions. If the discursive practices of mathematics education research are implicated in the construction of mathematical thinking within the community of researchers, and if different discursive practices are used to construct mathematical thinking in different ways in different communities (such as those of teachers or curriculum writers), what is the relationship between them? And how do some of these ways of constructing mathematical thinking become more prevalent, more acceptable, than others? Discursive psychology provides a powerful perspective from which to examine such questions. This paper offers a starting point for how such an examination might be conducted.
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This observation derives from a similar one originally made in sociology in the 1960s by Garfinkel (1967), which led to the development of ethnomethodology: “[…] Garfinkel demonstrates that social knowledge cannot be adequately characterized in the form of statistically countable, abstract categories such as scalar ratings of role, status or personality characteristics. He argues that social knowledge is revealed in the process of interaction itself and that interactants create their own social world by the way in which they behave. He then goes on to suggest that sociology should concentrate on describing the mechanisms by which this is done in what he calls ‘naturally organized activities’, rather than in staged experiments or interview elicitations” (Gumperz 1982: 158). Discursive psychology can be seen as the application of ethnomethodology to psychological questions.
See, for example, Edwards (1993, 1997, 2006), Edwards and Potter (1992), Wetherell and Potter (1992), Te Molder and Potter (2005).
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