Review by Paul Ernest

Brown, T. (1997) Mathematics Education and Language: Interpreting Hermeneutics and Post-Structuralism, Dordrecht, North Holland: Kluwer Academic Publishers, ISBN 0792345541, published in hardback only on 10 July 1997, price $105 / 63.

Theoretical discussions in mathematics education have for too long been inward looking. There has been much discussion of understanding, language, the construction of meaning, the classroom and the teacher, using home-grown ideas or out-of-date psychology. Of course the best work in the field has always looked to mainstream developments in sociology, psychology, philosophy, anthropology, linguistics, mathematics, etc. But this has been the exception rather than the rule. For example, theoretical discussions on radical constructivism over the past decade have focused on questions of individual minds and coming to know; ontology, the existence of the knowing subject and the world; and culture and the nature of knowledge; all too often in ignorance of the deep thought that has been directed at these issues, sometimes over many centuries, in philosophy, psychology, and sociology.

This book is potentially an antidote to this neglect. It draws on a number of deep theoretical domains outside of mathematics education, domains which are all too rarely used in the field. It is an ambitious book, for as the subtitle indicates, it seeks to weave together a "theoretical account of the individual confronting mathematics and representing her understanding to others" (p. 4). It draws on Hermeneutics, phenomenology, post-structuralism, and Saussurian linguistics. This constitutes part 1 of the book. Part 2 goes on to apply this perspective to learning of mathematics in the classroom; and Part 3 addresses the teacher's own perspective in the classroom, culminating in Part 4 which concludes by "combining and developing themes from all parts of the book the chapter point to an overarching framework for analysing the cultural functioning language in mathematical language and teaching." (p. 6)

Given the depth and breadth of its ambition, how is one to read and evaluate this book? Some of the questions one might ask are the following: What is its aim? What is the underlying desire or primal vision which drives the book? How well does it exploit the intellectual resources it draws upon to account for individual learning and the social/cultural/political context of the mathematics classroom? To what extent is a novel intellectual synthesis or theory derived? How well is this theory utilised to explain the phenomena and cases to which it is applied in the text? To what extent is it utilisable and generalisable to other contexts? What is genuinely novel about the insights it provides? Are these the right questions for evaluating the book, or should other perspectives drive a review?

Interestingly, the task of answering these questions, i.e., that of reviewing the book, is squarely a Hermeneutic one. It is a matter of raising questions, constructing an interpretation, returning to the book with this perspective, modifying the interpretation, and reiterating this process until some form of stable interpretation has evolved. This in no way guarantees the veracity of the account; such a notion is inappropriate in the context of reviewing a book; but it will provide a robust assay, a prolonged sortie into the terrain of the book, at least from this reviewer's perspective.

Let me begin, then, with my initial interpretation of the underlying aim, and the central (inferred) desire and primal vision which drives the book. It seems to me that central to the book is the desire to account for and respect the individual sense making, interpretation and reading of mathematical tasks and the classroom by both learner and teacher. I sense a deep valuing of the fragile, changing and fluttering meaning-in-the-moment as learners and teachers make sense of what is before and around them. Some people are driven by the desire to protect and comfort the emotional child present in others. Here I get a sense of the desire to protect and promote the untrammelled fresh awareness that persons potentially bring to learning situations: the budding growth and flowering of conscious awareness. The second element of my reading is concerns an awareness of the transitory nature of understanding, knowledge, social contexts. These are all the while in mid process, mid flow, coming into being and through change and impermanence moving into not-being. Speech, knowledge, the knowing subject, classroom understandings, are all everlastingly evolving, shifting, growing, always in mid-renegotiation. Knowing in the classroom is like surfing a giant wave of consciousness, all the while moving forward, all the while an exhilarating balancing act, leading the knower forward to fresh horizons.

So what does this reading mean and imply? It suggests to me that Tony Brown wants to recover the vision that radical constructivism seemed to offer; a progressive view of the learner's (and teacher's) individuality and subjective sense-making, but to go beyond the limitations of radical constructivism (to which he never subscribed). For radical constructivism seems to offer a rather limited and isolated cognising subject, epistemologically (and emotionally) divorced from its context. Furthermore, there are rigid (if changing) structures of the self and its environment in radical constructivism that belie the fluidity of the changing social surround. Tony Brown instead want a full human being - one that is active physically, intellectually, linguistically emotionally, and morally. He looks to phenomenology for the sensitive sense-making subject, active in the here and now. He looks to Hermeneutics for the sense of impermanence and growth of interpretation, and to post-structuralism for the shifting discourses and sea of language in which the individual is positioned and through which she grows and is modified. He also looks to post-structuralism, social studies of science and the sociology of knowledge to dethrone the rigid structures of society and knowledge, so that the shifting subject floats in a shifting surrounding sea - the social context.

Following through this reading, there are some weaknesses that emerge from this imputed position. First of all, the valorising of the knowing subject as in the ideology of progressivism (and radical constructivism) entails two weaknesses. There is first the sentimentalisation of the knower. But knowing is not always the sweet sense making of untrammelled reason and intelligence. Knowing is also about fighting for psychic survival in the face of emotional and other forms of threat. It is about interpreting hidden coercive meanings underlying the literal spoken word of a teacher or another. Knowing is about deviant scheming to compensate for personally sensed deprivations or inadequacies. Knowing is about desire for forbidden objects of gratification approached covertly under a patina of acceptable behaviour and discourse. Knowing is in the mind of the eagle as it sights and takes its prey. Knowing should not be sentimentalised.

Also associated with this view of the knowing subject is the ideology of individualism. Knowing subjects are individual sense-makers who understand the classroom and contexts independently; who have isolated and independent inner lives of thought, which emerge into the outer public arena as speech or actions. But usually we speak or act as we think, and how we act is ordered by public if tacit rules which help to define our selves in the context. We are not, in my view, the isolated and self-contained individuals that Tony Brown's account implies, on the reading given.

A second area of weaknesses concerns the politics of education. A focus on the activities and immediate surrounding context of the knowing subject ignores structural inequalities in terms of class, gender or race. It ignores the dominant ideologies of schooling. Ultimately it also ignores the fact that the main obstacle to individual sense-making and self-realisation is the active opposition to the political empowerment of the learner as a democratic citizen by most if not all forms of institutionalised education. Tony Brown's account claims to be social, but it is clearly not political in any sense of the word.

A problem which re-emerges periodically in Tony Brown's text is that of the character and the nature of mathematics. He rejects the notion that mathematics can be objective in the social constructivist sense of having been negotiated and accepted by the community of mathematicians because "Rather like the Platonic (sic) view where mathematics is created by the gods, existing independently of humans, here we have mathematics created by recognised experts. Whilst this perspective may have an unease with the notion of mathematics being absolute, it seems to accept the possibility of an absolute overview of mathematics at any particular point in time." (p. 13) There is an inconsistency in his account, because on the one hand he seems to credit social constructivism with acknowledging that mathematics has more than merely subjective existence, yet claims that "mathematical ideas, as located through notation, are not endowed with a universal meaning but rather derive their meanings through the way an individual attends to them." (p. 15) Is this the point that public texts elicit personal interpretations or that all the possible meanings are fluid and subjective? In the former case I and most others would agree - with the caveat that meanings are not generated in complete freedom, because we live in shared forms of life with associated language games that link meanings to actions, expectations, and shared and accepted discursive practices. Later (p. 84) he quotes Brookes with approval: "objectivity is achieved through the coincidence of interpreting, that is, agreeing". These conflicts are never resolved.

He does make the point that different persons in referring to mathematics may be referring to different discourses, and he indicates University lecturer's Platonistic discourse of research mathematics, the education bureaucrats association of mathematics with the discourse of school testing and assessment, the civil engineer's association of mathematics with the discourse of engineering applications, etc. His point, well made, is that mathematics means different things in different discourses, and that each is value-laden and ideological. He goes on (p. 67-68) to draw the analogy between Saussure's distinction between langue and parole and mathematics as practised versus mathematics as a system (respectively), which supports this point. However he never engages with the notion of mathematics as system, and critiques others who have done so for presupposing a fixed determinate subject. Tony Brown is not interested in philosophy of mathematics per se, but in challenging the misconception that there is a unique transcendental subject mathematics lurking behind our glib use of the term mathematics. And this is an important insight.

Now I must re-examine my reading, given the reality of the text before me. First of all, is it not unfair to accuse Tony Brown of sentimentalising an over-individualistic knowing subject as may be ascribed to radical constructivism, when he devotes so much time rebutting radical constructivism, other forms of constructivism, and even social constructivism?

I cannot fully concede this. Tony Brown may be critical of constructivism but he seems to continuously reassert the precedence of the thought over its utterance, of the experience before the text. He speaks of "freezing experience in time for the purposes of talking about it. Such fixing preserves experience whilst, at the same time, killing it off. Any assertion of how the experience was, is associated with a loss. The very act of reflection is a reorganisation of lived experience into a caricature of it" (p. 213). Now I am both in and out of sympathy with this. On the one hand, there is an authentic, near spiritual (and none the worse for that), and naive attention to the texture of subjectivity and lived experience. There is an authenticity in this. Tony Brown is very keen to separate the authenticity of a lived experience from the subsequent, second-hand reflection on and recounting of it. Such an impulse is doubtless true to the spirit of phenomenology.

On the other hand, there are some intellectual puzzles and contradictions involved. Derrida and Wittgenstein are drawn upon regularly in the book, and yet both have strongly denied the precedence of thought over language - whether in speech or text form. The book aspires to be post-modernist and post-structuralist and both of these loose groupings share a rejection of representationalism (which seems to be implied), and an insistence of multiple readings of any text. Tony Brown concurs with the latter - indeed it is one of his main points - but acts disappointed that his texts do not capture the meaning of lived experience. Finally, methodology in the interpretative research paradigm centres on the dangers of subjectivity and the means of safeguarding interpretations of educational experience (through triangulation, etc.). This is directly relevant to the account in the book, and its embedded reports of classroom episodes and interpretations of learner and teacher texts, but it is neglected. A handful of references are made to publications on action research, but there is no discussion of the methodology employed in the book, or of the problematic nature of the author's interpretations of events and texts. In the context of a book on the Hermeneutics of mathematics education this is unforgivable.

Earlier I asked how well the book exploits the intellectual resources it draws upon to account for individual learning and the social context of mathematics education, and to what extent it offers a novel theory able to explain the illustrative vignettes, and beyond the text, other contexts? On these issues I have to give critical answers. No explicit theory is formulated, used or tested. A number of the deep theoretical sources drawn upon are either misunderstood or described in such oversimplified terms as to be problematic. For example, the brief references to Peirce misrepresent Firstness (which could fruitfully be linked with phenomenology and Heidegger's philosophy), his theory of the sign (which is misrepresented as dichotomous) and his notion of the interpretant, which is controversially defined without further discussion as "the human experience of creating intelligibility" (p. 147). In mathematics education Tony Brown claims to draw seminally on the work of Walkerdine and Skovsmose, and although he periodically cites their work he seems to have missed their central point, namely that mathematics education is at root, political. Skovsmose has been developing a critical philosophy of mathematics education based on Habermas and Critical Theory for over a decade, and his work could have better informed the present book, especially given that Habermas is the second most cited author in the book (after Tony Brown himself). It is clear that Skovsmose's (and Niss's) notions of realised abstractions and formatting (which apply to the role of mathematics in invisibly structuring society) are misapplied when used to describe the effect of teacher's writings (p. 193). These are not minor criticisms given Tony Brown's claims.

I see myself building on the post-structuralist analysis of Walkerdine by building firmer links both with contemporary research in mathematics education and teacher education, by situating her work in a broader theoretical domain, and by referring to newer styles of teaching mathematics. I develop the theme of critical mathematics education as initiated by Skovsmose by developing his theoretical frame around the experience of individual students and teachers I seek to complement the work of social interactionists and socioculturalists by contextualising their theoretical themes in relation to the hermeneutic, critical and post-modernist traditions and so making stronger connections with contemporary philosophy and social theory. (p. 19-20)

Let me return once again, having waxed critical, to the hermeneutic task of interpreting this book. Trying to 'get inside' it, to adopt a more sympathetic perspective instead of imposing my own agenda on it, I think there is a way in which my criticisms can be side-stepped. Rereading the work in the light of my previous interpretation, I begin to see that Tony Brown is not trying to offer a novel theory or theoretical synthesis for mathematics education, putative task with which I find myself growing impatient. Instead, in a covert way he is writing autobiography. He cites all the authors he has drawn inspiration from in developing his personal philosophy and outlook, and undeniably engaged deeply with. He reports many episodes from his professional career working with students and teachers, episodes where he has drawn insights which he hopes to communicate and share with others. Instead of a textual theory, the book is a report of a developing connoisseurship, a professional delicacy of judgement. Perhaps this is why authenticity rather than theoretical consistency, thorough articulation or scholarly justification and argumentation, is the theme that runs through the text. Thus it is lived experience which is foregrounded in the text, linked with theoretical reflection. As such, I am sure the book will be well received by practitioners in education and schooling, who will be able to identify or aspire to the exploratory journey described. It is a sensitive and gentle journey through classrooms, libraries and the academy. Whilst I do not retract my critical evaluation, perhaps I can briefly see into another reality which values professionalism rather academic scholarship. Certainly there is much to be gained from the book in the way of insights.

There has been much discussion of understanding, language, the construction of meaning, the classroom and the teacher, using home-grown ideas or out-of-date psychology. Of course the best work in the field has always looked to mainstream developments in sociology, psychology, philosophy, anthropology, linguistics, mathematics, etc. But this has been the exception rather than the rule. For example, theoretical discussions on radical constructivism over the past decade have focused on questions of individual minds and coming to know; ontology, the existence of the knowing subject and the world; and culture and the nature of knowledge; all too often in ignorance of the deep thought that has been directed at these issues, sometimes over many centuries, in philosophy, psychology, and sociology.

However I cannot finish without returning to my initial discussion. I said that theoretical discussions in mathematics education have been too inward looking, often not informed enough by developments in other fields of thought. Unfortunately I cannot claim this book as the solution to this problem. It refers indeed to a number of deep theoretical domains but it cannot be said to do so definitively. However it may inspire others to develop a more systematic analysis.

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