Beginning mathematics teachers’ ways of knowing: the link with working for emancipatory change Hilary Povey Sheffield Hallam University email
[email protected] Abstract In this article, drawing on extended interview data collected as part of a larger study, a model of the ways of knowing of beginning mathematics teachers is elaborated. Central to the model is the construct ‘author/ity’ which links authoritative knowing and the authorship of knowledge. Four epistemological perspectives are described: silence, external authority, internal authority and the author/ity of self and reason. These are linked to two different traditions in mathematics classrooms - ‘school maths’ and ‘inquiry maths’ - and to the commitment of beginning mathematics teachers to working for emancipatory change in education.
Frances called in to use the Resources Collection at our Mathematics Education Centre and stopped for a chat. She did not yet know that I thought teachers’ ways of knowing were important in understanding their classroom practice and their commitment, or otherwise, to working for change in education. She had herself read science at University and had been a student on our two year post-graduate initial teacher education (mathematics conversion) course. She volunteered this comment. She said, ‘I’ve been teaching some science this year and doing it so badly. It shows that there really is something in teacher training. With science all the knowledge is here [gestures] in my head and I haven’t got a clue how to proceed. With maths, the maths is all out there in the classroom with them and I’m very happy with that.’ (based on personal research notes, May 1995)
This article is concerned with the elaboration of a model of the ways of knowing of beginning mathematics teachers. I start with this moment involving one of the participants in the research upon which this paper is based (Povey 1995) because this incident was something that I noticed, something that I saw differently, because of the model which I had constructed as a result of my enquiry. Here was someone saying that epistemology - how one knows and one’s beliefs about how one knows and about how other people know - makes a difference to teaching. She was also saying, something about which I am much more tentative, that initial teacher education can make a difference to teachers’ epistemologies.
Frances was one of a small number of new teachers whose classrooms I visited, whom I interviewed and/or with whom I corresponded as part of a personal research project which spanned five or six years. Through the interplay of reading, writing, data collection and data analysis I became convinced that teacher epistemologies matter in understanding how it is possible to become and to remain a teacher working for emancipatory change, to have a vision of a more just world and to strive for it. I am opposed to the perpetuation of inequality in and by mathematics classrooms and that was the impetus that drove the research and which led me to construct the model of teachers’ ways of knowing with which this article is concerned. That model takes ‘author/ity’ as a central construct, linking authoritative knowing and the authorship of knowledge; it associates the epistemological perspective of the teacher with two different traditions in mathematics classrooms; and it connects this with maintaining a commitment to education for social justice.
Two difficulties There are two difficulties to confront. First, the task of creating a model is fraught with contradiction when, as was the case in my research, it is part of a project informed by deconstructivist sympathies. (I follow here Jana Sawicki’s (1991, p113) interpretation: to deconstruct something is to give an account of the process of its construction. Carrie Paechter and Gaby Weiner note that ‘[d]iscourses ... are ways of thinking about the world that are so deeply embedded in our practices that we no longer notice that they exist. Researchers who focus on deconstructing everyday practices are, thus, able to challenge hegemonic sets of power relations which would otherwise be invisible’ (Paechter and Weiner 1996, p268.) ‘It’s not that everything is bad, but that everything is dangerous’ (Foucault 1983, p232 quoted in Sawicki 1991, p55) - where ‘everything’ in this case is ‘all constructions’. The invitation here is to use the model to make a (different) sense of one’s experiences: is the net effect, empirically, constraining or enabling? does using the model help one at all to advance one’s own emancipatory project? In other words, borrowing from Patti Lather, the validity of the model is centrally dependent upon whether or not it is catalytic (Lather 1986, 1991); that is, the extent to which each of us is able to use the model to help resist dominant discourses, to retain a capacity to act and to sustain a vision of ourselves as agents of change. [There are] formidable problems that one encounters in trying to develop a coherent integration of the aims and procedures of inquiry and the aims and procedures of a political agenda orientated to improving the conditions under which human beings live and carry out inquiry. Nonetheless, the effort must be made, just as the actions we undertake in everyday life must be carried out in the face of the ignorance, paradoxes, contradictions, and compromises that obscure, harass, and destroy emancipatory motives, aims, and ways of life. (Restivo 1983, p129) The second difficulty is that all the contributions to the development of the model have been located in, at least, white and Western perspectives. In particular, the main participants and I have been similarly (though by no means identically) positioned by the primary discursive fields of ‘race’, class, ethnicity, disability and sexuality. It must therefore be continuously read as narrowly perspectival; in this very important sense, the model needs to be read as local.
Introduction to the model Of what, then, is it intended to be a model? It is intended to model the ways of knowing of student and beginning teachers of mathematics. These ways of knowing relate both to their
pedagogy and also to mathematics. I was trying to make sense of, to draw out and offer some explanation for, the differences in thinking and in action amongst the students-becomingteachers who contributed to this study. I came to believe that only if certain ways of knowing are characteristic of a teacher will she promote equal opportunities in an unsympathetic climate, that is, be a motive force for emancipatory change. This will be so not only in the context of working to provide equal opportunities with respect to gender, which is where I started, but also with respect to other sites of systematic disadvantage, ‘race’, class, disability, sexuality and so on. The model also suggests a fundamental and intrinsic link between characteristic ways of knowing on the one hand and pedagogy on the other. In pointing to this I have borrowed the terms ‘school maths’ and ‘inquiry maths’ as conceptualised by Paul Cobb, Terry Wood, Erna Yackel and Betsy McNeal (1992).
Not least to help in its deconstruction, I describe the genealogy of the model. Early on, in a very exploratory stage of the research, I visited, in their schools for a couple of days each, three new teachers whom I had known as students in initial teacher education. Reflecting on the experience led me to suggest that the characteristic ways of knowing of the three students-becoming-teachers were noticeably different, as were the ways in which they related their pedagogy to ideas and the relationships which they set up between themselves and theoretical descriptions. I then read William Perry’s Forms of Intellectual and Ethical Development in the College Years (1970) and Women’s Ways of Knowing: the Development of Self, Voice and Mind (1986) by Mary Field Belenky, Blythe McVicker Clinchy, Nancy Rule Goldberger and Jill Mattuck Tarule. Both of them influenced my thinking and contributed to the development of the model of ways of knowing offered here.
I was concerned to understand better those ways of knowing exemplified by beginning teachers of mathematics who are working for emancipatory change, that is, by those who are working to establish a more liberatory pedagogy. I subsequently extended my empirical study to include two such teachers new to the research (Frances and Matthew) whom I studied in some depth and by revisiting the one such teacher from the first cohort of three (John). Listening to these three talk about their lives, their work and their understanding of mathematics was the grounding for the model of learning I offer here. I also obtained data from a variety of other sources which influenced the enquiry, albeit marginally.
Naturally, this oversimplifies the process. Of course, I kept in mind my experience of visiting my first three students-becoming-teachers throughout all the later stages. Indeed, I returned to the interview and observation notes I had made many times for a variety of purposes and listened to the tapes again occasionally. Similarly, the experience of the third phase field work, the re-reading of Perry’s accounts and of Women’s Ways of Knowing, the analysis of the new interview data and the creation of my model of epistemological positions all interacted with one another.
The underlying organisational principle behind the construction of the model has been the sifting out of those aspects of teachers’ epistemologies which make a difference to what, as teachers, they think, believe and do. I have sought to isolate and separate out those features which help us to understand these differences, maintaining distinctions where to do so is helpful in pursuit of this goal, conflating distinctions where it is not. The essential explanatory concept at the heart of the epistemological model offered here that of author/ity. Author/ity links back together again two words which have a common root but which have come to be read very differently from each other. An author is one who brings things into being, who is the originator of any action or state of things. Authority is concerned with power and the validity of knowledge. Linked together they lead to the construction of an epistemology which recognises each of us as the originator of knowledge.
I do not claim that any of us is fixed into only one outlook, one world view, one epistemological perspective; rather that it is characteristically human for us to be contradictory in our approach to and interpretation of the world. Experience and our reading of it are fragmentary. The model is not intended to be developmental although there will be links, transitions, interconnections between the epistemologies. I believe that an individual may have different ways of knowing about different aspects of their lives or have different ways of knowing about the same aspect of their lives at different times; that some movement back and forth between the perspectives is characteristic of human knowing; and that, because our lives become fragmented by both the social institutions and the discourses with and within which we interact, we can and do hold different perspectives on the same things at the same time until an opportunity comes to reintegrate (temporarily) our knowing as we reconstruct ourselves. (I invite you to verify this through introspection but see also, for example, Weiler 1988, Norquay 1990, Walkerdine 1990, Foucault 1991 especially chapter
3.3.) In addition, how we reconstruct our epistemological perspectives is influenced by the pedagogical practices in which we are engaged.
An author/ity model of teacher epistemologies In coming to an understanding of beginning mathematics teachers working for emancipatory change, it is helpful to separate out four different ways of knowing: silence, external authority, internal authority and the author/ity of self and reason. I shall say a little about each of these perspectives in turn.
The perspective of silence owes a great deal to the position of the same name outlined in Women’s Ways of Knowing. It was hardly heard from Perry’s elite, male undergraduates and is much more likely to be experienced by those who are not part of the hegemonic group. Silence It cuts off the knower from all internal and external sources of intelligence; the knower cannot see herself as developing, acting, learning, planning or choosing. She may have no vantage point from outside the self with which to view herself or she may see herself only as the object of such a gaze. She will feel ‘deaf’ because she cannot learn from the words of others and ‘dumb’ because she has no voice. ‘Reason’ is not known, is unacknowledged. By its very nature it is seldom heard in the context of teacher education, but it still can be experienced at moments of self doubt or as an underlying subtext in any of the other positions. It provides no defence against the critiques of others nor against unpleasant classroom events, both of which may provoke this perspective. Where this position departs significantly from that described in Belenky et al (1986) is that it is not irrevocably left behind when other ways of knowing are available. This perspective is significant in the context of working for emancipatory change when, as with most equal opportunities work, such change involves challenging dominant others: even though usually transitory, the perspective of silence is likely to be evoked when one is under threat. In addition, also significantly, it is immobilising.
An illustration was provided by Frances when she was remembering times past. She spoke of terror and voicelessness. Unsurprisingly from a feminist perspective, this had been evoked in a context of contemporary, Western arrangements for domestic childcare.
It was very very very scary living in West Camford because all the women around with me ... were very very confident and assertive about what they did and it was a very undermining society to move in ... [They] had such strong ideas about what you should and shouldn’t do with children ... I’d taken Grace round to play and endless awful things had happened. Grace had just done all the things that children shouldn’t do including throwing a chair down the stairs on top of this woman’s child ... I can remember very distinctly just how terrified I felt that afternoon and how unconfident and awful, I couldn’t say anything, my sentences all came out wrong, it was pathetic. (Frances, transcribed from interview tape) Here Frances felt she knew nothing and had no grounds for knowing anything. Inaccurately or otherwise, she felt herself to be the object of criticism from authoritative others and was silenced. Not dissimilarly, a new teacher (of mathematics) will be struggling to maintain and develop a sense of herself as a legitimate member of the profession, very often in the face of (at least apparently) hostile students and with a felt need to become established in the eyes and under the judgement of significant others. Both the critiques of others and also unpleasant classroom events are likely to provoke a perspective of silence, which perspective, of course, provides no defence against either. The perspective of silence also sounds like the ‘panic’ described by the participants in a study of adults’ experience of mathematics (Buxton 1981, especially p1-9) as ‘[p]ushing the button that cuts out logic ... “mind in chaos” ... nothing is stable, information “topples out of the mind” '. Such absence of knowing, where one does not even perceive oneself as a knower, is clearly not the intended outcome of ‘school maths’; but the demand of ‘school maths’ for the assimilation and regurgitation of received procedures without recourse to, and perhaps in despite of, the learner’s reason frequently provokes a collapse into silence. An alternative reading by the learner of ‘school maths’ does not deny the value of reason itself but interprets its role as being only the handmaiden for the inculcation of established procedural knowledge validated by external authority; this provides the label for the next epistemological perspective.
External authority Authority is external to the self and lies with ‘experts’. The voice of external authority is heard and is not subjected to independent critical judgement. Meaning is taken as given, knowledge is assumed to be fixed and absolute rather than contextual and changeable.
There is no sense of an inner voice to which heed might be paid. Internal authority is not heard. ‘Reason’ is valued because ‘They’ say so. It offers no defence against the conceptions, critiques, ideas and practices of significant others. Indeed beginning mathematics teachers experiencing this perspective can be expected to seek out just such manifestations in order to know what to think and how to behave. It can give some protection of the self against unpleasant classroom events. An epistemological perspective based on external authority offers strong support to the status quo whatever that happens to be, whether the voice of the tutor or of the mentor during training or, subsequently, the established view prevailing in the employing school. This means that, by definition, it is antithetical to working for change.
Beth provides an example. Four years after leaving the University, having taught by choice for several terms on a supply basis and now nearing the end of her first year in a permanent appointment, she seemed all at sea in terms of setting her own goals, making decisions about her teaching and following through such decisions. There’s no critical eye on my lessons, I need someone to say I’m not that bad ... I’ve started an appraisal book but I only kept it for two days ... The trouble is because there’s no probationary year, I’m not assessed ... The [initial teacher education] course sets you up for failure ... [because] it sets standards very high. I set myself high standards but it sets such a high marker to keep up with. (Beth, transcribed from interview tape) She seemed to require that external authority define her task for her: she did not seem pleased with a freedom that teachers working for emancipatory change might have enjoyed and she seemed to wish to believe that the constraint she sought (but did not find) was a given of schooling. Before my first term I sat down with Dick and he said here are some good topics you can start with, but I was working in terms of a month by month you know and so I never actually knew what had to fit in in what time, and it seemed pretty haphazard. I mean they’ve now put it all together in a year 9 scheme of work but it’s all very general ... I think some time on the [initial teacher education] course could be spent on things like ... how you can use [published textbook schemes], there was no guidance on following schemes of work that were already written and to be honest that’s what you get when you get into school really. (Beth, transcribed from interview tape) It is a perspective which cannot tolerate contingency. A contemporary of Beth’s, Sara, said, ‘How can I keep an open mind? I’ll just be convinced by whoever spoke to me last.’
This perspective can also be seen in response to the question ‘What would you do if you didn’t know something?’. Ask a colleague or several colleagues. Refer to the literature. (Simon, research correspondence) Look it up in a book, or ask one of my colleagues. I would try the former first as pride stops me from showing weakness. (Kevin, research correspondence) I show below how these responses contrast with those of other participants.
It seems at least plausible that beginning teachers of mathematics are particularly likely to be prone to experiencing the perspective of external authority. Knowledge of mathematics itself is commonly perceived as being based upon the authority of others, such a perception extending from those of our pupils who ask ‘Is it an add, miss?’ (Brown and Kuchemann 1976) to university professors of mathematics (Povey 1996). It also seems reasonable to suppose, therefore, that those who are drawn to the subject may typically be more comfortable with an epistemology based on external ‘guarantees’ and that such an epistemology will form part of the pervasive subject sub-culture found amongst student teachers (Lacey 1977, p58f and 63f). This would be congruent, for example, with the finding that post-graduate mathematics initial teacher education students had an external orientation to their own professional learning, ‘choosing to model their planning upon the work of others’ (John 1991, p316), in contrast to some other subject specialists who tended to have an internal orientation and ‘to view planning as a process of self guided discovery’ (John 1991, p316). (If these conjectures are correct, it would suggest that those entering mathematics teaching are also particularly prone to the perspective of silence, since a key feature in situations evoking this perspective is the presence of someone perceived as a significant authority figure who will make judgements about right and wrong.) Such an epistemological perspective is also highly congruent with ‘school maths’: rationales for practice may be offered but learning is ‘synonymous with acting in accord with the teacher’s expectations’ (Cobb at al 1992, p594) and the teacher acts as ‘the sole validator of her students’ interpretations and solutions as she [initiates] them into the realm of mathematical instructions that [are], for the most part, beyond explanation and justification’ (Cobb et al 1992, p594). There is no ‘community of validators’ (Cobb et al 1992, p594)
within which the teacher and students together construct knowledge; rather, in contrast to the ‘inquiry maths’ tradition, the basis for knowledge in ‘school maths’ is the external authority of the teacher.
In some ways the next perspective is an attempt to resist the loss of personal voice engendered by an epistemology based on external authority. Internal Authority External authority is not accepted. The voice of authority is still heard and provides absolute right answers but now the authority is the self. This internal authority is listened to but it is not subjected to critical appraisal and the knower is unwilling to submit to the analytical judgement of others. Meaning is taken as given (by internal authority) and fixed. ‘Reason’ is distrusted. It provides little defence against unpleasant classroom events and can involve the rejection of the feelings and worth of others. This perspective can make the knower impatient of dialogue. If internal authority vouches for the truth and the truth is known directly by the knower, what purpose is served by discussion, debate, analysis and the exchange of ideas? The strength of this perspective lies in its affirmation of the self but it provides little protection against unpleasant classroom events. This is because it does not support ‘seeing things from the other person’s point of view’. There is no theoretical perspective from which to understand the apparently hostile behaviour of others which is therefore likely to be both personalised and misunderstood. Hurt and bafflement are likely to precipitate a retreat into the status quo.
I think this perspective is less common than external authority amongst mathematics teachers - try ‘Well everyone is entitled to their own opinion and I think 2+2 is 5’- although I think you can hear it a bit sometimes in talk about teaching. [The purpose of initial teacher education is] to examine a range of teaching styles and approaches to see what suits you. (Simon, research correspondence) However, there do not appear to be strong links with either ‘school’ or ‘inquiry maths’. An epistemology based on internal authority will not motivate the acquisition of procedures validated by the authority of the teacher but neither will it support the meeting of minds and co-construction of knowledge implied by ‘inquiry maths’. There is personal authority here
but it is not the personal authority, honed through dialogue and the negotiation of meaning, which is characteristic of the fourth epistemological perspective. Author/ity of self and reason External sources of authority are listened to and evaluated as is the internal voice. Both external and internal authority are subjected to critical judgement. Knowledge is understood to be constructed and meaning to be negotiated; they are contingent and contextual and personal in the sense that they reflect the positionings of the knower. The innocence of ‘reason’ is questioned. It allows a critical distance from the critiques of others and from unpleasant classroom events. This perspective permits an understanding of how power relations shape what it is possible to think and to say. It is therefore able to comprehend a critique of contemporary gender relations in particular and unequal relations in general. It is able to stand outside received frameworks and to acknowledge the existence of hegemonic discourse and justificatory myths. It is in this sense a potentially emancipatory epistemology.
Of course, neither of the terms self or reason can any longer be regarded as unproblematic. All human ideas are situated. Thus there is no neutral, universal reason available as an arbiter of truth or knowledge. Similarly, there is no empirical, knowable object called the ‘self’ waiting to be discovered or observed. (Griffiths 1995, p227) The reason of this epistemological position is not that of ‘western rationality’ (Walkerdine 1994, p74) with its accompanying fantasy ‘ “Reason’s dream”, a dream of an ordered universe where things once proved stay proved forever’ (Walkerdine 1994, p74). It is a reason which permits, for example, metaphoric thinking, that acknowledges that knowledge is contextual and not universal (see Walkerdine 1988, especially p183-201, for a discussion of these attributes and ‘western rationality’); a reason that is interwoven with emotive thought (Belenky et al 1986, p134) rather than distanced from it; a reason that acknowledges practical engagement and is embodied (Confrey 1995); and, crucially, a reason that attempts to integrate personal knowledge with the knowledge of others. It recognises and embraces the need for negotiating shared meaning and for constructing ‘argument’ in a multiplicity of ways, attempting to interrupt the objectification, the formality, the disembeddedness and the
combativity of ‘Reason’ and helping one remain ‘paradoxically aware of one’s complicity in that which one critiques’ (Lather 1991, p10). (See also Ellsworth 1989 for a discussion.) The self is understood as constructed and reconstructed, not essentialised. It is ‘best understood as a subjectivity produced within the discourses in which it is positioned and positions itself’ (Griffiths 1995, p227, my emphasis); a subjectivity that recognises that power relations shape what it is possible to say and think, that ‘our basic assumptions about the nature of truth and reality and the origins of knowledge shape the way we see the world and ourselves as participants in it’ (Belenky et al 1986, p3, my emphasis).
As an example of this perspective, we can look at the way in which the teachers whom I characterized as working for emancipatory change answered the question about what you would do if you didn’t know something. I always ask someone [if it’s about school]. I suppose I’d ask them and they’d give me their answer and I might discuss it a bit, and then I’d go away and think about it and then I might get back to them because I’d have to sort of, yeah, sort of assimilate it and compare it with all sorts of other ideas and things (Frances, transcribed from interview tape) on points of information, what I call a technical question really, [I] look it up in a book or whatever, [and then use] somebody else, [like] how far on the national curriculum are they, the mark for a piece of coursework ... In terms of wider questions which aren’t technical ... I wouldn’t go and ask somebody ... I would have to think very carefully about what I thought about it in the first place, the issues, so with something like that I would be starting from not having a particularly fixed position, and I’d want to hear what other people have to say (Matthew, transcribed from interview tape) These teachers were also willing to change their minds about things. I might change my mind afterwards and change things but I don’t worry about them, once I’ve made a decision I don’t feel I have to stick to it, if it’s wrong it’s wrong (John, transcribed from interview tape) but then I’ve changed my mind about so many things to do with maths that maybe I’m going to change my mind about that one as well (Frances, transcribed from interview tape) What I think today, and I might not think it tomorrow, is ... (Matthew, transcribed from interview tape)
John illustrated the instability of author/ity of self and reason and the tendency to seek external authority as a guarantee, either the external authority of other individuals:
Well I sometimes find myself being fooled [laughs] by that kind of thing, where I would ask one person, I would be discussing something with one person and they’d put forward their view point and I think that sounds reasonable, and then I’d go and speak to somebody else and they’d say hmm I wouldn’t do it like that and then I have difficulty in thinking, well, which one is right. I agreed with that person and now I’m agreeing with this person, what do I believe? I sometimes have difficulty with that as well, actually knowing which of those two people’s advice I should follow - (John, transcribed from interview tape) or the authority of a doctrine:
- sometimes I feel to myself thinking which one should I agree with, I mean which one’s the socialist and which one’s - (John, transcribed from interview tape) but he was able to return to his own authority and concluded: - but that’s not always a good way to decide. Probably I’d think about it myself and come up with a third alternative, and it may be the alternative that I thought of in the first place if you know what I mean. It might be that speaking to two people about it would, I’d probably end up adapting it a bit and getting a mixture of what I first thought and what the other two people said (John, transcribed from interview tape) I offer one further illustration of this epistemology. Frances spoke of experimenting with alternative perspectives to see what they felt like, as it were, from the inside. Sometimes I’m listening to somebody else and they have a completely different view to me and I argue dramatically against them and think what they are saying is rubbish, and then I get myself into another situation where somebody holds my view and I argue the person’s point of view that I originally had the argument with and see how it feels arguing it from their view and then I tend to sort of come to some kind of middle, not always a middle ground because sometimes it’s not something you can have a middle ground on but I then sort of come to some kind of conclusion but I feel as though I have to hold, I have to try holding the different views before I can decide ... to say something totally outrageous and see what it felt like, and that’s what I kept wanting to do ... I wanted to kind of say things that were not the kinds of ideas we were supposed to have and see what it was in them that might be appealing or at the very least to reject them properly. (Frances, transcribed from interview tape)
The perspective of the author/ity of self and reason encourages the consideration of the views and respect for the vantage points of others whilst exposing those views and perspectives to critical judgement. As such, it provokes the epistemological climate upon which ‘inquiry maths’ tradition is predicated where students ‘create and manipulate mathematical objects in ways in which they can explain and, when necessary, justify’ (Cobb et al 1992, p598) and where their ‘relatively autonomous constructions [are used] to guide the constitution of takenas-shared mathematical meanings and practices (Cobb et al 1992, p595).
Conclusion I claim elsewhere (Povey 1995), first, that the ways of knowing available to pupils (in school) are constrained by the epistemological stance of their teachers and by the kinds of knowing that those teachers construct as feasible and possible; and, second, that the author/ity of self and reason supports the development of critical mathematics education. Here there is space only to hint at the connections between this model of teachers’ ways of knowing, their understanding of the nature of mathematical knowledge and their commitment to working for emancipatory change, within education in general and within mathematics classrooms in particular. An epistemology based on the author/ity of self and reason is a logical prerequisite for emancipatory education, although it is not a sine qua non for emancipatory commitment and practice nor does it guarantee, either logically or in actuality, such commitment and practice. As well as rational thought and intentions, myths and metaphors affect how we think and what we do, what we believe and how we act.
However, an epistemology based on the author/ity of self and reason opens up the possibility of challenging the ‘personal fatalism ... servility ...[and] negative self-esteem’ (Skovsmose 1994, p189) engendered by ‘school maths’ and of pursuing an emancipatory mathematics curriculum. The teachers whose epistemology is based on the author/ity of self and reason is encouraged to look for change to prevalent practices in contemporary schooling and to the taken-for-granted norms of many mathematics classrooms and thus to de-nature the present and to re-vision and re-envisage the future (Kenway et al 1994, p202). That re-envisaged future will include students whose personal authority allows them, within an ‘inquiry maths’ classroom tradition, to be the co-authors of knowledge.
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