Promoting Social Justice In and Through the Mathematics Curriculum ...

Mathematics Education Research Journal

2002, Vol. 14, No. 3, 190-201

Promoting Social Justice In and Through the Mathematics Curriculum: Exploring the Connections with Epistemologies of Mathematics Hilary Povey Sheffield Hallam University This article rehearses the argument that being a critical mathematics educator is associated with a particular epistemological stance, one which views the truths of mathematics as historically located, influenced by the knower and mutable. Case study data, collected in England, is offered which exemplifies this connection between epistemology and openness to equity issues in the thinking of some beginning secondary mathematics teachers. Teachers' responses are analysed around four themes: their beliefs about the nature of mathematics, how those beliefs affect their pedagogy, how they explain student failure, and their views on initial teacher education. These are linked to their commitment to social justice in and through mathematics. The links between subject studies in teacher education and equity issues in the classroom are discussed.

The Status of Mathematical Knowledge The conventional view of the nature of mathematics, which has an entrenched position in mainstream contemporary Western culture, rests on a taken-for-granted understanding of the nature of mathematical knowledge which accords it the status of absolute truth. The truths of mathematics are certain and unchallengeable, are objective, given, and unchangeable. Mathematics is free of moral and human values and its social and historical placings are irrelevant to its claim to truth (Ernest, 1991). Such views about the nature of mathematical knowledge have withstood wide ranging and varied challenges from the philosophy of mathematics (see, e.g., Benacerraf & Putman, 1964); from the sociology of mathematics (e.g., from Restivo, 1992); and from recent developments in mathematics itself (see Davis & Hersh, 1981, for an overview). In a classroom predicated on such an epistemology, the teacher will be encouraged to “constitute mathematics as the activity of following procedural instructions” (Cobb, Wood, Yackel, & McNeal, 1992, p. 573), such procedures being regarded as “ahistorical, unalterable norms ... [without] a specifiable source ... both fixed and self-evident” (Cobb et al., 1992, p. 588ff.). The model of learning is likely to be that of transmission which envisages items of content (or correct strategies reified as content), pre-defined and non-negotiable, being “delivered” to the head of the learner, reinforcing “teachers’ allegiance to limited bodies of content ... enshrined as ritual” (Noss & Dowling, 1990, p. 2). In this “banking” (Freire, 1972, p. 46), mathematics knowledge is seen as hierarchical, the teacher’s mental models are superior to those of the learner and education becomes the process of implanting those models in the mind of the learner. Knowledge is validated by external authority. Students are encouraged not to rely on their own experience but to deny it and to accept in its place the knowledge and experience of experts, through a hierarchical and competitive epistemology which undermines the validity of their own knowledge (Gerdes, 1985; Giroux, 1983).

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An alternative epistemology is based on a view of mathematics as being “coconstructed” (Cobb et al., 1992, p. 573) by teachers and students, a product of the human mind and, consequently, historically located, influenced by the knower and mutable. It is an epistemic strategy “not based on any claims about ultimate truth ... not clothed in guarantees of any kind” (Restivo, 1983, p. 141). Consequently, learner enquiry and the construction of meaning are valued and the focus shifts “from teacher delivery of ‘knowns’ to learner investigation of ‘unknowns’“ (Burton, 1992a, p. 2). This in turn allows for the development of the capacity to be critical: it allows for the possibility of hope and the belief that things might be otherwise than as they are (Giroux, 1992). Because we live in an unequal world, unequal in terms of power, opportunities, access to resources and so on, a curriculum based on transmitting received knowledge is epistemologically oppressive: it gives no opportunity for challenging the (unequal) status quo. The transmission model of teaching, in a traditional formal classroom ... is the opposite of what we need to produce learners who can think critically, synthesise and transform, experiment and create. We need a flexible curriculum, active cooperative forms of learning, opportunities for pupils to talk through the knowledge which they are incorporating. (Gipps, 1993, p. 40)

A curriculum focused on social justice involves helping students become aware that who is considered knowledgeable and who is not reflects the social positions and power of the knower; that “there are numerous hidden and uncritically accepted assumptions ... about human nature and the natural world ... [that] legitimate ways of experiencing and comprehending reality” (Restivo, 1983, p. 130); and that, unsurprisingly, those assumptions and ways of experiencing and comprehending are biased in favour of the current hegemony.

Empirical Work As part of a study investigating characteristics of mathematics teachers working for social justice (see Povey, 1995, for a full account), I asked a small number of beginning teachers to locate themselves on a chart of “one person's way of characterising different outlooks on maths and maths education” (Table 1) and then to use this and other prompts for reflections upon, amongst other things, the nature of mathematical knowledge. This was drawn from Paul Ernest’s typology of ideologies of mathematics educators. The responses from two teachers who came to be styled as working for emancipatory change – Frances and Matthew – are reported here, briefly, contrasting them, even more briefly, with responses from three other beginning teachers – Janet, Kevin, and Simon. (Beth, who features later in the article as one of the four teachers broadly unsympathetic to a critical perspective, was the first to take part in the research project and this task was missing from the data collection instruments at that stage.) The teachers were asked to mark on the chart “any of the sections that you think fits you well” and also to mark “any of the sections that you think is definitely not you”. The main purpose was to provoke the more extended reflections which follow but Table 2 gives an indication of the results.

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Table 1 Views of Mathematics and Mathematics Education 1

2

3

4

5

View of mathematics

Set of truths and rules

Body of useful knowledge

Body of structured knowledge

Process view, personalised maths

Socially constructed

Mathematical aims

Basics, numeracy

Useful maths to appropriate level and certification

Transmit body of mathematical knowledge

Creativity, selfrealisation through mathematics

Critical awareness and democratic citizenship via mathematics

Theory of learning

Hard work, effort, practice

Skill acquisition, practical experience

Understandin g and application

Activity, play, exploration

Questioning, decision making, negotiation

Theory of teaching mathematics

Transmission

Skill instruction, motivate through work relevance

Explain, motivate, pass on structure

Facilitate personal exploration, prevent failure

Discussion, conflict, questioning of content and pedagogy

Note. Based on Ernest, 1991, pp. 138ff.

Table 2 Individual Responses to Table 1 Responses to columns 4 & 5

Responses to column 1

Student

Positive

Negative

Positive

Negative

Frances Matthew Janet Kevin Simon

6 7 3 2 3

0 0 1 1 3

2 0 2 2 1

1 3 0 1 0

As expected, none of the teachers fitted neatly into just one of Ernest’s ideologies: yet there are clear patterns of difference already apparent between the two groups of teachers (with, perhaps, Kevin the most ambivalent). In the rest of the article, interview data and the teachers’ written reflections are drawn on in order to exemplify how this difference is borne out in the teachers’ thinking. These are organised around four themes: (a) the nature of mathematical knowledge, (b) how beliefs about the nature of mathematical knowledge affect pedagogical practice, (c) explanations of the roots of students’ failure, and (d) the role of initial teacher education.


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What is Mathematics? These first extracts from those discussions show the teachers who have been characterised by the researcher as working for emancipatory change grappling with the nature of mathematical knowledge. Their approach is mostly tentative but comes within a non-absolutist paradigm. The philosophical problem of the relationship between mathematical structures and the real world is unresolved but the existence of historical and cultural imprints on mathematics is recognised. Both see mathematics as something best described as a human product, either choosing to emphasise its personal construction or its social dimension. Frances:

Do triangles exist? [laughs] I started to think about the difference between maths and science ... You always hear it quoted that maths is a tool ... and starting the course ... neatly disposed of that idea and the maths became something in itself but it wasn’t science and it wasn't like science so what was it? Was it something that was objectively out there and you had to discover it, which I suppose is more like science, so do triangles exist in the universe and human beings have got to find them. Or is it just totally made up by people right from the start. And I think I’d been holding onto the idea that it’s totally made up by people right from the start for a very long time until I came across, back to pi, and then I found myself saying this is the only bit of maths that I think is like science because pi is just there and people found it and I don’t think I think that people made it up at all, it just seems to be there but that makes it stand out from the rest of maths. So maybe it isn’t maths, maybe we should get rid of pi out of the maths curriculum, it’s not allowed to count, so I still think about that because there are other things like pi really, geometric things and number patterns. But then I don’t know about number patterns because you wouldn’t have the number patterns if you hadn’t decided on the numbers in the first place which is quite, if you could get people to sit down and think about it, is quite an exciting thing really because if you start off by saying that we’re going to count because counting is useful and serves a lot of purposes so we devise a system of counting or tallying in some sense but you then discover that all these strange amazing things happen with these numbers that you’ve devised and where do the strange amazing things come from?

Matthew: What I think today, and I might not think it tomorrow, is: it’s a game. You play your game with symbols and ideas by moving symbols and ideas around, so I suppose that’s seeing it as something that’s independent of us but dependent on us as well. So it’s obviously got some independence, it’s an objective description of something and helps to describe the world around us, a language that helps to describe the world around us and that’s objective but at the same time that description has a particular bias and so on.

The other new teachers locate themselves differently with respect to their view of mathematics. Janet’s response is emphatically to deny that mathematics is socially constructed. Simon too rejects “socially constructed”: he considers mathematics to be a body of structured knowledge. He adds Simon:

I would once have argued more for the social construction of mathematics. I no longer believe this, it seems too wishy-washy.

For Kevin, too, mathematics is a body of useful knowledge, a body of structural knowledge and means “numbers and their manipulation, both in concrete and abstract terms”.

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How Do Beliefs about the Nature of Mathematics affect Pedagogical Practice? The teachers working for emancipatory change see their understanding of the nature of mathematics as inextricably linked to their aims, motivations, and practice: they see explicit links between epistemology and pedagogy. Matthew claims that his view of mathematics as both socially constructed and also as a way of describing the world affects what he does in the classroom. Matthew: It’s not like we have mathematics in there waiting for us to discover it, that isn’t the way it works ... the way my ideal about learning is moving ... is much more about kids or individuals building a mathematical structure around them which isn’t one that’s discovered and it isn’t one that’s either discovered in themselves or discovered out there but it’s a structure which is developed ... [that affects practice], it must do and it does at all sorts of levels. It affects it in terms of, where possible, … you’ve got some problem to solve and you get the maths along the way in the process of solving the goal so in that sense the kids are constructing the mathematics. But at the same time it’s to do with describing or trying to solve a problem in the world, the outside world. That’s on a very deep level. On a more basic level, I try and get them to puzzle, to think, answering questions with questions and so on.

He is aware that alternative perspectives amongst his colleagues are in turn related to their classroom practice and that changes in the former are necessary if one wants to achieve change in the latter. Matthew: ... actually in terms of changing people’s deep perceptions about, about mathematics, and so on, and about what we should be doing .…

These teachers speak of the need for their students to work in a way which permits the construction of meaning, of not having the curriculum broken down into pre-digested bits. Matthew: I’ve tried to hold on to not to spoon-feed, not give things piecemeal in tiny pieces. Frances:

I gave them a problem to do with fractions ... and I gave them two weeks to do and they kept coming and talking to me about it and that was a massive change ... One of the things I want to encourage is for them to do longer term pieces of work ... A lot of what they do is disjointed and quick and doesn’t come together.

Matthew: Start with the topic of say, suppose we’re doing some work on circles and I start with the topic of that and have some doing some work on, you know, just drawing circles, circle patterns, others finding out something about pi, others finding out something about the area of a circle, and have that full range and start with a topic and just go with it ... I think in terms of what I do in the classroom at best there’s discussion with them to try and motivate, discussion of targets and goals.

They relate this to letting the students develop ideas, of allowing the students to have some control over the agenda (Skovsmose, 1994). Flexibility and responsiveness to the students are valued and there is an openness to negotiation. This is linked with the rejection by these teachers of a transmission model of teaching and learning (Burton, 1992b; Skovsmose, 1994): Frances says “I don't think mathematics is a body of knowledge so I can't transmit it”. She adds,


Frances:

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I've had confirmed that standing up at the front of the classroom really doesn't do anybody any good whatsoever, [laughs] that it's a fairly useless teaching method to be honest and that I'm not sure that anybody really gets taught anything.

Janet too rejects a transmission theory of teaching but chooses in its stead “explain, motivate, pass on structure”. Kevin also feels happiest with this theory of teaching. He couples this with a statement about “discovery learning”. Kevin:

I feel that, where possible, children should discover mathematics for themselves. Having discovered it for themselves, I then explain that actually someone else discovered this before and here is the appropriate way to write it down.

But there is no sense here of a personal and creative act (see Papert, 1972, p. 236), nor of developing the capacity to criticise and produce classroom meanings. He claims that “investigational methods” are “one of his preferred ways of working” but adds “I also like ‘chalk and talk’ a lot”. His overall view might be encapsulated in the following remark: “I enjoy starting with a blank class and making it understand”. Both of the teachers working for emancipatory change place emphasis on, and value, student discussion and seek to develop it within their classrooms. Matthew:…within that, conversation and talk are really important ... In terms of teaching and how you actually teach in the classroom you might have what is a textbook exercise and you can turn that into a discussion exercise ... quite simply.

Student talk was not valued by all the new teachers. Discussion and questioning were marked by Simon as being definitely not representative of his outlook and Beth said, Beth:

[Group building?] I don't build any of my groups ... like I don't do much group work at all, the kids talk too much.

The quest for a curriculum which encourages critical thinking is linked by Matthew to a problem based approach to mathematics (Giroux, 1983): Matthew: [I try] giving problem solving tasks, goal orientated … the sort of taskorientated, what-do-you-need-to-know-in-order-to-get-to-a-particulargoal which is the way my ideal about learning is moving.

Frances, in promoting critical thinking, places emphasis on the work being such as to challenge the students and is aware that this is often lacking. Frances:

It seems to me that for a lot of them what happens, what their experience of maths is in secondary schools is that they get given things they can already do essentially because they get the satisfaction of doing it right and your classroom control is far, far easier, and that for a lot of them they get a diet all the way through of things they can already do, to the extent that the worksheets in year 11 are the worksheets in a sense of year 7 really.

Frances and Matthew want to foster in their students the same confidence and personal authority that they themselves feel where one grapples with difficulties, comes to conclusions which later have to be revised, where getting things wrong and changing one’s mind are fundamental to the process.

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Frances:

Some of the things I want to get across to them will be unsettling to them, that writing neatly isn’t what it’s all about, that I want to see their working, that I don’t want it to be hidden, or just the idea that when they actually get things wrong that that process of getting things wrong and then sorting it out is actually when they’re learning [laughs] that they’re learning by getting things wrong and it’s not something that they should tear the page out of their book and try to pretend it never happened ... I often say in the classroom when I’m presented with beautifully neat books, usually the girls’ with beautifully neat answers written and there’s a pile of stuff in the bin, that the stuff in the bin is their maths and I don’t want to see the beautifully neat answers so then they get cross with me. Because all the scribblings and the crossings out and the getting it wrong and doing it again is their maths, it’s the process of thinking it through and starting on a track and giving it up and going down a different track.

Because they wish to foster autonomy and personal authority in their students, their attitude to classroom management is not disciplinarian. Matthew: I’m thinking that as this pressure [of larger classes] increases, you can retreat but it isn’t a solution, a more discursive style is needed more or else you just get into a battle trying to get the kids to do things they don’t want to do, you end up rowing and shouting. Frances:

I’ve learnt to avoid confrontation whenever possible, if there’s a way to sort things out without confrontation.

This perspective is not shared by others. Simon:

Kevin: Beth:

There’s the problem of being a new, green teacher who cannot yet anticipate all the fiendish ways students will find to undermine you ... It has made me into a much more suspicious and cynical person than I used to be and I now understand why ‘real’ teachers on TP [teaching practice] used to demonise children ... I didn’t expect to become quite the authoritarian I am now. It became obvious to me though after a few months of teaching that I needed to feel in control. I am more of a disciplinarian than perhaps I thought I might be. I think I am quit e an authoritarian person to be honest. I mean I keep saying please and thank you [to the kids] and it really bugs me. I think, I mean [I want to say] ‘just do it’ you know. Like I’m really nice, I used to be really cold in the classroom on TP, I was told don’t tell the kids off so harshly. Like in a school like this, you need positive discipline, but what’s positive discipline or is it just weak discipline? ... There’s no school rules, the staff and the pupils agree, you know, agreement that they’ve come up with, that they shouldn’t chew in class, and they shouldn’t hit people in class and they shouldn’t be offensive to other pupils or staff. But you know there’s no real school rules and in terms of sanctions there’s no sanctions apart from your own sanctions that you impose. [It’s not enough.] ... I don’t think the system deals with [the kids] well. Like noone here will have a confrontation here, there’s real avoidance of confrontation in the school, like everything tries to be smoothed over.

Why Do Students Fail? The epistemological perspective of the teachers working for emancipatory change implies a respect for all human beings since they are the originators and final arbiters of knowledge. Consequently they do not blame their students for failure (see Lacey, 1977). Rather they look to structural reasons for failure:


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Matthew: As teachers you're in a hierarchical institution with kids at the bottom of the heap so again they fail because it ain't set up for them to succeed, that isn't the objective of school. I mean it is in one sense, it isn't in the sense of really wanting them to achieve because if it was then it would have to be a very different sort of place. Frances: We make it so hard for them to succeed. There's about 55 million ways they can fail, and it's an eye of the needle to succeed, it really is … This week I had two classrooms that were so alive with anger and fury 1 because I was told to set a test with my year 10 sets and it was GCSE questions on topics that they hadn't necessarily covered, maybe not even this year let alone recently, and it was real, photocopied GCSE papers which is quite threatening and it was the same test for [all the] sets and the same amount of time and they did appalling badly ... and each time when I gave the papers back it was just awful and it was even more awful because I was on their side and I understood entirely why they felt the way they did, I totally sympathise and yet from their point of view they just got at me with fury for the whole lesson. And I just felt very sad and thought why on earth are we doing this ... I wonder that any of them get anywhere.

Specifically on the issues of how students are grouped and organised within a school the two teachers voiced a commitment to not imposing groupings on students according to their attainment, this on the grounds of human dignity, linked to the idea that such groupings reflect feudal (Tahta, 1994, p. 25) notions of inherent ability. Nothing in the interview schedule referred to the issue of mixed ability teaching and no questions were asked about their views. Both, however, volunteered reflections which challenged, at times, the notion of inherent ability and, consistently, their schools' practice of setting the students in mathematics. Matthew: Certain principles I keep coming back to and that I think are worth defending, principles of fairness and equality, that every person's worthy of respect, those sort of ideas, then they really inform my motivation to carry on saying I don't want to see streaming and I don't want to see setting.

The other teachers speak differently. They locate the cause of failure in the students’ intellectual shortcomings, their moral turpitude, and so on. Beth:

Kevin:

Beth:

1

I try really hard with [my year 9 second set] 'cos you know they're not very bright but I think they're really enthusiastic about maths ... but like they're bursting out of their seats some of the time and they're the kids that can't do it, and I hate it. [Children fail] for any number of reasons: bad teaching, lack of motivation, social problems, expectations of parents (both too high and too low), lack of relevancy of courses. More importantly lack of ability some children will never succeed in academic life no matter how much support and help they receive. Some of the kids here are really horrible and they get away with it … I think the kids are quite lazy, quite laid back, don't do homework. I think they're a bit arrogant and lazy and I don't think the system deals with them well. Like ... everything’s by like appealing to their better nature. But kids just, like, just haven't got a better nature you know.

GCSE is the national examination taken in year 11 when the students are 16 years old .

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The Role of Initial Teacher Education The teachers working for emancipatory change seem to find their initial course of teacher education both relevant and moulding of their views, a perspective not shared by the other new teachers (McNamara, 1976; Cole, 1985). Matthew feels that the course equipped him to resist some of the pressures attendant upon work in school: Matthew: The path of least resistance is to revert to tried and tested, how you were taught, all that sort of stuff. I suppose in terms of the course, the course has given me a lot of options in terms of going back and saying hold on a minute this isn’t necessarily the best idea in the long run or even in the shorter for that matter.

Messages from the hidden curriculum are acknowledged as being central. Matthew: [The message from the course was] mathematics is something enjoyable, that it should be displayed ... You get a sense of there being some democracy in terms of the way things are decided ... I think within that, in terms of relationships, with students there was efforts made to give a sense of equality, equal status. [As a teacher] it was motivating and still is in the sense, in the sense that it’s important to know that there are, that it’s possible to work in that way ... and that general message of valuing mathematics.

Matthew and Frances talk about the course as though what they most feel they gained from it was that it had opened doors into a new vision for them, particularly a new vision of mathematics. What seems to be a key element in their experience has been pursuing subject studies as part of their initial teacher education. When asked to look back at the course and review it in the light of their experience now as practising teachers, these teachers both immediately speak about mathematics sessions and the ways the course has made them see mathematics differently. Matthew: [The course made me see things differently] certainly in terms of mathematics. A very different way of learning maths. For example, the tiling activity we did right at the beginning ... lots of other occasions. You can’t rely on that intellectual approach, a more intuitive approach has value. That has personal implications about infallibility and about valuing intuitions and so on. A very strong message ‘It’s OK to be wrong’. This did come from the course. Frances: I suppose there are times when, with particular issues, you remember a conversation or whatever ... [but] it’s more an opportunity to learn maths and to talk about learning maths with the group of people that I was with who were, in a way, who had a whole range of ideas and backgrounds which made it even better, our reactions to learning maths were very different so we were able to sort of talk about ... and I suppose ... well it did change my view of maths quite a lot and I’m sure that must and definitely does influence what I’m like in the classroom. Frances: When I embarked upon the [course] I imagined that the first year would be a straightforward revision of maths; a subject which I considered to be elegant, objective, and ‘issue free’ ... It is impossible to exaggerate the revolution that occurred in my thinking and outlook as a result of the teaching I received in the Maths Education Department ... it was particularly fascinating ... to debate whether maths can be said to have any objective reality ... I now believe that the neglect of the history of ideas is one of the greatest downfalls of current teaching. Children are


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too often presented with a fait accompli such as Pythagoras’ theorem without any mention being made of its development and significance in a variety of cultures ... Having discovered that maths was more than just elegant I was then to discover that it is far from ‘issue free’ ... the way we were taught, the resources we used, and the reading/reflecting we were encouraged to do all stimulated thought and I now see how many questions were planted during the first year which I have never stopped considering.

Commentary This article has offered empirical support for the claim that being a critical mathematics educator is associated with a particular epistemological stance with respect to the nature of mathematics. All the teachers except Kevin (who had a first degree in mathematics) had followed a graduate two year route of initial teacher education. This course is for those whose first degree contains some mathematics but not enough to allow them to enter the one year graduate route; therefore as part of their initial teacher education, they study mathematics for one year. Drawing on the thinking of these particular teachers, it is apparent that, for some, this experience of studying mathematics within the teacher education context, rather than outside it, was significant in shaping their understandings of the nature of mathematics and of teaching and learning. The claim here is not that pursuing subject studies within initial training guarantees the creation of critical mathematics educators: clearly this was not so for some of the research participants. Rather the claim is that it supports the effective development of those open to such a perspective and helps them in the process of connecting this with their practice. If these perceptions were to inform teacher education, we would no longer be happy with an initial preparation for teaching in which subject studies of an appropriate length and to an appropriate standard are undertaken disconnected from professional training.2 Within courses of initial teacher education which include a subject studies element, the integration of professional and subject studies can be used to provide rich opportunities for students to rethink their understanding of the nature of mathematical knowledge and their ways of knowing related to teaching and learning based on their own practices as learners. Few such opportunities exist within the dominant mode of initial teacher education in the United Kingdom. Similarly, many other countries, including some states in Australia, adopt these one-year end-on models. This is a one-year training route which takes “subject knowledge” as a prerequisite: as Burton notes, since prospective teachers mostly come from successful experience of didactic learning … [having] already gained a degree in mathematics, and their teaching practices are within the same mode, any questioning of its efficacy by the teacher education institution is more likely to fall on deaf ears. (Burton, 1992b, p. 380)

2

Such modes of initial teacher education are prevalent and often taken for granted in the United Kingdom. For example: “Once trainees have become secure in their subjects as we recommend, classroom skills are at the heart of teacher training” (National Commission on Education, 1993, p. 214) and, with respect to subject knowledge, “much might be covered through the use of supported self-study and through guided reading prior to the course” (Teacher Training Agency, 1998, p. 4).

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The single year is already too full adequately to allow students the opportunity to stand back from their practice and as such seems to discourage student reflection: there is simply no room to expand this curriculum to include a significant element of subject studies. On the data presented, the hypothesis that re-viewing subject studies as important to the epistemological perspective of the becoming teacher seems plausible. If these observations are sound, those who are concerned that initial teacher education should support a critical perspective and are concerned for social justice in mathematics education need to engage with what is happening in subject studies as well as pedagogical ones and, where possible, promote their inclusion in the initial teacher education curriculum.

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Skovsmose, O. (1994). Towards a philosophy of critical mathematics education. Dordrecht, The Netherlands: Kluwer. Tahta, D. (1994). Coming up to Russian expectations. Mathematics Teaching, 1(146), 5-9. Teacher Training Agency. (1998). Initial teacher training national curriculum consultation: Secondary mathematics. London: Teacher Training Agency.

Author Hilary Povey, Mathematics Education Centre, Sheffield Hallam University. Email: