‘Do triangles exist?’: the nature of mathematical knowledge and critical mathematics education
Hilary Povey
Sheffield Hallam University



After referring briefly to the argument that being a critical mathematics educator is associated with a particular epistemological stance with respect to the nature of mathematics, I offer case study data exemplifying this connection in the thinking of beginning teachers. I note that pursuing mathematical subject studies as part of one’s initial teacher education course allows students to explore alternative mathematical epistemologies than the ‘common sense’ one currently prevalent and suggest that placing subject studies within teacher education rather than separated from it may therefore have a role to play in supporting the development of critical mathematics educators.

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, chapters 1.1 and 1.2.).

In a classroom predicated on such an epistemology, the teacher will ‘constitute mathematics as the activity of following procedural instructions’ (Cobb et al, 1992, p573), such procedures being regarded as ‘ahistorical, unalterable norms ... [without] a specifiable source ... both fixed and self-evident’ (Cobb et al, 1992, p588f). The model of learning will 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 and Dowling, 1990, p2). In this ‘banking’ (Freire, 1972, p46) mathematics, knowledge is seen as hierarchical, the teacher’s mental models are superior to the learner’s and education becomes the process of implanting those models in the mind of the learner. Knowledge is validated by external authority.

An alternative epistemology is based on a view of mathematics as being ‘co-constructed’ (Cobb, et al, 1992, p573) 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, p141). 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, p2). 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).

As part of study investigating characteristics of mathematics teachers working for emancipatory change, I asked a small number of beginning teachers to locate themselves on a map of ‘one person's way of characterising different outlooks on maths and maths education’ (Figure 1) and then to use this and other prompts for a discussion amongst other things of the nature of mathematical knowledge. I report here, very briefly because of space constraints, on the responses from two teachers working for emancipatory change – Frances and Matthew – contrasting them, even more briefly, with responses from other beginning teachers – Janet, Kevin, Simon and Beth. (See Povey, 1995 for a full account).

Social group
Industrial Trainer
Technological Pragmatist
Old Humanist
Progressive Educator
Public Educator
View of mathematics
Set of truths and rules
Unquestioned body of useful knowledge
Body of structured pure knowledge
Process view: personalised maths
Social constructivism
Mathematical aims
'Back-to-basics': and social training in obedience
Useful mathematics and certification (industry-centred)
Transmit body of maths knowledge (maths-centred)
Self-realisation, creativity, via maths (child-centred)
Critical democratic citizenship via mathematics
Theory of learning
Hard work, effort, practice, rote
Skill acquisition, practical experience
Understanding and application
Activity, play, exploration
Active, questioning, empowerment 
Theory of teaching mathematics
Authoritarian transmission, drill, no 'frills'
Skill instructor, motivate through work-relevance
Explain, motivate, communicate, pass on structure
Facilitate personal exploration and prevent failure
Discussion, conflict, questioning content and pedagogy.

Figure 1: Views of mathematics and mathematics education
(Based on Ernest, 1991, p138f)

What is mathematics?

In these first extracts from those discussions, the teachers working for change grapple 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 of them see mathematics as something best described as a human product, either choosing to emphasise its personal construction or its social dimension.

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 its 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? (Frances)
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 (Matthew)

New teachers whom I have not characterised as working for change 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

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

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’.

How do beliefs about the nature of mathematics affect pedagogical practice?

The teachers working for change see their understanding of the nature of mathematics as inextricably linked to their aims, motivations and practice. Matthew claims that his view of mathematics as both socially constructed and also as ‘a language that helps to describe the world around us’ affects what he does in the classroom.

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 trying, where possible, giving problem solving tasks, goal orientated so 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 (Matthew)

He is also 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.

... actually in terms of changing people’s deep perceptions about, about mathematics, and so on, and about what we should be doing ... (Matthew)

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, and 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.

I’ve tried to hold on to not to spoonfeed, not give things piecemeal in tiny pieces (Matthew)

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 (Frances)

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 (Matthew)

This is linked with the rejection by these teachers of a transmission model of teaching and learning (Burton, 1992, Skovsmose, 1994): Frances says ‘I don't think mathematics is a body of knowledge so I can't transmit it’. Janet too claims to reject 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’;

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 (Kevin)

but there is no sense here of a personal and creative act (see Papert, 1972, p236), 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’.

All of the teachers working for change place emphasis on and value student discussion and seek to develop it within their classrooms.

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 (Matthew)

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

[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 (Beth)

The quest for a curriculum which encourages critical thinking is linked by Matthew to a problem based approach to mathematics (Giroux, 1983):

[I try], where possible, giving problem solving tasks, goal orientated so you’ve got some problem to solve and you get the maths along the way in the process of solving the goal … the sort of task-orientated, what-do-you-need-to-know-in-order-to-get-to-a-particular-goal which is the way my ideal about learning is moving (Matthew)

and also by Frances who places emphasis on the work being such as to challenge the students and is aware that this is often lacking.

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)

They 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.

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 (Frances)

Because they wish to foster autonomy and personal authority in their students, their attitude to classroom management is not disciplinarian.

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 (Matthew)
I’ve learnt to avoid confrontation whenever possible, if there’s a way to sort things out without confrontation (Frances)

This perspective is not shared by others.

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 t.p. 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 (Simon)

I am more of a disciplinarian than perhaps I thought I might be (Kevin)

I think I am quite 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 no-one here will have a confrontation here, there’s real avoidance of confrontation in the school, like everything tries to be smoothed over (Beth)

The role of initial teacher education

The teachers working for 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 (cf McNamara 1976, Cole 1985). Matthew feels that the course equipped him to resist some of the pressures attendant upon work in school:

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 (Matthew)

Messages from the hidden curriculum are acknowledged as being central.

[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)

They 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 all immediately speak about mathematics sessions and the ways the course has made them see mathematics differently.

[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 (Matthew)
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 PGCE 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 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. (Frances, final Education assignment of PGCE, sent to me after the interviews)

Their perception seems to be that studying mathematics within the teacher education context, rather than outside it, has been 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: rather 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 the mode of preparing to teach, prevalent and often taken for granted in the United Kingdom, of subject studies of an appropriate length and to an appropriate standard being undertaken before embarking on professional training. For example, the National Commission on Education writes:

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, p214)

and the intended National Curriculum for initial teacher education, whilst allowing that some subject knowledge might be taught alongside pedagogical knowledge, nevertheless recommends ‘much might be covered through the use of supported self-study and through guided reading prior to the course’ (Teacher Training Agency, 1998, p4).

Within courses of initial teacher education which already include a subject studies element (two year PGCE, two, three and four year undergraduate courses), present national structures allow, although they do not require, the integration of professional and subject studies. This 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. Students can also be encouraged to mine the practices of their tutors for practical help (Sikes 1993) in implementing a liberatory mathematics curriculum.

No such opportunities exist on the one year training route which takes ‘subject knowledge’ as a prerequisite. 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 but, as Leone Burton has noted, if prospective teachers

come from successful experience of didactic learning, very likely for example in the case of the Postgraduate Certificate of Education (PGCE) student who has 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, 1992, p380)

If these observations are sound, then it will not be surprising if initial teacher education through the one year route does not have much success in enabling students to become teachers working for change.

Correspondence: Hilary Povey, Centre for Mathematics Education, Sheffield Hallam University, 25 Broomgrove Road, Sheffield, S10 2NA, UK. E-mail: h.povey@shu.ac.uk