1. The Discourse Strand: Whose Mathematics? Which Mathematics?
Uniting theme construction of maths and maths learners hence title.

2. Morwenna reminded us of this quote
Suspend your belief in the innocence of words and the transparency of language as a window on an objectively graspable reality.
Maggie Maclure (2003)
This is the over-arching theme in the discourse strand. Morwenna reminded us of it and Tansy picked this quote up too.

3. Major themes in this strand
The discourse of mathematics itself
Construction of mathematics as a subject
Subject positionings
These all deal with issues of power and control
I’m going to organise this overview along these themes – the presentations in the discourse strand tend to have focussed on one or other of these themes but of course what unites them is a concern with power and control. By the way, I should say that what I’m going to say about what people have said is of course my interpretation – I might well be misrepresenting them of course but I hope not! I’ve also used their slides where I can, editing them sometimes in order to fit everything in.

4. Steve Lerman – using Bernstein to understand power and control in pedagogy
Suppose you had a garden this shape and you were in a helicopter right above your garden looking down on it.
Which of the following shapes would be like yours?
Starting with Steve. In the Manchester seminar on pedagogy he focused on issues of access to the mathematics discourse and the discourse of mathematics pedagogy. He showed us this example from research with Robyn Zevenbergen. Of course, interpreting this question in the way intended requires more than an understanding of shape. Steve pointed out tht Bernstein provides a theoretical framework for what is going on when children engage with this kind of question.

5. Typical incorrect answers
Robyn: Why did you take that shape [the square]?
Girl: Because it looks like the shape of my garden.
R: Is your garden at home like that?
Girl: Yes.
Boy: None of those.
R: Why aren’t any of them the same?
Boy: My garden goes like that [draws a semi- circle in the air].”
So of course what happens is that some children produce answers like these. It’s reminiscent of work by Barry Cooper on social class and maths assessment, and also of Valerie Walkerdine’s very early work on the use of the shopping analogy to teach subtraction that she talked about in Cardiff. Getting the ‘right answer’ involves access to the discourse of maths and specifically maths as it appears in pedagogic contexts.

6. The production and reproduction of disadvantage
Forms of control (invisible/visible pedagogy)
Schools are middle class by definition: disadvantage reproduced
Language: elaborated and restricted codes
Principle of recontextualisation – play of ideology
So why do they make the error? Steve pointed out how Bernstein gives us theoretical tools to explain this and how disadvantage is reproduced. So seeing maths lessons in terms of visible and invisible pedagogies, issues of language and Bernstein’s idea of elaborated and restricted codes and the connection with social class is useful. In this clip he talks about how, when the task is made clear, children can demonstrate the knowledge that they have. These kinds of contexts don’t however give them opportunity to do this.
Steve also refers to Bernstein’s principle of recontextualisation - the creation of imaginary discourses from real discourses via the embedding of an ‘instructional discourse’ in a ‘regulative discourse’. Instructional discourse refers to specialised skills and regulative discourse to a moral order. For example, carpentry—a ‘real discourse’ is recontextualised in pedagogic discourse as ‘woodwork’—an ‘imaginary discourse’ and physics as done by university researchers is recontextualised as school physics.

7. The shopping sum – ‘You have 10p, a yacht costs 2p, how much change do you have?’ - illustrates the issue
Valerie Walkerdine made the point about the cross-over between mathematics and shopping practices in Cardiff. The main point is that although they are similar, they are fundamentally different. I think this might be an example of recontextualisation although it seems rather more complicated than that because we are also introducing metaphor here.

8. But there are limits to Bernstein’s analysis
Identities are multiple and overlapping in discourses, and his analysis helps us analyse pedagogic discourses, not others. For students in the classroom, how they appear to their peers, which behaviours will gain acceptance from important others, techniques to avoid being noticed by the teacher, issues associated with race, gender, ethnicity, sexuality, and so on are much more important than engaging in the pedagogic discourse.
Going back to Steve’s talk, though, he went on to talk about the project of the self in late modernity. As researchers we need to make sense of how different voices are produced. This is where I see the discourse strand taking us.

9. Tom on the curriculum: an analysis of textbooks
Are learners able to see self or their interests represented within the text book?
Broad political categories: gender, race, social class, sexuality, special needs – for example:
Do they favour boys rather than girls?
How stereotypical are they?
Is the textbook gender neutral?
Are there appropriate male and female role models?
What careers/ occupations are represented?
Other self-identifications related to inclusion or exclusion: cool, nerdy, logical, expressive, creative, clever
Text
Spacing & style
Colour/ B&W
Language
Images
Cartoon characters
Clip Art
Photographs of real people/ artefacts
Moving on, Tom and Morwenna talked about the curriculum in Edinburgh, focussing firstly on the language of textbooks themselves and how they related – or not – to student identities. Tom asked these questions. What part do textbooks play in the experience of studying mathematics and ‘framing’ of the curriculum?
These are the issues that Tom was concerned with. Steve also raised similar issues in referring to Cooper and Dunne’s work on assessment – Dowling also looks closely at maths textbooks and unpicks the issues of what maths curriculum is being presented to different groups of learners.

10. Morwenna’s interviews: Constructions of mathematics - people who can do it are…..
The art teacher:
Logical. Logical thinkers, perhaps absolutist.
The technology teacher:
Me because I have an engineering background
People who want to understand more about the world around them.
[There is] mathematics for thinkers and mathematics for doers.
Vocational maths I suppose would be engineering.
That’s how is maths framed in the textbook. Morwenna moved on to look at how maths is actually constructed, and she showed us interviews with teachers talking about maths. She drew these classic contrasts. The technology teacher identifies himself as able to do maths, but then goes on to draw a distinction between different kinds of maths for different kinds of thinkers – this is a pervasive discourse about maths of course.
Tansy illustrated a similar discourse about maths in her talk on teachers at Sheffield. She showed how assumptions about maths versus english come up in the official discourse of the teacher training report.

11. Construction of mathematics as a subject: Heather on “maths is hard!”
Independent research shows that Mathematics is the most challenging subject at A-level. Nationally, last year's AS results in maths were far worse than any other subject.
If you don't really enjoy Maths and if you're not genuinely good at it, don't do it! Two years of struggling and constantly being 'stuck' is not an experience we would wish on anyone.
Success at A-level Mathematics usually depends on:
Positive attitudes. Do you enjoy solving problems? Do you like Maths?
Persistence. Do you give up easily and ask for help? Or do you prefer to get the answer for yourself?
Independence. Do you need spoon-feeding every step of the way? Can you learn it by yourself?
Tansy’s observation on the perceived difference between maths and other subjects comes up again in Heather’s talk on the uniqueness of maths as ‘hard’, taken from school information for students who think they might take maths at A-level

12. Heather: ‘maths is hard’ relies on various discourses
hard vs. easy subjects…
knowledge is separable into different subjects that have stable identities and that can be arranged in a hierarchy
values and ideas of what knowledge is hardest, best, purest, most rational…
slippage into discourses around masculinity and sexuality, hard has unspoken opposites – easy, soft and yielding…
And so on to Heather herself.
Maths is hard:
relies on the discourse of hard vs. easy subjects…
relies on the discourse that knowledge is separable into different subjects that have stable identities and that can be arranged in a hierarchy, links to value and to ideas of what knowledge is hardest, best, purest, most rational…
relies on discourses of hardness, slippage into discourses around masculinity and sexuality, hard has unspoken opposites, the obvious one is easy but there’s also soft and yielding…
And so where does this leave the learner? Drawing on Foucault, Heather put her case for why a discursive approach helps us to understand the force of a statement ‘X doesn’t do their maths homework’. She argued that discourses do people, not round the other way – discourses construct the subject. They are the only way in which we can be, and they limit what we can be.

13. Candia on subject positioning in official discourse: the QCA example
Once the teacher has established what the pupil is to achieve and how the pupil can achieve it, the pupil is in a position to guide their own learning. The pupil can seek help from suitable sources such as books, other learners and the teacher. When they know what they are trying to accomplish they can forge ahead without reference to the teacher if that is appropriate. This frees the teacher to provide help where it is really needed. When pupils take responsibility in this way their performance standards can rise across the board. It is true that some pupils will resist this, wanting to blame the teacher rather than themselves for their lack of learning, but such methods are surprisingly successful if persisted with.
QCA (2003) Using assessment to raise achievement in mathematics at key stages 1, 2 and 3
And so to Candia, who also talked about how subjects are positioned and position themselves. She uses this example from the QCA to show how discourses construct subject positions and ascribe these to participants. Candia unpacked this piece of writing in detail.
NB: Discourses construct subject positions and ascribe these to participants.
Individuals participate in multiple discourses with different resources and sets of subject positions available. They constrain but do not determine individuals’ positioning.
But individuals also position themselves. This occurs through the choices (conscious or unconscious) made between the available resources as they seek to account for their practices.

14. Construction of ‘truths’ about the learner and the teacher
Resisting pupils…
Are deviant
Do not learn
Are to blame for their lack of learning
Falsely blame teachers
Will not succeed in their deviance
Teachers…
Are not to blame for pupils’ lack of learning - if they use the approved methods
The teacher prescribes the content of learning but at the same time there is a high value on independence. But pupils who resist are positioned as deviant, and to blame for their failures. Lack of learning is a nominal phrase – as a nominal phrase it exists. It’s a coercive discourse, not only on the students but also on the teachers. Teachers aren’t to blame for failure IF they use the approved methods. There is a lot of strong language which constructs this as an absolute truth.

15. Tansy: construction of the primary teacher - confidence and competence
What was done well in Number:- pupils are becoming more confident at solving problems involving missing numbers. (QCA 2004)
Here Tansy draws attention to another part of the maths discourse – this issue of confidence caused a lot of discussion at Sheffield. Taking a discursive approach allows us to question our understanding of the very process through which we know what we know about others, in this case primary teachers teaching mathematics.
This is what Tansy said:
How, from a written national test, can pupils be judged as more confident? There is a suggestion in the same report that further improvement is possible if teachers “encourage pupils to persist when solving problems, for example, checking that the answer makes sense in the context of the question”. Whilst I have no difficulty with advice that promotes the usefulness of this strategy, I am struck by the invisibility of the presumption that confidence can be implied from pupil responses that indicate ‘checking an answer makes sense’.
Discourse (including language, texts and practices) is seen as the resource through which we come to know ‘what we think we know about’ (Lather, 1991, 2004)

16. Neoliberalism, choice and mathematics education – Valerie Walkerdine
Nikolas Rose: in advanced liberalism we are compelled to be free - consumption brings the idea of freedom of choice – how is this played out in education?
Choice (for instance when shopping) is a product of a complex set of relationships that are not necessarily mathematical
These practices produce us as self- regulating and self-managing choosing subjects but the practices which produce that possibility and the discursive positions we can take up become obscured
Finally, Valerie Walkerdine’s presentation at the seminar on choice in Cardiff. Again, she is talking about subject positioning, and the idea of choice of who we want to be – the idea of choice makes us look as though we are self-regulating, so obscuring the kind of discursive positions that everyone in the discourse strand has talked about.

17. How ‘choice’ might obscure lack of choice
So she argues here that choice is a fantasy.
Choosing which ketchup to buy, the complex discursive organisation of that choice - everyone in the family’s tastes, a big bottle or a small bottle – all are discursively organised. But seeing this as freedom of choice obscures the discursive relations and the ideological relations through which people are enjoined to buy a particular brand. So everything is obscured to apparently produce for us a subject who chooses of their own free will. In relation to education it’s important in thinking about curriculum practices that are apparently based on choice.

18. The last word
So then the choosing subject is equally both a fiction in Foucault’s sense, produced through self management in Rose’s but also routinely assembled in relational dynamics, which involve fantasy positions (the winner, the rich shopper, the perfect mum) which are created in the social relations of everyday practices
So the last word to Valerie, in her own words. So the possession of power is not the possession of any individual, but is produced in social relations which we fail to see because they are constantly backgrounded.