Students’ positionings with respect to cultural models of mathematics: a socio-cultural analysis

1. Students’ positionings with respect to cultural models of mathematics: a socio-cultural analysis Laura Black

2. Analysis of PREMA data 2 Key Concepts: • Positioning in relation to Cultural Models (Gee 1996, 1999) which draw on wider Discourses • States of Mind (Kleine, Bion – c.f. Waddell (1998)

3. Methodological Stuff… • Interviews as narrative biographies (Sfard & Prusak 2005) outlining self identity - the reflexive 'story of the self', that we tell ourselves and sometimes others • Students storying themselves within the interview & in response to the interviewer • But students are boundary crossers from other social practices and so these biographies are inter-textually connected to prior experience

4. The Discourses of the two students • Gerry – liberal humanist – wants to do art to express her creativity & individualism • Becca – also liberal humanist – enjoys maths and science but has not decided on future • N.B. in our sample of 50 students from the TLRP project this kind of discourse was evident in interviews with students from higher SES neighbourhoods – suggesting it maybe classed!

5. Cultural Models • “Everyday theories' which are situated in social and cultural experiences and which inform action (behaviour).” (Gee 1992) • Distributed threads • ‘cultural’, Discourse, Ideal • Elements that are used to construct/narrate one’s self – position the self in a ‘figured world’

6. What cultural models are drawn on in the narrative to construct this imagined leading identity? • Gerry on maths: • Maths lacks opportunity for creativity & individualism • Maths is easy and non-stressful • Being good at maths comes naturally for me • Doing maths successfully involves properly understanding • Maths is competitive • Gerry’s other cultural models: • I need to be challenged & challenge = being creative • Education for enjoyment & interest • Resistence to exchange value of educational qualifications

7. Maths lacks scope for creativity ….but I do, I really like to be challenged… um and I do enjoy writing the creative aspect… as you can see of the other subjects. But I still like, yeah, I still found maths quite… like almost relaxing you know its not stressful really because… its just, you don’t have to um you don’t have to sort of create. So that would sort of appeal to some people.

8. Doing maths involves understanding • I have always found it much easier in maths and its probably why I’m good at it, I have to understand why something works because then I can work it… out whereas if I’m just memorising a formula it means nothing to me. So I’ve always like it when teachers can really explain something so you think then. And that’s the best way. • I asked questions a lot when I didn’t understand ??? which may have made me look more stupid a bit sometimes but actually normally no one else would understand so it was good to ask questions.

9. Education for Enjoyment • like I said I am not one to take something just because I’m good at it. Even if art history I know is quite a big challenge because it’s a public school subject… and we only have maybe three people in our class but um, [J:yea] you know it’s a difficult exam but because I’m really interested in it I’m sure I’ll do much better than I would otherwise… [J:yea] because I don’t [J:yea] you know [J:yea]. I think it’s more important to do what you enjoy, that’s all. [J:yea][J:yea]

10. Resistance to exchange value of maths • I suppose you get pushed more into academic… if you’re good at something like maths I think… it’s pretty good these days but even if its rubbish people like the idea of you doing something that’s more academically rigorous [J:yea] IF you’re good at maths [J:yea] you’re you’re encouraged that way …um… so that is a difference. [J:yea] You’re taken a bit more seriously. [J:yea] • J: So talk to me about why doing well in art, art history is important? C: Um, because I enjoy it, that’s why it’s important to me.

11. What cultural models are drawn on in the narrative to construct this imagined leading identity? • Becca on Maths • Doing maths successfully involves the need to understand & connect ideas • Maths as disciplined & ordered • Maths as enjoyable • Maths has exchange value • Maths isn’t hard

12. Doing maths involves understanding • J: What makes YOU good at Maths? So you got encouragement then? • Yeah. Erm, just just wanting to know what was going on, how things are, how things are found out. …I wanted to know the reasons for things I guess. ... There’s a formula, someone gives you a formula and there like here’s pi it like why, why would you use pi? There must be a reason… its like Pythagoras, well how does that work why does that work? And just just working out how it works which is really interesting.

13. Maths as disciplined & ordered • Erm… most interesting… it’s a very hard question… probably formulas just the way the formulas work and… now I am on A level we learn about theorems like… looking at people’s theorems about…and we look into those it’s just amazing how someone can sit there… for I don’t know how long… and work out these things and give a specific number which works out every time… it’s just its just amazing how they do that… and its good that we get we get shown it and… then shown how it works and its just just amazing… seeing how it works and then see it and then we can use it now, still from years and years ago when it was made, I mean I love learning that kind of thing.

14. Maths as Enjoyable • J: So… Maths, Physics, Further Maths… was that… an easy choice to make? • Yes I think it was… em… maths is something I have always really enjoyed so I knew I always wanted to take maths and also an extension of maths, is something I would love to do… and physics… was… because… of the subject I enjoyed as well. I wanted to take two things I enjoyed … [J:yea] just to make sure that then I’d…mm… hopefully um want to do one of them .???eventually [J:yea]

15. Maths has Exchange Value • I enjoyed maths and wanted to get good GCSEs I did it at A level and I wanted to work hard just…just to be good at it just to say I’ve got an A level in maths. Just to show people… and then further maths is like well that’s even further into the maths world [laughs] hopefully that will be [J: yea] like something people [J: yea] will be really [J: yea] impressed with. • J: Would being good at art and humanities, is that important to you? • Not as a, ..not as an A level or degree, - as a hobby… if I enjoyed art I’d love to have something to do, on a CV thing I could say that ‘Oh I got an A in my er A levels in maths and physics and further maths… and I enjoy art and things like that, but then… I don’t know how much of a difference it would be if I had an A in art and things like that and then…say I enjoyed mathematics… it wouldn’t be the same thing would it? [J: yea] It would be kind of [laughs] yeah ok. [J: yea]

16. Why do these students position themselves in alignment/disalignment with these cultural models? • Socio-cultural explanation might focus on mediating influences (e.g. parents) • Discursive approach might focus on power relations • Psycho-analytic offers an account of positioning which recognises internal ‘states of mind’ characterised by defenses, anxieties and types of relationship (Margot Waddell) • This offers an account which blurs the boundary between our historical and socio-cultural experience with the external world and the notion of the mind • States of mind are rooted in the past and encompass a possible future

17. Psychoanalytic themes Similarities across both accounts: • A need to understand & its association with maths • Reparation – fulfilling parents incomplete dream (altruism) Differences between the accounts: • Becca – self preservation (egoism) – ‘you can’t say you enjoy maths as a hobby’ – hence the need to draw on exchange value of maths • Gerry – self preservation – avoiding competition & resisting exchange value

18. The need to understand • Giving shape/form to sense data of experience through use of symbolic terms (alpha function) • We do this throughout life & it is the means by which we come to make sense of ourselves in the world • As such, we develop a desire to understand rather than a need to know • Understanding is about extension of the capacities of the self rather than merely adding to a stock of knowledge (or qualifications)

19. Reparation & Redemption Becca • … and also when I was little my Dad… um he was brilliant at maths, well he still is.. and he always WISHED he’d taken it further, because he didn’t go to university. So he’s like ‘you know you’re good… you can keep doing it’, so he encouraged me a LOT with my maths, a lot… • He looks back now and he did an open university course in maths and got a brilliant grade on that and he’s like I wish I did it so he’s like just ‘Oh do it otherwise you’re just going to look back and be like oh I wish I did it now’. And em for a long time he wasn’t really in the job he wanted to be in and now he is… and I guess for me he wants me to go into something that…that I’m going to enjoy for my whole life so I guess… yeah, I guess I think that if I get good grades, then I’ll… even if I don’t know what I want to do I will have opportunities to do things… with a good grade.

20. Reparation & Redemption Gerry mmm…Probably because both of my parents are very artistic um they were… sort of you know when they were younger they were…obviously art was not the thing to do and they were pushed away from it and they never even did it until they took it up about fifteen years ago [and made their own business, which is nice. And now my Dad paints a lot… and I’ve always grown up... with a lot of classical music from my dad and a lot…the art cupboard was our playground not a computer [C:laughs]

21. Self Preservation • Becca - I can have art as a hobby but not maths • Enjoyment of maths as threatening to the self (egoism) • Gerry – the need for individualism & avoiding competition • Self preservation afforded by a liberal humanist discourse

22. Avoiding competition • Erm… it’s a bit different for… art based subjects I think because…there’s there’s more variation in art subjects, [J:yea] which areas you can be good at and things. Stuff like maths and I suppose English and sciences its more clear cut the route to achieving well. I think its more competitive… because you’re doing exactly the same things at GCSE. • I suppose it’s more clear you know if there’s always one person who’s always really fast at doing all of their …sums but then there are other people like me who go more sort of slowly and get good marks. …English is a bit more vague I suppose and art is just down, its just…art is quite free in our school when everyone develops their own…[J:yea] sort of personality in that subject. Which is not to say maths is bad its just more clear to see.

23. Final Thoughts… • “….States flicker and change with nuancies of external forces and relationships – forever shifting between egoistic and altruistic tendencies” (Waddell 1998, p9) • Altruistic state of reparation invoked by relationship with parents • Becca’s self preservation motivated by positions available in a wider discourse?? • Gerry’s self preservation afforded by her liberal humanist discourse • This begs the question - which comes first? positioning or state of mind? A constant dialectic.