Keeping open the door to mathematically demanding F&HE programmes Laura Black Pauline Davis Paul Hernandez-Martinez Graeme Hutcheson Maria Pampaka Su Nicholson Geoff Wake Julian Williams

2 Aim We aim to understand how cultures of learning and teaching can support learners in ways that help widen and extend participation in mathematically demanding courses in F & HE.

3 Programme effectiveness Classroom practices Learner identities

4 Jan 06 March 06 Preparation Sept 06 Programme effectiveness Classroom practices Learner identities Questionnaire design Pilot case studies June 07 Sept 07 Dec 07 (i) initial questionnaire (ii) post test (ii) delayed post test Case studies in UoM and traditional AS Follow up case studies (i) initial interviews (ii) interviews round 2 (ii) follow-up interviews Oct 06 Conferences Feb 07 June 07

5 Outcomes Knowledge about how mathematics teaching and learning cultures can support better participation in mathematics Knowledge about how mathematics teaching and learning cultures can support better participation in mathematics Measurements of the effectiveness of two distinctive programmes of mathematics on learning Measurements of the effectiveness of two distinctive programmes of mathematics on learning Development of theories of learner identities in maths contexts Development of theories of learner identities in maths contexts

6 Disposition to study more maths

7 Disposition to enter HE

8 Identity Questions What kind of maths learner identity are there? What kind of maths learner identity are there? How are identities ‘narrated’? (Bruner, 1996) How are identities ‘narrated’? (Bruner, 1996) What resources/CMs do students use in their identity work? What resources/CMs do students use in their identity work?

9 T & L Classroom culture Mathematical learner identity Programme institutional culture Technology rules of assessment problem solving Cultural models discourses

10 Cultural models ‘Story-like chains of prototypical events that unfold in simplified worlds.. (including) metaphor’ (HQ,’87) ‘they allowing humans to master, remember and use … knowledge requireed in everyday life’ “Everyday theories' which are situated in social and cultural experiences and which inform action (behaviour).” (Gee?) More?

11 Examples Is the pope a ‘bachelor’? (Fillimore) Is the pope a ‘bachelor’? (Fillimore) US campus ‘dating’ scene: ‘jocks’ ‘bitches’ ‘nerds’ etc (Holland & Skinner) US campus ‘dating’ scene: ‘jocks’ ‘bitches’ ‘nerds’ etc (Holland & Skinner) ‘Coffee’ (Gee) ‘Coffee’ (Gee)

12 CMs are: Distributed threads Distributed threads ‘cultural’, Discourse, Ideal ‘cultural’, Discourse, Ideal Elements that are used to construct/narrate one’s self Elements that are used to construct/narrate one’s self Fragmented Fragmented In our model: boundaries between classroom and storying of the self. In our model: boundaries between classroom and storying of the self.

13 Gemma’s story Draws on some positive ‘maths’ cultural models, but others that one might hope for do not seem to be available Draws on some positive ‘maths’ cultural models, but others that one might hope for do not seem to be available Cultural models can be ambivalent, i.e. used to tell opposite stories Cultural models can be ambivalent, i.e. used to tell opposite stories Models of maths learning may be influential but not necessarily ‘leading’ the story Models of maths learning may be influential but not necessarily ‘leading’ the story

14 Lee’s story Lee arrived in AS with stronger grades at GCSE but got ‘dropped’ Lee arrived in AS with stronger grades at GCSE but got ‘dropped’ Claims maths is hard and ‘boring’ Claims maths is hard and ‘boring’ He appears to have been marginal in his AS maths classes He appears to have been marginal in his AS maths classes

15 Some CMs evident in interviews Maths is hard but challenging versus maths is hard and dull Maths is hard but challenging versus maths is hard and dull Maths is black and white versus ‘your own’ Maths is black and white versus ‘your own’ Maths is ‘on your own’ versus ‘learning with others/ sociable’ Maths is ‘on your own’ versus ‘learning with others/ sociable’ etc etc

16 Does pedagogy make a difference? One notable contrast between Lee and Gemma: the sociability of maths for them One notable contrast between Lee and Gemma: the sociability of maths for them Might different pedagogies offer different CMs, or different positionings in relation to CMs? Might different pedagogies offer different CMs, or different positionings in relation to CMs? Over to Pauline Over to Pauline

17 Identity We build our identities (i) in practice and (ii) discursively using cultural models; We build our identities (i) in practice and (ii) discursively using cultural models; What models are there of ‘ways of being a mathematician/learner of mathematics?’ What models are there of ‘ways of being a mathematician/learner of mathematics?’ How can mathematics learner identity be mediated by mathematics classroom social practice? Can we expand the repertoire of cultural models? How can mathematics learner identity be mediated by mathematics classroom social practice? Can we expand the repertoire of cultural models?

18 Classroom Discourse/Practice

19 We often find student identities are double- discoursed in a genre adopting a dual register, one of student and the other (sometimes more quietly spoken and ‘hidden’) of everyday teenage talk, indicating tensions between these opposing voices. We often find student identities are double- discoursed in a genre adopting a dual register, one of student and the other (sometimes more quietly spoken and ‘hidden’) of everyday teenage talk, indicating tensions between these opposing voices. The micro data shows an alternative discourse where there is flip flopping between themes, maths and non-maths (every-day teenage) talk; The micro data shows an alternative discourse where there is flip flopping between themes, maths and non-maths (every-day teenage) talk; The tenor (or voice) remains broadly the same. …crazy 20 The tenor (or voice) remains broadly the same. …crazy 20

20 Cultural models associated with this classroom Maths as negotiable, not black and white Maths as negotiable, not black and white Maths as fun Maths as fun Maths as hands on/practical Maths as hands on/practical Maths as sociable Maths as sociable

21 KAnd like not only you think for yourself but like we can ask other people why they got that and it’s not just like black and white, like you get to a different way to work it out. KAnd like not only you think for yourself but like we can ask other people why they got that and it’s not just like black and white, like you get to a different way to work it out. J … it sounds daft but you’re having fun while you’re doing it cos you can sit and you can talk to people but… talk about the work but you can… it’s not a thing where you come down and sit in silence and you do it, you can talk to people and can, you know, do practical things J … it sounds daft but you’re having fun while you’re doing it cos you can sit and you can talk to people but… talk about the work but you can… it’s not a thing where you come down and sit in silence and you do it, you can talk to people and can, you know, do practical things

22 A : …more fun than just doing examples all the time and we have the whiteboards and like all the games that …[the teacher].. makes us play and like… it’s just fun, rather than just textbooks and notebooks all the time… K…so it’s quite good, you always know the faces and stuff like that, where in other lessons you don’t even know them, you don’t even know they’re in your lesson, so it’s really good.

23 Ownership, understanding/conceptual, fun, joint activity No, it isn’t just about having fun. I mean that obviously, it’s part of it, because, just number crunches on data can be sort of boring, can’t it? So, the fun element is a part of trying to make stats a bit more fun. It’s not my favourite topic of maths, I have to confess…But I also think that if you just write some numbers on the board and put a couple of extreme values for example, then well, what’s the point of them? [ ] So there’s an understanding of why there are these sort of extreme values, so even though it’s been …you saved yourself of data collection…And also I think there is, it’s ownership as well, which I think just makes it… ‘Yeah, OK we haven’t got the full purpose, we haven’t got a comparison, I am not going to do much work afterwards, but it’s their data, they’ve done something with them, they’re finishing of by…you know make it look nice… and using it.

24 1. The teacher wanted to construct a 'sociable' pedagogy, and part of this involves accepting 'where the kids are coming from' (her identity...). 2. This is arguably an attempt to construct a new/alternative 'cultural model' of 'being a maths person/learner'. … ‘you’re a mathematician…’ 1. The teacher wanted to construct a 'sociable' pedagogy, and part of this involves accepting 'where the kids are coming from' (her identity...). 2. This is arguably an attempt to construct a new/alternative 'cultural model' of 'being a maths person/learner'. … ‘you’re a mathematician…’ 3. There is data from the students interviews that suggests they at least in part buy into this; they like maths and when thinking of going to university maths "I don’t see why not" (and also it seems 'being accepted socially' might be an important factor in this). 4. There is evidence that the 'outside school' peer discourse is accepted in the classroom, possibly even encouraged. These interactional features (tenor etc) then facilitate mathematical interactions: that is talking mathematics becomes an accepted part of the banter of peer talk (in the classroom).

25 Hypothesis: This acceptance of mathematics into the peer discourse/sociality of the students may be the first sign in a chain of acceptance of a 'mathematical identity'. In other words, the discourse is a sign that, perhaps, they are accepting "being a maths-person" as part of their self/identity... themselves. This acceptance of mathematics into the peer discourse/sociality of the students may be the first sign in a chain of acceptance of a 'mathematical identity'. In other words, the discourse is a sign that, perhaps, they are accepting "being a maths-person" as part of their self/identity... themselves.