Keeping open the door to Keeping open the door to mathematically demanding F&HE mathematically demanding F&HE

programmesprogrammesLaura Black Laura Black Pauline DavisPauline Davis

Paul Hernandez-Martinez Paul Hernandez-Martinez Graeme HutchesonGraeme Hutcheson

Maria PampakaMaria PampakaSu NicholsonSu NicholsonGeoff Wake Geoff Wake

Julian WilliamsJulian Williams

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AimAim

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

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Programme effectiveness

Classroom practices

Learner identities

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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

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OutcomesOutcomes

Knowledge about how mathematics teaching and Knowledge about how mathematics teaching and learning cultures can support better participation in learning cultures can support better participation in mathematicsmathematics

Measurements of the effectiveness of two Measurements of the effectiveness of two distinctive programmes of mathematics on learningdistinctive programmes of mathematics on learning

Development of theories of learner identities in Development of theories of learner identities in maths contextsmaths contexts

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Disposition to study more mathsDisposition to study more maths

Uo

MA

ST

rad

Co

urs

e

4.002.000.00-2.00-4.00

MHEdisposition

15

12

9

6

3

0

Fre

qu

en

cy15

12

9

6

3

0

Fre

qu

en

cy

7

Disposition to enter HE Disposition to enter HE

Uo

MA

ST

rad

Co

urs

e

3.002.001.000.00-1.00

HE Disposition

7

6

5

4

3

2

1

0

Fre

qu

en

cy7

6

5

4

3

2

1

0

Fre

qu

en

cy

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Identity QuestionsIdentity Questions

What kinds of maths learner identity (and What kinds of maths learner identity (and trajectory) are there?trajectory) are there?

How are identities ‘narrated’? (Bruner, How are identities ‘narrated’? (Bruner, 1996)1996)

What resources/CMs do students use in What resources/CMs do students use in their identity work?their identity work?

How does pedagogy impact?How does pedagogy impact?

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T & L

Classroom culture

Mathematical

learner identity

Programme

institutional culture

Technology

rules of

assessment

problem solving

Cultural

models

discourses

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Cultural modelsCultural models

‘‘Story-like chains of prototypical events that unfold in Story-like chains of prototypical events that unfold in simplified worlds .. (including) metaphor’ (HQ,’87)simplified worlds .. (including) metaphor’ (HQ,’87)

‘… ‘… allowing humans to master, remember and use … allowing humans to master, remember and use … knowledge required in everyday life’knowledge required in everyday life’

“ ‘“ ‘Everyday theories' which are situated in social and cultural Everyday theories' which are situated in social and cultural experiences and which inform action (behaviour).” (Gee)experiences and which inform action (behaviour).” (Gee)

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ExamplesExamples

Is the pope a ‘bachelor’? (Fillimore)Is the pope a ‘bachelor’? (Fillimore)

US campus ‘dating’ scene: ‘jocks’ ‘bitches’ ‘nerds’ US campus ‘dating’ scene: ‘jocks’ ‘bitches’ ‘nerds’ etc (Holland & Skinner)etc (Holland & Skinner)

‘‘Coffee’ (Gee)Coffee’ (Gee)

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CMs are:CMs are:

Distributed (threads, networks?)Distributed (threads, networks?) Cultural: Discourse, IdealCultural: Discourse, Ideal Elements that are used to construct/narrate one’s Elements that are used to construct/narrate one’s

selfself Providing a figured worldProviding a figured world Sometimes fragmentedSometimes fragmented In our model: boundaries between classroom In our model: boundaries between classroom

discourse and ‘storying’ discourses of the self.discourse and ‘storying’ discourses of the self.

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Gemma’s storyGemma’s story

Draws on some positive cultural models, but others Draws on some positive cultural models, but others that one might hope for do not seem to be that one might hope for do not seem to be available: e.g. graduate role modelsavailable: e.g. graduate role models

Maths hard but challengingMaths hard but challenging Models of maths learning may be influential but not Models of maths learning may be influential but not

necessarily ‘leading’ the story: Orcasnecessarily ‘leading’ the story: Orcas

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Lee’s storyLee’s story

Lee arrived in AS with stronger grades at GCSE but Lee arrived in AS with stronger grades at GCSE but got ‘dropped’got ‘dropped’

Claims maths is hard and ‘boring’Claims maths is hard and ‘boring’ He appears to have been marginal in his AS maths He appears to have been marginal in his AS maths

classesclasses

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Some CMs evident in interviewsSome CMs evident in interviews

Maths is hard but challenging versus maths is hard Maths is hard but challenging versus maths is hard and dull: cultural models can be ambivalent, i.e. and dull: cultural models can be ambivalent, i.e. used to tell opposite storiesused to tell opposite stories

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/ Maths is ‘on your own’ versus ‘learning with others/

sociable’sociable’ etcetc

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Does pedagogy make a difference?Does pedagogy make a difference?

One notable contrast between Lee and Gemma: the One notable contrast between Lee and Gemma: the sociability of maths for them: for Gemma and her sociability of maths for them: for Gemma and her classmates maths is sociable, for Lee maths is classmates maths is sociable, for Lee maths is isolatingisolating

Might different pedagogies offer different CMs, or Might different pedagogies offer different CMs, or different positionings in relation to CMs?different positionings in relation to CMs?

Over to PaulineOver to Pauline

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IdentityIdentity

We build our identities (i) in practice and (ii) We build our identities (i) in practice and (ii) discursively using cultural models;discursively using cultural models;

What models are there of ‘ways of being a What models are there of ‘ways of being a mathematician/learner of mathematics?’mathematician/learner of mathematics?’

How can mathematics learner identity be mediated How can mathematics learner identity be mediated by mathematics classroom social practice?by mathematics classroom social practice? Can we Can we expand the repertoire of cultural models?expand the repertoire of cultural models?

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Classroom Discourse/PracticeClassroom Discourse/Practice

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We often find student identities are double-discoursed in We often find student identities are double-discoursed in a genre adopting a dual register, one of student and the a genre adopting a dual register, one of student and the other (sometimes more quietly spoken and ‘hidden’) of other (sometimes more quietly spoken and ‘hidden’) of everyday teenage talk, indicating tensions between these everyday teenage talk, indicating tensions between these opposing voices. opposing voices.

The micro data shows an alternative discourse where The micro data shows an alternative discourse where there is flip flopping between themes, maths and non-there is flip flopping between themes, maths and non-maths (every-day teenage) talk; maths (every-day teenage) talk;

The tenor (or voice) remains broadly the same.The tenor (or voice) remains broadly the same.…crazy 20…crazy 20

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

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KK And like not only you And like not only you think for yourselfthink for yourself but but like we can ask other people why they got that and like we can ask other people why they got that and it’s it’s not just like black and whitenot just like black and white, like you , like you get to a get to a different way to work it out.different way to work it out.

J J … it sounds daft but … it sounds daft but you’re having fun you’re having fun while you’re doing itwhile you’re doing it cos you can sit and you can cos you can sit and you can talk to people but… talk about the work but you talk to people but… talk about the work but you can… it’s not a thing where you come down and sit can… it’s not a thing where you come down and sit in silence and you do it, in silence and you do it, you can talk to people you can talk to people and can, you know, do practical thingsand can, you know, do practical things

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

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Ownership, understanding/conceptual, Ownership, understanding/conceptual, fun, joint activityfun, 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.

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1. The teacher wanted to construct a 'sociable' pedagogy, and part of 1. The teacher wanted to construct a 'sociable' pedagogy, and part of this involves accepting 'where the kids are coming from' (her this involves accepting 'where the kids are coming from' (her identity ...).identity ...).

2. This is arguably an attempt to construct a new/alternative 'cultural 2. This is arguably an attempt to construct a new/alternative 'cultural model' of 'being a maths person/learner'.model' of 'being a maths person/learner'.… ‘you’re a mathematician…’… ‘you’re a mathematician…’

3. There is data from the students interviews that suggests they at 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 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 to university maths "I don’t see why not" (and also it seems 'being accepted socially' might be an important factor in this).accepted socially' might be an important factor in this).

4. There is evidence that the 'outside school' peer discourse is 4. There is evidence that the 'outside school' peer discourse is accepted in the classroom, possibly even encouraged. These accepted in the classroom, possibly even encouraged. These interactional features (tenor etc) then facilitate mathematical interactional features (tenor etc) then facilitate mathematical interactions: that is talking mathematics becomes an accepted part of interactions: that is talking mathematics becomes an accepted part of the banter of peer talk (in the classroom).the banter of peer talk (in the classroom).

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Hypothesis:Hypothesis:

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