How pedagogy can help provide resources and motivation for/against mathematical identity work: a socio-cultural perspective
Julian Williams and Pauline Davis, University of Manchester
The Research Project
We draw on our ESRC-TLRP project Widening Participation project “Opening doors to mathematically-demanding programmes in Higher Education” (www.lta.education.manchester.ac.uk/TLRP.html), where among other things we are exploring ways in which students in case study AS maths (and Use of Maths) classrooms, especially students with low GCSE grades and often from 1st generation to Higher Education (HE) families, tell of their experiences of learning maths and their dispositions to undertake further study in HE and particularly to undertake courses in which mathematics may be demanding. We will then discuss how sociocultural theory has informed our understanding of the dynamic relation between mathematical identity and classroom mathematics practice.
Sociocultural Theoretical Framework for Identity
We draw on Gee (1999) and analysis of students’ positioning with or against mathematical cultural models in order to explain differences in their dispositions towards mathematics. Gee (1999) refers to the everyday theories (i.e. storylines, images, schemas, metaphors and models) that people use to make sense of their lives as cultural models. ‘Cultural models are not static …and they are not purely mental but are distributed and embedded in socio-culturally defined groups of people and their texts and practices (Gee, ibid., p. 23). Thus, cultural models tell us what is ‘typical’ or ‘normal’ and mediate our actions, not universally, but from the perspective of our experiences. According to Holland , Lachicotte, Skinner and Cain (1998) it is this ‘stuff of existence’, what is real to people and has meaning for them, which ‘grant shape to the co-production of activities, discourses, performances and artefacts’ (p.51).
We take the view that identity processes are no longer seen as connecting individuals in homogenous or fixed ways; our identity work is never ‘done’, it is always on-going’ (Holland et al, op. cit.). Although a person’s identity is not determinable, neither is the meaning-making involved in identity-work entirely free but, instead, is mediated by the discourses and practices of people’s social activity systems (Engestrom, 1995). Situated human creativity exists not despite, but because of social structures and concrete activities (Marx, 1867) with particular tools, social rules and division of labour (Engestrom, 1987; Engestrom and Cole, 1997; Leontev, 1978) Thus, talk about identity in social terms does not deny individuality but views the very definition as something that is part of the practices of specific communities. ‘From a social practice perspective it is through cultural practices as people ‘do life’ that social identities are constructed (Nasir and Saxe, 2003).
To inform our analysis of classroom practice/discourse and classroom talk we draw on Bernstein and Systemic Functional Linguistics SFL (Halliday and Hasan).
The key theoretical sources here are:
Cultural-historical activity theory (CHAT): Vygotsky, Leont’ev, Engestrom, Cole, Bakhtin.
Cultural models, Discourses and figured worlds: Holland & Quinn, Holland et al., Gee.
Pedagogy, discourse and Systemic Functional Linguistics: Bernstein, Halliday & Hasan.
Different kinds of maths learners
In our widening participation in mathematics research we identify cultural models used by mathematics students in the stories they told us about their experiences of learning mathematics and their disposition to study mathematics in the future. Each model provides a tool for forming a way of identifying or dis-identifying e.g. maths is ‘challenging’ versus maths is ‘hard’. The main result reported here is that students can tell their imagined life story to represent or reveal a positive disposition to mathematics or a negative disposition using the same models. Thus maths is ‘hard’ can be rewritten as ‘maths is challenging’ if a student’s narrative demands it. We will illustrate this with two contrasting narratives of identity: in one, the motive may be shifting ‘for/towards’ a mathematical identity and in the other it has clearly become against.
Significance for pedagogy
Our case studies reveal two key elements. First, pedagogy can provide a narrow or a broader range of models and Discourses of ‘being a mathematician’ than is traditionally available. Thus, we have examples where students say of their experience of mathematics that they find it fun, sociable, negotiable, etc. We can say that this pedagogy involves a relatively flexible classification (Bernstein) of mathematics. Second, the way the classroom activities are organised can encourage different kinds of mathematics classroom talk. When students work together engaged in common/joint activity this can encourage their normative, everyday, peer Discourses. In this way we believe that some students can come to talk mathematics and be more active ‘mathematicians’ through their classroom practice. We will illustrate this with an example of group talk that seamlessly switched from ‘pooh talk’ to ‘mathematics’ and back again – it is seemless in the sense that consistency of the tenor (Halliday and Hasan, 1976) of the discourse is maintained. Thus mathematical- and peer-identity work reciprocally to legitimise each other within the classroom.
Three of a possible many key references
Holland, D., Lachicotte, W., Skinner, D., & Cain, C. (1998). Identity and agency in cultural worlds. Cambridge: Harvard University Press.
Gee, J.P. (1999) An introduction to discourse analysis: Theory and method. London Routledge.
Halliday, M.A.K. & Hasan, R. (1976) Cohesion in English. Harlow: Pearson Education
Limited.
Biography
Julian Williams taught mathematics in High Schools and became involved in research and development before entering Teacher Training at Manchester University x years ago. Research interests focussed on mathematics education have progressively included research in curriculum (e.g. mechanics, modelling and SMP/MEI/Nuffield 16-19), pedagogy and assessment (ITAM, MaLT), and theory, including Cultural-historical activity theory (CHAT: see BERA SIG ) and social- and cognitive- linguistics. Among other things, Julian is Professor of Mathematics Education, directs the School of Education’s Post-Graduate Research Students and is PI on the ESRC–TLRP research project in widening participation in mathematics.
Pauline Davis taught mathematics in the Further Education sector for some years in the late 1980s/early 1990s, following life as a student also reading maths. Having rather disengaged with maths during my degree, following a positive and proactive maths identity in school (when I swore I would never to teach), I saw being a maths teacher as something very different to the academic world of mathematics and it is only in recently that I have returned to maths in relation to research in education. It was also about that time that sociocultural perspectives progressively began to inform my work. Pauline Davis is Senior Lecturer in Education, is a member of the ESRC-TLRP widening participation in maths team and PI on a research project in financial education. She previously researched in the area of special and inclusive education.
Williams and Davis Seminar Data
Kieran and Eleanor are taking AS Mathematics.
K I just want to check that … please do not destroy my calculator.. ..I I pinched this!
E its annoying.
K No Its NOT like YOUR calculator
E its not like my calculator
K …. No its not like your bloody cheating…bloody Im gonna-cheat- in- exams calculator,
Ester my calculator is something amazing…
pause
E never diss the calculator!
E I can diss what I wanna diss …
E as if.. ..
K
O, Eleanor smells, Eleanor smells, Eleanor is a smelly bint /bitch(?)… (singing /rhyming a la footie/primary school?) do dah …do dah … Eleanor smells of pooh …Eleanor smells of pooh ….Eleanor is a smelly poo .. Smells of pooh… pants …
[E itstrue -I do
K … pooh .. pooh
E AhA.. you see it rhymes (laughs)
K Im angry now… [does not sound angry- its ironic]
There is a switch here to maths with no change in ‘voice’
E did I just do divide? No times ..
K yeah , times it
Then back to the argument over equipment- change of theme BUT NO change of tenor
E I’ve forgot’n how much I like my pencil (sarcastic)
[Teacher: have you finished with the poster- (background]
K yeah its nice to write with….. what am I’m not allowed it now … just as I’ve actually got to work out ..
Switch back to maths
E ……… Do you just divide by 7?
K NOOOOO you divide by the the total number
E you divide that by that don’t you?
K ….right thats just about it ?
E is that a 13 or a 15 I don’t know
K name of teacher!... name of teacher… (calls teacher over to help)
pause
E do you get 119?
K…I got my crazy estimated mean there
72 minutes: calls the teacher over to show their different mean-estimates; she looks at Kieran’s answer, Eleanor’s makes more sense, she diagnoses that Kieran made arithmetic error, and they discuss )
K to Teacher I got this crazy 20 here …
Teacher crumbs
E I added all them there and then divided by total frequency … got that
S yeah that makes more sense doesn’t it?
E yea!!!! (pleased with self)
(more chat)
Teacher whats half way between 100 and 150 Karl?
(he has 25 on a scrappy bit of paper- on which is written “Karl smells” – I think I recall from the video?]
…
Teacher you’ve lost one of the 100s
….
Teacher I told you, you should have a nice table, didn’t I… who was right?…[Kieran’s o god] O gosh we’ve recorded this now haven’t we? (some laughter)
(Eleanor gets a sensible answer and the teacher says it makes more sense than Kieran’s…and diagnoses the problem.)
K to Teacher you’re not always right…I won before
S just this once I’m right (as she walks off.. more laughter)
K you’re not always right…I won before…
K I won the from on the first day //I still remember that first day / I got a mars bar for it (Julian says ‘what?’)
K I proved her wrong – what she wrote on the board … and I got a mars bar…
K to E don’t DO THAT!
….
E How do you get that?
K I DONT KNOW
E why do you divide by 20?
…
K how come you divide by 20 and you get that …
74mins 15
E and K now all out because they got different answers…
K How come..?
E because your CRAZY…
K yes I’ve managed to redeem myself and prove myself right
Williams and Davis Seminar Data
gemma’s story
Gemma will be the first generation from her family to go to university. In fact she cannot name anyone she knows in her family circle who has been to university. But there is no question in her mind that she WILL go, she says “I’ve been going to uni since I was 8”). She has lived ‘locally’ all her life in a community that has all the ‘poorest’ social indicators. Her principal and teacher described, with almost ironical pride, the local community as sitting regularly at the bottom, or near the bottom, of every league table of performance and social index of deprivation. Gemma tells us that her mother’s work as a cleaner and shop worker is stressful, which has helped to motivate her as “And I see my mum like working in a shop and cleaning and I don’t want to do that, so that’s kind of influenced me in my own work not to follow that path ‘cos she gets stressed out and stuff”. Gemma tells us several times that her mum has been very supportive of her and encouraged her ambitions all her life (as has her mother’s partner).
She did well at Primary school: “I was always into books at school and I was always levels ahead”. She said that getting level 5 at age 11 National tests was an important marker for her. She experienced her Catholic primary school as relatively – compared to secondary – ‘inspiring’.
At age 8 she decided she wanted to become a marine biologist so she could work with Orca whales: “I’ve just always taken a fancy to Orcas, .. Killer whales, .. Free Willy is my favourite movie (laughs).” Even though her mother thought she would ‘get bored’ of this particular ambition, Gemma has stuck with it and her mum has continued to support her; she got advice during secondary school from the ‘connections’ service and knows exactly what she has to do in her AS and A level grades in science and maths to get to university and then to do a PhD in Marine Biology. She knows she will spend 6 years at uni and which one she wants to go to for her studies, as it has a connection with research into Orcas. In fact she tells us that the specialist field she will need to follow to get to work with Orcas is more specialised, those who study big sea animals are called marine mammologists, and “you have to be one of the top ones” to get into it.
Her experience of secondary school was very mixed, with classrooms being boring and classroom behaviour off-putting. The teaching was often uninspiring and she lost interest or a while:
From when I went to secondary school I lost interest in quite a lot of my study … at (Primary school) there was more passion in it while at secondary school it was just “you’ve got to get through this..”
In contrast to her self-ranking in mathematics at Primary school, she says now “I wouldn’t class myself as that good but maybe a bit above average”. However, she describes maths as being ‘challenging’ rather than hard: “… there was a lot of noise in the class … [disruptive?] yes; but I enjoyed it and it was a challenge as well …”