What Can We Learn From: Identity As It Relates Mathematics Learning?

What Can We Learn From: Identity As It Relates Mathematics Learning? Gaye Williams Deakin University Education: Burwood Campus gaye.williams@deakin.edu.au

Identity • Participating practice (Cobb et al,in press) Or • Stories we tell about ourselves that include stories others tell about us (Sfard & Prussak, 2005)

Different Perspectives on Identity: Commonalities “Human beings in action and … mechanisms underlying human actions” (Sfard & Prusak, 2005, p. 14). Identity: “Man made and constantly created and recreated in interactions with people” (Sfard & Prusak, 2005, p. 14).

Each Lens On Identity • Has a present state And • A state yet to be realised

Change in Identity Seen as closing the gap. May be useful methodologically

Participation in Practice Cobb, Gresalfi, & Hodge (in press) Identities that children develop in mathematics classrooms Cobb, P., Gresalfi, M., & Hodge, L. (in press). A design research prspective on the ientities that sudents are dveloping in mthematics cassrooms. In B. Schwarz, T. Dreyfus & R. Hershkowitz (Eds.), Transformations of knowledge in classroom interaction. Amsterdam: Elsevier.

Participating in Practice • Normative identity: practices a competent student is considered to undertake in that classroom • Personal identity: How an individual participates in that classroom (they may: resist, comply, or engage) More than developing the language (Catherine Beavis) can be developing argumentation

Change in Identity Narrowing the gap between student’s own way of participating in the classroom and the ways a student seen as competent participates in that classroom

Is this change in identity always productive? (The ways in which people perform in different context, Catherine Beavis)

Identity As Narative (Sfard & Prusak) Constructs developed to study differences between the persistence of students who were recent immigrants from Russia and local Israeli students Sfard, A., & Prusak, A. (2005). Telling identities: In search of an analytical tool for investigating learning as a culturally shaped activity. Educational Researcher, 34(4), 14 - 22.

Sfard & Prusak (2005) Identity: A set of reifying, significant, endorsable stories about a person. Learning: closing the gap between actual and designated identity.

Identity Constructs • Actual: present perceived state of affairs • Designated: what is expected for some reason to be the state of affairs in the future (like projective identity, Catherine Beavis on Gee: “Seeing the virtual character as one’s own project in the making”)

Present tense: actual • Future tense: designated

Words That Are Taken Seriously and that Shape One’s Actions Influence of the significance of the story teller to the subject

Usefulness of Identity as a Construct • “[i]dentity talk makes us able to cope with new situations in terms of our past experience and gives us tools to plan for the future” (Sfard & Prusak, 2005, p. 16)

Linking These Ideas With My Research? Not all students are inclined to explore, some resist working with unfamiliar mathematical ideas (Williams, 2005). In terms of identity constructs: They resist participating in the normative activity of inquiry classrooms (Cobb et al, in press). The stories they tell of themselves in the present tense do not include a capacity to explore unfamiliar ideas (Sfard & Prusak, 2005).

Not Inclining to Explore Wanting to remain within the confines of what is already known and achieve success through memorisation and repetition.

Linking Inclination to Explore with Resilience/Optimism An optimistic child sees ‘failures’ as • temporary • specific • possibly influenced by some external factors. They see successes • permanent • pervasive • personal Seligman, M. (with Reivich, K., Jaycox, L., Gillham, J.). (1995). The Optimistic Child. Adelaide: Griffin Press.

Optimism (Seligman, 1995) ‘Inclination to explore’ associated with orientation to successes and failures (Williams, 2003). Optimistic students are inclined to ‘step into’ unknown territory and explore new ideas because they see: • ‘not knowing’ as temporary and • able to be overcome through personal effort by • looking into the situation to find what can be changed to increase chances for success.

Optimism (cont) They see their successes as: • Permanent • Pervasive • Personal

Pervasiveness as Key to Identity-Building / Optimism-Building • “turning properties of actions into properties of actors” (Sfard & Prusak, 2005, p. 16, naratives) • Building the pervasive nature of success through repeated successes in ‘flow’ situations (Seligman, 1995) “I am good at this, I can …”

Optimism Building Flow: State of high positive affect occurring when students set intellectual challenges and overcome them by developing new knowledge. (Developing themselves in new sports rather than familiar ones can produce conditions for flow; Katherine Meldrum) Seligman found: Optimism builds when students experience flow / gain pleasures associated with overcoming self-set challenges. Question: Is it just wanting to replicate pleasure or is there a building of pervasiveness as well as described by Sfard & Prusak? Csikszentmihalyi, M. (1992a). Introduction. In M. Csikszentmihalyi & I. Csikszentmihalyi (Eds.), Optimal experience: Psychological studies of flow in consciousness (pp. 3-14). New York: Cambridge University Press.

This is an important area for future study. Not only in mathematics education but in all areas of education. What builds student ability to think for themselves rather than remain within the confines of what they have been taught? How can we build such capacity in school students, prospective teachers, and teachers? Gaye Williams gaye.williams@deakin.edu.au

Dean: Failure as Temporary 19. Int How do you learn something like … that lesson? 20. Dean Um … how do I learn it? 21. Int What helps you? 22. DeanWell … I write it down … 23. Int I see … yeah … [soft] 24. Deanin my book and then 25. Int yep 26. Deanwhen I got like … when he’s talking or … something that I have already known or something … then I just look over it … again. 294. DeanWell when I first get a … um … sheet … which I’ve never done before … then I um … I get a bit … stressed [small laugh seems to be at self rather than anxiety] … 295. Int Oh okay. 296. Dean Cause I k- I- cause the first time … um … I do stuff um … I always don’t get it … at first- it takes me like … a little while- that’s why I go over it and over it … Key: … pause. [text] Researcher comment

Dean: Success as Permanent, and Personal 746. DeanI’m going alright but um … if … like … here like the teacher or somebody asks me a question I just remember what I have done … like in my books and … what I’ve known and stuff like that and then I just … put it all together … and then … work it out I guess yeah. Dean perceives his past experiences of looking in his books and thinking about what he knows will lead to him being able to answer more questions in the future (Success as Permanent and Personal).

Dean: Failure as Specific 451. DeanI always put … I didn’t know … where the corners went … in it … like I did- I thought … um- see when he told me you put the … corners facing in 452. Int Yes? 453. Deanbut I was- I was doing it all different- I was facing them out… and up… but the … corner has to be facing in the middle 454. Int So when you had those little pieces of paper- you recognise the corners did you? 455. DeanYeah … that’s why you rip em… instead of cut em When he was not succeeding, Dean was able to look into the problem to identify what was causing problems.

Success as Internal 506. Dean// No I thought it up- yeah I thought it up by myself

What Can We Learn From: Identity As It Relates Mathematics Learning?