1. Exploring Relational Identities
Jo Boaler, Sussex University

2. Sociocultural Perspective
Barbara Rogoff (1995). ‘Observing Sociocultural Activity on three planes: participatory appropriation, guided participation, and apprenticeship.’ in Wertsch, del Rio & Alvarez, Sociocultural Studies of Mind, Cambridge: UK.

3. The Study
Examining the impact of different teaching approaches upon learning in 3 Californian high schools
Following students 4 years different mathematics approaches, 700 students, ‘traditional’, ‘integrated’ & ‘Railside’
600 hours classroom observations, interviews, questionnaires, group and individual assessments

4. Understanding Learning
Individual
Interpersonal / Social
Community / Institution
Cultural

5. From Rogoff Planes of focus:
Apprenticeship ..individuals participating with more experienced others
Guided Participation .. the mutual involvement of individuals and their social partners as they participate in shared endeavours
Participatory Appropriation .. people changing through their involvement in activities.

6. Identities Emerged as important differences. Subject specific.
Considered changes in identity as “participatory appropriation”
This is a hard thing to say clearly, I feel like we are going to have to do some careful explaining about what we mean. Tell me if you think this is right: what we are trying to understand is the relationship that students develop to mathematical knowledge—how they approach new ideas, how they view mathematical knowledge in relation to themselves (i.e. something that they control vs. something that controls them (I think at one point you said “submit to,”)

7. Conceptualising Identity
Nature & purpose of mathematics
Interest & enjoyment
Agency
Authority
Methods: Classroom observations, Interviews with 80 seniors - coded and categorized.

8. 3 Identities
Active
Passive
Relational

9. The Railside Students Low income, racially diverse
Working together, responsibility and respect, the communication of mathematical ideas.

10. Staying Together
Int: Is learning math an individual or a social thing?
G: It’s like both, because if you get it, then you have to explain it to everyone else. And then sometimes you just might have a group problem and we all have to get it. So I guess both.
B: I think both - because individually you have to know the stuff yourself so that you can help others in your group work and stuff like that. You have to know it so you can explain it to them. Because you never know which one of the four people she’s going to pick. And it depends on that one person that she picks to get the right answer. (Gisella & Bianca, Railside, Y2)

11. Responsibility
N: I feel it was a responsibility because if you know something you have to, and somebody doesn’t know, I feel it’s your right, it’s, you have to teach them how to do it. Because it’s only fair to them that they get the most out of it as you’re getting out of it because you’re both in the same classroom. (Neil, Railside, Y4)

12. Responsibility
I just kind of tell them “well are you going to act serious now? Because you need to! There’s a person – I won’t say no names – but who is in our group right now and he just kind of lies around. And it seems the other guy he just likes doing stuff by himself, and I go over and talk with him and once I’m done talking to him I go over and I sit by the one who really doesn’t care and I tell him you know – “are you understanding this stuff?”(Caroline, Railside, Y3)

13. Advanced forms of communication and help
K: I think the biggest help is just to stop with the problem, stop doing it and kind of step back from it and start asking questions, start asking thought-provoking questions about the problem. (Jon, Railside, Y4)

14. Multidimensionality
What does it take to be successful in mathematics class?
asking good questions,
rephrasing problems,
explaining well,
being logical,
justifying work,
considering answers, and
using manipulatives.

15. Multidimensionality
Back in middle school the only thing you worked on was your math skills. But here you work socially and you also try to learn to help people and get help. Like you improve on your social skills, math skills and logic skills (Janet, Railside,Y1)

16. Because there were many more ways to be successful, many more students were successful.

17. Respecting Ideas
T: You got everyone’s perspective on it, ‘cause like when you’re debating it, a rule or a method you get someone else’s perspective of what they think instead of just going off your own thoughts. That’s why it was good with like a lot of people.
C: I liked it too. Most people opened up their ideas. (Tanita & Carol, Railside, Y4).

18. Build the arrangement of LabGearTM blocks, and find the perimeter of the arrangement.1 x 1

19. Identities: Interest & Beliefs
Interest: high
Beliefs: Mathematics is a form of communication.
I think of it like a language class kind of. Math seems like a second language or another language that we’re learning—because it is something that you can use to communicate to others through math. (Jon, Railside, Y4).

20. Identities: Agency Active, but also relational:
Cause I want, like people that doesn’t know how to understand it I want to help them. And I want to, I want them to be good at it. And I want them to understand how to do the math that we do. (Amado, Railside, Y1)

21. Identities: Authority
How would you know when you had found the maximum volume?
‘When it would make sense to everyone’

22. Relational Identity & Choice
41% of Railside seniors were in advanced classes of pre-calculus or calculus compared to approximately 27% of seniors in the other two schools.

23. Future in mathematics
Did not want to see math again or take another math course
Prepared to take math courses if needed
Planning a mathematical future

24. Teaching approach vs future in mathematics

25. How important are these relationships?
Interest v future in mathematics .746
p<0.01
Agency v future in mathematics .491

27. Passive
C: I like formulas. That helps me. Like, the actual formulas, if it’s like just formula, put this and this here, I’m fine.
Greendale, Algebra 2 (coded as 1)

28. Active
Like, um, I don't know. If nothing else, it's just breaking out of the pattern of just taking something that's given to you and accepting it and just, you know, going with it. (…) It's just looking at it and you try and point yourself in a different angle and look at it and reinterpret it….It's like if you have this set of data that you need to look at to find an answer to, you know, if people just go at it one way straightforward you might hit a wall. But there might be a crack somewhere else that you can fit through and get into the meaty part. (y4 student, coded as 3)

29. Interest & enjoyment scale- +
OK with
math
(2)
Hates math
(1)
Likes math a lot
(3)

30. Agency scale Passive Active Needs to remember standard Methods (1)
Somewhat
willing to
adapt
& apply (2)
Willing to adapt & apply (3)

31. Authority scale Dependent On book / teacher Sense Making
Needs book or teacher to move forward (1)
Somewhat
willing to trust sense
making (2)
Commitment
to sense making (3)

32. Nature & purposes of mathematics scale
Useful
for the world
- for solving problems
- for communicating
Procedures
only
Conceptual
Domain
nature
purpose
Grades only

33. Knowledge relationships
Interest, agency, authority, nature & purpose
How do the domains relate to each other?
How do the different domains impact students’ choices about further study?

34. Results

35. Teaching Approach vs. Interest

36. Teaching Approach vs. Interest

37. Teaching Approach vs. Agency

38. Teaching Approach vs. Authority

39. Relationship between authority & agency
Significantly correlated, 0.704, p <0.01

40. Future in mathematics
Did not want to see math again or take another math course
Prepared to take math courses if needed
Planning a mathematical future

41. Teaching approach vs future in mathematics

42. How important are these relationships?
Interest v future in mathematics .746
p<0.01
Agency v future in mathematics .491

43. What is Learning?
Learning is both about students’ knowledge and about their relationships with knowledge
Interest & enjoyment
Nature & purpose
Agency
authority

44. Authority through sense making
Int: So when you’re doing a math problem, how do you know when you’re right?
J: Um…probably I know I’m right when I can explain myself. When someone else has another answer, and I come up with a different answer, and I can tell them why my answer is correct, and explain all the steps that I took, and go even further to explaining where they messed up, if they did mess up. OR if I’m wrong, I can see where I messed up, and see where they did something right. SO I think that kind of helps too. (coded as 3)