Changing Participation and Identity through Instructional Scaffolding that Promotes Transactive Discussion 2008 AERA An

1. Changing Participation and Identity through Instructional Scaffolding that Promotes Transactive Discussion2008 AERA Annual Meeting, New York, NY Maria L. Blanton University of Massachusetts Dartmouth mblanton@umassd.edu Despina A. Stylianou City College, The City University of New York dstylianou@ccny.cuny.edu • The research reported here was supported in part by the National Science Foundation under Grant # REC- 0337703. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.

2. two instructional scenarios (Boaler, 2002) • In one class students are learning mathematics using their textbook • In a second class students are learning the same topic in mathematics through an active discussion • In both classes students potentially learn some mathematics • But, can we distinguish the activity through which mathematics is learned from the knowledge that is eventually developed?

3. The ways in which students participate in mathematics classrooms have real implications for the nature of the mathematical knowledge they produce - (e.g,. Wertsch & Toma, 1995;Cornelius & Herrenkohl, 2004) “Different pedagogies are not just vehicles for more or less knowledge, they shape the nature of the knowledge produced and define the identities students develop as mathematics learners through the practices in which they engage” (Boaler, 2002, p. 132).

4. An important question to consider:Do certain types of pedagogy shape different mathematical identities and, subsequently, lead to the development of certain forms of mathematical knowledge?

5. Goal:To examine how a classroom pedagogy that focused on scaffolding students’ transactive reasoning shaped the ways in which students participated in mathematical discussions, and hence, the nature of their mathematical thinking.

6. identity and participation as constructs of learning • learning • can be understood as "legitimate peripheralparticipation in communities of practice" and is characterized as movement from peripheral to full participation in the practices of a community • can be traced as the learner's identity within a community of practice shifts from "newcomer" to "old-timer" status • (Lave & Wenger, 1991) "identities are lived in and through activity and so must be conceptualized as they develop in social practice" (Holland, et al, 1998, p. 5)

7. discourse as an agent of learning We use discourse analysis as a tool to understand classroom participation and the development of identity and how these connect to students’ mathematical learning

8. discourse as an agent of learning "utterances are not indifferent to one another, and are not self-sufficient; they are aware of and mutually reflect one another….[and] every utterance must be regarded primarily as a response to preceding utterances”. (Bakhtin, 1986, p. 91)

9. discourse as an agent of learning "utterances are not indifferent to one another, and are not self-sufficient; they are aware of and mutually reflect one another….[and] every utterance must be regarded primarily as a response to preceding utterances”. (Bakhtin, 1986, p. 91) if we accept this claim as an axiom of discourse, it points us to the reciprocal influence conversation has on its participants and raises the issue of how teacher utterances in classroom conversation impact student learning by changing the ways in which students participate in those conversations

10. transactive reasoning criticisms, explanations, justifications, clarifications, and elaborations of one's own or another's ideas Berkowitz, et al, (1980, presented in Kruger, 1993)

11. questions (1) How does instructional scaffolding that promotes transactive reasoning in whole class instruction shape the ways in which students participate in mathematical discussions and the identities they develop? (2) How do the resulting forms of participation shape the nature of students’ mathematical thinking?

12. background Part of a larger study on the development of proof One year teaching experiment in a college classroom where the instructor introduced students to proof using instructional scaffolding that promotes transactive reasoning. www.theproofproject.org

13. framework for coding transactive discussions Because our underlying question concerned instructional scaffolding whose purpose was to promote transactive discussion, our first task was to establish a framework that would detail both that and how this occurred.

14. framework for coding transactive discussions Transactive prompts - defined as teacher requests for critique, explanations, justifications, clarifications, elaborations, or strategies. Facilitative - Teacher utterances that revoiced or confirmed students’ ideas or served to structure classroom conversation. Didactive utterances - teacher utterances on the nature of (mathematical) knowledge. Directive utterances - defined as utterances that give either immediate corrective feedback or information towards solving a problem. Other - non-transactive - statements intended to elicit or clarify information that required a low level cognitive response (e.g., recall) from students.

15. Preliminary analysis of instructional scaffolding of transactive reasoning Type of Teacher Utterance Percentage Transactive prompts 40 Facilitative utterances 47 Directive utterances 8 Didactive utterances 5 Thus, when the teacher participated in the classroom discussion, we maintain that her role was primarily to direct students towards transactive reasoning about mathematical proof.

16. Students as ‘newcomers’ - September whole class discussion 1 Teacher: Will some of you tell me, anybody, how did you, what convinced you that that is true? You drew a picture? Mac, you drew a picture? What did your picture look like? 2 Mac: Eh… it was like little lines. 3 Teacher: Little lines? 4 Mac: Then I coupled them off in pairs of two. [inaudible] 5 Teacher: Can you come draw it quick on the board. I am having trouble imagining what little lines look like that convinced you of this. [Mac goes to the board; he describes his idea as “common sense”] O common sense. 6 [Tuning to another student] What did common sense tell you John? 7 John: Well basically what it broke down to was that the umm odd number could be separated into an even number and one and so two even numbers would add up to an even number and you would still have that one left over. 8 Teacher: The odd number… 9 John: But I had no idea how to put that into an equation.

17. Students as ‘newcomers’ - September whole class discussion We would argue that, mathematically speaking, students were ‘newcomers’ in how they could discourse about mathematics, including how they were able to initiate, critique and analyze complex mathematical ideas. They were also newcomers in how they viewed themselves as participants in the group: Mac seemed to hold a personal view that he was not qualified to offer a legitimate mathematical idea for the class to consider because his idea was just “common sense”. Mac seemed to feel more comfortable on the periphery of this mathematical community, as a new-comer. Although John had an informal idea that could be developed into a more rigorous proof – indeed, he already recognized the inadequacy of testing examples as a rigorous form of proof – he drew back from the conversation by asserting that “I had no idea how to put that into an equation”

18. The teacher’s goal was to use transactive prompting to scaffold John’s and Mac’s participation in the socio-cultural practices of this community – to get them to share their thinking publicly, to move into the cognitive (and even physical) space of the community by making their work public and visible. • In short, transactive prompting was a means to change how students participated in the community.

19. Students as ‘Oldtimers’ - Moving Beyond the Teacher’s Guidance Second Semester: Prove hat the center H of group G, where H = {aag=ga, gG}, is a subgroup of G.

20. Students as ‘Oldtimers’ - Moving Beyond the Teacher’s Guidance 18 Thom: If 'a dot g' is in there and 'a dot g' equals 'g dot a' then 'g dot a' is in there too… [inaudible] (transactive – explanation) 19 Atram: [inaudible] if you know the original…the original operation must be in there. (transactive response) 20 John: OK, how does it look? (transactive question) 21 Atram: [inaudible] you say something then you say it again. [Writing on a piece of paper] You know that, you know, 'a dot g' is 'g dot a'. If it didn’t, [a] would not be in H. (transactive response) 22 John: Right. (confirmation – nontransactive) 23 Thom: But we don’t know that 'g dot a' is in H. (transactive question) 24 Atram: No we don’t. You’re right. Not yet. (transactive response) 25 Atram: So you know that the operation 'a dot g' equals 'g dot a' which has to be in H [inaud.] 26 Thom: We know that a is in H. We don’t know that g is H. 27 Brot: Well if that’s the case then g has to be in H. (transactive response)[inaudible] 28 Mac: The way that I read it, I don’t think that g actually exists in H [inaudible]. (transactive response) 29 John: [points to something on the table] No, we have definitely established that they are both in there. (transactive response) 30 Thom: H? (transactive question)

21. Students as ‘Oldtimers’ - Moving Beyond the Teacher’s Guidance 18 Thom: If 'a dot g' is in there and 'a dot g' equals 'g dot a' then 'g dot a' is in there too… [inaudible] (transactive – explanation) 19 Atram: [inaudible] if you know the original…the original operation must be in there. (transactive response) 20 John: OK, how does it look? (transactive question) 21 Atram: [inaudible] you say something then you say it again. [Writing on a piece of paper] You know that, you know, 'a dot g' is 'g dot a'. If it didn’t, [a] would not be in H. (transactive response) 22 John: Right. (confirmation – nontransactive) 23 Thom: But we don’t know that 'g dot a' is in H. (transactive question) 24 Atram: No we don’t. You’re right. Not yet. (transactive response) 25 Atram: So you know that the operation 'a dot g' equals 'g dot a' which has to be in H [inaud.] 26 Thom: We know that a is in H. We don’t know that g is H. 27 Brot: Well if that’s the case then g has to be in H. (transactive response)[inaudible] 28 Mac: The way that I read it, I don’t think that g actually exists in H [inaudible]. (transactive response) 29 John: [points to something on the table] No, we have definitely established that they are both in there. (transactive response) 30 Thom: H? (transactive question)

22. Students as ‘Oldtimers’ - Moving Beyond the Teacher’s Guidance • We take the amount of transactive reasoning and the influx of new ideas that mark this episode as evidence that practices of argumentation, scaffolded by the teacher through transactive prompts, were being internalized into students' ways of participating in classroom discussions. • The structure of the conversation as a flow of transactive reasoning seemed to reflect the instructional focus on scaffolding transactive discussions. • Students were transactively interpreting other student utterances, without prompting by the instructor. They had, it seemed, moved beyond the periphery of the community, where utterances sometimes reflected a perception that ideas were not sufficiently mathematical or adequate for sharing publicly and where full community membership was not a personally held identity, to fuller participation in a rich web of transactive discussion where participants held each other accountable for establishing valid thinking and interpreted each others’ utterances as ideas to be questioned, analyzed, and then accepted or rejected.

23. Conclusion • We maintain that students' capacity to engage in the conjecturing and negotiation of mathematical ideas seemed to be scaffolded mainly through the teacher's transactive prompts and facilitative utterances as opposed to directive or didactive utterances. • In contrast, classrooms where the pedagogy at play is largely directive or didactive (as, we argue, undergraduate math classrooms often are) do not draw students into full participation as a community of learners because they do not provide a rich, sustained context for publicly and actively interpreting students’ thinking.