Moana continued to monitor closely how different students i n the classroom participated i n interactions. She viewed and discussed video excerpts o f the Maori and Pasifika girls' emerging communication and participation patterns. She closely monitored how less confident or less able students managed within the heterogeneous grouping. Moana specifically focused on strategies to build these students' mathematical confidence. She actively positioned specific students. For example, as she listened to a group discussion she
overheard a quiet comment by a less able student and responded by directi ng the other students to listen to him. She then said: good thinking Tama, something to get you all going. At another time she li stened to the exchange of ideas then commented loudly: wow Teremoana see how you have made them think when you said that? Now they are using your thinking. She regularly halted expl anations as they were bei ng shared to draw attention to how a low achieving or unconfident students' reasoning had contributed. The following vignettes il lustrate how Moana li stened carefully as small groups interacted and then when required stepped in and positioned specific students to participate and contri bute their
reasonmg.
Positioning the students to access and own the mathematics talk
Moana observes Anaru noddi ng her head in agreement so she questions Anaru directly: Moan a
Beau
You are sayi ng yeah . . . w hat are you sayi ng yeah to?
She's just been going . . . making up suggestions . . . dumb ones . . . she's just. . . Moana without responding to Beau repositions Anaru as a valued group member:
Moana Well it' s not dumb ones. I have been li stening and she ' s maki ng you think because she is using the problem and making sense of it. She knows that you have to l isten to each group member, listen to their thinking and make sense of what they think [turns and asks Anaru] So, what was your way? Can you explain it please?
(Term 3 Week 8)
A group is discussing partitioning a line segment i nto fractional pieces. Moana quietly questions a passive on-looker to ensure that he is accessing the reasoning.
Moana Beau Caliph
Do you understand it Haitokena? Come on people talk about it. So that one in the middle is half.
[points at mid-point] Yeah you can just like draw a line through there. Moana positions Haitokena so he can sense-make and then provide an explanation.
Moana But we [turns to i nclude Haitokena] are asking what are each of these bits? What is the fraction of those bits? Then Haitokena will be able to explain them to you.
Changing student status, shifting beliefs, changing roles
Through direct pedagogical actions Moana repositioned the low achievers, the passive and shy students, and Maori and Pasifika girls from their previous social and academic status. She assigned competence (Boaler, 2006b ) to them through explicitly drawing other students' attention to the intellectual value of their reasoning. Moana recognised how her own actions and expectations for different students had shifted when she commented to me informally: I have realised I did have low expectations because why would I be so excited when they say something that is relevant to what is happening. I thought I had high expectations for my class; well I had high expectations about certain people .. . so at first when someone would say something and I thought did you just say that? Now I am hearing what they all say and its relative and it 's linking in . . . so when I hear them talk out loud I help them to be heard.
Moana regularly reviewed previous video records and discussed the shifts she observed in the students' communication and participation patterns. She voiced specific concern about the tensions and contradictions the shifts in interaction patterns had caused for the Maori and Pasifika girls. She included the changes in her expectations of them, and their own expectations they now held, of their role in the classroom mathematical community. She voiced her observations of how they had changed their view of what 'doi ng' mathematics meant: when you go back and look at the videos of them working in the groups at the beginning they were just pretending to be working in those groups. They were just parroting the words . . . but now you see them really on track. But then as they got hooked in I can see the changes, it 's the girls who are really changing like A naru. I have seen Anaru 's behaviour . . . well it is .. . it is coming to an uneasy uncomfortable place. She is taking herself seriously now in maths. She 's a Pacific Island girl and she has come to {1 cross road and she is making a choice. ( Term 3 Week 8)
Within the sociocultural approach of this study, to understand Moana' s perspective requires explicating the relationships between her actions and the "cultural, institutional and historical situations in which this action occurs" (Wertsch et al., 1 995, p. 1 1 ). To explain the many internal tensions i ndividuals i n a community contend with Forman and Ansell
(200 1 ) u se the notion of "multiple voices" (p. 1 1 5). These multiple voices match the many communities individuals belong to, but also incorporate a historical di mension in which "memories of the past and anticipation of the future affect life in the present" (p. 1 1 8) .
Moana, a s the old-timer (Lave & Wenger, 1 99 1 ) was required t o i nduct the students i nto the mathematical community but in doing this a historical voice included her own memories of her learning experiences in mathematics as a Maori girl. Her expectations for the possi ble future in mathematics of the Maori and Pasifika girls coloured her present interactions. Forman and Ansell drawing on soci al cultural theories explain this "discursive mechanism by which the past and future are drawn into the present as prolepsis" (p. 1 1 8); a form of anticipation of the existence of something before it actually does or happens. Moana's statements and interactions with the Maori and Pasifika girls can be understood as part product of her own experiences and also her future expectations for them.
7.4.4 PROVIDING E XPLANATORY JUSTIFICATION FOR MATHEMATICAL