SUMMARY OF THE FIRST PHASE OF THE STUDY

In this section I have outli ned the many hurdles Moana encountered as she laid the foundations of the i nteraction norms of an inquiry community. Moana had appeared to support the use of the communication and participation framework to plan out shifts in the classroom community' s i nteraction patterns but the novice status of both her and her students in the inquiry environment meant that initially they lacked knowledge and experience of the many roles they were required to take in this culture. Many researchers (e.g., Ball & Lam pert, 1 999; Franke & Kazemi, 200 l ; Huferd-Ackles et al ., 2004; Mercer, 2000; Rittenhouse, 1 998; Sheri n, 2002b; Wel ls, 1 999) report similar chal lenges when teachers shift their classroom culture from a conventional to an inquiry classroom. In an i nterview at the end of the research Moana described what this first phase was like for her. She explained the effect of the shifts i n classroom communication and participatio n patterns: I was slowly coming awake . . . in those initial stages thinking and I suppose when you are a teacher and you do professional development and you think I will try this .. . then I started seeing .. . and I was more impressed not by their maths but by their talking, how they were talking. Then I didn 't feel so harsh on myself for focusing on the participation norms because they were actually talking. So I just started to settle down to the maths. The reconstruction of the communication and participation framework i nto smaller more incremental steps supported a gradual change in the classroom i nteraction patterns. This i n

turn appeared to provide Moana with confidence to conti nue making shifts in the learni ng culture.

The students now acti vely engaged in construction and examination of conceptual explanations-an important shift because according to Cobb and his col leagues ( 1 997) bei ng able to provide explanatory reasoning is an important precursor for supporti ng development of explanatory j ustification and argumentation. Moreover, the close attention Moana had placed on active listening, questioning and rethinking shi fted the students towards sense-maki ng within zones of proximal development (Forman & McPhail, 1 993). They had begun to view their reasoni ng from the perspective of others which potentially provided an important fou ndation for future mutual engagement (Wenger, 1 998) in collaborative activity. The classroom culture Moana had constituted had shifted toward what Wood and McNeal (2003) define as a strategy reporting discussion context.

7.4 FURTHER DEVELOPING THE COMMUNICATION AND

PARTICIPATION PATTERNS OF A COMMUNITY OF MATHEMATICAL INQUIRY

Moana' s pedagogical actions m the first phase had focused on establishing the socio­ cultural norms which supported the students to provide mathematical explanations. Our joint discussion of the students' current communication and participation patterns prompted us to establish the next steps on a trajectory towards developing reasoned collective discourse. Moana considered that the students were ready to learn how to engage in mathematical inquiry and argumentation to justify their reasoning. However, she voiced her key concern that the students retain their growing mathematical confidence e xplaining that she thought that: this was risky stuff moving them, upping the ante; I'll be hanging out there as much as they will be hanging out there but they are ready. Together, we carefully analysed the communication and participation framework and constructed small specific steps (see Appendix K) to support the shift towards student use of mathematical i nquiry and argumentation.

7.4. 1 COLLECTIVELY CONSTRUCTING A N D MAKING MATHEMATICAL EXPLANATIONS

In the first i nstance Moana focused on further developing how the students worked collaboratively. She restructured the dai ly lesson format so that she was no longer leading mathematical activity from a central position. Now in each lesson, after an initial short di scussion the students worked in smal l heterogeneous problem solving groups and then returned to the larger group situation to conclude with a shari ng session.

Moana's previous focus had been to develop individual student capacity to actively li sten, explain their reasoning and make sense of the reasoning of others. Now Moana wanted to press group behaviour towards increased col laborative interaction. She stated that she wanted the students to di scuss, negotiate, and construct a collective solution strategy. In shaping their interactions Moana emphasised their personal responsibility to engage and u nderstand the reasoning used by other members. She established a pattern where the students began their small group activity with a mathematical problem which they were directed to read and think about individually, then discuss, interpret and together negotiate a solution strategy. Moana gave each group one s heet of paper and pen to use. She moved from group to group listening to their talk and only intervened to ensure all i ndividuals contributed and could explain the developing reasoning. She explicitly establi shed with them that they cou ld only bring questions or problems to her if the whole group agreed that they required assistance. When a group member requested help she discussed with the group their prior actions, drawing from them ways they might solve their problems together. She explained that she wanted to assist the students to recognise and use their col lective strengths to engage more autonomously in mathematical activity.

Although Moana required that the students develop a joint explanation for the larger sharing session she encouraged them to explore and discuss a range of ideas, then select the one they agreed they could all understand and explain. She guided their negotiation and selection of a shared strategy solution through establishing a set of ground rules (see Appendix J) for talk loosely based on those developed by Mercer (2000) but pertinent to this group of students. When she observed that they had developed a shared explanation she

asked them to predict which sections their audience might find difficult, discuss and explore questions they might be asked, and rehearse ways they might respond. She said that she wanted to maintain their confidence but also to i ncrease their s hared understanding and prepare them to respond appropri ately to questions. The following vignette illustrates how Moana facili tated discussion which supported the students to question and probe an explanation for sense-maki ng but at the same time ensured that ownership of the reasoning remained with the explainer.

Scaffolding exploratory talk to explore an explanation

Moana joins a small group and is a listening participant. Anaru is making an explanation for the problem4.

Anaru Moan a Anaru Moan a Wiremu Anaru Donald Anaru

I drew six cakes and my way was halving it. Can you show us?

[draws four squares and divides each one i n half] And then I halved it. Has anybody got any questions they want to ask Anaru?

Why did you cut them in halves?

Oh because they said that there were four people and she was to half it so they have equal amounts.

What did you do next?

Each person took three parts and there was one whole cake left and I hal ved it and then I halved another half and they took one each. That's four pieces they took.

Aporo watches closely and moves his hands as he counts the pieces. Aporo

Donald Anaru

How could she [referring to Mrs. Dotty] half it again? If it was al l gone?

Yeah. There was one whole cake left and then I halved it into fourths. The listening students count the halves and then look at Anaru questioning.

Moan a Donald Anaru Rona Moan a

Do you all agree? You have listened to her but remember you have to agree. [points at the last two cakes] Can you tel l us what you said in that bit? I said that I halved it and each person took three.

I know. So you are saying they got a w hole and a half. There are two cakes left that are halves so they got a whole cake and a half.

Do you agree Anaru? Is that what you mean?

4 Mrs Dotty had just baked 6 chocolate cakes and suddenly she heard a knock on the door. Guess what? 4 friends had come to visit i n their flash car! They were really hungry so she cut up the cakes and they shared them between them. She didn't eat any because she had already had lunch. If they each ate the same amount how much did each person eat? What fraction of the six. cakes did each person eat?

Anaru No because I j ust found out that there was halves of a cake. I said i n quarters but I meant in halves.

Hone So if she cut all six cakes i nto halves and gave them three halves each that means one and a half or three hal ves each eh?

(Term 3 Week 2)

Facilitating the talk, increased student autonomy, gaining consensus through exploratory talk

The pedagogical actions Moana took to establish collaborative group skills are simi lar to those used by Lampert (2002). The direct attention Moana gave to the group processes, as she stepped in and out of a range of roles, meant that her students learnt that both social and academi c outcomes result from group work.

Evident in the data is a shift in positioning of all the members of the classroom. Moana analysing her actions in a video excerpt of a lesson observation recorded: more korero (discussion) less teacher talk and I am really moving into a facilitating role. Her more facilitati ve approach could be attributed to her direct attention to group processes. Like other researchers (e.g., Rojas-Drumrnond & Mercer, 2003; Rojas-Drummond & Zapata, 2004; Wel ls, 1 999) who explicitly scaffolded a talk-format with diverse students, this resulted in a progressive shift toward increased exploratory talk.

The changes evident in Moana' s classroom resonate with those Hufferd-Ackles and her col leagues (2004) maintain are important in the growth of "a math-talk learning community" (p. 87). These researchers described the developmental growth of a classroom community i n four dimensions. These included questioning; explaining mathematical thinking; the source of mathematical ideas; and responsibility for learni ng. As Moana' s students gained greater agency, i ncreasingly they initiated questioning. Similarly, their mathematical thinking became an i mportant source for m athematical discussion. Moana noted the confidence the students showed in explaining, elaborating on, and defending their reasoning when she recorded after a l esson : A naru shows confidence and she has a definite sense of trust about her way.

The classroom observations reveal that explanations had become mathematical arguments as a direct result of Moana' s expectation-the negotiation of a collective view. Through the close examination of mathematical explanati ons the criteria for what the students considered acceptable as mathematical explanations was established. Not only were explanations required to be experientially real and relevant but the listeners also expected elaboration or an alternative explanation.

7.4.2 PROVIDING A SAFE RISK-TAKING E NVIRON M E NT TO SUPPORT