LEARNING TO MAKE MATHEMATICAL EXPLANATIONS

As described previously, Moana had adapted the Numeracy Development Project (Ministry of Education, 2004a) to better match the pedagogical beliefs she held. As a result, the students had learnt a different form of mathematics than that intended-one i n which the use of concrete materials, and recording and practising rules and procedures prevai led. Initial classroom observations revealed the students' immediate response when given a problem was to compute an answer. Moana wanted to address this behaviour but she was concerned that her students' growing mathematical interest and confidence be maintained. In the first instance she redesigned a set of problems and used a family of television cartoon characters which she considered would better engage student attention. Using these she addressed their persistent answer seeking behaviour, directly outli ning her expectations: it 's

not about the answer. It 's about how you solve it. You need to be talking about it. Discuss it, what you are doing and then what you are doing all the way.

Moana was aware that the students had many difficulties explaining their reasoning fully. In accord with a revised section of the communication and participation framework (see Appendix K) Moana explicitly scaffolded ways to extend their explanations. In their pairs she emphasised need for extensive exploration and examination of each of the sequential steps. Through these actions the students began to recognise what needed to be included in an explanation to meet their audience's need. They were also learning how to respond to questioning with clarification of their reasoning.

In shari ng sessions Moana closely attended to their verbal explanations and stepped i n to scaffold those who needed support and to address students' use of everyday language and short utterances. The following vignette illustrates how Moana revoiced to name solution strategies or to press the students to explain their actions and solution strategies.

Scaffolding mathematical explanations Aporo uses counters to model how his group sol ved the problem2 . Aporo Two, four, six, eight, ten, twelve.

Moana So that ' s cal led skip counting because you are skippi ng across the numbers. Tere We kept adding like two more. We counted in twos.

Moana Counting in twos. Yes that is skip counti ng.

(Term 2 Week8)

Faa J ays out an array of three rows of three counters to explai n a solution for the problem3 . Faa

Moan a Faa

Moan a Tui

We went three times three equals nine people.

B ut there might be people in here who are not sure . . .

Yes. Because three plus three equals six and plus another three equals ni ne. Did you see how he solved that and explained it? What he said?

He said three plus three equals six plus another three equals nine or three times three equals nine.

(Term 2 Week8)

2 Mrs Dotty has baked some muffins. She puts them on the bench to cool in two rows. Each one of the six

Dotty chi ldren sneaks in and takes two of them so when Mrs Dotty returns they have all gone. How many muffins did Mrs Dotty bake?

3 Mrs Dotty has a car which has her driving seat and then three rows of three seats in the back. How many Dotty children can she fit in her car?

Teacher dilemmas, changing scripts to focus on the reasoning, teacher revoicing

In i nquiry c lassrooms the teacher role is a complex one with many challenges (Yackel, 1 995). Moana described these in her reflections of her lesson video: as a teacher I am put in a position of 'sculpting ' an outcome without the proper tools. The long silences and I feel quite lost on when to jump in and when to let the children struggle. These children are just not used to me letting them struggle. In informal discussions with me Moana outlined her personal conflict at allowing her students to struggle or be confused. Similarl y, she described problems she had scaffolding student development of viable explanations while also maki ng sense of them herself. She also described the challenge of knowing when to question and challenge, when to insert her own ideas, and when to lead discussion back to the mathematical understandings under consideration. Similar dilemmas have been described by other researchers (Chazan & Bal l , 1 999 ; Lobato et al . , 2005 ; Schwan Smith, 2000) when teachers shift the communication and participation patterns towards inquiry.

Previously the students had experienced maki ng mathematical explanations through calculating objects arithmetically in an i nstrumental manner (Skemp, 1 986). Constructing conceptual explanations posed immediate and on-going di fficulties for many students. Some students continued to interpret explaining as providing procedural steps; others had difficulties knowing what details were required for sense-making and what they could assume as taken-as-shared. These difficulties are simi lar to those other researchers have described as students learn to explain and j ustify their reasoning in i nquiry classrooms (Cobb, 1 995 ; Cobb et al. , 1 993; Kazemi & Stipek, 200 1 ; Yackel, 1 995; Yackel & Cobb, 1 996).

To transform how the students had interacted previously in mathematical activity required that Moan a directly address their previous scripts (Gallimore & Goldenberg, 1 993 ). These involved establishing a range of socio-cultural norms identified by many researchers (e.g., B lu nk, 1 998; Lampert, 200 1 ; Sullivan et al., 2002) as supporting rich mathematical activity and discussion. The small i ntermediate steps added to the communication and participation framework mediated a gradual shift in the communication and i nteraction patterns. Moana

used specific pedagogical practices to establish the foundations for peer collaboration (Forman & McPhail, 1 993). These included use of what O' Connor and Michaels ( 1 996) term revoicing-an interactional strategy used to socialise students into mathematical situations. Moana' s revoicing subtly repositioned students to extend their explanations.

7.3.5 LEARNING HOW TO QUESTION TO MAKE SENSE OF