COLLABORATIV E CONSTRUCTION OF MATHEMATICAL EXPLANATIONS IN SMALL GROUPS

Ava stated that in line with the NDP she regu larly used small acti vity groups m her mathematics lessons. These groups of 3-4 students were required to construct solution strategies for explaining at a larger sharing session. However, in reviewi ng of videotapes with me A va recognised that the students worki ng in their s mal l groups predominantly engaged in use of either cumulative or disputational talk (Mercer, 2000) . Cumulative and disputational talk, as we saw in Chapter 3, has been characteri sed as an unproductive form of talk which limits how group members explore each other's mathematical reasoning. Ava acknowledged that her students needed more specific guidance on how to work collaboratively.

In the first i nstance, A va focused on how the students participated together in smal l group acti vity. She outlined to the students her requirement that they actively engage in listeni ng, discussi ng, and making sense of the reasoning used by others. To develop their skills to work col lectively she stressed that all group members needed to engage in construction of mathematical explanations and be able to explai n them to a wider audience .

In accord with the pathway she had mapped out, Ava initiated an immediate shift towards establishing that the mathematical explanations the students constructed should be well reasoned, conceptually clear, and logical . She explicitly scaffolded how they were to provide an explanation : Talk about what you are doing . . . so whatever number you have chosen don 't just write them. You say I am going to work with . . . or I have chosen this and this because . . . and this is what I am going to do. S he outlined not only how these explanations needed to make sense for a listeni ng audience but also how listeners needed to make sense of the explanations offered by others. To develop their skill in the examin ation and analysis of explanations she provided opportunities for the small group members to construct, explain, and, in turn, question and clarify explanations step-by-step. The

fol lowing vignettes from three separate observations early in the study illustrate how Ava inducted the students into public construction and evaluation of conceptual explanations within collaborative zones of proxi mal development (Vygotsky, 1 978).

Collaborative interaction and sense-making

The students had individual time to think about a solution strategy then Ava said:

Ava You are going to explain how you are going to work it out to your group. They are going to l isten. I want you to think about and explain what steps you are doi ng, each step you are doing, what maths thinking you are using. The others in the group need to listen carefully and stop you and question any time or at any point where they can ' t track what you are sayi ng.

(Term 2 Week5)

The students were asked to construct i ndividual conjectures then Ava directed them to examine and explore each solution strategy:

Ava They might say I think it is 59. That's cool but they have to back it up, explain how they came up with it. They have to say why. So I want you before you even begin to go around in your group and actual ly talk about it. Someone in your group may ask you a question. For example, that' s an interesting solution . Why do you think that? Could you show us how you got it?

( Term 2 Week 6)

Ava explained and explored the group roles then directed them:

Ava Argue your maths. Explore what other people say. Listen carefully bit by bit and make sense of each bit. Don ' t just agree. Check i t all out first. Ask a lot of questions. Make sure you can make sense that you understand. What' s another i mportant thing i n working in a group?

Alan Share your ideas. Don't just say I can do it m yself that adds on to teamwork. A va That's right. We do need to use each other' s thinking . . . because we are very

supportive and that's the only way everyone will learn. So we have to be discussi ng, talking, questioning, and asking for clarifi cation. Whatever i t takes to clarify what you understand i n your mind.

(Term 2 Week 5)

Interaction scripts, peer collaboration, making and clarifying mathematical explanations

Ava had assumed that the students would construct appropriate knowledge because they were asked to interact cooperatively in a small group. Thi s is a common mjsconception (Mercer, 2000). The i nitial examination of lessons in Ava' s classroom confirmed what

Mercer and other researchers report (e.g., Irwin & Woodward, 2006; Rojas-Drum mond et al., 2003 ; Wegerif et al., 1 999): the students had many difficul ties e ngaging in and using mathematical talk in the smal l group situation

To transform how the students interacted in mathematical activity in their small groups required expl icit renegotiation of both what the task required and the scripts for conduct­ the ground rules which shape the interaction (Gallimore & Goldenberg, I 993). A va's use of clear directives to outline her expectations gave the students a working knowledge of what their obligations were. It also provided them with understanding of the learning potential of small group interactions, offeri ng them a motive to participate appropriately and develop ownership of the ground rules. She laid the foundations for peer col laboration (Forman &

McPhai l, 1 993) as she structured the group activity . The students were i ntroduced to and practised skills for explaining their reasoning and examining the reasoning of others. Many of these actions paral lel the pedagogical actions taken by Lampert (200 1 ) when she i nstituted small group collaboration. These included : establishing with the students recognition of themselves as valuable sources of knowledge; emphasising mutual responsibility for sense-making; and the requirement of i ndividual responsibility for understanding, thus removing any possibility that a lack of understanding cou ld be attributed to others.