Report: Scaffolding Small Group Interactions

Mathematics: Essential Research, Essential Practice — Volume 1
Roberta Hunter – Massey University
Proceedings of the 30th annual conference of the Mathematics Education Research Group of Australasia
J. Watson & K. Beswick (Eds), © MERGA Inc. 2007
Pages 430 – 439

Pg 430 To achieve such learning communities teachers are required to establish ways in which students can engage in multiple forms of interaction. These include whole class discussions and also small cooperative problem solving groups.

Proponents of collaborative grouping maintain that through providing the individual students with opportunities to articulate their thinking not only do they learn to exchange mathematical ideas – but also they make available their reasoning for examination and critique (Artzt & Yaloz-Femia, 1999; Rojas- Drummond & Zapata, 2004). In addition, through opportunities to explain and justify reasoning, explainers are able to review and reconstruct their mathematical thinking, and extend and build stronger arguments (Whitenack & Yackel, 2002). Other advocates who support teacher use of small groups propose that this structure better meets the needs of the diverse or at-risk students (Baxter, Woodward, Voorhies, & Wong, 2002; Boaler, 2006; Rojas-Drummond & Zapata, 2004; White, 2003).

Pg 431

Within the small supportive groups it is the peers who provide an important forum for the diverse students to develop and extend their mathematical reasoning. In turn, through listening and making sense of their peers’ explanations they are able to integrate their reasoning with that of others.

…Irwin and Woodward’s close examination of groups working independently revealed a predominant use of competitive talk both student to student, and between the boys and girls. Although the teacher had directed them to work cooperatively in these groups she had provided no specific guidance.

…In disputational talk the students rather than trying to reach joint agreement work through cyclic assertions and counter-assertions as they struggle for control and status. In the cumulative form a collective view is reached but without evaluative discussion.

…if students are to engage in productive small group activity teachers need to scaffold specific interactional strategies that support equitable outcomes for all participants.

…The teachers use a set of ground rules that emphasise sharing of information, a need for group agreement and responsibility for decisions. But the ground rules also focus on challenge and justification of the collective reasoning.

pg 432

…use of heterogeneous grouping and open-ended problems to draw multiple ways to value student
contribution. … group roles for students and responsibility for each others’ learning.

…from a sociocultural perspective. From this perspective mathematical teaching and learning are inherently social and embedded in active participation in communicative reasoning processes (Lerman, 2001)

Developing a Shared Perspective in Small Group Interactions

…a focus on [students] need to engage actively in listening, discussing and making sense of the reasoning used by others.

pg 434 …emphasised their responsibility to develop understanding of the reasoning from the perspective of each member of the group. … the roles of members in the group and placed particular importance on the need for justification and reasoning to develop a collective view.

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

Ava was aware that different students had different status in her class. Although she focused on their need to consider the reasoning used by all the participants in the group she also actively positioned specific students.

Pasifika student making an explanation to the small group she began the large group sharing by asking:
Ava: Aporo do you mind if we kick off with you because you were doing some really good talking and explaining to your group and I think this will be a really good opportunity for you to show your maths thinking.
When Aporo began his explanation in a quiet voice Ava requested that the other students listen closely. Then when another student began to prompt him and he hesitated she told the student:
Ava: He knows. He knows. You don’t have to prompt him because he knows where his thinking is going

Learning Ways to Disagree and Challenge Politely

Engaging in questioning and inquiry involved considerable challenge to how many of these diverse students had experienced mathematics previously…use of open-ended tasks and problems

These supported the notion that there were multiple ways the students in their small groups could develop and support each other in the construction of explanatory reasoning and justification.

436

Ava recognised the social and academic risks students took when they disagreed or challenged the reasoning of others. Therefore she carefully structured ways in which the students in their small groups could approach disagreement and challenge. She would watch the students working together in their small groups and then she would ask specific members if they agreed or disagreed with the reasoning being used. She also consistently required that they provide justification for the specific stance they took.

Ava would also place herself as a participant in small group activity and model behaviour that tuned the students into becoming more aware of other participants responses revealed in their body language. She would actively prompt and probe for agreement or disagreement when she noted a frown on participants’ faces or a querying shift in their bodies. Her active prompts to voice agreement and disagreement were appropriated by the students when they worked independently. They would explain a solution strategy step by step, watching the other group members carefully. When they saw a hesitant or querying look on a peer’s face the explainer would halt the explanation and respond by asking:
Rachel: Tama you look confused? Do you need to ask some questions?
Tama: Well three times three? Isn’t it three plus three plus three not the times way?

Learning the Practices of Mathematics

437

Ava consistently interacted with the students, exploring and discussing with them interactions that supported them learning the practices of mathematics

…at regular intervals during the year she introduced a different set of questions and prompts. She began with a set of questions that the students could use to elicit more information about mathematical explanations. They included such questions as “what”, “where”, “is that”, “can you show us”, “explain what you did”.

She actively modelled the use of these questions and prompted the students to use them as she participated in their small groups. She also displayed them on charts on the wall. When she heard a student use a different form of one of the questions she would halt the group and draw their attention to the question and how it was being used. Then she would add it to the wall chart.

Conclusions and Implications

Ava used to scaffold the students in small group interactions. Over the year, Ava implemented a wide
range of interactional strategies that focused the students’ attention on the development of a collective view…. the importance of open-ended problems and tasks that supported a range of ways to contribute to the group processes.

pg 438… central to the group responsibility was the requirement for the students to justify and provide valid reasoning for their solution strategies.

Effecting change in the small group interactions was a lengthy process. It required ongoing attention by Ava of the discourse used in the groups. It also required her active participation as a model of the interaction patterns in the group and her highlighting student behaviour to demonstrate valued interaction patterns. Further research is needed to examine other factors that are important in enacting and maintaining diverse learners’ use of productive discourse.