LEARNING HOW TO QUESTION TO MAKE SENSE OF MATHEMATICAL EXPLANATIONS

Owing to the previous u nivocal patterns of discourse (Knuth & Peressini , 200 1 ) used in the classroom the students were inexperienced at listening to, and making sense of, each others' explanations. Observing their novice behaviour in video records of classroom observations Moana noted their need to learn to talk and to listen to each other. She stated that she planned to address a former interaction pattern in which interjections were an accepted norm. Observations had provided evidence that the interjections maintained a focus on conect answers, affected self-esteem negatively, and detracted from other students having ti me or space to actively listen, thi nk, and examine the reasoning.

In the first i nstance, Moana began to halt mathematical explanations at specific points in the large group discussions. She directed the students to take time to think about what had been explained, and then to ask questions. To focus students on responding appropriately to enoneous reasoning she required that they ask questions and cause the explainer to rethink. For example, when an enoneous explanation was presented she halted discussion, withheld her own evaluation and asked the students to frame a question: Who has got a really good question they can ask Wiremu to make him rethink. I like the way you are thinking about the question. Would you like some help with your question ? Moana drew on a set of questions (see Appendix E) which had been developed in the study group context and used these as models for the students, of how to question to extend explanatory reasoning. She would listen c losely to explanations and regul arly halt the explainer to provide space for the other students to ask questions, if none were forthcoming she would often ask questions herself as illustrated in the fol lowing vignettes. These show how Moana adopted a number of different roles as she participated in discussions. These i ncluded her directly modeling aski ng questions, scaffolding student questioning and shaping how and what questions

were asked, or acting as an observer and directi ng attention to student models of active listening and questioning.

Questioni ng mathematical explanations

Moana as a participant in a discussion listening to an explanation asks a question.

Moana What have you actually done there Mahine? Mahine I have plussed 1 0 onto 47.

Moana So you have added 1 0 onto 47 ? Are there any questions ? Questions like where did you get the ten from?

(Term 2 Week 7)

Moana stops an explainer and asks the listeners.

Moana Do you want to ask a question? You need to start with what.

(Term 2 Week 8)

Moana halts an explanation and directs student attention to an example of active listening Moana Donald is real ly li stening. He is actually listening and watching what has

been taking place. He is not only listening to the person, he is watching when they write things down. His questions are really specific to what the person is doing.

(Term 2 Week 1 0)

Active listening, questioning to provide space for rethinking

In the i nteraction patterns previously established Moana had assumed that the students were learning through listening. This was evident in a statement she recorded after a lesson when she observed the outcomes of the new i nteraction patterns : good to see different thinking coming through, blows me away because before they never had a chance to explore . . . / just thought that they understood by me talking all the time. Now she had begun the gradual process of inducting students into an i nquiry culture which placed value on active listen i ng and questioning. Within this changed learning climate there was a discernible shift i n how the students listened to each other with i ncreased respect.

Evident in the data is how errors had become learning tools, valued as a means to examine and analyse reasoning, rather than cause loss of self esteem . Moana' s approach to errors as a way to widen discussion and questioning is similar to the approach used by teachers i n

Kazemi and Stipek' s (200 1 ) and White ' s (2003) studies. The previous focus of the classroom on provision of answers to predetermined solutions had established a learning culture i n which knowing mathematics was "associ ated with certainty: knowing it, with bei ng able to get the right answer, quickly" (Lam pert, 1 990b ) . Moan a wanted to address this pattern and so she emphasised a need for the students to stop, think, and reconsider the reasoni ng as she explai ned: I was reinforcing rethinking because calling out has been the prevalent means but they need to learn to hold back and do some thinking first. Moana' s actions scaffolded the students to use a more 'mindful' approach to listening and using specific questions to understand the mathematical reasoning. As Lampert illustrated "teaching is not only about teachi ng what is conventionally called content. It is also teaching students what a lesson is and how to participate in it" (p. 34).