G EN E RALISING MATHEMATICAL REASONING

In conjunction with a focus to shift the students towards generalising Ava chose to use a series of problems which required the students to engage in active and extended discussion of their reasoning. These problems2 were intentional ly devised in the teacher' s study group to support the use of early algebraic reasoning. They required exploration of patterns and relationships and the construction of a rul e as an unexecuted number sentence to describe the relationship. The problems began as single tasks but col lecti vely they provided a means to scaffold the students to construct and voice generalisations. Duri ng task enactment A va emphasised the physical representation of the problems to i l lustrate the recursive geometric patterns. However, discussion with me led to Ava varying the task parameters so that the students were also encouraged to use analytical approaches. As the problem contexts were extended A va pressed the students to look beyond the concrete constraints of the immediate situation. S he directed them to make and test their conj ectures on a range of numbers and explore whether they worked on all numbers. She began teaching sessions with discussion and direct modeling of the use of diagrams, ordered pairs and tables of data. These operated as a structure in which the students were able to explore and develop systematic strategies to test out the patterns that produced the data.

As I described in Section 6.3.4 A va carefully selected groups to provide explanations to the larger group. In the fol lowing vignette she had observed functional reasoning in one group. She used this group' s explanation as a way to shift the wider groups' thinking towards generalising. Continuing from the vignette in section 6.3.4 Ava takes a different role, recognising that this group are presenting an argument which many of the listeners may have difficulty accessing. The explainers had established and recorded a functional relationship between the number of sticks needed to develop a triangular pattern and the number of triangles. She faci litates a prolonged discussion of their model and how they came to develop it. She participates in questioning and directs attention to their notations. Her questioning prompted the l isteners to reflectively analyse their reasoning. Her questions drew further explanation of how the group had reflectively constructed the

generali sation together through persistent pattern seeki ng and exami ning and re-examining the reasoning they were worki ng with.

Rachel

Ava Sandra Rachel

Generalising a functional relationship

[recording] We started off with a table and we stuck a picture of a little triangle and we wrote sticks. We know that one triangle has three sticks but from then on it is two. So two triangles have five sticks . . . We put a sign that said plus one here then we went from one to ten plus one.

Put the lid on the pen for now. Anybody got any questions? Why did you do that?

Well it helps us with our rule. We put a sign up here that says times two. Rachel explains and records x 2 and + I , records from 1 to 1 0 and records the value of each number x 2 + 1 underneath Ava Wang Ava Sandra Ava Sandra

Ruru can you follow what Rachel ' s group have done so far? Can you explain what you see happening here? Wang what about you?

The first one is the one triangle makes up to three because they are addi ng two on.

[poi nts at the ordered pairs] Why though have Rachel and their group . . . done this? Why have they put these numbers down here as well as these two numbers? You are right in that they are adding two on. But why have they done it like this? Why? Maybe somebody else can help?

To help them know what their strategy is.

[asking the explaining group] You will accept that?

[interjecting] M ost of us haven't done the two times and the one plus. Ava facilitated further discussion and questioning related to the use of the systematic recording of the table and ordered pairs. Then she asked the explainers to continue. Aorangi Wel l six times two equals twelve . . . plus one equals thirteen. Then Rachel

when she was thinking of this, that' s when s he saw Pania writing it. She thought of this [points at the recorded rule ] . She thought six times two plus one equals thirteen . . . Then what we did, we did brackets around the six and the two. We started thinking it was times two inside and plus one outside. A va Can you stop there just for a moment. Think about this. All of you think

about your own e xplanations. Wang said they are adding two on . . . is this group addi ng two . . . w here are the two? We need to think about what they are doing?

Alan Instead of plus two, plus two . . . they are timesing by two. So they didn't have to go long . . . it was six times two plus one. Or it could be one hundred times two plus one.

Open-ended problems, example spaces, reflective pattern finding

Through construction and exploration of problems in the study group, Ava was i ntroduced to possible ways she could mediate her students' early development of algebraic reasoning. However, it was the informal discussions after teaching sessions which shifted Ava ' s u nderstanding that the students' attention o n use (and enjoyment) o f manipulatives can i n some i nstances potentially hinder their development o f underlying mathematical u nderstandi ngs. Moyer (200 1 ) identified the tension many teachers have when using manipulati ves as one of balancing student confidence and enjoyment with maintai n i ng a press towards more generalised mathematical understandings. In the lesson the shift in emphasis from the use of manipulatives and the concrete situation extended the problems so that they became more open-ended. Ava provided the students with example spaces (Watson & Mason, 2005 ); these became i ntellectual spaces in which they then had opportunities to search for and test examples and counterexamples of numerical patterns and rel ationships.

Usi ng student explanations in which there was evidence of a ' mindful' approach (Fuchs et al., 2002) to reflective pattern seeking provided a foundation for i nducting them into the use of more "intellectual tools and mental habits" (RAND M athematics Study Panel, p . 3 8). The reasoning contributed by the explainers provided a platform for other participants to make connections and to build from . The use A va made of student explanation, her modeling of careful listeni ng, and the positioning of other students to access the reasoni ng, are i mportant actions which influenced student beliefs about themselves as mathematical doers and users.