EXPLORING RELATIONSHIPS AND PATTERN SEEKING

Through the specifical ly designed problems and the search for multiple forms of justification the students began to tentatively di scuss numerical patterns they observed. Moana had participated in discussions in the study group of generalisations students may use but she had given them little attention at the initial phases of the study. Now, the need to provide multiple levels of explanatory j ustification led to i ncreased student recognition and voicing of numerical patterns. Moana began to use these with the students, often as position statements used to explore and extend numerical connections and patterns. In the vignette Moana extends discussion of a student-voiced observation to press the students to explore and examine patterns they observed in fractional numbers.

Shifting reasoning from justifying to pattern-seeking and exploring

The students in groups have discussed and explored the statement Aporo m ade "the bigger the denominator the smaller the bit". Moana began the large group discussion by positioning Aporo and his group to validate their conjecture.

Aporo [uses two segmented lines with their fraction equivalents recorded as symbols] Because that number is big [ 1 3/ 1 3] and this number is little [5/5] and you can tell the pieces because the five ones are bigger and the thirteen o nes are smaller.

Moana Questions? What have you got there on your. . . what is your fraction there Wire mu?

Wiremu/Pita Four out of four, four quarters.

Moana Okay. In comparison to five out of five, fi ve fifths which bit is bigger? But what happens if you have thirteen thirteenths? Is what Aporo said true? Pita you j ust said 'different' why? Why? Why do you think it was different?

Rona Moan a Rona Aporo Moan a Rona

I know why [points at segmented li nes and notation for 5/5 and 4/4]. That one [4/4] is a bit smaller that that one [5/5] .

S o what are you sayi ng Rona?

Li ke that' s . . . just a little number [4/4] and this is a kind of a little one [5/5 ] . But that one ' s got like the bigger piece [4/4] and this one has got the little pieces [5/5 ] .

Because you have got to cut this into fi ve pieces and you have to cut that into four pieces so smal ler number, bigger piece.

So if I had a chocol ate bar and I said to you that you can have five fifths or thirteen thirteenths or twenty twentieths . . .

That will be the same.

Beau They are all the same, because it' s just smaller but sti ll one whole piece. Moana So how does that work? Can anyone see the pattern? Is there any ru le we

can use?

Beau Yeah. If the top and bottom are the same then you just have one whole doesn ' t matter what they are if they are the same.

( Term 3 Week 8)

Pattern seeking and exploration, a collective zpd, importance of validating conjectures Evident in the classroom observational data is how a shift from explaining to justifying increased the community li stener-ship. Moana acknowledged the value of student contribution to progress collective reasoning noting after the lesson: this session was interesting because it put me in the position of a sponge board or sponge board/spring board. I am revoicing, not in a negative or condescending way, what they are making sense of I am not taking their words and changing them, but adding to a shared understanding for them and me. It helps me view things from their perspective. Lerman (200 1 ) describes how i ncreased participation in discourse and reasoning practices pulls all participants i nto a shared zpd. This was evident i n this classroom.

Moana's press on the students to validate conjectures scaffolded potential development of more generalised reasoning. Blanton and Kaput (2003, 2005 ) and Carpenter and his col leagues (2004b) note the i mportance of teachers using student validation of conjectures as a tool to mediate generalised reasoning.

7.4.6 USING MATHEMATICAL LANGUAGE

Although the students had developed an i ncreased repertoire of questions to inquire and chal lenge they often still used short utterances and informal colloquial language to explain or respond to questions. We agreed that they needed richer ways to share their reasoning but Moana voiced concern that a push toward more extended responses might cause loss of confidence and withdrawal from participating. A research article7 mediated Moana's next shifts as she used the teacher' s actions in the article to map out her next steps. She increased her explicit models of mathematical talk, activel y participating and describing the mathematical actions using informal, then formal descriptions. She listened carefully to their explanations and then revoiced and extended what they said using multiple layers of meaning. For example, after a student described cutti ng a chocolate bar into: six bits Moana responded with: yes six bits, sixths and six of them, six equivalent pieces all the same size, six sixths of the one whole chocolate. Then she asked : is there anyone else who can model another equivalent fraction ? Good Rona for taking a risk like this. Just go ahead and construct another fraction which is the same, equivalent. Gradually Moana' s phrasing of questions and responses were appropriated. The students used Moana' s models, often rephrasing and using terms and concepts she had previously introduced.

Increasing and extending fluency in mathematical discourse

The discourse and communication patterns had been appropriate for their former situated classroom context (Gee & Clinton, 2000; Moschkovich, 2003 ; Nasir et al. , 2006) . Moana mode led her actions on Khisty and Chval ' s (2002) description of teachers who inducted students i nto more fluent forms of mathematical talk through use of specific models of rich multiple layers of mathematical words and statements. Moana' s actions were designed intentionally to shift students from using colloquial talk which had been accepted previously when the answer was the focus but w hich now limited how they explai ned and justified their reasoning. Meaney and Irwin (200 3 ) describe how informal and i mprecise 7 Khisty, L. L., & Chval, K. B. (2002) . Pedagogic discourse and equity in mathematics: When teachers' talk matters. Mathematics Education Research Journal, 14(3), 1 54- 1 68 .

use of a mathematical di scourse eventual ly restricts students' mathematical reasonmg. Simi larly, Latu (2005 ) emphasises the importance of extending student understanding of mathematical concepts beyond the exact context in which they are learnt. Moana recognised student growth of communicative competence (McCrone, 2005) when she informal ly commented: when we come to maths it 's like these little antenna go up. They go right we are in maths what 's the language and they start to think about the language, the strategies, like they always talk about what strategies are you using or why. Like when I was watching them working in a group the other day and one of them said 'just prove that then ' and it was said so naturally, just part of what they say all the time.

7.4.7 SUMMARY OF THE SECOND PHASE OF THE STUDY

Threaded through this research study is the influence of Moana' s own past experiences and beliefs she had constructed about doi ng and learni ng mathematics. Her observations of the positive outcomes which emerged for her students were key factors which convi nced her to continue the shift in communication and participation patterns towards inquiry and argumentation. My collaborative support, study group activity, the communication and participation framework, research articles and video observations supported each tentative step. These occurred in conjunction with Moana' s careful analysis of their effect on the students' self-esteem and the growing confidence in doing and using mathematics. Influenci ng how Moana scaffolded i nteractions were the presence of multiple voices, past and future, but also a present ' new voice' Moana had constructed as she guided development of the sociocultural norms of an i nquiry community.

Moana's many pedagogical actions to scaffold student engagement in more proficient mathematical practices parallel those described by other researchers who studied teachers worki ng with diverse learners (e.g., Boaler, 2006b ; Khisty & Chval, 2002; White, 2003). Through these carefully considered actions Moana laid the foundations for a community of inquiry. Increasingly, the students participated i n exploratory talk (Mercer, 2000) as they explained and justified their reasoning in classroom interaction patterns which had begun to resemble more closely what Wood and McNeal ( 2003 ) defined as an i nquiry culture. The

students had increased agency within a more balanced intellectual partnership (Amit & Fried, 2005 ).

7.5 TAKING OWNERSHIP OF MATHEMATICAL PRACTICES IN

A COMMUNITY OF MATHEMATICAL INQUIRY

The gradual development of a community of mathematical inquiry had been a long, steady, change-process. At this poi nt Moana noted her facilitative role and that the students: expect everybody to make sense of what they are saying . . . they ask lots more questions all the time of each other, they just expect that they have to justify what they are thinking and that they can use words and other ideas to back up what they are saying. Participation in communication of mathematical reasoning had become an integral part of what it meant to 'know and do' mathematics in the classroom community. Moana outlined to me that she considered being able to participate in mathematical di scourse a fundamental right of her students. She metaphorical ly li nked the way in which Maori are privileged when they know how and when to speak on a Marae to her facilitative role in mathematics lessons : they all have the right to have that privilege. I am making sure that these children all know that they have got the right to talk and be heard in maths. Moana wanted to sustai n the press on mathematical i nquiry and argumentation but also ensure ownership of it was vested i n the contributions and reasoning of all community members.

7.5 . 1 USING M ODELS O F C ULTURAL CONTEXTS T O SCAFFOLD STUDENT ENGAGEMENT IN INQUIRY AND A RGUMENTATI ON

Whilst maintaining an expectation that individual students engage actively i n i nquiry and argumentation Moana directed attention to their responsibility to each other. She re-vi sited notions of the whanau (family) and emphasised the strengths inherent in being a member. She made direct l inks to the students' famil y context, for example emphasising that family members take different roles to make a Cook Island haircutting ceremony successful. When she observed groups working together she drew attention to their collaborative actions, paralleling these with the actions of a Kapa H aka group (Maori cultural group) who i nduct, support and challenge members unti l they ach ieve similar levels of expertise. She drew attention to similarities in the role of the tuakana (elder brother or sister or cousin in a