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Using students' ideas to shape the lesson

See B1; A; T2; F1; F2; M1; M2; M3; N1.

In all lessons teachers used students' ideas, their examples, their questions, their ideas, and sometimes their outside-school and previous experiences to shape discussion. In A students' ideas are used to ensure that the context is meaningful. In B1 examples are used to motivate thinking about remainders. In F1 students pose their own questions while in F2 the teacher compares students' work to show that different representations are possible. In M1 and M2 the teacher uses students' ideas to orchestrate a shift from additive to multiplicative thinking, and then in M3 collects their suggestions for proportional situations. In T2 the teacher encourages everyone to explore the validity of one student's conjecture, which is generally false. In N1 students' physical engagement leads them to contribute requirements for precise instructions. All of these clips illustrate how classes which are broadly-grouped rather than narrowly- set provide rich mathematical environments for all students through student-contributions