A cultural study of classroom discourse and its impact on students' initiation of geometry proofs
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Proof is essential in learning mathematics. However, it is hard to teach and often students have difficulties initiating proofs. The purpose of this qualitative case study is to investigate classroom discourse and its impact on how some tenth grade students learn to begin proofs in one school in Iran that is intended for students who are mathematically gifted. Data collected through classroom observation, student and teacher interviews, and students' proofs of nine geometric problems were analyzed to identify cycles of classroom discourse. The data also showed strategies students use to begin proofs, difficulties they encounter in doing so, and how the difficulties are overcome. Analysis of the classroom discourse identified not only a number of intertwined cycles students undergo in order to understand, plan, and write proofs; but also the classroom's didactical contract. There is evidence that this contract was set mutually by the teacher and his students, and possibly by the principal and some of the parents as well. This contract demanded that the teacher create an environment that promotes students' construction of knowledge. Students were required to create at least three proofs for each problem, and to focus on the shortest and most elegant ones. Five strategies are identified that students use to begin proofs. They may express a geometric problem algebraically and use algebraic manipulations to complete the proof. They may use three strategies to find how the given information is connected to what they want to prove. If none of these strategies produced results, students would use a strategy that, as was taught and practiced, would never result in complete or valid proofs. Often proof of the problems requires students to identify theorems, relationships, or auxiliary lines that help them with the proof. Different approaches were identified that students use to do so. One of the students' difficulties with initiating proofs was due students losing their focus on the problem. Three of their difficulties were linked to the presence or absence of a diagram with the problem. The last difficulty was associated with the invalid strategies that students were taught.