Analysis of the cognitive unity or rupture between conjecture and proof when learning to prove on a grade 10 trigonometry course

We present results from a classroom-based intervention designed to help a class of grade 10 students (14–15 years old) learn proof while studying trigonometry in a dynamic geometry software environment. We analysed some students’ solutions to conjecture-and-proof problems that let them gain experience in stating conjectures and developing proofs. Grounded on a conception of proof that includes both empirical and deductive mathematical argumentations, we show the trajectories of some students progressing from developing basic empirical proofs towards developing deductive proofs and understanding the role of conjectures and proofs in mathematics. Our analysis of students’ solutions is based on networking Boero et al.’s construct of cognitive unity of theorems, Pedemonte’s structural and referential analysis of conjectures and proofs, and Balacheff and Margolinas’ cK¢ model, while using Toulmin schemes to represent students’ productions. This combination has allowed us to identify several emerging types of cognitive unity/rupture, corresponding to different ways of solving conjecture-and-proof problems. We also show that some types of cognitive unity/rupture seem to induce students to produce deductive proofs, whereas other types seem to induce them to produce empirical proofs.     

我国少数民族教育公平的文化考量

一、引言 《国家中长期教育改革与发展纲要》把促进教育公平提升到前所未有的高度,把促进公平作为国家基本教育政策,把促进公平作为国家未来十年教育改革与发展的重点.温家宝总理最近在全国教育工作会议上指出,教育公平是社会公平的重要基础,是最基本最重要的公平,是实现社会公平"最伟大的工具".少数民族教育(以下简称民族教育)是我国教育不可缺少而又具特色的重要组成部分,民族教育的公平问题是我国教育改革与发展的应有之义.