学生接受假设的认知困难与课程及教学对策

Cognitive Difficulties of Students’ Accepting Assumption and the Strategies of Curriculum and Teaching

运用质性研究方法揭示了学生在数学概念、原理和证明的认知过程中接受假设的认知困难,并就数学教学和教材编写提出了解决问题的对策.学生面临的认知困难主要表现在：难以接受抽象的定义和公理化的数学原理;难以接受数学定义、原理和证明中的假设条件;学生证明了一个命题与相信该命题是两回事.克服接受假设的认知困难的对策：一是分别按照概念形成的教学模式和由例子到原理的教学模式来教抽象的数学定义和数学原理;二是按照数学概念、数学原理发展的历史来设计教学;三是数学证明的理解与接受离不开实用性证明的帮助;四是教材的编写要尽量降低学生接受假设的门槛,否则教师要为学生搭建理解的＂脚手架＂.文科生和理科生都可以学习曲线与方程、数学归纳法的内容.

Using qualitative methods to reveal the cognitive difficulties of students＇ accepting assumption in the cognitive process of mathematical concepts, principles and proof, and put forward the countermeasures to solve the problems in the teaching of mathematics and the compiling of teaching materials. Students faced cognitive difficulties in： first, it was hard to accept abstract definition and axiomatic mathematical principles; second, it was hard to accept prerequisites in mathematical definition, principles and proof; third, there were differences between proving a statement and believing one. Solutions for the students＇ cognitive difficulties in accepting assumption： first, teaching abstract mathematical definitions and principles in the way of the teaching model of concept formation and the teaching model from examples to principles respectively; second, designing teaching process in the way of the development of mathematical concepts and principles; third, combining with the practical proof; forth, the compilation of teaching materials should try to reduce the threshold for students to accept the assumption, otherwise, the teacher should build an understanding scaffold for the students. Liberal arts students and science students could learn the contents of curves and equations and mathematical induction.