Students' Conceptions of the Mathematics of Change and Variation'...

本研究项目考察了小学生、高中生和大学生如何发展微积分的直观概念——在这种情况下，微积分被广泛定义为变化和变异的数学。核心假设是，从很小的时候起，人们就形成了关于事物如何变化的模型。数学教学必须建立在这些直观的模型之上。调查方法包括一系列的教学访谈，在这些访谈中，个别学生被提出关于变化情况的问题，如汽车的运动、水的流动或植物的生长。他们得到了实物材料和软件工具来帮助他们实验这些问题。研究人员观察并录像随后的讨论，材料的使用，以及学生对解决方案的表达。随后的分析集中在学生对问题的构建，指导他们预测的想法，他们使用的资源，以及他们产生的验证预测和解释的原则。该项目的研究结果将帮助数学教育工作者重新思考目前微积分的教学方式，以及引入关键微积分思想的年龄水平。研究结果还将以“变化”课程链的形式，直接提供给主要的小学数学课程。

This research project examines how elementary, high school, and college students develop intuitive conceptions about calculus--which in this case is broadly defined as the mathematics of change and variation. The central assumption is that, from an early age, people develop models about how things change. Mathematics instruction must build upon these intuitive models. The method of investigation involves sequences of teaching interviews in which individual students are presented with problems about situations of change such as the motion of cars, flow of water, or plant growth. They are given physical materials and software tools to help them experiment with the problems. Researchers observe and videotape the ensuing discussion, use of materials, and student-constructed representations of the solution. Subsequent analysis focuses on students' constructions of the problem, the ideas that guide their predictions, the resources they use, and the principles they generate to verify predictions and explanations. The project's findings will help mathematics educators rethink the ways in which calculus is currently taught, as well as the age levels at which key calculus ideas are introduced. Results will also feed directly into a major elementary mathematics curriculum, in the form of a curriculum strand about "change."