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Students’ understanding of Riemann sums for integrals of functions of two variables
The Journal of Mathematical Behavior Pub Date : 2020-09-01 , DOI: 10.1016/j.jmathb.2020.100791
Rafael Martínez-Planell , María Trigueros

Abstract In this paper we report on a study of how students understand some of the most fundamental ideas of Riemann integrals of functions of two variables. We apply Action-Process-Object-Schema (APOS) Theory to pose a preliminary genetic decomposition (GD), conjecturing mental constructions that students would need to relate Riemann sums to integrals of functions of two variables over rectangles. The genetic decomposition is informed by the researchers’ classroom experience, findings of a previous study that applied semiotic representation theory, and by a study on integrals of functions of one variable. We pay particular attention to the case of an integral of a continuous function over a rectangle and the simplest partition possible, that consisting only of the rectangle itself. We then explore students’ geometrical understanding of the relation between the single term f a , b Δ x Δ y , where a , b is a point on the rectangle, and the double integral over the rectangle. We tested the GD by performing student interviews with 10 students who had just finished taking a lecture-based multivariable calculus course. The findings underscore the importance of each of the mental constructions described in the genetic decomposition and suggests that students have difficulty in some mental constructions that may commonly be assumed to be obvious during instruction.

中文翻译:

学生对两变量函数积分的黎曼和的理解

摘要 在本文中,我们报告了一项关于学生如何理解两个变量函数的黎曼积分的一些最基本思想的研究。我们应用动作-过程-对象-模式 (APOS) 理论来提出初步的遗传分解 (GD),推测学生需要将黎曼和与矩形上两个变量的函数积分联系起来的心理结构。遗传分解的依据是研究人员的课堂经验、先前应用符号表示理论的研究结果以及对一个变量的函数积分的研究。我们特别注意连续函数在矩形上的积分情况和最简单的可能划分,即仅由矩形本身组成。然后,我们探索学生对单项 fa , b Δ x Δ y 与矩形上的二重积分之间关系的几何理解,其中 a , b 是矩形上的一个点。我们通过对 10 名刚刚完成基于讲座的多变量微积分课程的学生进行学生访谈来测试 GD。研究结果强调了基因分解中描述的每种心理结构的重要性,并表明学生在某些通常被认为在教学过程中很明显的心理结构方面存在困难。
更新日期:2020-09-01
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