Aspects and “Grundvorstellungen” of the Concepts of Derivative and Integral

Abstract

This paper discusses aspects and Grundvorstellungen in the development of concepts of derivative and integral, which are considered central to the teaching of calculus in senior high school. We will focus on perspectives that are relevant when these concepts are first introduced. In the context of a subject matter didactical debate, the ideas are separated into two classes: firstly, more mathematically motivated aspects such as the limit of difference quotients or local linearization within the concept of derivative, as well as the product sum, antiderivative, and measure aspects of integration; secondly, the Grundvorstellungen associated with the concepts of derivative and integral. We consider finding a comprehensive description of aspects and Grundvorstellungen to be an important objective of subject matter didactics. This description should clarify both the differences and the relationships between these perspectives, including both mathematically motivated aspects and Grundvorstellungen which are central to the students' perspective. The primary objectives of this paper include a specification of the concepts of aspects and Grundvorstellungen in the context of differentiation and integration and a discussion of the relationships between the aspects and Grundvorstellungen associated with the concepts of derivative and integral. We begin by presenting the characteristic properties of aspects and Grundvorstellungen, including an account of related concepts and the current state of research. These two concepts are then analyzed, based on a subject matter didactical analysis of the concepts of derivative and integral. We conclude with an account of how these insights can be beneficially exploited for introducing differentiation and integration in real-life environments, within the framework of a theory of concept understanding and subject matter didactics.

Students’ Understanding of the Derivative Concept in the Context of Mathematics for Economics

“…A second model in the literature to describe an understanding of the derivative was put forward by Greefrath et al (2016). It is based on the notions of aspect and Grundvorstellung.…”

“…A detailed description of this notion and its development can be found in vom Hofe and Blum (2016). Greefrath et al (2016) define a Grundvorstellung of a concept as "a conceptual interpretation that gives it a meaning" (p. 101). Grundvorstellungen, as normative notions, should be part of the student's concept image (Greefrath et al 2016, p. 103).…”

“…Grundvorstellungen, as normative notions, should be part of the student's concept image (Greefrath et al 2016, p. 103). Greefrath et al (2016) newly added the construct of aspect, which they call "a subdomain of the concept that can be used to characterize it on the basis of mathematical content". Each aspect can be converted into a possible concept definition (p. 103).…”

“…Concerning the derivative, two relevant aspects are mentioned by Greefrath et al (2016): (1) limit of the difference quotient, and (2) local linearization. To give these aspects a meaning, Greefrath et al (2016) mention the following Grundvorstellungen of the derivative: (1) local rate of change, (2) tangent slope, (3) local linearity, and (4) amplification factor.…”

Relevant Content for a Scientific Collaboration in Mathematics and Physics Education Research - A Comparative Content Analysis of Handbooks and Conference Proceedings in Germany and Vietnam

“…On the basis of the mathematical content, possible ideas are developed which the teaching is intended to convey in this field. Examples in the field of analysis can be found in Greefrath et al (2016). In contrast, there are many descriptive perspectives on the concept of imagination.…”