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

Abstract

The derivative concept plays a major role in economics. However, its use in economics is very heterogeneous, sometimes inconsistent, and contradicts students’ prior knowledge from school. This applies in particular to the common economic interpretation of the derivative as the amount of change while increasing the production by one unit. Hence, in calculus courses for economics students, learners should acquire an understanding of the derivative that is mathematically acceptable and connected to their prior knowledge, but which also takes into account its practical use in economics. In this paper we first develop a theoretical model describing such an understanding of the derivative for economics students. We then present an exploratory study investigating the extent to which economics students have such an understanding after their calculus course. The results indicate that many of them might not have acquired this kind of understanding, in particular concerning the common economic interpretation of the derivative. The study furthermore yields possible gaps in students’ understanding and possible misconceptions.

Students’ Understanding of the Economic Interpretation of the Derivative in the Context of Marginal Cost

“…This claim has also been backed up by our own research. An analysis of tasks from economics students' final calculus exam at the University of Paderborn suggests that many students did not have a deep understanding of this connection (Feudel & Biehler, 2021). In one task, for instance, in which they had to interpret the derivative of a profit function, 51.4% of the students stated an interpretation as an additional profit, but the interpretations of only 26.1% were fully adequate.…”

“…Its theoretical basis is the construct of concept image by Tall and Vinner (1981) that they define as "the total cognitive structure associated with the concept that includes all mental pictures, associated properties, and processes" (p. 152). Details of the framework's development can be found in Feudel and Biehler (2021). The framework is displayed in Fig.…”

“…As already mentioned, our model was based on the theoretical construct of concept image by Tall and Vinner (1981), since the model describes associations that bridge the way the derivative is understood in mathematics to the way it is practically used in economics (for details see (Feudel & Biehler, 2021)). However, it can also be related to other constructs from the education literature.…”

“…A second point concerns the notion of marginal unit used by the two students who were able to make a connection between the derivative C' (x) and its common economic interpretation by themselves (and many students used it to interpret the derivative in their final calculus exam (Feudel & Biehler, 2021)). It could therefore be fruitful to investigate what the students actually meant with this notion to judge if the use of this notion might help students to better understand the connection between the mathematical concept of the derivative and its economic interpretation.…”

“…We investigated in an interview study to what extent this is achieved in current teaching. This study is part of the first author's Ph.D.-project about economics students' understanding of the derivative, under the supervision of the second author at the Centre for Higher Mathematics Education in Germany (khdm) (Feudel, 2019;Feudel & Biehler, 2021). In this project, we first developed on the basis of didactic and economics literature a framework that aimed at describing an understanding of the derivative that might be desirable for economics students after their calculus course.…”