Basic mental models of integrals: theoretical conception, development of a test instrument, and first results
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ORIGINAL PAPER
Basic mental models of integrals: theoretical conception, development of a test instrument, and first results Gilbert Greefrath1 · Reinhard Oldenburg2 · Hans‑Stefan Siller3 · Volker Ulm4 · Hans‑Georg Weigand3 Accepted: 17 November 2020 © The Author(s) 2020
Abstract A basic mental model (BMM—in German ‘Grundvorstellung’) of a mathematical concept is a content-related interpretation that gives meaning to this concept. This paper defines normative and individual BMMs and concretizes them using the integral as an example. Four BMMs are developed about the concept of definite integral, sometimes used in specific teaching approaches: the BMMs of area, reconstruction, average, and accumulation. Based on theoretical work, in this paper we ask how these BMMs could be identified empirically. A test instrument was developed, piloted, validated and applied with 428 students in first-year mathematics courses. The test results show that the four normative BMMs of the integral can be detected and separated empirically. Moreover, the results allow a comparison of the existing individual BMMs and the requested normative BMMs. Consequences for future developments are discussed. Keywords Basic mental model · Grundvorstellung · Integral · Empirical evidence · Approaches in textbooks
1 Understanding the concept of integral In many countries, integral calculus is established as a key component of mathematical education at school during the higher stages of secondary school and in calculus courses at university. There are various aspects to the concept of integral, each emphasizing different facets. For example, the definite integral—to which we restrict our attention here—is often interpreted as the area under a graph or the total variation of a dynamic process (Kouropatov and Dreyfus 2013). * Hans‑Georg Weigand [email protected]‑wuerzburg.de Gilbert Greefrath greefrath@uni‑muenster.de Reinhard Oldenburg [email protected]‑augsburg.de Hans‑Stefan Siller hans‑[email protected]‑wuerzburg.de Volker Ulm volker.ulm@uni‑bayreuth.de 1
University of Münster, Münster, Germany
2
University of Augsburg, Augsburg, Germany
3
University of Würzburg, Würzburg, Germany
4
University of Bayreuth, Bayreuth, Germany
For university students, Jones (2013) identified different facets or conceptual schemas of the integral: the ideas of ‘adding up pieces’ and ‘infinite addition’ involve thinking similar to that involved in a Riemann sum; the ‘perimeter and area form’ involves the conceptualization of the definite integral as the area of a fixed region, and the ‘function matching form’ is closely linked to the antiderivative process. The main thesis of this paper by Jones is “that student difficulties might not necessarily arise from lack of knowledge, but from the activation of less-productive cognitive resources over others” (p. 138). However, empirical studies also have repeatedly shown that students at school and at university experience fundamental difficulties in understanding the conce
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