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.

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The concept of derivative is characterised with reference to four basic mental models. These are described as theoretical constructs based on theoretical considerations. The four basic mental models—local rate of change, tangent slope, local linearity and amplification factor—are not only quantified empirically but are also validated. To this end, a test instrument for measuring students’ characteristics of basic mental models is presented and analysed regarding quality criteria.Mathematics students (n = 266) were tested with this instrument. The test results show that the four basic mental models of the derivative can be reconstructed among the students with different characteristics. The tangent slope has the highest agreement values across all tasks. The agreement on explanations based on the basic mental model of rate of change is not as strongly established among students as one would expect due to framework settings in the school system by means of curricula and educational standards. The basic mental model of local linearity plays a rather subordinate role. The amplification factor achieves the lowest agreement values. In addition, cluster analysis was conducted to identify different subgroups of the student population. Moreover, the test results can be attributed to characteristics of the task types as well as to the students’ previous experiences from mathematics classes by means of qualitative interpretation. These and other results of students’ basic mental models of the derivative are presented and discussed in detail.

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.