Aspects of functional thinking in applied mathematics education at tertiary level

Similar to Germany (Wolter et al., 2013), in Switzerland the transition to university is a hurdle in the field of scientific and technical studies. This is reflected in high dropout rates in the first year of study. The most frequently mentioned reason for dropping out is insufficient performance in the entry courses (Neugebauer et al., 2019).

At the University of Life Sciences (HLS) of the University of Applied Sciences and Arts Northwestern Switzerland (FHWN), high failure rates in the introductory course "Grundlagen der Mathematik - Analysis I" illustrate this trend. The content of the course consists of arithmetic basics and predominantly the concept of functions.
Difficulties in dealing with functions during school or university studies have been pointed out several times (Carlson et al., 2002; Nitsch, 2015). Students in STEM studies need to acquire further context-specific skills in addition to mathematical skills in order to be able to completely understand functional relations in their applications, which creates further requirements for the students. In chemical kinetics, a subfield of physical chemistry in which functions and a variety of mathematical representations are used to model, students' problems dealing with mathematical functions and connecting mathematical and chemical knowledge have already been pointed out (Rodriguez et al., 2019).

One main reason for the difficulties is often a lack of conceptual understanding of functions. According to Vollrath (1989), a conceptual understanding of functions includes three aspects that are “typical for dealing with functions” and thus for functional thinking.

• The aspect of correspondence emphasizes the unambiguous mapping of an independent quantity to a quantity dependent on it.
• The aspect covariation (cf. Malle, 2000) focuses on questions, how a change of the independent variable affects the change of the dependent variable.
• The aspect function as a whole considers the functional relations as a new object, i.e., the focus is no longer on single pairs of values, but the total set of all pairs of values. This turns the function into a new object that can be manipulated.

When learning functional thinking, the focus is often on the correspondence aspect, which is used by most definitions. However, there is an increasing emphasis on the changing nature of a function, i.e. covariation, which is described "indispensable for practical work with functions" (Malle, 2000, p. 8)

Furthermore, for a sophisticated conceptual understanding it is necessary to grasp functional relations in different representations, i.e., table, function graph, algebraic equation, verbal description) and to be able to switch flexibly between these representations.

When asked how to promote functional thinking, Lichti (2019) elaborated the benefits of digital simulations over real experiments in her dissertation. The simulation provides a simple way to vary the variables involved. Thus, the covariation aspect became more accessible, allowing learners to subsequently formulate reasoning involving the covariation aspect more frequently. Of the various forms of representation, there is evidence that the function graph is most efficient for learning (Rolfes et al., 2021).

Based on this, a digital learning environment was developed consisting of exercises that build on each other and analyze the development of concentration in various chemical reactions, and suitable GeoGebra applets. The exercises and applets are intended to trigger engagement with the function graph and the covariation aspect.
The learning environment is implemented in the course "Grundlagen der Mathematik - Analysis I" at HLS during the chapter "Functions". Using a pre-posttest control group design, quantitative differences in learning gains will first be evaluated. Then, the students' written submissions will be examined guided by the methodology of a qualitative content analysis to get insights into the use of the three aspects of functional relations and translation processes of mathematical and chemical knowledge.