Lynn Knippertz

• Learning processes of linear functions in mathematics and physics are widely unclear. In this work, we replicate and extend existing results on learning processes of linear functions in mathematics and physics and use triangulation of gaze data and interview data for a deeper understanding of cognitive processes. Furthermore, we utilize mathematical structures such as networks to algorithmically detect solution strategies from gaze data and to distinguish between correct and incorrect solvers. Although there are validated tests of linear functions in mathematics and physics, they are not sufficiently validated and remain unused across countries and age groups. Furthermore, these tests are paper-based, making it impossible to investigate visual attention processes. There is also a lack of research on analyzing students’ visual data when solving math and physics problems using algorithmic, network-based approaches. In this thesis, we improve tools in learning systems that can guide learning without the help of teachers. A three-step approach was taken to fill these research gaps: 1. We conducted a large-scale study using the paper-based test instrument on linear functions in mathematics and physics by Ceuppens et al. (2019), which was validated for 9th grade students in Belgium. We evaluated this test instrument for German students in the same grade with a sample size of N = 249. In addition, we used the test instrument in the upper school (grade 11) to check whether the challenges identified by Ceuppens et al. (2019) persist across school years (N = 298). Results show that German 9th graders perform significantly worse than Belgian 9th graders. However, the challenges surprisingly identified by Ceuppens et al. (2019) remain, and are even more serious for German students. 2. To explicitly understand the cognitive processes behind the students’ problems, we conducted an eye-tracking study in schools (N = 131), in which we were able to cognitively resolve transfer difficulties in adequately applying mathematical content in a physical context using gaze- and interview data for two forms of representation: formulas and graphs. Here, we showed that cognitive processes are context sensitive. 3. In a final step, we use network analysis methods to detect solution strategies based on geometric patterns (for example slope triangles) from gaze data. It was investigated which network metrics are suitable for distinguishing correct from incorrect solvers. There exist suitable network metrics—a first step toward the development of a predictor to distinguish between correct and incorrect solvers. In summary, based on three extensive studies with samples ranging from 131 (eye-tracking studies) to about 250 people (paper-pencil tests), this thesis uses various different methods (paper and pencil, eye-tracking, interviews, network analysis) to empirically analyze learning processes for linear functions regarding mathematics and physics. The results show that students face challenges in interpreting mathematical content adequately in a physical context. The analysis of the cognitive processes form a basis for the sustainable improvement of learning in the future. A teacher training course on this topic has already put this into practice.