Denkformen des formalen Denkens : eine qualitative empirische Studie zur spezifischen Kognition von Studienanfängern im Fach Mathematik

  • Modes of formal thinking : a qualitative empirical study on the specific cognition of first-year mathematics students

Hänisch, Corinna; Walcher, Sebastian (Thesis advisor)

Aachen : Publikationsserver der RWTH Aachen University (2011, 2012)
Dissertation / PhD Thesis

Aachen, Techn. Hochsch., Diss., 2011


The transition from elementary mathematics taught in schools and highschools to mathematics taught in universities is often problematic for first-year students. This is evident from high drop-out rates during the first academic year, for instance (cf. e.g. Dieter, M., Brugger, P., Schnelle, D., Törner, G. (2008): "Die Zahlen rund um das Mathematikstudium" in den Mitteilungen der DMV, 16). However, there are only a few studies that investigate possible reasons in a systematic manner. This is precisely the starting point of the semi-structured interview study "Denkformen des formalen Denkens" ("Modes of Formal Thinking"). The objective of the study is to investigate the transition to formal thinking of first-year students while taking into account the characteristics of higher mathematics. It is presumed that sources of the recurrent difficulties lie in the subject matter as well as in the methods of teaching and learning at Universities (cf. Fischer, A., Heinze, A., Wagner, D. (2009): 5.2 Mathematiklernen in der Schule - Mathematiklernen an der Hochschule: Die Schwierigkeiten von Lernenden beim Übergang ins Studium, in: Heinze, A. und Grüssing, M. (Hrsg.), Mathematiklernen vom Kindergarten bis zum Studium, Waxmann, Münster, S. 258). The study focuses particularly on the problems of first-year mathematics students, since its subject matters as well as its general lines of approach differ considerably from other courses dealing with mathematics such as engineering courses. Accordingly, the difficulties are persumed to be different as well. The study was conducted during the winter term 2009/2010. Both students pursuing a Bachelor of Science degree in Mathematics and students pursuing a teaching certificate in Mathematics took part. Since both groups attend the course "Mathematische Grundlagen" (principles of mathematics) at the very beginning of their graduate school programmes, the course's topics are taken as a basis for questions in the semi-structured interview. This study comprises and analyses four interviews covering set theory, mathematical induction, functions, and binary relations. Taking into account existing approaches to explain acquiring mathematical concepts as well as taking into account the results of an initial pilot study, three categories for the analysis could be identified: misconceptions; formal skills; generic and exemplary illustrations. The second part of the thesis focuses on gender specific differences. As students change their way of mathematical thinking from an elementary to a more abstract level, attention is given to possible correlations with regard to predicative or functional ways of thinking. The terms predicative and functional thinking were introduced by Schwank to distinguish certain cognitive patterns. Several studies already showed manifest gender specific differences with respect to understanding elementary mathematics in school. Initially the study identifies preferences for one of these two ways of thinking for each participant and thus it is possible to relate this classification to traces of predicative and functional thinking in the semi-structured interviews.