Invertieren als Fundamentale Idee der Mathematik : Stoffdidaktische Begründung der Fundamentalität und Anwendung als Analyseinstrument in der Stochastik
- Inverting as a fundamental idea of mathematics : Justification of fundamentality in the tradition of subject matter didactics and application to analysing stochastic examples
Wiernicki-Krips, Tobias; Heitzer, Johanna (Thesis advisor); Cramer, Erhard (Thesis advisor)
Aachen : RWTH Aachen University (2021, 2022)
Dissertation / PhD Thesis
Dissertation, RWTH Aachen University, 2021
In this dissertation, considering the idea of Inverting is motivated by the following observations: First, Inverting is a candidate for a Fundamental Idea in mathematics which has not yet been sufficiently investigated and highlighted, although determining an initial state based on a given process and final state is a frequently recurring motif inmathematics, not only for inverse functions. This is in contrast to the fact that reversing mental processes through inverse problems with different didactic objectives is a teaching principle in mathematics education that is fundamental, generally accepted and well-founded in learning psychology. This led to the following question which is the main focus of this doctoral thesis: In what whay is Inverting a Fundamental Idea in mathematics? Second, the instructional relevance of Fundamental Ideas is sometimes critically discussed in didactics. This thesis takes the position that adapting teaching to Fundamental Ideas fosters a deeper understanding of concepts and increases mathematical competency. Third, paying attention to the idea of Inverting is particularly relevantto stochastics didactics: For one thing, the few preliminary works on the Fundamental Idea of Inverting contain no links to stochastics, and for another, this area of stochastics education has been dominated by stochastics-specific Fundamental Ideas. The procedure of this thesis is designed as follows to elaborate on the research needs outlined above and the resulting tasks: First, the underlying understanding of both inverting mathematical processes and a Fundamental Idea is presented. This is followed by the description of the state of research, which essentially consists of an analysis of existing approaches in mathematics education regarding Inverting as a Fundamental Idea. On the basis of this, the originality of the thesis with respect to theories of Fundamental Ideas becomes evident. Subsequently, the fundamentality of Inverting is established in terms of subject matter didactics (‘Stoffdidaktik’), using the criteria of a Fundamental Idea by Schubert and Schwill. With respect to the relevance of teaching practice, this work emphasizes three goals of mathematics-didactic theories of Fundamental Ideas in the tradition of Bruner: learning concepts in an understanding-oriented manner, providing an adequate picture of mathematics and its characteristics and also stimulating didactic reflections. On the basis of the two stochastic examples probability distribution of a random variable and quantiles in descriptive statistics and probability calculation, links with the idea of Inverting are worked out in didactically oriented content analyses (‘didaktisch orientierte Sachanalysen’). Both examples contain aspects of Inverting that are unknown to both high school students and freshmen students. The resulting proposals for teaching at school and university arise not only from the content analyses but also from the experience of workshops with interested and high-performing high school students which were designed to be close to university mathematics. The first important result of the thesis is the justification of the fundamentality of the idea of Inverting. This includes the content analyses regarding mathematical contexts and processes in which Inverting plays an important role. Relationships especially between the mathematical terms inverse function, inverse element, inverse operation,inverse image mapping and quantile function are representated. Teaching the idea of Inverting is possible at school, for example by showing exercises from primary school to high school level. The second focus of the dissertation is the analyses of the stochastic examples with regard to the idea of Inverting. These outline how topics can be made accessible in teaching both at school and university level; they also yield suggestions for further research. The proposals with respect to the distribution of the random variable refer primarily to the central role of the inverse image mapping of a random variable. The proposals with respect to quantiles relate to determining quantiles based on cumulated distribution functions in both descriptive statistics and probability calculus.