Konzentrationsmessung und Lorenzkurven - Darstellung, Analyse und Modellierung im Mathematikunterricht

Hausmann, Hannah Lea Octavia; Kamps, Udo (Thesis advisor); Cramer, Erhard (Thesis advisor)

Aachen (2017, 2018)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2017

Abstract

The present thesis outlines central ideas and methods of concentration measurement, i.e. capturing and presenting disproportionateness in e.g. economy or social demography, focusing on didactics in mathematics classes with view to the goals of a modern, current curriculum. In particular, it will be analyzed which mathematical content that is relevant to the curriculum can be found in the various aspects of concentration measurements, and which key competences can be developed in pupils in its applications. Concentration measurement and its applications in school will be put into context of the educational standards of the Federal Republic of Germany in Mathematics, of general principles of technical teaching, as well as exemplary in connection with federal key curricula. Furthermore, suggestions will be provided on how to develop said aspects in class, and concepts will be presented and assessed for the implementation of these ideas in daily practice. Specifically, the following aspects will be evaluated:First, typial applications of concentration measurements such as the distribution of market shares or income will be discussed with view to their transferability as an example of use, and their suitability as framework for tutorials in teaching. Piecewise linear Lorenz curves as well as the Gini coefficient as the most common descriptors of concentration will be presented, and subsequently be analyzed regarding their didactic potential and accessibility to students at junior high school level. Subsequently, options will be discussed towards deepening insights into these topics and basic approaches, with the aim to refresh and enhance these in the further curricular course, particularly in the advanced secondary school (Sekundarstufe II).On the one hand, some options will be discussed that are accessible by means of scholar mathematics to determine smooth Lorenz functions in the form of a polynomial or power function to estimate the concentration for a given data set. In this context, approximative methods such as systematic sampling or regression will be broached. On the other hand, geometrical approximations for the Gini coefficient that represent modeling and optimization problems will be presented as an alternative approach to deepen the learning matter. The discussion outlines approaches that differ substantially in level of difficulty and demands on the pupils' a priori knowledge, and whose level of comlexity is suited for classes at junior high school level, for advanced secondary school ("Qualifikationsphase") or for projects outside the regular scholar curriculum.Based on the selected topics, a one-day workshop was designed for students of advanced secondary school that was initially held in the framework of the "Schüleruni Mathematik" in 2015 and repeated in 2016 in an adapted version. The thesis finishes with a documentation and critical analysis of the practical implementation of this workshop, and concludes with constructive suggestions for further specific ways of implementation in the form of teaching units.