# Normative Modellierung im Mathematikunterricht : Bildungspotenzial, exemplarische Sachkontexte und Lernumgebungen

- Normative modelling in teaching mathematics : educational potential, exemplary authentic contexts and learning environments

Pohlkamp, Stefan; Heitzer, Johanna (Thesis advisor); Lengnink, Katja (Thesis advisor); Maaß, Jürgen (Thesis advisor)

*Aachen : RWTH Aachen University (2021)*

Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2021

Abstract

General pedagogical objectives such as the promotion of civic engagement and individual empowerment apply to teaching mathematics. Heinrich Winter has combined the social expectation of the formation of fully-fledged citizens with mathematical contents, as there is a reciprocal relation between universal educational and subject-oriented objectives. One aim of the present thesis is to concretise this ideal of mathematical literacy by means of exemplary issues and to offer adequate learning environments. Given the significance of the mathematical and contextual dichotomy, mathematical modelling becomes crucial for an enlightenment via mathematics. In particular, normative modelling is of interest, being defined as a procedure where mathematics is used to shape reality – in distinction to descriptive modelling where existing phenomena are described with the help of mathematics. But you can only pay taxes in reality, if a tax rate has been mathematically predetermined. This use of mathematics is ambiguous and normally biased as well as a subject of negotiation. These characteristics are rather rare in the context of learning mathematics, but even more instructive for a public discussion and judgement. German curriculums focus on – or are even limited to – descriptive modelling. This thesis exposes the educational value inhering in teaching normative modelling, even from a content-related perspective. The result is a catalogue of concrete learning targets which decisively add to the existing competencies in mathematical modelling and that rounds off the vision of mathematics. Apportionment methods for proportional representation are a paradigmatic example for normative modelling. On the one hand, normative decisions in the modelling process reflect classical problems of the mathematics of apportionment, on the other hand the German debate about the current increase in parliament’s size is an authentic occasion to analyse the reciprocity between mathematics and context. An associated analysis starts with a comparison of apportionments as algorithmically defined functions, which leads to the Balinski–Young theorem as the mathematical reason for a normative approach, followed by the mathematics behind the German election law. This content-oriented, didactically motivated illustration constitutes one main impulse of the present thesis: By discussing apportionment methods students should become aware of normative modelling. A designed and evaluated workshop consists of retracing the modelling process from the actual votes cast to the composition of parliament. After general questions concerning mathematics of elections, the apportionment on the German Bundestag represents a concrete case study. The students’ activities contain interest awakening authentic situations open for a dynamic exploration of the mathematical background. In order to complete the proposed learning targets, this thesis presents three other contexts and the linked mathematical insights to specify some aspects: learner-centered modelling of business valuation, the reciprocity of normative and descriptive aspects in a modelling with respect to the Arctic sea ice and the advantages of digital tools for promoting citizen empowerment illustrated by the example of income taxation models. As a learning result, students formulate beliefs about (normative) modelling and mathematics in general. These beliefs offer insights into the productive tension premised by this thesis as there are didactically constructive interdependencies between mathematical literacy, normative modelling and the role of mathematics. In total, this thesis addresses following main issues:– the analysis of the dimensions of mathematical literacy in reference to normative modelling– the establishment of learning targets for a broader understanding of modelling– the mathematical and contextual depiction of apportionment methods as a prototypical normative modelling– the didactical implementation of a workshop regarding this topic as well as the design of additional learning environments for further foci– the disclosure of ideal and real consequences for students’ beliefs about mathematics and resulting possible citizen empowerment. The didactical composition of relevant subject matter in exemplary contexts results in concrete leaning environments. These enable the discovery of normative modelling on the basis of mathematical contents. The accessibility to important civic and yet complex issues has increased – also by means of digital visualization: New insights and hence the reflection about the way in which mathematics is applied in socially relevant areas characterise the emerging educational value.

### Identifier

- DOI: 10.18154/RWTH-2021-08443
- RWTH PUBLICATIONS: RWTH-2021-08443