Axiomatisches Denken und Arbeiten im Mathematikunterricht

• Axiomatic Thinking and Working in Teaching Mathematics

Hock, Tobias; Heitzer, Johanna Maria (Thesis advisor); Schwank, Inge (Thesis advisor); Jahnke, Hans Niels (Thesis advisor)

Aachen (2018)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen University, 2018

Abstract

Theorems that are proven within the framework of mathematical theories enjoy an especially high degree of security and stability. This reputation is intrinsically tied to the axiomatic method: Starting with propositions whose truth is indubitable, it is possible to logically deduce further results that exclusively rely on already accepted theorems. This leads to a comprehensive theoretical structure. Since in the 3rd century BC Euclid’s Elements provided an axiomatic-deductive description of geometry for the first time, this form of presentation has been paradigmatic of exact sciences. Over the course of the centuries, the picture of mathematics and with it the view on the role of axiomatics have fundamentally changed: At the beginning of the 20th century an abstract-formalistic position prevailed, according to which mathematical terms are seen as variables for entities that are not further determined with regard to their content. Against this background, axioms describe the relations that exist between these terms, thus defining them implicitly. As a consequence, the classical claim to truth of mathematical statements, as it manifested itself in the supposed self-evidence of the axioms, was dispensed with in favour of the formal concept of consistency. This paradigm shift paved the way for a type of mathematics in which topics that differ in content but are structurally similar can be subsumed under and organized in abstract theories. Despite its fundamental importance for the self-conception of mathematics as a discipline of proof the axiomatic method plays hardly any role in contemporary mathematics education. Reform programmes comprising among other things a greater consideration of axiomatic aspects in curricula are considered to have failed. Reasoning skills, which are included in all educational standards, may comprise different levels of rigour; however, axiomatic considerations are consequently excluded in them. The main goals of this thesis are to give the reader a comprehensive overview of the significance of axiomatics in mathematics and to provide teachers with a didactic basis for a well-reflected decision as to whether and to what extent they wish to treat axiomatic topics with their pupils. Eventually, concrete ways of integrating axiomatic thinking and working methods into learning environments are illustrated. To achieve these goals,• the main characteristics of the axiomatic method are depicted from a historical and philosophical perspective;• the arguments for and against dealing with axiomatics in mathematics education are analysed and evaluated from a contemporary didactic point of view;• overarching learning objectives and didactic guidelines for a comprehensive treatment of axiomatics at Sekundarstufe II are set up;• successful teaching concepts on selected axiomatic aspects are presented;• a teaching concept of my own is provided that includes extensive learning materials for treating axiomatic topics in supplementary courses at Sekundarstufe II.