The role of mathematical didactics, insofar as it is committed to the demands of science, includes comprehensibly describing the teaching and learning of mathematics and lesson planning. This role can be fulfilled on many different levels; in any case, the complexity of mathematical-didactical considerations is not connected with the complexity of the mathematics concerned. Mathematical-didactical issues related to primary education can be just as ambitious as those arising from a university environment. These also include issues relating to the education of students for teacher training programmes. In order to be able to provide what is required, a theoretic foundation is needed to help provide orientation as regards the view of potentially present data, which may involve the activities of students.
As part of these considerations, the critical examination and adapted application of theories from related disciplines in mathematical didactics is required. One example is given below: these approaches may be selected from cognitive sciences and educational psychology. We recall the observations of Jean Piaget and the corresponding adaptations with Hans Aebli and Erich Wittmann. In any case, you benefit from the fact that an accessible and generally carefully-formulated theoretical reference structure is provided. It is essential to carefully adapt these theoretical approaches to the requirements of mathematical didactics.
Mathematics as a symbol activity
Mathematics has less to do with everyday things and far more to do with symbols, and particular uses of these symbols. In the words of David Hilbert: “In the beginning (…) is the symbol”[R.1] . Anyone learning mathematics is forced to think about how the symbols of mathematics are to be used, and is confronted at an early stage with the fact that mathematics often contains a variety of symbols for situations that are seemingly identical.
[R.1]Please note I’ve been unable to find this exact quote
The problem of the relationship between the symbol and what is designated is exacerbated by the introduction of computers to the teaching of mathematics. The variety of inscriptions now possible and their rapid exchange, along with the opportunity to ‘experiment’ with mathematical configurations is making insights into the correlation of uses of different symbols and their relation to mathematical issues increasingly problematic.
In terms of findings and learning theory, it is important that the possibility of a finding is always relative to a perspective, meaning that it is itself mediated by symbols. To ‘understand’ something means to be able to present and represent it, both internally and externally. Seen this way, the entire process of our thinking is carried out through symbols. Symbols are therefore not only a subject of mathematics, but they are also a means of knowledge and learning.
Specifically in the classroom, the problem of representation arises as to which role materials play as a means of visualisation, communication, and representation of knowledge.
Colleagues in the study group
Prof. Willi Dörfler
a.o. Prof. (ret.) Hermann Kautschitsch
From teaching practice:
Dr Martin Brunner
Dr Felix Poklukar