Didactics of mathematics

Due to the growing influence of science on modern society, changing our view on the world, it is necessary to improve working on and with the mathematical, scientifical and technical branches of knowledge. Central points for the science of didactics of mathematics or mathematics education are the different ways of development, depiction and explanation in mathematical and scientific knowledge. In this context, the term 'development' has to be understood as the communication of knowledge in the radius of school, university and furthermore the public.

But additionally, the term 'development' does represent the process of the constitution of theories inside mathematics and science, and even broader in our whole cultural history. Thus the development of mathematical knowledge in the light of the examination of the philosophy of science is an important branch of the didactics of mathematics.

In general one can diversify two different approaches to working on the development of mathematical knowledge. One is to observe today's learners, and empirically come to conclusions about the development of mathematical knowledge from the child to the grown up. But there is also the way of examining how mathematicians throughout history coped with mathematical problems, and to transfer these results on today's learners of mathematics. Phylogeny (development of people) and ontogeny (development of individuals) have often enough shown substantial parallels allowing to draw conclusions from one to the others:
It can be proven that students of higher levels have a more empirical understanding of mathematics, coming from concrete vision of mathematical (e.g. geometrical) objects. This genuine empirical approach makes it difficult for the students to make the step to abstract mathematical terms as functions or limits which are central for modern calculus.
If we then look i.e. at the development of the calculus from Leibniz to Euler, we see that in history of mathematics the same problem needed a great deal of energy to overcome. Now, the science of the didactics of mathematics offers tools and a theoretical background to analyse the individual stages of mathematical thinking in the focus of historical development of mathematical knowledge in general.

Mathematics needs an embedding into philosophy and history to be understood as a creative and lively branch of science, and not just as a dead manual-science. New approaches in the German educational system have clearly shown that the teaching of mathematics at universities in a philosophical and historical context have improved the motivation, understanding and attitude to the subject dramatically in a positive way. Hilbert’s revolution at the beginning of the last century, turning away from truth in mathematics to consistency, should not lead to an abandonment of what has been before altogether, especially if communicated to students. Here, didactics of mathematics sets theoretical foundations for the development of mathematical knowledge in general and pleads for an approach to the process of learning mathematics, integrating the philosophical and historical aspects.

Ingo Witzke