Abstract:

There are two different ways to teach mathematics. Their extreme cases could be characterised as follows: One is based on strictly formulated definitions and theorems, with the following strict proof. Only then follow examples, illustrations, models to explain the essence of the definitions and theorems. The second path is grounded in discoveries based on gained experience and previous understanding of mathematics. Naturally, we would prefer the second path mentioned, but we must also see that all the mathematical discoveries that have crystallised for centuries in the consciousness of mathematicians cannot be rediscovered with students after a few teaching hours of mathematics. Therefore, we will try to at least show the way, to show such an angle of view from which the goal is more visible.

Constructivism occurred in education theory as an opposite to transmissive teaching, which predominated in the school-teaching up to the middle of the 20th Century. From the second half of the 1950s, there were many attempts to make mathematics lessons more creative and investigative instead of the previously prevailing rote learning or memorising.

Von Glasersfeld (1989) characterised the constructivist teaching by the following principles; firstly, knowledge is not passively received but actively built up by the student, secondly, the function of cognition is adaptive and serves the organisation of the experiential world, focuses the attention on the importance of context in the creation of knowledge. The second principle, in particular, emphasises the importance of context as individuals create their knowledge about either mathematics or the teaching of mathematics. Creating knowledge means also creating a mental image about the relationships among particular concepts and ideas - this process is the so-called visualisation. The result of the visualisation process are mental representations, physical objects (geometric illustrations, pictures, diagrams, 3D models), and cognitive processes in which physical or mental visualisations are interpreted.

In the article, we show some examples from different parts of subject matters of mathematics which illustrate how visual approaches can improve a deeper understanding of the essential relationships in the given field.

Keywords:

Constructivist teaching of mathematics, visualisation, problem-solving.