Investigation can play a vital part in the learning of mathematical concepts and in problem-solving. At all stages the teacher has an essential part to play. He sets the scene, providing real materials or a challenging problem when necessary. He observes what his pupils do with these and asks questions which will help their learning.

There are different stages in learning by investigation. All are important for sound and lasting learning.

(1) Free exploration. A problem arises, or the teacher asks a question. The children use materials which they find for themselves or which are provided by their teacher. When materials are not necessary pupils explore the problem for themselves. Very often they will discover relations which the teacher has not thought of. These should not be rejected but postponed if the teacher wants to pursue a particular concept.

(2) Directed discovery. Here the teacher wants the pupils to investigate a particular idea which has developed during the first stage. This is probably the idea which the teacher originally planned to develop. His questions are now focused on the concept he has in mind. Since pupils vary in background, in ability and in attitude, it is useful to group them so that questions can be appropriate to the needs of each group.