The most interesting part of the course was the first day, when we heard about the theoretical basis and pedagogical strategies, because those things are suitable for every level of mathematics teaching and learning. The example from Piaget, the one where children were "asked" to put sticks in length-order was in my opinion very strongly of principle. It tells that if a person have readiness to learn something it can happen very easily and if one is on the lower thinking level somebody or something must first help him to reach the suitable level. In teaching the helping can often happen by using more concrete "materials".

For example, when teaching abstract algebra only few students are on the level where it is possible to learn just by reading or listening lectures. Most students need "concretisation" and those exercises which we did with computer is a good example of how the helping can be done. But its the same thing for example at primary school, when learning for example multiplication. Some pupils have in their minds already the idea of multiplying and for them its is enough to tell that for example 4*7 = 28 and do some exercises, but for the others, teacher must first use concretisation and help them to reach the readiness.

The problem is to know on what thinking-level someone is and how to do the helping. I think those genetic decomposition are good assistance when trying to recognise someone's thinking level and so it is very important to research these genetic decompositions and the ways students could be helped to reach higher readiness.

Working cooperatively is a good and nice way to learn but with people who are not used to it, it can be stiff in the beginning - like it was during our course. I think for those "class" situations students must be more active and more ready to tell their ideas.

Maybe there could have been such kind of exercise, where the groups could have tried to analyze those interviews and "find" on what level each of the interviewed student was.

Thank you for the course!