Multiplying after the turn

In education multiplying is usually viewed as repeated joining together and dividing as repeated taking away or, which comes to the same thing, as an equal distribution. This presentation springs from Antiquity, when thought was mostly concrete. In modern mathematics we have relation-numbers instead of, image-numbers and likewise multiplying is a facet of relational thinking. The view that children merely can learn through the concrete is often biassedly understood in the sense that the concrete has to be abstracted, which characterizes substantial thinking. However, in the case of relational thinking, learning through the concrete means that to achieve insight the mathematical activities have to be applied to reality, a crucial point, for most people have difficulties with applying multiplication, much more than with, the inherent algorithms. It appears that they do not really know what multiplication is, particularly not its space structure. The more general the structure the more and wider the applications. This thesis infers that multiplying as multiple of classes is much less useful than multiplying as space form. Questing for the essence of multiplication is the major topic of this paper. Which changes has its structure undergone and how can education deal with them? At the end it is illustratively explained why probability, based on the established multiplication, is usually such a tough domain.