Insight into Cognitive Mathematics
Mathematics is about ideas. One of the most basic is the use of natural numbers. What enables us to understand them? Born with the cognitive ability to subitize, i.e., to distinguish totals of multiples of one, two or three directly from one another - but not, for example, four out of six - we were born with a fundamental sense of exact quantitative understanding. As infants, without having words or symbols as cognitive tools at our disposal, we can easily distinguish between the situation of a couple "this and that" compared to that of a trio "this, that and that". In order to be able to differentiate between the most diverse multiplicity and to deal with them operatively, a coding is required that fades out irrelevant aspects such as form or color and captures the mere numerical difference. Over the millennia, with the creation of the place-value notation, people have succeeded in getting a grip on the progressive becoming one more, which leads from one multiplicity to the next multiplicity, in writing in such a way that not only can the quantity of any number of multiplicities be referred to exactly, but above all simpler written calculation procedures, accessible not only to experts, have become possible. By means of a place value system, each quantity can be unambiguously indicated as the sum of certain multiples of powers of a basal quantity (for example ten in the decimal place value notation). However, only the most necessary is noted: a list of the multiples, which vary according to the size of the associated powers [from right (small) to left (large)]. There are considerable difficulties in trying to introduce young people to this cultural achievement in mathematics education. The constructs of "functional-logical thinking" and "predictive-logical thinking", which are differentiated in the field of cognitive mathematics, help to shed light on these problems and provide an opportunity to develop mathematical game worlds that are more targeted and easier to understand. If teachers do not succeed in introducing children to arithmetic thinking during primary school, the damage is great, since they lack a decisive cognitive tool to cope with more complex mathematical ideas such as number range expansions or the use of the symbolic language of algebra.