Structuring the subject matter of arithmetic I

Milosav Marjanović, Aleksandra Mandić, Marijana Zeljić

In the case of construction of the block of numbers up to 100 (block $N_{100}$), all processes that lead from observation to the creation of abstract concepts are traced and didactically shaped. Sums with summands in the block $N_{20}$ having the value exceeding 20 are used to extend this block to the block $N_{100}$. Then addition and subtraction of two-digit numbers is treated and, for the sake of understanding, all intermediate steps are expressed in words and symbols. But when these operations are performed automatically these steps are suppressed and the expressing in words is reduced to its inner speech contraction. The block $N_{100}$ is a natural frame within which multiplication is introduced and where the multiplication table is built up. In the school practice the meaning of multiplication is established through examples of situations having the structure of a finite family of finite equipotent sets which we call multiplicative scheme. Some suitable models of multiplicative scheme (as, for example, boxes with marbles) are used to establish main properties of multiplication. Let us also add that we build multiplication table grouping its entries according to the way how the corresponding products are calculated and we find that these ways of calculation should be learnt, instead of learning this table by rote.