Jedan pristup izvodjenju nastave aritmetike, III

Milosav M. MarjanoviÄ‡

This is the third part of our paper An Approach to Learning and Teaching Arithmetic, where the building of the number block up to 20 is sketched and discussed. Using consistently some early algebra techniques, the numbers 11, 12, ... , 20 are introduced to be the sums 10 + 1, 10 + 2, ... , 10 + 10 (whose meaning has already been established in the number block up to 10). Then, the first cases of calculation of the type: 13 + 5 = 10 + (3 + 5), 18 - 5 = 10 + (8 - 5) are treated and followed by the methods of addition and subtraction when the ten line is crossed. When learned in the classroom, these procedures are relied on iconic representations in the form of number images and they are applied to compose addition and subtraction tables. But a thorough didactical analysis shows that they are also relied on the following algebraic rules: l + (m + n) = (l + m) + n, (l + m) - n = l + (m - n), l - (m + n) = (l - m) - n. Of course, these rules can be derived from the axioms of Abelian group, but in didactics of mathematics they should be considered as independent and should be intuitively established by equating different denotations for the same number. In particular, the role of associative law is seen as a means of reducing sums of three (or more) numbers to double sums. Thus, all properties of addition and all entries in addition tables are expressible in terms of double sums.