TITLE:
A Survey of the System N of Natural Numbers Assigned to Primary Teachers
AUTHORS:
Milosav M. Marjanović
KEYWORDS:
Expressions and Relations in the System N, Distinction between Syntactic and Semantic Concepts, Derivation of the Rules in N and the Peacock’s Principle of Invariance of Form, Formulae Expressing the Number of Dots (Circlets) of Regular Patterns
JOURNAL NAME:
Open Access Library Journal,
Vol.4 No.6,
June
28,
2017
ABSTRACT: From our standpoint, a school teacher should be acquainted, in a deeper way, with the content of teaching themes which are on school curricula. In case of primary teachers and their preparation for teaching mathematics, they should have a solid knowledge of the properties of the system N of natural numbers and an understanding of its position as being a basis upon which all other number systems are built. Up to some degree, these teachers should also be acquainted with further extensions of number systems going along the line as it is done in school: natural—positive rational—integer—rational—real numbers. These extensions are enlightened by Peacock’s principle of invariance of the form—a rule derived for natural numbers, when expressed in general form (as a literal relation) continues to hold true in all extended systems. In Section 2 of this survey, a precise terminology is fixed which is needed for the study of the system N and in particular, for making a difference between syntactic and semantic concepts. The Cantor principle which expresses the dependence of conception of number on perception of set is also formulated and largely exploited in this paper. In Section 3, several rules are derived when different expressions denoting two different groupings of elements of a set are equated. Forgetting that the variables are bound to N, all these rules also express the properties of the extended number systems, as well as they are algebraic laws or their derivatives. At the end, discovering of rules of correspondence of sequences given by a number of their initial terms is considered as a type of exercises which help the development of the idea of variable. Some cases of finding formulae for sums of consecutive natural numbers are also included. This paper is intended to be a paradigmatic example how a mathematical content has to be elaborated to serve best the school teachers to deepen their knowledge of subject matter.