Volume XXII , issue 1 ( 2019 )back
Structuring systems of natural, positive rational, and rational numbers1$-$16
Milosav M. Marjanović and Zoran Kadelburg
Serbian Academy of Sciences and Arts, Kneza Mihaila 35, 11000 Beograd, Serbia


In this paper, we are concerned with the extensions of the system $N$ of natural numbers with $0$ to the system $Q_+$ of positive rational numbers with~$0$ and then, the extension of this latter system to the system $Q$ of rational numbers. Structuring these systems, we start with the system of natural numbers forming the list of its basic operative properties (i.e\. the properties of operations and the order relation) and using only these postulated properties, we deduce from them a series of properties of proportions and equalities that relate differences. While deducing, we suppose that all involved ratios and differences are defined in~$N$, i.e. they have the value which is a natural number.\par Keeping in mind these properties, we define equivalence of arbitrary ratios (without supposing their values in~$N$) and then, we extend the meaning od addition, multiplication and order relation to the equivalence classes of these ratios. Thus, the system of positive rational numbers with $0$ is constructed and the list of its basic properties is accomplished.\par Comparing this new list and that with properties of~$N$, two lists are identical except that the variables are denoted by different letters and the new list contains an item more: the existence of multiplicative inverse. Omitting this item, two systems become formally identical. Therefore, they have the same postulated properties, as well as those deduced from them. This is a precise formulation of the Peacock's principle of permanence, telling which properties are transferable and it is also the way of its logical justification.\par Similarly, $Q_+$ is extended by equivalence classes of formal differences in $Q_+$ and addition, multiplication and the order relation are defined in this set of equivalence classes. Thus, the system $Q$ of rational numbers is constructed and the list of its basic properties accomplished. Taking the operative properties of the systems $N$, $Q_+$ and $Q$ as axioms, the $N$-structure, $Q_+$-structure and $Q$-structure (standardly called the ordered field) are defined, respectively. Since the axioms of $N$-structure are least restrictive, it follows that the systems of positive rational and rational numbers and of real numbers are examples of $N$-structure. Thereby, all operative properties of $N$ and those deduced from them are also valid in these number systems, when they are transcribed writing corresponding variables. This is the necessary precision and logical justification for the Peacock's principle of permanence.\par At the end, we prove that each system satisfying axioms of $N$-structure contains an isomorphic copy of the system $N$ with $0$, what characterizes the system of natural numbers with $0$ as the smallest system satisfying the axioms of $N$-structure.

Creative Commons License

Keywords: Natural numbers; positive rational numbers with zero; rational numbers; Peacock's principle of permanence; characterizing properties of the system of natural numbers with zero.

MSC Subject Classification: 97F40

MathEduc Subject Classification: F43

Formalising the use of the centre of mass method in mathematical problems17$-$32
Meirav Amram, Miriam Dagan, Sagi Levi and Artour Mouftakhov
Shamoon College of Engineering, 84 Jabotinski St., Ashdod 77245, Israel and Shamoon College of Engineering, Beer-Sheva 84100, Israel.


In this paper, we adopt from physics the concept of centre of mass and use it in the solution of classical geometric problems. Such uses include finding both the incentre and orthocentre of a triangle. We define the concept of centre of mass mathematically and show how to formalise the intuitive solution from physics into a formal mathematical solution. This concept is easy to teach because it requires only basic knowledge of vectors; it can help in solving many complicated geometrical problems.

Creative Commons License

Keywords: Vectors; mass point; centre of mass.

MSC Subject Classification: 97G70, 97D40, 97M50

MathEduc Subject Classification: G74, D44, M54

Influence of TIMSS research on the mathematics curriculum in Serbia: educational standards in primary education33$-$41
Djordje M. Kadijevich
Institute for Educational Research, Dobrinjska 11/III, 11000 Belgrade, P.O.B. 546, Serbia


Since 1995, achievements in mathematics and science have been assessed worldwide every four years by TIMSS (Trends in International Mathematics and Science Study), whose outcomes have influenced the development and (re)design of mathematics and science education curricula in a number of countries. This paper examines how TIMSS has influenced changes in the mathematics curriculum in Serbia in primary education. The paper first briefly presents TIMSS results of Serbian students that call for curricular improvements. It then gives a summary of these changes regarding educational standards for the end of primary education. The paper ends with a critical examination of these standards and suggestions for their enhancement.

Creative Commons License

Keywords: Educational standards; primary education; TIMSS.

MSC Subject Classification: 97B70

MathEduc Subject Classification: B72