Volume XXII , issue 2 ( 2019 ) | back |

Number systems characterized by their operative properties | 43$-$48 |

**Abstract**

In our paper Structuring Systems of Natural, Positive Rational and Rational Numbers [The Teaching of Mathematics 22, 1 (2019)], we have studied operative properties of number systems (i.e., the properties of operations and the order relation). In the same paper we have selected a number of operative properties of the system $N$ of natural numbers with $0$ which we called the basic operative properties of $N$. \par Let $\{S,+,\cdot,<\}$ be a structure, where $S$ is a non-empty set, ``$+$'', ``$\cdot$'' are two binary operations and ``$<$'' is the order relation. We called provisionally such a structure $ N$-structure, when its axioms are basic operative properties of $N$ taken abstractly and we proved that the system $N$ of natural numbers with 0 is the smallest $N$-structure}. \par Here we rename the $N$-structure and call it the ordered semifield. Adding to the axioms of the ordered semifield the axiom: $(\forall a)(\exists b)\, a+b=0$, then such a structure we call the ordered semifield with additive inverse and adding to the same axioms, the axiom: $(\forall a\ne0)(\exists b)\, a\cdot b=1$, we call such a structure the ordered semifield with multiplicative inverse. When both of these axioms are added to the axioms of the ordered semifield, then such a system of axioms coincides with the axioms of the ordered field. \par In this note we prove that the system of integers is the smallest ordered semifield with additive inverse and that the system of positive rational numbers with 0 is the smallest ordered semifield with multiplicative inverse. \par The fact that the system of rational numbers is the smallest ordered field is well known. At the end of this note we also include a proof of this fact.

**Keywords:** Number system; ordered semifield; characterization of number systems as the smallest ordered semifields.

**MSC Subject Classification:** 97F40

**MathEduc Subject Classification:** F43

Participation in online cooperative professional development: Factors to consider and activities to practice | 49$-$57 |

**Abstract**

This study deals with relevant components of a technology innovation framework suitable for the examination of online collaborative professional development. By using a sample of 55 lower-secondary mathematics teachers and 155 primary school teachers, this study examined the relationships among teachers' intention to participate in online collaborative professional development (OCPD), their perspective taking, and their computer self-concept. It was found that while this intention was positively related to computer self-concept, perspective taking could positively relate to this intention only indirectly through computer self-concept. It was also found that, among three OCPD activities used to describe the intention, the activity of cooperatively analyzing videos of lessons given was preferred least by all these teachers. Implications for research are also included.

**Keywords:** Computer self-concept; online collaborative professional development; perspective taking;
technology innovation framework.

**MSC Subject Classification:** 97C60, 97D40

**MathEduc Subject Classification:** C69, D49

A short proof of Hölder's inequality using Cauchy-Schwarz inequality | 58$-$60 |

**Abstract**

The aim of this note is a to give a new and short proof that the Hölder inequality is implied by the Cauchy-Schwarz inequality.

**Keywords:** Young's inequality; Cauchy-Schwarz inequality; Hölder's inequality.

**MSC Subject Classification:** 97H30

**MathEduc Subject Classification:** H35

A general method of proving some classical inequalities using AM-GM inequality | 61$-$70 |

**Abstract**

In this paper, a general method is presented of proving some inequalities of Cauchy-Hölder-Carlson type using AM-GM inequality.

**Keywords:** Cauchy inequality; Hölder inequality; Carlson inequality.

**MSC Subject Classification:** 97H30

**MathEduc Subject Classification:** H35

Generalized associative and commutative laws | 71$-$76 |

**Abstract**

School algebra is not an abstract mathematical discipline but its variables denote the numbers from a number system which is under elaboration. Basic properties of a number system are in the same time basic rules of algebra. Which other rules have to be deduced is a matter of concern to those who research problems of teaching algebra in school. \par The commutative and associative laws are often seen formulated in math school books. But they have a full effect when they are used to define numerical value of sums and products of three and more members and when it is proved that this value is independent of the way how summands and factors are associated and ordered. To accept these proofs with understanding, students have to be prepared for deductive reasoning and to be acquainted with all different ways how elements of a set can be ordered and with the method of mathematical induction as well. Hence, it is a student of a serious secondary school of age 14 or more.

**Keywords:** Associative law; Commutative law; Questions of their generalization.

**MSC Subject Classification:** 97F40

**MathEduc Subject Classification:** F43

New impulses for research in mathematics education | 77$-$85 |

**Abstract**

We report about results of the scientific conference ``Research in Mathematics Education'' held in Belgrade under organization of the Mathematical Society of Serbia, May 10--11, 2019. A survey of the presented investigations is followed by identification of common issues. Reports are classified according to the topics of research. Our discussion is accomplished with personal impressions about potentials for selected contributions to be implemented in schools and to serve as impulse for further research in the field.

**Keywords:** Problem solving; problem posing; assessment; representations; reformed curriculum; research methods.

**MSC Subject Classification:** 97D99

**MathEduc Subject Classification:** D90

Corrigenda: Bimatrix games have a quasi-strict equilibrium: an alternative proof through a heuristic approach | 86$-$90 |

**Abstract**

Our paper published in this journal [Bimatrix games have a quasi-strict equilibrium: an alternative proof through a heuristic approach, The Teaching of Mathematics, 21, 2 (2018), 97--108] contains at least two errors, and the proof is too sloppy in dealing with a really subtle problem. The purpose of this note is to correct those errors and to give more explanation, and accordingly to present slightly different economic interpretations of our method as well as to show calculated results, suitable to our corrected proof, in the numerical example.

**Keywords:** Bimatrix game; Kakutani fixed point theorem; multivalued map; quasi-strict equilibrium.

**MSC Subject Classification:** 97M40, 91A05, 91B62

**MathEduc Subject Classification:** M45