Volume XI , issue 1 ( 2008 ) | back |

The Mathematical Society of Serbia---60 years | 1$-$19 |

**Abstract**

The Mathematical Society of Serbia was founded in January 1948. It had wide activities during previous 60 years in many fields---scientific, educational and in popularization of mathematics. It publishes five journals and a lot of other publications, mostly intended for young mathematicians and programmers. It is also the organizer of all competitions in mathematics and informatics in Serbia and it sends Serbian teams to international competitions. Finally, the Society represents Serbian mathematicians in international associations. This article is an attempt to give a brief description of these activities.

**Keywords:** Mathematical society, Mathematical congress, Mathematical olympiad, Olympiad in informatics.

Convexity of the inverse function | 21$-$24 |

**Abstract**

This note answers the following question: Having an invertible convex real valued function $f\:A\rightarrow R$, what can be said about convexity of $f^{-1}$?

**Keywords:** Convex function; inverse function; continuity; first derivative; second derivative.

Geometrical definition of $\pi$ and its approximations by nested radicals | 25$-$34 |

**Abstract**

In this paper the length of the arc of a circle and the area of a circular sector are defined in an elementary way. From this we derive the geometrical definition of the number $\pi$, in two equivalent ways. Formulas for the area and the perimeter of a circle are proved. Also, the number $\pi$ is represented as a limit of several sequences involving nested radicals. From this some approximations of $\pi$ are obtained. One of them is in terms of Golden ratio.

**Keywords:** Circle; circular sector; area; length.

Why is it not true that $0.999\ldots<1$? | 35$-$40 |

**Abstract**

The contribution describes three basic obstacles preventing students from understanding the concept of infinite series in teaching of mathematics and provides means to their removal.

**Keywords:** Infinite series; obstacle; limit of a sequence; potential and actual perception of the infinite limit
process; the relation of phylogenesis and ontogenesis.

Is mathematics teaching developing learner's key competences? | 41$-$52 |

**Abstract**

The goal of this paper is to try to answer following questions: To what extent does mathematics teaching develop key competences of students? Are trends of student's key competences development implemented in current education so that our school graduates stand the European competition? We searched for the answers in a survey and in this work we want to present the results of the study. In the introduction theoretical resources are described, the term 'key competences' is defined and system of mathematical key competences is outlined. In the next part of this paper we follow the survey offering answers to asked questions. Goal of the survey is presented, hypotheses are set and applied methods are characterised. We deal with survey realisation and quantitative and qualitative evaluation and we introduce possible improvements.

**Keywords:** Key competences; categories of mathematical key competences; pedagogical survey.