| On polygons and polyhedra | 1$-$14 |
Abstract
This paper is written for those Gymnasium students with an intensive interest in mathematics who are often in search for some extra reading matters not being on school curriculum. We have chosen to elaborate here a number of interesting and beautiful geometric topics: the diagonals of polygons and the triangulations, the art gallery theorems, the Euler formula, regular polyhedra, that could satisfy their curiosity and instigate their further reading.
Keywords: Diagonals of polygons, triangulations, art gallery theorems, Euler formula, regular polyhedra.
| Interchanging two limits | 15$-$29 |
Abstract
By the use of convenient metrics, the ordered set of natural numbers plus an ideal element and the partially ordered set of all partitions of an interval plus an ideal element are converted into metric spaces. Thus, the three different types of limit, arising in classical analysis, are reduced to the same model of the limit of a function at a point. Then, the theorem on interchange of iterated limits, valid under the condition that one of the iterated limits exists and the other one exists uniformly, is used to derive a long sequence of statements of that type that are commonly present in the courses of classical analysis. All apparently varied conditions accompanying such statements are, then, unmasked and reduced to one and the same: one iterated limit exists and the other one exists uniformly.
Keywords: Interchange of two limits, uniform convergence, definite integral as a limit.
| Inequalities of Karamata, Schur and Muirhead, and some applications | 31$-$45 |
Abstract
Three classical general inequalities---those of Karamata, Schur and Muirhead---are proved in this article. They can be used in proving other inequalities, particularly those appearing as problems in mathematical competitions, including International Mathematical Olympiads. Some problems of this kind are given as examples. Several related inequalities---those of Petrović, Steffensen and Szegö---are treated, as well.
Keywords: Relation of majorization, divided difference, Karamata's inequality, Petrović's inequality, Steffensen's inequality, Schur's inequality, Muirhead's inequality.
| Educational technology standards in professional development of mathematics teachers: an international study? | 47$-$52 |
Abstract
Designing and implementing technology-based professional development of mathematics teachers is the key to fundamental, wide-ranging educational reforms. This development should be based on some suitable educational technology standards. In order to understand the extent to which the integration of technology in day-to-day teaching/learning has taken place in terms of such standards, we need to search for critical variables influencing their attainment. By adopting the ISTE Technology Foundation Standards for Students, this study used a sample of 134 mathematics teachers from Finland, Serbia and Slovakia---three countries at considerably different levels of technological development---to examine the subjects' interest to achieve these standards in relation to their computer attitudes and the received professional support concerning the standards. For these students, who studied at institutions that did not offer any explicit instruction on the utilized or other ET standards, three important findings were obtained. First, the interest was higher than the support: while on average the interest was medium, the support was rather small. Second, both the interest and the support for the Finnish subjects were lower than that for the Serbian and Slovak subjects. Third, the interest was primarily influenced by computer attitude. Implications for professional development of mathematics teachers and further research are included.
Keywords: Educational technology; Technology Foundation Standards.