Volume IX , issue 2 ( 2006 ) | back |

Tverberg's theorem and shadows of simplices and convex bodies | 1$-$17 |

**Abstract**

Helge Tverberg published more than forty years ago his original proof of the theorem which has been widely acclaimed and today bears his name. This beautiful result has been one of the most celebrated results of discrete geometry and, together with its relatives, still remains a central and one of the most intriguing results of geometric combinatorics. Here we give a reasonably non-technical presentation of this result having in mind a larger mathematical audience, particularly school teachers and their talented students, hoping that it may raise their interest for this very attractive area of mathematics. In the remaining part of the article we briefly visit some of other branches of convex geometry and outline how ``smashing'' and `slicing'' of convex bodies offers a deep insight into their structure and behavior.

**Keywords:** Simplex; join of sets; colorful Carathéodory theorem;
Radon's theorem; Tverberg's theorem.

Critical look at dynamic sketches when learning mathematics | 19$-$29 |

**Abstract**

Among neglected topics in the development of mathematics education is the frequently mentioned use of modern technology, especially hypermedia-based learning environments. This paper examines the use of computer technology for mathematics teaching on tertiary level, especially for distance learning and assessment. It focuses on dynamic sketches, which were used to offer students opportunity to recognize and represent elements of functions and binary operations within a distance education management. It seems that moving from old study culture towards a modern one brings many kinds of cognitive, emotional and social problems. Our experiences do not support the view that using interactive JAVA applets, for example, would bring special advantages without an appropriate pedagogical framework connected to reflective tutoring.

**Keywords:** Assessment, JAVA applet, conceptual
knowledge, distance education, hypermedia, interactive,
mathematics, procedural knowledge, tertiary education.

Dualitätsprinzip und musikalische Interpretation der komplexen Zahlen | 31$-$40 |

**Abstract**

Although expressed in the same way in different sciences and arts and in the case of nature, the principle of duality resumes various phenomenological forms. That fact is demonstrated in this paper by a comparative consideration of complex numbers and their conjugate values in mathematics and of big and small terza and the relation ``major-minor'' in music, including also the teachings about Jin and Jang which represent the formulation of the ancient Chinese ``Book of Change -- Jih Djing''. Using Weber-Fechner's Law, a musical interpretation of complex numbers and operations with them is developed, particularly observing musical chords. On the other hand, chords of classic tonality (dyatonal quadritones) correspond to the eight trigrams in the Book of Change. Finally, without entering some deeper area of mathematics, elements of combinatorics and interpretations by means of regular polygons can be applied to the whole Book of Change, what closes this sort of considerations.

**Keywords:** Duality principle; Chinese Book of Change; musical interpretation of complex numbers.

Developing trustworthy TIMSS background measures: a case study on mathematics attitude | 41$-$51 |

**Abstract**

This study, which used a sample of 197,707 students from 46 countries that participated in the TIMSS 2003 project in eight grade, examined whether, for a large number of the TIMSS countries, trustworthy TIMSS measures of several dimensions of mathematics attitude can be developed. By focusing on self-confidence in learning mathematics, usefulness of mathematics, and liking mathematics, it was found that both factor validity and reliability of the measures of these three dimensions derived from the raw data was only attained for the students from the United States. However, when scores concerning the utilized attitudinal statements of all subjects were transformed into Guttman's image form scores, the factor validity and reliability of the three measures utilizing such transformed data was attained for thirty-three countries ($N=137{,}346$). It was found that for all these thirty-three countries mathematics attitude was mostly saturated by either usefulness of mathematics or self-confidence in learning mathematics. A higher mathematics achievement was found for countries where mathematics attitude was mostly saturated by self-confidence in learning mathematics.

**Keywords:** TIMSS, mathematics attitude, factor
validity, reliability, Guttman's image theory.