Volume V , issue 2 ( 2002 ) | back |

Extremal problems---past and present | 59$-$69 |

**Abstract**

The evolution of a phragment of the theory of extremal problems, the necessary conditions of extremum, is explained. Four problems of Fermat, Lagrange, Euler and Pontryagin are presented and four classical examples of Euclid, Kepler, Newton and Bernoulli are solved.

**Keywords:** Extremal problems, problems with and without
constraints, Lagrange multipliers, calculus of variation, optimal control,
Pontryagin's maximum principle.

**MSC Subject Classification:** 00A35

**MathEduc Subject Classification:** 01Axx

Mathematical education and society (an outlook from Russia and into Russia) | 71$-$80 |

**Abstract**

The author gives surprising examples of mathematical ignorance of modern society. One can say that people nowadays not only are ignorant of or do not understand elementary mathematical concepts but that they are afraid of mathematics. Mathematicians are not last who should be blamed for this. Among other problems that impede the development of mathematics education, the author singles out the following three: Americanisation, dollarisation and computerization.

**MSC Subject Classification:** 00A35

Suwnost\cprime\ i nekotorye tipizacii matematiqeskih zadaq | 81$-$89 |

**Abstract**

The purpose of the paper is to present the authors' idea concerning the mathematical problems' nature and the nature of the training mathematical problems in particular, to consider their structure and classification according to their type, based on their components and to point out the role of the types of training problems in the process of teaching mathematics. All types of problems considered are illustrated by appropriate examples.

**Keywords:** Classification according to the type of
training mathematical problems, components of mathematical problems.

**MSC Subject Classification:** 00A35

Are quantitative and qualitative reasoning related? (A ninth-grade pilot study on multiple proportion) | 91$-$98 |

**Abstract**

The major objective of this study was to determine whether quantitative and qualitative reasoning are related, and, if so, which kind of instruction promotes their relation. The study had a pre-test/post-test design with two parallel groups. Both groups solved quantitative and qualitative problems, but while the QN group was only taught how to solve quantitative problems, the QL group was exclusively taught how to solve qualitative problems. The study used a sample of 68 ninth-grade Gymnasium (high-school) students of average mathematical abilities. The students solved multiple proportion problems. The study showed that: (a)~the examined problems were hard for most students; (b)~even in the QN group, quantitative reasoning was not improved; (c)~qualitative reasoning was improved in both treatment groups and the QL group scored better; and (d)~only the QL treatment related quantitative and qualitative reasoning.

**MSC Subject Classification:** 00A35

Didactical analysis of primary geometric concepts | 99$-$110 |

**Abstract**

Inspired by Freudenthal's didactical phenomenology, in this and a series of subsequent papers, didactical analysis of geometric concepts in primary school programmes is undertaken. In this paper, spontaneous concept of shape is reflected historically, covering shortly handworks and decorations of human beings from the far archaeological past. Some samples of inherent geometrical ideas from ethnomathematics are also included. At the end, the birth of the first Greacian schools is touched and the beginnings of science are reflected.

**Keywords:** Handworks and ceramics, ground designs,
incommensurable magnitudes.

**MSC Subject Classification:** 00A35