Volume II , issue 2 ( 1999 ) | back |

Selected chapters from algebra, III/1 | 65$-$80 |

**Abstract**

This paper is the third part of the publication ``Selected chapters of algebra'', the first two being published in the previous issues of the Teaching of Mathematics, Vol\. I (1998), 1--22, and Vol\. II, 1 (1999), 1--30.

**Keywords:** Set, subset, one-to-one correspondence, combinatorics.

**MSC Subject Classification:** 00A35

A broader way through themas of elementary school mathematics, II | 81$-$103 |

**Abstract**

In this paper we start to analyze the first topics of the school arithmetic. To help the reader recognize the fundamental position and role of the system of natural numbers, a historical glance at the evolution of the number idea is given. Then, the block of numbers 1--10 is seen as a conceptual structure which dictates a series od didactical steps and procedures. As it is easy to observe, many textbooks contain rashly gathered groups of arithmetic problems without bringing before the mind of cognizing subject the effects of these steps. Since at this stage, numbers and operations as well as all their properties are perceptual entities and experiences, the role of drawings, from those showing a piece of reality to the schematic ones which condense the carrying meaning with an elegant simplicity, is particularly emphasized. A bad practice of ``proving'' the arithmetic rules by means of calculation of values of a few related numerical expressions is also criticized. Though, we have given the chief points only, we expect that the sketch of this block will reflect some good practice in a pragmatic way.

**Keywords:** Didactical blocks of numbers, addition and
subtraction schemes, rules of interchange and association of summands.

**MSC Subject Classification:** 00A35

Die Entdeckungsgeschichte und die Ausnahmestellung einer besondered Zahl: $e=2,71828182845904523536\dots$ | 105$-$118 |

**Abstract**

Mathematics pupils first learn about the irrational numbers $\pi$ and $e$ at the secondary school level. While the appearance of $\pi$ in simple geometrical formulas makes it easy for pupils to grasp its special importance, the significance of $e$ is less clear. The nature of $e$ can best be understood from a historical perspective, In the sixteenth century, work on various mathematical problems led, along different paths, to the discovery of~$e$. In this article, I will outline these paths, and propose that describing them to pupils is the best way to help them understand the uniqueness of~$e$.

**Keywords:** Irrational numbers, $e$, Napier's logarithms.

**MSC Subject Classification:** 00A35

<\`Erlangenskaya programma> Feliksa Kle\uı na i ee vliyanie na reformirovanie matematiqeskogo obrazovaniya | 119$-$129 |

**Abstract**

The article is devoted to the 125th anniversary of the Erlangen program of F. Klein. Its outstanding importance for development of mathematical science and for formation of the modern mathematical concepts is pointed out. The influence of the Erlangen program on creation of natural-scientific representations, first of all physical, is analyzed. The special attention is given to the role the Erlangen program has played in the history of mathematical education in schools and universities.

**MSC Subject Classification:** 00A35