I. Khavkin
Institute for Advanced Science in Arad

To Mathematics through Computer and through Mathematics to Programming

The first part of this paper deals with methods of teaching modern applied information science with an accent on Microsoft technology. The case in point is about teaching technique of making graphic images by means of Paint, Power Point, Excel, etc. In so doing one is tempted to print various texts, draw pictures and even amuse children with potentialities of the computer.

The author has preferred another way, which, to her mind, is more promising. It rests on the assumption that any operation we perform on the computer is, in any way or another, connected with mathematics and thus the problem facing children should be ingenious, constructive, interesting, beneficial and sometimes even amusing. For instance the author gives little children such a joke-problem about the triangle and the square. The triangle is wicked but the square is kind. At night the wicked triangle cut the angles of the kind square and there appeared the octagon. All this should be drawn and children devise a kind of a play and elder children even make use of animation.

One more example is associated with the well-known chess legend. We draw in Power Point (and this is not an easy task for little children) the chess board and start counting and writing down seed-numbers in the cells: 1 in the first cell, 2 in the second cell, etc. The condition is to count without a calculator but when the numbers become too big some children think of the Windows calculator and keep on counting up to the bell. We would like to assure you that these children will remember this chess legend for life and understand how rapidly the number 2 to the power n grows.

Further we just mention briefly: Pythagorean theorem, making up figures and solid bodies, Tangram puzzle and a great deal of many other things are very good to develop mathematical and constructive abilities of children. Let us take for instance quite a common problem. All the given nine significant digits should be put into the circles on the triangle in such a way that the sum of them on each side makes 20 (Fig. 1).

Fig. 1

This problem can brilliantly reveal editorial work and logical thinking of children.

Various geometrical figures, e.g. pyramids, cones, cylinders, balls, prisms, etc. are learned by children not in the 12th form, but in the 3rd and 4th forms of our computer classes, and as this takes place all the editorial work is being done quite naturally imperceptibly. It is recommended to practice various textual problems in algebra to make children acquainted with tables. Here a correct and skillful arrangement of data demonstrates the usefulness of the table, its simplicity and obviousness. Let us take for instance the textbook on geometry by Beni Gorin. In the section “Triangle” you would not see any picture on its pages. And we believe it to be an excellent work in Word or Power Point when a new page of the book is made, when every definition and every theorem is illustrated. Of course it is quite possible to build up the system of coordinates and to solve graphically equalities and inequalities.

The second part of this paper deals with programming. The idea of teaching is just the same, i.e. methods of programming using various and even famous problems where a search of the solution idea, constructing algorithm and, at last, writing and adjusting the program are necessary. Functions, number theory (p , e, Pythagorean numbers, Pascal’s triangle, etc.) factorials, series, combinatorial analysis, number system, random numbers, vectors, various algorithms and formulas are examples of the problems recommended for application at school. For this aim there is a very good drawing operator –Draw having its lettered instructions (Fig. 2).


The most striking is the fact that even little children of the 2nd and 3rd forms draw with vectors and not with the mouse. By way of illustration we give some most characteristic problems.

1. Is it possible to position eight segments on the plane so that each of them intersects with other three? (Fig.3).

Fig. 3

2. Find out and show on the screen the set of the points the sum total of distances of which to the sides of the square with the side 1 equals 4.

3. Calculation experiment: which number is bigger e to the ? power or ? to the e power?

4. The sum total of figures of the book pages equals 3684. Find out the number of pages of the book.

It is this kind of problems the solution of which clearly shows the relation between mathematics and modern programming.

To sum up we would like to remark that the methods of teaching we apply yield good results, ensure good progress of children at school, good results at various mathematical Olympiads, competitions, etc.