When the going gets tough, the tough gets going problem solving in Hungary, 1970–2007: research and theory, practice and politics

Abstract

In the 1970s significant research was conducted concerning the development of methods for teaching mathematics. The most outstanding of these projects, led by the late Tamás Varga, and which had a major influence on teaching mathematics in Hungary, was called OPI. This project comprised research based on experiments aiming at the complete renewal of methods and content in mathematics teaching. In 1978 a centralized and compulsory new curriculum was introduced that was based on the results of the Varga’s research. In the following decade development aimed at adopting and realizing the research results within practice. Research mainly aimed at examining the effects of the newly introduced curriculum by looking into the development of children’s problem-solving skills. Other research was associated with international studies such as SIMS, TIMMS, and PISA. Additional research and development into different aspects of problem solving, summarized here, was conducted by various research groups around the country.

Appendices

Appendix 1

Table 3

Table 3 Part of the National Curriculum (2003) Management and solution of problems

Appendix 2

Table 4

Table 4 Education is compulsory up to the age of 18

Appendix 3

Problem 4 (Radnai-Szendrei 1977)

Each package on the following picture contains ten matchboxes.

• (a) How many matchboxes are on this shelf?

• (b) 52 boxes were sold.

Using this picture, write down the number of boxes remained on the shelf: ………

Appendix 4

(Kosztolányi, 2007 p 16)

Strategy: if the nature of the problem lets one guess, make up a reasoning, use recursions or complete induction by substituting natural numbers in.

Problem: is it true that all triangles can be cut into n isosceles triangles for each positive n that is not smaller than 4?

Questions

• 1. Try to cut an optional triangle into four isosceles triangles. (Helping questions: 1. Can all triangles be cut into two right triangles? 2. Can all right triangles be cut into two isosceles triangles?)

• 2. How could we cut an optional triangle into 7; 10; 13; …; 3k + 1 (k ≥ 1) isosceles triangles by using the result of the previous section? How can it be generalized? What other cases should be proved in order to say yes to the problem’s question?

• 3. Try to cut an optional triangle into five isosceles triangles. How could our previous results be used? How should it be cut if the triangle is an equilateral one and if the triangle is not equilateral?

• 4. Try to cut an optional triangle into six isosceles triangles. How could our previous results be used?

Appendix 5

Baccalauréat at 9 May, 2006.

Part I (11 problems for 45 min)

Problem 1

The ratio of the inner angles of a triangle is 2:5:11. How many degrees does the smallest angle measure? (2 points)

Problem 9.

A company of four members keeps in touch through emailing. Each of them writes at most one mail to each of the others. At most how many letters could have been written by the four people all together within a week? Choose of the possibilities listed. Give an explanation to your answer (2 points).

Part II./B (135 min. Two arbitrarily chosen problems have to be sold of problems 12, 13, and 14.)

Problem 13

Five questions are posed in a TV game. If the contestant answers the first question correctly he or she wins 40,000 forint. Before each following question he or she must decide what percent of his or her money won to that point to risk; 50, 75, or 100. If the contestant answers correctly, the money bid is doubled. Otherwise all money bid is lost but the rest can be taken and the game is over.

• (a) How much money can be won by a contestant who answers all 5 questions correctly, risking the most possible amount (100%) each time? (4 points)

• (b) How much money can be won by a contestant who answers all five questions correctly and plays carefully, bidding the least possible (50%) each time? (4 points)

• (c) During the contest one of the players answer the first four questions correctly. He bid 100% before the second question, and 75% before questions 3, 4, and 5. He missed the last question. How much money could he take home? (5 points)

• (d) A contestant answered all five questions correctly and bids one of the choices with the same probability. What is the probability that he wins the highest prize possible? (4 points)

Appendix 6

The School of Thinking

(Life and science 21 November 2005)

As it happens every year András this year also participated in the traditional autumn cross-country race. By the end on the rather demanding ground the field considerably drew apart, and there was no dead heat at all. András is not satisfied with his achievement and he is rather tight—lipped about the whole thing and what he says is as much as this: He did not expect that he would be outrun by as many people, but at the same time it is a comfort that he also outran exactly as many of his rivals as the ones who ran home preceding him. One of the competitors left behind was Béla, who is happy with his tenth place, as Csaba, who most of the time beats him, had only the sixteens place.

Where was András placed in this race?