When the going gets tough, the tough gets going problem solving in hungary, 1970 2007 research and t

ZDM Mathematics Education (2007) 39:443–458 DOI 10.1007/s11858-007-0037-0

ORIGINAL ARTICLE

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

Accepted: 1 May 2007 / Published online: 23 June 2007 FIZ Karlsruhe 2007

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 Tama´s 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.

Keywords OPI project Tama´s Varga Mathematics Journal Olympiads

1 Hungarian landscape The unique taste of Hungarian wines is determined by the ‘‘territoire’’, by the soil, by the sunshine in this region and by the wine casks made of black oak grown in Erd}oha´t (a hilly region in North Hungary). Similarly, our mentality J. Szendrei (&) Department of Mathematics, University Eo¨tvo¨s Lora´nd, ´ FK, Kiss Ja´nos altb. u. 40, 1126 Budapest, Hungary TO e-mail: Szendrei@kincsem.tofk.elte.hu

has been formed by the history of this region—it is history that has contributed to its development and has added its special Hungarian features.

Our topic (development of problem-solving thinking in the teaching of mathematics) is based to a great degree on the traditional culture of the Carpathian basin and on the experience of the people living here. This region has been the scene of devastating wars every 50 years on the average.

We were defeated by the Mongols (Tartars) in the thirteenth century, and we were occupied by the Turkish (Ottoman) Empire in the sixteenth–seventeenth centuries. There have been numerous uprisings and revolutions; different political systems have come and gone. As a result, human tragedies have always been characteristic of our region.

A recent example in which our cultural identity was damaged more than ever before is the period between 1945 and 1948 when 10% of the Hungarian population was forced to change its social status, and during which the institutions and top layers of the aristocracy, of the substantial middle class and of the administration were liquidated. Consequently, the remaining intact parts of society were left, in the sociological sense, without any pattern to follow and were deprived of value orientation. One had to start over from scratch, building on the ruins that were left behind after devastation. Defeated, one had to carry on living amidst the rules of an obscure, unknown system.

Unless you are resourceful and ingenious, you will not survive. Life here means solving problems on a daily basis.

It follows that in principle it is not too difficult to make people understand the term ‘‘problem solving’’ whether it is used in its abstract scientific form or in its culturally ‘‘domesticated’’ form which can be taught in schools.

The same issue can be approached from another point of view: the culture of the nation as a whole as opposed to the strategy of the individual person. From that perspective, it

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can be stated that civilization in this part of the world is more fragmented than continuous. One must start defining and identifying his/her values over and over again. After this has been done, these values must be turned into materials, buildings, everyday practices and customs.

2 Teaching of mathematics: institutions and workshops Compulsory public education was introduced in Hungary by the National Schools Act in 1868, at the same time as in the major developed countries of Europe. In the years that followed, hundreds of standard schools and teacher-training colleges were built throughout the country. Primary schools with eight school-year classes became compulsory in 1945; the scope of secondary schools was widened in the 1970s. At present education is compulsory until the age of 18 and Hungarian educational policy now focuses on the widening of tertiary education.

In Hungary the teaching of mathematics has always been esteemed, ranking second only to Hungarian language and literature and Hungarian and world history, which play a decisive role in the development of children’s national identity. This is true even today when mathematics is one of the four compulsory subjects to be taken at the Baccalaure´at (like the British GCSE, a test of competency in important secondary content areas); this policy is approved by 95% of the adult population of Hungary despite the fact that there are many of them who do not cherish memories of their mathematics studies at school. We are proud of the fact that many outstanding figures in teaching mathematics or doing research in mathematics were born in Hungary and are thus of Hungarian origin. (Among them one finds Farkas Bolyai, Ja´nos Bolyai, La´szlo´ Ra´tz, Gyo¨rgy Po´lya and Ja´nos Neumann.) For the purpose of this study, however, we are focusing on the average level of teaching of problem solving in public education.

Thanks to the ingenuity and initiative mentioned above, which can be regarded as a national characteristic feature of the Hungarian people, and also of the highly organized school system, the level of mathematics teaching is quite high in Hungary. Unfortunately, this only means conveying the very basics of the essentials at a high level. The development of knowledge in a sequenced process, the examination and placement of more special and/or more recent areas, the comparison and introduction of teaching experience and the establishment of contacts with life outside school would require: (a) more stable and coherent working conditions, and (b) a steadfast and well grounded educational policy, which must be built upon an ethos based on general consensus.

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This was lacking in the relevant time frame. Between 1949 and 1989 it was only by coincidence that there was occasionally some kind of common ground, given the nature of centralized political dictatorship on the one hand, and the desire for the freedom of scientific research on the other. In addition, in the current system of our parliamentary democracy (since 1989) educational research institutes, as a rule, are routinely re-organized after the General Elections held every 4 years. As a result there is not much point in making consistent plans for a period longer than 1 or 2 years. For the most part, teachers have grown absolutely indifferent to the Central Recommendations on the Curriculum, which are also re-written every 4 years. They hardly have time to learn about new initiatives, let alone distinguish between productive (from the professional point of view) developments and initiatives that have been pushed in by some irresponsible lion-hearted soldiers of fortune arriving from the world of politics.

To sum up, it can be seen that there is little chance for professional innovation, in the teaching of mathematics in particular (including the chances of widening the teaching of a problem solving approach) and more generally in public education as a whole. The reason is that the cultural conditions for transforming this fragile and sensitive system into a coherent and well-structured organization are still lacking.

What has been achieved and what we can be proud of is not the result of the contribution made by the social and political systems or of the effective measures taken by the educational policy. No, not at all. The major advances are the result of the activities of a few steadfast committed teachers and a few devoted and gifted researchers who would not be deterred.

3 A closer look at mathematics teaching One of the characteristics of the Hungarian school system is that teaching and the compulsory learning of mathematics starts in the first grade elementary level and ends at the last year of secondary school. In this article I wish to describe the presence and role of problem solving in teaching mathematics. It is important to emphasise it because problem solving plays an important role in teaching physics, chemistry, etc., in Hungary as well. Thus, we cannot state explicitly that the development of an individual child’s problem-solving skills is a result of learning mathematics. Teaching mathematics has a long tradition in Hungary. The practice of problem solving moved from generation to generation for centuries but hardly any written documentation of the details is left to us.

However the idea that all teachers at all levels would have to teach problem solving, despite its universality, did

When the going gets tough, the tough gets going problem solving in Hungary

not spread easily. Mano´ Beke, professor of mathematics (one of the leading proponents of Hungarian reformist ideas) gave a talk at a meeting of the School Doctors and Health Educators Committee (which was actually founded to ease anxiety over the belief that reforms proposed by mathematicians would place a greater burden on students) in November 1907 (!): ‘‘To put it briefly I will tell you what the reform committee is up to. It wants mathematics to be more practical by getting it nearer to everyday life and it wants to make it more integrated, as the committee intends to treat the currently isolated chapters according to a uniform viewpoint and a standardised method’’ (Beke & Mikola 1909, p. 10).

In spite of these reform ambitions, only the best teachers took special care to give their students more than typical or artificially constructed (for educational purposes) problems, but mathematical problems that originated in the real world as well.

During the years following World War II Ro´zsa Pe´ter1 and Tibor Gallai2 wished to make progress in secondary mathematics education, both content- and methodologywise. They wrote a textbook for first-year secondary school students that was based on real applicable problems, suggested explanations, and was mathematically correct.

Endre Ho´di and Jen} o Tolnai were co-author of the textbooks over the following years. They worked hard for decades to spread a view of teaching mathematics based on these ideas. The book was introduced—according to the education policy of the era—in all secondary schools.

Introducing ‘‘the new’’, however, did not mean that all teachers could or would adopt the new ideas in their teaching. Changes in content were not followed by the application of the suggested methods. In spite of regular in-service courses the problem-posing method, which starts a mathematical problem by posing a problem that raises the students’ interest, lets the students flounder for some time, and encourages the students to do some research to make progress on the problem, did not gain ground. All in all one can say that through several decades many areas of secondary level mathematics education were renewed. And this set the ground for further thoughts about mathematics education (Szendrei 2005, p. 427).

1

Pe´ter Ro´zsa (1905–1977) mathematician, corresponding member of the Hungarian Academy of Sciences.

2 Gallai Tibor (1912–1992) mathematician, corresponding member of the Hungarian Academy of Sciences.

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4 Problem solving at the curriculum level Starting from the 1950s problem solving has been explicitly mentioned in the mathematics curriculum as one of the major goals of mathematics education. ‘‘Mathematics education has to contribute to a deeper understanding of the natural and social environment and the development of the mental skills of the students by exploring the simple quantitative correlation of the surrounding world, and by understanding and solving problems concerning these’’ (Mathematics syllabus in 1956, p. 117).

But in the realization of the curricula mainly special types of problems were taught. One of the optimistic misconceptions of teachers was that by solving typical problems the students would learn to solve any kind of problems themselves. And so called ‘‘talented’’ students could do so, indeed. But this achievement was not the result of traditional schooling. These students either participated in special out-of-class mathematical activities or studied with private tutors. Some excellent teachers could develop high level problem-solving knowledge for almost all of their students, but these teachers did not restrict themselves to typical problems during their teaching activities. There were of course deep differences between students and between teachers.

In Hungary there was a centralized education system until the 1990s. There was only one curriculum and only one series of books for each class. The quality of teaching in a particular class was supervised by the director of the school and by the inspectors from the central authorities.

Since the 1990s, there has been a continuous compulsory five-scale grading system of the achievement of the students by their teacher. But there has been no centralized measurement of achievement. The baccalaure´at given at the end of the secondary school at approximately age 18 is the only central milestone. The baccalaure´at is not compulsory but is required for entering universities. Taking it is seen as part of becoming an adult. It has a much greater importance than a simple exam: it is obligatory for becoming a middle-class adult. Moreover it acts as a degree and usually ensures a higher category of salary for employees. Mathematics has been one of the four compulsory exam subjects (mother tongue and literature, mathematics, history, foreign language).

Since 1978 mathematics has been taught as a school subject, starting in primary school. There were no independent subjects like arithmetic, algebra or geometry.

Arithmetic, algebra, geometry, logic, functions, probability, statistics, and other mathematics content areas are all part of the integrated mathematics school curriculum.

These areas grow in relationship with each other through

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the curriculum. Consequently the problems given to the students at any level can be solved using different kinds of methods. That means that students in the same class will find algebraic, geometric, functional solutions of the same problem at the same time; 1978 was the year when new mathematics topics officially entered the primary-school curriculum.

In addition to the political changes some other factors influenced mathematics education. Apart from the public schools, private and church schools were founded. The number of students in higher secondary level increased from approximately 15–20% of the age cohort to 50%. This has changed the character of the centralized educational system. A democratic school system was built by legislation. This included the National Core Curriculum, and school programs were based on it. The National Curriculum (1995) specified the content of the education in detail. The schools prepare or choose their curriculum using a national accreditation system. The requirements of the baccalaure´at changed from a problem-oriented system to a strictly documented system of requirements. From 2003 the National Curriculum has been transformed from a content-oriented system into a development and competency-oriented system. Appendix 1 contains the part of the National Curriculum which is most closely related to problem solving.

We shall now examine in more detail how research results, the application of these results, and their adaptation into the schools transformed the arithmetic and measurement school subject of the 1950s into an integrated mathematics school subject even on the primary level. Moreover the typical style of mathematics teaching—that is, the teaching of the concrete fragments of mathematics—was slowly transformed into a style focusing on the development of mathematical problem-solving ability.

5 Research in the 1970s

matematikatanı´ta´si kı´se´rlet’’ (composite method, Varga 1965) into a nationwide schema. Dozens of teachers, inspectors and educators formed the research group he led.

The project covered mathematics education in the general school (age 6–14). Problem solving was the central activity of mathematics education at all levels.

The research project which, was called the OPI project in international discussion, was one of the deepest research studies on mathematics education in Hungary. (OPI was the name of the formal National Institute of Education in Hungary) The project investigated mathematics education in the general school. (See the table describing the Hungarian school system in Appendix 2).

The aim of the project was to determine the situation needed to create a new environment for mathematics education, to determine when society is ready to change mathematics education, and to determine the actions (that is, changes in in-service and pre-service teacher training, etc.) that would be necessary to move forward.

The aim was also to create a success-oriented environment for the learning of mathematics. Participants wanted to give students the freedom to make mistakes, help them develop a greater need to achieve, let them achieve on their own through their own independent actions with the teacher handling individual student differences, and create a friendly learning climate for the students where teacher and students were partners. Algebra and geometry were well represented in the curriculum as were elementary ideas of topology, information theory, sets, logic, probability and statistics–from the beginning of the primary school because that allowed a deeper and longer period in which mathematical concepts could ripen in the mind of the learner.

Naturally there was a need to change the traditional subject matter as well.

The selection of new subject matter was based partly on the following: •

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5.1 The OPI project As a result of the United Nations Educational, Scientific and Cultural Organization (UNESCO) symposium in Hungary (The International Symposium on Mathematics Teaching) held in Budapest from 27 August to 8 September 1962, several small research projects for the improvement of mathematics education at primary level (basic school) were initiated in Hungary.

Tama´s Varga4, starting with a small size mathematics project in the early 1960s, developed the ‘‘komplex 3

OPI was the name of the formal National Institute of Education in Hungary.

4 Varga Tama´s (1919–1987) mathematician.

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• •

•

• •

It must be fundamentally important from the point of view of the whole of mathematics.

It should be easily accessible to children through problems and through play.

It should be presented if possible using the children’s tactile sense. (This is an advantage rather than an absolute condition.)

Subject areas must be closely related. They must reinforce each other and thus help to construct an organically developing system.

The subject areas included should prepare and introduce the teaching of other important subject areas.

The subject areas included should be used in modern mathematical applications.

The essence of the project was to ensure the continuous development of each child on affective and cognitive levels

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Table 1 The cognitive objectives of the OPI Mathematics project (After Varga cited by Klein 1987)

Analytic

Receptive–reproductive

Productive

Understanding Distinguishing

Assessing (forming an opinion, defending or refuting) if:

Seeing relationships:

a statement makes sense

Sameness (in one respect or another)

a statement is true

Order (larger, less etc.)

a problem is clearly defined:

Appreciating a pattern (e.g. symmetry)

if it contains enough data

Understanding a statement: (being able to illustrate it, to give instances, to reword, recode, translate it)

if there are superfluous or contradictory data or condition in it

Following a train of thought, a reasoning

a symbol, a definition, a suggested way of solution, etc., is suitable, appropriate, or promising a reasoning is correct a solution satisfies the conditions a solution is reasonable, if it satisfies the practical requirements, certain standard, etc.

Synthetic

Knowing, doing, using

Constructing

terminology, symbols, graphical devices

Formulating a problem

mathematical statements

Stating hypotheses, educated guesses

routine problems, algorithms manual skills

Planning a solution Finding tools for a solution Finding some objects (concrete or mathematical) which satisfy given conditions Finding all such objects Finding, formulating a definition for a concept Developing a proof Generalizing, extending by analogy

to the extent in which school mathematics can have an impact Table 1.

OPI research was based on a deep collaboration between teachers and researchers. Actually the teachers of the first period of the research were teacher–researchers (if we apply the current terminology).

The leaders of The Mathematics Department of the National Institute of Education in Hungary, the late Andor Cser and Endre Ho´di, provided enormous academic help and offered a humanistic background for the research. However, the educational environment was not friendly at all. The following elements of the background philosophy of the new ideas were considered problematic: • •

•

Routine work, which was formerly typical in the first 4 years of primary school, was de-emphasized.

The teacher was no longer considered the sole authority in the classroom, the only source of the knowledge and the truth.

Instead of dead silence, noise was allowed in the classroom.

•

New signs were used (e.g. ‘‘/’’ as the sign of partition), and problem-solving tasks included the concepts of negative numbers and fractions, for example.

These and other unusual elements shocked many teachers, inspectors, parents and journalists. There was a fear of failure. The shadow of ‘‘New Math’’ in the world was too near.

But some positive effects arose from this somewhat hostile climate. The climate forced the researchers to develop a deep and detailed research project to be sure, on one hand, that the teaching methods proposed did not require specially selected or talented teachers or students to succeed, and to prove at the same time that students taught by the new methods could achieve the same or better results than those taught by the traditional one. Moreover, students were not tired and stressed, but are more willing to work on problem solving (Sherbrooke Problem Facing Questionnaire, Klein 1987, p. 408).

There was no textbook–only the series of worksheets which were carefully developed. The worksheets (the so called ‘‘Munkalapok’’) were continuously revised as

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teachers reflected on their work. The designers tried to avoid the atomism which could arise from the many parallel mathematics topics covered in the classes.

The research was well-known in the international community of mathematics educators. The strongest channel for the relationship was the CIEAEM5 Communication was oral, because it was not possible for Hungarian mathematics educators to publish their results internationally at the time. The international community invited Varga and his colleagues to give lectures at several universities and compensated them for this work by offering them books and articles. The best international innovations were available for designing the curriculum.

The designers tried to adopt many of them, while avoiding extreme solutions and mere formalism. The research was evaluated at international level as follows: ‘‘The excellent Hungarian arithmetic books, in my view the best textbook series of the world, avoid the Venn diagram completely…’’ (Freudenthal 1973, p. 244) ‘‘A counter example to this radical atomism are the Hungarian innovation of T. Varga’s Munkalapok, Emma Castelnuovo’s work for the Scuola media, and in the last few years effort such as made by our IOWO’’ (Freudenthal 1980, p. 97).

One major element of the research was the careful use of concrete materials in the classroom. The majority of the artefacts and teaching materials were new for the teachers.

This was not chalk-and-blackboard mathematics. This was the first curriculum in mathematics education in Hungary which prescribed the method, the content, the manuals, the manipulatives, the teacher guides and the in-service training courses together.

The in-service courses tried to teach at least the modest use of certain manipulatives such as a Hungarian version of Cuisinaire rods, Dienes multi-base blocks, the Gattegno pegboard, and a bead-and-block modeling set for teaching geometry and the elementary ideas of probability.

Naturally the manipulatives themselves did not change education. They are not a panacea for mathematics teaching (Szendrei 1996). But a growing percentage of teachers were able to use the new methods and new devices productively, as showed by ongoing measures of achievement.

Teachers competed with each other to get into the in-service courses. Counties competed with each other to open classes where teachers would be able to use the new methods and local schools ensured the necessary manipulatives for each child.

5 Commission Internationale pour l’E´tude et l’Ame´lioration de l’Enseignement des Mathe´matiques.

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The number of classes participating in the experimental phase of the research was gradually increased as more teachers were trained in the new method. Teachers were not allowed to teach without this training. In fact, the project leaders wanted to slow down the expansion of the project so that teachers would not be forced to use the new methods without fully understanding them. As they explained it: ‘‘This was a great opportunity and I felt I had to avoid taking much risk. ‘‘I decided therefore not to impose anything upon the teachers. They had to be convinced of the potential usefulness of what they were going to do and never to jump into darkness. Both teachers had much good will and they were well qualified. Neither of them clung rigidly to their views concerning either the syllabus or the teaching methods, but they would not have given up willingly their feeling of security developed during their years of school teaching’’ (Varga, 1965).

5.2 Research on the success of the OPI mathematics project Soon after its introduction the Ministry of Education ordered a study of the effectiveness of the OPI project. The evaluation was undertaken by Ministry of Education with the support of OPI and the Pedagogical Department of the Technical University of Budapest. A short description of the methods and results was presented in several conferences. The results were impressively in favor of the experimental classes (Radnai-Szendrei 1977, p. 97). Students using the new curriculum surpassed the others even in the solution of traditional problem-solving tasks.

(Appendix 3 contains one of the problems).

The quantitative development of the Project was also impressive. In 1971 there were 140 experimental classes (out of approximately 4,000 classes in Hungary in one age cohort at that time). These 140 classes used the so-called ‘‘temporary mathematics curriculum’’. OPI researchers wanted to restrict the project only to those teachers who were ready for the change. As Varga said: ‘‘Gradual extension is the key word for us. We are against conclusions of the kind that the results of our experiments show that our way of teaching mathematics is better than the usual; therefore it should be introduced into mass education by school year ‘‘n’’. ‘‘By what reason and right would we impose the reform on teachers if we are against imposing mathematics on children?

When the going gets tough, the tough gets going problem solving in Hungary

‘‘What we would like is:

1.

2.

‘‘To acquaint as many teachers as possible, as deeply as possible, with our way of teaching mathematics, or rather some version of it, and ‘‘To attain that every teacher be given the opportunity (including the right and material aid) to practice an upto-date teaching of up-to-date mathematics.

(Varga 1965 quoted by Klein 1987 p. 37).

Despite this caution, educational authorities urged the introduction of a new curriculum based on the OPI project.

They did not accept the protest of the researchers. Even though gradually year by year (starting with grade 1) a new curriculum based on the temporary one became the compulsory mathematics curriculum in the general schools in Hungary (in 1978), the teachers’ organization was not ready for this radical change. As could have been predicted, project objectives were not completely realized. But the style of problem solving was changed in the general school.

We should conclude that this movement somehow touched the spirit of mathematics education in Hungary. The types of problems that had only one solution lost their privilege. Teachers realized the value of solving problems with many solutions or none at all. Therefore the game of ‘‘arguing for the completeness of a solution’’ entered the primary grades. Solution methods like trial and error–trying all the possible solutions and selecting the good ones– became ‘‘legal’’. As a matter of fact it was much easier to get primary teachers to accept these methods than to gain acceptance from teachers who had received their mathematics qualifications at the university level. Three decades were not enough to make the majority of upper secondary schools accept heuristic methods or the importance of allowing students to generate their own hypotheses before seeing a result or a proof.

5.3 The SIMS6, TIMMS7 and subsequent studies in Hungary Hungary participated in IEA’s Second International Mathematics Study8. The results for the 8th graders (14year-old students) were exceptional. In each comparison (that is scores, subtests, subtopics, etc.) Hungary was in one of the top three spots, along with Japan and The Netherlands (Szendrei 1991, p. 99). This shows that the types of 6

SIMS 1980: Second International Mathematics Study.

TIMSS 1995: Third International Mathematics and Science Study.

8 IEA: International Association for the Evaluation of Edicational Achievement.

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problems to be solved were similar to the problems used in the Hungarian mathematical classes.

Since, within the population tested, it was possible to create a sub-population of 8th graders who used the traditional curriculum and a sub-population of 8th graders who used the new curriculum, the advantages of the new curriculum could be seen in each dimension of the study (Radnai-Szendrei 1983, p. 157).

In 1986, after the SIMS, the National Institute of Education started the Monitor studies. These studies periodically examined several elements of students’ problemsolving achievement. They covered cohorts at grades 4, 6, 8, 10 12. The aim was to measure advanced mathematical skills and the level of problem solving in problems with two or more steps. Studies were done in 1991, 1995, 1997 and 1999. The results showed that students living in the capital (Budapest) reached the highest level in each age cohort. In the case of the secondary schools there was an enormous difference by the type of the secondary school (See the table of the Hungarian school system in Appendix 2.) The students of the gimna´zium always surpassed students in other types of higher secondary schools (Va´ri 1997. p. 124, 131).

The results of TIMMS and the subsequent IEA studies flattered Hungary’s vanity.

Other analyses drawing on Hungarian data and data from other countries in the TIMMS project identified the stages in the educational system where most growth had occurred in student performance and where student performance had declined or been maintained following students’ exit from the compulsory education system. In this regard, the findings in the Hungarian system demonstrating that the examinations for entry to post-primary schools prompted a spurt in student performance that was not maintained in secondary level education were most telling (Pertl 2000).

The TIMSS studies showed nonetheless that the Hungarian results were excellent.

Eighth grade students in the Republic of Korea, Singapore, Hong Kong SAR, and Chinese Taipei had significantly higher achievement than in the other participating countries. After these four top-performing countries, Japan had significantly higher achievement than all the rest of the participating countries. Estonia, Belgium (Flemish), and Hungary also performed very well (Mullis et al. 2005, p. 23).

Nevertheless the careful analysis of SIMS data showed that the percentage of good answers was radically lower on problems that required the application of mathematical knowledge (Szendrei 1991. p. 87). These studies contained elements that allowed judgments of the level of problem solving and the development of cognitive factors. Fundamentally, however, achievement was still analyzed based

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on the requirements of the students’ mathematics class in school.

The PISA9 was the first international study which did not measure the level of knowledge transmitted by the school, but tried to assess the problem-solving ability of the student. PISA looked at the students’ ability to use their knowledge and skills to meet real-life challenges, rather than their ability to master the facts of a specific school curriculum.

5.4 Hungarian results in the PISA studies The PISA problem-solving model was entirely different from the previous studies which concentrated only on one subject. Still the items on the final tests corresponded with the problems taught in the mathematics classrooms of the individual countries (Dossey et al. 2000). The theoretical background of the study focused on measuring four basic thinking methods: qualitative, quantitative, analogical and combinative ways of thinking. The problems and results of the PISA study–which measured trouble shooting, decision making, system analysis and design–shocked Hungarian teachers and educational decision makers. Instead of realizing the teachers’ hope that Hungary would be one of the top achievers in problem solving, as had been the case in the IEA studies, the PISA studies showed that the achievement of Hungarian students in complex problem solving corresponded to the international average. Thus, international surveys prove that Hungarian students perform well if questions are presented in the form they have grown accustomed to at school, but that the students are less successful when problems require the application of knowledge in realistic problems. This was the result of the PISA-survey in 2000 as well (Va´ri 2003, c. p. 98). Because of PISA, teachers at last had to face results which clearly indicated that it was an illusion to think that solving a large number of typical problems would make the children able to solve real problems, make decisions, and design and analyse situations.

J. Szendrei

1.

During the slow introduction of the new mathematics curriculum there was massive resistance to the changes from parents, teachers, and politicians. The courses for teachers also needed concentrated efforts. Public television, National radio, journals like E´let e´s Tudoma´ny (Life and Science), and daily newspapers tried to convince people that these efforts would not hurt their students or their children. Moreover, the case also had to be made that the new curriculum could avoid the extremities which were criticised during the New Math movement.

This was not the time for quiet research work. Leading university educators tried to facilitate the realisation of the goals of the new curriculum with their lectures, articles, and courses.

2.

3.

In the early 1980s the school inspectorate system was eliminated. Therefore the usual channels for quick exchanges of information and feedback did not work anymore. The didactical research group of the Mathematical Research Institute of the Hungarian Academy of Sciences was eliminated. Moreover the National Institute of Education (OPI) was closed by the government. The new units in the Ministry of Education had either no responsibility for educational research or no desire to do research in mathematics education. It was declared that research in education was the responsibility of universities. Moreover financial support for such research was not guaranteed. The combination of rapidly changing educational policy and the continuous restructuring of institutions broke down the connections between researchers.

Scientific literature about mathematics education has not been easily accessible in Hungary. Until now libraries have even lacked the money to buy journals or books published in Hungary. The prices of international journals and books make it impossible to access the international literature. Educators and researchers depend mainly on international library exchange, which enables them to obtain specific books from the nearest library for only one month.

5.5 Background research in recent decades 5.6 Recent studies Hungarian research in the domain of mathematics education has been mainly conducted from the point of view of (general) educational sciences. Only a small number of mathematics educators put their energy into research on mathematics didactics. There are some major reasons for this situation.

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Programme for International Student Assessment.

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5.6.1 ‘‘Szeged Workshop’’ One of the most prestigious and esteemed Departments of Education in Hungary today is the ‘‘Szeged Workshop’’.

This department does empirical research on a broad scale of samples, focusing on the key questions of public education such as reading comprehension, operational level in basic mathematical operations or the development of logical abilities.

When the going gets tough, the tough gets going problem solving in Hungary

According to the PISA 2000 Survey, Hungarian students perform at the average level or below in their ability to apply their knowledge of mathematical and natural sciences and in complex problem solving. In addition, the findings of Szeged University Educational Studies revealed that although children in Hungary acquire immense amounts of knowledge in schools, they cannot apply this knowledge very well. They may solve tasks that require the mere reproduction of the skills taught by the school curriculum but they just cannot put these skills into practice outside school and in common everyday situations. As a rule, they make decisions based on their naive ideas or misconceptions. A lively description of this phenomenon can be found in the study by Csı´kos (1999) and Csı´kos (2003, p. 44) where a comparative analysis of the performance of fourth-grade children is provided.

This investigation used 20 mathematical word problems from Verschaffel, de Corte and Lasure’s (1994) work. The Hungarian version of this test contained the same ten standard and ten parallel tasks as the original test and was administered to students aged 10–11 years. The results were similar to those of earlier international studies. Even if students solved the standard problems well, their achievement declined radically when the problem was embedded into a real world context Table 2.

5.6.2 Mathematics education tradition of Europe project Recently the European Union (EU) changed its financial politics and began to support international research studies.

Under the umbrella of the EU a research group at Eo¨tvo¨s University participated in a classroom study as part of the METE (Mathematics Education Tradition of Europe) project (Andrews 2005a, b).

Table 2 The solution of the 20 tasks compared with the ‘‘parallel task’’ results of Verschaffel et al. (1994) Problem context

‘‘Friends’’ ‘‘Deszka´k’’

Hungarian study (2002) N = 280

Hungarian study (2002) N = 280

Standard problem (%)

Problem with realistic text (parallel tasks) %

98

18

11

71

14

14

‘‘Water’’

96

17

17

‘‘Bus’’

89

36

49

‘‘Running’’

67

2

3

‘‘School’’

92

7

3

‘‘Balloon’’

37

82

59

‘‘Age’’

85

0

3

‘‘String’’

46

4

0

‘‘Container’’

52

1

4

451

The METE project was a five-way, EU-funded, comparative study of mathematics teaching in Flemish Belgium, England, Finland, Hungary and Spain. These countries represent well the socio-economic diversity of the continent and diverse attainment in recent international comparisons of mathematical attainment like TIMSS, PISA and their reiterations. The project collected data on the teaching of key mathematical topics by means of video recordings of sequences of four or five lessons taught by teachers perceived by their local team to be representative of the better practice in that country. ‘The topics, which were thought to be representative of the breadth of school mathematics, concerned the teaching of percentages (a topic of arithmetic applicability) in grades 5 or 6, polygons (a routine geometrical topic) in grades 5 or 6, polygons (not only a routine geometrical topic but an opportunity to examine curriculum continuity) in grades 7 or 8, and linear equations (an early topic of formal algebra) in grades 7 or 8. Data collection was preceded by a year of live observations in each country to develop, in a bottom-up, grounded manner, a framework for describing and analysing mathematics lesson activity. The coding scheme comprised three sections: 1.

2.

3.

The first, mathematical focus, addressed the observable generic objectives or outcomes of a given episode.

The second, mathematical context, focused on the conception of mathematics underlying the tasks presented in an episode.

The third, didactics, considered the observable strategies employed by teachers in their classroom activity.

(In project terms, an episode was that part of a lesson in which the teacher’s didactic or managerial intention remained constant.)’ (To¨ro¨k 2007, p. 3).

Further analysis of the Hungarian data (in process by Judit To¨ro¨k and E´va Szeredi) produced promising results for Hungary in the domain of problem solving.

In the past decade Hungary’s educational policy has been to increase the number of university places for students. This is one of the reasons why this population became the target of some of the problem-solving research.

5.6.3 Problem-solving skills of pre-service primary teachers A recent study by Erzse´bet Zsinko´ (University Eo¨tvo¨s Lora´nd) examined the problem-solving skills and the attitudes towards mathematics of pre-service primary teachers (Zsinko´ 2006).

The aim of the study was to determine whether the students have the type of problem-solving skills that they will have to develop in their students when they become

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primary teachers. The problem-solving part of the study was combined with a questionnaire whose purpose was to determine the students’ interest in mathematics and their opinion of its importance. Some of the questions were aimed at determining whether pre-service teachers are satisfied with their own problem-solving knowledge. Onethird of all students were satisfied with this knowledge.

Among students who solved all the (5) problems there were some who considered their own knowledge to be poor and among students who gave only one correct solution, there were some who considered their own knowledge to be good. More than 40% of the students who considered own knowledge to be good performed well, in fact, 70% of those who thought their knowledge satisfactory gave satisfactory or better performance. The study indicates that it is the responsibility of the university to ensure that students have a good understanding of their own knowledge.

The greatest challenge for university education is to determine whether a well designed course can increase the problem-solving abilities of the pre-service mathematics teachers.

5.6.4 Research on teaching problem-solving strategies A study to meet that need was conducted by Jo´zsef Kosztola´nyi (2006). The aim of the study was to examine whether a well designed and conducted course can greatly improve the problem-solving ability of students in the application of strategies. Research by Kosztola´nyi joined that of Alan Schoenfeld on the effectiveness of different problem-solving methods (Schoenfeld 1980, 1983, 1985, 1994).

Kosztola´nyi used a typically Hungarian problem-solving environment. A pre-test which contained 5 problems measured the extent to which students recognized and applied strategic ideas in solving a given problem.

Kosztola´nyi then gave a course using approximately 30 problems. The students worked individually to solve the problems. The problems were then discussed both mathematically and in terms of the methodology used by the whole class. Appendix 4 contains one problem from the course. At the end of the study a post-test was administered to measure the extent to which the problem-solving ability of students improved with regard to the application of different kinds of strategies.

The results of the study demonstrated that there is a place for teaching heuristic problem-solving strategies on the teacher-training track in mathematics. Solving problems with the help of orienting questions seems to be a great deal more effective for students in helping them acquire strategic ideas than receiving a lecture on the strategies.

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5.7 Secondary schools with a greater percentage of university bound students reach higher achievement levels Problem solving at the higher grades of secondary schools in Hungary was strongly influenced by the entrance examination system. Until 2005 there was a special, mainly written examination in mathematics for all universities specializing in mathematics, physics, technology and economics.

Because of enrolment limitations at the universities (with 25% of the age cohort attending in the 1980s and gradual increases to 50% within the past few years) the entrance examination served as a gateway. The exam’s last two problems (of 7 or 8) were especially difficult. This system did not give enough flexibility to secondary schools. Many of the teachers catered to the top level students in the mathematics classroom. There was not enough time for nice and easy problems. Students had to spend time outside school to prepare for these examinations.

In 2005, the separate entrance examination was eliminated. Eligibility is determined by the results of the baccalaure´at, secondary school grades, and knowledge of foreign languages.

Since that time, the baccalaure´at has been given two levels: medium and high level. In principle either level is acceptable for university entrance. The mathematics problems in this examination touch on everyday problems but still focus on the types of problems taught at school.

Appendix 5 gives some examples of medium level baccalaure´at problems. Naturally the mathematical content of the high-level exam requires continual study on the part of the teachers. There are a number of problem books that help such as the series by Imre Ra´bai.

5.8 Out-of-school forums for problem solving 5.8.1 Mathematical Journal for Secondary Schools Problem solving is not learned only in classes. A monthly periodical was started in 1894 by a science teacher Daniel Arany for students talented in mathematics. The Ko¨ze´piskolai Matematikai Lapok (Ko¨MaL, the Mathematics Journal for Secondary Schools) became the source of problem-solving knowledge as well (Radnai 1988). Several generations of mathematicians and scientists developed their problem-solving skills through Ko¨MaL. As Hersh and John Steiner noted, ‘‘When George Po´lya (1887–1985) was asked to explain the appearance of so many outstanding mathematicians in Hungary in the early twentieth century, one of the three explanations was the

When the going gets tough, the tough gets going problem solving in Hungary

existence of Mathematics Journal for Secondary Schools which stimulates interest in mathematics and prepared students for competitions’’ (Hersh & JohnSteiner 1993, p. 15).

In 1994 Sa´ndor Ro´ka established the ABACUS, a mathematics monthly for students aged 10–14, their teachers and for those interested in mathematics. The periodical communicates broad knowledge in a professionally exact but accessible way for young readers. It is published punctually, and its layout is aesthetically pleasing. The authors of the columns are teachers and university students from different parts of Hungary (Szendrei & Szita´nyi 2007).

5.8.2 E´let e´s Tudoma´ny (Life and Science) For the past 30 years the journal E´let e´s Tudoma´ny (Life and Science) has been staging a competition in mathematics’’, The School of Thinking’’ (A gondolkoda´s iskola´ja), in which competitors must provide the solutions to ten different problems in written form. The entries are published along with the correct solutions and additional remarks.

The text of the problems is not a ‘‘fabricated’’ mathematics problem. On the contrary, in many cases a well defined mathematical problem is rephrased in order to obtain an interesting text problem. Over 1,000 competitors take part. The mathematics teachers Ja´nos Herczeg, Istva´n Reiman and Ja´nos Pataki run this competition. A typical issue is attached in Appendix 6.

5.8.3 National competitions One of the nationwide vehicles for problem-solving activities in Hungary is the system of mathematics competitions, Olympiads (Burkhardt et al. 1984, p. 106). At the very beginning the competitions served as an extra source of development for those students who were highly talented in mathematics. What does this extra opportunity mean? In hopes of success students and teachers devote extra energy and time to the teaching and learning of mathematics. Naturally not only those students who are able to compete successfully in mathematical competitions are talented in mathematics. To win in a competition, students need the ability to achieve under high stress and to keep up high-level thinking under the pressure of time restrictions even more than the necessary creativity and high level knowledge. Some students even do better in a competition than in a less stressful situation. But there are many talented students who need more time and deep work to be successful in problem solving.

The majority of Hungarian nationwide competitions contain problems which need long periods of concentrated

453

work and participants must write down the solutions in detail, giving proofs of the validity of their solutions.

Sometimes the rules of the competition even allow using mathematics books during the competition. The students’ solutions are evaluated by at least two independent members of the competition committee. But the work of the proposed winners is usually read by all members of the committee. (This is the working style of the Arany Da´niel Memorial Mathematics Competition for example.)

The problems and the solutions of such competitions are published from time to time in books in order to expand the circle of students and teachers who can learn from the experience of each particular competitions.

One of the highest levels in the competition hierarchy is the ‘‘Miklo´s Schweitzer Memorial Mathematics Competition’’, which lets the high school and university students solve the problems at home. The community of mathematicians does not help the students in their work.

Because the number of secondary-school students is increasing, it is important to increase the number of competitions so that children will have more chance of being successful. The competitions are mainly aimed at 14– 20 year olds.

After the mathematics curriculum of the general schools was changed in 1978, competitions were created for 10 to 14-year-old students. Later the Matematika´ban Tehetse´ges Gyerekeke´rt Alapı´tva´ny (Mategye) (Foundation for Talented Children in Mathematics) created a competition for students aged approximately 8–14. This competition uses multiple choice problems. But there is also a competition for the 10–14 age group that requires a detailed solution for each problem. (There are 5–7 problems which must be completed in 3 h).

5.8.4 Mathematics interest groups in the past, mathematical camps of the present Mathematics education in Hungary for the gifted and talented looks enviable. This is because the teachers have always put extra energy into the work of interest groups for talented students. In recent years the efforts of Lajos Po´sa and his students, in mathematics camps, have opened up possibilities for more average students.

6 Summary In this study, we have analyzed the developments of the past 40 years in Hungary, in particular the research and innovation that contributed to the development of problem-solving thinking in mathematics. Of the numerous remarkable results, some of which have also been acclaimed abroad, the

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most significant achievement by far is the experiment in the complex teaching of mathematics in schools, which was developed and conducted by Tama´s Varga. It is significant in both its content and its didactical approach.

What is of particular importance in Varga’s project is that it helped improve mathematical thinking for a great number of people. Models of mathematical thinking can provide a means of solving real-life situations. By including several areas from the science of mathematics that had never been taught before in school mathematics, Tama´s Varga not only vastly expanded the number of problemsolving models but also made them available to the masses in public education. This diversity could have easily resulted in children’s developing chaotic, incoherent feelings and ‘‘knowledge’’ of mathematics. Varga managed to avoid this by describing the diversity as a rainbow, whose colors run side by side in the sky along the common arch of mathematics.

Another element of Varga’s experiment was the ingenuity of the teacher. Varga was knowledgeable about the latest achievements in pedagogy and psychology and actually superseded them in his special field. He considered the teaching of mathematics, which has always been a touchy and delicate issue, to be a source of joy for everyone. He would work out many playful exercises and make witty and pertinent remarks as he taught. His mathematical didactics was (at that time!) partly based on the type of group work that we call today ‘‘co-operative learning’’. He emphasized the importance of developing social abilities in group work—this is built into our current curricula under the definition of ‘‘joint competency’’.

Similarly, what is called ‘‘differentiation’’ today was an inevitable part of his approach in classes with as many as 40 children. In addition, when developing his innovative exercises and his worksheets, which are useful even today, Varga applied the basic idea of today’s constructive pedagogy, namely: each and every student has different abilities in different disciplines and each of these abilities must be developed.

Varga himself estimated that it would take some 100 years, on the one hand, for teachers to be completely committed to these new methods in the teaching of mathematics and, on the other hand, for society to be ready to accept these methods. The new problem types have found their way into practice with little or no difficulty at all. In everyday teaching, the ‘‘new mathematics’’ was accepted and applied by most primary school teachers, by a majority of lower secondary school teachers, but only by a small minority of upper secondary school teachers. At this upper

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secondary school level one can hardly say that education is based on the spirit of the new concepts. It is here that the dream cherished by Gyo¨rgy Po´lya, Tama´s Varga and many others—a dream where problem solving is not fear inducing but rather a sound intellectual sport enjoyed by everyone involved and a thrilling success story for much more than the favored few—has not come true (Po´lya 1945, 1954, 1957, 1981). In today’s teacher training system, which has been changed to meet the requirements of the new National Curriculum in Hungary, teacher training for the specialized junior and senior classes has been adapted to the new expectations of mathematical didactics, but there have been few changes in the training of the upper secondary school teachers. There are still many who believe, erroneously, that the only pre-condition for teaching mathematics in an efficient way is that the teacher himself/ herself be good at problem solving. This is why even today there is much less time spent on the teaching of mathematical didactics than on the teaching of mathematics itself.

We are happy to note that the new GCSE exam tasks, which contain not only pure mathematical elements embedded in a more colorful context and which raise several problems at the same time, definitely have had a favorable effect on secondary school mathematics teaching. A solid background for such innovation in the teaching of mathematics has been provided by the Bolyai Ja´nos Society for Mathematics, the joint organization of mathematicians and mathematics teachers. The possibility of putting the findings of scientific research in didactics into practice has been enhanced by the Ph.D. College in Mathematics Didactics established some ten years ago.

This college has done a great deal to help didactics, whose scientific validity is consistently underestimated, become a more respected field. And, last but not least, it is encouraging to note that it has been 30 years since these new approaches to mathematics teaching were introduced into public education. This means that Varga was right: our plans will be realized in as little as 70 years.

Acknowledgments The author is grateful to Jane Schoenfeld for her proofreading and editing the article. I had the great opportunity to show my previous version to Alan Schoenfeld and benefit from his observations and suggestions. Furthermore I thank the help of Katalin Fried, Miklo´s Somogyi and Rozy Brar in English translation and bibliographic help.

Appendix 1 Table 3

When the going gets tough, the tough gets going problem solving in Hungary

455

Table 3 Part of the National Curriculum (2003) Management and solution of problems TABLE III Grades 1 to 4 Grades 5 to 6 Grades 7 to 12 Realisation of a problem (experiencing of a problem situation); sensibility to problems.

Understanding of a problem read or verbalised in situations and stories; application of devices helping understa nding (acting out in natural situations, creation of pictures, acting with setting, talking about the given situation, verbalisation of questions, making conciuous of data known or significant from the point of view of the problem, its separation.from the unsignificant).

Debate about the known and the unknown elements; verbalisations of conjectures and questions. Overview of simple problems.

Ability in problem solving and undertaking the problem.

Trial with a new solution after unsuccessful trial of a solution. Exploring the reason of the failiure (e.g. skipping a condition Searching for a simpler (already solved) problem similar to the given problem.

Choice, seachring and creation of a mathematical model corresponding to the problem.

(Taking th e problem apart; overview of a complex problem. Rephrasing to other more known problem; looking for analogies.)

Solution within mathematical schemes. Knowledge of mathematical schemes (e.g. open sentences, graphs, sequences, functions, graphing of functio ns, computer programs, statistical analysations), methods and limits of its applications (accuracy, applicability).

Self-checking; taking the responsibility for the results Searching for several methods of solutions, comparison of alternative solutions.

Choice of method(s) of solution most corresponding to the problem; justification.

Relating to the result to the original problem. Comparison of the result with the preconditions and pre-projected result and reality.

Discussion. (Counting the possibilities. Making students conscious how and in what way the conditions influence the result during the comparison with the conditions. How the solution is changed if omit or change one of them.)

Answering in oral and later in written form.

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Appendix 2 Table 4

Table 4 Education is compulsory up to the age of 18 TABLE IV The Hungarian System of Education (2007) Type of the education Age of starting Name of the institution

Preschool education

Compulsory education

3

6

Kindergarten (Óvoda)

Basic school (Általános iskola)

18-

10

12

14--18

Post secondary education After baccalauréat

Upper secondary school (Középiskola) General lower and upper secondary (8-years long) (Gimnázium) General lower and upper secondary (Gimnázium) (6years long) General upper secondary (Gimnázium) (4years long) Vocational secondary school (Szakközépiskola) Vocational training school (Szakiskola) University, High School

Vocational studies may not be commenced before the age of 16, up to which students are to acquire a fundamental education

Appendix 3

Appendix 4 (Kosztola´nyi, 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

Problem 4 (Radnai-Szendrei 1977) 1.

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: ………

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2.

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?)

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?

When the going gets tough, the tough gets going problem solving in Hungary

3.

4.

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?

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

Appendix 5 Baccalaure´at 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

457

(Life and science 21 November 2005) As it happens every year Andra´s 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. Andra´s 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 Be´la, who is happy with his tenth place, as Csaba, who most of the time beats him, had only the sixteens place.

Where was Andra´s placed in this race?

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