creating a culture of problem solving

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CREATING A CULTURE OF PROBLEM SOLVING

THE HUNGARIAN APPROACH TO EDUCATION

KRISTF HUSZR

PURE MATHEMATICS BSC

ETVS LORND UNIVERSITY

BUDAPEST,HUNGARY

EVANSVILLE,INDIANA,2011


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TABLE OF CONTENTSAcknowledgements ...................................................................................................................................................................................... 3Introduction ...................................................................................................................................................................................................... 4

Notes ........................................................................................................................................................................................... ...................... 5The Brief History of Education in Hungary ........................................................................................................................................... 6

The Beginnings ....................................................................................................................................................................... ...................... 6The First University .............................................................................................................................................................. ...................... 6Reformation and Counter-Reformation ............................................................................................................................................ 7Ratio Educationis ................................................................................................................................................................... ...................... 7The Statute of Jzsef Etvs on Public Education ......................................................................................................................... 8Development between the World Wars ............................................................................................................................................ 9The Soviet Influence ............................................................................................................................................................. ...................... 9Notes ........................................................................................................................................................................................... ...................... 9

The Education System today .................................................................................................................................................................... 10Notes .............................................................................................................................................................................................................. 11

Four Pil lars of Math emat ica l Education ....................................................................................................................................... 121. Grea t Educators in the Pas t and Pres ent ..................................................................................................................... 12

Lszl Rtz (1863-1930) ................................................................................................................................................................. 12JzsefKrschk (1864-1933) ........................................................................................................................................................ 13Lipt Fejr (1880-1959) .................................................................................................................................................................. 13Gyrgy Plya (1887-1985) ............................................................................................................................................................. 14Tams Varga (1919-1987) .............................................................................................................................................................. 15Pl Erds (1913-1996) ..................................................................................................................................................................... 16

2. Diverse Primary and Secondary School Opportunities ...................................................................................... 17Advanced Math Programs ............................................................................................................................................................... 17KMaL Mathematical and Physical Journal for Secondary Schools .......................................................................... 18Math Competitions ............................................................................................................................................................................. 18Mathematical Camps, Math-Weekends................................................................................................................................. 19

3. Exte nsive and Deep Aca demic Curriculum ................................................................................................................. 224. A Very Strong Community ..................................................................................................................................................... 24Notes .............................................................................................................................................................................................................. 24

Challenges Today ........................................................................................................................................................................................... 26Notes .............................................................................................................................................................................................................. 28

Picture Gallery ................................................................................................................................................................................................. 29Sources of Figures ......................................................................................................................................................................................... 37


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ACKNOWLEDGEMENTS

This paper would not have been possible without the support of many people. I would like to

thank my hosts MARGARET MCMULLAN andPATRICK OCONNORfor their very kind invitation to Evans-ville. I am excessively grateful for their support. I would like to express my gratitude to the Depart-

ment of Mathematics, University of Evansville, especially DR.DAVE DWYER, chair of the department.

Working with him is a great honor and pleasure for me.

In Hungary,LAJOS PSA provided deep and beautiful insight into the adventurous world of Ma-

thematics for many of us. In his math camps I made a lot of friends and gained life experience. He

also shared some photos of his exceptional lectures which are included in this work. I am highlygrateful for them. Ilearned a great deal about Mathematics and Mathematics History from the deep

conversations with DR.DM BESENYEI,Etvs Lornd University. I am grateful for his critical andhelpful comments on my paper, as well. I would like to thank GYRGY MARCZIS,Erkel Ferenc Second-

ary School, Gyula for his inspiring lectures at the Nagy Kroly Mathematical Students Meeting and

for sharing some photos from his collection.

I very much appreciate the help of my English language instructor and academic advisor CHRISTI-

NA EDDINGTON, Beloit College, who read through the manuscript carefully and corrected my mis-

takes. I have already gained invaluable experience at Beloit College about different cultures, liberal

education and more. Particularly, I would like to thank DR.DAVE ELLIS, Beloit College, for his very

inspiring lectures in Chaotic Dynamical Systems and Topology and for the highly motivating discus-

sions out of the classroom. I owe an enormous debt of gratitude to DR.ANDRAS BOROS-KAZAI and

MARY BOROS-KAZAI for their helpfulness.

Without the continuous support of my family I could not have achieved so much in my life. My

parents DR.ZOLTN HUSZR, University of Pcs, and RENTA H.PRIKLER, Zipernowsky Kroly Technic-al High School, made every effort to ensure a solid education for my brother, GERG,and me. Theyhave always encouraged us, taken care of us, and shared their experiences with us.

I would also like to say thank you to all of my friends and m athematics teachers who I met atvarious math camps, competitions or other events. And last but not least, I would like to express the

highest gratitude to my very good friend and former mathematics teacher, VERA LNYIwho intro-

duced me to the wonderful world of Mathematics during my high school studies and gave me life-

long inspiration.


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INTRODUCTION

Hungary has always been famous for her many renowned scientists,

particularly mathematicians. This small country in Central Europe, nowa-

days with slightly less than ten million inhabitants, has thirteen Nobel-Prize

laureates1 (despite the fact that there is no Nobel-prize in mathematics, I

think it is important to mention it), three each of Wolf-Prizewinners2(PlErds, 1983; Pter D. Lax, 1987; Lszl Lovsz, 1999) and Leroy P. Steelelaureates3(Paul R. Halmos, 1983; Peter D. Lax, 1992; Endre Szemerdi;2008). In addition, John von Neumann won the Bcher Memorial Prize4in1938, which is only awarded by the American Mathematical Society every

five years. But there have been recent prize-winning occasions, as well.

Jnos Kollr, specializing in algebraic geometry received the prestigiousCole Prize5in 2006 and Lszl Lovsz (of the Etvs Lornd University, Budapest) the Kyoto-Prize6in2010. These are only a few examples from a very long list of Hungarian mathematical highlights. And

the Hungarian combinatorial school, which is famous around the world, has not been mentioned yet.

Not only the professional researchers are successful. The Hungarian team is a regular participant in

the International Mathematical Olympiad(IMO), which is the most prestigious contest for high school

level competitors. According to the cumulative results, Hungary occupies third place (with 77 Gold,

143 Silver, 80 Bronze medals, and 5 Honorable Mentions) in a list that includes more than 100 coun-

tries.7In second place is the USA, and in first place is China. On the other hand, the team of the EtvsLornd University finished first in the International Mathematics Competition(IMC) in 2008. This is anannual contest for undergraduates. There were ninety universities represented, including Moscow

State (Russia), Princeton (USA), Sharif (Iran), cole Polytechnique (France) and many other universi-ties known world-wide.8

So far it seems that the mathematical education is Hungary is perfect. Unfortunately, this is not the

case. The Programme for International Student Assessment (PISA), a survey given every three years,

showed that, in general, the Hungarian (Math) education faces serious problems.9In 2009 in the sub-

ject of Math, Hungary finished twenty-ninth out of sixty-five countries in the field of Mathematics. The

United States finished thirtieth. Many of the problems have financial as well as structural origins. The

regional IMO Training Centers (TC) have been closed recently because of the lack of financial re-

sources.10These TCs used to provide mathematics courses, which enhanced the problem-solving skills

and the knowledge of the students living in the countryside and in smaller cities. Now, if someone

would like to participate in Olympiad Training, this person would have to travel to the capital city,

Budapest. The trip there and back takes at least six hours by train and the tickets are getting more andmore expensive. Many families cannot afford this, so the talented and motivated youngster quits the

training after a while and, maybe in some extreme cases, decides s/he does not want to deal with math

in the future at all.

In this paper, I would like to give a brief outline on mathematics education in Hungary; it is strongly

based on my own experience. In addition, I will try to answer the question: Why are (some) Hunga-

FIGURE1 : Pter D. Lax;Abe l-Prize, 2005.


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rians so good at math? Furthermore, I would like to point out some of the main problems of education

in Hungary and, what we could learn from the education system in the United States.

NOTES

All online material accessed on March 7, 2011.

1BECK,MIHLY: A Nobel-dj s a magyar Nobel-djasok. KFKI, Csillebrc(http://www.kfki.hu/~cheminfo/hun/teazo/nobel/nobeldij.html)2Wolf Foundation, Mathematics (http://www.wolffund.org.il/cat.asp?id=23&cat_title=MATHEMATICS)3Leroy P. Steel Prizes, American Mathematical Society(http://www.ams.org/profession/prizes-awards/ams-prizes/steele-prize)4Bcher Memorial Prize, American Mathematical Society(http://www.ams.org/profession/prizes-awards/ams-prizes/bocher-prize)5Frank Nelson Cole Prize in Algebra, American Mathematical Society(http://www.ams.org/profession/prizes-awards/ams-prizes/cole-prize-algebra)6Kyoto Prize (http://www.kyotoprize.org/news/pressrel/pressrel_111010_lovasz.htm)7International Mathematical Olympiad, Cumulative Results by Countries(http://www.imo-official.org/results_country.aspx?column=awards&order=desc)815thInternational Mathematics Competition for University Students 2008, Team Results(http://www.imc-math.org.uk/imc2008/results2008teams.htm)9OECD (2010), PISA 2009 Results: Executive Summary(http://www.oecd.org/dataoecd/34/60/46619703.pdf)10MIHLY CSORDS,TIBOR NAGY(editors): Cserepek a magyarorszgi matematikai tehetsggondoz mhelyekbl.Bolyai Jnos Matematikai Trsulat, Budapest, 2010; p. 180(http://www.mategye.hu/download/cserepek/cserepek.pdf)


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THE BRIEF HISTORY OF EDUCATION IN HUNGARY11

First of all, I would like to highlight some important milestones in the history of Hungarian edu-

cation that are not directly related to the mathematics education in the 20 thcentury; however, they

determined the education over the past thousand years.

THE BEGINNINGS

In 996, the first Hungarian Benedictine

monastery was founded on Saint MartinsHill in Pannonhalma.12 This date can be

regarded as the beginning of school edu-

cation in Hungary. The establishment of

the monastery was made possible by the

open-minded and progressive politics of

Prince Gza13and his son, Saint StephenI,14the very first king of Hungary, was

crowned in the year 1000. Under his rule

Hungary became a Christian country,

which significantly affected the education

in the following millennium. King Stephen

I founded ten episcopates across the coun-

try and chapter schools were established

everywhere. In the monastery of Pannonhalma the main aim used to be the education of cleric intellec-

tuals, but this has changed over time. The Benedictine Order in Hungary has been dealing with public

education since 1802. Nowadays, the Benedictine Secondary School of Pannonhalma is one of the mostexcellent secondary schools in Hungary. This is a boarding school for boys only.

THE FIRST UNIVERSITY

In 1367, Pope Urban V founded the very first university of Hungary in

Pcs (my hometown). It was the forty-fourth university in the world. Thefinancers were King Ludwig I of Hungary and Bishop William of Pcs. In theend of the 14thcentury, the university had only about 800 students. Nowa-

days, this is a state university with ten different colleges. In addition, the

number of people learning here is over 30,000.15 On the left you can see the

seal of the university with the Latin description and in the center there is the

blazon of the institution.

FIGURE2: The Arch Abbey of Pannonhalma

FIGURE3: The seal of

the University of Pcs


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REFORMATION AND COUNTER-REFORMATION

The Protestant Reformation and later Counter-Reformation brought mentionable innovation into

the education system in Hungary in the 16-17thcenturies. Calvinist Colleges and Lutheran Lyceums

were founded with very high standards in teaching. In a school like a Lyceum a student could learn

from the age of 6 until the age of 28; from the basics to advanced academic knowledge. Theseschools provided education at all levels and they had also a great tutoring system. (In my opinion,

the lack of a well-working tutoring system and the lack of a close master-and-pupil-relationship is a

big problem to be solved in the Hungarian education system today.)

On the other hand, the Jesuits institutions played and continue to play a relevant role in educa-tion even today. In 1599, the Jesuits issued the Ratio Studiorum (Latin: Plan of Studies), whichprovided new structural concepts in education and a new curriculum, as well. This booklet became

so popular that even other Christian orders used it and implemented these regulations in education.

In 1635, Pter Pzmny the leader of the Counter-Reformation move-

ment, founded a university in Nagyszombat (today Trnava, Slovakia) whichwas the predecessor of the Etvs Lornd University (ELTE) in Budapest,Hungary.16 The ELTE is one of the largest universities in Hungary with

eight colleges and about 32,500 students. In addition, this is the oldest uni-

versity in Hungary that is in continuous operation. In our case, the most

relevant aspect is that generations of excellent mathematicians have been

educated here, especially since the beginning of the 20thcentury. Today the

Institute of Mathematics at ELTE has eight departments; 110 professional

mathematicians work here teaching the next generation and doing diverse

research on various topics.17The director of the Institute is Lszl Lovsz, former president of the

International Mathematical Union and one of the most renowned mathematicians today.18

FIGURE 5: The ELTE, Lgymnyos -Campus on the coast of the Danube. In these two buildings there arethe colleges for nature sciences, social sciences and information-technology.

RATIO EDUCATIONIS

FIGURE 4: The seal of

the ELTE


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In 1777 one year after the United States Declaration of Indepen-dence was issued - Maria Theresa (member of the House of Habsburg)

the Queen of Hungary issued the first public education act in Hungary,

the so-called Ratio Educationis.19This was an important reform in the

Hungarian educational system. The most relevant parts of the act are

the following:

Taking part in education became compulsory for everyone be-tween ages 6-12.

The act broke the hegemony of the Catholic Church in the fieldof education. All levels of education became the responsibility of

the state.

Ratio Educationisaffected the entire education system. The par-tition of the school system was as follows:

four years of elementary school three years of junior secondary school five years of senior secondary school two years of academy four years of university

THE STATUTE OF JZSEF ETVSON PUBLIC EDUCATION

Another milestone in the history of education was the 38thStatute in

1868, introduced by Jzsef Etvs, the Minister of Religion and Educa-tion in Hungary from 1867 to 1871. Etvs was a dominant statesmanand writer in Hungary in the second half of the 19 thcentury. He was a

member of the first government after the Hungarian Revolution in1848, as well. His son Lornd Etvs was a renowned physicist and alsoa minister of religion and education.20 The brief summary of the 38th

Statute from 1868 is listed below.

Taking part in education: compulsory for everyone between theages 6-12.

Freedom of learning and freedom of education; equality of allreligions in education.

Everyone should get educated in his/her mother tongue. Ethnicities could found their own schools and get education in their mother tongue. Founding of upper public schools in settlements with more than 5,000 inhabitants is man-

datory.

Higher standards in education; better equipment and surroundings; more (financial) bene-fits for teachers.

Despite the early death of Etvs in 1871, in the following decades a remarkable improvement inthe education in Hungary was seen. The number of teachers increased quickly and so did the num-

ber of public schools, from approx. 10,000 to 16,000.

FIGURE 6: Maria Theresa,

Queen of Hungary in 1762(by Jean-tienneLiotard)

FIGURE 7: JzsefEtvsin1845, by Mikls Barabs


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DEVELOPMENT BETWEEN THE WORLD WARS

The Treaty of Trianon (June 4, 1920) ended the First World War, which was extremely disadvan-

tageous we can say catastrophic for Hungary. The countrysterritory was reduced from 283,000km2to its current area today of 93,000 km2, and we lost sixty percent of the population, too. (It was

18.2 million before the treaty, and became 7.2 million after it.) All this was accompanied by terribleinflation and an economic crisis. The Minister of Religion, Culture and Education, Kuno Klebelsberg,

saw the only solution in the development of the sciences. In this difficult period, between 1926 and

1930, eight to eleven percent of the state budget was spent on education and culture: 5000 elemen-

tary schools were built. It was the period when the best Hungarian secondary schools were founded.

These schools are still operating and have given outstanding scientists to the world.

THE SOVIET INFLUENCE

After 1945, Hungary became the part of the so-called Soviet zone, and it remained so until 1990.

After the suppression of the 1956 Revolution in Hungary a great number of intellectuals left the

country. A lot of them found a new home and new life in the USA, and hopefully they added to thescientific achievements of this great country.

NOTES11This chapter summarizes the history of education in Hungary very briefly. It is based on the following com-prehensive book: ISTVN MSZROS,ANDRS NMETH,BLA PUKNSZKY: Nevelstrtnet. Bevezets a pedaggia saz iskolztats trtnetbe.Osiris Tanknyvek, Budapest, 2003(Source of the facts is this book, unless otherwise stated.)12Today this is the Arch Abbey of Pannonhalma, member of the UNESCO World Heritage. Next to the monas-tery there is the Benedictine Secondary School of Pannonhalma which is one of the most distinguished boysboarding schools in Hungary. (http://www.bences.hu/en)13Gza, Grand Prince of the Hungarians (c. 945-997). During his rule Christianity began to spread among theHungarians. Today this is the Arch Abbey of Pannonhalma, member of the UNESCO World Heritage. Next tothe monastery there is the Benedictine Secondary School of Pannonhalma which is one of the most distin-guished boys boarding schools in Hungary. (http://www.bences.hu/en)13Gza, Grand Prince (http://en.wikipedia.org/wiki/G%C3%A9za,_Grand_Prince_of_the_Hungarians)14Saint Stephan I (c. 970-1038). (http://en.wikipedia.org/wiki/Stephen_I_of_Hungary)15The First University of Hungary (http://www.pte.hu/menu/21); Homepage (http://english.pte.hu/) andWikipedia article (http://hu.wikipedia.org/wiki/P%C3%A9csi_Tudom%C3%A1nyegyetem)16Homepage of the Etvs Lornd University (http://www.elte.hu/en) 17Source of facts: Homepage of the Institute of Mathematics (http://www.cs.elte.hu/index.html?lang=en)18Wikipedia article about Lovsz (http://en.wikipedia.org/wiki/L%C3%A1szl%C3%B3_Lov%C3%A1sz) 19The first issue of Ratio Educationis in 1777 from the Hungarian Digital Library (MEK)(http://mek.oszk.hu/06500/06559/)20In 1895 Lornd Etvs founded the Etvs Jzsef Collegium(EJC) in Budapest, named after his father. Withvery high standards in education this institution has been an important workshop of talent nurturing since itsformation.


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THE EDUCATION SYSTEM TODAY

After mentioning the most

important stations in the histo-ry of education, I would like to

continue by outlining the main

characteristics of the Hunga-

rian education system today.

Our education system is no less

complicated than the most

education systems in the world.

Actually, a whole dissertation

written by an expert would be

necessary to show all the de-

tails. Using the following table, I

have tried to capture the es-

sence of the system. The Pre-Primary and Higher Educa-tion levels seem to be clear,but the General School andSecondary school levels are alittle bit confusing. We have

three different systems at these

levels, but in general there are

only minor differences. Basical-

ly, the curriculum is every-

where the same.23Education is

mandatory between the ages of

five and eighteen.

The 8+4-system is themost common. After four or six years of General School , the student has the opportunity of chang-ing to Secondary School. Six- or eight-year long secondary school programs are usually offered byspecial institutions. In Hungary twelve secondary schools have a six-year long Advanced Mathemat-

ics program (at secondary school level), which is the most important thing in our case.24

The veryfirst Spec-Math (according to the Hungarian description) class was started in 1962 in the famousFazekas School in Budapest, which is, according to many surveys, one of the best secondaryschools in Hungary.25From the early sixties Fazekas has been an important talent-nurturing cen-ter, especially in mathematics. Some members of the first math class were: Zsolt Baranyai (), IstvnBerkes, Mikls Laczkovich, Lszl Lovsz, Jzsef Pelikn,Lajos Psa andKatalin Vesztergombi. All ofthem have become renowned mathematicians.

AGE LEVEL21 DESCRIPTION

3PRE-

PRIMARYISCED0

3-4 years of kindergarten4

5

6

8+4 6+6 4+8

7

GENERALSCHOOL

ISCED1-2

8

910

11

12

1314

15SECONDARY

SCHOOL

ISCED3

16

17

18

FINAL EXAM (BACCALAUREATE)19

HIGHER

EDUCATIONISCED4-6

B. A. / B. Sc.(undergraduate)

Exceptions:medical,

law, archi-

tecture

VocationalSchool20

21

22 M. A. / M. Sc.(graduate)23

24Ph. D. / D. L. A.

(post-graduate)2526

[27,) LIFELONG LEARNING

Based on: EURYDICE22


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Obviously, this is only a very short outlook of todays system, but in the following, I will providemore information that is strongly connected to this section.

NOTES

21

ISCED refers for the UNESCO International Standard Classification of Education levels(http://www.uis.unesco.org/ev.php?ID=7433_201&ID2=DO_TOPIC)22Organization of the Education System in Hungary 2008/2009; EURYDICE(http://eacea.ec.europa.eu/education/eurydice/documents/eurybase/eurybase_full_reports/HU_EN.pdf)23This is the so called Nemzeti Alaptanterv, which means National Core Curriculum. The document can bedownloaded from the homepage of the Ministry of National Resources:(http://www.nefmi.gov.hu/kozoktatas/tantervek/nemzeti-alaptanterv-nat)24Specilis matematika tagozatos iskolk s tanraik. Matematika Oktatsi Portl(http://matek.fazekas.hu/portal/rolunk/spectanarok.html)25Fazekas Mihly Primary andSecondary School and Teacher Training Centre, Budapest. History of the Faze-kas(http://www.fazekas.hu/iskolank/iskolatortenet)


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FOUR PILLARS OF MATHEMATICAL EDUCATION

We arrive at the main part of this paper. During my secondary school years (2005-2009), I had

the opportunity to take part in several mathematics competitions and students programs, whichgave me knowledge, memories and experience for life. In my opinion, these diverse secondary (and

primary) school programs lead by great teachers are the most important places of talent nurturing

in mathematics in Hungary at this level. I will explore this topic in the second section. The third sec-

tion will contain the consequence of the second one. In all of these programs most participants make

friends, not jealous rivalries and, therefore, these are the birthplaces of a very strong community

where people help each other, not hinder. I will specify my thoughts about this in the fourth section.

Initially, I would like to write about the first pillar of the mathematical education in Hungary: its

great educators.

1. GREAT EDUCATORS IN THE PAST AND PRESENT26If I have seen further it is by standing on the shouldersof giants. (Isaac Newton)

At this point I think I am in both a very lucky and unlucky situation. I am lucky because there are

many hugely important teachers, especially from the 20thcentury, who created the culture of prob-

lem solving not just in Hungary, but also in other countries where they lived during their lives. On

the other hand, I am unlucky because I cannot mention all of these great teachers due to the limited

length of the paper. More precisely, I can mention only a few examples.

LSZL RTZ(1863-1930)27

He was a legendary mathematics teacher of the famous Fasori

Evanglius Gimnzium(Fasori Lutheran Secondary School)in Budapestfrom 1890 to 1925. During these thirty-five years of teaching, Rtz re-formed mathematical education with his excessively efficient teaching

methods and with his personality. He dealt with his students as equals

and as colleagues. He often invited them to his home and into the compa-

ny of his university colleagues, both of which could be very motivating for

a talented young person. When he realized that he could not teach more

to a student, he then requested another university professor to take

him/her over to teach. For his work in education he was awarded the

prestigious Officer d'Acadmie award at a 1910 Paris congress. From

1896 to 1914 he was editor-in-chief of the journal KMaL which hasplayed a very important role in the mathematical talent nurturing in Hun-

gary since 1894.28

NOTABLE STUDENTS

Eugene Wigner, Nobel-Prize laureate in Physics, 1963 John von Neumann

FIGURE 8: LS ZL RT Z;after the painting of

Kunwald Czr


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According to a nice anecdote, Eugene Wigner was asked in the late 1970s 'Do you remember

Rtz?' to which he answered, 'There he is!' and pointed to a picture of Rtzon his office wall.29

Nowadays, the LszlRtzLife Work Awardis one of the most prestigious awards for secondaryschool teachers. In addition, the annual national math teachers meeting of the Bolyai JnosMathe-

matical Society is named after LszlRtz.

JZSEFKRSCHK (1864-1933)30

outstanding mathematicians such as [Eugene] Hunyadi, Julius Knig,Krschk and [Gusztv] Rados have contributed to the high standard ofmathematical education at the Technical University [of Budapest]. Their

scientific and teaching activity affected mathematical life in the whole

country and laid the foundation of the internationally recognized mathe-

matical school in Hungary.(Pter Rzsa)31

Krschk was an outstanding mathematician (especially in algebraand geometry). He was also a great teacher at the turn of the century. At

the beginning of his career Krschk was a teacher in secondary schools. Besides teaching, he worked on the national mathematics curriculum. In

1891, he started teaching at the Technical University of Budapest and

continued to work there the rest of his life. He made important contribu-

tions to mathematics, and he was a nurturer of talent, too. Dnes Knigand John von Neumann were both students of Krschk. He was one of themain organizers of the Etvs Lornd Mathematics Competition for secondary school graduates,which was started in 1894. (In 1949, this competition was renamed the Krschk Jzsef Mathemat-ics Competition in his honor.) This is the very first modern mathematical competition of the world!

Since it was started, the Jnos Bolyai Mathematical Society has organized the competition every year(during the World Wars there were some exceptions). One of his most important works is the so-

called Hungarian Problem Book, issued in 1929. In this work he summarized the challenging prob-

lems and solutions of the first thrity-two Etvs Competitionsand Krschks extensions. Today thisis a widely known four-volume book series and it was translated into English, Russian, Romanian

and even to Japanese, Arabic, and Korean.32

LIPT FEJR (1880-1959)33

It was not given to him to solve very difficult problems or to build vastconceptual structures. Yet he could perceive the significance, the beauty,

and the promise of a rather concrete not too large problem, foresee the

possibility of a solution and work at it with intensity. And, when he had

found the solution, he kept on working at it with loving care, till each detail

became fully transparent. (Gyrgy Plya on Fejr)34

"Lipt Weiss has again sent in a beautiful solution."(Lszl Rtz)35

He was born in Pcs, a southern Hungarian city where I am from, asFIGURE 10:Lipt Fejr

FIGURE 9: JZSEF

KRSCHK


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well. Fejr studied mathematics and physics in Budapest and Berlin. In Budapest he was taught by,among others, Julius Knig, Jzsef Krschk, Man Beke and Lornd Etvs. In 1911, he became theleader of the mathematics institute at the University of Budapest (later: Etvs Lornd University)and held that position until his death. During these nearly fifty years he created a very successful

school in mathematical analysis. He had many students who later became great mathematicians.

NOTABLE STUDENTS

Pl Erds Gyrgy Plya Gbor Szeg Tibor Rad John von Neumann Lszl Kalmr Mihly Fekete Tibor Bakos Pl Turn Marcel Riesz Kornl Lnczos

Except for Kornl Lnczosand Tibor Bakos, all of the listed people were doctoral students ofFejr. Today the mathematical competition in the county Baranya (the county seat is Pcs) is namedafter him.36

GYRGY PLYA(1887-1985)

If you can't solve a problem, then there is an easier problem you cansolve: find it. (Gyrgy Plya)37

There is no doubt that Gyrgy Plya was one of the greatest math edu-cators in the 20thcentury. He was born in Budapest and after graduating

from the University of Budapest, he became a professor of mathematics at

the top Swiss university, ETH Zrich. He held this position from 1914 to1940, but due to World War II he had to leave Europe. From 1940 until

his death, Plya worked at Stanford University in theUnited States.

In his own words, Plyabecame a mathematician because the journalKMaL, the Krschk Jzsef Mathematical Memorial Contest, and FejrLipt affected him so much.

During his long life Plya worked in various fields of mathematics:number theory, analysis, geometry, algebra, combinatorics and probabili-

ty theory.38He also made great contributions to mathematical education and the methodology of

math teaching. Plyas books are classic masterpieces in this field. Probably his most popular bookis How to Solve It: A New Aspect of Mathematical Method, issued in 1945 by the Princeton University

Press. The fact that it has been in print continuously since 1945 and has been translated into twen-

ty-three different languages tells much about the reputation of the book. 39In this book Plya ex-

plores the ways of mathematical thinking and makes suggestions (The Four Principles40) in problemsolving. He emphasizes the heuristic thinking methods, too. The book is used by many teachers, stu-

dents, problem solvers and mathematicians today, as well. I also clearly remember reading this

Plya-book when I was a secondary school student. Another classic work of his is Problems andTheorems in Analysis I-II with another mathematician, Gbor Szeg.41This is a very detailed, deepand unique introduction to analysis through exercises and problems. Today this book is often re-

ferred only as The Plya-Szeg, because itis simply one of the best textbooks ever written on thistopic. The structure of this book strongly affected Lszl Lovsz when he wrote his famous book

FIGURE 11 : Gyrgy Plya(G. L. Alexandersons

collection)


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Combinatorial Problems and Exercises.42Plya was an exceptional figure (not only) in mathematicseducation. Thanks to his extensive social network, he could present and share his ideas with a very

broad audience.

TAMS VARGA (1919-1987)43

Tams Varga was an outstanding figure of mathematics educationeven from an international perspective. He caught worldwide attention

by elaborating on a complex teaching method in mathematics during the

1960s or early 70s. With his colleagues he was working carefully on the

details and, as a result of their hard work, a completely new, integrated

mathematical curriculum was first introduced in primary schools in

1978. Todays primary and secondary level math education in Hungary ispartly based on Vargas main ideas:44

Mathematics classes from the very first years in school. Instead of Arithmetic and Geometry classes there is oneMathematics class with various topics. Stimulating creativity, mathematical thinking. Calculation techniques are important, as

well, but they should not be overemphasized.

The introduction of abstract concepts should be based on the experience and explora-tions of the students. This introduction should not be very quick.

Flexible teachers who are capable to make responsible decisions and regularly trainthemselves. They are familiar with many areas in mathematics.

Various activities in classes. Playing (mathematical) games is very important.Although everyone agrees with the high efficiency of Vargas method, only a few hundred teach-

ers use it in practice for some reason. Other countries have already adopted this idea successfully,but in Hungary the process is very slow (not only in regard to the legislation, but in peoples atti-tudes as well). Sometimes even the direction of the development is not obvious.

Tams Varga was aco-editor of an internationally respected book on teaching mathematics.45Today the national mathematic competition for middle-school students is named after him.

FIGURE 12: Tams Varga


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PL ERDS (1913-1996)

Never, mathematicians say, has there been an individual like PaulErds. He was one of the century's greatest mathematicians, who posed andsolved thorny problems in number theory and other areas and founded the

field of discrete mathematics, which is the foundation of computer science.He was also one of the most prolific mathematicians in history, with more

than 1,500 papers to his name. And, his friends say, he was also one of the

most unusual.(G. Kolata)46

The quotation above is a good characterization of Pali bcsi (UnclePaul). At that time, the internet and email had not been invented yet, but

he had traveled so much during his life that, some people said he was a virtual substitute for elec-

tronic mail. There is a quotation, which describes this very kindly: "Want to meet Erdos? - mathema-

ticians would ask. Just stay here and wait. He'll show up."47With no home and no permanent job,Erds was the most prolific mathematician ever (he had 1,525 publications, more than LeonardEuler had).48Although he had no special teaching methodology when he met somebody, even atalented child, he often told him/her unsolved problems the first time he was probably one of themost inspiring mathematicians of all time. He dealt with everyone as colleagues. In his life Erdswon several mathematical prizes with high cash rewards, but he didnt need much money due to hisstrange and puritan lifestyle. He donated most of his fortune to support talented students and he

offered cash prizes for solving problems he posed. During his career Erds worked withso manypeople (511 co-authors49), that some of his friends introduced the concept of Erds-number, whichdescribes the collaborative distance between a person and him. The definition is the following:

Definition: Pl Erds has the Erds-number of zero. Someones Erds-number is , where is the lowest Erds-number of any co-authors. If there is no suchk, then the persons Erds-number is infinite.50

Erds has become a legendary figure of the mathematical folklore, which is proven by manyprojects (e. g. The Erds Number Project51), books, films, jokes and anecdotes related to him. One ofmy favorite examples is the Collaboration Distance Calculator52of AMSs MathSciNet:

FIGURE 14: One can calculate the Erds-number directly with the Use Erds -button.

FIGURE 13:Pl Erds


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2. DIVERSE PRIMARY AND SECONDARY SCHOOL OPPORTUNITIES A childs brain is like a sponge.(Proverb)

Some people say that talented children should be recognized as early as possible and they should

get special treatment from a very young age. Other people say that we only have to show the oppor-

tunities, and then, let students choose freely from them. I personally do not know which way is bet-

ter. Probably, there are more than these two choices. Nevertheless, I cannot agree more with the

quotation above.

In Hungary there are many mathematical talent-nurturing programs. This is the consequence of

having so many great educators, partly shown in the previous section. There are math contests even

for third grade primary school pupils and, from the age of twelve, students who are interested in

them have various [mathematical] opportunities.

I started dealing seriously with math when I entered secondary school. The personality and atti-

tude of my math teacher VERA LNYIplayed a huge role in my decision of mathematics. I remember

that during the first class she showed us the journal KMaL and encouraged us to think on theposed problems in it. She is also one of the main teachers of the Erds Pl School for MathematicalTalents,53which is one of the most relevant mathematical workshops today.

The following section is strongly based on my own experience. Therefore, it is a more subjective

than comprehensive introduction to the important building bricks of mathematical talent nurturing.

ADVANCED MATH PROGRAMS

Today twelve secondary schools offer a six-year long advanced mathematics program54(recall

the 6+6 system on p. 10) from the seventh grade up to the twelfth. In the higher grades the curri-culum is usually divided into more parts (algebra, geometry, analysis, etc.) and students have seven

to eight math classes per week. If this is not enough, the students can take part in extracurricular

classes. Probably the most renowned advanced mathematics program is provided by the Fazekas

Mihly Primary and Secondary School and Teacher Training Centre in Budapest. The first class wasstarted in 1962 under the leadership of Gyula Komls and the teacher of mathematics was ImreRbai. Some prominent members of the class were: Zsolt Baranyai (), Istvn Berkes, Mikls Lacz-kovich, Lszl Lovsz, Jzsef Pelikn, Lajos Psaand Katalin Vesztergombi. In the past forty years,many students graduated from these schools and became successful scientists.


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KMALMATHEMATICAL AND PHYSICAL JOURNAL FOR SECONDARY SCHOOLS55

This monthly journal for secondary school students is unique

in the world. It was founded by a mathematics teacher, DnielArany in 1893. (At that time the name of the journal was

Kzpiskolai Mathematikai Lapok.) During its 118 years of exis-tence (its publication was paused during the World Wars), many

generations grew up solving and thinking on problems posed in

the journal, which have always offered a great opportunity for

students to read about exciting, extra-curricular topics in math

and physics and to train themselves by participating in the con-

tests. It is also an extensive source of classroom materials for the

teacher.

Many great scientists (mainly mathematicians) of the past and

present, when they are asked about it, mention the KMaL as one

of the main sources of motivation at the beginning. There is notenough space in this paper to list all these scientists. I also have

very good memories related to KMaL. While thinking on prob-lems in the journal, one can gain problem-solving experience for

life.

Nowadays, the KMaLhas four math (A, B, C and K), two physics (P and M), and two computerscience (I and S) competitions at different levels. In addition, you can participate in the online-

competitions, as well. Most contests are year-long challenges. In most cases, there are nine rounds

in a school year; the problems are posed at the beginning of each month and the contestants have

one month to solve and hand them in via mail or on the Electronic Workbook. One can find a de-

tailed description of the contests on the homepage of the journal. 56Another hugely motivating re-

ward: the pictures of the best contestants are included to the legendary Photo Archives of the jour-

nal.57According to the idiom Practice makes perfect, in the past century KMaL has been theplay-ground of practice not only for mathematicians, but also for many other excellent scientists.

MATH COMPETITIONS

We have already explored the contests of KMaL. Now, I would like to address some other con-tests. Due to the limitations of time and space, I will mention only national competitions. For more

detail I would suggest the comprehensive publication of the Jnos Bolyai Mathematical Society.58I

will write briefly about the style of some secondary-school level competitions. Most of them havetheir analogues at lower levels. The historical background can be found in the references.

PROOF-BASED COMPETITIONS

KRSCHK JZSEF MATHEMATICAL COMPETITION59The oldest mathematics competition in the world and the most prestigious one in Hungary.

FIGURE 15:The issue 2006/3 of

the KMaL


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It was started in 1894. Even university freshmen are allowed to participate. Three problems

are posed to be solved in four hours.

ARANY DNIEL60MATHEMATICAL COMPETITIONThis is THE national competition for 9-10thgrade students.

NATIONAL COMPETITION FOR SECONDARY SCHOOL STUDENTS (OKTV)

61

This is THE national competition for 11-12thgrade students. With five hours of thinkingtime, this is the longest exam-type competition in Hungary.

MULTIPLE CHOICE TESTS

MATH KANGAROO62An international speedy competition. The contestants have only seventy-five minutes tochoose their answers.

GORDIUSZMATHEMATICAL COMPETITIONVery similar to the Math Kangaroo, but the students are given ninety minutes of thinkingtime.

MATHEMATICAL CAMPS,MATH-WEEKENDS

These are the mathematical programs in which I took part during my secondary school years.

There are many more opportunities (see in 58), but I do not have experience about all of the pro-

grams.

UNIVERSITY OF PANNONIAS ERDS PL SCHOOL FOR MATHEMATICAL TALENTS63

This school was founded in 2001 by some enthusiastic math teacherswho realized the importance of an institution that can provide intense

extra-curricular education for those students who are interested in ma-

thematics. Since its beginning, the ErdsPl School has been in operationunder the auspices of the University of Pannonia,64which provides finan-

cial support to the school. Therefore, the admitted students do not have

to pay a tuition fee, but they have to pay for other expenses (travel costs,

room and board).

Every student has to apply annually. The judgment of the applications

is based on the applicants former success in mathematical competitions.

The very first time (when most students dont have a record in math con-tests yet) almost everyone is admitted (obviously a teachers recommendation is necessary). Eachyear fifty to sixty students (per grade) get the opportunity to pursue deeper knowledge in mathe-

matics between the walls of the Erds Pl School.

The curriculum of the school is deep, extensive and focuses on enhancing the problem-solving

skills of the students. The venue of the education is a vocational school in Veszprm, which is a me-dium size city in Hungary. There are five long weekends in a school-year, when teachers and stu-

FIGURE 16: Seal of the

University of Pannonia


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dents meet there. On each weekend students take part in seven 90-minute long interactive lectures

about various topics presented by the best math teachers in the country. In most cases, a guest

speaker (a famous scientist) gives an additional talk. At every math-weekend teachers organize a

problem solving contest. They pose two problems for each grade. Students are free to think on them.

The solutions have to be handed in on the last day . The successful competitors rewards are beauti-

ful books on diverse topics.

The Erds Pl School for Mathematical Talents has a special place in my memory. I made manyfriends and met a lot of inspiring teachers there. I continue to have a connection with them. I expe-

rienced my first success in mathematics in Veszprm, as well. I regularly took part in the problem-solving contest and often stayed up at night thinking about the problems. I cannot describe the feel-

ing I felt when I won my first book. I am proud of the fact that I was there for all twenty weekends

during my secondary school years.

THE MATHEMATICAL CAMPS OF LAJOS PSA

SHORT BIOGRAPHY

Lajos Psa has been already mentioned, as the member of the very first advanced math class atthe Fazekas School in Budapest. Firstof all, I would like to write about him and, just after that,about his unique math camps.

He was born in 1947 in Budapest. After graduating as a mathematician from the Etvs LorndUniversity he started teaching at the Department of Analysis. However, he started doing mathemat-

ics much earlier. Psa met Pl Erds when he was just twelve. I would like to quote Erdss famousanecdote from 1969 about their first meeting:

"I will talk about Psa who is now 22 years old and the author of about 8 papers. I met him beforehe was 12 years old. When I returned from the United States in the summer of 1959 I was told about a

little boy whose mother was a mathematician and who knew quite a bit about high school mathemat-

ics. I was very interested and the next day I had lunch with him. While Psa was eating his soup I askedhim the following question: Prove that if you have n +1 positive integer less than or equal to 2n, some

pair of them are relatively prime. It is quite easy to see that the claim is not true of just n such numbers,

because no two of the n even numbers up to 2n are relatively prime; Actually I discovered this simple

result someyears ago but it took me about ten minutes to find the really simple proof. Psa sat thereeating his soup and then after half a minute or so he said "If you have n + 1 positive integers less than

or equal to 2n, some two of them will have to be consecutive and thus relatively prime." Needless to say,

I was very much impressed, and I venture to class this on the same level as Gauss' summation of the

positive integers up to 100 when he was just 7 years old." (Pl Erds)65

This little story tells much about Psas talent. He was only 15 years old when he caught theworlds attention by giving a sufficient condition for the existence of a Hamiltonian circuit in agraph. Today this is known as Psas Theorem.66But after a while Psa was not really interested inmathematical research anymore. He completely turned to teaching mathematics. Actually, he

started teaching very early. He was still a secondary school student when he had the opportunity to

give lectures in front of his class. As a freshman at the university he was requested to teach at his


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former secondary school, the Fazekas. At that time, he had many students who later became rele-vant mathematicians, e. g. Lszl Babai, Gyrgy Elekes(), Pter Komjthand Imre Z. Ruzsa.67Dur-ing the seventies, he also worked together with Tams Varga on the reform of mathematics educa-tion in Hungary. On March 14, 2011 Psa received the Szchenyi-Prize, which is one of the mostprestigious awards is Hungary.

HIS MATHEMATICAL CAMPS

Since 1988, Lajos Psa has organized more than 200 math camps for pupils from the age of 12 to18. These programs are very different from other mathematics camps. Psa deals in parallel withmany groups of students. In a group there are usually 15-30 youngsters, and he meets with each

group twice or three times a year. These meetings are whole-weekend programs with a very intense

curriculum.

The main idea of his methodology is teaching mathematics through discovery.Basically, thismeans that the student should figure out as many proofs as possible, preferably without help. How-

ever, the problems are selected very carefully in order to push the students gently in the right direc-tion. In most cases the problems are strongly related to each other, but these connections are often

hidden and they seem to be very tricky at first look. Then, as the student solves more problems, the

connection is gradually revealed and there is a magical moment when the last piece of the puzzle

finds its place, too. It is amazing and respectable that despite having 40-years experience in teach-

ing, Psa still works very hard to prepare for his next camp. This system works very well. Psasmany former students who became successful prove the efficiency of his method.

It is extremely important that the connections between Psa and his students are not interruptedafter they become too old to take part in his camps. He often invites his former students to work

with him as teaching assistants in the camps.

I think I am very lucky that I could and can still learn from him.

NAGY KROLY MATHEMATICAL STUDENTSMEETING

This annual conference for secondary school students and teachers was started in 1991 by Gyr-gy Olh (Gyuri Bcsi), an exceptional mathematics teacher from Komrno, Slovakia. His main aimwas to organize a conference for all Hungarian students who are interested in mathematics. They

must also be living within the borders of the historical Hungary (as they were before the Treaty of

Trianon). This Students Meeting has a double purpose. Obviously, the firstone is the nurturing ofthe mathematically talented. The second one is to provide the opportunity for students, who are

interested in math, to meet each other, no matter where they come from.

I met Gyuri Bcsi in a mathematical summer camp where he invited me to the next Nagy KrolyStudents Meeting. In the following years, I took part in this program five times. Three times as astudent and two times as a lecturer. When I graduated from secondary school, Gyuri B csi re-quested me to give a presentation the following year. It was a huge honor for me and Gyuri Bcsisgeneral attitude toward his students was highly motivating, as well. I think this is the most impor-


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tant thing for young math-candidates: to get motivation and, of course, knowledge from theirteachers.

ABOUT THE MEETING

The venues are the Selye Jnos Secondary School and the Vocational School, both located in

Komrno, Slovakia. The Nagy Kroly Students Meeting is a weekend-long conference with 90-minute lectures about various topics in mathematics held in different sections. All in all, there are

approximately 30-35 lectures during the weekend, but a student has the opportunity to take part

only in six lectures. Lectures are often interactive: in some cases the teacher gives handouts to the

members of the audience and assigns some tricky problems that will be discussed at the end of the

lecture. Traditionally, on the day of arrival there is always a cultural evening organized by the hosts.

The Nagy Kroly Mathematical Students meeting is a unique opportunity for most everyone. Al-though I am not a secondary school student anymore, every year I look forward to the Nagy KrolyDays very much.

OLYMPIAD TRAININGS

A successful participation in the International Mathematical Olympiad (IMO) is the consequence

of very good preparation. Although I have never taken part in the IMO, I was a regular visitor of the

trainings in Budapest and Pcs. In Budapest there is the central training center, and in the biggercities there used to be the regional training centers. (Due to financial problems the Jnos Bolyai Ma-thematical Society suspended the regional trainings.)

Surprisingly, I have more pleasant memories about the regional training in Pcs, which was heldat the University of Pcs. The instructor was Jnos Ruff, an enthusiastic assistant professor of theUniversity. We understood each other very well and, therefore, I really looked forward to the meet-

ings with him.

Once I was the only student who attended the training. Jnos asked the question: Do you knowcomplex numbers? My answer was a short No. If no, then we dont have time to waste! and thefollowing afternoon Jnoss lecture was an introduction to complex numbers. His explanations wereso effective that in the following days I already could use this tool to solve an A-problem from the

journal KMaL.

3. EXTENSIVE AND DEEP ACADEMIC CURRICULUMTo know is to know that you know nothing. That is themeaning of true knowledge. (Socrates)

In this section I will try to introduce how math education works at the Etvs Lornd Universitywhere I study. Nowadays, the Institute of Mathematics of the ELTE is a renowned workshop of high-

er mathematics with internationally recognized professors and research fellows. As it was men-

tioned earlier, the Institute of Mathematics has eight departments and 110 professional mathemati-

cians who are teaching the next generation and doing research on the cutting edge in various topics.


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Due to the financial structure of the education system in Hungary (which means the institution gets a

determined grant per student from the state), many universities admit all of their applicants. At the

Etvs Lornd University approximately 200-250 freshmen major in mathematics. When you apply youdont have to worry about your grades. Nearly everybody gets in. It is completely clear that 250 peoplecannot be taught it the same way. The ELTE has a striking solution for that. On the very first day at the

university all freshmen have to take a test which measures their mathematical backgrounds. If you fail thetest, you have to take an additional basic mathematics course that provides a revision of the secondary

school curriculum. People who achieve 75% or more get an outstanding certificate. But the most impor-tant thing is that in the first year all (!) math courses are offered at three different levels (regular < ad-

vanced < intense). Those students who took part regularly in competitions and math programs are sup-

posed to choose the intense level, but this is not mandatory. At the end of the first academic year every-one has to choose one of the four specializations (Mathematical Analyst, Mathematics Teacher, Applied

Mathematics, and Pure Mathematics). Instead of highlighting the differences between the U. S. and the

Hungarian higher education system (there are too many to list here, and my knowledge in the topic is not

enough to do it), I would like to summarize the courses of Mathematics BSc Pure Mathematics major,

which is an undergraduate-level curriculum. However, I have to tell some differences anyway. In the tablebelow you may notice that there are only six columns. Instead of four years, the undergraduate programs

in Hungary are generally three-years long, but most graduate programs need at least two years to com-

plete. As you can see the curriculum is enormous and often it is not easy to keep up with it.

SAMPLE CURRICULUM1. 2. 3. 4. 5. 6.

ElementaryMathematics

Geometry 1 Geometry 2 Geometry 3DifferentialGeometry 1

DifferentialGeometry 2

Analysis 1 Analysis 2 Analysis 3 Analysis 4FunctionalAnalysis 1

FunctionalAnalysis 2

Algebra 1 Algebra 2 Algebra 3 Algebra 3 ComplexAnalysis

FunctionSeries

NumberTheory 1

ComputerAlgebraSystems

NumberTheory 2

ComplexAnalysis

(Extension)

Finite Mathe-matics 1

Finite Mathe-matics 2

OperationsResearch 1

OperationsResearch 2

DifferentialEquations

Partial Diffe-rential Equa-

tions

Introductionto Information

Technology

Introductionto Computer

Programming

ProgrammingLanguage

(JAVA, C++)

ProgrammingLanguage

(JAVA, C++)

NumericalAnalysis

ComputerScience

Writing

MathematicalPapers

Introductionto Topology AlgebraicTopology Fourier-Integral

ProbabilityTheory 1

ProbabilityTheory 2

Statistics

Set TheoryMathematical

Logic

Based on the official curriculum.68

LEGEND: compulsory; compulsory elective; elective, but recommended.


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4. AVERY STRONG COMMUNITY...A community needs a soul if it is to become a truehome for human beings. You, the people must git it this

soul.(Pope John Paul II)

Studying mathematics means not only theory and beautiful thoughts, but it has also meantfriendships and lifelong experiences for me. Hungary (and even the world) is small enough, that I

can say: there is only one big community in mathematics. Although there are some separate islands,

the collaborative distance between them is probably less than we think.

Thanks to the many math camps and weekends that I participated in, I already knew almost the

entire class when I entered university. My friends were there, my former roommate in a camp was

there, and everyone [I met at camp] was there. During the Nagy KrolyStudents Meetings I madefriends from the neighboring countries, too. Since I started my studies at Beloit College I have con-

tinued to meet many people mathematicians and non-mathematicians, alike. This is what really

counts: making friendships through mathematics, a subject of study that does not know borders ordistance.

NOTES26Partly based on the following: KOLLEGA TARSOLY ISTVN(ed. in chief): Magyarorszg a XX. szzadban I-V.IV. ktet (Mszaki s termszettudomnyok). (http://mek.niif.hu/02100/02185/html/613.html)

27KENYERES GNES(editor): Magyar letrajzi Lexikon 1000-1990. (Rtz Lszl), Magyar Elektronikus Knyvtr.(http://mek.oszk.hu/00300/00355/html/ABC12527/12777.htm)28LSZL KOVCS: Hungarian Traditions in Talent Support.(http://talentday.eu/content/hungarian-traditions-talent-support)29NORMAN MACRAE:John Von Neumann: The Scientific Genius Who Pioneered the Modern Computer, Game

Theory, Nuclear Deterrence, and Much More.AMS Bookstore, 2000; p. 69.30KENYERES GNES(editor): Magyar letrajzi Lexikon 1000-1990. (Krschk Jzsef), Magyar ElektronikusKnyvtr. (http://mek.niif.hu/00300/00355/html/ABC07165/08989.htm)31R.PETER:200 years of teaching mathematics at the Technical University of Budapest. Internat. J. Math. Ed. Sci.Tech. 25 (6) (1994), 805-809.32JNOS SURNYI:A 100-adik Krschk Jzsef Matematikai Tanulverseny.Matematika Oktatsi Portl, 2004.(http://matek.fazekas.hu/portal/feladatbank/adatbazis/Kurschak_Jozsef_verseny.html)33KENYERES GNES(editor): Magyar letrajzi Lexikon 1000-1990. (Rtz Lszl), Magyar Elektronikus Knyvtr.(http://mek.niif.hu/00300/00355/html/ABC03975/04236.htm)34GYRGY PLYA: Leopold Fejr. J. London Math. Soc. 36 (1961), p. 501-50635Lipt Fejr Biography.MacTutor History of Mathematics, 2010.(http://www-history.mcs.st-andrews.ac.uk/Biographies/Fejer.html)36Lipt Fejr Mathematics Competition. in: MIHLY CSORDS,TIBOR NAGY(editors): Cserepek a magyarorszgi

matematikai tehetsggondoz mhelyekbl.Bolyai Jnos Matematikai Trsulat, Budapest, 2010; p. 9293.37GYRGY PLYA: How to Solve It: A New Aspect of Mathematical Method.Princeton University Press, 1945.38http://en.wikipedia.org/wiki/George_P%C3%B3lya39Princeton University Bookstore, web shop (http://press.princeton.edu/titles/669.html)401. Understanding the Problem; 2. Making a plan; 3. Carrying out the plan; 4. Looking back our work. (Fromthe book How to Solve it).41GYRGY PLYA,GBOR SZEG:Problems and Theorems in Analysis I-II, Springer, 1976.(http://www.springer.com/series/3838)42LSZL LOVSZ: Combinatioial Problems and Exercises. Elsevier, 1993.


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43Wikipedia article on Tams Varga.(http://hu.wikipedia.org/wiki/Varga_Tam%C3%A1s_%28tan%C3%A1r%29)44Matematika/Mdszertani alapelvek.Tantk a Hln, 2008.(http://alsos.fazekas.hu/wiki/Matematika/M%C3%B3dszertani_alapelvek)45W.SERVAIS,T.VARGA (editors): Teaching School Mathematics.Penguin Books UNESCO, 1971.46

G.KOLATA: Paul Erdos, a Math Wayfarer at Field's Pinnacle, Dies at 83 . New York Times, Sep. 24, 1996.47Bruce Schechter, in My Brain Is Open : The Mathematical Journeys of Paul Erdos(1998), p. 1448Publications of Paul Erds. The Erds Number Project. (http://www.oakland.edu/enp/pubinfo/)49The Erds Number Project. (http://www.oakland.edu/enp/)50Wikipedia article on the Erds number (http://en.wikipedia.org/wiki/Erd%C5%91s_number)51The Erds Number Project. (http://www.oakland.edu/enp/)52Collaboration Distance (http://www.ams.org/mathscinet/collaborationDistance.html)53Erds Pl Matematikai Tehetsggondoz Iskola (http://www.mik.vein.hu/erdosprog/index1.html)54Specilis matematika tagozatos iskolk s tanraik. Matematika Oktatsi Portl(http://matek.fazekas.hu/portal/rolunk/spectanarok.html)55NAGY GYULA: Tudomnyok kataliztora, a KMaL.in: Magyar Tudomny, 2003/11, p. 1455.(http://www.matud.iif.hu/03nov/016.html)56KMaL Contest descriptions (http://www.komal.hu/verseny/verseny.e.shtml)57

KMaL Photo Archives (http://www.komal.hu/tablok/)58MIHLY CSORDS,TIBOR NAGY(editors): Cserepek a magyarorszgi matematikai tehetsggondoz mhelyekbl.Bolyai Jnos Matematikai Trsulat, Budapest, 2010(http://www.mategye.hu/download/cserepek/cserepek.pdf)59Summary about the history of the competition and the collection of the posed problems.(http://www.batmath.it/matematica/raccolte_es/ek_competitions/ek_competitions.pdf)60BRA ESZTER: Ismeretlen ismersnk: Arany Dniel. In: Termszet Vilga, 2009/10.(http://matek.fazekas.hu/portal/kutatomunkak/Bora_Eszter/AD_ismeretlenismeros.pdf)61The OKTV on the Wikipedia(http://hu.wikipedia.org/wiki/Orsz%C3%A1gos_k%C3%B6z%C3%A9piskolai_tanulm%C3%A1nyi_verseny)62Mathematical Kangaroo on Wikipedia (http://en.wikipedia.org/wiki/Mathematical_Kangaroo)63Homepage of Erds Pl School for Mathematical Talents (http://www.mik.vein.hu/erdosprog/index1.html)64University of Pannonia (http://englishweb.uni-pannon.hu/)65

ROSS HONSBERGER: The Story of Louis Psa.University of Waterloo(http://www.math.uwaterloo.ca/navigation/ideas/articles/honsberger/index.shtml)66Weisstein, Eric W. "Psa's Theorem." From MathWorld--A Wolfram Web Resource.

(http://mathworld.wolfram.com/PosasTheorem.html)67Lajos Psa on Wikipedia (http://hu.wikipedia.org/wiki/P%C3%B3sa_Lajos_%28matematikus%29)68ELTE Mathematics BSc. Curriculums of the specializations.(http://www.cs.elte.hu/~ewkiss/ujbsc/ujhonlap/BSc_Mat_halo_2010.html)


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CHALLENGES TODAY

Remember, the storm is a good opportunity forthe pine and the cypress to show their strength

and their stability. (Ho Chi Minh)

In the last section of my paper I would like to achieve two aims. My first objective is to highlight

the main problems of the mathematical education in Hungary today. Secondly, I would like to sum-

marize my impressions about the education system of the USA and compare them with my Hunga-

rian experiences. In my opinion, both systems have advantages and disadvantages.

PROBLEMS TO SOLVE

In the introduction I mentioned the results of the PISA survey. In 2009, Hungary finished twenty-

ninth out of 65 countries in the field of mathematics, which is not a performance to be proud of. This

small country, which gave so many scientists to the world, performed modestly. Although we have a

world-class talent-nurturing system, especially in mathematics, there are only a few workshops of-fered. Unfortunately, most secondary schools are falling behind. Although the teaching methods of

Tams Varga, recognized world-wide are part of the National Core Curriculum, only a few hundredteachers actually follow them.69Theoretical knowledge is emphasized instead of creativity. Therefore,

secondary school graduates have many algebraic tools in their hands but, in a new situation, they

wont be able to use their knowledge. In the most cases there is not enough time to deal with problemsfrom everyday life. The less talented students lose the thread and will carry bad memories from math

classes. These people might later get into high governmental positions and make decisions about fund-

ing science and education. The bad experiences with mathematics could be detrimental. According to

one of my professors at the university, we have to face similar problems. When a mathematician gra-

duates from the ELTE he/she is familiar with many branches of mathematics and knows lots of theo-ries, however, he/she is unable to solve a simple operations research problem.

Another problem is the shrinking number of teachers (especially science teachers). In my opinion,

one of the main reasons for this shrinkage is that teachers are seriously underpaid. An average sec-

ondary school teacher with twenty years teaching experience earns about $10,000 per year.70Most

young people, who are interested in mathematics, choose specialization applied math and after grad-uation they might find a job in the financial sector or at a computer science company. Fortunately,

there are some exceptions; some teachers still work enthusiastically 10-12 hours a day (for free) to

share their knowledge with pupils and to nurture talents, but no one is able to do it for a whole life.

Those secondary school teachers who are active now are getting older and you cant see who will be

the next generation of teachers. This is a very serious problem to solve in the Hungarian educationsystem.

Many university professors and secondary school teachers have been complaining about the Bo-logna process in higher education (the renowned mathematician Mikls Laczkovich wrote a toughcriticism about the system71), which roughly means that the education is split into two parts: three

years in undergraduate education and two years in a graduate school. Today in Hungary a Bachelor sDegree in teaching is virtually worthless; therefore, teacher candidates have to go to a graduate school


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anyway. Then why is the education process split? A lot of people think that a well-structured five year

program for teachers would produce much better results.

USAAND HUNGARY:ASHORT COMPARISON

There are so many differences between the two systems in higher education, that the title of this

section A Short Comparison sounds paradoxical. However, I wi ll try to sum up my experiences(which reflect the viewpoint of a math student from Hungary). I can easily imagine that other people

see things differently.

The very first and maybe the biggest difference is that in Hungary most universities are owned

by the state and the first degree is free for all (of course your grades have to be high enough to get

into a university). More precisely, the tuition fee for twelve semesters is financed by the state (six

semesters in undergraduate school, four semesters in graduate school and you get two extra seme-

sters, too). Compared to the American education system this seems to be very comfortable from the

students viewpoint. Parents do not have to pay thousands of dollars annually to send their children

to the university. However, most institutions are struggling with very serious financial problems(even the Etvs Lornd University which is atop-ranked university in Hungary) and, according tomy experiences, the circumstances at an American university are generally much better than at a

Hungarian one. In addition, Hungarian university professors are underpaid. On the other hand, I

remember my first days at Beloit College. I was flabbergasted by the venues of education. The Col-

leges sports center is bigger than the municipal gymnasium in my hometown, Pcs (which is a quitelarge city in Hungary with its 160,000 inhabitants). And the Center of Sciences at Beloit is simply

amazing. In my opinion, sooner or later the Hungarian government will have to introduce a mod-

erate tuition fee with a fair grant and scholarship system to keep the high quality in education and

to avoid financial bankruptcy of the institutions.

Secondly, in Hungary (and in the most European countries) everyone has to declare their majorbefore applying to university. Even in their freshman year students have to take courses mainly

from their chosen field of interest. Actually, if you are a math major, you have the opportunity to

take, for example, history classes if you are interested in history, but the mandatory courses are so

demanding that you simply do not have enough time to deal with other topics. Compared to a liberal

arts school (e. g. Beloit College), which provides a broad education, the curriculum at Hungarian

universities is greatly/intensely focused. An average student at the Etvs Lornd University hasmany more courses per semester (recall the chart on page 10), than someone who studies at Beloit

College. In addition, we (people who are specialized in pure mathematics) deal with topics even in

the first semester, which are regarded as graduate level topics in the United States.

WHAT WE COULD LEARN

Introduce moderate tuition fee with a fair grant and scholarship system Lower student/faculty ratio Create/nurture closer professor-student relationships Improve our professional look (compare the homepages of the American Mathematical So-

ciety (AMS)72and the Jnos Bolyai Mathematical Society.73There is a world of difference.)


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WHAT THE USCOULD LEARN

Unfortunately, I am not an expert on mathematical education in the United States. Therefore, I

personally do not know what Americans could learn from the Hungarian educational system. I know

that young people who are interested in math have various opportunities here. The homepage of

AMS has a whole section dedicated to high school students and teachers. Among the subsections Icould see Summer Programs, Mathematics Help, Local Math Clubs and Events, Math Competitions,

etc.

CONCLUSION

The Hungarian (math) education, especially its talent nurturing aspect, has long and great tradi-

tions that resulted in the education of many renowned scientists in this small country in Central

Europe over centuries. Today the gifted students still have many opportunities, but we also have to

face several problems. The future of the system is in the hands of our political leaders, and I cantagree more with the quotation below, which, I think, applies both to the American and to the Hunga-

rian approach:

Education reform should focus on getting children out of pover-ty, not finding the bad teachers.(Dianne Ravitch)74

NOTES69Matematika/Mdszertani alapelvek.Tantk a Hln, 2008.(http://alsos.fazekas.hu/wiki/Matematika/M%C3%B3dszertani_alapelvek)70Kzalkalmazotti brtbla.TUDOSZ (http://www.tudosz.hu/2010_bertabla.html)71MIKLS LACZKOVICH: Bologna and the Teachers Education.(http://bolyai.cs.elte.hu/~laczk/bol.pdf)72http://www.ams.org/home/page

73http://www.bolyai.hu/en/index.html74from: The Daily Show with Jon Stewart.Guest: Dianne Ravitch; Comedy Central; March 3, 2011(http://www.thedailyshow.com/watch/thu-march-3-2011/diane-ravitch)


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PICTURE GALLERY

FIGURE 17:Ferenc Pintr, director of the Erds Pl School is opening the newschool year. Next to him is Orsolya Ujvri, secretary of the institution. (2005)

FIGURE 18:Vera Lnyi while teaching (Erds Pl School, 2011)


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FIGURE 19:Antal Kubatov talks to me in class (Erds Pl School, 2006)

FIGURE 20:Ferenc Csorba. He can draw perfect circles by hand (Erds Pl School, 2005)


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FIGURE 21:Jzsef Szoldatics and Vera Lnyi while verifying the students solutionsfor the posed problems (Erds Pl School, 2005)

FIGURE 22:GzaKiss has just started his lecture (Erds Pl School, 2009)


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FIGURE 23: Me and my very good friend Jnos Wolosz focusing on class(Erds Pl School, 2006)

FIGURE 24:Gyrgy Marczis giving his l ecture(Erds Pl School, 2006)


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FIGURES 2530:TYPICAL PHOTOS OF LAJOS PSA IN THE COMPANY OF HIS STUDENTS.

FIGURE 25

FIGURE 26


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FIGURE 27

FIGURE 28


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FIGURE 29

FIGURE 30


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THE NAGY KROLY MATHEMATICAL STUDENTSMEETING

FIGURE 31: Vera Lnyi while giving a talk

FIGURE 32:Me and Dvid Nagy , a friend from Kaposvr. Behind us there is Zoltn Veszelka and Pter Barabs (from left to right),

also very good friends from my former secondary school. (2008)


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FIGURE 33:I am giving a presentation on Fractals on the XX. Students Meeting in2010. Right at the moment I am explaining something about Peanos Curve.

SOURCES OF FIGURES

All sources accessed March 8, 2011

Figure 1: http://www.abelprisen.no/en/prisvinnere/2005/images/peter_lax.jpg .................................... 4Figure 2: http://www.keneditravel.hu/kepek/336/pannonhalma2.jpg .......................................................... 6Figure 3: http://upload.wikimedia.org/wikipedia/hu/6/6e/PTE_cimer_kicsi.jpg ...................................... 6Figure 4: http://www.kutatokejszakaja.hu/2010/data/userfiles/image/ELTE_logo.jpg ......................... 7Figure 5: http://upload.wikimedia.org/wikipedia/commons/1/15/ELTE%2C_L%C3%A1gym%C3%A1nyosi_Campus.jpg...... 7Figure 6: http://upload.wikimedia.org/wikipedia/commons/4/41/Maria_Theresia11.jpg ................... 8Figure 7: http://upload.wikimedia.org/wikipedia/commons/9/9d/Barabas-eotvos.jpg ........................ 8Figure 8: http://www.kkmk.hu/onszolg/eletrajz/kepek/nagy/ratzlaszlo.jpg........................................... 12Figure 9: http://www.omikk.bme.hu/archivum/angol/kepek/kurschak_jozsef.jpg ............................... 13Figure 10: http://members.iif.hu/visontay/ponticulus/images/szemelyek/fejer_lipot2.jpg .............. 13Figure 11: GERALD R.ALEXANDERSON: Random Walks of George Plya.MAA, 2000 ...................................... 14Figure 12: http://mathdid.elte.hu/pic/vtcikk/vt.jpg ............................................................................................. 15Figure 13: http://www.wolffund.org.il/admin/user_files/paul_erdos.jpg ................................................... 16Figure 14: http://www.ams.org/mathscinet/collaborationDistance.html ................................................... 16Figure 15: http://komal.elte.hu/lap/2006-03/cimlap.jpg ................................................................................... 18Figure 16: http://universitas.uni-pannon.hu/index.php?option=com_docman&Itemid=53 ................ 19Figure 17: Archive of the Erds Pl School (http://www.mik.vein.hu/erdosprog/index1.html) ....... 29
http://c/Users/Husz%C3%A1r%20Krist%C3%B3f/Documents/Beloit%20Spring%202011/Presentations/Creating%20a%20Culture%20of%20Problem%20Solving_2.docx%23_Toc287319937http://c/Users/Husz%C3%A1r%20Krist%C3%B3f/Documents/Beloit%20Spring%202011/Presentations/Creating%20a%20Culture%20of%20Problem%20Solving_2.docx%23_Toc287319937http://c/Users/Husz%C3%A1r%20Krist%C3%B3f/Documents/Beloit%20Spring%202011/Presentations/Creating%20a%20Culture%20of%20Problem%20Solving_2.docx%23_Toc287319938http://c/Users/Husz%C3%A1r%20Krist%C3%B3f/Documents/Beloit%20Spring%202011/Presentations/Creating%20a%20Culture%20of%20Problem%20Solving_2.docx%23_Toc287319938http://c/Users/Husz%C3%A1r%20Krist%C3%B3f/Documents/Beloit%20Spring%202011/Presentations/Creating%20a%20Culture%20of%20Problem%20Solving_2.docx%23_Toc287319939http://c/Users/Husz%C3%A1r%20Krist%C3%B3f/Documents/Beloit%20Spring%202011/Presentations/Creating%20a%20Culture%20of%20Problem%20Solving_2.docx%23_Toc287319939http://c/Users/Husz%C3%A1r%20Krist%C3%B3f/Documents/Beloit%20Spring%202011/Presentations/Creating%20a%20Culture%20of%20Problem%20Solving_2.docx%23_Toc287319941http://c/Users/Husz%C3%A1r%20Krist%C3%B3f/Documents/Beloit%20Spring%202011/Presentations/Creating%20a%20Culture%20of%20Problem%20Solving_2.docx%23_Toc287319941http://c/Users/Husz%C3%A1r%20Krist%C3%B3f/Documents/Beloit%20Spring%202011/Presentations/Creating%20a%20Culture%20of%20Problem%20Solving_2.docx%23_Toc287319942http://c/Users/Husz%C3%A1r%20Krist%C3%B3f/Documents/Beloit%20Spring%202011/Presentations/Creating%20a%20Culture%20of%20Problem%20Solving_2.docx%23_Toc287319942http://c/Users/Husz%C3%A1r%20Krist%C3%B3f/Documents/Beloit%20Spring%202011/Presentations/Creating%20a%20Culture%20of%20Problem%20Solving_2.docx%23_Toc287319943http://c/Users/Husz%C3%A1r%20Krist%C3%B3f/Documents/Beloit%20Spring%202011/Presentations/Creating%20a%20Culture%20of%20Problem%20Solving_2.docx%23_Toc287319943http://c/Users/Husz%C3%A1r%20Krist%C3%B3f/Documents/Beloit%20Spring%202011/Presentations/Creating%20a%20Culture%20of%20Problem%20Solving_2.docx%23_Toc287319944http://c/Users/Husz%C3%A1r%20Krist%C3%B3f/Documents/Beloit%20Spring%202011/Presentations/Creating%20a%20Culture%20of%20Problem%20Solving_2.docx%23_Toc287319944http://c/Users/Husz%C3%A1r%20Krist%C3%B3f/Documents/Beloit%20Spring%202011/Presentations/Creating%20a%20Culture%20of%20Problem%20Solving_2.docx%23_Toc287319944http://c/Users/Husz%C3%A1r%20Krist%C3%B3f/Documents/Beloit%20Spring%202011/Presentations/Creating%20a%20Culture%20of%20Problem%20Solving_2.docx%23_Toc287319945http://c/Users/Husz%C3%A1r%20Krist%C3%B3f/Documents/Beloit%20Spring%202011/Presentations/Creating%20a%20Culture%20of%20Problem%20Solving_2.docx%23_Toc287319945http://c/Users/Husz%C3%A1r%20Krist%C3%B3f/Documents/Beloit%20Spring%202011/Presentations/Creating%20a%20Culture%20of%20Problem%20Solving_2.docx%23_Toc287319946http://c/Users/Husz%C3%A1r%20Krist%C3%B3f/Documents/Beloit%20Spring%202011/Presentations/Creating%20a%20Culture%20of%20Problem%20Solving_2.docx%23_Toc287319946http://c/Users/Husz%C3%A1r%20Krist%C3%B3f/Documents/Beloit%20Spring%202011/Presentations/Creating%20a%20Culture%20of%20Problem%20Solving_2.docx%23_Toc287319947http://c/Users/Husz%C3%A1r%20Krist%C3%B3f/Documents/Beloit%20Spring%202011/Presentations/Creating%20a%20Culture%20of%20Problem%20Solving_2.docx%23_Toc287319947http://c/Users/Husz%C3%A1r%20Krist%C3%B3f/Documents/Beloit%20Spring%202011/Presentations/Creating%20a%20Culture%20of%20Problem%20Solving_2.docx%23_Toc287319947http://c/Users/Husz%C3%A1r%20Krist%C3%B3f/Documents/Beloit%20Spring%202011/Presentations/Creating%20a%20Culture%20of%20Problem%20Solving_2.docx%23_Toc287319947http://c/Users/Husz%C3%A1r%20Krist%C3%B3f/Documents/Beloit%20Spring%202011/Presentations/Creating%20a%20Culture%20of%20Problem%20Solving_2.docx%23_Toc287319947http://c/Users/Husz%C3%A1r%20Krist%C3%B3f/Documents/Beloit%20Spring%202011/Presentations/Creating%20a%20Culture%20of%20Problem%20Solving_2.docx%23_Toc287319947http://c/Users/Husz%C3%A1r%20Krist%C3%B3f/Documents/Beloit%20Spring%202011/Presentations/Creating%20a%20Culture%20of%20Problem%20Solving_2.docx%23_Toc287319947http://c/Users/Husz%C3%A1r%20Krist%C3%B3f/Documents/Beloit%20Spring%202011/Presentations/Creating%20a%20Culture%20of%20Problem%20Solving_2.docx%23_Toc287319947http://c/Users/Husz%C3%A1r%20Krist%C3%B3f/Documents/Beloit%20Spring%202011/Presentations/Creating%20a%20Culture%20of%20Problem%20Solving_2.docx%23_Toc287319947http://c/Users/Husz%C3%A1r%20Krist%C3%B3f/Documents/Beloit%20Spring%202011/Presentations/Creating%20a%20Culture%20of%20Problem%20Solving_2.docx%23_Toc287319947http://c/Users/Husz%C3%A1r%20Krist%C3%B3f/Documents/Beloit%20Spring%202011/Presentations/Creating%20a%20Culture%20of%20Problem%20Solving_2.docx%23_Toc287319948http://c/Users/Husz%C3%A1r%20Krist%C3%B3f/Documents/Beloit%20Spring%202011/Presentations/Creating%20a%20Culture%20of%20Problem%20Solving_2.docx%23_Toc287319948http://c/Users/Husz%C3%A1r%20Krist%C3%B3f/Documents/Beloit%20Spring%202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