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This page is being developed and will be added to when abstracts are available.
There are some problems which seem to be hardly avoidable by traditional means in the area of mathematical competitions ( and not only there ):
There are also other problems.
Clearly they can not be solved in a short time. Nevertheless, the state- investment project " Latvian Education Informatization System " ( LEIS ) has provided some tools to start.
The project started in the June 1997. It consists of 5 parts.
Informatization of
are three of them. It includes also
The informatization of the content of education is considered as the most important of these parts. Within it, a large number of electronical teaching aids are developed in various disciplines and made accesible through INTERNET and CD-ROM`s free of charge in all regions and schools of Latvia and also to individuals. The main idea of LEIS is that the needs of EACH student must be served; so needs of talented students also must be taken into account. Therefore there are many teaching aids developed also for the needs of those preparing to mathematical competitions (we must remember that mathematical olympiads are recognized as an essential part of mathematical education system in Latvia and are very popular between students, their teachers and their parents ):
Note that in all cases we are not restricted by allowed number of pages as when publishing books !
These materials and a fact that they can be used free of charge makes the teacher in some sense independent from the restrictions his salary makes on his professional growth, and makes the student in some sense independent from his teacher`s qualification and eagerness to work with gifted boys and girls. ( Note that at this moment only appr. 50% of Latvian schools have INTERNET connection but at the end of this year this will grow up to 100% . )
We want to especially stress that among authors of more than 3/4 of these materials are also teachers from high schools, often from outside Riga, the capital of Latvia. It was possible because of convenient use of INTERNET between the authors in the working process. This fact allows us to take into account the real needs of students and teachers and to make the experience of each strong teacher known to all others.
We want also to stress that we do not think that the computer will replace teachers willing to work with talented students preparing them to competitions; we are sure that it is providing substantial help to such teachers.
The other effect of INTERNET on mathematical competitions in Latvia is the widening of math competitions by correspondence from newspapers also to WEB. We are able now to publish solutions in much more detail, to make suitable references to WEB which can be used by the reader at once, etc. As the result, the number of participants of these competitions has increased for appr. 40%. Of course, publishing the problems and solutions in newspapers continues also.
At present we are working on introducing new kinds of problems into the " curricula " of math competitions by correspondence- e.g., problems in which the interaction with a database is assumed.
In the period of using INTERNET in math competition system in Latvia the number of participants of the competitions and also the quality of papers have increased. So we are sure that this practice should be developed further.
In this paper, some of the major mathematical competitions being organised all over the world are described with special reference to India. The scheme is as follows:
In conclusion, some suggestions for strengthening mathematical competitions in India are given.
The International Scene
The idea of mathematical competitions originated in Hungary in 1894. The example of Hungary was followed by USSR, USA, UK, Netherlands, South Africa and several other countries. The IMOs started in 1959 in Romania. Other than IMOs, there are several multinational competitions in which several groups of countries participate. For example Nordic MOs, Balkan MOs, Asia-Pacific MOs, Ibero-American Olympiad, and Baltic Way Contest. We have described some of the major mathematical competitions being organised in some of the countries of the world.
This will be a presentation of the American Mathematics Contests (AMC 8, AMC 10/12), the American Invitational Mathematics Examination (AIME), and the USA Mathematical Olympiad (USAMO). Many examples and hand-outs will be provided.
The Third Junior Balkan Mathematics Olympiad took place in Plovdiv, Bulgaria, from 23 till 27 June 1999. The first issue of the Balkan Olympiad held in Belgrade, Yugoslavia, on June 1997. The member countries are: Albania, Bulgaria, Cyprus, Former Yugoslav Republic of Macedonia, Greece, Republic of Moldova, Romania, Turkey and Yugoslavia.
Each team consists of five 14-year-old students. The aims of the Olympiad include:
The competition held in one day. Students were given 4 problems for 4 hours and 30 minutes. The problems covered the following topics: Algebra, Number Theory, Geometry, Combinatorics (on elementary level).
Each problem was scored of 50 points maximum. During the presentation I will show the problems and will comment them from mathematical and didactical point of view.
The final results were the following:
Bulgaria 191 points Former Yugoslav Republic of Macedonia 152 points Romania 150 points Yugoslavia 148 points Turkey 140 points Republic of Moldova 121 points Greece 112 points Cyprus 76 points Albania 53 points
At the Iberoamerican Math Olympiad of 1997 was proposed by Spain a geometrical problem, inspired by a non-solved problem, more general, published at the Romanian journal Gazeta matematica. At my presentation, two solutions of this more general problem will be presented.
The aim of this paper is to report on the mathematical competitions climate in Romania during the 20th century, the most important period in the history, culminating with the extraordinary idea of organizing the IMO and with the effective organization of the first two IMO's in Romania.
The results of the Romanian IMO team are also presented.
The presentation consists of 7 paragraphs
In the paper a famous and important group, named KOVB, of six geometric transformations is considered.
The ISOGONAL CONJUGATES, INVERSION, RECIPROCATION, FIRST and SECOND PEDAL TRANSFORMATIONS together with the IDENTITY form a group.
Some corollaries for everyone of the elements of the group KOVB are proved and some characteristic examples are given.
Five THEOREMS about : THE EXISTENCE OF THE GROUP KOVB WITH THE BINARY OPERATION "PRODUCT", ITS GENERATING ELEMENTS, SUBGROUPS, INVARIANTS, NORMAL SUBGROUP OF GENERATING ELEMENTS and about THE THIRD DEGREE OF THE FIRST PEDAL TRANSFORMATION are proved.
For the illustration of the power of the elements of the group KOVB some NICE COMPETITIONAL PROBLEMS are given and solved.
At the end it can be underlined the the paper :
In this paper is solved the following problem :
"LET ABC BE AN ARBITRARY TRIANGLE WITH SIDES a, b, c. IF F(ABC) IS THE AREA OF THE TRIANGLE ABC AND F(OIH) IS THE AREA OF THE TRIANGLE WITH VERTICES THE CENTRES OF THE CIRCUMCENTRE, INCENTRE AND ORTHOCENTRE, THEN PROVE THAT :
16 F(ABC).F(OIH)= |(a+b+c)(a- b)(b-c)(c-a)| ."
The solution of the problem is interesting not only with its smart complicated algebraic calculations and its connection with "THE FUNDAMENTAL TRIANGLE INEQUALITY" but with the given possibility for obtaining new results in the field of geometric inequalities by using the method of transformations.
Some nice examples of competitional problems are given.
Island School (HK) is a coeducational comprehensive school that follows the national curriculum of the UK. The breadth of the curriculum has inevitably made the content of the mathematics syllabus simpler than its counterpart in HK. As a result, the Island School mathematics team had found the competitions against local HK schools most challenging. In spring 1999, regular training classes were first run after school to prepare the team members for these competitions. Since then the school won the Inter-school Mathematics Contest twice in a row.
In the course of these two years, a number of difficulties were encountered. First of all, the mobility of students has made it difficult for the team to maintain a consistent standard. Secondly, the mathematics training classes have to compete against over 100 other extra-curricular activities offered by the school, the tuition classes of students arranged by their parents and other temptations available in the metropolitan city of HK. Thirdly, the extra time invested on learning and practising more advanced mathematical techniques is challenged by the need to diversify students’ interests and experience. Fourthly, the abstract concepts that are tested by competition questions are usually isolated from the daily experience of students who cannot even see the connection between these concepts and their schoolwork. Fifthly, due to the location of HK, there are lots of training resources written in Chinese. But the corresponding resources are not usually available in the English language. Sixthly, the induction of junior members of the team has slowed down the progress of the senior members. This has inevitably reduced the motivation of some students.
In a nutshell, training for mathematics competitions is a time-consuming exercise that serves only a minority of students but is certainly a rewarding experience.
![[John Dowsey]](dowsey.jpg)
John Dowsey
The Mathematics Challenge for Young Australians seeks to foster mathematically talented youngsters in years 5 to 10 and encourage their continuing involvement with mathematics. It aims to develop their mathematical skills and knowledge in general and in some cases to the point where they may become candidates for the IMO.
Many of the problems have been inspired by known research problems; others have led to extension material which in some cases have no known solution. In this paper, we will discuss some of these problems.
Mathematics competitions are not an end in themselves. Their justification at school level lies in what they have to contribute to mathematics, to the education of those who take part, and to the support of mathematics in schools. This means that those involved with mathematics competitions have to be permanently sensitive to the impact of the events they administer. In particular, one might reasonably expect a flourishing national mathematics competition scene to have an impact on the number of students studying mathematics at undergraduate and postgraduate levels. We consider some positive and some negative evidence on this front.
Purpose and Meaning
With the great upsurge of culture and knowledge studying keeping rising, more and more competitions or contests have been appearing in all fields since the reform and opening in China. In order to succeed in contests, quite a few schools give groups of selected excellent students some special training. However, these students are burdened not only with much more school work but also with much greater pressure from schools, families and the society than common students. And this undoubtedly has an effect on their mental health. This text tries to inquire the anxiety of some students in mathematics contests so as to collect some materials and data for the research of students' mental health problems and also to make schools' psychological consulting more comprehensive and purposive.
Subjects and Methods
Subjects
The students who took part in China's 14th Mathematics Winter Camp in 1999 have been chosen to take a test. 123 students of them are boys and 13 are girls.
Materials
Mental Health Test (MHT), which was revised by the Department of Psychology, South China Normal University, is adopted in the research. The predecessor of this book is Uneasiness Tendency Test (UTT) written by ??? and others of Japan. This mental scale includes eight anxiety tendencies such as study anxiety, anxiety towards others, solitude, self-blame, allergy, health symptoms, terror and impulse, and it also includes 8 anxiety scales and I reliability scale. In this inquiry, 138 questionnaires were given out, and valid papers are 127, of which 114 are from boys and 13 from girls.
Methods
After the Winter Camp Mathematics Contest ended on January 14th, 1999, we sent questionnaires directly to the tests on February 1st and began to count up the result on February 25th.
In order to do some comparative research, we chose an ordinary school named No.4 High School in Lengshui Tan, Hunan Province. In this school, we chose 125 boys and 15 girls at will from senior 3 and gave them the same scale questionnaires to have a test. Finally we collected 118 valid questionnaires from boys and 14 from girls, adding up to 132 in all.
We marked the questionnaires of the two different kinds of students by the same standard and made qualitative analysis.
Results
Number Comparison between Two Different Kinds of Students in anxiety
Comparing the number proportion of the high-mark students in anxiety, we found obvious difference in anxiety-troubling number proportion between two different kinds of students. See Table 1.
Type N Number Proportion Z P
in Anxiety
Competitors 127 41 31% 3.4 P<0.01
Common Students 132 23 17%
Table 1
Number Distribution of Different Kinds of Anxiety
There is no difference in anxiety number distribution between two different kinds of students. See Table 2. x2=11.64 df=14, p>0.05
Type Number Distribution of Anxiety
Study Anxiety Solitude Self- Allergy Health Terror Impulse
Anxiety Toward blame Symptoms
Others
Competitors 23 15 17 9 10 10 11 13
Common Students 11 8 8 7 7 8 7 9
Table 2
Average Score Comparison in Anxiety Distribution between Two Different Kinds of Students
Average score and standard deviation in 8 anxiety distribution scales of the two kinds of students can be seen in Table 3.
Item Type N X S Z P?
Study Competitors 127 7.612 3.84 3.51 p<0.01
Anxiety Common 132 5.723 3.04
Students
Anxiety Competitors 127 8.140 4.20 4.73 p<0.01
Towards Common 132 6.518 3.17
Others Students
Solitude Competitors 127 7.199 3.76 3.55 p<0.01
Common 132 5.327 3.07
Students
Self- Competitors 127 5.191 3.42 1.83 p>0.05
blame Common 132 4.452 3.23
Students
Allergy Competitors 127 4.967 3.40 1.12 p>0.05
Common 132 4.331 3.27
Students
Health Competitors 127 4.101 3.47 1.47 p>0.05
Symptons Common 132 3.201 3.01
Students
Terror Competitors 127 5.671 3.51 2.69 p<0.05
Common 132 3.477 3.05
Students
Impulse Competitors 127 5.982 3.76 2.77 p<0.05
Common 132 4.001 3.26
Students
Total Competitors 127 48.863 3.76 3.13 p<0.01
Score of Common 132 36.030 3.14
Anxiety Students
Table 3
From Table 3, we can find that between the students of two kinds there is obvious difference in average score among study anxiety, anxiety towards others and solitude. There is some difference between terror and impulse and also much difference in total average score of anxiety.
In view of great disparity in proportion of boys to girls, there is little value in comparative research into anxiety of boys and girls. Therefore, we don't make any statistical analysis about it in this text.
From Table 1, of the common students 17% possess the serious problems of anxiety, while of the competitors 31% possess so, which shows that the mental health problems are serious of both the common students and the competitors, esp. the latter. Such thing should arouse much attention of the educational departments of all levels.
General inquiries reveal that the great anxiety towards learning(ATL) of the middle school students exists widely. Not only ATL, but also anxiety towards others the solitude and impulse can be shown in Table 2 and 3.
The learning tasks of the competitors outside class are much harder than those inside class, such as plane geometry, combinatorial mathematics and number theory, which is very hard for them to cope with. They have to take great pains to deal with the hard tasks in order to meet the needs of their parents to recommend them to enter into famous universities by means of winning the competition of mathematics of all kinds, which can also win the honour for their schools and coaches as well.
The successful competitors come to form the habit of indulging in self-admiration caring for themselves without co-operating with others or properly getting along with classmates, teachers and parents, because they always receive awards and rarely receive criticism even if they have shortcomings. As a result, solitude and anxiety were formed in their mind, hence the syndromes of headache, diarrhoea, cold sweat, vomit, sleeplessness and unconsciousness sometimes appear which are harmful to their health.
We only inquires 100 top students who are competent and manage to go in winter camp and they are only a few of thousands and thousands of student-competitors including in junior and senior middle schools as well as in primary schools. If our educational departments neglect the psychological health problems there will exist bad results. According to the author's long-time experience, therefore, the suggestions are offered for reference and given to policy-making departments concerned.
Making competitors increase interest and abilities as far as coaches are concerned.
Interest plays an very important role not only in modern educational theories . but also in the Olympic mathematics competition. However, a lot of coaches paid no attention to it. On the contrary, they usually taught the students to work out the extremely difficult mathematics in problems quite a strange and new way, so as to make them feel tired easily. And the tiredness or anxiety produced prevent them from giving full play to their mathematics knowledge or skill. Arousing the students' interest should be associated with their knowledge, competence and moderate degree of teaching content.
The Olympic mathematics represents number theory, combinatorial mathematics, inequality, function, polynomial, geometry and their poly-foundational knowledge and mathematics methods of thinking. Therefore, the common mathematics should be combined with competition mathematics, and the relationship between the two mathematics should be got along with quite well. In order to work out the competitive mathematics problems, the competitors must get rid of the limit from the old and fixed mould, surpass the limit by imitation, break the already-made frame of procedure and creatively use the thinking method of observation, analysis, induction, analogy, classification, procedure, conversion, research and construction. Furthermore, the competitors must develop the self-study habit; must study, research and develop all alone; must receive the modern mathematical and cultural education in advance for the purpose of either laying solid foundation for further study and development or assuring them of wisdom and competence.
Attention should be attached to the mental health of the students in all schools and super-departments concerned during the competition coaching.
In the past 20 years, some thousand kinds of competition books have been published and about 20 kinds of mathematical journals open the competition column. However, of so many kooks and journals which the students really have not enough time to go over, many are not suitable to use owing to non-system, unimprovement and the too-hard-to-understand contents. And it's very difficult for coaches and students to find out a suitable textbook for reference. Indeed, it's necessary for the departments concerned to call on the specialists to write about 2 or 3 kinds of books with scientific and practical contents in them for the textbook of competition coaching.
The nation associational competition examination (NACE) papers are set in turn by different provinces once a year, with the difficult extent of different years being greatly different. Sometimes, the papers set by some provinces are so difficult that the students' marks got from these paper are very low, which makes students feel uneasy to do during the competition coach. If the NACE papers or the winter camp competition examination(WCCE) papers are set by the departments concerned all alone just like the examination for entering colleges, with a suitable degree of difficulty or easiness in them, the students may have a definite aim in their mind and feel easy during the competition coach.
At present, the NACE and the WCCE are symbols for successful competitors to be recommended into some famous universities, the more successful the competitors, the more famous universities or colleges they will enter. For achieving the profitable aim, the student competitors, take an active part in the coach even if in the bad condition of their health, which is really against the initial aim of the mathematics competition. The competitions should not linked with entering college or university, but should be the steps towards the scientific palace for the youngsters, and as a result, to find out and then to train the hopeful scholars and youngsters with brilliantly scientific prospects.
Quite a few schools run many mathematics competition training classes during which attention should be paid to the students' abilities to put up with the courses, not teaching too much to the students but giving them pressure-relaxing activity training such as Spring outseeing or Autumn outing, social practice activities, sports games and recreational activities, in which they can communicate with each other and build up friendship better, at last attaining the aim of opening their mind or of moulding their temperament. All this should be done by special psychological teachers so as to promote the students to grow up smoothly psychologically.
Trying to cause their sons (the competitors) to develop very well in an all-round way so far as the parents are concerned.
Most competitors or students spend a great deal of time in mathematics competitions and less time on other subjects only for the purpose of recommending them into famous universities if their competition achievements are wonderful; and they would otherwise fail to be entitled to the universities, in which they would disappoint. Even if they managed to be entitled to recommend them into the famous universities through attaining great achievements in the competition, they would meet with difficulties in some other subjects on which they spent deficient time.
The substantial content of the Olympic competition possesses the qualities of advancement and development, with its connotation of mathematics being profound and modern as well as being educational function. The competitions aim at arousing the students to increase the strong interest of science, to develop and widen their knowledge areas, to enlarge their mental abilities or intelligence and to keep on researching and finding out scientific truth in spite of setbacks. For this reason, the competitors and their parents should attach great importance not to the result but to the procedures of the competition.
The history of the competitions together with the main characteristics will be presented.
The problems from the last edition (Rousse, February 2000) of the Mathematics Competition for high-school students from 8-11 grades will be presented.
The areas, ideas, difficulties of the given problems, the personalities of the proposers and their messages for today and for the future are discussed.
Some real statistics shows the current level of the mathematics abilities and knowledge expected from the participants.
In this paper we present information about the hobbies and career aspirations of high school students who have excelled in mathematics at school by winning a medal in the Australian Mathematics Competition [AMC]. In particular, we examine whether career aspirations have changed over the more than two decades since the AMC was introduced, to what extent the career intentions of younger and older medallists differ or overlap, and whether the hobbies and leisure time activities of this group of exceptionally able young mathematicians is narrowly linked to mathematics and related activities or representative of a broader range of interests.
Note: Unfortunately Mrs Mukherjee can now longer attend ICME and this talk will not be given, but we are leaving the abstract available.
Mathematics competitions have become an important component of curriculum innovations. Sometimes competitions are confined within the school boundary, some are held at regional or at state level, yet there are quite a few conducted at national level . A number of international events in mathematics have become quite popular among the secondary and senior secondary students. Diagnosis of a child's mathematical ability and aptitude is the objective of some competitions while talents are searched by some , some competitions are held to hammer the intellect of both gifted and ordinary student and maximise their mathematical potential . May what be the objective of the respective competitions , they indeed contribute a great extent to a child's mathematical learning , a teacher's enrichment of knowledge and allows a child an exposure which is otherwise not provided by the day to day school curriculum.
Creating motivation and a spirit of participation is the significant role played by a mathematics competition. A competition if so designed can also attract the slow learners or mathematically backward child or those with 'mathematics-phobia' in the classroom. Experience reveals that 'out-of-school' competitions sometimes acted as a joyful experience for them and created motivation for them to do extra learning in mathematics beyond the school syllabus. Probably 'team' based participation plays the key role in helping them getting over their usual complex they show in the classroom. There are lot of encouragement for those who conduct these competitions as both the parents and other members of the society provide whole hearted support for such ventures. But there are some constraints for which the goal of this competition cannot be fully achieved . Such problems are discussed in this paper.The author expects that through exchange of ideas with people involved in mathematics contests all over the globe, effective modifications will be possible. And a day will come when mathematics competitions will be able to create spontaneous motivation for learning for not only the gifted or talented but the 'weaker ones' also.
Australian students who enter the Australian Mathematics Competition are voluntary entrants and in that sense are "self-selected".
This paper investigates whether the students who enter from each school are the most able, mathematically. Data from 60 schools in New South Wales from 1994 to 1999 were analysed, the variable being the number of questions correctly answered out of 30 (the students' "score"). Twelve analyses were carried out, corresponding to the 12 combinations of gender with school year (years 7 to 12), with the mean score for a school as the dependent variable.
The estimate of the change in the school mean resulting from a ten-fold increase in the number of entries from a school from one year to the next ranged from -3.1 and -1.6. A theoretical analysis, based on the assumptions that the score is a normally distributed variable and that self-selection is upper truncation selection, produced an expected change in the mean from a ten-fold increase in the number of entries from a school of approximately -5.7.
These results are in agreement with the notion that the students who enter tend to be the most able, a factor which would need to be taken into account when comparing school or gender means across schools and/or across years.
In an earlier paper it was noted that over the 10 year period to 1992 performance differences between females and males had decreased. In this study it is shown that, for the period 1992 to 1999, the performance differences between males and females have continued to decreased but at a slower rate than in the earlier study.
For three decades mathematical fights are very popular in Russia's schools with mathematical bias. A mathematical fight is a competition between two teams in solving non-standard difficult problems selected by the jury, in oral explaining the solutions, in checking the solutions of the other team. Mathematical fights may be conducted between several teams as tournaments. The experience shows that mathematical fights (between classes of one school, between different schools and even between cities) are very efficient way of developing critical and inquiring mathematical thinking, of fostering and polishing mathematical talents, especially in problem solving and posing.
General description
Mathematical fights have been conducted in Soviet mathematical schools for more than three decades. I. Ya. Verebeychik, a mathematics teacher of school No. 30 in Leningrad, invented and organized them in his school since 1965. They became popular all over Russia since the beginning of 70-s, when the report about the mathematical fight that took place in Chelyabinsk during the All-Union Mathematical Olympiad, appeared in the ěKvantî magazine (Fedotov, 1972). Almost two decades later, a more detailed account of the rules was published in the journal ěMatematika v Shkoleî (Deryagin et al., 1990).
A mathematical fight is a competition between two teams in solving non-standard difficult problems selected by the jury, in oral explaining the solutions, in checking the solutions of the other team. It consumes usually one dayís time (from the morning till the evening). Mathematical fights have well-developed and rather complicated rules.
The rules
Two teams consisting of equal number of members (usually from 6 to 12), receive from the jury a set of problems (the number of problems is approximately the same as the number of members in a team) and try to solve them sitting in different rooms for some time (usually from 2 to 5 hours, sometimes much less). After the lunch, the main fight begins. First, two captains of the teams compete in answering a mathematical puzzle posed by the jury. The winner of the captainsí contest decides which team will make the first move. Then the chosen team makes the first move (moves in mathematical fights are called ěchallengesî), choosing one problem and challenging the second team to present a solution of it. Two cases are possible:
If the second team accepts the challenge, it must set one of its members to present the solution. In this case, the first team must set somebody to refute the solution found by the second team. If the second team rejects the challenge, the teams exchange their roles, the first team must present the solution of the problem chosen, and the second can refute the solution. In this case, if the first team does not itself have a solution and so refuses to present it, the challenge of the first team is declared incorrect. However, if he first team presents the solution and the second team proves it to be false, the challenge is also declared incorrect.
Each problem ěcostsî 12 points (thus, the difficulty of the problems should not be indicated). During one turn, the challenged team gets all 12 points if the solution presented is correct and full. On the other hand, the challenging team gets 6 points for the full and correct refutation. Moreover, in this case the roles also (as in the case of the rejection of the challenge by the second team) become inverted, and the first team, presenting its own solution, can get up to 6 additional points for their solution (the second team becoming the opponent). If the first team can not present own solution, the jury gets the rest of points (or the part of the rest if the solution is partial). During one turn, the roles can be inverted only once (in cases of the rejection or false solution by the second team).
After the first turn, if the challenge was admitted to be correct, the second team makes the move with another problem and everything is repeated in similar way.
In some moment, one of the teams may admit that it does not have solutions of the problems left. In this case, it may give up challenging. In case of such refusal, the other team has the right to present all other solutions it has got. The team that gave up challenging has the right only to refute. It gets 6 points if the other team refuses to present its solutions.
The fight finishes when there are no more undiscussed problems, or one of the teams gives up challenging in some moment and the other refuses to present the solutions of the rest of the problems. The jury sums the points of each team and names the winner. Usually, if the difference between suns of points does not exceed 3, the fight ends in a draw (neither side wins). If there is time left, the jury demonstrates the solutions of problems not solved by the participants.
Mathematical fights have some peculiarities that distinguish them from usual mathematical competitions.
Peculiarities
Conclusions
The experience of participating in mathematical fights will help participants in the future: they are acquiring the skills to present a scientific report, to listen to and understand the work of colleagues, to pose pertinent and clear questions ń all these skills will be useful for participating in conferences and seminars, for reviewing articles and monographs, for joint scientific work (Deryagin et al., 1990, p. 56).
In our Pedagogical University, we have organized a tournament of mathematical fights between eight student groups (classes) of various years of study. Participants were extremely excited about this opportunity to test and train their abilities to solve difficult entertaining problems. The hidden talents of (mostly male) students, usually not getting good marks at the examinations in regular courses, have been revealed. After this tournament, our students, who are pre-service mathematics teachers, organized mathematical fights during their practice at schools. It is important because currently, there are very few school teachers able to conduct mathematical fights in particular and to promote creative mathematical thinking of their pupils in general.
Appendix: Examples of problems.
For a captains' contest:
It is known that the fraction equals to an integer number. Different characters denote different digits, and between them the signs of multiplication are placed. What is the numeric value of the fraction?
20 matches are placed on the table. Each of two captains may take one or two matches in one turn. The person who takes the last match wins.
For a fight:
References
In this paper the importance of geometrical inequalities in acquiring deep knowledge and understanding of geometry as a whole is stressed. Examples of experience gained in teaching mathematics to 14 to 16 year old pupils are presented. Some examples connected with teaching geometry to the gifted students are also given.
Inscribe in a unit square r squares, which have no interior points in common. Denote by f(r) the maximum of the sum of the side lengths of the r squares. (We allow side lengths of squares to be zero.) The problem is to evaluate the function f(r): For every positive integer r find the value of f(r).
In 1932 Paul Erdös formulated the following $50 conjecture that has so far been proven only for k=1: For any positive integer k, f(k2+1)=k. From this conjecture, I will take you on a journey through other problems, conjectures and results.
The use of mathematical competitions for raising interest for the subject and for recognising mathematically gifted young people seems togo back about 100 years only. - interest for mathematics is also raised by the mathematical periodicals for school childrens (where such publications exist), by their articles and the problems to solve, whose solutions, whether there is one or there are more, are published in later issues.
In fact, problem solving is one though not the only important feature of mathematical activities. Many other abilities are required to succeed in contests, e.g. good nerves. Not all gifted children are good at competitions and failures may discourage them. One should avoid discoragement but this is no indication against contests. Discouragement is a disadvantage , which, however is outweighed by the pleasure of problem solving. Whoever discovers its beauty enjoys it more and more. Problem solving as discribed in Polya's "How to solve it" creates an attitude which is not easily lost and is helpful in many situatins other than mathematical.
Sometimes teachers are mistakenly assessed by their students' success at competitions.
Mathematization consists of different components like the ability to abstract; to formulate mathematical ideas orally and in writing; the ability to apply one's knowledge; to find mathematical connections; to recognise mathematical connectins of a situation; to recognise the structural build-up of mathematics; and also the ability of problem solving. The development of all these and similar features requires sound preparation and attention from the teacher - and a social and financial reward for teachers.
People often find it chic not to like mathematics, even to understand nearly nothing of it. Mathematicians are sometimes characterized as unpractical, awkward persons. Quiz games organized repeteadly by the Hungarian television during the 60's on different subjects (among them on mathematics) proved surprisingly useful also in reducing this general repugnance to mathematics. People who hardly understandeven the question, not mentioned the solutions, watched watched the contesting paires of students with great excitement. Seing that normal, lively children, far from the "dry-as-dust" type have found the solution in 2-3 minutes, this people have lost their proudness for being ignorant to mathematics. [1] (Unfortunately, these quiz games have been discontinued ever since.)
Mathematical competitions organized yearly partly for raising interest in mathematics and partly for recognizing mathematically gifted young people, date back for somewhat more than 100 years [2]. According to Freudenthal's report, the first one was the Hungarian Eoetvoes, later Kürschák competition, first held in 1894, and becoming widely known outside of Hungary as well [3].
Contests improve the capacity of problem solving, and whoever recognized the pleasure of it and discovered its beauty enjoys it more and more.
Pólya writes in the introduction of his excellent book "How to solve it?" [4]: "A great discovery solves a great problem but there is a grain of discovery in the the solution of any problem. Your problem may be modest; but if it challenges your curiosity and brings into play inventive faculties, and you solve it by your own means, you experience the tension and enjoy the triumph of discovery. Such experiences at a susceptible age may create a taste for mental work and leave their imprint on mind and character for a lifetime."
Problem solving as described in in Pólya's book creates an attitude which is not easily lost and is helpful in many situations other than mathematical ones.
However, not all gifted children are good at competitions and mainly not all of them can become winners. Good contest results also depend on good nerves, quick reaction time and other psychological components besides talent, but failure can discourage the participants not only from competing but from mathematics at all.
Problem solving is an essential component of mathematization but far not the only one. Moreover, emphasis is given to some topics appropriate for good contest problems and for some types of problems, whereas other fields do not fit to competitions. These are, however, only warnings which must be taken attenttively into consideration in order that competitions play their beneficial role. It is a difficult task for the teacher to compensate for this effect.
In the case of two-round competitions, the problem of discouragement can be essentially lessened by giving more (say 8) problems in the first round among them sufficiently easy ones, but also others wich are hard enough to select the partcipants for the second round. It is not necessary to solve all problems, the participants can choose to some extent according to their interest.
An overgrowth of competitions - and we see this symptome in Hungary - and generally an overfeeding of talented youngsters with mathematics can turn them in disgust towords their first interest and this danger must not be neglected.
The emerging different difficulties desire skills and efforts from the teacher. Without going in details, I will illustrate this by an example and a counterexample that traditional teaching metodes must be changed too [5].
By traditional teaching, the teacher gives the definition, supports it by by some examples, rquires the pupils to repeat it and believes they kept the notion.
Let us take the natural numbers as results of counting. Children use it seemingly well, add and subtract in the range of natural numbers. However, an experiment shows that the situatin is not completely encouraging. (I must admit I was sceptical initially wether the experiment will give difficulties to the students at all.) Five match-boxes were set on a table, and before each one a coin was placed. This was shown to 7-8 year old children, one at a time, and the question was asked, were there more boxes or more boxes or more coins. - Naturally the answer was: the same. Aferwards the boxes were drawn apart before the pupil while the coins remained unmoved, and a great part of the children told that there were more boxes. some of them tried even to guesshow much more.
Similarly, children often belive that the space object occupy influences the number of them.
An example in the opposite direction shows how childreen develop real notions. First form pupils have discovered negative numbers while practising addition and subtraction mounting and descending on stairs. They numbered each step and as stairs led to the cellar called the new numbers cellar numbers and denoted them by c1, c2, etc. They have long conserved this notation. They excersized addition and subtraction in this domain long before the syllabus prscribed it and the usual notation became introduced, which reminded them to the subtraction sign, where "cellar numbers" came from.
Test-type competitions became a wordwide movement nowadays, which is to my mind a completely abortive attempt. The short time left for the solution of one problem does not give the possibility for what the major aim and benefit of these competitions shoud be: creative thinking, instead it reduces the competition to a game of trial and error.
Nowadays more and more disciplines demand different domains of mathematics, which threatens with a fear that that the use of mathematics is reduced to a formal introduction of procedures. It would be necessary instead to train pupils to read cleverly mathematical texts which would enable them to aquire later even knowledge not yet taught at present. (for more details see [5].)
Demanding teachers can also supplement their collection of exercises whith prpoblems and types of problems set on competitions.
Competitions must be and are independent from the teaching practice inthe sense that their results should not influence the marks obtained at school. This is however, not the case by the assessment of the efficiency of the teacher's work by some headmasters and even by the teacher himself. Moreover, having the children attend mathematics competitions cannot replace the teachers work and demand to himself at the classroom: a low level of of mathematics teaching seldom produces good mathematicians and good competitors.
As for sufficintly well prepared teachers, an additional danger is that it is not difficult at all, or can even be comfortable for them to train only the best students and pay not much attention to the rest. The oposite case occurs, too: the teacher does not inform his pupils about competitions and student journals for for mathematics, because student raise difficult problems, and to answer these questions gives him much work and trouble.
I mentioned several problems which teacher have to solve and to overcome besides their everyday work: preparation for the lessons considering also the difficulties the pupils can have in understanding, correcting their papers, etc. All this assumes thorough grounding and much attention from the teacher. This would demand appreciation, social as well as material, which is often lacking.
Significant tools for out-of-school mathematical education are journals of maathematics for students - if and where such periodicals exist. they arouse interest for mathematics very well. In Hungary, the first journal of this type has been published regularly (10 issues a year) since 1896 whith some breaks due to the two World Wars. After the Second World War I was a long time Editor of this periodical myself.
The articals of the journal give a broad view of old and new fields of mathematics. It also publishes problems to solve for different age-groups and the solutions (different ones if several solutions exist) are published in later issues This also helps the preparation for contests.
The collection of the problems of the journal appeared from time to time. The contest problems of the Eoetvoes and later Kürschák competition were published in 4 volumes already - and some of them in foreign laguages too - with solutions and with comments which partly shed light upon the background of some problems, and partly expound some fields of mathematics in connection with the problem.
I returned repeatedly to different tasks and difficulties of the teacher. To supplement their work, methodological periodicals for mathematics teachers are also published. They contain besides articles from experts and teachers on different mathematical and didactic questions, problems to solve as well. (Solutions by pupils are not accepted here.)
Mathematical competitions are also organized in Hungary for university students. They have 10 days for home work on usually 10 problems from very different branches of mathematics [6]. Not even the suspicion of consultation with others (either colleagues or experts) emerged since 1950. Here, however, work of secondary school students is accepted too, and from time to time, some of them also obtain prizes.
![[Nob]](nob.jpg)
Nob Yoshigahara (right) with Andy Liu at a conference in Miami in early 2000
On a transparent sheet, draw a development of a cube as shown (I apologise for not showing the diagram here).
Paint the black part with opaque black, or past black paper. (Or copy this on OHP sheet.) On all 17 holes should be covered with polarized sheet. The "diagonal direction" of polarization is shown as a hatch.
Make a cube. But do not forget to put a square sheet (also diagonally polarized) inside. Its size is a little smaller than the cube. That's all.
Shake it well and throw it on OHP. The probability is 1/6 each face.
Ambushes: