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Materiāla nosaukums Autors
1993./94. m.g. matemātikas olimpiāžu uzdevumi ar atrisinājumiem Andžāns Agnis,Ambainis Andris,France Inga
1994./95. m.g. matemātikas olimpiāžu uzdevumi ar atrisinājumiem Andžāns Agnis,Kluša Julita
1995./96. m.g. matemātikas olimpiāžu uzdevumi ar atrisinājumiem. Andžāns Agnis,Kluša Julita
1999./2000.m.g. Latvijas matemātikas olimpiāžu un konkursu uzdevumi un atrisinājumi pamatskolām Andžāns Agnis,Čerāne Aelita
2000./2001. m.g. matemātikas olimpiāžu uzdevumi 9.–12. klasei līdz ar ievaduzdevumiem un atrisinājumiem Andžāns Agnis,Broka Elvīra,Cauka Anita,Ramāna Līga
Activities and programs for gifted students Autoru kolektīvs


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Kategorija: Matemātika
Materiāla nosaukums: Activities and programs for gifted students
Autors: Autoru kolektīvs
Formāts: Online (tiešsaistes) materiāls,FTP servera fails
Datums: 2004


www.cmeegs3.rousse.bg; , www.icme-10.com, www.icme-10.dk




    • EDWARD BARBEAU, Department of Mathematics, University of Toronto, Toronto, Canada
    • HYUNYONG SHIN, Department of Mathematics Education, Korea National University of Education, Korea
    • EMILIYA VELIKOVA, Department of Algebra and Geometry, Centre of Applied Mathematics and Informatics, Faculty of Education, University of Rousse, Rousse, Bulgaria
    • ALEX FRIEDLANDER, Weizmann Institute of Science, Rehovot, Israel
    • SHAILESH SHIRALI, Rishi Valley School, Rishi Valley, India
    • AGNIS ANDŽĀNS, University of Latvia, Riga, Latvia


It is my great privilege and honor to greet the participants of the 10th International Congress on Mathematical Education. Gifted students of mathematics today are not simply the gift of Nature neither unpredictable prodigies. They are trainees of concrete scholars. They are coming together from big and not so big states however always they are coming from regions with developed schools of science tradition and math education.

As rector of the University of Latvia I am proud that scholars and students of our country are able to present our local culture and especially that of math education in this forum. In Latvia the Extramural School of mathematics of the University as well the School of Professor “Cipher” is well known for students starting from the age of 12, and has the history of more than 30 years. In the Open Contest of math 2004 e.g., they were able to collect more than 4000 participants. It is great, that today always running between too railway stations, two TV serials or even two sandwiches we are reasoning how to stimulate ability of not traditional thinking or thinking at all. Scholars and teachers must come together for shearing experience of best approaches how to select the talent and how to stimulate its growth.

Today we are not allowed to left teachers alone and scholars of the university must be in close relationship with them. We must work together and as close as possible.

On behalf of the University of Latvia, Riga I wish you productive meeting and number of difficult math problems for tomorrow!

Rector of the University of Latvia:

Professor Ivars Lācis, PhD


As the Rector of the University of Rousse it is my honor to congratulate the participants of the 10th International Congress on Mathematical Education, attending the activities of Topic Study Group 4: Activities and Programs for Gifted Students.

The University is currently offering 32 different bachelor courses in the field of engineering, management, law, teaching and informatics. In the Faculty of Education are running four of them. Master courses devoted to preparing high level teacher trainers are included in the teaching program of the Faculty of Education as well.

Throughout recorded history and undoubtedly even before records were kept, people have always been interested in persons who have displayed superior abilities.

In nowadays the encroachment of new information technologies and the continuous integration of mathematics with other sciences have a great need of:

    • gifted mathematicians able to discover and summarize diverse conclusions within the information flow and to generate new ideas;
    • personalities able to develop their creative potential, enrich both their knowledge and experience and apply them in socially useful areas and activities.
Those factors and the conditions of contemporary global problems define that the society needs new models of educational and upbringing activities. Based on the positive experience they ensure opportunities for developing gifted and talented students not as consumers or destroyers but as creators of knowledge and future builders of a peaceful world.

From this point of view the TSG4 and many of the Congress activities are aimed to dive further enlightment of these important problems.

At the University of Rousse work many holders of awards of national and International mathematics competitions and some of them are the creators in Rousse of one of the earliest schools in Bulgaria for training of gifted and talented students in mathematics and informatics.

I sincerely wish you successful and beneficial participation at the Topic Study Group 4: Activities and Programs for Gifted Students!

Rector of the University of Rousse:

Assoc. Prof. Eng. Marko Todorov, Ph.D.



Alex Friedlander

Abstract: I would like to assert here that regular mathematics classes, with a heterogeneous student population, can provide a suitable learning environment for higher-ability students.I will give several examples of tasks, and responses provided by higher-ability students in the context of two projects. Next, I will discuss some characteristics of these tasks and their potential for advancing the learning of mathematics by these students.


Mathematics educators have presented various models designed to cater to the needs of high-ability students in mathematics. Most of these models (for example, enrichment lessons, extracurricular math clubs or teams, accelerated or advanced math classes or special schools,) are based on a homogeneous population of high-ability students. However, for a wide variety of reasons, most of the relevant student population are not reached and hence, do not participate in any of these learning environments.

I would like to assert here that regular mathematics classes, with a heterogeneous population, can also provide a suitable, and frequently, even an effective learning environment for higher-ability students. I will attempt to support this claim by presenting and discussing several episodes from two learning projects, basically aimed at a general student population.

The Compu-Math project created a technologically based learning environment that systematically covers the entire mathematics syllabus for grades 7-9. As described by Hershkowitz and her colleagues (2002), the project is based on the following principles:

    • Investigation of open problem situations;
    • Work in small heterogeneous groups, where the problem is investigated and discussed;
    • Consolidation of the mathematical concepts and processes that arose in group work, and were discussed by the wholeclass;
    • Investigations that utilize computerized tools to facilitate operations within and between various mathematical representations, to reduce the load of formal algorithmic work, to enable the construction of mathematical concepts and processes, to provide feedback regarding hypotheses and solution strategies, and to resolve a real need to explain important processes and products;
    • Interactions between students in a group or in the class as a whole, between students and computerized tools, and between students and the teacher;
    • Reflections on the learning process;
    • Development of students’ mathematical language and their ability to provide convincing arguments.
The second project – Discoveries, presents a collection of activities for grades 1-6. The activities were originally designed for higher-ability math students, but the classroom implementation demonstrated that the activities provide good learning opportunities for a wider range of student abilities.

First, I will give several examples of tasks presented in the context of these two projects and the responses to these tasks provided by high-ability students. Next, I will discuss some characteristics of these tasks and their potential for advancing the learning of mathematics by high-ability students.


Example 1. The sum of a polygon’s exterior angles (Grade 9).

The issue of exterior angles was raised by the teacher after investigating of the sum of an n-sided polygon’s interior angles. Two (higher-ability) students presented the following arguments:

sum for the (n+1)-sided polygon is (n+1)· 180° - so that this transition adds 180° to the sum. We can see, however, that all these additional 180° were “taken up” by the interior angles – and that “leaves us nothing to add” to the exterior angles. Thus, the sum of the exterior angles does not increase and remains the same through the whole process of increasing the number of sides. At the initial stage of this process, we have the triangle, and the sum of its exterior angles is 3 · 180° (the sum of the interior and exterior angles) minus 180° (the sum of the interior angles) – that is 360°.

Example 2. Calendar patterns (Grade 7).

In an introductory course to algebra, the class found patterns in the dates of a monthly calendar and gave reasons for their generality. Next, the teacher asked her students to find and justify patterns in squares of dates (see Fig. 3).

The teacher was aware that an algebraic argumentation for the “diagonal products” required that two binomials be multiplied [(x + 1)(x + 7)], and therefore this was not within the reach of her students at this stage of the course. However, she asked whether somebody could justify this pattern. The students made various attempts to prove the pattern. Some provided numerical examples, and others gave incomplete algebraic explanations. The same student that raised the issue provided the following argument:

Let us start with the product x· (x + 8) = 

If we increase the first factor by 1, the product will grow by (x + 8).

Thus (x + 1)· (x + 8) =  + x + 8.

Now, if we take away 1 from the second factor, the product will decrease by (x + 1).

Thus (x + 1)· (x + 7) =  + x + 8 – x – 1.

So we see that this product is larger by 7 than the first one.

Example 3. Groups of matches (Grade 3)

The students had to put 21 matches into two boxes, so that the numbers of matches relate to each other in various ways. One of the conditions required that the number of matches in one box equals half of the matches in the second. Besides the trial and error method employed by many, two of the more talented students offered the following solutions:

Ben: "I took all twenty-one matches in my hand and put two on one side… and after each two on this side, I put one on the other side… two on one side and one on the other side…and so I continued and put on one side fourteen, and on the other seven."

Tomer: "I divided the 21 by 3 and got 3 times 7 … added up two sevens and got 14 and 7… and then checked."


I would like to discuss some common characteristics of the situations described above.

    • The tasks were taken from a longer sequence of questions that focused on the same problem situation. The tasks that belong to the same sequence were of various levels of cognitive difficulty, and were usually presented in an increasing order. The seventh graders, for example, were first required to look for other, simpler patterns in monthly calendar sheets in general, and in squares in particular.
The sequencing of tasks allows all students to contribute to the solution process. Our observations show that sequenced tasks provide effective learning opportunities for higher-ability students as well, and enable them to realize their potential. Their work in heterogeneous groups provides higher-level solutions to the more basic questions and has a leading role at the more advanced stages of the problem. Frequently, they are required to provide explanations to other group members, and this activity has the additional benefit of improving their communication skills and raising their own level of understanding.
    • The tasks were presented before, or at an early stage of learning a mathematical concept. The lack of the necessary tools did not allow for an algorithmic solution, or for the employment of previously encountered methods or formulas. The same task, however, can be solved routinely at later stages of learning. In our case, for example, the third graders were not familiar with the method of dividing a quantity into two parts, according to a given ratio, and the seventh graders did not learn how to multiply two binomials in order to prove the discovered pattern.
A lack of possibilities in applying an algorithmic or a routine solution raises the awareness of the need for learning a concept or a solution method, encourages students to employ higher-level thinking skills, to make connections to other domains and to be creative. A learning environment that does not nurture these skills is particularly disadvantageous for higher-ability students, and consequently creates a need to separate these students from the general student population.
    • Each task can be solved in various ways and at various mathematical levels. I presented here only a few solutions that were provided by higher-ability students. Many other solutions (some of them incomplete and others incorrect) were given for the same tasks. I presented here only those solutions that were particularly original or required an unusually high level of thinking for that particular age group. Frequently, the use of a numerical example as a prototype for a more general case or working by trial and error prove to be successful for solutions or explanations, but we consider them to be at a lower mathematical level.
My main purpose in presenting the three episodes, was to show that learning in heterogeneous classes can provide opportunities to employ higher-level thinking, and hence can be favourable for higher-ability students as well.

To conclude, I would like to emphasize that the tasks and the students described here constitute segments of a larger model. Moreover, the problems are a part of a cyclic learning model, where each cycle includes the following components:

    • an open problem situation, which is solved by informal methods, and increases the need for learning a particular concept or method,
    • consolidation of mathematical concepts and processes,
    • additional investigations of the problem or other applications,
    • exercises and tasks that involve the learned concepts and processes in a mathematical context.
Our classroom observations show that this model provides effective learning opportunities for students of various mathematical abilities. In the team investigation of open and unknown problems, there were authentic interactions among students, and these do not create unreasonably large gaps among students' contributions to the solution process. On the other hand, the examples described above indicate that in this learning environment, higher-ability students have the opportunity to realize their potential and to provide qualitatively better solutions. Moreover, these students raise the general level of their group's work by playing a larger role in the solution process of the more difficult parts of a problem, and by ensuring that all team members understand the solution of the task at hand.


1. Hershkowitz, R., Dreyfus, T., Ben-Zvi, D., Friedlander, A., Hadas, N., Resnick, T. & Tabach, M. Mathematics curriculum development for computerized environments: A designer-researcher-teacher-learner activity. In L.English (Ed.). Handbook of International Research in Mathematics Education (pp. 657-694). Mahwah, NJ: Lawrence Erlbaum, 2002.


Alex Friedlander

The Weizmann Institute of Science



E-Mail: Ntfried@wisemail.weizmann.ac.il



Alexander Soifer


New Olympiad problems occur to us in mysterious ways. This problem came to me one Summer morning of 2003 as I was reading a never published 1980s manuscript of a Ramsey Theory monograph, while sitting by a mountain lake in Bavarian Alps. It all started with my finding a hole in the manuscript, which prompted a construction of a counterexample (part b of the present problem). I do not think that this Olympiad problem would have been born without the manuscript’s authors allowing this minor mistake!

Chess 7x7 (21st Colorado Mathematical Olympiad, April 16, 2004, A.Soifer).

    a. Each member of two 7-member chess teams is to play once against each member of the opposing team. Prove that as soon as 22 games have been played, we can choose 4 players and seat them at a round table so that each pair of neighbors has already played.
    b. Prove that 22 is the best possible; i.e., after 21 games the result of (a) cannot be guaranteed.

I found three truly marvelous solutions, which are too long to fit in the marginal time allowed me in this section of the ICME-10 Congress. The good news is, you have thus an opportunity to hunt for a solution on your own. Good hunting!


Alexander Soifer

Princeton University, Mathematics, Fine Hall, Princeton, NJ 08544, USA

DIMACS, Rutgers University, Piscataway NJ, USA &

University of Colorado at Colorado Springs, USA








Alexandr Chumak, Vladimir Chumak


Solutions of main mathematical problems are based on using of algorithms. Contemporary computer technologies give the possibilities to create interactive algorithms for mathematics education.

We present basic and algorithmic didactic materials based on using the symbol- graphic language. They are compact, visual and free of the international barriers. The symbol-graphic language is developed for creation of these didactic materials.
The Word service abilities allow to make them interactive for using in the Internet technology. Basic didactic materials (BDM) contain symbol-graphic formulations of basic definitions, theorems, formulas from the instruction materials, which have been chosen for learning. BDM are used for algorithmic didactic materials creation. They contain algorithms of proofs of basic theorems and solutions of key problems. We suggest to develop three forms of these algorithms: Speed up, Linear, Forked algorithms.

Samples of such algorithms you may see on the Mathematical Education web site:

http://www.mycgiserver.com/~chumak/index.jsp (Main page/demo Testing/Algorithms).


Alexandr Chumak, Assoc. Prof., Ph.D.

Department of Mathematics

Kharkov National University of Radio Electronics

Lenin Avenue 14, Kharkov, 61166, UKRAINE

Tel.: +38 057 702 13 72

Fax : +38 057 702 10 13

E-mail: chumak@kture.kharkov.ua

Vladimir Chumak, Second year student

Artificial Intelligence Chair

Computer Science Faculty

Department of Mathematics

Kharkov National University of Radio Electronics

Lenin Avenue 14, Kharkov, 61166, UKRAINE

Tel.: +38 057 702 13 72

Fax : +38 057 702 10 13

E-mail: vchumak@ukr.ne




Anatolii Chasovskikh, Yury Shestopalov

Abstract: The contribution outlines the history of creation, activities, and main objectives of one of the leading educational enterprises in Russia for gifted students in the field of natural sciences (mathematics, physics, computer science, chemistry, and biology), The Advanced Education and Science Centre of the M.V. Lomonosov Moscow State University---The Kolmogorov college. The Centre was established in 1963 on Academician Andrei N. Kolmogorov initiative and from the very beginning was set up by the special Governmental decree as a boarding school. In 1988 the school was incorporated into the structure of the Moscow State University. More than 6000 pupils finished the College. Many students of the Kolmogorov college won international competitions in mathematics, physics and computer science. The College with enrollment of about 350 students and a two-year educational cycle has elaborated a unique system of testing and selecting gifted high-school pupils through all over the country.


On the 23rd of August 1963 the USSR Council of Ministers adopted a resolution “On establishing specialized boarding schools in the field of physics, mathematics, chemistry and biology”. According to this resolution four boarding schools were opened in Moscow, Novosibirsk, Leningrad and Kiev. The Moscow school was set up under Moscow University. Many outstanding Soviet scientists, physicists and mathematicians like Andrei N. Kolmogorov, Isaac K. Kikoin, Ivan. G. Petrovsky took an active part in its establishing.

In October 1988 by a Governmental decree the Advanced Education and Science Centre of the Moscow State University was set up. It consists of Moscow boarding school, Research Department and the teaching staff departments.


Nowadays AESC is one of the leading secondary educational establishments in Russia. Its main goal is to select and educate senior pupils from different regions of Russia, those who show special interest in the field of mathematics, physics, computer science, chemistry and biology. The educational process is conducted by the professors and teachers of the five departments: Mathematics, Physics, Computer Science, Chemistry, and Humanities. Now the problem of establishing a department of biology is being considered, the initiative was introduced by the faculty of bioinformatics and bioengineering.


According to the AESC Regulations the principal body that governs the Center is its Academic Council. The Head of the Council now is Moscow University Vice-Rector, professor Alexander V. Sidorovich. The Council consists of five Deans of Moscow University faculties, three members of Russian Academy of Sciences, corresponding members of Russian Academy of Sciences, professors and seven University faculties administration representatives.


The system of education at the Center is based upon lectures and seminars, it is very close to the system of Russian universities. When studying the Center students attend practical work classes and take credit tests and exams in the end of each term. For that reason the Center graduates can adapt to the student life quite easily. The curricula and education programmes are composed in such a way that they not only provide the necessary preparation for the future study at University but also (and that is the main thing) promote the development of student’s creativity in the future.


The influx of senior pupils who show disposition to studying science subjects is provided by the unique admittance system introduced by Kolmogorov school founders. The system still exists and is very effective. Every spring more than 40 Russia regional centers receive the Center Admittance Board representatives.


Besides the compulsory courses the pupils attend some special courses and take part in the work of special seminars. The Center’s students win the prizes of different science Olympiads of a very high level and deliver the reports at science conferences. Every May beginning from 2001 the Center organizes a Pupils science conference “Kolmogorov Readings”. The conference show the results of the different Russia regions pupils’ creative work in the field of science. III Kolmogorov Readings (2003) were devoted to the centenary of Andrei N. Kolmogorov. An intellectual competition of pupils, the young physicists tournament was founded at the Center. Now it’s a prestigious international competition, its 2003 final was held in Sweden, 22 best teams of all the world took part in its work. Some preparatory work is assumed for the tournament participation. This work is supposed to be connected with the solving of research problems, learning how to conduct a scientific discussion, learning English, etc. All that turns the pupils participation in the tournament into a continuous creative enthusiasm.


Among school teachers there are both very experienced teachers and very young scientists, Philosophy Doctors and Full Professors. Many of them also teach at Moscow University science faculties. The Center teachers are the authors of manuals, books, original methodical findings. For more than ten years the Center has been carrying out the research work in the field of the Russian language under the supervision of professor A. N. Kachalkin. Beginning from 2003 this direction of research has a financial support within the Federal Programme “The Russian Language”.

Department of computer science has worked out a new educational course “Mathematical background of computer science”. It is of an interdisciplinary character and is oriented for the pupils major in physics and mathematics of a secondary school. The course’s authors Associate professors E. Andreeva and I. Falina got a grant of the National foundation for the scientific personnel preparation. The grant was received on the competitive basis within the Programme of Russian Government.

The Center teachers have also contributed to the creation of multimedia manuals. One of the most popular of them is a book of problems in computer science. The use of this book allows to increase intensity of the educational process, to rise the teachers’ professional return and provides the pupils’ motivation as to the studying of the computer science problems. The inculcation of new computer science teaching methods at the Center and the usage of the scientific results give the pupils the opportunity to successfully take part in different competitions in this field of science. Thus in 2003 our pupils won 6 first prizes out of ten, among the victors were A. Lakhno and M. Ivanov. At All-Russia Computer Science Olympiad the Center’s pupils got one diploma of the first degree, four diplomas of the second degree and one diploma of the third degree. In 2003/2004 academic year the Center team has become Russia’s champion on programming.


Recently, special programs of cooperation and exchange were initiated on international level, in particular the agreement with the school for gifted children established in Pusan, in the Republic of Korea.


1. Kolmogorov v vospominanijah (Kolmogorov Memoires), A.N. Shirjaev, Ed., Moscow, Fizmatlit, 1993.

2. Javlenie cherezvychajnoe. Kniga o Kolmogorove (A book about Kolmogorov), Moscow, FAZIS, MIROS, 1999.

3. Est’ FMSh …: Sbornik fol’klornyh proizvedenij (There is a college: A collection of traditionals and lyrics), Moscow, Moscow University Press, 1995.

4. Sbornik statej ko dnju rozhdenija A.N. Kolmogorova (A collection of papers devoted to A.N. Kolmogorov’s birthday), Moscow, Nauchno-technicheskij centr “Universitetskij”, 2003.


Anatolii Chasovskikh

Yury Shestopalov




Andrejs Cibulis, Ilze France

Abstract: The paper deals with students` achievements in the investigation of polyforms, including the compatibility problem for polyominoes and polyiamonds as well as problems for tetratans. Attention is focused on a surprising result in constructing convex shapes from tetratans.

Key words: Polyomino, Polyiamond, Compatibility, Polytans, Tetratans


The problem of compatibility of polyforms is attractive, however, very difficult in general and it has been solved only in a few cases. This problem is a very good theme for the gifted pupils and students to carry out research. The author’s first findings on compatibility of pentominoes (a registered trademark of Solomon W. Golomb) were announced in the third congress of WFNMC (China, Zhong Shan, 1998). Having joined the efforts of several authors the work Polyomino number theory has appeared in three parts, see [1-3]. The classic reference book on polyominoes is [4].

Notions.Polyominoes are connected plane figures formed of joining unit squares edge to edge. A polyominoA is said to divide another figure B if B may be assembled from copies of A. We also say that A is divisor of B, B is divisible by A, and B is multiple of A . If two figures have a common multiple, they are said to becompatible. Aleast common multiple of two compatible figures is a common multiple with minimum area. Analogously when a square (being the generating element for polyominoes) is replaced by the equilateral triangle or regular hexagon we obtain polyiamonds and polyhexes respectively. As far as it is known, the first findings of hexiamonds’ compatibility were published in Rodolfo Kurchan’s “Puzzle Fun” [5].


They are briefly described in [6]. Some new information has been added in the further text.

Valdemārs Plociņš and Margarita Lukjanska (Latvia) investigated compatibility of hexominoes and n-iamonds (when n < 6) respectively in their papers for young scientists’ contests (2002, 2003). In 2004 the contest paper of Alīna Česnovicka was dedicated to compatibility of n-hexes (when n < 5). She has stated compatibility of 379 pairs of polyhexes by constructing common multiples of a small size. The largest ones consist only of 6 copies of pentaxehes. M. Lukjanska`s contest work on compatibility of polyiamonds was highly appreciated in the 15th European Union Contest for the Young Scientists, Budapest, Hungary in 2003. She was awarded Honorary Prize and was selected to represent the European Union Contest for the Young Scientists at the 46th London Internat ional Youth Science Forum, 28 July - 11 August, 2004. In 2003 V. Plociņš succeeded in finding a 360-mino being the common multiply for P- and A-hexominoes shown in Fig. 1.

Tetratans (superTangrams) are the polyforms obtained by combining four unit isoscel es right triangles snugly in every possible way. [8]. Several convex t-shapes (i. e. assemblable from different tetratans) can be found on the Internet. The perfect contest paper “Analysis of t-polytans” for young scientists was elaborated by Jurijs Bedratijs in 2002.

Hexagon containing all tetratans is shown in Fig. 2. Heptagon shown in Fig. 3 is assembled from 13 tetratans. It has a very large number of solutions – 16821.

One can observe origins of tetratans in the well-known puzzle “Tangram” containing two tetratans among its seven pieces. There are two sources stimulating investigation of tetratans: the manuscript of A. Liu [7] and the webpage of Henry Picciotto [8]. There are only 8 convex t-shapes assemblable from all 14 tetratans. This result (including the number of assembles) coincides with that one given on the Internet [9]. According to J. Bedratijs there are 297 convex t-shapes (including 5 tetratans). He also found 14 t-shapes with 4 axes of symmetry. J. Bedratijs came to an unexpected conclusion that later became a theorem on an inseparable pair. Moreover, such an inseparable pair is unique. He had analysed a large amount of solutions (of convex shapes) obtained by the computer programme elaborated by A. Blumbergs.

Colouring and parity

Colouring principles and parity are of a great importance in solving many olympiad problems. Often they help us also to solve more difficult problems. Generally speaking colouring and parity are a powerful combinatorial toosl.

To investigate polytans we pay attention to two types of colouring, namely A-colouring and B-colouring, shown in Fig. 4 and Fig. 5 respectively.

There are four colours used in these two colourings of the plane. Each tetratan covers 8 triangles. We represent 8 as the sum of four numbers (c1 , c2, c3, c4 ), where ck is the number of triangles of k colour. We define a tetratan to beeven with respect to A-colouring (or B-colouring) if all the differences c 1- c2, c2 - c3 , c3 - c4 are even. Analogously if all these differences are odd tetratan is said to be odd. In the same fashion we can count the number of triangles covered by an arbitrarily polytan and define its parity.

Let us note that each tetratan keeps its parity (independently of its position) for A-colouring. The same refers to B-colouring. Moreover, only two tetratans, namely the ones shown in Fig. 6 and Fig. 7, change their parity when we pass from A- to B-colouring. The tetratan T1 (Fig. 6) is even with respect to A-colouring. It covers the following number of red, grey, yellow and blue triangles: (2, 2, 2, 2). This tetratan is odd with respect to B-colouring. The number of the covered triangles is as follows: (1, 4, 1, 2), or (2, 3, 0, 3). These numbers may vary only in the cyclic order that keeps parity. In its turn, tetratan T2 (Fig. 7 changes its parity from odd to even with respect to A- and B- colouring. These preliminary statements allow us to obtain the main result.

Theorem (on the inseparable pair).

Each convex t-shape contains either two tetratans or none shown in Fig. 6 and Fig. 7.

Proof . The key to the proof is the following lemma.

Lemma . A convex t-shape keeps parity in both the colourings.

Let us first observe that the unit square is of the same parity (actually even parity) in the both the colourings. This immediately implies Lemma for rectangles, moreover, for shapes consisting of unit squares. Now let us prove the Lemma for convex t-shapes.

Remark . The even number of unit triangles is not a sufficient condition to keep the parity of convex polytan simultaneously in both the colourings, e. g. the trapezoid in Fig. 8 is odd for A- and even for B-colouring.

As tetratan consists of four unit triangles the number of unit triangles of any t-shape is multiple of 4. The same refers to the number of triangles touching the boundary of t-shape. There are four positions (see Fig. 9) for triangles touching the boundary of polytan. Irrespective of position each pair of triangles shown in Fig. 8 keeps parity. Let us suppose that we have excluded all such pairs. Then the boundary of convex t-shape may contain four unit triangles shown in Fig. 9. As a set of these four triangles keeps parity Lemma has been proved.

Some problems for research
    • Hexomino P (see Fig. 1) is compatible with each of the remaining 34 ones. Determine all others hexominoes sharing this property. At present 7 such hexominoes are known.
    • Investigate the similar problem for hexiamonds.
    • Prove or disprove a compatibility of the pentominoes (X, Z) and (X, W).
    • Prove or disprove a compatibility of the hexiamonds shown in Fig. 11.
    • Is the polytan shown in Fig. 3 the one having the maximum number of solutions?

In Latvia the learning content of mathematics is determined by the subject standard. The transition to new primary education standards is taking place. It is planned that they will be in force as of 2005. In the new standard both mathematical and problem solving skills are equally important in developing students` research skills in everyday learning process.

Until now a greater attention was paid to the acquisition of certain knowledge of mathematics and algorithms. It is welcome to give an inspiration for the gifted students already in everyday learning process so that students would be able to carry out serious research, for example in the field of polyforms, when finishing a primary or secondary school. Pupils of Latvia acquire two separate subjects – algebra and geometry starting from Form 7. It gives the possibility to carry out different research projects from simple tasks in a primary school to scientific research in a secondary school. The aims of geometry are to develop understanding of figures, to investigate and to classify them, etc., therefore just geometry develops the skills needed for various research. The standard prescribes to acquire the skills to investigate and analyse, for instance, the tetragons, but in the learning and teaching process such figures as pentominoes and others are also dealt with.

The education process orientated to pupils` research can be broken down into three levels:

The first level is work during the lesson.

A lesson is a basis for creating interest in mathematics. It is very important to provide pupils with tasks of different levels that can be found in the textbooks. For instance, the following tasks are given in the textbook of geometry [10] meant for Form 8:

    1. A rectangle 6 x 10 consists of equal squares. Cut it along square lines into 12 figures so that there are no equal ones among them.
    2. Solve a similar task for rectangles consisting of 3 x 20, 4 x 15, 5 x 12 equal squares.
    3. A square consists of 8 x 8 squares. One of them is cut out. Is it possible to cut the rest of the square into isosceles rectangular triangles each consisting of halves of two squares (see Fig. 12)?

The solutions of these tasks require creativity and give the opportunity for pupils to work with ‘untraditional’ geometrical figures and solution methods.

Pupils get an idea about figure variety and their common features by solving such tasks already in lessons, and thus pupils` interest in research is developed.

The second level is mathematics Olympiads. The preparation work is done usually after lessons and the content is not included in the standard of the subject. Pupils get additional mathematical knowledge and skills helping them to develop various solution approaches.

The third level is scientific research carried out by pupils of Forms 9-12. School determines who will carry out this research - either all pupils or those who want it themselves. Pupils choose a subject and his/her own topic or a topic suggested by a teacher. As distinct from olympiads this independent research sometimes yield more profound mathematical results. Such a research is carried out within one year or in a longer period of time.


This paper was partially supported by a state–investment project “Latvian Education Informatization System”.


        1. Cibulis A., Liu A. and Wainwright B. Polyomino number theory (I ), Crux Mathematicorum, v. 28, No. 3, April 2002, 147-150.
        2. Barbans U., Cibulis A., Lee G., Liu A., and Wainwright B. Polyomino Number Theory (II ), in the book "Mathematical Properties of Sequences and other Combinatorial Structures" published by Kluwer Academic, 2003, 93-100 pp.
        3. Barbans U., Cibulis A., Lee G., Liu A., and Wainwright B. Polyomino Number Theory (III ), Gathering for Gardner, Atlanta, April, 2002.
        4. Golomb S. W. Polyominoes: Puzzles, Patterns, Problems and Packings , Princeton University Press, NJ, 1994. (First edition: Polyominoes, New York: Scribners Sons, 1965)
        5. Puzzle Fun by Rodolfo Marcelo Kurchan (Parana 960 5”A” (1017) Buenos Aires, Argentina, N6, August 1995.
        6. Cibulis A. Common Multiples of Polyominoes and Polyiamonds: Theoretical, Practical, Learning and Teaching Aspects , Proceedings of the Third International Conference “Creativity in Mathematics Education and the Education of Gifted Students”, Bulgaria, Rousse, August, 2003, 223-227.
        7. Liu A.Super-Tangram , Andy Liu’s Exchange Puzzle, 20th International Puzzle Party, Los Angeles – August 2000.
        8. http://www.picciotto.org/math-ed/puzzles/index.html
        9. http://alpha.ujep.cz/~vicher/puzzle/polyform/tan/tan.htm
        10. Andžāns A., Falkenšteine E., Grava A. “Geometry for Forms 7-9, Part V. Squares”, Riga, Zvaigzne ABC, 1997, pp. 82. (in Latvian)

Andrejs Cibulis, Ph.D., Assoc. Prof.

University of Latvia

29 Rainis boulevard, Riga, LV-1459


Phone: ++371 7211421

E-mail: Andrejs.Cibulis@mii.lu.lv

Ilze France, Mg.Math.

Ministry of Education and Science of the Republic of Latvia,

Centre for Curriculum Development and Examinations

2 Vaļņu Street

Riga, LV-1050


Phone: ++371 7814354

E-mail: ilze.france@isec.gov.lv




Bettina Dahl

Abstract: The purpose of this paper is to argue that gifted students need special programmes to avoid, for instance, psychological disturbances and/or being turned of school and furthermore the paper argues that successful students learn in a qualitative differently way from less successful students and that even among the successful there are differences in how they learn.

Keywords: Gifted, successful, analysing, generalising, problem-solving procedures, memory, learning theories, learning strategies.


In this paper I discuss if gifted students can take care of themselves, how successful students learn mathematics compared to lower-achieving students, and if there are similarities or differences in how successful high school students learn mathematics.


There are many synonymous for ‘gifted’ such as ‘talented’, ‘able’, ‘successful’, ‘capable’ ‘high-achieving’ etc. Basically these terms fall into two groups. Some describe the state of actually being “good” (yet another expression) namely: ‘successful’ and ‘high-achieving’ while others more describe a person who has the potential for being ‘successful’ or ‘high-achieving’. These terms are: ‘gifted’, ‘talented’, ‘able’, and ‘capable’. By introducing this distinction I at the same time argue that there is not necessarily a direct link between having the potential to become successful and actually being or becoming successful in either daily life or the classroom. Some might argue that gifted students can take care of themselves and helping gifted students is perceived as taking resources from weaker students. I will argue that it is a question of equality of opportunity to provide for the needs of the gifted. The focus of the education system should be on meeting every student where he is and help him to reach his full potential. Special emphasis on the needs of the gifted is for instance seen in the United Kingdom where Ofsted (Office for Standards in Education) considers the needs of able students as part of equality of opportunity. The DfEE (Department for Education and Employment) has furthermore in two Circulars (14/94 & 15/94) in 1994 recommended that in primary and secondary schools all School Prospectus should include details of arrangements to identify and provide for exceptionally able students ([2], pp. 16-17). One can argue as follows:

If we accept that it is the duty of society ... to provide educational opportunities for all children appropriate to their individual abilities and aptitudes, and if one further accepts that some children are exceptional … then the issue is settled. … For children to receive specialized educational treatment in such circumstances is not for them to get more than their fair share; they are simply receiving what, in their individual circumstances, is appropriate. ([6], p. 4)

UNESCO’s Salamanca Statement, 1994, declares that “The guiding principle that informs this Framework is that schools should accommodate all children regardless of their physical, intellectual, social, emotional, linguistic or other condition. This should include disabled and gifted children, street and working children, children from remote or nomadic populations, children from linguistic, ethnic or cultural minorities and children from other disadvantaged or marginalized areas or groups“([10], p. 6) and further: “every child has unique characteristics, interests, abilities and learning needs; education systems should be designed and educational programmes implemented to take into account the wide diversity of these characteristics and needs ([10], p. viii ). Hence, it is not “un-just” to help successful and gifted students; they too have a right to receive what fits them. Furthermore: “all children are born as unique individuals, each different from the other, and in developing them we need to make them more equal by overcoming whatever inabilities they may have and more different from one another by developing their abilities and propensities” ([12], p. 31). In that sense, special education (for both weak and strong students) both improves inabilities and develops the person’s talent(s).

No student can progress towards the limit of his capacity unless he has an opportunity to learn: “Mozart might have had an extraordinary aptitude for music, but this could hardly have been realized unless his parents possessed a piano. It is at best inefficient to rely on nature or chance to develop talents, while for potentially gifted children in homes with limited cultural horizons it borders on neglect” ([6], p. 5). Studies have furthermore shown that some gifted students are underachieving and sometimes suffer psychological disturbances including poor concentration, exaggerated conformity, excessively inhibited behaviour, anxiety, social isolation and aggressiveness, or the opposite such as extreme passivity ([6], p. 6). Other studies have shown that if gifted students are held back or bored in school, some of them will be ‘turned off’ by school, achieve far below the level of which they are capable, drop out, fail, or even become delinquent ([6], p. 14). Another study showed that children who could read before beginning in school do not develop new competencies if they are just being taught what they already know, and many of the early readers later loose interest in reading. These students therefore need special attention and need to be challenged ([5], p. 6). It is further stated that some gifted students deliberately hold themselves back:

Some able students receive a shock when they move on to university. The leisurely study habits which had ensured reasonable grades in the mixed ability classes in secondary schools prove to be inadequate for the more intellectually demanding environment of the university. … there are too many students of high ability who wastefully drop out. … it is very probable that many gifted children ‘learn to be average’ or deliberately hold themselves back in order to have a quiet life in school: this is the phenomena of ‘faking bad’. ([6], pp. 14-15)

Gifted students therefore need adequate stimulation. Studies suggest that association with other students of high ability raises a student’s level of performance. One study showed that the “overall intellectual level within a group had an effect on the development of the level of individuals within the group - contact with clever people tended to raise the level of ability of the less clever” ([6], p. 13). Another study showed that “down to an IQ of about 65, mentally retarded students taught with normal peers achieved better than those who were taught in self-contained classes” ([6], p. 13). And further “that students of high ability were penalized academically by being taught with students of lesser ability” ([6], pp. 13-14). Hence, it might seem as a Catch-22 situation: when each student seems to do better when taught together with more gifted students, and suffer from being with less gifted students, there will always be a “looser” in the “game”. However, it does not have to be this way if it is the teachers’ duty to stimulate the students according to their abilities, which is also what is argued below:

Refusing to make special provision for the unusually able, on the grounds that they are necessary for the optimal development of the other children, means that adults shrug off the task of promoting the development of less gifted youngsters onto the shoulders of clever children. Naturally, educators should be looking at the needs of the less gifted, but not at the expense of the gifted and talented. ([6], p. 14)



When gifted students work on a mathematical problem they perceive the mathematics of it analytically , which means that they isolate and assess the different elements in its structure, systematise them, and determine their ‘hierarchy’. At the same time they perceive the mathematical material synthetically , and here combine the elements into complexes and investigate the mathematical relationships ([4], pp. 227-228; [9], p. 15). Gifted students perceive problems as a composite whole, while average students see a problem in its separate mathematical elements. It is only through analysing the problem that the average students are able to find the connections of the mathematical elements. Lower-achieving students have great difficulties in establishing these connections, even when they achieved help. The speed of the analytical-synthetic process in the gifted student is so fast that they see its ‘skeleton’ at once. It is often impossible to trace the process. The fast grasping of a problem’s structure has been observed to be the result of exercises, but gifted students need only a minimal number of exercises to make the analytical-synthetical perception arise ‘on the spot’ ([4], pp. 228-232).

Example 1 ([4], p. 230)

A 6th grade class gets the following problem:

A jar of honey weighs 500 g, and the same jar, filled with kerosene, weighs 350 g.

How much does the empty jar weigh?

A gifted 3rd grader (V.L.) answers (E is the experimenter):

V.L.:And then?
E:That’s the whole problem.
V.L.:No, that isn’t all. I still must know how much heavier honey is than kerosene.
V.L.:Without that, there could be many solutions. There are two unequal quantities, connected by the fact that some of their parts are equal. There could be very many of these parts. To limit their number, we must introduce one more quantity, characterizing the ’remainder’.

A less gifted 6th grader was not able to solve this problem, even when he got the hint: ”honey is twice as heavy as kerosene”.


The ability to ‘grasp’ structural relationships in a generalised form is a central feature for the productive thinking ([4], p. 234). The gifted students do this on the spot whereas lower-achieving students need a lot of practice and exercises covering all possible cases and levels before an elementary level of generalization is possible ([4], pp. 240-242). Gifted students can analyse one phenomenon and generalise from this by separating the essential features from inessential. Their method is to infer “the features’ generality from their essentiality. … to be essential means to be necessary and, consequently, it should be common to a number of phenomena of this type, that is, it should inevitably be repeated” ([4], p. 259). Lower-achieving students perceive the generality of features by contrast.

Example 2 ([4], p. 241)

A gifted student, O.V., had previously solved just a single example using the formula of the square of a sum: (a+b) = a + b + 2ab. Then he got the problem: (C+D+E) × (E+C+D).

(E is the experimenter.)

O.V.:What’s this? Here it’s not by the formula – we must simply multiply the polynomials. ... But that will be 9 terms. That’s a lot. But we can use the formula – that is a square [quickly writes: (C+D+E) ]. Right. Now any two terms can be combined [writes: (C+[D+E]) )].
E:But can you do that? The formula applies only to the square of a binomial, but didn’t you have a trinomial?
O.V.: As soon as I combined D and E into one term, I got a binomial – look [shows]. A ’term’ can be any expression. ... [Solves it, repeating the formula aloud. Writes:

C + 2C(D+E) + (D+E) = C + 2CD + 2 CE + D + 2DE + E )]

Procedures for problems-solving

The trials for problem-solving for lower-achieving students are blind, unmotivated, and unsystematic. On the contrary gifted students have an organised plan of searching ([4], p. 292). Gifted students switch easily from one mental operation and method to another, they have great flexibility and mobility in their mental processes in solving mathematical problems, and it is therefore easy for them to reconstruct established thought patterns. For average students it is much harder to switch to a new method of problem-solving. Lower-achieving students experience even greater difficulties in that ([4], pp. 278-282). For the gifted students the trials are a way to thoroughly investigate the problem through extracting information from each trial. Without having finished the trial, gifted students seem to know if they are on the right track. This is owing to the existence of an acceptor, which is a psychological control-appraisal mechanism, where ‘line-of-communication’ is received from each mathematical operation. Under this acceptor lies a generalised and concentrated system of past mathematical experience ([4], p. 293). The gifted students thoroughly investigate the problem, which may suggest that they enjoy working with mathematics. The emotional factor is seen in that they often try to solve the problem in a more simple way or improve the solution and they show satisfaction when the solution was economical, rational, and elegant ([4], p. 285), which is seen in the example below.

Example 3 ([4], p. 279)


Gifted students do not have a “better” memory than lower-achievers, but gifted students usually remember the general character of a problem-solving operation and not the problem’s specific data. On the contrary, lower-achieving students usually only remember the problem’s specific facts. The mathematical memory of gifted students is selective and only keeps the mathematical information that represents generalised and curtailed structures. This means that the brain is not loaded with extra information which makes it possible to retain the information longer and use it more easy ([4], pp. 299-300).

Example 4 ([4], pp. 298-299)

A lower-achieving student, I.G., solved the problem: 113 – 112 with the experimenter’s help. After one week she had forgotten the mathematical relationship (difference of squares) but remembered that the problem had used the numbers 112 and 113.

The figure below ([4], p 297) shows the forgetting-curve of generalized relations, concrete data, and unnecessary data for gifted students.


I have previously done research ([1]) in how ten successful high school students (aged 17-20) explain how do they come to understand a mathematical concept new for them. Four students were Danish (Z, Æ, Ø, Å) and six were English (A, B, C, D, E, F). They were interviewed in pairs and fours: Z-Æ-Ø-Å, A-C, D-E, B-F. All studied mathematics at the highest level possible in each of their school system and their teachers selected them as successful. The study rests on the assumption that successful students have a metacognition, which means that they have knowledge about and regulation of their cognition. Knowledge of own learning means that one has relatively stable information about own learning processes. This knowledge develops with age and there is a positive correlation between the degree of one’s insight into own learning and one’s performances on many tasks. Regulation of own learning is the planning before one begins to solve a problem and the ongoing evaluation and control while one learns something new or solves a problem ([8], pp. 138-141). I asked general explorative questions to not be leading. In their own words, the students, among other things, describe the relationship between visualization and verbalization and the individual and the social side of learning. I used the learning theories of Ernest, von Glasersfeld, Hadamard, Krutetskii, Mason, Piaget, Polya, Sfard, Skemp, and Vygotsky in the analysis. The ten students fall in different groups regarding their preference for learning style. For language reasons the examples below are from the English interviews.

Visualization and verbalization

Regarding verbalization, Student A, C, Z, and Æ tell that an oral explanation helps the one that is talking. For instance Student Æ tells that very often if she tries to explain the mathematics to a person, then when she is explaining it, she understands it herself. Student C adds that verbalization (saying things out load) helps the visualization:

C:If you just read it in your head, I just read it and I don’t understand. If you come across a phrase which sounds really awkward, like the one here: “in the plane whose interior intersects the diagram in one of the configurations involved”, you just read it over, but you don’t think about it, but if you read it out, then you think about the disc, and then you visualise the disc, and then you visualise the plane.

If one focuses on visualization, Student Z describes that she does not need to see pictures but to see things in writing. Another group consists of the students being moderately positive to visualization (A, D, E, F, Æ, Å). They describe that pictures sometimes makes it easier. The two examples below are from two interviews, but both students talk about some knot theory that I had given them to work on, to initiate a discussion:

F:The first thing to do [when explaining to classmates] would be to draw some knots and then show the respective oriented diagrams whatever they are called, these graphs.
E:Actually the graphics are a big help, I know I was relieved when I got to this first graphic [D & E laugh], it showed, it kind of showed what they were.

A third group (C, Ø) perceive visualization as something that is very important for learning. Student Ø says for instance that visualisation can be a great advantage for instance in terms of cosine equation. Student A, C, and Æ argue that the relationship between visualisation and verbalisation depends on the mathematics. For instance Student A:

A:Sometimes if you just READ a sentence like the first here “A link is a finite collection of mutually disjoint disjointed simple closed curves”, if you just take that as the words it is difficult for you to seeany other way that it can work. If you start to visualize what it is talking about then you can SEE all the different ways in which it can happen.
I: So is it important that the words come first and then you can see what the words are about, or do you want to see it first and then be explained and then get the words?
A:Well. I don’t know really. It probably depends on the sort of problem. If it is a very visual problem where you have to think it through maybe in 3 dimensions, I think it is probably better to have pictures first and maybe dealing with graphs as well. If you are doing vectors it may be better to have the picture first, and then the words explaining how it works. But if it is more a linear methodological process it might be better to have the words first and then pictures to help you understand, cause it is the words you are trying to understand. But where there is something visual like this or graphs or vectors it’s the pictures that you are trying to understandso the one supports the other depending on what it is you are trying to learn, I think.

The students group as below. They do not all express something in this connection.

1. Oral explanation helps the one that is talking: Student: A, C, Z, Æ

2. Mainly verbal: Student Z

3. Relatively visual: Student A, D, E, F, Æ, Å

4. Mainly visual: Student C, Ø

5. Depends on circumstances, for instance the type of mathematics: Student A, C, Æ

Individual or social learning

It seems that the students argue that learning has both a social and an individual side. The value of the social side is mainly when the students experience problems with self-learning. After input from the outside, they can continue on their own. Particularly Student C, D, E, F, Z, Æ, Ø, and Å argue this. Conversely, Student A and B express that they learn more through the discussions than through self-activity. I will now investigate some of these students further. First an example from the interview with Student D and E who both emphasize the individual side:

E: I know people do er differently but it’s all very individual even if you work on something together er and you’re both aiming to solve the problem, I mean you’ll do it completely differently from someone else and quite oftenly I find I don’t like other people’s styles, you know, you always get your own.
D:Yea, but that’s the whole point in learning why you do something rather than how, because a lot of the times there are some things I do my own little way, I don’t necessarily follow the textbook example and it, that doesn’t matter cause I understand what I’m doing, what I want to achieve by doing it, and I can get the right answer by doing it a different way to the textbook, but, you know, it just works for me.

Student Z tells that basically the teacher cannot help her learn mathematics. She has to work for herself, she has to take the responsibility for her learning.

The social side seems to be most important to Student A and B. They tell that they learn more through the discussions that by sitting for themselves. Student A tells that it is better to work for oneself through looking at examples than being told “this is wrong, try again”. This might sound like support of individually aimed theories such as Piaget’s ([7], pp. 16-19), who argued that the students through own activities in interaction with the surrounding world construct their knowledge. In that case, it would be a contrast to what Student A also tells, namely that he learns best through the discussions. However, what Student A actually says is that it is better to “work for himself” instead of “being told”. To “be told” is not necessarily the same as a discussion, perhaps rather the opposite. Student A therefore priorities as follows: 1. discussion, 2. work for himself, 3. be told. Student A is therefore probably more a “Vygotsky-student”. Vygotsky ([11], pp. 56-57) argued that internalisation happens through activity and communication in social interaction. Student B supports this (I is the interviewer):

B:I think it certainly helps if you can discuss it with someone else. Two brains are better than one.
B:Er, one person can have one idea which should trigger another idea in the other person’s head which the first person wouldn’t have had, and then the second person having said that thing, and then one thing leads to another if you got two people to think.

Student C and Æ do on own initiative use words such as “combination” or “two-way thing” to describe the relationship between the individual and the social side of learning, however still with emphasis on the individual side. An example:

C:It all boils down to the teaching method and the teacher. It’s a two-way thing you see, it’s more about you learning, you being able, no, you learning as well you being taught properly. If you are taught in a way that you can fit in, you know, then it is good.

Student Æ and Z tell that one can learn through discussions with oneself, perhaps even better than in groups. The students group as follows:

1. Individual: Student C, D, E, F, Z, Æ, Ø, Å

2. Social: Student A, B

3. Combination (emphasis on the individual side): C, Æ

4. Learn from discussing with oneself: Student Æ, Z

Relating to the discussion above about if it is best to work with students on the same level or not, Students A, C, and Z argue that if they have problems learning, it is best to discuss with someone who also has not understood. The reason is that they do not want to be bullied. Student Å says that discussion with “equals” is good in any case, and Student B argues that it is more helpful to discuss with someone who knows.

The students’ learning styles

It seems that the students divide themselves into different groups that either support individual or social learning, or support visualization or verbalization with a preference for one, or describe a kind of combination between these factors. The preferences are independently of nationality or gender (D, Z, Æ, Ø, Å are girls; A, B, C, E, F are boys). The students therefore learn in different ways. These students constitute a rather homogeneous group of gifted students, which might suggest that if one looks at the whole spectrum of students, even more learning strategies or preferences might emerge. It is therefore vital with variation in the teaching. Teaching after a one-sided pedagogical theory or idea will lead to that some students are lost.

Furthermore Student A says that the learning strategies one uses are connected with the ways one has been taught to do things. Other students explain the following in two of the interviews (I is the interviewer):

D:When I first came here [to the new school], the first couples of weeks I found math very difficult because it is kind of hard to adapt to a different teaching style.
I: I noticed when you talked about presenting it to the class you wanted to give them examples and you also mention, while you were talking, that it would be nice with examples. Why?
B: That’s because the way that we’ve always been taught is using examples thoroughly to explain, so that’s the way we think the people in our class will understand it easiest, explain through examples.

This phenomena might be explained by that the teaching methods must be part of, what I would express as a zone of proximal teaching (ZPT) , inspired by Vygotsky’s ZPD. According to Vygotsky, the potential for learning is limited to the “zone of proximal development (ZPD)”, where ZPD is the area between the tasks a student can do without assistance, and those, which require help ([11], p. 86). Hence, if a (new) teacher uses teaching methods that are too “far away” from teaching styles the students are used to, the students might have difficulties in learning.


Gifted students need special programmes not just to make sure that they, for instance, avoid psychological disturbances, but there are also major differences in how successful and less successful students learn mathematics and even among the successful students there are some differences in learning styles. Also Hadamard wrote about different mathematical minds and “even among men who are born mathematicians, important mental differences may exist” ([3], p. 11). Successful students furthermore experience problems when facing a new teaching style but they do nevertheless seem to overcome these difficulties. This is to some extent supported by [4]’s investigations that showed that successful students switch easily from one mental operation and method to another.


        1. Dahl, B. (2004). 'Analysing Cognitive Learning Processes Through Group Interviews Of Successful High School Pupils: Development And Use Of A Model'. Educational Studies In Mathematics. (Forthcoming)
        2. Eyre, D. (1995). School Governors & More Able Children (The National Association for Able Children in Education (NACE)).
        3. Hadamard, J. (1945). An Essay on The Psychology of Invention in the Mathmematical Field (New York, Dover).
        4. Krutetskii, V. A. (1976). The Psychology of Mathematical Abilities in Schoolchildren (Chicago, The University of Chicago).
        5. Kyed, P. (ed.) (2002). Undervisning Af Elever Med Særlige Forudsætninger [Teaching Of Gifted Students]. (Vejle Denmark, Krogh).
        6. McLeod, J. & Cropley, A. (1989). Fostering Academic Excellence (Oxford, Pergamon).
        7. Piaget, J. (1970). Genetic Epistemology. (New York, Columbia University).
        8. Schoenfeld, A. (1985). Mathematical Problem Solving. (London, Academic).
        9. Tall, D. (Ed.) (1991). Advanced Mathematical Thinking (Dordrecht, Kluwer).
        10. UNESCO (1994). The Salamanca Statement and Framework for Action on Special Needs Education. Adopted, Salamanca, Spain, 7-10 June 1994.
        11. Vygotsky, L. S. (1978). Mind in society: The Development Of Higher Psychological Processes. (Cambridge Massachusetts, Harvard University).
        12. Young, P. & Tyre, C. (1992). Gifted or Able? Realizing Children's Potential (Buckingham, Open University).

Bettina Dahl (Soendergaard), Ph.D.

Norwegian Centre for Mathematics Education

Norwegian University of Science and Technology (NTNU)

Realfagbygget A4, 7491 Trondheim


E-mail: bdahls@math.ntnu.no



Bharath Sriraman

Abstract: It is a well-established fact that the curriculum at most public schools do not provide any explicit enrichment for gifted students in the mathematics classroom. In the United States, researchers in the curricular division of gifted education have suggested parallel curricula, curriculum compacting and curriculum differentiation [9] as a mean to provide enrichment to the regular school curriculum. The shortcoming of these approaches is that they require considerable professional development of teachers and subsequent support from the school administration for successful implementation in the classroom. In fact numerous studies have shown that this approach is only implemented in a small fraction of the K-8 classrooms and is a statistical rarity in the 9-12 classrooms. In this paper I present a radically different approach to differentiation, one that is specifically content driven and allows for creative mathematics to manifest in the classroom. In particular I describe the structure of two teaching experiments, their implementation and the fascinating outcomes with 9th grade public school students at a rural American high school. The paper concludes with an argument for the use of novel problem-solving situations with an underlying structure as a means of both differentiating and enriching the regular mathematics curriculum.

Key words: Abstraction;Curriculum Differentiation; Combinatorics; Generalization; Heterogeneous-Classrooms; Pedagogy; Mathematical giftedness; Problem Solving.


Today, in the U.S., in spite of the availability of alternative research-based curricula, the traditional mathematics sequence of Algebra-Geometry-PreCalculus-Calculus dominates the curricular landscape. Calculus is the utopian dream and thought of as a capstone class by most college bound students. As a full time teacher in a rural Midwestern U.S. public school, I was responsible for teaching various sections of " traditional" Algebra to 9th grade students (13-14 year old). These classes consisted of students of varying mathematical abilities and included some mathematically gifted students, who had received glowing recommendations from their previous middle school teachers and counselors as having the potential to do "accelerated" work. The 9th grade Algebra curriculum focused on the study of the properties of the real number system with tangential excursions to analytic geometry. Research indicates that most gifted students have already mastered up to 60% of the mathematics content encountered in high school [8]. This posed the conundrum of potentially boring these students in my classroom and perhaps even robbing them of utilizing their mathematical capabilities. The solution to this conundrum was found in an often quoted survey article [4] which called for increased emphasis in enumerative or non-continuous mathematics because of its independence from Calculus.

Each teaching experiment was approximately 3 months long and was structured as follows. The students in my Algebra classrooms had to maintain a mathematics journal in which they solved problems from Combinatorics and Number Theory, topics rarely covered in the traditional high school curriculum. Students were given cues that asked them to reformulate the problem in their own words, devise an explicit strategy prior to solving the problem, and reflecting on the solution. The pedagogical goal was to pose problem situations that had an underlying structure or a principle which could be potentially discovered by the gifted students in the classroom, but simple enough for all students to have the opportunity to create representations, think abstractly and create generalizations at varying levels.


The first experiment consisted of assigning 8 problems over the course of 3 months. The solutions of four of the eight posed problems were characterized by the construction of Steiner Triple Systems (STS), a useful combinatorial structure. For instance, the following STS problems (Fig. 1) were posed un-sequentially in the order 1-3-6. The five mathematically gifted students in the class were able to uncover this sequence and gain an insight into the structure of Steiner Triple Systems. They were successful in devising an efficient strategy for counting triples in these problems and making a substantial attempt at tackling more general questions.

Gifted students who discovered the underlying structure of these three problems were given the following additional challenging questions.

    1. In an arrangement of “n” objects in triplets, how many ways can each pair of objects appear in a triplet once and only once? How many pairs are possible?
    2. How can you create problems like the ones above, which are always solvable (or unsolvable)?


In the second experiment, five problems which represented diverse situations but whosesolutions were characterized by the application of the pigeonhole principle were assigned sequentially over the course of 3 months. In this experiment, the four students (out of the 9) who were identified as mathematically gifted were able to discover the pigeonhole principle by abstracting the structural similarities from the assigned problems.


Based on these experiments of using combinatorial problem-solving situations as means of differentiating the mathematics content the following general commendations are provided. As reported here carefully chosen classes of problems which involve the discovery of a combinatorial principle [5] or a combinatorial structure [6] in a heterogeneous setting along with the use of journals as a means of establishing communication with students, led to the positive outcomes of gifted students abstracting structural similarities, conceptually linking related problems, utilizing their creativity pursuing general solutions and creating theoretical generalizations [2]. Typically, empirical generalizations are the result of being unable to discern the structural features of a problem and are based on the superficial aspects of a task or problem. A theoretical generalization on the other hand is characterized by the discernment of the essential elements of a task or a problem [1], [2]. The distinction between empirical and theoretical generalizations is illustrated by using the notion of ‘roundness’ [1]. That is, "roundness" can be empirically abstracted from a dish, a wheel, etc., but this empirical, everyday notion of circularity does not reveal the underlying theoretical construct of circularity as a locus of points at a constant distance from a fixed point which is not readily apparent in the mere appearance of roundness [1]. This does not mean that empirical generalizations do not have any pedagogical value. The teacher can use the students’ empirical generalizations and hopefully facilitate the discovery of the theoretical generalization.

In conclusion, I highly recommend problems that result in the discovery of a combinatorial structure or principle. One research implication would be construct a "metric" that sorts a class of related combinatorial problems into increasing levels of difficulty, which can then be given to teachers to use in the regular classroom. The novelty of this approach of using combinatorial problem-solving situations in the heterogeneous classroom is that they are easy to understand which allows weaker students to somehow model such problems by trial and error strategies thereby resulting in the positive outcomes of problem-solving, systematizing thinking, and empirical generalizations [2]. The use of a class of contextually different but structurally isomorphic problems can be used in the heterogeneous classroom to foster independent thinking as students initially tackle the problem. The teacher can also encourage flexibility and sharing of solutions as students begin to explore simpler contexts and alternative representations. Ultimately it is hoped that teachers will focus on problem structure as students generalize across dimensions, and further present problem-posing opportunities as a culminating activity [7].


        1. Davydov, V.V. (1988). The concept of theoretical generalization and problems of educational psychology. Studies in Soviet thought, 36, 169-202.
        2. Davydov.V.V. (1990). Type of generalization in instruction: Logical and psychological problems in the structuring of school curricula.” In Soviet Studies in Mathematics Education (Vol. 2), edited by Jeremy Kilpatrick, National Council of Teachers of Mathematics, Reston: VA.
        3. Gardner, M. (1997). The Last Recreations. New York: Springer-Verlag.
        4. Kapur, J. (1970). Combinatorial analysis and school mathematics. Educational Studies in Mathematics, 3, 111-127.
        5. Sriraman, B. (2003) . Mathematical giftedness, problem solving, and the ability to formulate generalizations. TheJournal of Secondary Gifted Education . 14(3): 151-165.
        6. Sriraman,B (2004) . Discovering Steiner Triple Systems via Problem solving. The Mathematics Teacher, 97 (5), 320-326.
        7. Sriraman, B. & English, L.D. (2004). Combinatorial mathematics: Research into practice. To appear in the Connecting Research into Teaching section of The Mathematics Teacher.
        8. U.S. Department of Education, office of Educational Research and Improvement. (1993).National Excellence: A case for developing America's talent. Washington, DC: U.S. Government Printing Office.
        9. Van Tassel-Baska (1998) . Curriculum for the gifted: Theory research and practice. In J. Van Tassel-Baska, J. Feldhusen, K.Seeley, G. Wheatley, L. Silverman and W. Foster (Eds.), Comprehensive curriculum for gifted learners , (pp.1-19). Boston, MA: Allyn and Bacon.

Bharath Sriraman, Ph.D., Asst. Professor of Math & Math-Education

Department of Mathematical Sciences

The University of Montana, Missoula, MT 59812


E-mail: sriramanb@mso.umt.edu



Borislav Lazarov

Abstract: The concept of cognitive activeness of students could be accepted as a criterion for the success of different strategies for students dealing with mathematicsbeyond the ordinary curriculum. A qualitative description of the change of student’s activeness could help to understand the impact of a significant activity on the learning process. Under consideration are several models of change of activeness and an idea for operationalization of the qualitative concept of activeness.


The concept of motivation is crucial in understanding the psychology of a significant student studying mathematics, participating in competitions and setting as a personal goal a high level of math knowledge. Different points of view focused on students’ motivation to study mathematics are highlighted in large number of investigations. But even a simple analysis of student’s motivation is a difficult task for the experts and an impossible mission for an ordinary teacher. More sophisticated is the case dealing with group of students, where individual motivating factors should be overviewed and generalized.

But the behaviour of a group of students could be studied, predicted and changed during the process of education without deep analysis of motivation factors. An operationalization of this concept could be the category students’ activeness. The concept of students’ activeness (some authors use the notion creative activeness [1]) was in focus mainly of Russian and Bulgarian scientists for a long period. Egorov [2] pointed some sources considering students’ activeness from the late 19th century. A descriptive definition of students’ activeness is given by Aristova [3], p 32. Further we will introduce a qualitative variable and represent it in a quantitative expression for measuring students’ activeness and its change caused by different activities.


Following Lazarov [4] we introduce the variable situative activeness (SAC) of students as a synthetic characteristic of students’ behaviour during a short period in the process of math education including the attitude of a student to mathematics, student’s motivation to deal with mathematics alone and to take part in extracurricular activities. The SAC can be factorized in the following directions:

    • student’s interest in mathematics during the lessons;
    • time that a student spends for a significant homework;
    • problem solving as a part of everyday preparation for school;
    • problem solving as a preparation for an extracurricular activity;
    • usage of math books extra school textbook, journals etc.
The above list should be considered as a flexible frame and could be extended according to the specifics of age, type of education or purpose of the math study.

It is convenient to introduce also the notion integral activeness (IAC) for the average value of the SAC for a longer period. Since SAC is approximately a day-long characteristic, IAC is a reflection of student’s behaviour during a couple of weeks or even during a term.

The changes in SAC are provoked by activators (ACF). An ACF can be a test, an exam, a competition etc.


The hypothetic graph of SAC is given below.

The period in which a significant ACF acts could be separated into three phases. Phase 1 includes the preparation; phase 2 is the period of performance; phase 3 is the period of fade. The amplitude of the graph corresponds to the change in SAC. It depends on different parameters such as initial and final values of SAC, the priority range of ACF, i.e. how important for the student is a good performance at this event etc. The same ACF has different amplitude for different groups of students: as a rule extracurricular activities cause bigger amplitude for more able students; classroom tests generate bigger amplitude for students with no special interest in mathematics. The change in SAC depends also on student‘s self-estimation for good performance. It is smaller for students with too high or too low level of self-estimation. On the contrary the amplitude of the graph is bigger for the students that are challenged by the ACF.


The graph of SAC caused by consecutive ACFs could be different. The shape of the graph depends mainly on priority range of each ACF and their accommodation on the time-axes. Let us first consider the hypothetic graph of SAC caused by two ACF: A1 and A2 and let the earlier be A1. We will show by graphs some effects that are possible.


The effect of superposition appears when the first phase of A2 overlaps the third phase of A1 but the second phases of A1 and A2 are separated by a small interval. In such a case the resulting SAC is higher than the SAC caused by any of ACFs taken one by one.


The effect of shadow appears when the second phase of A1 touches the second phase of A2. In such a case the ACF of higher priority range dominates. On the picture A1 has higher priority range and the resulting SAC is higher than that of A1 but is lower than the SAC caused by A2 if A2 acts independently.


The effect of depression appears when the second phase of A2 overlaps the second phase of A1. In such a case the resulting SAC is lower than the SAC caused by any of ACFs taken one by one.


The SAC can be registered and measured by the quantitative variable spectrum of activeness . To define the spectrum of activeness we first declare a list of indicators which characterize the SAC of a student.

    1. Interest during the lesson to the currently considered topics.
    2. Taking part in discussions during the lesson.
    3. Time which the student spends dealing with mathematics outside school.
    4. Solving problems as a part of everyday preparation for school.
    5. Solving problems for itself or as a preparation for an extracurricular activity.
    6. Interest to math magazines, journals, web-sites etc.
    7. Interest to math books.
    8. Discussing math topics outside classroom.
    9. Willingness for attending extracurricular activities.
    10. Level of self-estimation about own problem solving abilities.

The above indicators are specific for more able students at the secondary school. It is clear that some of them correlate, e.g. both 6 and 7 point to similar features of student’s behaviour but there is some difference between book-readers and magazine-readers. Some indicators overlap but none of them contains another one. A scale Ai assigned to the indicator i ( i=1,2,…,10) and a parameter ai, taking its values in Ai, gives us the oportunity to calculate the value of a numeric variable S=a 1+a 2+...+ a10, which yelds a quantitative expression of SAC .

The chart below shows statistics from a case study held in the 119th Secondary school in Sofia with students of 10th grade. There was a CAF on April 2 and an EAF on April 5. The average value of S for the group of students with high SAC reaches its maximum on March 31, bu t the group of students with medium SAC attains its maximum on April 2.


The above effects could be implemented to rise the SAC and to keep it on a high level for a long time. The key factor in designing a strategy for teaching more able students is the proper accommodation of ACF during the scholastic year. Since there are compulsory ACF such as tests and term examinations (call them CAF), it is crucial to place the extracurricular ACF (call them EAF) in a way to avoid the negative effects as depression and particularly shadow and to gain benefit from the positive effects as superposition. The next picture shows an exemplary accommodation of CAFs and EAFs during a term.


It is easy to see that the above model refers to the McClelland’s Trichotomy of Needs [5]:

Need for achievement is a need to accomplish and demonstrate competence or mastery; a person who continuously asks for and masters increasingly difficult tasks demonstrates a need for achievement.


A math teacher can apply different strategies in teaching only in a class where the level of activeness of the major part of students is at least medium. In such an environment the teacher can explain more details during the lesson and leave students a part of routine exercises; teacher can recommend students some books or journals for additional preparation, or to state topics for discussions. But if the level of activeness is law the first task of the teacher should be rising of students’ activeness in an appropriate level. Extracurricular activities are very reliable tool for such a purpose. For example in Bulgaria more than 30 math competitions are held during the scholastic year and any teacher can include the most appropriate of them in students time schedule to keep their SAC high.

But it is delicate task to do such a selection. An easy competition does not change the SAC of more able students and a too hard competition may have even negative effect on students with low preparation.


The list below contains some problems which the author considers as important, but has not find satisfactory solution of them.

    • How to establish benchmarks of indicators to distinguish low from medium and medium from high SAC?
    • How to recognize real from pseudo SAC?
    • Is it useful sometimes to consider an EAF as a classroom test, i.e. to change an EAF to CAF?
    • How to explain the phenomenon: the students with the best scores from math competitions do not have the highest SAC?

The author thanks Australian Mathematics Trust and Prof. P.J. Taylor for the support that made this presentation possible.


        1. Shchedrovickij,G. Psychological features of creative activeness of students. Moscow, 1962 (in Russian)
        2. Egorov, S. The problem of activeness and self-dependence of students at thedidactics of 19 th century and the beginning of 20th century. Moscow, 1968 (in Russian)
        3. Aristova, L. Activeness in students’ learning. Moscow, 1968 (in Russian)
        4. Lazarov, B. Student’s activeness as basic criterion for effectiveness of extracurricular activities. RIK-I-S, Sofia, 2003. (in Bulgarian)
        5. McClelland, D. The Achieving Society, NJ, Princeton,1961.

Borislav Lazarov

Dept. of Mathematics

Higher School of Transport

158 Geo Milev Str.

1574 Sofia, BULGARIA

National Institute of Education

E-mail: byl@abv.bg




Brenda Bicknell

Abstract: This paper describes the state of play in New Zealand for the identification and provision of gifted and talented students. The results from a national survey on the extent, nature and effectiveness of planned approaches for identifying and providingfor gifted and talented students specific to the intellectual/academic ability area of giftedness are presented.The purpose of this paper isto combine recent national data on gifted education and policy development with current practice in mathematics classrooms.

Key words :Gifted and Talented,Policy, Curriculum Initiatives, Identification, Provision


In New Zealand the National Education Guidelines establish a common direction for state education and include National Administration Guidelines, policy statements, and national curriculum statements. Schools are managed by elected boards that have responsibility for meeting both general policy objectives for all schools and specific policy objectives applying to that school. Until very recently there was no national statement that made explicit the obligations for Boards to address the needs of gifted and talented students in the school system.

However, in 2002 a commitment was made by the government to support the achievement of gifted and talented students by commissioning research, developing an online learning centre, appointing school advisers and funding education programmes targeted at gifted and talented learners. To further strengthen the position for gifted and talented learners an amendment to policy was announced in March 2004. This was a change to one of the National Administration Guidelines so that “from Term 1 2005, schools will be required to identify their gifted and talented learners, and develop and implement teaching and learning strategies to address their needs” [1]. The Minister in announcing this change acknowledged that gifted and talented learners are found within any group in society and that the initiative “explicitly recognises that schools are powerful catalysts for the development of talent” [1].


In 2003, a national investigation of current identification and provisions for gifted and talented students was conducted by a team of researchers led by Dr Tracy Riley from Massey University [2]. It was premised on the acknowledgement of a “somewhat limited research” base [3]and driven by the need to identify strengths and gaps in provision so that future directions in gifted and talented education may be informed by both theory and practice. It was recognised that “giftedness and talent can mean different things to different communities and cultures … and there is a range of appropriate approaches towards meeting the needs of all such students” ([4], p. 2).

The questionnaire probed schools’ identification and provisions for gifted and talented students. All schools in New Zealand were invited to participate in the survey; there was a 48% response rate (n=1285). An analysis of the respondent schools’ demographics indicates that the sample is representative. Sixty per cent of the schools reported that gifted and talented students had been formally identified over the last 12 months. The areas of giftedness included intellectual/academic, visual/performing arts, creativity, physical and sport, social/leadership, and culture-specific abilities.Of the schools reporting formal identification (n=768), the most frequently identified area of giftedness was in the domain of intellectual/academic giftedness (96.9%). The intellectual/academic area of giftedness was described as referring to students with exceptional abilities in one or more of the essential learning areas (i.e., mathematics, language and languages, technology, health and physical education, social sciences, science, and the arts).

Although the survey did not specifically address mathematics, the findings under the intellectual/academic ability area of giftedness can be used to give insight into formal identification procedures, and provisions for mathematically gifted students. These findings can be linked to the literature in terms of effective practice and outcomes for students.


This is one of the critical issues in gifted education and most publications related to identification recommend the use of multiple methods of identification. It is not surprising therefore that only 4.1% of the schools formally identifying intellectual and academic abilities reported a reliance on one method. Almost half the respondent schools (49.1%) indicated use of between two and four methods of identification. It is acknowledged that the multi-method approach is more likely to identify those with mathematical promise [5] and a multi-method approach is also more likely to be inclusive rather than exclusive so that there is representation of students who may otherwise be overlooked in the identification process [6].

The most often utilised methods for formal identification by the 768 schools that identifiedintellectual/academic giftedness was teacher observation (94.1%) and achievement tests (89.7%). Teacher observation may have been the most strongly supported method of identification, but issues are raised about potential ineffectiveness. Effectiveness is known to be variable and is attributed to the level of formality of the identification process and the teacher’s professional knowledge and expertise [6]. Teachers may not have proficient knowledge about the mathematically gifted and may be focusing on a narrow set of skills such as computational ability. Teacher bias and stereotyping may also contribute to ineffectiveness.

In New Zealand teachers commonly use results from a nationally standardised mathematics test, the Progressive Achievement Test (PAT) (published by the New Zealand Council for Educational Research) for identifying students as gifted and talented when they score above the 90th percentile. These tests assess a student’s recall, computational skills,aspects of understanding, and application of various mathematical operations and concepts. One of the suggested applications of the test is to assist teachers in selecting able students requiring special mathematics programmes [7]. However the tests are not designed specifically for gifted and talented students; they are age and class norm referenced. It is highly likely that gifted students will reach a ceiling and this effect is not taken into account.A study that used Receiver Operating CharacteristicAnalysis showed that, independent of any chosen percentile, the PAT was 78% accurate in identifying mathematically gifted and included errors of commission and omission [8].Multiple choice tests such as the PAT are limited in that the results can only be considered summatively. It is important in mathematics for teachers to be able to gain insight into aspects such as the student’s problem solving abilities [9], communication of mathematical ideas, attitude and interest towards mathematics, and application of mathematical understandings [10].

Other methods reported as being used (in order of preference from highest to lowest) include teacher-made tests, teacher rating scales, portfolios, parent nominations, self-nomination, IQ tests, peer nomination and whanau (extended family) nominations. Teacher-made tests provide the opportunity for more open-ended and divergent questions so that a student can reveal creative problem solving approaches or elegant solution paths. Student portfolios can provide a rich variety of samples of student work showing evidence of levels of achievement, interest and perseverance. They can provide useful information not evidenced in test results. Examples include reports from a statistics or measurement investigation and geometric models. Self-nomination is recommended by the Ministry of Education as part of the multi-method approach [11]. Individual student behaviours and characteristics, including culture, age and self perceptions are likely to influence the potential effectiveness of self nomination. Only when a class climate or school culture permits students to acknowledge openly that they are gifted are they likely to nominate themselves for gifted programmes [12]. The Ministry of Education recommends parents and whanau as valuable sources of information in the identification process [11]. Parents can provide information that may not be available from classroom observations and tests. They may also recognise mathematical talent at a young age and advocate for their children to be challenged mathematically by the classroom programme [13].


The survey probed schools’ provisions for gifted and talented students and included their preference for enrichment and acceleration, school-based provisions, and provisions within classrooms. The majority of schools (61.4%) reported a preference for a combination of acceleration and enrichment approaches to provision for their gifted and talented students. Of those schools not preferring a combined approach, enrichment was more favourably viewed(35%) than acceleration. The most frequently reported provision was in the area of intellectual/academic abilities. Of the765 respondents that provided school-based programmes for intellectually and academically gifted and talented students the most frequently reported strategy was withdrawal or pull-out programmes (67.6%). Over half the schools reported the use of cross-age grouping (52.7%), competitions (54.4%), and external examinations (50.7%). Of all the reported school-based provisions, early entry was the least frequently provided strategy.

Classroom-based provisions were reported by the majority of schools (82%) in the survey.These classroom-based provisions included ability grouping, independent study, teacher planning, learning centres, individualised education plans, curriculum compacting/diagnostic-prescriptive teaching and the use of a consulting specialist teacher. Of the 1049 schools reporting classroom-based provisions 87% indicated that they used ability grouping. There is certainly confusion in the literature about the use and interpretation of the term ‘ability grouping’ and debate continues on whethergifted and talented students should be homogeneously grouped (by ability) or heterogeneously grouped (mixed ability).Given that grouping is merely an organisational strategy it is not necessarily the type of grouping that should be of concern rather the quality of the programme. Research shows that grouping students in a class by ability especially where the curriculum is accelerated as well as enriched is an effective strategy [14]. Curriculum compacting/diagnostic-prescriptive teaching was only used by 30% of the schools providing for gifted and talented students within the regular class. Curriculum compacting makes sense for teaching mathematically gifted and talented students in that material that students have already mastered can be identified and replacement strategies can be provided that allow for more meaningful and productive use of time. Although this study enabled us to document the type and frequency of classroom provision it gives no indication of the quality or effectiveness of the classroom-based programmes. This is an area recommended for further research.

In the learning domain of mathematics what are the options in New Zealand for providing for the mathematically gifted?The national curriculum statement‘Mathematics in the New Zealand Curriculum’ document [15] aims to support teachers in developing enrichment programmes for the more able students. This curriculum statement is organised in to strands (mathematical processes, number, measurement, geometry, algebra, and statistics) and levels (1-8) to cater for the range of students (ages 5-18) in the school system. The expectation is that students work at a level that best suits their needs and abilities. “It is not expected that all students of the same age will be achieving at the same level at the same time, nor that an individual student will necessarily be achieving at the same level in all strands of the mathematics curriculum” ([15], p. 17). The document provides teachers with additional learning experiences labelled as Development Band activities. This feature was introduced to encourage teachers to extend and enrich the programmes of the more able mathematics students and to support their efforts in that direction.

The intention of the development band is to encourage teachers to offer broader, richer and more challenging mathematical experiences to faster students. Work from the development band should allow better students to investigate whole new topics which would not otherwise be studied and to work at a higher conceptual level. ([15], p. 19)

Throughout the curriculum document for each strand and level there are suggested topics and activities for development band work. These include number bases, history of mathematics, cryptography, modulo arithmetic, and topology. To date these are presented as topics with limited support provided for teachers. For our primary teachers this represents a real challenge as many do not have the content knowledge to be able to take a topic and present it in a suitably challenging way for the gifted student. These teachers usually need to access additional support from mathematics specialist teachers.

The New Zealand Association of Mathematics Teachers (NZAMT) provides a Development Band Certificate Course for students from Years 4 to 12. The material is presented in modules and based on themes incorporating mathematics from a variety of related content areas. The modules are based on an investigative approach with some structure for the students but also pose questions that give opportunity for flexible pathways and more open-ended investigations. It is this type of investigation that helps maintaina student’s interest and stimulates intellectual development [16].

Competitions are used by 54% of the schools and are available for students at a variety of levels. These include local, national and international competitions, team and individual entries. Competitions give gifted and talented students an opportunity to put their talents to the test, a chance to showcase their special abilities and to receive recognition and acknowledgement [17]. Competitions are available for students from about 8 years of age. These include team problem solving competitions at the local level, and national and Australasian competitions for individuals. Competitions are also provided through the NZAMT website, and Otago University and Auckland University. Some of these competitions are completed online and engage students in problem solving skills through more interactive ways.

One of the fifteen Ministry funded Talent Development Initiatives includes the New Zealand Mathematical Olympiad. A series of activities involving many students culminates in the selection of the team for the International Mathematical Olympiad. However, leading up to this all potential students are invited to complete a set of problems, the solutions are submitted for marking and from this 24 students are chosen to attend a training camp. Students can access further problems from a website especially set up to provide practice problems and support.

Other provision options in New Zealand schools for students gifted and talented in mathematics include early entry to intermediate, secondary and tertiary study (early entry to primary school is not a legal choice), dual enrolment with the New Zealand Correspondence School, grade skipping, cluster grouping, pull-out programmes, virtual schools and mentoring. Eleven per cent of the schools in the survey utilised mentoring which is one of the strategies recommended by the Ministry of Education [11]. The mentorship involves a partnership with a more experienced older student or adult who can provide guidance and help develop expertise as an appropriate role model [18]. Mentoring is recommended as a culturally-appropriate strategy for our gifted and talented Maori students [19].Mentoring builds upon the Maori tradition where a tohunga, a master works with a child to nurture his or her gifts and talents.


The national survey demonstrated a growing awareness of the need to provide for gifted and talented students. As the Minster of Education stated; “it is an exciting time for the education of gifted and talented students” ([8], p.1). The government is committed to supporting the achievement of gifted and talented students and has made amendments to national requirements so that these students will have to be explicitly identified as a group and schools will need to make appropriate provisions. As a consequence of the Ministry’s mandatory requirements forgifted and talented students, it is more likely that the mathematically gifted will be identified and provided with an education matched to their individual learning needs.

Despite this growing awareness of the need to provide these students with an individualised and appropriate education,schools reported that the implementation of suitable programmes was hindered by a lack of teacher professional development in gifted education.Professional development was cited as a critical factor in developing effective practices. With a reported preference for implementing a combined approach of enrichment and acceleration the implementation of this approachneeds to be supported by teacher professional development to ensure that programmes are informed by a sound understanding. Future studies could consider the evaluation of the effectiveness of teacher professional development and subsequent identification and provisions for mathematically promising students. Other factors identified as barriers to effective identification and provisions includedteacher confidence and competence, funding, resources, and time.

Gifted and talented learners are found in every group within society yet our Maori students are under-represented; they are not being identified and culturally appropriate provisions are not being planned, implemented or evaluated. It means that Maori perspective and values (and those of other ethnic minority groups) must be embodied in all aspects of the education of gifted students. This provides a real challenge for those teaching in bicultural and multicultural schools.

The school is a powerful catalyst for the development of talent and a responsive school will consider a range of potentially effective approaches that are differentiated and individualised. In New Zealand there is a lack of reported evaluation of the effectiveness of provisions. Likewise the international literature abounds with advocacy and descriptive articles but with limited empirical studies outside of the United States. Programmes themselves for students gifted and talented in mathematics should be based upon sound research. The curriculum should be rich in depth and breadth and at a pace commensurate with their abilities. It should not only meet academic needs but address social and emotional needs.

There are valuable implications to be drawn from the research. The extensive literature review revealed a paucity of national and also international research validating recommended approaches. The questionnaire results further indicated that even when approaches have been validated in the literature they were not being implemented and conversely some practices unsupported by research were being implemented. The message has to be that practices in both the identification and provision should be grounded in reputable theory and research.

The survey reported in this discussion paper raises as many questions as it provides answers. The questionnaire did not allow for any indication of the quality of reported practices and so the effectiveness of the school practices remains an unknown. Effectiveness is contingent upon how programmes are designed and delivered, hence the need for ongoing evaluation and for the evaluation to be used to inform changes in practice.

The research demonstrates that New Zealand schools are making an effort to identify and provide for the education of gifted and talented students but as one respondent wrote;“It’s along journey and we ain’t there yet!”


        1. Mallard, T ., 2004, Research to help schools support gifted students . Retrieved April 12 2004 from http://www.beehive.govt.nz
        2. Riley, T. L., Bevan-Brown, B., Bicknell, B., Carroll-Lind, J., & Kearney, A., 2004, The extent, nature and effectiveness of planned approaches in New Zealand schools for providing for gifted and talented students . Wellington: Ministry of Education.
        3. Ministry of Education, 2001, Working Party on Gifted Education. Report to the Minister of Education. Retrieved December 12 2003 from http://www.executive.govt.nz/minister/mallard/gifted_education/index.html
        4. Office of the Minister of Education , 2002, Initiatives for gifted and talented learners. Wellington: Ministry of Education.
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        6. Davis, G. A., & Rimm, S. B. (1998). Education of the gifted and talented (4th edition). Needham Heights, MA: Allyn & Bacon.
        7. Reid, N. A., 1993, Progressive achievement test of mathematics , Wellington: New Zealand Council for Educational Research.
        8. Neiderer, K., Irwin, R. J., Irwin, K. J., & Reilly, I. J., 2003, Identification of mathematically gifted children in New Zealand. High Abilities Studies, 14(1) 71-84.
        9. Niederer, K., & Irwin, K. C., 2001, Using problem solving to identify mathematically gifted students. In M. van den Heuvel-Panhuizen, (Ed.), Proceedings of the 25th Conference of the International Group for the Psychology of Mathematics Education (pp. 431-438). Utrecht, The Netherlands: Freudenthal Institute.
        10. Wieczerkowski, W., Cropley, A. J., & Prado, T. M. , 2000, Nurturing Talents/Gifts in Mathematics. In K. A. Heller, F. J. Monks, R. J. Sternberg, & R. F. Subotnik (Eds.), International handbook of giftedness and talent (pp. 413-425). Oxford: Elsevier.
        11. Ministry of Education, 2000, Gifted and talented students: Meeting their needs in New Zealand schools . Wellington: Learning Media.
        12. Gross, M. U. M., & Sleap, B., 2001,Literature review on the education of gifted and talented students . Retrieved April 14 2004 from: www.aph.gov.au/senate/committee/eet_ctte/gifted/submissions/sub032.doc
        13. Lupkowski-Shoplik, A. E., & Assouline, S. G., 1994, Evidence of extreme mathematical precocity: Case studies of talented youths. Roeper Review, 16(3), 144-151.
        14. Kulik, C. C., & Kulik, J. A., 1992, Meta-analytic findings on grouping programs. Gifted Child Quarterly, 72 - 76.
        15. Ministry of Education , 1992,Mathematics in the New Zealand Curriculum. Wellington: Ministry of Education.
        16. Eddins, S., & House, P., 1994, Flexible pathways: Guiding the development of talented students. In C.A.Thornton & N.S. Bley (Eds.), Windows of opportunity: Mathematics for students with special needs (pp. 309-322). Reston, VA: National Council of Teachers of Mathematics.
        17. Riley, T. L. & Karnes, F. A ., 1999, Forming partnerships with communities via competitions. Journal of Secondary Gifted Education, 10(3), 129-134.
        18. Casey, K. M., & Shore, B. M., 2000, Mentors’ contributions to gifted adolescents affective, social and vocational development. Roeper Review, 22(4), 227-231.
        19. Bevan-BrownJ., 1996, Special Abilities. A Maori Perspective. In D. McAlpine & R. Moltzen (Eds.), Gifted and talented: New Zealand perspectives (pp. 91-110). Palmerston North: ERDC Press.

Brenda Bicknell





Carmel M. Diezmann, James J. Watters

Abstract: Generating interest in mathematics for gifted students starts in the early years of schooling. In this paper we summarise our research undertaken over a number of years, which has adopted a lateral approach to challenging the mathematically gifted. The paper argues that teachers can provide rich mathematical learning opportunities within the regular classroom through strategies that capitalise on the essentials of the regular curriculum but which provide that extra challenge needed by gifted students. These strategies enhance both mathematical knowledge and motivation towards mathematics.

Keywords: mathematically gifted, elementary students, instructional strategies, regular classroom.


In this proposal, we focus on instructional strategies to support mathematically gifted elementary students in the regular classroom. Mathematically gifted students are distinguished from their non-gifted peers by their mathematical reasoning, their capacity for learning, and their mathematical orientation [10]. Although strategies such as acceleration, enrichment programs, curriculum compacting, competitions can support gifted elementary students, they may be inaccessible to some students, or fail to cater for particular types of gifted students (e.g., underachievers, minority groups). Additionally, students participating in specialist gifted programs usually spend some time in their regular classroom. Therefore, of particular importance is ways to develop rich mathematical learning opportunities within the regular classroom. Many teachers have limited knowledge, resources and time to provide challenging experiences for mathematically gifted students. However, in less than optimal environments, gifted students are “at risk” and may demonstrate boredom, loss of interest in or commitment to mathematics, limited metacognition, and poor behaviour [3], [13]. It is not surprising that teachers often attempt to cater for mathematically gifted through higher-level work in addition to normal classroom work. However it is fallacious to assume that mathematical giftedness relates solely to speed, accuracy and proficiency in set tasks. Our research has focussed on developing instructional strategies for general classroom use that differentiate the curricula in ways that enrich learning experiences laterally rather than sequentially. These approaches undertaken in the regular classroom enhance gifted students’ reasoning, accommodate their capacity for learning, and foster their interests in accordance with best practice in gifted education [12].


Lateral strategies are designed to be challenging but interconnected to the regular curriculum. Underpinning these strategies is the view that tasks should be a mechanism for empowering students as mathematicians rather than discrete activities to be completed. Examples of four lateral strategies drawn from a range of studies with variously aged elementary students follow. Common to these strategies is the nexus between what is advocated in research on gifted students and contemporary approaches to teaching mathematics.

1. The level of challenge in regular classroom tasks can be increased be problematising mathematical tasks [8], [6]. Problematisation includes inserting obstacles to the solution, removing some problem information, or requiring students to use particular representations or develop generalisations. For example, after easily calculating the sum of the numbers from 1 to 10, a young gifted student used a generalisation to calculate the numbers from 1 to 100 [4]. This task was appropriately challenging for the gifted student but still connected to the regular curriculum focus of summing sequential numbers.

2. Implementing mathematical investigations requires students to apply and create mathematical knowledge in posing and solving novel problems. Investigations are central to the reforms advocated internationally to develop children’s mathematical power [1]. For example, after some guided investigations with Smarties [sweets], young children formulated investigable questions about Smarties and explored these: “How many regular Smarties weigh the same as one giant Smartie?” (See Figure 1) and “What is the most popular coloured Smartie?” [9]. The open-ended nature of this task enabled gifted students to explore their mathematical interests and to develop their capabilities.

3. Extending manipulative use capitalises on visual-spatial or kinaesthetic representations to support higher-level thinking. For example, young gifted students’ construction of a number-line to represent the distances of the ten brightest stars required the application of knowledge of large numbers, relative magnitude and scale [2]. This application goes beyond the more typical use of a prepared number line to represent a simple set of numbers.

4. Modifying educational games can provide rich mathematical and social learning opportunities for the gifted. For example, in a simple place value card game, students are dealt three cards in order to represent a three-digit number. They then take turns to select one card at a time from the pile of remaining cards with the goal of making their number larger. For example, if Student A was dealt “345”, and picked up an “8” they could make “845” (See Figure 2). This game becomes more challenging if students select the card to be replaced before they see their new card (See Figure 3). As shown by Students B and C’s responses, the selection of the card to be replaced results in substantially different outcomes (See Figure 3). This game proved to be a particularly useful thought-revealing activity [11] because it uncovered gifted students’ propensity for calculated risk taking, their erroneous reasoning, and indicators of their metacognitive processes.


The adoption of more lateral instructional strategies expands gifted students’ mathematical knowledge through challenging experiences that are connected to the regular curriculum. These lateral strategies have six particular advantages. Firstly, lateral strategies are not add-ons or extensions but take the existing curriculum and problematise, adapt and enrich the experiences for gifted students. Secondly, these approaches lend themselves to both collaborative and independent learning [5]. Thirdly, through strong linkages to the regular curriculum, lateral strategies provide underachieving gifted students with opportunities to oscillate between regular activities and more challenging activities according to their capability, confidence and motivation. Fourthly, lateral strategies address the mathematics reform agenda, for example through attention to learning through problem solving [7]. Fifthly, because lateral approaches capitalise on elementary teachers’ pedagogical knowledge, for example in using games, teachers have opportunities to develop their confidence and competence in working with gifted students. Finally, lateral strategies are thought-revealing activities [11] that provide opportunities for the identification and development of gifted students’ mathematical ability.


    1. Baroody, A., & Coslick, R. T. Fostering children’s mathematical power: An investigative approach in K-8 mathematics instruction. Mahwah, NJ: Lawrence Erlbaum, (1998).
    2. Diezmann, C. M., & English, L. D. Developing young children’s multi-digit number sense. Roeper Review, 24(1), (2001), pp. 11-13.
    3. Diezmann, C. M., & Watters, J. J. Bright but bored: Optimising the environment for gifted children. Australian Journal of Early Childhood, 22 (2), (1997), pp. 17-21.
    4. Diezmann, C. M., & Watters, J. J. Catering for mathematically gifted elementary students: Learning from challenging tasks. Gifted Child Today, 23(4), (2000), pp. 14-19.
    5. Diezmann, C. M., & Watters, J. J. The collaboration of mathematically gifted students on challenging tasks. Journal for the Education of the Gifted, 25(1), (2001), pp. 7-31.
    6. Diezmann, C. M., & Watters, J. J. The importance of challenging tasks for mathematically gifted students. Gifted and Talented International, 17 (2), (2002), pp. 76-84.
    7. Diezmann C, Thornton C., & Watters J. Meeting special needs. In F. Lester & R. Charles (Eds.), Teaching mathematics through problem solving, Reston, VA: NCTM.(2003), pp. 169-182.
    8. Diezmann C. M, Watters J. J., & English, L. The needs of mathematically gifted Learners: Raising the challenge of academic tasks. Paper presented at the International Congress of Mathematics Education, Tokyo, Japan, 31 July - 6 August 2000.
    9. Diezmann, C. M., Watters, J. J., & English, L. D. Difficulties confronting young children undertaking investigations. Proceedings of the 26th Annual Conference of the International Group for the Psychology of Mathematics Education.Utrecht, Holland: PME. (2001), pp. 353-360
    10. House, P. (Ed.) Providing opportunities for the mathematically gifted K-12. Reston, VA: NCTM. (1987).
    11. Lesh, R. Hoover, M., Hole, B., Kelly, A., & Post, T. Principles for developing thought revealing activities for students and teachers. In A. Kelly & R. Lesh (Eds.), Handbook of research design in mathematics and science education Mahwah, NJ: Lawrence Erlbaum. (2000), pp. 591-646.
    12. National Association for Gifted Children Pre-K–Grade 12 gifted program standards. Washington, DC: NAGC. (1998).
    13. Sheffield, L. J. Developing mathematically promising students. Reston, VA: NCTM. (1999).


Dr Carmel Diezmann

Centre for Mathematics and Science Education, Faculty of Education

Queensland University of Technology

Victoria Park Road, Kelvin Grove,

Brisbane Q4059


+61 7 3864 3803 (Phone)

+61 7 3864 3989 (Fax)

e-mail c.diezmann@qut.edu.au

Dr James J Watters

Centre for Mathematics and Science Education, Faculty of Education

Queensland University of Technology

Victoria Park Road, Kelvin Grove

Brisbane Q4059


+61 7 3864 3639 (Phone)

+61 7 3864 3643 (Fax)

e-mail j.watters@qut.edu.au

url: http://education.qut.edu.au/~watters/




Dace Bonka, Agnis Andžāns

Abstract: the uses of the methods of mathematical induction, invariants, extremal element and interpretation in problem solving contests for 5-8 Grade students are dis cussed. Examples of corresponding problems from Latvian contests are provided.

Key words: Math Contests for Junior Students, Mathematical Induction, Invariants, Extremal Element, Mean Value, Interpretation.


Latvia is a small country with app. 2 300 000 inhabitants in North Eastern Europe. Only 10 years ago it gets rid of Soviet occupation which has enormously damaged Latvian nation, economy and culture.

Latvia has no rich natural resources as oil, iron, diamonds etc. So almost only its resource to restore the country is well-educated people. With such a small population each talent is of great importance. Therefore serious and constant efforts are made to develop the abilities of each child in a best way and to help the teachers in this work.

Advanced mathematical education starting from 3-4 th Grades is considered as one of the most essential tools for at least two reasons:

A. There are four main roads of inquiry discovered in the world: rational, empiric, emotional and modeling. The main representatives of theseroads are respectively mathematics, natural sciences, literature and informatics. Thus mathematics represents one of these roads and is an important tool for development of skills for two others – empiric and modeling.

B. A mathematical way of thinking – the creation and analysis of amathematical model, the inner need to fortify own judgment andconclusions, the creation of a strong deductive view of the world is an important element of forming the personality.

Mathematics serves the pupil as an example of truth, objective and independent from daily needs and individual desires.


Since late seventies the educational system of Latvia is being reformed almost in a non-stop way. The recent changes are aimed to switch from acquiring large amount of knowledge to acquiring large amount of skills, including information search and evaluation skills. So the general methods of reasoning should be considered as having great educational value. Mathematics provide examples of such general methods. Among most important of them are mathematical induction (MIM), invariant method (IM), extremal element method (EEM), mean value method (MVM) and the method of interpretation (MI). They are even not only purely mathematical methods, but rather general principles discovered by the mankind in a long period of time.

Accordingly to the new standards of education [1] the acquaintance with these methods should be welcome for all middle school students, adding new uses of them in high school. Nevertheless, though they are common in various contests and a lot of teaching aids are prepared and published (see, e.g., [2]), little effort is made to incorporate them into the official school curricula (see, e.g., [3]).


The abovementioned methods are described in a number of papers and monographs (see, e.g., [4]-[6]). Nevertheless, our opinion shows that only part of their uses is suitable for middle grade students. Namely, the classes of problems best for introductory examples are following:

    • for MIM – inductive constructions, inductive algorithms, the recurrent relations;
    • for MVM – the simple form of Dirichlet principle, the uses of it in finding the extremal values;
    • for IM – the invariance of the result of counting, the idea of parity and its uses;
    • for EEM – the greatest and the largest element of a set of numbers, the convex hull;
    • for MI – uses of geometry in algebra, physical interpretations.
At the beginning of studying these methods it is important that the general idea of the method must not be hidden in the formal manipulations connected with writing down the solution "for thegeneral case". Therefore the so called "general special cases" are ofgreatest importance; the analysis is done only for some values ofparameters which contain all the essential features of the general case.Such a concept has been derived from the concept of "full system ofexamples" in program testing.

The studying of the abovementioned methods has alsofollowing positive general pedagogical effects:

    • the demonstration of the unity of mathematics,
    • aesthetically considerations,
    • the possibility to use the underlying ideas of the methodsoutside mathematics,
    • the broadening of the concept of proof.

There are two main classes of competitions, mainly in problem solving.

A. Mathematical Olympiads

They are organized at three levels:

    • school Olympiads, often supported by universities; they are usually held in November,
    • regional Olympiads held in 39 different places in Latvia each year in February,
    • Open math Olympiad held each year in April. This competition is a very large one; morethen 3000 participants arrive in Riga.
All these competitions are open to everybody who wants to participate.

Other present-way competitions are organised at schools, at summer camps etc.

B. Corresponding contests

There are many students who need more than some 4-5 hours (usually allowed during math Olympiads) to go deep enough into the problem. For such children a system of correspondence contests has been developed:

    • “Club of Professor Littledigit” (CPL) for students up to the 9th Grade. There are 6 rounds each year, each containing 6 relatively easy and 6 harder problems. Problems are published in the newspaper “Lat vijas Avīze” (having the largest circulation in Latvia), and on the INTERNET.
    • “Contest of young mathematicians” for students up to 7th Grade, originally developed for weaker students than the participants of CPL, especially in Latgale, the eastern region o f Latvia. The problems are published in regional newspapers and on the INTERNET, and today it has become popular all over Latvia.

General principles established in Latvia for constructing a contest problem set are described in [7].

From the previous it is clear that math contests should cover broad spectrum of mathematics, as more as better. It is particularly important alsobecause olympiad and contest problems from previous years arebroadly used afterwards in everyday teaching practice. Main criteria accordingly to which the set should bewell – balanced follow:

    • it should cover main areas of school mathematics: algebra, geometry,number theory and combinatorics. Analysis is included into algebra here,and combinatorics is understood in a broad sense including not onlycounting but also existence and non-existence of combinatorial objects.Particularly, the general combinatorial methods (mathematical induction,invariants, mean value, extremal element, interpretation) must be reflected;
    • it should contain both problems of deductive nature and problems ofalgorithmical nature;
    • there should be problems of "prove it!" type along with problems in which the answer must be found by the solver;
    • "discrete" mathematics and "continuous" mathematics both are to be represented.
It is clear that all these desires hardly can be implemented in a small set of five or six problems. Indeed, there are very few sets of problems which can be calledsatisfactory from all mentioned points of view. Good balance can and must beachieved during the whole school year.
    • Mathematical induction
There are few problems for middle grades that use the full version of it. Usually inductive constructions consisting of 5-6 steps are a good choice.

Example 1. Does there exist a 10-digit number consisting only of digits 2 and 3 and dividing by 1024?

The construction is done by adding digits one by one to the end of the number and ensuring that n-digit number is dividing by 2n.

    • Method of invariants
    • in geometry problems.
Correspondence contests are organized mainly for those grades where only basic concepts of geometry are studied. Therefore the main part of geometry problems there are of combinatorical nature, e.g., problems of dissection.

Example 2. A floor in a room of dimensions 6 m ´ 10 m must be covered by tiles of two types each consisting of 4 squares (see fig.1).

Can it be done if we have 5 tiles of type a) and 10 tiles of type b)?

Appropriate colouring and the use of the fact that the number of elements in a finite set doesn’t change if the counting order is changed solve the matter.

    • in algebra and arithmetic’s problems.
Invariants connected with a divisibility, mainly, parity, are the most common tools here.

Example 3. There are 6 cats and 7 dogs drawn on the blackboard. You can erase one dog, or you can erase two cats and draw a dog instead of them. Prove that the last animal on the blackboard will be a dog.

Replace each cat by number 1 and each dog by number 2, and investigate how does the sum of all numbers on the blackboard change.

    • Extremal element method
The main challenge here is to realize what kind of extremality should be considered.

Example 4. Does there exist a rectangle consisting of whole equal quadratic cells which can be dissected into “squares” and “crosses” (see fig.2) so that at least one part of each type is obtained?

Suppose it is possible, and consider the “cross” being situated no lower than any other one.
    • Method of interpretations
In solving problems by means of this method we translate a problem into an “appropriate language” in which its solution is much easier or even trivial. It can also be said that we build a model of the problem, solve the corresponding problem for the model and then translate the solution back to the original language. The classical example is analytical geometry. Clearly the effectiveness of this method increase together with the knowledge of various branches of mathematics. So it is not very appropriate for younger students. The main interpretations here are those using graphs or modelling one game by another.

Example 5.Let A1A2...A100 be a regular 100-gon. Each two vertices are connected with a straight road. There are two players at A1 and A3 correspondingly. The move consists ofgoing from one vertice to another using only one road. It is forbidden to go to the vertice where there is another player at the moment; it is forbidden to go along the road that has been already used. Who has made the last move is the winner.

Let's pay the attention to the fact that the game with initial positions of players at A1 and A2 is an easy exercise, but the game on the 101-gon is an unsolved problem.

The algorithm for the game of example 5can be developed from the isomorphism similar to that between the hexagons in Fig. 3.

    • Mean value method
The pigeonhole principle (Dirichlet principle) is the undoubtfull leader here for all grades.

Example 6. What is the largest number of natural numbers between 2 and 120 inclusively such that no one of them is a prime but each two of them are coprime?

The answer “4” is obtained by considering the least prime divisor of each of selected numbers. Note that together with pigeonhole the extremal element method is also presented in this solution.

A large number of sorted problems with solutions can be found in teaching aids published in [2].


This paper was prepared partially with the support of the state – investment project “Latvian Education Informatization System”.


        1. The Regulations of the State Standard of Basic Education. The Cabinet of Ministers of Latvia, Riga, 2000 (in Latvian).

        2. http://www.liis.lv
        3. I.France, L.Ra māna. The Strategies of Content and Teaching of Mathematics: Necessity of Changes. – Proc. Int. Conf., Liepāja: LPA, 2002, pp. 67-70 (in Latvian).
        4. A.Engel. Problem-Solving Strategies. Springer, 1998.
        5. J.Tabov, P.Taylor. Methods of Problem Solving. AMT, 1996, 2003.
        6. T.Andreescu, R.Gelca. Mathematical Olympiad Challenges. Birkhauser, 2000.
        7. A.Andžāns, L.Ramāna. What Problem Set Should be Called Good for a Mathematical Olympiad. – Matematika ir matematikos destymas – 2002. Kaunas, Technologija, 2002, pp. 5-8

Dace Bonka, Mg.Math.

University of Latvia

19 Rainis Boulev. Riga, LV-1586


Phone: +371 7034498

E-mail: dace.bonka@lu.lv

Agnis Andžāns, DSc., Prof.

University of Latvia

19 Rainis Boulev. Riga, LV-1586


Phone: +371 7034498

E-mail: agnis@lanet.lv



Djordje Kadijevich, Zora Krnjaic

Abstract: This study examined the relation between cognitive style and link between procedural and conceptual mathematical knowledge. It used a sample of 34 mathematically talented eleventh-grade students. A significant positive correlation was found between the students’ achievements on the administered Embedded Figures Tests (where “field-dependence-independence” cognitive style has a very specific perceptual connotation) and the measures of link between their scores on procedural and conceptual mathematical knowledge. The same relation was again found in a group of particularly talented students who participated in mathematical competitions (N = 16), but not in the control group comprising other talented students (N = 18).

Key words: Cognitive Style, Procedural Knowledge, Conceptual Knowledge, Linking Procedural and Conceptual Knowledge, Talented Students, Mathematics Education, Upper Secondary Education.


As regards cognitive style, academically gifted students may be more field independent than their counterparts involved in the regular education program [1]. A previous analysis of the structure of test achievements (including EFT test [2]) shows that highly gifted students in mathematics and technical sciences, who are scholarship holder candidates in Serbia, are characterized by a form of general fluid intelligence contained in figurative tests [3].

Linking procedural and conceptual mathematical knowledge is an important yet neglected goal of mathematics education, the attainment of which is a complex but achievable enterprise [4]. When the effects of developing procedural and conceptual knowledge and establishing links between them are examined, cognitive style should be included since some students, because of their less flexible (say more field dependent) cognitive style, may demonstrate unbalanced gains in these knowledge types resulting in missing or poor links between them [5].

Having in mind the presented research context, the objective of this study was to examine the relation between cognitive style and link between procedural and conceptual mathematical knowledge. The rest of this paper presents how these variables were operationalized, what were the main results, and why such results might be obtained.


The study used a sample of 34 mathematically talented students who came from two eleventh-grade classes of Matematička Gimnazija - the specialized high school in Belgrade for mathematically talented students.

The study had a correlative design. The variables were: class (1 – control group comprising self-financed students, 2 – target group comprising students financed by the state; all these students passed the school entrance examination, but free education is reserved for those who achieved better total scores at that examination), cognitive style, procedural knowledge, conceptual knowledge, and link between procedural and conceptual knowledge (hereafter denoted by P-C link).

Cognitive style was measured by the last (perhaps the hardest) 16-item subtest of Bukvi ć’s modification of Embedded Figures Tests [2] standardized for Yugoslav population. This instrument was administered under a group setting (one class at a time; both classes within 45 minutes) in exactly 10 minutes by a psychologist (the second author of this report). The subject’s cognitive style was represented by the first principal component factor score obtained from the subjects’ answers. The factor score reliability (Lord-Kaiser-Caffrey) was .83.

Procedural and conceptual knowledge were measured by scores given to different solutions of the following task taken from [6]: “In the square below, M and N are midpoints of the corresponding sides. Determine the numerical value of sin a.”

Answers to this task were written in a questionnaire administered under a group setting (one class at a time; both classes within 45 minutes) in exactly 20 minutes by a mathematician (the first author of this report) who also precisely scored the students’ answers. For each correct solution, student received 1 point for conceptual knowledge and 1 point for procedural knowledge. Partial credit was given when: (1) student wrote a solution plan (how the task can be solved) that was partially or fully correct (for conceptual knowledge), and (2) some of the required calculations (plan implementation) were performed correctly (for procedural knowledge).

P-C link was measured by formula 2PC/(P2 + C2 ) introduced in [7], where P and C denote total scores on procedural and conceptual knowledge, respectively. For those students where PC equaled 0 (when one or both types of knowledge was (were) not demonstrated), P-C links were equal to 0. Note that such a defined link takes values from interval [0, 1], where a bigger number indicates a stronger P-C link.

The data collection was realized at the end of the fall semester in January 2004 during regular school lessons. The authors told the subjects that this study would examine their problem solving performance and the subjects willingly provided the requested data.


The correlations among procedural knowledge, conceptual knowledge, P-C link, cognitive style, and class are presented in Table 1.

The correlations between cognitive style and P-C link for the two classes are given in Table 2.


Two important findings emerged from this study.

First, there was a significant positive correlation between cognitive style and P-C link.

Second, while this relation also held true for the target group of particularly talented students participating in mathematical competitions, this was not the case for the subjects of the control group.

As Table 1 evidences, there were no significant differences between the two classes with respect to cognitive style, procedural knowledge and conceptual knowledge, which is acceptable (not expectable) as all these students belong to the same highly selected student sample (two students solved the problem in 3 ways and two in two ways; three students pursued a specific way of solving the problem ¾ not listed among 16 different solutions summarized in [6] ¾ and one of them succeeded; an easier variant of this solution is given in the appendix). But, while particularly talented students were competition oriented (even in 20 minutes, two students solved the problem in 3 ways), most students in the control group were not so directed. So, the competitors, who obtained more unbalanced scores on procedural and conceptual knowledge than those in the control class (recall that correlation was -.38), may in general, compared to other talented students, be more prone to procedural errors or calculation ignorance when conceptual knowledge is correct, which was evidenced by the student’s answers (P-C link for 8 students, a half of the group, was 0; in the control group just 3 students had such a link). However, the more competitor’s cognitive style was field independent, the stronger P-C link he/she established, which is an important finding that, to our reading, has not been reported so far. This was, however, not the case for those in the control group, which, among others, might be caused by a less competitive approach to day-to-day learning requiring a more relaxed cognitive processing. Another reason was suggested by the examined data: while no significant differences between the classes were found for the variance of the link measure (.20 vs. .11; Levene's Test: F = 2.90, p = .098), the variance of the cognitive style measure was higher in the competitor class (1.43 vs. .61; Levene's Test: F = 6.81, p = .014), which was a less homogenous sub-sample. Despite such plausible explanations, further investigations are still needed, which should also include ordinary (not mathematically talented students) from gymnasium or vocational school.


The authors wish to express many thanks to the students for their successful participation and the school staff for their professional support and assistance.


As the hypotenuses of the shaded right-angled triangles (Fig. 2) are perpendicular (one triangle can be rotated into the other by angle of 90?), AD is an altitude of triangle ABM. If AB = 2, the area of triangle ABM (or BMA) is 2, and, since BM = , AD is equal to . As = , the numerical value of is (obtained from the relation ).

The above-mentioned successful solver utilized the fact that MD is an altitude of triangle ANM, whose area, compared to that of triangle ABM, cannot be obtained at a glance.

However, when the length of MD is known, the numerical value of can easily be found (from the relation = ).


    1. Terrell, S. R . The Use of Cognitive Style as a Predictor of Membership in Middle and High School Programs for the Academically Gifted . Paper Presented at the Annual Meeting of the American Educational Research Association, April 1-5, 2002, New Orleans, Louisiana. Available at: http://www.gifted.uconn.edu/siegle/aera/NewOrleans/SteveTerrell.pdf.
    2. Witkin, H. A. Oltman, P. K., Raskin, E., & Karp, S. A. A Manual for the Embedded Figures Tests. Consulting Psychologists Press, Palo Alto, 1971.
    3. Krnjaic, Z. Intellectual Giftedness in Young People (in the Serbian language). Institute of Psychology, Faculty of Philosophy, Belgrade, 2002.
    4. Kadijevich, Dj. & Haapasalo, L. Linking procedural and conceptual mathematical knowledge through CAL. Journal of Computer Assisted Learning, 17, 2, 156-165, 2001.
    5. Kadijevich, Dj., Maksich, S. & Kordonis, I. Procedural and conceptual mathematical knowledge: comparing mathematically talented with other students. In Velikova, E. (Ed.), Proceedings of the Third International Conference Creativity in Mathematics Education and the Education of Gifted Students (pp. 103-108). V-publications, Athens, 2003.
    6. Barry, D. An Abundance of Solutions. Mathematics Teacher, 85, 5, 384-387, 1992.
    7. Kadijevich, Dj. Impact of mathematical-self concept on linking procedural and conceptual mathematical knowledge (in the Serbian language). Presented at 8th Scientific Meeting “Empirical Research in Psychology”. Abstracts (p. 21). Institute of Psychology & Laboratory of Experimental Psychology, Faculty of Philosophy, Belgrade, 2002.


Djordje Kadijevich, Ph.D.

Graduate School of Geoeconomics

Megatrend University of Applied Sciences

Makedonska 21

11000 Belgrade


Cell phone: +381 11 3373 796

E-mail: djkadijevic@megatrend.edu.yu

Zora Krnjaic, M.S.

Institute of Psychology, Faculty of Philosophy

University of Belgrade

Cika Ljubina 18-20

11000 Belgrade


Cell phone: +381 11 639 724





Elena Koublanova

Abstract: In Developmental Mathematics courses, the variation in the abilities of students can be recognized by comparing them in a learning environment such as work in study groups. Capable individuals soon become group leaders, whose role is not only to solve a problem but also to explain a solution to other students. Working in study groups is particularly effective in solving word problems. Ancient word problems that implement elements of games serve well for in-group studies. Other problems, which are useful for collaborative learning, come from introductory topics in Logic. Examples of word problems appropriate for in-group studies are presented.

Key words : Capable students, collaborative learning, study group, old word problems.


In two-year colleges, a number of students need introductory and developmental courses to refresh their basic knowledge in Mathematics. Most of such students are adults who took Mathematics years ago. Success in studying Mathematics is particularly important for adult students because it increases their confidence and self-esteem and creates necessary conditions for further education. A population of students in Developmental Math classes is non-homogeneous and includes individuals that either failed or never could complete a school program as well as capable students who studied Mathematics a long time ago. While chances are high that an instructor would concentrate on weaker students in order to bring them to an average level, students with higher potentials can also benefit from such classes. Learning specific topics in study groups could be effective for both average and strong students. This approach is particularly productive in solving word problems. Ancient word problems that contain elements of puzzles serve successfully for in-group learning.


The abilities of students in introductory mathematics courses can often be recognized by comparing students in the learning environment. When working in study groups, capable students become group leaders, whose role is not only to solve a problem but also to explain a solution to others. Such students would demonstrate an interest in the subject, creativity in problem solving, and capacity to properly present material. Apt students usually enjoy “teaching” and tend to use an adequate mathematical language and logical reasoning. Leadership in study groups gives capable students satisfaction and confidence and results in better understanding the subject.

Even a simple word problem, if it requires writing a linear equation in one variable or a system of linear equations, can be difficult for average students in developmental courses. Ancient word problems, which describe a problem with humour and implement an element of a game, help students to relax and overcome a mathematical anxiety. Many ancient word problems can be found in literature, and it is always possible to find a problem suitable for a given audience. Some problems can be solved by reasoning, whereas others require writing equations or systems of equations. In this paper, we present examples of old Greek, Arabian, Hindu, and Russian problems. These problems are suitable for collaborative work in Algebra classes or can be used as “warm-up” exercises in any Math courses.


Sinbad and Hinbad [5; p. 12]. Sinbad and Hinbad each owned the same number of horses. How many should Sinbad give Hinbad so that Hinbad has six more than Sinbad? This problem can be solved more easily by reasoning than by writing an equation. Problems of this type, although rather simple, may be challenging and serve as an indicator of a student’s ability for logical thinking. After such simple puzzles, problems that require writing and solving linear equations are offered. Many ancient problems, including the well-known “Diophantus Tomb Problem” [2; p. 25], involve linear equations. Writing an equation is the most important step in solving word problems. Isaac Newton, in his book “The Universal Arithmetic”, wrote that, in order to answer a question related to quantities, one just needs to translate a problem from a native language to the language of Algebra. He gave examples of how such a translation is performed. Not much has changed from Newton’s time, and contemporary textbooks in Beginning and Intermediate Algebra use exactly the same language to describe steps in solving word problems. Some problems that require writing a linear equation are given below.

Ancient Greek Puzzle [5; p. 26]. Demochares has lived one fourth of his life as a boy, one fifth as a youth, one third as a householder, and has spent thirteen years beyond that. How old is he?

Apples Thief [5; p. 33]. While three watchmen were guarding an orchard, a thief slipped in and stole some apples. On his way out he met the three watchmen one after the other, and to each in turn he gave a half of apples he then had, and two besides. Thus he managed to escape with one apple. How many had he stolen originally?

Old French Problem [2; p. 22]. A man spent one third of his money and lost two- thirds of the remainder. He then had 12 coins. How much money did he have at first?

Father and Son [4; p. 54]. A father was 32 years old, and his son was 5 years old. In how many years will the father be ten times as old as the son? Even though this problem is a simple one in terms of writing equation, it has a negative solution that needs an explanation such as “it was two years ago”.


Horse and Mule [4; p. 35]. A horse and a mule each carried a heavy load, and the horse complained about its weighty pack. The mule said to the horse: “Why are you complaining? If I take one bag from you, my burden would be twice as heavy as yours. But if you take one bag from my back, your load would be equal to mine.” How many bags did each of them carry?

Starlings and Trees [3; p. 12]. Flying starlings saw some trees on their way. When the starlings sat one on a tree one starling remained without a tree. When two starlings sat on each tree one tree remained without a starling. How many starlings and how many trees were there?

Hindu Age Problem [2; p. 42]. I am twice the age that you were when I was your age. When you get to be my age our ages will total 63 years. How old are we?


Another set of problems, in which gifted students can benefit from in-group studies, comes from Logic. Maurice Kraitchik, who gathered a wonderful collection of ancient curious problems and puzzles, wrote in the introduction to his book “Mathematical Recreations” [2; p. 13]: “…there are problems…whose solutions are obtained by the direct exercise of the powers of reasoning, without the intervention of formulas and computations. For mathematics is applied logic in its simplest and purest form”. Although such a characterization of Mathematics looks quite simplified, many mathematicians as well as students in Math classes can appreciate the charm of ancient logic problems. Logic problems and puzzles may be offered to start collaborative work in any Math course. They would provide an opportunity for group members to better recognize each other, without the pressure of “serious” mathematical problems, and allow gifted students to show their potential in Mathematics and logical reasoning.

Two Medieval Problems [1; p. 27, 28].

1. A man goes to a well of water with two jars, of which one holds exactly three pints and the other exactly five pints. How can he bring back exactly four pints of water?

2. Three men robbed a gentleman of a vase, containing 24 ounces of balsam. While running away they met a glass-seller, from whom they purchased three vessels. On reaching a place of safety they wished to divide the booty, but found that their vessels contained 5, 11, and 13 ounces respectively. How could they divide the balsam into equal portions?

What are Their Ages ? [5; p. 72]. Iskandar once asked his friend Kamar the ages in years of his three children. The following conversation ensued:

K: The product of their ages is thirty six.

I: That does not tell me their ages.

K: Well, by coincidence, the sum of their ages is your own age.

I: (after several minutes of thought) I still don’t have enough information.

K: Well, if this will help, my son is more than a year older than both his sisters.

I: Oh good! Now I know their ages.

What were their ages?

The Problem of the Pandects [2; p. 28]. A hungry hunter came upon two shepherds, one of whom had three small loaves of bread, and the other five, all of the same size. The loaves were divided equally among the three, and the hunter paid eight cents for his share. How should the shepherds divide the money? Versions of this problem can be found nearly in every collection of old problems.

Three Greek Philosophers [2; p. 15]. Wearied by their disputations and by the summer heat, three Greek philosophers lay down for a little nap under a tree of the Academy. As they slept a practical joker smeared their faces with black paint. Presently they all awoke at once and each began to laugh at the other. Suddenly one of them stopped laughing, for he realized that his own face was painted. What was his reasoning?

Robbery [5; p. 45]. A man was being tried for robbery. Three witnesses came forth and made the following statements. First witness: “The defendant has committed over a dozen robberies in the past!” Second witness: “That is not true!” Third witness: “He certainly committed at least one robbery!” As it turned out, only one of the witnesses had told the truth. Is the defendant guilty of robbery or is he innocent?

In general, it is easier to solve a logic problem or puzzle than to explain the solution to others. For students, who solved a problem by intuition, the following step of justification and reasoning provides a good possibility for learning logic rules of inferences. The knowledge of such rules is important for studying Math courses in which not many proofs are presented.


Capable students can benefit from collaborative learning in Math courses of all levels. In Developmental Math courses, working in study groups is particularly useful for solving word problems. Ancient word problems can help average and capable students to overcome mathematical anxiety and enjoy learning Mathematics. As leaders of study groups, students with higher potentials can enhance their abilities in Mathematics, improve their skills in logic and critical reasoning.

In the Middle Ages, competitions in solving word problems were carried out, and participants had to present solutions in a verbal form. A similar approach is applicable to the work in study groups. Using the instructor’s advice, apt students can thoroughly work up all the steps of translating verbal expressions into the language of Algebra and explain solutions to other students. Our experience shows that solving word problems by an instructor is less effective, no matter how many examples are given, than students’ collaborative work and independent discussion.


        1. Ball, W.W.R. Mathematical Recreations & Essays, The Macmillan Co., NY, 1963.
        2. Kraitchik, M. Mathematical Recreations, Dover Publications Inc., NY, 1942.
        3. Olechnik, S.N., Nesterenko, U.V., Potapov, M.K. Old Recreational Problems, Nauka, Moscow, 1988 (In Russian).
        4. Perelman, J. Recreational Algebra, Nauka, Moscow, 1976 (In Russian).
        5. Smullyan, R. The Riddle of Scheherazade, Harcourt Brace & Co., NY, 1998.

Elena Koublanova, Ph.D., Associate Professor

Department of Mathematics, Community College of Philadelphia

1700 Spring Garden Street, Philadelphia, PA 19130,


Office phone: 1-215-751-8928, E-mail: ekoublanova@ccp.edu



Emiliya Velikova

Abstract:The paper presents:
    • a new Model of Joint and Independent Creative Work (MJICW) between a leading teacher (or a team) and a“creative-productive gifted” student;
    • original authors methods (a system of mathematical problems,a system of transformations, idea for creating new transformations and problems, teaching methods), corresponding to the MJICW and also authors experience with gifted students in Bulgaria, England, Greece.
Key words :Creative work, Transformations, Methods of Instructions, Students’ problems


The encroachment of new information technologies and the continuous integration of mathematics with other sciences in the contemporary world call for:

    • gifted mathematicians, able to discover and summarize diverse conclusions within the information flow, and to generate new ideas;
    • personalities, able to develop their creative potential, enrich their knowledge and experience and apply them to socially useful areas and activities.
Those factors define the search for new models of educational and upbringing activities, which, based on the positive experience, ensure opportunities for developing thestudent not only as a person who studies his/her lessons or a consumer, but also as a creator of knowledge, a person who uses the methods of creative mathematical activity for acquiring creative application of information and mental processes in an integrated and inductive way, directed to real-life problems, in conformity with his/her interests and abilities.


The organized combination of activities, directed towards developing the productive potential of students and creating students’ products, is viewed as joint and independent creative work between a leading teacher (or a team) and a“creative-productive gifted” student. There are three inter-related parts to it[1], [9], [13], [14]:

    • independent research and creative activity of the teacher himself;
    • joint work between the leading teacher and the gifted student with the purpose of preparing the latter adequately for independent creative activity;
    • independent creative activity of the gifted student, through which he satisfies his needs for achievements and their expression, by creating, formulating and presenting the product of his creation in front of an appropriate audience.
The model (MJICW) is principally based on the Schoolwide Enrichment Model of J.Renzulli and S.Reis [6], [7] but also includes original authors methods (a system of mathematical problems that includes geometrical inequalities [13], a system of transformations [2], mathematical methods for creating new transformations and new problems [13], teaching methods [1], research methods [13])and also authors experience[10], [14].

The main principles, set in the model, are:

    • the volunteer principle – a strong desire of both student and teacher to participate inMJICW;
    • principle of creative applicability – application of abilities, knowledge and skills for generating a creative product, meant for a specific audience.
The main goals are
    • stimulating students for mathematical creativity in a specific area of mathematics;
    • providing conditions for realization of creative activity.
Activities of Type I (general preparation, general exploratory activities).

The main goals are:

    • forming a strong interest in students towards a specific mathematical area;
    • developing dedication to the task set, i.e. converting the interests into internal motives for achievement.
Contents of activities Type I. Through those activities students get the opportunity to enter the world of science, according to their knowledge and interests, to experiment with their ideas without being evaluated. Those activities are: intensive group or independent discussion of scientific questions and problems, which can be more easily solved through transformations, lectures and meetings with experts in the respective area, etc.The activities of this type strongly influence the natural curiosity and activity of students, bring satisfaction, develop their interest, initiative and independent thinking. The leading teacher is responsible for planning and implementing those activities.

Activities Type II (preparation for creative work, group training activities). They reveal to students the charm of research andprepare them for self-creative work.

The main goals are:

    • developing the creative abilities, producing research skills, acquiring knowledge to a degree of applicability in the field of the student’s interests ;
    • acquaintance with the methods of creative work in the area;
    • achieving an optimal motivation level of creative work.
Contents of activities Type II. Acquainting with the methods of creative mathematical work in the area of transformations – the leading teacher and the mathematician present to the students their experience in creating new problems and transformations, generating new dependencies,generalized relationships, etc.A particular attention is given to investigation, analysis and decision-making, acquisition of new knowledge, formation and presentation of results. The example set by the members of the leading team is extremely important.

Activities of Type III (creative work) comprise of independent choice and realization of creative mathematical work in and through which the student expresses himself/herself as a creator.

The main goals are to provide conditions for mathematical creativity and presentation of creative products to an appropriate audience.

Contents of the activities of Type III. The student follows his/her own ideas for as long as he/she needs, tries different ways of solving the problem chosen by him/her for achieving the goal set.The methodological help of the teacher in the collective planning of activities,determining the appropriate reference and target audience, discussing the ideas, additional training if necessary, proofreading of the paper, forming and presentation of the results, etc. The leading teacher organizes contacts with experts, provides access to scientific reference and technologically new information sources, audience for the presentation, etc.

MJICW is a completepsychological and pedagogical system, which displays the characteristic elements of pedagogical and research, as well as creative activity in the specific area of mathematics [13].


The program correspond to the MJICWincludes [2], [13]:

    • Developing a system of mathematical problems that includes or can be solved by geometrical inequalities;
    • Developing a system of transformations;
    • Developing methods for creating new transformations and problems;
    • Developing teaching methods.
    • Creative training of students.
The method of instruction corresponding to the activities of Type I has been divided into three topics:
    • Examples of interesting problems that include geometrical inequalities.
    • The Best Low and Upper Bound of Function.
    • Specific Inequalities.
The topics have been divided into several parts in respect to problem solving methods such as:
    • Geometry helps Inequalities;
    • Geometry helps Algebra;
    • Vectors help, and others.
The problems have been chosen according to the following criteria:
    • to include mathematical methods, both known to the students, and nonstandard;
    • to be diverse and to meet the requirements at a competitive level;
    • to have been applied to similar purposes at home or abroad;
    • to serve as grounds for student mathematical creative process.

Problem 1. If x, y, z are positive real numbers, prove the inequality

(1) [5].

Solution. Weconstruct (Fig. 1). Thenbyapplying the cosine theorem for the triangles ADB, BDC ? ADC respectively, we get

(2) .

The inequality (1) follows from the existing of , i.e. from the inequality , which we apply in (2).

The equality occurs in (1) when the point .


(3) , i.e.

(4) or

(5) .

Problem for self work

For any prove the inequalities:

(6) [4],

(7) [3].

When do the equalities occur?

Thetransformationofalgebraicproblemintogeometric oneisapplied into the topic “The best low and upper bound of a function”.

Problem. Find the best upper bound of the function

(8) ,

where .

The problems on view are developed, the areas of work or the given conditions are changed, and new problems are created.

New problem. Prove the inequality

(9) ,

where and ? > 0.


    • Individual work on the basis problem with known methods (one month before).
    • Discussions on students solutions.
    • Showing very attractive solutions by the method of transformations.
    • Solution of closely problems.
    • Creating new problems by enrichment, analogies, etc.
The teaching method, corresponding to the activities of Type II is developed in the next three topics:
    • Method of Transformations. Application of Transformations for Solving Geometrical Inequalities.
    • Dual Geometric and Algebraic Inequalities;
    • GeometricandMixed Inequalities for a Triangle;
    • Discovering New Transformations and Their Application for Creating Original Hard Problems.
Example of mathematical transformation:

Theorem 1.Let ABC be a triangle with the usual elementsa, b, c, F, , R, r, . Then there exists a triangle ABC1with elements (Fig.2) [2]:

(10) , , , , , , , , , , , , .

Consequence. For any true triangle inequality (equality)

(11) ,

there exists another true inequality (equality) in the form

(12) ,

where the elements are given by (10), i.e.

(13) .

Definition. Activityduringwhichfromtheinequality(11) weget intoinequality (11) (12), i.e. we carry out a transfer (13), is called “Parallelogram Transformation” in respect to the side c of the triangle .

The creation of a transformation requires from a given triangle ? to construct a new one ?' and to find mathematical relations in the form (10) between the elements of these triangles. It is possible, in specified conditions, to construct many new triangles ?' and to create and prove many new transformations.

The transformation has a lot of applications:

    • Creating of new asymmetric inequalities:

(15) .

    • Creating of “Generators” of inequalities:
(16) , ,

where an equality occurs when and only when .

For every in (16) different new inequalities are created or well-known inequalities, such as (15) if are proved.

    • Creating of combination of transformations.
    • Others.
    • Examining a transformation, whose formulae are closer to students’ knowledge; applying those transformations for the solution of competition problems and creation of new ones.
    • Show the methods of creative activity of mathematicians – the paths for the creation of some transformations, their summaries, combinations and scientific applications, for the creation of new problems.
Theprocessofsolvingeveryproblemincludesthethree important stages of the general empiric model of the Method of Transformations:
    • transformationof theproblemintoan easyone - transformation (for example, algebraic problem is transformed into easier geometric problem);
    • solving the dual problem by known methods (for example, geometric methods);
    • transformationof the results in respect to the given problem (for example, into algebraic solution) – inverse transformation (if it exists).
Problem for self work. Prove the inequality

(17) ,

wherex, y, z are positive real numbers. The condition for x, y, z is given. When does an equality occur in (17)?

New Problem.Prove the inequality

(18) ,

wherea, b, c aresidesofanytriangle. The condition for a, b, c is given. When does an equality occur in (18)?

New Problems. Prove the inequalities:

(19) ,

(20) ,

(21) ,

whereA, B, C are angles of an arbitrary triangle. When do equalities occur?

The teaching method, corresponding to the activities of Type III depend on the students’ interest, ideas, speed of work, etc. It can include:

    • meeting of students and creator of some transformations proper for high school education of mathematics;
    • presenting some transformations andcombinationoftransformationsand the creative methods of the authors for their creation;
    • finding two suitable triangles using geometry or algebra methods, depending on the interests and abilities of the students;
    • determining the transformational formulae in a “beautiful” appearance;
    • choosing prominent inequalities from the scientific sources in the field;
    • application of transformations;
    • choice of “beautiful” inequalities;
    • designing of problems;
    • presentation of results in front of a proper auditorium;
    • assessment of one’s ownpresentation.

One of the main tasks is to research the role of the application of MJICW for creative character of the students’mathematical problems.

Students from 9th-12 th grade are the object of the research.

The subject of the research isthe creative character of the problems created by the students.

For example, through the comparative analysis of the creative character of students’ products (Fig.3÷5, tables 2, 3) it was found that:

    • the problems of the students from the experimental group have the properties of creative products in the field of geometrical inequalities;
    • the theoretical and application instruments based on the method of transformations, form the foundation for student creativity.

From the survey conducted and the observation it was concluded that in order to achieve the goals of MJICW it is important to:
    • acquaint parents and students with the experiment;
    • have a high-achieving leading teacher;
    • plan activities in advance;
    • develop study materials in advance;
    • supply enough reference material;
    • have a well-organized teaching team with good communication;
    • have a specialist in transformations participate in the experiment.
In conclusion, the comparative analysis, factor analysis, correlation analysis and regression analysis of the experiment results showed that the application of MJICW as a set of stimuli, provides conditions for the development of the productive potential of students and for making creative products.


Problem of A.Velikov [12]. Prove the asymmetric inequality

(22) ,

wherea, b, care sides of a triangle with perimeter 2.

Problem of T.Petrov [11]. Find one or more solution of the equation

(23) .

ProblemofJ. Nikolov.Prove the inequality

(24) ,

where , and ?, ?, ?, r, R the usual elements of an arbitrary triangle

Problem of K. Ivanova. Prove the inequality

(25) ,

where a , b, g , r, s, R are elements of an acute triangle.

Problem of N.Velikova. Prove the inequality

(26) ,

where a, b, c, s are elements of a triangle and .

Problem of D.Lorenov. Prove the inequality

(27) ,

where a, b, c, R, ma, mb, mc are the usual elements of an arbitrary triangle.

Problem of M.Penchev. Prove the inequalities:

(28) ,

where s, r, R are the usual elements of an arbitrary triangle.


MJICW is specified for extracurricular work only in mathematics for students in grades IX-XII and requires the participation of a leading teacher and a mathematician, who have some achievements in the area specified. The model includes:

    • activities which can be applied to extracurricular or regular studies in each area of mathematics;
    • specific activities, which can be applied only to the field of transformations.
MJICW can be applied to:

a) extracurricular forms of education in mathematics for “gifted” students belonging to different age groups, as follows:

    • determining of an appropriate topic or area in mathematics and complementing MJICW with new activities;
    • development of appropriate lessons and aids;
    • completing the methods of education in accordance with the requirements of the working area and the age features of the students;
b) regular education of students in grades IX – XII, which should:
    • create a strong interest in students towards different problems and areas of mathematics;
    • plan where to apply the products created by students;
c) training of future teachers of mathematics by:
    • carrying out of theoretical analysis of the teacher’s psychology and his/her willingness to work with gifted students;
    • testing of problems created and lessons developed, as well as their presentation by the future teachers.
        1. Bilchev, S.J.&E.A.Velikova Jointly and Independently Creative Work Of Talented Students and There Tutors I: The Process Of Learning Of Mathematics, Mathematics and Education in Mathematics, Proceedings of the Twenty Fourth Spring Conference of the Union of Bulgarian Mathematicians, Svishtov, April 4 - 7, 95, Sofia, Bulgarian Academy of Sciences, (1995), p. 88 - 108, plenary report (Bulgarian).
        2. Bilchev, S.J. & E.A. Velikova On an Generation of Parallelogram Transformation, Mathematics and Education in Mathematics, Proceedings of the Fifteenth Spring Conference of the Union of Bulgarian Mathematicians, Sunny Beach, April 2-6, 1986, Sofia, Bulgarian Academy of Sciences, 1986, pp.525-531.
        3. Klamkin, M. S. Problem 805. CRUX Math., 1, 1983.?.22.
        4. Klamkin, M. S. Problem 1394. CRUX Math., 10, 1988.?.302.
        5. Problem ? 1090 ?vant, 2, 1988.p.26
        6. Renzulli, J.S. & S.M.Reis The Schoolwide Enrichment Model: A Comprehensive Plan for the Development of Creative Productivity. In: Hanbook of Gifted Education, N. Colangelo (Ed.), Boston, MA: Allyn & Bacon, 1991. pp.111-141.
        7. Renzulli, J.S. & S.M.Reis The Schoolwide Enrichment Model: A How-to Guide for Educational Excellence. (2nd ed.), Mansfield Center,CT: Creative Learning Press, 1997.p.423.
        8. Renzulli, J.S. & S.M.Reis. Research Related to the Schoolwide Enrichment Traid Model. Gifted Child Quarterly, 38, No1, Winter, 1994. pp.7-20.
        9. Velikova, E.A. Models and Programs for Diagnostics and Training of Gifted Students in Mathematics, Proceedings of the Union of Scientists - Rousse, Bulgaria, Ser.5, Mathematics, Informatics and Physics, Vol.2, 2002, pp.15-21.
        10. Velikova, E. A., P.Vlamos Creative Education on Mathematics for Gifted Students, Proceedings of the Second International Conferences “Creativity in Mathematics Education and the Education of Gifted Students”, 15-19 July 2002, The University of Latvia, Riga, Latvia, 2002.pp.106-107.
        11. Velikova, E. A., S.J.Bilchev & P.Vlamos Student’s Hard Geometry Problems, Geometry & Mathematics Competitions, A. Rejali and F. Sajadi (Eds.), Isfahan University of Technology & Isfahan Mathematics House, Iran, Paper Contributed to Topic area 1, 4th WFNMC Conference (4-11 Audust, 2002), Melbourne, Australia, 2002.pp.227-234.
        12. Velikova, E. A., S.J.Bilchev & P.Vlamos Hard Problems Created by Gifted Students, Digital Journal “Revista Escolar De La Olimpiada Iberoamericana De Matematicas” , Spain, June, 2002. www.oei-campus.org/oim/revistaoim
        13. Velikova, E.A. “Stimulating the Mathematical Creativity of Students (9-12 Classes)“, Dissertation thesis, State Specialized Scientific Council of the High Attestation Commission of the Republic of Bulgaria, Sofia, 2002, 369 pages
        14. Velikova, E. A. One Bulgarian Experience in Identifying and Developing Mathematically Talented Students,In:Developing Mathematically Promising Students,L. Sheffield (Ed.), Published by the National Council of Teachers of Mathematics, USA, 1999, pp 203-206.

Emiliya Velikova , Ph.D., Faculty Socrates and International Relations Co-ordinator

Department of Algebra and Geometry, Centre of Applied Mathematics and Informatics,

Faculty of Education, University of Rousse

8 Studentska str., 7017 Rousse


Cell phone: +359/889 625 222 , Fax: +359/82/845 708


emily@ami.ru.acad.bg, emivelikova@yahoo.com


- The Third International Conference “Creativity in Mathematics Education and the Education of Gifted Students”, August 3-9, 2003, Rousse, Bulgaria


– TSG4: Activities and Programs for Gifted Students, ICME-10, July 4-11, 2004, Copenhagen, Denmark




Emiliya Velikova, Svetoslav Bilchev, Marga Georgieva

Abstract: The paper presents a diagnostic technology for finding “creative-productive gifted” students based on thecombined theoretical and experimental research on the problem from the aspect of the sciences: psychology, pedagogy, methods of teaching in mathematics, statistics.

Key words : Giftedness, Characteristics of Gifted Students, Mathematical abilities, Diagnostic technology


Every society is interested in gifted persons who are able to develop their creative potential, enrich their knowledge and experience and apply them to socially useful areas and activities.

The formation of future creators is a long and hard process, which starts at school. Hence, the teachers play the most important role in the identifying process.

The emphasis on clarifying the notion “creative giftedness” is placed on the basic characteristics of the gifted and talented students in the contemporary theories and research and J.Renzulli’s three-ring conception of “giftedness”.

The “schoolhouse (academically) gifted” students quickly adapt to the school environment, have a high IQ, numerous manifestations of it, etc. The “creative-productive gifted” students possess or can develop a cross-section of three clusters of characteristics: above-average general and specific abilities, ahigh level of creativity and a high level of task commitment [4]. They have a high creative quotient (CQ),often take a risk, etc. In J.Renzulli’s conception “creative-productive giftedness” is looked upon not as a “magical gift”, a genetically determined fact, because the qualities from the three classes are encountered in all normal people, but as a chance of developing a combination of qualities in an environment close to the real creative work [4], [5]. G.Piryov notes that those qualities get a higher level and more specific development; they are more successfully combined in creative personalities’ characters [3]. In this sense J.Renzulli postulates: “our orientation must be redirected towards developing “gifted behaviours” in certain students (not all students) at certain times (not all the time) and under certain circumstances” [4]. Giftedness is a condition that can be developed [4].

J.Renzulli emphasize [4]:

    • Both types are important.
    • There is usually interaction between the two types.
    • Special programs should take appropriate provisions for encouraging both types of giftedness as well as the numerous occasions when the two types interact with each other.

The comparisons of general and typological characteristics in Mathematical work (Table 1, Table 2) of these two types of giftedness express many research results on creative-productive people that although no single criteria can be used to determinate giftedness,

Persons who have achieved recognition because of the unique accomplishments and creative contributions process a relatively well-defined set of three interlocking clusters [4], [2]:

1) Well above average abilities (general and specific)

    • general abilities: high level of abstract thinking, verbal and numerical reasoning, spatial relations, memory, and word fluency; adaptation to the shaping of novel situations encountered in the external environment; actualization of information processing, rapid accurate, and selective retrieval of information.
    • mathematical abilities:
2) High level of creativity

3) High level of task commitment


The main aim of the diagnostic technology is choosing “creative-productive gifted” students for joint and independent creative work with a leading teacher (or a team) in Mathematics (Fig. 1) [1], [8], [9]. The main goal of the work with gifted students is stimulating students for mathematical creativity in the specific area of mathematics.

The first step of the diagnostic technology: identifying creative-productive gifted students:

    1. Measuring above average general abilities by IQ. As the students’ intelligence is growing up it is necessary to apply adapted IQ test [6]. IQ has to be more than 80%.
    2. Measuring creativity by standard CQ test (Mensa). CQ has to be more than 70%.
    3. Measuring above average mathematical abilities by [7]:
    • teachers’ assessment - Y1 – teacher in regular mathematics training;
    • teachers’ assessment - Y2 – teacher in extracurricular work. Teacher training for assessment the expression of students’ mathematical abilities is necessary. The teachers observe: activity character and results; verbal expression of thinking process; explanations and answers in the process of problem solving.
    • experts’ assessment using Mathematical test (CMA).
The sum of Y1, Y2 and CMA has to be more than 50%.

The second step of the diagnostic technology: defining the students’ and their parents’ longing.


The diagnostic technology was applied as a part of an experiment in finding creative-productive gifted students and stimulating their mathematical creativity for real science work [8]. 69 students (9-12 classes) that showed interest in Mathematics took part in it. The analysis of the results shows that the research instruments have a high level of reliability. Some of the results are present: Fig.2 ÷ Fig. 6, Table 3.

Remark: Every index in Table 3 [8] stands for a mathematical ability [2]


Diagnostic technology can be applied to newly developed research on the extracurricular work in many areas of mathematics, or developed for other age groups as well


        1. Bilchev, S.J.&E.A.Velikova Jointly and Independently Creative Work Of Talented Students and There Tutors In TheProcess Of Learning Of Mathematics, Mathematics and Education in Mathematics, Proceedings of the Twenty Fourth Spring Conference of the Union of Bulgarian Mathematicians, Svishtov, April 4 - 7, 95, Sofia, Bulgarian Academy of Sciences, (1995), p. 88 - 108, plenary report (Bulgarian).
        2. ?rutetskii, V. A. Psychology of mathematical abilities of Students., M., Prosveshtenie, 1968 (Russian).
        3. Piryov, G. Psychology in Your Life. Horizonts, S., Partizdat, 1975. p.301 (Bulgarian).
        4. Renzulli, J.S. The Three-Ring Conceptions of Giftedness: A Deve lopmental Model for Creative Productivity. In Conceptions of Giftedness, R.J.Sternberg & J.E.Davidson (Eds.), Cambridge, London, New York, New Rochelle, Melbourne, Sydney, Cambridge University Press, 1986. pp.53-92.
        5. Renzulli, J.S. & S.M.Reis The Schoolwide Enrichment Model: A Comprehensive Plan for the Development of Creative Productivity. In Hanbook of Gifted Education, N. Colangelo (Ed.), Boston, MA: Allyn & Bacon, 1991. pp.111-141.
        6. Serebryakov, V & S.Langer Check The Intelligence of Your Child by IQ Tests for Children From 7 Till 17 Years Old, S., LIK, 1999. p.124 (Bulgarian).
        7. Velikova, E.A. & M.Y.Georgieva, Diagnostic of Mathematical Abilities,Proceedings of the Union of the Scientists – Rousse, vol.1, R., Bulgaria, 2001. pp.67-73 (Bulgarian).
        8. Velikova, E.A. “Stimulating the Mathematical Creativity of Students (9-12 Classes)“, Dissertation thesis, Government Specialized Scientific Council of the High Attestation Commission of the Republic of Bulgaria, Sofia, 2002, 369 pages
        9. Velikova, E. A. One Bulgarian Experience in Indentifying and Developing Mathematically Talented Students,In DEVELOPING MATHEMATICALLY PROMISING STUDENTS,L. Sheffield (Ed.), Published by the National Council of Teachers of Mathematics, USA, 1999, pp 203-206.

Emiliya Velikova, Ph.D., Faculty Socrates and International Relations Coordinator

Department of Algebra and Geometry, Centre of Applied Mathematics and Informatics,

Faculty of Education, University of Rousse

8 Studentska str., 7017 Rousse


Cell phone: +359/889 625 222 , Fax: +359/82/845 708


emily@ami.ru.acad.bg, emivelikova@yahoo.com


- The Third International Conference “Creativity in Mathematics Education and the Education of Gifted Students”, August 3-9, 2003, Rousse, Bulgaria


– TSG4: Activities and Programs for Gifted Students, ICME-10, July 4-11, 2004, Copenhagen, Denmark

Svetoslav Bilchev, Ph.D., Associate Professor, Dean

Department of Algebra and Geometry

Centre of Applied Mathematics and Informatics,

Faculty of Education, University of Rousse

8 Studentska str., 7017 Rousse


Cell phone: +359/886 735 536, Fax: +359/82/845 708



Marga Georgieva, Ph.D., Professor, Vice-Dean

Faculty of Education

“St. Ciril and St. Methodius” University of Veliko Turnovo

2 T. Tarnovski str., 5000 Veliko Turnovo,


Cell phone: +359/888 409 044, Phone: +359/62/ 649 832, Fax: +359/62/649 834

E-mail: myg@uni-vt.bg , tyg@vali.bg



Eugenia Meletea

Abstract: The presentation includes:
    a. The issue of the education of Gifted/Talented students in Greece,
    b. The results from 2002, since the Scientific Association – Non Profit Organization “ ?PLOUN”, (Development of Pedagogical Reasoning to Support Gifted/Talented Students) appeared,
    c. “IASONAS”: Educational Networks, Programs, Activities and Studies for Gifted/Talented Students,
    d. “APLOUN” and “IASONAS”, as an “Arch”, beginning and introducing a Specific Issue (as the Education of Gifted/Talented Students in Greece), and the use of Models of Mathematical Knowledge to develop:
    e. Educational Networks,
    f. Proposals of Holistic and Specific Strategies,
    g. Quality Developing Activities,
    h. Heuristic of Sophisticated Mental Models of Mathematical Knowledge.

For the first time in Greece, a non-profit organization (APLOUN), and the “IASONAS”, Educational Open Networks, Programs, Activities and Studies, deal with the issue of the Educational Needs of Gifted/Talented students. Participation in the above and my experience as an independent Researcher during the last 10 years, looking for teaching quality indicators, appropriate education materials and methods is considered.


In recent years, an important research initiative into matters of special education has been underway in Greece. However, while measures have been taken by the Greek state for the education of individuals belonging to specific special categories, such as children with physical, mental or learning problems et al., there is no such welfare concerning the special Gifted/Talented children/young adults.

Several of these Gifted/Talented children/young adults are not discovered and properly treated from the pre-school age; they demonstrate poor performance at school, subsequently being referred to for control of learning difficulties or problems in behavioral and social adaptability.

In Greece, during the last 10 years schools start to accept and support students with Talent in Music (20 Secondary Public Schools). For other category of Talent there do not exist schools to support and educate students. Only in Big Cities there are supporting classes for Athletes and underway is the program for the schools to support Talents in Fine Arts.

Many of competitions are taking place and are organized by the Greek Ministry of Education, several Public and Scientific Associations, for students in the Secondary Education. The Greek Mathematical Association (EME) organizes several competitions in Mathematics: there are National (Panellenic) and International Competitions and Olympiads, see [3]. Other competitions also are taking place (Chemistry, Computers, etc).

In the Greek Public elementary schools, there is no any Talent support program. Only in some Private Schools they use basic programs supporting Talent, and there is the philosophy of the Enrichment Model. Greek Schools are not allowed to apply Acceleration programs; there is no way for Greek Students to Grade Skipping, if this is necessary for them.

From multi-source research it has been estimated that about 50% Gifted/Talented children abandon their studies by their adolescence. The absence of welfare and special planning to meet the special needs of these children results in a large part of them, though possessing outstanding abilities, demonstrate poor academic performance and have difficulties in adapting to their environment. Furthermore, insufficient or misconceived pedagogical approaches serve to undermine the fragile emotional world of these children/young adults, due to the suppression or obstruction of the development of their high mental abilities, with direct consequences to their regular emotional development.


The Greek Scientific Association – Non Profit Organization called “APLOUN”, consists of people from various fields, who, out of their sensitivity to the particular issue, have set the proper concern for these children as their goal.

The foundation of “APLOUN” aims to the development of a framework to support Gifted/Talented children/young adults, hereby setting as its basic objectives the following [5]:

    • Informing the general public about the special needs facing Gifted/Talented pupils/students.
    • Timely distinction and recognition of those children who possess outstanding abilities, so that they are properly treated, initially by family, and later by their broader educational and social environment.
    • The initiation of research regarding the education of Gifted/Talented pupils/students in our country, aiming to locate those factors that contribute to their regular mental as well as emotional development.
    • The investigation of the concept of differentiability and also of the special needs facing Gifted/Talented children/young adults.
    • The encouragement of collaborative learning within an environment providing ‘different’ individuals with equal opportunities for educational group effort, with specialized material based on recent educational methods and means of teaching.
    • The production of specialized educational material based on models already in application in other countries, to meet the special needs facing Gifted/Talented pupils/students, namely the Heuristic Model and the School-wide Enrichment Model, which may also be taken advantage of by pupils of standard classes. [2].
    • Special training and reinforcement of educational staff in new methods and productive procedures.
    • The development of a communication network aiming to inform the general public, the educators, the specialized scientists and parents.
    • The connection of the Society with international organizations involved in similar endeavors.
    • The participation and reinforcement of parents in their effort to support and meet the needs of Gifted/Talented pupils/students.
    • The Laboratory of new Technologies in Communication, Culture, Education and Media Studies of the National & Capodistrian University of Athens, Dpt. of Communication & Media Studies, Athens, Greece.
    • The Medical Office for Developmental Paediatrics at the 1st University Clinic of the University of Athens (Juvenile Hospital “Aghia Sophia”); it is appointed as President.
    • The Department of Physical Education and Athletics of the University of Athens.
    • Professors from several Departments and Universities in Greece and Oversees.

In an initial effort aiming to the internal organization of the Association so as to achieve the fulfillment of its goals, was unanimously decided, following discussion, the formation of twelve (12) committees, that is, the creation of twelve (12) fields of activity, which will operate autonomously but will also communicate among each other. Association members will have the opportunity to declare, in accordance with their personal interests, in which of these fields they would be interested to contribute. There will be no remuneration at the present phase for the members’ involvement in the Association’s activities, owing to the Association’s financial standing, which does not allow for that.

More specifically, the administrative board unanimously decides to form the following twelve (12) Committees, while also defining the responsibilities there of and appointing the President of each Committee:

Committee A.: The central Communication Network of the National Association “APLOUN”.

Committee B.: Centre for timely distinction and recognition of those children Deemed to have outstanding abilities.

Committee C.: Centre for the Psychological Support of the Gifted/Talented children/young adults.

Committee D.: Center for Assessment of Motor Behavior and Athletic Performance of Gifted/Talented Children/teenagers.

Committee E.: Education and Teachers Training - Collaboration of Teachers for the Production of educational material aimed to educate Gifted/Talented pupils.

Committee F.: Educational Programs – Parents/Family Counselling.

Committee G.: Development of Communicative, Cultural, Educational and Scientific Activities.

Committee H.: Able Underachievers.

Committee I.: Global Curriculum Development, Extra Curriculum Activities – Preschool Education.

Committee J.: Global Curriculum Development, Extra Curriculum Activities – Primary Education.

Committee K.: Global Curriculum Development, Extra Curriculum Activities – Secondary Education.

Committee L.: Fine Arts: Gifted/Talented Pupils/Students


Committee A.: The central Communication Network of the National Association “APLOUN”.

The responsibilities of this committee are:

    a. The creation and development of a communication network for the information of the general public, educational staff, specialized scientists as well as parents/families
    b. The development of a communication network among the members of the administrative board and the presidents of the Association’s committees, the trustees, the representatives and the society members
    c. The information of the general public about the special educational needs of Gifted/Talented pupils/students
    d. The downloading of material on the Internet and the promotion of the Association through it
    e. The connection of the National Organisation with other International Organisations involved in similar endeavours, with international research centres, relevant Associations and Bodies abroad, and
    f. The creation of a database, which will emerge out of the collection of all the material from the Committees’ activities, as well as those of the trustees and the Association members, e.g. Communicational and/or educational material, relevant bibliography, research and programs.

Committee B.: Centre for timely distinction and recognition of those children deemed to have outstanding abilities

The responsibilities of this committee are the timely distinction and recognition of those children deemed to have outstanding abilities, so that they are properly treated, at first on the part of their family and then, by their educational and broader social environment.

Committee C.: Centre for the Psychological Support of the Gifted/Talented children/young adults

The responsibilities of this committee are the following:

    a. Psychological Support of the Gifted/Talented children/students, and
    b. Research regarding Gifted/Talented children/students, involving the location of factors, which contribute to their regular mental and emotional development.

Committee D.: Centre for Assessment of Motor Behavior and Athletic Performance of Gifted/Talented Students

The responsibilities of this Committee are the following:

    a. The timely distinction and recognition of those children/young adults deemed to have outstanding abilities, so that they are properly treated, at first on the part of their family and then, by their educational and broader social environment.
    b. The Sports training of Gifted/Talented pupils/students.
    c. The educational support and social integration of gifted and talented students, by means of sporting activities, and
    d. The support for research into the above-mentioned issues

Committee E.: Education and Teachers Training - Collaboration of Teachers for the Production of educational material aimed to educate Gifted/Talented Pupils, D.3) Research on teacher professional development [4].

Committee F.: Educational Programs – Parents/Family Counselling

The responsibilities of this Committee are the following:

    a. The recording of educational programmes, for Gifted/Talented pupils/students, which have already been developed in Greece.
    b. The participation in, and reinforcement of, parents and family in the active support of Gifted/Talented children/young adults, and the reinforcement of cooperation with specialists and educators.

Committee G.: Development of Communicative, Cultural, Educational and Scientific Activities

The responsibilities of this Committee are the following:

    a. Special categories of Gifted/Talented pupils/students (in Maths, Music, Sciences, Technology) investigation of the concept of differentiability and the special needs facing Gifted/Talented pupils/students, to develop their special Association skills and abilities.

Classroom Strategies in Primary and Secondary Education.
    b. Development of Communicative, Cultural, Educational and Scientific activities, aiming to fulfill the Association’s goals by means of the following: organisation of seminars, lectures, conferences, programmes in Mass Media, publication of printed matter such as magazines and newspapers or posters and leaflets.

Committee H.: Able Underachievers

Committee J.: Global Curriculum Development, Extra Curriculum Activities - Primary Education

The responsibilities of this Committee are the following:

    a. Global Curriculum Development to Support Gifted/Talented Students.
    b. Extra Curriculum Activities in Secondary Education.

Committee K.: Global Curriculum Development, Extra Curriculum Activities – Secondary Education

The responsibilities of this Committee are the following:

a) Global Curriculum Development to Support Gifted/Talented Students.

b) Extra Curriculum Activities in Secondary Education.

Committee L.: Support Gifted/Talented Students: Fine Arts

The responsibilities of this Committee are the following:

    a. Special Educational activities, to discover children Special Talents in Fine Arts, liberate imagination, ability to perform and harmonic development
    b. Multicultural Music-Theater activities

    • OnedayConference on the subject: “Intellectual and Talented Students: Pedagogical Approach”, that tookplace on February 2003, in the Central Hall of University of Athens, under the aegis of Ministry of Education.
    • In a meeting of ours with the Minister of Education, we submitted our proposals and were given his promise that the issue of Gifted/Talented students will be dealt with in Greece as well. The appropriate legislative regulation is already underway.
C. “IASONAS” - Educational Open Networks, Programs, Activities and Studies (


It provides Access and Services in Open Network for Advanced Studies of Developing Pedagogical Reasoning Goals, helping all students to growth, support, educate and develop their personal interests and talents.

In order to cover the needs that usually some students have and to motivate them to discover, create, develop and perform their personal talents and needs the Educational Open Networks, Programs, Activities and Studies “IASONAS” decide the following activities:

1. Framework: “Plaisio Dialogue Workshops on Network - Framework – Databank

Development of Pedagogical reasoning in order to Support HarmonicDevelopment of Gifted/Talented Students.

Questions, Opinions and Dialog are the following issues for Gifted/Talented students’ Support and Education needs. The dialogue is necessary at the time in Greece because the society needs to accept that it is a very important mistake not to respond to the educational needs of Gifted/Talentd children, and not to give them the possibility to improve their personal characteristics.

The areas for discussion are organized to create an interactive network.

2. Production of Educational Materials that identify, discover, create end develop Students Personal Talents, in cooperation with educators, writers, publishers and computer specialists using Multimedia Managers and Content Management systems. The following databases are under consideration:

    a. “Argo and Argonautes”

Creative & Interactive Educational Network

With issues from the Greek Classical Literature, History, Philosophy and Ancient Greek Mathematics

    b. “SXHMA”

Interactive Databank with Software and Multimedia applications to developand cultivate Mathematics

3. Development of Multicultural and Multidisciplinary Educational Activities

Program “ELLAS” - Expanding Learning Laboratory Activities and Studies.

Library on Network in Cooperation with Organizations and People.

Collected Materials, Programs, Events, and Activities.

    a. Development of Communicative, Cultural, Educational and Scientific Activities
    b. Use of the Information and Communication Technologies in the teaching and learning process.
    c. Connecting Schools with the Local Communities and Industry
    d. Connecting Schools between themselves
    e. Multicultural Communication between Schools
    f. Enrichment of the School Curriculum
    g. Development of the interactive School Materials
    h. Development of the Global Curriculum
    i. Extra Curriculum Activities
    j. Active Learning
    k. Cooperation between Students
    l. Harmonic Human Development Patterns

4. Music – Theater Group “Spheres Harmony” to discover, create, develop performance students’ personal talents and needs.

5. Program “Aristotle’s” Teachers Training Programs and Seminars

6. Group “Odyssea and Odysseus”: Teachers/Parents of Gifted/Talented Students

Activities and seminars to Support Parents/Family

7. Villages “Orpheas”, “Apollon” etc:

Summer Schools for Gifted/Talented Students

Supporting Programs for Students, Educators, Specialists and Parents

Activities Performance Center

Congress Center


Questions, Opinions and Dialog in the following issues for Gifted/Talented Students Support and Education are needed [4]. The dialogue is necessary because the society needs to accept that it is a very serious mistake not to give to the gifted children the ability to improve the Educational possibilities they have.

The following areas for discussion are organized to create an interactive network:

Workshop A: Identification of Gifted/Talented Children/Students

    • Definitions
    • Timely recognition of children with outstanding ability.
    • Gifted/Talented Children/Students: School performance - Academic achievement - Special or Specific Learning Difficulties
Workshop B: Opinions and understanding about Gifted/Talented Students Support and Special Education needs
    • Public
    • Parents/Family
    • Educators
    • Pupils/Students
Workshop C: Educational programmes and methods for Gifted/Talented Students
    • Use of the educational and communicational technology
    • Development of educational, communicational and cultural activities
    • Extracurricular activities
Workshop D: Global Curriculum development for Gifted/Talented Students Special Education needs
    • Educational Materials
    • Production of appropriate educational material
Workshop E: The educator of Gifted/Talented Children/Students
    • Professional education
    • Training programs
    • Support
Workshop F: Psychological support of Gifted/Talented Children/Students

Workshop G: Support Parents/Family of Gifted/Talented Children/Students


To develop multicultural communication networks, we need universal communicating language, like music [4]. The emotions and the multidimensional discipline Music creates are educational, cultural and multicultural tools. Mathematics is next to the music, the second universal language. The concepts and skills that belong to mathematics can be a multicultural communication language. Mathematics is embedded into knowledge communication systems and organizing information that expresses decision making processes, explicit encoding of knowledge, qualitative understanding. Starting from Basic Concepts, Machine learning techniques and the Artificial Intelligence , we design reflective educational materials expanding mathematics.

Selected issues from Classical Literature also appear in this project.

The interactive simulation educational network to communicate heuristic and reflective mental models for the special educational needs of Gifted/Talented Students needs:

    1. Pedagogical principles for coaching, individual differences in learning, personal learning plans and self-improving Experts/Tutors Systems.
    2. Developing Framework for pedagogically organized Global Curriculum
    3. Using new Technologies and advanced Technology to develop a Technology – based learning environment, using also the Distance Learning.
    4. Sophisticated mental models to teach Basic Concepts, expanding their meaning to the level that encourages the Gifted/Talented Students to the success.


In state to prepare teachers to perform in a professional and creative way in teaching the gifted and talented pupils/students we scheduled the following activities [6]:

    a. The support for research, employing the Action Research method, related to the investigation of the competencies of Teachers concerning the efficiency of their educational work by means of the creation of a communication network to reinforce the collaboration and interaction among teachers and other educational staff.
    b. Collaboration for the production of educational material aimed to educate Gifted/Talented pupils.
    c. The education and training of teachers, organisation and realisation of educational seminars to promote teachers to use new educational methods as well as productive procedures.
    d. The research in teaching quality indicators and the impact on the students’ reaction [4].
    e. The production of specialized educational material based on models already in application in other countries, to meet the special needs facing gifted and talented students, namely the Heuristic Model and the School-wide Enrichment Model, which may also be taken advantage of by pupils/students of standard classes [2].
    f. The encouragement of collaborative learning within an environment providing “different” individuals with equal opportunities for educational group effort, with specialized material based on recent educational methods and means of teaching.


Taking under consideration the relation and influence between Education & Culture, it is obvious that Family, School and the broader Community have a common responsibility for developing young people's discipline and skills. In order to creat inderdiciplinary materials and applications in teaching & learning, we develop cooperation of scientists and performers [6]. Holistic educational programs to support gifted, with (as Plato also point) Music, Mathematics, Gymnastic, Art History, Archaeology, Geography and other subjects included, will be developed.

In cooperation with Professors from several Departments of the University, and also in cooperation with the Scientific Associations, like the Greek Mathematics Association, we assume that we will improve pedagogical reasoning, creativity and development of teachers’ and students’ personal talents, developing a human world.


        1. Plato, Republic, book VII.
        2. Renzulli, J. & Reis, S. (1986) The enrichment triad/revolving door model: A schoowide plan for the development of creative productivity. J. Renzulli, Systems and models for developing programs for the gifted and talented. Mansfield Center, CT: Creative Learning Press.
        3. Dimakos, G. (2003) The Greek Mathematical Competitions and the Mathematical Olympiads, Proceedings of 1st APLOUN Conference, 09.02.04
        4. Meletea E. (2003) (Paper on the 1st APLOUN one Day Conference 09.02.04)
        5. Thomaidou L., Meletea E. (2003) Research and Action for the Education of the Gifted/Talented students in Greece: Vol. 25, Dec-Mar 2003, Patakis Editions, Athens.
        6. Meletea E. (2004) (Paper on the 1st APLOUN’ s Congress 13 – 16 May)

Eugenia T. Meletea

Mathematician – System Analyst

President of “? P L O U N ”

The Greek Scientific Association – Non Profit Organization - Development of Pedagogical Reasoning to Support Gifted/Talented Students

Director of “ I A S O N A S ” -

Educational Open Networks, Programs, Activities and Studies for Gifted/ Ta lented Students

17, Panagi Kyriakou,

Amaroussion, 15124, Athens


Tel. Fax: (30) 210 80 68 563

(30) 210 96 74 019

e-mail: emel@tellas.gr





Abstract: This paper takes into consideration the state of instruction for making students master a high level mathematical thinking through mathematical problem solving. The attention is paid to the quality of the reasoning in problem solving processes by looking at the problem that can be solved in different ways with different mathematical concepts in each as an example. Based on some levels being assumed, the design for the lesson and unit for acquiring the new concept creatively is shown.

Key words: Cognitive entity, Creativity, Mathematical structure,Problem solving.


Problem solving and creativity have been one of the most important themes since the 1980s. The relationship between these two is not so clear, especially when it comes to the effective way of enhancing students’ creativity. Because of the complexity and variety of the concept of the word “creativity”, this term has become associated with various aspects of creative behaviour and mental functioning that range along a cognitive-emotive continuum.

In this paper, the focus is upon the process of creativity, not the definition of creativity with a great source of apprehension. First, the three-stage model of development of mathematical thinkingwith regard to cognitive entity is shown for the basis of the following.Second, an example that can be solved in different ways is considered.Then the conformity of the model with the example and the assignment of further research is considered.

1. The stages of development of mathematical thinking

In mathematics education, the concern about problem solving has not decreased after the 1980s. And one of the most important themes in problem solving research is how to make students master higher-order-thinking ability, which brings them flexible and creative ways of reasoning towards non-routine problem.

By taking a somewhat closer look at the aspects of different kinds of problem solving as a heuristic, there seems to be three stages that can be hypothesized as follows:

    • Stage1: The empirical (informal) activity
    • Stage2: The algorithmic (formal) activity
    • Stage3: The constructive (creative) activity
Stage1: The empirical (informal) activity

In this stage, some kind of technical or practical application of mathematical rules and procedures are used to solve problems without a certain kind of awareness.

Stage2: The algorithmic (formal) activity

In this stage, mathematical techniques are used explicitly for carrying out mathematical operations, calculating, manipulating and solving.

Stage3: The constructive (creative) activity

In this stage, a non-algorithmic decision making is performed to solve non-routine problem such as a problem of finding and constructingsome rule.

It has been hypothesized that the learning activities are to be developed in this order. That is to say, the context of problem solving is prepared by previous experiences and set by a preparatory stage in which mathematical procedures become interiorized through action before they can be the objects of mathematical thought. However, it is quite clear that there is a qualitatively great difference between Stage2 and Stage3. Especially, little research on developing, teaching and evaluating problems which require the reasoning with various conceptual entities and the insight of the underlying problem structure has been done.

2. Course materials

Taking three stages described above into consideration, it can be said that the ultimate goal of mathematical problem solving is to acquire not only knowledge and techniques but also the attitude that enables students to think about the main structure of the problem without adhering to things they have learned before.

Therefore, an example that contains many elements of approach corresponding to each stage described above is considered.


Stage1: The empirical (informal) activity

Maybe almost all the students who haven’t met a similar problem will have difficulties with this one. For them, there are no rules or techniques available except “try and error” method as all line segments are measured. In that case some students will be able to find the property;

However, this will happen by chance, without the awareness of its’ theoretical (mathematical) foundation.

Stage2: The algorithmic (formal) activity

We can use the assumption given by the problem and write down formulas like

Then we’ll get the formula;

Stage3: The constructive (creative) activity

Here is a little different way of manipulating formulas;


“What does this mean?”

“Let’s draw!” (Figures are omitted)

    • Add the point F under the conditions of
    • Make parallelogram BCEF (add the point E)
    • Connect B to E
    • Make parallelogram MNGH (show )
    • Connect A to E
“Pay attention to triangle ABE”



“Show it”

“Can you find any relation to the givenproblem?”

Now we can invent the following solution (Figures are omitted).

    • Construct parallelogram ABF’D (F’: an internal point of the quadrangle ABCD)
    • Construct parallelogram BCE’D (BC//F’E’ and BC=F’E’)
    • Connect B to E’, A to E’ respectively (construct triangle ABE’)


3. Discussion

The quality of problem solving approach is categorized into approximately three stages in accordance with the way of using mathematical knowledge. And the final stage (Stage3) requires a higher-order-mathematical thinking that is more difficult to be mastered.

Based upon rationales and examples, it can be said that one of the most effective ways of reaching Stage3 is to emphasize the relation between Stage1 and Stage2 because both deductive and empirical reasoning are essential to discover some mathematical rule of nature.

Therefore, it is important to reacknowledge the educational value of naive manipulation anddevelop the way of teaching and evaluating problems that can be solved in different ways with different mathematical concepts.


        1. Gotoh, G. Lesson Design for Reconstructing Mathematical Knowledge. In Konno, Y(Ed)., The Reconstruction of School Knowledge. Chap.3, pp.204-217. Gakubunsha, 2002. (In Japanese)
        2. Tall, D. Advanced Mathematical Thinking. Kluwer Academic Publishers. 1991.
        3. N. Mashiko. The difference of problem solving processes between higher order thinking tasks and standard tasks in introductory algebra. Naruto University of Education. Vol.9, 1994, pp.151-166 (In Japanese)
        4. N. Mashiko. Some problem characteristics of nonroutine problems in school mathematics. Naruto University of Education. Vol.11, 1996, pp.175-185 (In Japanese)

George GOTOH

Media Network Center

Waseda University

1-104 Totsuka-machi




E-mail: gotojoji@mnc.waseda.ac.jp




Gregory Makrides,Emiliya Velikovaand partners

Abstract: In many European schools the mathematics curriculum is designed to serve the average and special needs students without identifying and supporting potentially talented/competent students in mathematics. The aim of this project is to develop methods and educational tools, which will help the educators to identify and motivate talented students in mathematics as well as to support their development within the European Community without any discrimination. The project intends to merge forces and establish a network through the Mathematical Societiesand universities in the European area to support the aims of the project as well as to use new technologies in the support, dissemination and sustainability of the developed structure of cooperation.

MATHEU is a new approved project under Socrates-Comenius 2.1 action with characteristic activities that are expected to enhance the learning of mathematics in the European region as described above. The presentation will discuss the philosophy, aims and objectives and the work plan of the project and will invite interested individuals and organisations to support the aims of the project.


ThedecisionoftheEuropeanUnion, COM/2001/678,says, «Inasocietyofknowledge, Democracyrequires the citizens to have scientific and technological knowledge as part of the basic competence».

The future objective aims of the European Educational Systems, which were agreed on 12 February 2001 from the Education Council in Stockholm, identify Mathematics as one of the major priority subjects. The basic objective is the increase of interest in mathematics from early age and the impulsion of youth to follow careers in these subjects, more specifically in the research in these fields.

The types of students who will be able to contribute in the research of these fields are more likely to be students who are talented in these fields and more specifically in mathematics.

Certain activities towards this objective are already taking place in some countries. The aim of MATHEU is to bring together experts from the partner countries and to exchange ideas, background knowledge and experience and to develop together a system that will work for the whole of Europe.

Talented students in mathematics have to be discovered in early stages and in a systematic way. The usual method for identifying such students is through competitions but it is generally acceptable that many talented students in mathematics are never discovered simply because they do not participate in competitions or simply because they were not among the top ten during the competition process, or they are talented students who cannot perform under strict time limits.

European countries have to find ways to keep their talents and brains in Europe. In order to accomplish this, mathematicians, academicians and educators have to work together in a European dimension and to design a programme, which will change attitudes of governments, universities and foundations in favour of supporting the gain of mathematical talents in Europe and decreasing the brain drain outside the European Community. Talented students need attention, love, support, training, recognition, identification. MATHEU promises to offer solutions to all these for the development of European talented students through their teachers, educational administrators and other bodies-organisations-institutions-government, as well as through the direct links via the Internet.


The aim of MATHEU is the development of methods and supporting material for the identification of talented students in mathematics in European Schools and their development and support. The project aims to establish a network of sustainable support through universities, mathematical societies and foundations of the partner countries at first and later throughout the European community at large.

Teachers in Europe need to be trained with methods and to be provided with material in order to be able to identify and support such talented students.

The main objective is to help Europe to gain the maximum contribution to and from these students, who will become the backbone for the scientific and technological knowledge necessary to make Europe the major technology developer and economic power in the years to come.

These aims and objectives will be accomplished by:

    • Analysis of the flexibility of existing mathematics curricula in European Schools with emphasis in the partner countries focusing on the aspect of talented students
    • Analysis of methods and tools used in European countries for the identification, motivation and support of talented students in mathematics
    • Design methods and tools for identifying potentially talented students in both primary and secondary education levels and for training teachers so that they can bring the students to express their 'talent' in mathematics (talent as ability to face and solve problematic situation and to appreciate the role of theoretical thought)
    • Design special pedagogical methods and subject material for the development and promotion of talented students in European schools
    • Develop methods/solutions and a programme for changing attitudes within government , universities and foundations in providing fellowships and support in order to keep mathematical brains in Europe
    • Design a special Web-site devoted to this purpose which will enable the sustainability of the project aims
To achieve the above a team of European mathematicians, experts in either Didactics or Subject or Technology has been assembled to address the problem in an integrated and coherent manner.

Developing a unified European training programme that offers tools for identifying, motivating and supporting talented students in mathematics is a new approach, which can be used complementary to the existing curriculum systems in the European community without any discrimination. In addition, using new technologies to provide sustainable support for a group of students with high competence in a particular topic is a recent development. Keeping mathematical talents in Europe through a “brain-gain effect” is also a new philosophy and approach. Also, part of the work will investigate the problem of mathematics talented students with learning disabilities, which currently lacks investigation.

We see mathematics educators as educators able to practice the teaching in either a mixed ability environment or in a selective environment. Regardless of the environment, mathematics educators need to have the necessary tools and skills in order to evaluate the competence of their students and to be able to encourage, motivate and support the development of those who appear to be strong in subject areas such as mathematics. Educators with such developed skills will achieve higher methodological standards. The project is also investigating the problem of “talented students with learning disabilities”.

Specifically, MATHEU seeks to develop a programme emphasizing the role of the teachers in:

    • Identifying mathematical talent through a range of measures that go beyond traditional standardized tests. Measures should include observations, student interviews, open-ended questions, portfolios, and teacher-, parent-, peer- and self-nomination. Recognition should be made of the fact that mathematical talents can be developed; they are not just something with which some students were born.
    • Presenting interesting tasks that engage students and encourage them to develop their mathematical talents.
    • Improving opportunities for mathematics learning and a much more challenging, nonrepetitive, integrated curriculum which is needed to help students develop mathematical talents. Students must be challenged to create questions, to explore, and to develop mathematics that is new to them. They need outlets where they can share their discoveries with others.
    • Encompassing a variety of methods including differentiated assignments, a core curriculum, pull-out programmes, in-class programmes, magnet schools, and extracurricular activities such as after-school or Saturday programmes, mentorship programmes, summer programmes, and competitions.
    • Improving the ways in which students learn mathematics. Teachers must become facilitators of learning to encourage students to construct new, complex mathematical concepts. Students must be challenged to reach for ever-increasing levels of mathematical understanding.

All mathematics educators at all educational levels (primary and secondary) associated with the partner institutions will directly benefit. Educators in other areas may also benefit as the methods and tools for identifying mathematical talented students could also be used to identify talented students in other subjects. Through the dissemination process individual educators from all countries in the European Community and beyond as well as educator staff of educational authorities will benefit from the outcomes of MATHEU through their participation in the training course to be offered under Comenius Action 2.2. Finally, the largest group to benefit is the European Community’s potentially talented/competent students in mathematics.

Target group 1: Teachers/Teacher trainers/Teacher trainees

MATHEU will allow the individual non-gender biased, to develop knowledge on pedagogical methods, use of tools appropriate to support different levels of students.

Target group 2: Educational administrators/Inspectors

In addition to the above listed impacts this target group will become better curriculum developers, will raise the quality of their teacher support, improve their own background in the needs of the topic and will gain an educational tool.

Target group 3: Educational Psychologists/Counsellors

Nowadays, educational establishments involve educational psychologists and counsellors and therefore MATHEU has to provide for these educators as they play a very important role in the educational development of all types of pupils and students. Counsellors could become the catalysts for the “brain-gain effect” in Europe through their counselling to the students.


The outputs of the MATHEU project are listed below:

    • A Tool that identifies talented/competent students in mathematics at two different age levels. Methods/Activities for motivating potentially talented/competent students in mathematics
    • A European Manual and CD-Rom, which will contain the tool above together with material needed to support the development of such students. The Manual will be initially translated in seven languages, English, German, Greek, Italian, Bulgarian, Czech and Romanian.
    • A Course Design in English for teacher trainees and teacher trainers for primary and secondary levels for the target age levels using the tool, methods and the manual mentioned above
    • An Information and Dissemination Symposium for administrators/government decision makers, for university enrolment managers/deans/representatives, for presidents/ representatives of Foundations and Societies
    • The MATHEU Web-site, designed to provide sustainable communication between the partners and to provide support to talented students in mathematics as well as to mathematics educators of different levels. The site will initially support the languages of the partner countries.

The recent developments of the project suggest that Identification, Motivation and Support (IMS) are divided in two age groups, the 9-14 age group and the 15-18 age group. For each age group a number of topics were agreed as a basis for IMS development. It was agreed to use the idea of a curriculum and level of difficulty ladder for each topic. A group of experts is now developing these ladders which will then be evaluated and will constitute the basis for Identification. Motivation elements have been discussed and are circulated among the partners of the project to evaluate in their own countries and to try to finalize them by the end of 2004. Support materials have been collected for different topics and will be developed further in connection with the final form of the ladder in each topic and at each age level.


Electronic articles

1. John F. Feldhusen, Talent Development in Gifted Education,ERIC Digest E610, 2001. http://searcheric.org/digests/ed455657.html

2. Dana T. Johnson, Teaching Mathematics to Gifted Students in a Mixed-Ability Classroom , ERIC Digest E594, 2000. http://ericec.org/digests/e594.html

3. Richard C. Miller, Discovering Mathematical Talent, ERIC Digest E482, 1990. http://ericec.org/digests/e482.html

4. Joan Franklin Smutny, Teaching Young Gifted Children in the Regular Classroom , ERIC Digest E595, 2000. http://searcheric.org/digests/ed445422.html

General Websites

5. www.nfer-nelson.co.uk

6. www.math.bas.bg/bcmi/

7. www.excalibur.math.ust.hk

8. www.unl.edu/amc

9. www.math.scu.edu/putnam/intex.html

10. www.mathleague.com

11. www.olympiads.win.tue.nl/imo

12. www.olemiss.edu/mathed/problem.htm

13. www.mathforum.com/library

14. www.geom.umn.edu

15. www.problems.math.umr.edu

16. www.math.fau.edu/MathematicsCompetitions

17. www.schoolnet.ca

18. www.mathpropress.com/mathCener.htm

19. www.enc.org/topics/inquiry/ideas

Specific Websites

20. MATHEU website: http://www.matheu.org

21. The National Research Center on the Gifted and Talented (NRC/GT) http://www.gifted.uconn.edu/nrcgt.html

22. Johns Hopkins University: The Center for Talented Youth (CTY) http://cty.jhu.edu/

23. Northwestern University’s Center for Talent Development (CTD) http://www.ctd.northwestern.edu/

24. The Education of Gifted and Talented Students in Western Australia http://www.eddept.wa.edu.au/gifttal/gifttoc.htm


Gregory Makrides , Ph.D., Project Coordinator

Dean and Associate Professor of Mathematics


President of the Cyprus Mathematical Society

E-mail: makrides.g@intercollege.ac.cy


Emiliya Velikova, Ph.D., Faculty Socrates Coordinator

Department of Algebra and Geometry

Centre of Applied Mathematics and Informatics

Faculty of Education

University of Rousse,

8 Studentska str., 7017 Rousse


Cell phone: +359/889 625 222, Fax: +359/82/845 708,

E-mail: emily@ami.ru.acad.bg emivelikova@yahoo.com






Hanhyuk Cho, Hyuk Han, Manyoung Jin, Hwakyung Kim,

Minho Song

Abstract: The purpose of this paper is to introduce a microworld designed for mathematical creativity and gifted education. We have developed a microworld named JavaMAL by combining LOGO and DGS microworlds. Using JavaMAL, students can make and manipulate semi-dynamic objects “tiles” to explore mathematics. We have tested JavaMAL for creative mathematics program and gifted education, and we have confirmed that there are positive educational results from the teaching experiments.

Key words: Tile, Semi Dynamic, LOGO, JavaMAL, Microworld, DGS, Creative environment, Gifted education


In his Enrichment Triad Model Theory, Renzulli[10]considers three different levels of learning for mathematically gifted students: (1) surveying and understanding the given problems, (2) developing their strategic skills in solving the problems, (3) producing the final solutions and products through relevant project activities. In this sense, it is important to investigate and design educational environment for the gifted learners to explore problems and make creative products. Accordingly, this paper is on the design of a mircoworld [4], a playground for creative mathematical education. In brief, microworlds are primarily exploratory learning environments where the learners can manipulate or create objects and test their effects through discovering spaces and constraining simulation of real-world phenomena [8].

The LOGO [1], [5], [2] the representative microworld, is an environment where students can create and produce mathematical figures using the most basic commands fd and rt . The Fig. 1 shows the textual command and the figure drawn by the turtle executing the command. As a result of this textual command in the LOGO environment, the learners would be able to produce a more visually and creatively oriented configurations. This is the most salient feature of the LOGO mircroworld. Here, the basic commands fd and rt are easily understood, and by appropriate arrangement and operation of these two commands, the learners are able to make more creative and diverse commands to draw figures.

The previous researches on LOGO opened the gates of new possibilities in creative mathematics education. As new computer techniques and technology are being developed, however, the LOGO environment needs to be revised. Based on its fundamental philosophy, constructionism, we have improved the LOGO environment to be more dynamic and diverse for creative education.

Meanwhile, there have been researches on the Dynamic Geometry System (DGS) along with Cabri and GSP and its application to overcome the static aspect of “usual” school geometry. The DGS is an experimental environment in which the learners hypothesize, justify, and confirm geometric and algebraic properties through mouse exploration, and the DGS help students overcome the static features of the diagrams and mathematical figures in the textbook. In short, the DGS system plays an important role by providing an excellent environment for conjecturing, testing and confirming.

Sherin [12] proposed yet anotherup-graded DGS by adding textual commands to the system. We consider Sherin's attempt educationally desirable and have tried to combine the LOGO and DGS microworlds, keeping the features of both microworlds but adding new educational functions.In this environment, named JavaMAL microworld, “making” is emphasized and the “making” comes firstfollowed by experimenting and confirming. For this attempt, we intoroduce textual command system using the keyboard as well as the mouse-clicking sequence command system in addition to the already existing pull-down menu system. In sum, this paper proposes the designing of new functions that can connect the LOGO with the DGS using newly invented command systems, and it will demonstrate some examples of creative mathematical education programs that are possible in this new environment.


In the previous section, we surveyed the strengths of the LOGO and the dynamic function of the DGS. In our effort to connect the two, we have provided dynamic feature to the LOGO on one hand and have created a function that makes the DGS more static on the other hand.

Such a complementary is needed in mathematics education for the following reasons: First, let’s observe the following procedure of making creative products in the LOGO environment. The left hand side of t he Fig. 2 shows a command ? for drawing basic figure using the primitive commands fd and rt . The command ? in the middle of Fig. 2 shows that the previous process has been “internalized” and “condensed”, and the right side of Fig. 2 shows a function command ? that can be used to draw more complicated figures when one encapsulates the process of drawing as an basic object.

The three processes in the Fig. 2 are similar to those of Sfard’s [11] concept formation scheme: interiorization, condensation and re ification. In fact, the Fig. 2 can be regarded as the prime representaion in the LOGO environment that shows the necessary steps of concept formation development. Following the Circulation Theory in cognitive development, we design a “tile” command so that a semi-dynamic object “tile”is created by the commands fd and rt , and the object can be manipulated again by the commands fd and rt This is the basic idea of the tile command.

Thefollowing Fig. 3 shows a sequence of making tiles. That is, a student draws a basic figure using the Logo commands fd and rt, and makes tiles using a tile command. Then the student makes another tile by copying and reducing the tile, and the student moves and rotates the tiles, again using fd and rt tile commands. Hence, the student can control not only the turtle’s movements of the turtle, but also those of the tiles constructed by the turtle using the “fd tile” and the “rt tile” commands. Again, as shown in the Fig. 3, a tile is an object made by fd and rt commands, and it is a basic object moved freely by mouse dragging or by fd and rt commands.

Laborde [9] showed that the DGS contains a rich learning context for proof since the “dragging” sequence in the invariant geometric properties stimulate students’ thinking. The DGS helps the learners to experiment with moving figures on the screen, which is an impossible task for the generation whose tools were consisted of pen-and-paper. Be that as it may, even though it is possible for students to conjecture certain geometric pr operties by dragging his or her mouse in the DGS, these properties need to be justified. In the following Fig. 4, we can see that the tile ACE is made by mouse-clicking sequences, and is being rotated to confirmed the conjecture D ACE ? D DCB.

The pur pose of the DGS environment is to explore the invariant geometric properties by mouse dragging when the situation is changed. The DGS is good for providing an appropriate environment for exploration and experiment of various properties. The reasoning of properties begins, however, with a fixed instance of the dynamic situations. That is, mouse dragging dynamic representations assure the invariability of the figure, but the deductive reasoning takes a step-by-step processbetween the two static situations. For this purpose, we introduce a “tile command” to take snapshots of the dynamic movements, and then the tiles are manipulated to check the figure’s properties such as congruence and similarity properties. In part, the tile is expected to be a medium for deductive reasoning. In the Fig. 4 one is able to assume that the triangles are congruent and check one’s assumption bymanipulating the tile. The Fig.5 shows the tile’s geometric operationto check the congruency and similarity of the figures. By allowing the blue triangle semi-dynamic, that is movable but keeping the same shape and size, and by using the tile commands, one can transform it in many ways (translation, reflection. rotation. and dilation).

As a student moves the tile with the LOGO commands on the keyboard, student can also manipulate it with the mouse. Student can drag a tile, just as student drag a point. Unlike a point, however, student can reflect and rotate the tile, which is semi-dynamic with a shape and size. There are two ways to move a tile: using textual commandson the keyboard and the mouse-clicking sequences to connect the LOGO and the DGS.

From the Fig. 6, we can see that the Pythagoras trees are given not only by the

LOGO's turtle command, but by the DGS construction command. We can also see that the tiles are created by a turtle command and manipulated by mouse dragging.

The JavaMAL microworld is designed not only to combine the LOGO and the DGS, but also to introduce bridges between the two such as “tiles”. From our teaching experience, we find that the tiles in JavaMAL can provide a rich environment such that students can make their own creative mathematical products and explore mathematical properties from the given mathematical premises. Since JavaMAL is written in
JAVA language, it can be used on the internet (

Communications between the students and the teachers in cyber space are also available on the internet, the unified environment of democratic access [7] and internet board. It is accessible to anyone, anytime and anyplace. Now some new examples of creativity education will be given in the next two chapters that the JavaMAL microworld can provide.


In creativity education, a small group as well as an individul project is encouraged. Social constructivism emphasizes the social construction of knowledge. Group activities can stimulate the group members so that they can produce more creative results after they share ideas through communication. Group activity in cyber space as well as in classrooms should be emphasized,and theJavaMAL microworld is designed for this purpose. The JavaMAL microworld is available on the internet, and the group activity can be realized through the internet board using its textual command system. The Fig. 7 shows an example of how a small group members work together not only in the classroom but also in the web. The group members draw a sketch of “The Garden,” after exchanging their ideas in the classroom. The garden is divided into the ‘sun’, ‘tree’ and ‘butterfly,’ and each module is made by each members for movements, which, in turn, comes together and make the whole project complete. The feedback from the group members for an individual module can be exchanged for further discussion not only in the classroom but also in the cyber space.

The function of tile here is, therefore, to add the semi-dynamic features for the modules in the sketch and initiate them to move. It is the tile that makes the ‘sun’ blazing, the ‘tree’ growing, and the ‘butterfly’ flying. Also, each tile comes together to construct a creative product as a whole. This kind of module based group activity is a good example of creativity education in a microworld environmentsince it foster both an individual and a small group projects.


So far, we have talked about the function of tile as a snapshot in the dynamic DGS. The tile provides certain functions such as testing the congruence and similarity (of figures), and confirming and verifyingtheconjectures. The tile is expected to contribute to the creativity education in many ways. For example, imagine a space filled with many pieces of tile, or a semi-dynamic tile expecting to play a role as teaching aids, tessellation and tangram. It is common to use ready made tiles in most activities such as LEGO activities and tangram puzzles. Meanwhile, in the JavaMAL environment the learners are able to “make” tilesby themselves, and design and solve the puzzle with the tiles of their own making. In an advanced level of manipulation of the given objects, the learners may able to make the objects themselves, and then manipulate them to solve the puzzle. The Fig. 8 shows tessellation and tangram in which a student can ‘make’a basic piece by mouse-clicking sequence, then by using the tile piece, he or she copies it and fills the space.

While tessellation and tangram focus on puzzle-solving,JavaMAL also focuses on making a piece itself. In JavaMAL environment, making is emphasized prior to manipulation, and making is possible thanks to tile.


We surveyed the JavaMAL microworld as a playground for creativity education. In order to connect the static LOGO with the dynamic DGS, we introduced a new function, “tile”.We mentioned a couple of examples for creativity education to apply what we studied. From our pilot experiments, we confirmed that the JavaMAL microworld is well-designed for mathematical creativity and education for the gifted. This year, we plan to do well-structured teaching experiments using well-organized curriculums and instructional designs based on Renzulli's Enrichment Triad Model Theory. We will modify the JavaMAL microworld by monitoring the reactions from students, and we will try to obtain desirable educational outcomes by designing not only the JavaMal microworld but instructional strategies to provide an enriched learning environments based on the constructivism philosophy.

In the future, we plan to up-grade the JavaMAL environment so that it can be used to represent three-dimensional geometric objects. As we were able to fill the space with the basic pieces, we’llattempt to design the environment where the pieces will make a polyhedron. We are going to add some new functions, that is, drawing the folding net of a polyhedron using the LOGO commands, folding it to make a polyhedron, and manipulating it. Here the tile acts as a medium that can make the folding net to move. The folding net of a polyhedron is drawn by the basic LOGO command, fd and rt . The folding net is, in turn, folded and becomes three-dimentional, and a couple of polyhedrons merge together to make some creative product. The Fig. 9 shows the process of the students’ activity to make decahedron. First, the students figure out what the folding net might be look like. Second, they draw the folding net using fd and rt . Then, they make the folding net to move and to fold using the functions of tile. Finally, they can translate or rotate the complicated polyhedron. Furthermore, the complicated polyhedrons can be changed and connected to each other, to make another creative product such as a doll and a train.

This paper is primarily based on Constructionism that emphasizes the active construction of the external artifact. We studied on the creativity education in different microworlds, and considered its educational values for creativity and education for the gifted. We propose the necessity of the tile functions and proposed some programs using the tile for creativity education. The designing of a microworld for creativity education as well as its application in the given situation will be an important issue for the gifted and creativity education.


        1. Abelson, H., & diSessa, A. A. (1980). Turtle Geometry: Computation as a Medium for Exploring Mathematics, Cambridge, Mass : MIT Press.
        2. Clements, D. H., & Battista M. T. (2001). Logo and Geometry, Reston, Virginia: NCTM.
        3. diSessa, A. A. (2000). Changing Minds, Cambridge, MA: MIT Press.
        4. Edwards, L. D. (1995). Microworlds as Representation. In A. A. diSessa, C. Hoyles, R. Noss, & L. Edwards (Eds.), Computers and Exploratory Learning , Berlin: Springer.
        5. Hoyles, C., & Sutherland, R. (1999). Logo mathematics in the classroom, London and New York: Routledge.
        6. Kafai, Y. & Resnick, M. (Eds.) (1996). Constructionism in Practice: Designing, Thinking, and Learning in a Digital World, Lawrence Erlbaum Associates, Publishers.
        7. Kaput, J. (1999). The Mathematics of change and Variation from a Millennial Perspective, In C. Hoyles, C. Morgan, &G. Woodhouse (Eds.), Rethinking the Mathematics Curriculum, Falmer Press, 155-170.
        8. Jonassen, H. D. (1996). Computers as Mindtools for Schools, Prentice Hall.
        9. Laborde, C. (2000).Dynamic Geometry Environments as a Source of Rich Learning Contexts for the Complex Activity of Proving, Educational Studies in Mathematics, 44, 151-161.
        10. Renzulli, J. S. & Reis, S. M. (1991).The Schoolwide Enrichment Model: A comprehensive plan for the development of creative productivity. In N. Colangelo, & G. A. Davis (Eds.) Handbook of Gifted Education , Needham Heights, MA: Allyn and Bacon, 111-141.
        11. Sfard, A. (1991). On the dual nature of mathematical conceptions: reflections on processes and objects as different sides of the same coin, Educational Studies in Mathematics,22, 1-36.
        12. Sherin, B. (2002). Representing geometric constructions as programs: A brief exploration ,International Journal of Computers for Mathematical Learning 7(1), 101-115.
        13. Wilensky, U. J. (1993). Connected Mathematics- Building Concrete Relationship with Mathematical Knowledge , Thesis of doctor of philosophy at the Massachusetts Institute of Technology.

Cho, Hanhyuk; Han, Hyuk; Jin, Manyoung; Kim, Hwakyung; Song, Minho

Department of Mathematics Education

College of Education, Seoul National University

Shinlim-dong, Kwanak-gu

Seoul 151-748, Korea

?-mail addresses:






Supported by Korea Research Foundation KRF-2003-015-C00011





Hye Sook Park, Kyoo-Hong Park

Abstract: We study on the mathematical disposition of mathematically gifted students in the middle of KOREA. For this purpose, we use the tools of psychological test of disposition of math disliking which was developed by Kim et al.(2001) to analyze the mathematical disposition of gifted students and investigate the characteristic of it.

Key words: Mathematical Disposition, Gifted Students, Math Disliking Factor


The education in the school must help each student to show their ability of latent faculties. But there are many places of bad educational surroundings in Korea, such as the entrance examination, poor facilities of the school, parents' tendency of distrust to the school education and enervation of teachers on the educational guidance in the daily life. These are the causes of the stumbling block to the education which bring up student's ability of latent faculties in Korea.

Recently, the government of Korea gave emphasis to the education of gifted students in the school. The law of promotion of the gifted students' education was established by presidential law and ordinance at April, 2002 in Korea. By this law and ordinance, the education centres of gifted students were established in the 15 universities all over the country. And some of science high schools carry out the program of gifted students. Also each offices of educational district established gifted students’ class and training the teachers who take part in the education of gifted students.

In this research, we study the mathematical disposition of mathematically gifted students in the middle school before we study the teaching methods of mathematically gifted students. For this study, we use the tools of psychological test of disposition of math disliking which was developed by Kim and 7 others(2001) to analyze the mathematical disposition of mathematically gifted students and investigate the characteristic of it.


The mathematical disposition is a mind of good feeling and interest in the mathematics and it is the same as mathematical cast of mind which was defined by Krutetskii (NCTM 1987). We can see the self confidence, good feeling and the tenacity on the homework for mathematics in the mathematical disposition.

The disposition of math disliking which was developed by Kim et al(2001) has several domains of factors such as psychological factor, environmental factor, academic trait of mathematics factor. In the prestudy, we studied the disposition of math disliking of underachievers(Park and 4 others 2004). And we use the result of prestudy to analyze the mathematical disposition of mathematically gifted students.

The factors causing math disliking are divided into 11 groups as follows:

    1. Psychological and environmental domain contains 4 factor groups, i.e., cognitive factor(mt1), mental ability factor(mt2), teacher related factor(en1) and math perception related factors(en2).
    2. Academic trait of mathematics domain contains 7 factor groups, i.e., comprehension related factor(com), hierarchy factor(rl1), connection related factor(rl2), operation related factor(rl3), analysis reasoning related factor(rl4), basic application factor(ap1) and composite application factor(ap2).

When we evaluate the cumulative percentage for each factor of math disliking, the increasing appearance of the point of disliking factors means the factor of math disliking is weak. On the contrary, the decreasing appearance of the point of disliking factors means the factor of math disliking is strong. So the factor of math disliking point excess more than 70% is defined the mathematical disposition and the factor of math disliking point shortage less than 30% is defined the mathematical disposition of math disliking.

By using the definitions of mathematical disposition, we investigate the relations between each factor and the mathematical disposition of mathematically gifted students. From this result, we will find the desirable teaching methods for the mathematically gifted students.

We already have analyzed the properties of math disliking disposition of underachievers of middle school(Park et al., 2004).

The following (Table 1) shows the means of the cumulative percentage of the math disliking factors for each level of the achievement and the table 2 shows the number and percentages of math disliking students-who have more than or equal to 3 math disliking factors- of each math disliking factor for each level of achievement.

From the above tables, we can find that mathematical underachievers show remarkable difference in the math disliking factors of cognitive factor(mt1), teacher related factor(en1) hierarchy factor(rl1), connection related factor(rl2), operation related factor(rl3) and basic application factor(ap1).

The above result is the analysis of disposition of math disliking focused on the underachievers(Park et al, 2004). It may be quite different from those of gifted students. So, it is needed to survey the difference between the mathematically gifted students and the others. For this purpose, we made about 100 mathematically gifted students of middle school an object of investigation by questionnaire, and we are analyzing the data presently.


In this study, we expect to develop of the textbook for mathematical gifted students, to develop the program for teaching the mathematical gifted students after analyzing their mathematical disposition.


        1. Gagne, F. Constructs and Models Pertaining to Exceptional Human Abilities, International Handbook of Research and Development of Giftedness and Talent , edited by Heller, K. A. etc., Oxford: Pergamon Press, 1993, pp.69-87.
        2. House, P. A.(Ed.) Providing Opportunities for the Mathematically Gifted, K-12 , Reston, Virginia: NCTM, 1987
        3. Kirk, S. A. Educating Exceptional Children , 2nd ed., Rev., Boston: Houghton Mifflin Co.. 1972
        4. Kim et al. Studies on Exploring Math Disliking Factors and Devising Tools to Analyze Students’ Disliking Trends about School Mathematics, J. Korea Soc. Math. Ed. Ser. A : The Mathematical Education, Vol. 40 No. 2, 2001. pp.217-239.
        5. Renzulli, J. S. & Reis, S. M. The Schoolwide Enrichment Model: A Comprehensive Plan for Educational Excellence, Creative Learning Press, INC, 1985.

Prof. Hye Sook Park, DSc.

Dept. of Mathematics Education

Seowon University

Cheongju, Chungbuk 361-742,KOREA

E-mail: hyespark@seowon.ac.kr

Prof. Kyoo-Hong Park, Ph.D.

Dept. of Mathematics Education

Seowon University,

Cheongju, Chungbuk 361-742, KOREA

E-mail: parkkh@seowon.ac.kr



Kyoko Kakihana, Suteo Kimura

Abstract: The Super Science High school (SSH) project was set up by the Ministry of Education, Cultural, Sports, Science and Technology (MEXT) in 2002 to develop a high school science and mathematics curriculum for gifted students. Every year twenty-six high schools are selected to take part in the project for a period of three years. In this article, we will explain what the Super Science High school project is as well as categorize the new curricular activities involved.

Key words: Curriculum, Motivation, gifted students, Creative Process,


These days everything changes so fast that things learnt in school often become outdated in a very short time. In Japan, despite a policy over the last 50 years of a standardized education for all, in the last 10 years, we have seen a widening gap in achievement between top-level and low-level.Many students complained that they did not like mathematics or science. As the result, the Ministry of Education, Cultural, Sports, Science and Technology (MEXT) set the guidelines for mathematics education at the below average level in an attempt to reduce the burden for students. However, many university professors complain that the achievement level of students in mathematics classes is decreasing, even to the level where they cannot compute basic calculations such as fractions (Okabe, T. and others, 1999). On the other hand, rapid development in the science and technology make it essential to produce creative students with a high-level ability in science and mathematics. Therefore, MEXT recognized the need to promote comprehensive research and development in order to accomplish the highest creative achievements in worldwide comparisons by nurturing the merits of science, technology, and scientific research, and by seeking harmony and balance between science and technology on the one hand and scientific research on the other (Japan's Science and Technology Policy, 2001). Along the formulation of the "Science and Technology Basic Plan" at the end of March 2001, MEXT set a plan to develop students based on the creativity of science and technology, including curriculum development for gifted students in science and mathematics. Then, the Super Science High School (SSH) project started in 2002.


The purpose of this project is to develop a curriculum in mathematics, physics, biology, chemistry, and earth science for gifted and mature students. The three-year project was inaugurated in 2002 and each high school participates for 3 years. MEXT selects 26 high schools every year. Seventy-seven schools applied for the project in the first year. The budget for the project was 727,000,000 yen (about 6,000,000 dollars) for 26 schools in 2002, and 1,186,000,000yen (about 9,000,000 dollars) for 52 schools in 2003. Selected high schools are expected to develop curricula based on science and mathematics in cooperation with universities or research institutes. Specialists in each field and researchers in education for each subject make up the research group that will examine and analyze their activities and curricula.


We referred to reports from the SSH and the data from the websites of these high schools to categorize the curricular topics and their activities.

(1) Lectures by university professors to introduce new areas in mathematics

All schools planned lectures by university professors. In scientific topics (physics, biology, chemistry, and earth science), there were many lectures by people from companies, but few lectures on mathematical topics. Nineteen mathematical topics were lectured by eleven high schools. They were beyond the government guidelines in mathematics, such as, “Fractal science and the Logalism,” “A Polyhedron and Geometry,” “The Forefront of Modern Mathematics: Chaos Theory.” These topics matured students about new areas in mathematics. In two high schools, students investigated the lecture topic before the lecture and continued to study it after. For example, Benoit Mandelbrot gave a lecture on fractals at a high school attached to the Education Department of Kyoto University. This is a very new topic in high school curricula. To prepare for listening to the lecture, students studied affine geometry, sequences, logalism, and such beforehand. After the lecture, they investigated fractals on the computer and reported what they learned.

Students in Nagaoka High School in Niigata prefecture took lectures “Chaos Theory” to introduce modern mathematics by the Professor Kawamura. Results of a questionnaire showed 43% of students were practically able to take an image what chaos theory is and 55% of students were interested in the lecture. Forty-eight percent of students became motivated to learn mathematics as a result of the lectures.

(2) Constructing New Curricula that Combine Mathematics With Other Subjects

Six schools tried to combine mathematics with other subjects. For example, Honjyo High School which is attached to Waseda University, constructed a curriculum that combines mathematics with physics and chemistry.At first, students learn what a vector is, the reason why you need a vector, and calculation with vectors in their mathematics class. Then they learn about projection from a slanting direction, relative speed, and the composition of forces in their physics class. As table 1 shows students take the mathematics class and physics class in turn, and the classes are called “corroboration classes”.

Students in Daiichi Girls High School in Miyagi prefecture took a lecture “The Roles of Mathematics in Economics”. In their impression of the lecture, some students wrote that they understood from the lecture the importance of mathematics even for students who are not majoring in science, engineering or mathematics and other students were interested the relation between mathematics and the economy

Students in Chiba High School took lectures entitled “Fractal Science”. The goal of these lectures was to connect mathematics to physics and biology. After the lectures students visited a university laboratory and observed by microscope fractals in the pattern of bacterial development and crystallization. It was written in students’ visiting reports that they understood fractals more clearly after visiting the laboratory.

(3) Trial of New Curricula and New topics for High School Mathematics

Five schools held lectures about fractals by university professors. They held extra classes to understand the topic, and some of the classes used computers.

Other schools tried new materials such as “the structure of ciphers,” “learning mathematics and science in English,” “statistics.”. In Japan, text books which are used in school are usually published by MEXT. In this project, six schools produced their own text book. Some of them changed the order of teaching. Some of them added new topics connected with other subjects like modeling, more advanced materials like epsilon-delta logic or introduction of symbolic logic.

(4) Using Technology

Five schools conducted classes with technology. Three of them used Mathematica, and two of them used a programming language. In the high school attached to the Education Department of Kyoto University, students made a program to draw Sierpinski triangles and explored the characteristics of these triangles. Two of them planned to use a graphical calculator. At Kaiho High School in Okinawa, students used the calculator to explore the foci of quadratic curves and the functions of sounds. A software for learning functions, Grapes, produced by a high school teacher, Katsuhisa Tomoda, and offered in his website as a freeware ( http://okumedia.cc.osaka-kyoiku.ac.jp/~tomodak/grapes/volume. html) is widely used in high schools.

(5) Increasing Mathematics Classes and Enhancing the Topics in Standards.

Most of the SSH schools increased mathematic classes by two or three per week and went beyond government guidelines by teaching more advanced and challenging material. In other subjects, they tried to make scientific experiments beyond the government guideline material. Some of them made experiments at the university laboratory. Experiments in other subjects motivated students in learning mathematics. Increasing mathematics classes must be effective for all science. One student comments, "it was too difficult to understand the university professor’s lecture in a physic class, without basic mathematics knowledge. So, I should study mathematics more" in his report.


New trials of mathematics classes lead to motivating students to learn mathematics more effectively. Moreover experiences in other subjects also motivated them to learn mathematics. In this research, systematization of these trails in the SSH for a curriculum for gifted students is left.


We deeply appreciate the twenty-six SSH 2001 schools for allowing us to view their reports and for cooperating with our research.This research is supported by the science research fund of MEXT 14022101.


        1. Japan's Science and Technology Policy (2003) http://www.mext.go.jp/english/org/ science/07a.htm
        2. Okabe, T., Tose, N. and Nishimura, K. (1999), University students who are not able to calculate fractions, Toyo Keizai Shinpou

Kyoko Kakihana, Ph.D., Professor

Department of Computer Science

Tokyo Kasei Gakuin Tsukuba Women’s University

3-1 Azuma Tsukuba

Ibaraki 305-0031


Cell phone: +81 29 858 6292

E-mail: kakihana@cs.kasei.ac.jp

Suteo Kimura, Ph.D.

University of Meijyo




Kang Sup Lee, Dong Jou Hwang, Woo Shik Lee

Abstract: In this study, we developed teaching and learning contents of an enrichment program for the mathematically gifted high school students focused on the conic section. Class activities in the contents were selected and organized based on topic-centered, activity oriented, open, interdisciplinary and student-selected approaches, and also designed according to the Triad Enrichment Model developed by Renzulli.


The gifted education should be composed of various educational contents and methods by considering the characteristics of the gifted. In Korea, since 1999 which is the origin year of the Gifted Education Act, the gifted education program can be characterized by variety of approaches: special purposed high schools for the gifted and talented, after-school enrichment programs in elementary and junior high schools, acceleration and enrichment programs provided by gifted education centers affiliated with school boards and universities, and cyber education systems for the gifted.

The Korean Education Research Institutes (KEDI) developed the general curriculum for the gifted and talented [7], [4] and the curriculum for the mathematically gifted and talented [6]. Based on the curriculums mentioned above, practical programs are in progress of development. For the area of mathematical enrichment program, after-school activities in elementary and junior high school were studied [16], [8], [9], [3], [13], [14]. However, the mostly developed programs were for the gifted in middle school. We have the only material ‘Mathematics III’, the textbook of science high schools, which is developed from KAIST 1999 (revised 2003). In their study, Shin, etc [24] pointed out that the current system for the enrichment programs of special purposed high schools does not efficiently provide know-how on how to guide the mathematically gifted students to become creative knowledge producers. For these reasons, teaching and learning materials for the science high school and general high school’s mathematically gifted class students are needed.

The other hand, exploration of ERIC from 1981 to 2003 showed 233 documents related to the mathematically gifted/talented program and of those found there were 131 (56%) acceleration program documents, 89 (38%) enrichment program documents and 16 (6%) acceleration/enrichment program documents. As the result, mathematically gifted/talented program should be implemented for intensive learning through acceleration and enrichment [12].

The purpose of this study is carried out to develop an enrichment program with conic section for the mathematically gifted students who are identified by and participate in the enrichment programs provided by the science high schools in Korea. This study is based on the characteristics of the mathematically gifted, nature of enrichment programs for the gifted, principles of developing program for the gifted.


1. Development Trends on Enrichment Program for Mathematical Gifted

The National Council of Teachers of Mathematics (NCTM) [17] recommends that all mathematically talented and gifted students should be enrolled in a program that provides broad and enriched view of mathematics, which holds them to higher expectations. NCTM endorses use of Renzulli’s Traid Enrichment Model in providing opportunities for the mathematically gifted K-12. NCTM, however, contents that in almost all cases, gifted students benefit more from enrichment than acceleration. For a limited number of extremely talented and productive students, NCTM supports accelerated programs enhanced by enrichment. Any program for mathematically talented students should be expected to measure up on the following essential components [11]. First, the mathematical content of the program must be of a high quality. These differences should be reflected in the difficulty, cognitive level, breadth, and depth of the curriculum. Second, programs for the mathematically gifted and talented must nurture high-level thinking processes. Third, efforts must be made to include applications of mathematics to real-world situations as well as the examination of standard topics in greater depth. Fourth, the ability to communicate is essential in learning mathematics. Fifth, mathematics, with its unique content characteristics, provides an effective vehicle for developing study skills and work habits. Sixth, the program must provide opportunities for students to explore mathematical ideas in a creative fashion. Seventh, gifted students need frequent and imaginative use of manipulative materials and other instructional aids. Mathematics should be related to other content areas of school program.

As we plan programs for these top students, we first examine the purposes behind our programs. There are many reasons why we might want to help students develop their mathematical abilities [23]. These include: First, Helping students become deep mathematical thinkers. Second, Developing an informed citizenry. Third, allowing students to experience the joy and the beauty of mathematics. Forth, enabling students to be competitive at the university level and beyond. Fifth, developing world leaders in our increasingly technological world.

Other developments that had their origins in special programs are currently being examined for general practice, These developments include: a focus on concept rather than skill learning, the use of interdisciplinary curriculum and theme-based studies, student portfolios, performance assessment, cross-grade grouping, alternative scheduling patterns, and perhaps most important, opportunities for students to exchange traditional roles as lesson-learners and doers-of-exercises for more challenging and demanding roles that require hands-on learning, first-hand investigations, and the application of knowledge and thinking skills to complex problems [20].

Gu, etc [6] and Hwang, etc [12] have picked out some points on concerning the methods of teaching and learning for the mathematically gifted in order to maximize the mathematically gifted considering their properties. Their suggestions are as follows:

First, the curriculum is plotted to develop not only the intellectual area but also the emotional and social area of the children and emphasizes interdisciplinary approach.

Second, the curriculum should be composed of various educational content and method by considering the characteristics of the high schools gifted.

Third, the curriculum emphasizes the enhancement of creative thinking and higher-order thinking through enrichment than acceleration.

Fourth, the curriculum should be composed of various modern mathematics content, which will lead future society as well as education program, which enables the children to prepare for the future information and creativity society.

Fifth, the curriculum is an educational program that enables student to perform an expertise study on the area where they are interested in and specialized in, inclusive of individual project or study project to reinforce the communication function in the mathematical thinking.

Sixth, the curriculum provides an education program, which has good connectivity to the elementary school, middle school, university and other educational agencies.

KEDI [14], Kim [15] and Han [9] have suggested teaching-learning materials as well as a direction in development of the teaching-learning material for the gifted students. On their research, when developing materials, followings should be considered; creative problem solving ability in concerned subject, developing high-order thinking power, interest in the concerned subject content and activity, concentration on assignment, developing self-confidence, developing self-directed learning attitude, focusing more on reinforcing than speed renovating study, individualized education, diverse degree of difficulty and presenting reinforcing activity assignment, including content from various area, connecting with problem situation which can occur in actual life, composing of sub-subject, interdisciplinary subject, including content or activity, organizing various group, various teaching and learning material, application on location, emphasizing on creative products.

2. Contents of Enrichment Program Focused on the Conic Section

In this study, all activities were organized according to the Triad Enrichment Model developed by Renzulli [19], which consists of Type I, Type II and Type III enrichment activities. Type I activities are planning and implementing process. Type II activities training consists of process that should be taught in connection with a Type III activity. Type III activities are individual and small group investigations of real problems.

Type I activities are mainly organized to strengthen motivation of students to study on a topic. Each activity devised and organized to increase motivation and required students to exercise the variables conic section activities and mathematical thinking and to produce products at the professional level. It also required providing free and rich educational environments for the mathematically gifted. The goal of Type II activities is to help students to have deep understanding and master skills required to carry out mathematical investigation studies. Each activity was organized to exercise the property of the conic section and to find the cryptosystem on elliptic curves. The goal of Type III activities is to provide students with opportunities of carrying out research projects on the topic being studied.

The developed teaching and learning material is composed to the first step as making plan, the second step as acquiring the knowledge and function, the third step as performing them and the fourth step as presenting, evaluating and reviewing. Each step contains various activities. The teaching and learning materials of this study is what reinforced the unit of ‘quadratic curve’ in the ‘Mathematics III’ of high school textbook. This teaching and learning material is developed for gifted high school students in mathematics as well as for the general high school students who are advanced and strongly interested in mathematics. Sixteen class activities are composed as project performing procedure in four steps so it can be utilized efficiently when required a group instruction for the mathematically talented students or a focused teaching.

First step: It calls interests and active participation of students into the subject through the various activities; finding the conic section used in real life, paper folding of conic section using rectangle, expressing the activity of paper folding of conic section mathematically, proving conic section using the Belgian mathematician Dandelin (G. P. Dandelin)’s sphere and etc.

Second step: This is to provide the students sufficient knowledge concerning conic section through activities; studying on the property of focus reflection of the conic section, making inequality in the shape of parabola, finding the relationship between the unit complex number and trigonometric function using the polar form of the complex number, finding the relationship between orthogonal equation and polar equitation, finding the property of conic section. Also it enables students to study on the discrete logarithm problem which is basic of RSA cryptosystem using the conic section property and algorithms for simple signature scheme.

Third step: This enables students to express the conic curve as a polar equation based on the knowledge and function acquired from the performance, and to study on cryptosystem and crypto experiment on the elliptic curve as well as the algorithms. And it instructs them to compose a subject report based on the properties of elliptic curve.

Fourth step: In this presentation and evaluation/reflection step, students are to present, discuss on the their subject report and evaluate it according to the objective followed by the overall review and modification on project procedure and products (subject report or idea suggestion) based on the evaluation.

3. Teaching and Learning Plan

The topics, main contents and activities in the proposed program are shown as in the following Table 1.


We showed the new directions for development the enrichment program in the high schools for the mathematically gifted.

We could think about mathematics students along a continuum or hierarchy as shown in the following diagram. Sheffield [22] showed that illiterates ? doers ? computers ? consumers ? problem solvers ? problem posers ? creators. The notions of fluency, flexibility and novelty were adapted and applied in the domain of mathematics by Balka [1], who asked subjects to pose mathemati cal problems that could be answered on the basis of information provided in a set of stories about real world situations. Problem posing, along with problem solving, is central to the discipline of mathematics and the nature of mathematical thinking [25]. When mathematicians engage in the intellectual work of the discipline, it can be argued that the self-directed posing of problems to be solved is an important characteristic [18].

Also, Runco and Chand [21] explains that problem finding is the starting point and key to producing creative products. Given the "creating a problem" ; characteristic of problem posing and the "bring into being" nature of creativity one might see problem posing as a kind of creativity. In fact, problem finding has sometimes been considered as a creative process in itself [5]. Studies in mathematical creativity were reviewed [10] and one may see problem posing ability as a creative ability.

We assume the teacher's belief about mathematics and a role of textbooks are connected to fostering mathematical creativity, and role of textbooks is important. "Problem finding and problem posing" can be never seen in Korean high schools textbooks. To make textbooks contain problem finding and problem posing we have a proposal of Three-Stage Mathematical Creativity Enrichment Model for mathematically gifted and regular students.


        1. Balka, D. S. (1974). Creativity ability in mathematics. Arithmetic Teacher, 21(7). 633-636.
        2. Bang, S., Hong, J., & Hwang, D. (2001). Development of Curriculum for the Gifted in the High Schools. Journal of the Korea Society of Mathematical Education Series F: Studies in Mathematical Education,6. pp.223-245.
        3. Bang, S., Lee, S., & Lee, W. (2002). Development of Enrichment Programs for the Mathematically Gifted: Focused on the 9th grade. Journal of the Korea Society of Mathematical Education Series F: Studies in Mathematical Education,7. pp.103-119.
        4. Cho, S., Kim, H., Kim S., Bang., S. & Hwang., D. (2000). Development of Curriculum for the Scientifically Gifted . Seoul: Korean Education Development Institute.
        5. Dillon, J. T. (1988). Levels of problem finding vs problem solving. In: questioning exchange 2(2), 105-115.
        6. Gu, J., Cho, S., Kim, H., Soe, H., Jang, Y. Lim, H., Bang, S., & Hwang, D. (2000). Development of Curriculum for the Gifted - Fundamental Study of development the curriculum for the gifted in the high schools . Seoul: Korean Education Development Institute.
        7. Gu, J., Cho, S., Kim, H., Soe, H., Jang, Y., Hwang, D., & Lim, H. (1999). Development of Curriculum for the Gifted - Fundamental Study of development the curriculum for the gifted in elementary and the middle schools. CR 99-20. Seoul: Korean Education Development Institute.
        8. Han, I. (2000). A study of development the Enrichment Program Materials: Focused on a problem for construction. Journal of the Korea Society of Mathematical Education Series F: Studies in Mathematical Education,5. pp.221-232.
        9. Han, I. (2001). The direction for development the program in middle and the high schools for the mathematical gifted. Proceeding of the Korean Society for the Gifted.
        10. Haylock, D. W. (1987). A framework for assessing mathematical creativity in schoolchildren in Educational Studies in Mathematics 18, pp. 59-74.
        11. House, P. (1987). (Ed.) Providing Opportunities for the Mathematically Gifted, K-12. Reston, VA: National Council of Teachers of Mathematics.
        12. Hwang, D., Hong, J., & Seo, J. (2002). An Analytic Study of Mathematics Gifted/Talented Education Program of U.S.A. by ERIC Search. Journal of the Korea Society of Mathematical Education Series F: Studies in Mathematical Education, 7. pp. 121-131.
        13. KEDI (2002). Development of Instructional Materials for the Gifted Students in the Regular Schools. Seoul: Korean Education Development Institute.
        14. KEDI (2003). Development of Instructional Materials for the Gifted Students in the Regular Schools. Seoul: Korean Education Development Institute.
        15. Kim, S. (2001). The direction for development the program in elementary schools for the mathematical gifted. Proceeding of the Korean Society for the Gifted.
        16. Lee, S., & Han, I. (2000). A study of development the Geometrical Enrichment Program Materials: Focused on a analogy. Journal of the Korea Society of Mathematical Education Series F: Studies in Mathematical Education,5. pp.165-174.
        17. NCTM (2000). Principles and Standards In Mathematics. Reston, VA: National Council of Teachers of Mathematics.
        18. Pólya, G. (1954). Mathematics and plausible reasoning. Princeton, NJ: Princeton University Press.
        19. Renzulli, J. S. (1978). The enrichment triad. Mansfield Center, CT: Creative Learning Press.
        20. Renzulli, J. S., & Reis, S. M. (2000). The Schoolwide Enrichment Model. In: Heller, K. A., Monks, F. J., Sternberg, R. J., & Subotnik, R. F. (Eds.), International Handbook of Giftedness and Talent (2nd Ed.). Oxford: Pergamon Press.
        21. Runco, M. A., & Chand, I. (1995). Cognition and creativity. Educational Psychology Review , 7, 243-267.
        22. Sheffield, L. J. (Ed.) (1994). The Development of Gifted and Talented Mathematics Students and the National Council of Teachers of Mathematics Standards. Research-Based Decision Making Series. Mathematics. The National Research Center on the Gifted and Talented.
        23. Sheffield, L. J. (1999). Developing Mathematically Promising Students , Reston, VA: National Council of Teachers of Mathematics.
        24. Shin, H., Lu, G., & Han, I. (2000). A study on the Gifted Education for the Mathematical Special Class in the Special High Schools. Journal of the Korea Society of Mathematical Education Series F: Studies in Mathematical Education,5. pp.125-140.
        25. Silver, E. A. (1994). On mathematical problem posing. For the learning of mathematics , Vol14(1). 19-28.

Kang Sup Lee

Professor and Dean of College of Education, Dankook University and

President of Korea Society of Mathematics Education

Dept. of Mathematics Education

Dankook University

Hannam-dong, Youngsan-Ku, Seoul 140-714


E-mail: leeks@dankook.ac.kr

Dong Jou Hwang

Graduate School of Dankook University and

Division of Electronic & Info-Communication

Yeungjin Junior College

218 Bokhyun-dong, Buk-ku, Daegu 702-721


E-mail: hdj0719@chol.com

Woo Shik Lee

Dept. of Mathematics

Gyeongbuk Science High School

418-1, Yongheung-dong, Buk-ku, Pohang City

Gyeongsangbuk-do 791-170


E-mail: lws54@chol.com



Kathy Gavin, Linda Sheffield

Abstract: The following proposal is designed to address questions of interest to the participants in Topic Study Group 4 (TSG4): Activities and Programs for Gifted Students by presenting findings from Project M3 , Mentoring Mathematical Minds that focus on:

a) What activities and programs are useful for identifying gifted students and assessing their potential?What are the cognitive processes of gifted students? What are characteristic features of talent in mathematics?

b) What are special learning environments for gifted students, and how does one work with them there? What happens when they interact with peers?


In 1980, the National Council of Teachers of Mathematics (NCTM) made a bold statement, "The student most neglected in terms of realizing full potential, is the gifted student of mathematics." As test scores indicate, progress since that time for gifted students in the United States has been slow or nonexistent in this area. This is especially true for underrepresented students from economically disadvantaged backgrounds.

Project M3 , Mentoring Mathematical Minds, is a five-year collaborative research effort of faculty at the University of Connecticut, the University of Northern Kentucky, and Boston University and teachers, administrators, and third through fifth grade students in ten schools of varying socioeconomic levels in Connecticut and Kentucky.

As part of Project M 3, a team of national experts in the fields of mathematics, mathematics education, and gifted education are creating a total of 12 curriculum units of advanced mathematics (four units per grade level) accompanied by professional development modules. A mathematics talent pool of students is identified in each of the ten schools (total N = 800) and the units are being implemented in a variety of settings. We will also modify some of these units to use with all students across ability levels and backgrounds in differentiated classroom settings. Pre and post achievement and attitude data are being gathered using standardized and criterion referenced tests. To enhance the effectiveness of these units, we provide extensive professional development for a total of 40 teachers, including yearly summer institutes, school year in-service, and an Internet portal for continuous communication and dissemination of resources.

Research questions focus on three items: measuring the changes in mathematics achievement and attitudes for talent pool students after exposure to the intervention model; measuring the difference in mathematics achievement and attitudes between the experimental and comparison groups; and measuring the changes in mathematics achievement and attitudes of students exposed to modified units in differentiated classroom settings. The research questions will contribute to the summative evaluation component. The formative evaluation will include an annual assessment of the delivery of training using classroom observations, teacher interviews and surveys, and student focus groups.


In order to identify, create and serve students with mathematical promise, especially those in economically disadvantaged areas, a variety of measures are being used. These include traditional measures such as achievement tests and teacher recommendations as well as other instruments such as a nonverbal ability test and measures of creativity. Students in the program have widely divergent scores on a variety of measures. Data indicate that no single measure is sufficient to identify the majority of students from diverse backgrounds with mathematical promise.


The initial group of students selected for Project M3 : Mentoring Mathematical Minds are third graders who began the program in Fall 2003. One of the most critical aspects of the program is the use of a student-centered inquiry approach that encourages students to think like mathematicians, asking questions that enable them to make sense of mathematics. Students study four units per year that were developed to add depth and complexity to the typical elementary mathematics curriculum following recommendations from the National Council of Teachers of Mathematics Principles and Standards for School Mathematics and based on best practices in gifted education. Each lesson has “Think Deeply” questions and a Mathematician’s Journal that students use to develop and organize their mathematical reasoning. Students who are ready for more challenge are presented with “Think Beyond” questions that encourage them to delve more deeply into the mathematics. “Hint Cards” are available for students who need more information to get started on an investigation. Students frequently work with a partner and in small groups that provide stimulating and necessary dialogue to foster conceptual understanding. This is often followed by whole class discourse giving students an opportunity to further develop and consolidate their own mathematical reasoning and questioning skills as they work with classmates to develop and analyze complex skills and concepts.

Additional information on Project M3 can be found on our website at http://www.projectm3.org.


Dr. Kathy Gavin , Project Director

E-mail: kathy.gavin@uconn.edu

Dr. Linda Sheffield, Project Co-director

E-mail: sheffield@nku.edu



Elena Levit, Larisa Marcu, Orna Schneiderman

Abstract: Mofet Science classes start in 7th grade. All students who want to study in a science class have to take a preparation course [1] (“Mechina”).


The transition of a child from elementary school to middle school occurs simultaneously with the transition from childhood to adolescence.

According to Piaget, at ages 11-12 children are able to understand the world surrounding them, and its application in their lives [2]. An adolescent understands the role of independence and the importance of taking responsibility.

The preparatory course offers the adolescent an opportunity to develop skills in decision-making and accepting responsibility, similar to the way an adult would [3].

Moreover, the preparatory course helps develop the adolescent’s process of self-evaluation, and attempts to further independent identity [1], [3]. 

According to the educational/psychological literature, the assessment of individual intellectual capability is very important in utilizing the creative processes to forecast, plan and set up targets and make decisions [4], [5], [6].

The course is aimed to reduce as far as possible, the conflict between the adolescent’s inner values and those of the society in which he or she lives.The balance between these two sets of values inspires the development of the new skills and capabilities of the adolescent to create new relationships and take responsibility for his or her decisions [1], [3].

Middle schools accept students based on their grades from elementary school, and other tests. Mofet accepts children based on theirexpected ability to improve their grades by teaching them new knowledge, and establishing learning habits and motivation in this transition to middle school.

The demands of the preparatory course are very strict. It is an elite course and the students are proud to graduate. Indeed, the course eases the transition from elementary to middle school. The student studies in the course with his or her peers, which is helpful and diminishes pressure and anxiety. At the end of the course, we obtain high quality students, with outstanding learning skills, who believe in their ability to succeed academically.

The preparatory course has ameliorated the performance of students with average achievements, and their integration into the science classes.

Knowing each student and his unique needs enables the teacher and the staff to cope with his problems efficiently. This is based on a diagnostic process aimed to reveal the potential in every individual. In this process, both teacher and student obtain a tool; a direct personal relationship is necessary for building common trust [7].

The development of intellectual processes is dependent upon the nature of instruction and the overall abilities of the student [7].


            1. Students interested in science.
            2. Students without any exceptional behavior problems.
            3. Students motivated to study.Motivation, according to Mofet’s method, is the key to success. It is important to note that motivation is the engine of development of any child. Mofet believes that with motivation, event the child who is considered “weak” at the onset of study will be able to attain significant results.
            1. Learning the work patterns in Mofet science classes.
            2. Acquiring effective learning methods and skills.
            3. Filling gaps in knowledge.
            4. Exploiting natural curiosity of children as a stimulating and propelling factor in learning.
            5. Increasing self-confidence and establishing the ability to cope with exam anxiety because of past failures.
            6. Creating a supportive learning environment [1], [7], [4], [5].
            7. Expanding in depth the material studied in elementary school.
            8. Developing mathematical, logical and analytical skills [1], [6], [7], [8].
    1. Subjects studied: Mathematics, Physics and English, total of 60 hours.
    2. The preparatory course is given in 20 sessions of 3 academic hours each or 15 sessions of 4 hours each.
    3. The sessions are partly based on frontal teaching and partly on group work, giving emphasis and attention to the unique problems of each student.
    4. The special structure of the course gives the students optimal training for the beginning of their studies in the Mofet class:
        1. Learning different approaches to problem solving.
        2. Learning models of elimination and educated guessing.
        3. Learning methods to solve problems with the aid of evaluation and appraisal.
        4. Repetition of basic subjects in mathematics in a deeper and broader view.
    5. Teaching English with a special method, fulfilling the potential of each student and assuring better ability in middle school.
    6. Quizzes are used to monitor the personal progress of each student, and are used for feedback to each student personally.
    7. A workshop is conducted, aiming at:
        1. Linking the students who come from different schools.
        2. Differentiation among the terms leadership, leader, and public.
            1. The importance of job definition, authority delegation and responsibility within a work team.
            2. The workshop includes the following activities:
            3. Acquaintance exercise.
            4. Correlation of expectations and an agreement.
            5. Determining leadership.
            6. Cooperation versus competition.
            7. Organizing a team.
    1. The first stage of this process is a day of interviews with the student and his/her parents, the school principal, the school advisor, the class coordinator of the school and a representative from Mofet.
    2. During the interview we can tentatively assess the interaction of the child within the family and comment on family interaction in the presence of the child and his parents.
    3. Examples from the student’s questionnaire completed in the interview:
    4. What do you know about Mofet classes?
        1. Why do you think you are a suitable candidate for a Mofet class?
        2. What are your expectations from a Mofet class, academically and socially?
        3. How do you intend to fit into the Mofet class? What ideas can you bring up for the bonding of the class?
        4. Do your parents want you to study in Mofet class? Yes/no (Afternoon activities: sports (hours per week) Arts (hours per week), any other, hobbies.)
        5. Parents and other family members are very significant factors, whose involvement or lack of it will influence the educational team’s success. The acquaintance of the educational team with the parents and the family members will lead to constructing common work patterns, and to an intensification of their participation in their child’s education process (Fig. 1).

Mofet, as an external body, has no commitment to accept a child to a Mofet class because of parental pressure. The talks with the students are most significant: a student who does not find the program suitable can consider other alternatives.

The home has an extensive influence on the commitment to study; therefore all children and their parents will sign, at the beginning of the course, a binding agreement in the following areas:

        • Academic demands (preparing homework, successfully complying with work in class)
        • Complying with disciplinary regulations (one breach, two instances of tardiness or missing a course twice will result in immediate suspension).

In a modern school, the teacher should know how to plan the intellectual development of each and every student [6], [8]. Therefore, the studies in Mofet are a means for developing the intellectual capability of each student, i.e. the importance of studying Math as a tool for developing mathematical thinking [8], as well as a tool for developing creative potential, reason and general learning skill [2], [4], [5].

    1. Designating a suitable person to coordinate the whole project of the preparatory course, from the stage of planning until the end.
    2. An excellent teachers team, professional and experienced.
    3. Availability of teaching material, teaching aids and equipment needed for the course.
    4. Regular meetings of the team
    5. Building cooperation with the parents, and establishing a mutual channel for dealing with all the problems (discipline, studies, motivation, anxiety).
    6. The teachers’ multidisciplinary teamwork is based on the principle that the team should work in interaction, focusing on the student they undertook to promote. The educational team learns to work in cooperation and takes total responsibility for the education program. The core of teamwork is mutual fertilization, and taking a professional stance towards the student as an individual. Feedback from the student enables each team member to deal accurately with each of the students’ needs. In this way an encouraging atmosphere is created, supporting initiative, creativity, and the search for new ways and approaches in teaching.

Mofet's teachers are creative enough to use a beneficial variety of activities in the classroom, to be able to adapt a course book to meet the needs of the learners, and to design materials and activities tailored to specific classes.

Our approach to the role of the teacher in the learning process is represented in the Fig. 2.


    1. A first quiz aimed to assess the initial level of each student, and that of all the group of students, in order to select suitable teaching material for any gaps that have been identified.
    2. Forming small learning groups, which enables the teacher to identify problems and deal with them immediately.
    3. Establishing learning habits: each student will receive a booklet with the curriculum, explanations, working exercises and reading material, so that he or she can plan towards defined targets.
    4. Establishing work practices of a high-school level, starting with an orderly copying from the blackboard, taking notes during lectures, and preparing homework.
    5. A daily check of the student’s output, locating problems and immediate reaction.
    6. Planning free time: studying, social activity and hobbies.
    7. Lesson Structure:
        1. Frontal lessons, independent learning and group teaching, simulations, competitions, viewing and discussing educational films, research and discovery activities, activities for enhancement and elaboration on subjects from the studies, construction of mathematical and physical models (demonstrations)
        2. Mathematics and Physics - presentation of data, analysis, solving exercises and problems in math, geometry, logic, comprehension of math through physics.
    8. English – grammar, reading texts, vocabulary, composition, understanding.
    9. After several lessons, the work is done in groups or in couples, when the teacher knows the students’ level quite well, and can match them in learning groups.
    10. During the lessons the focus is on:
        • Emphasizing multiple possibilities – there is no one right answer.
        • An atmosphere of love (trust, understanding, confidence, tolerance towards the other) and support, enabling each individual to access in-depth knowledge, to locate platforms to express his/her creativity, and to be able to take risks.
        • An atmosphere that supports the strength of effort. When there is no room for mistakes or failures, one does not tend to take risks, and a large part of one’s creativity is not expressed.
        • A transition to a research method of study in which the student defines what interests him/her and in what method, what time and place, he/she chooses to study. The teachers will take into consideration the interpersonal aspects between the students.
The following illustrates a sample lesson for mathematics students:

What digit is located in a particular location?

1. Students make a table (searching for the last digit in each column).

2. What can be deduced from the table?

3. Question 1 : Prove that 1 + 2 + 3 + …+ n cannot end with the digits 2, 4, 7, 9.

Question 2 : Prove that no three whole numbers (x, y, z) exist whereby

x4 + y4 =3 z4.

4. Explain: Filling in the table leads the students to draw the following conclusions:
    • the square of a whole number cannot end with the digits 2, 3, 7, 8.
    • the fourth power of whole numbers and the doubled squares of four can end in 1, 5, 6.
    • any power of numbers ending in 1, 5, 6 will end in 1, 5, 6, respectively.
    • the third power of any whole number can end in any digit.
    • the multiple of two consecutive natural numbers n (n+1) = n2 +n can end in the digits 0, 2, 6 only.
Solution to Question 1:

According to the table, n2 +n can end in 0,2,6, therefore can end in 1, 2, 3, 5 only.

Solution to Question 2:

According to the table, x4 + y4 can end in 0, 1, 2, 6, 7 only. Therefore 3z4 can end in 3, 5, 8 only.

Further expansion

Question 3: Prove that there are no whole numbers, a, b, c, whereby the sum of
8 + b8 + c8 will not end in the digit 9.

Solution to Question 3: According to the table, each of the variables in the equation can end in only 1, 5, or 6. The sum of the three variables can end in 1, 2, 3, 5, 6, 7, or 8.

Question 4 : Determine the last digit of the number 212 +243 and check whether the next to last digit is even or odd.

Solution to Question 4: 212 + 243 + 46 + 43 63 + 43 (43 +63) + 64 …0 = …0

Summary : these are activities designed to enrich the student’s knowledge of the subject. Students learn to ask and answer questions, as well as to draw conclusions from the activity for the purpose of solving new problems.


Students are assessed in the following procedures:

1. Observation and analysis of events in real time

2. Pedagogic monitoring

3. Exams and quizzes

4. In each lesson, an exercise or a quiz or a dictation is given, to assess the student’s relative progress.

5. A feedback form is circulated mid-course to receive student reaction to course progress. The students are requested to answer the form fully. On this basis the pedagogic team can learn and draw conclusions how to improve the second half of the course.

Examples of questions from the feedback form:

        • Is the material you study in the course difficult to understand or easy?
        • Is the material interesting?
        • Are the lessons presented in an interesting way?
        • My feeling in general in the course (organization, teachers, attitude)
The following is a sample of the type of questions used to gain understanding of the student's attitudes toward the learning process, in this case their views on homework:

6. At the end of the course each student receives a self-assessment sheet , to sum up his or her participation in the course.

Some questions are:

        • How do you think you will cope with material you have not yet mastered (that will demand a lot of effort to understand)?
        • How did you feel about your struggle with homework?
        • Are you interested in joining the Mofet class now that you have experienced the course, and to what extent?
        • Evaluate the extent to which the Mofet plan is suitable for you?
        • In doing the homework - did you use your personal knowledge, or you did you need additional knowledge from parents/friends/books/Internet?
        • Are the students you met in the Mofet course appropriate classmates?
        • Does your decision to study in Mofet depend on whether or not one of your classmates will join? If yes, what is his name?
7. A personal assessment is given to each student, consisting of discipline, ability to cope with difficult targets, teamwork, independent work, curiosity, motivation to study and scientific orientation.

8. At the end of the course each student receives an achievement diploma written as follows:

”In the short period of the Mofet course you have shown high motivation and wonderful diligence; you’ve succeeded in learning new terminology and coping with difficulty, finding new friends and getting to know the teachers. We wish you success wherever you go.”

9. At the end of the course, the Decisions Committee – a professional steering team of the middle school, whose members are the principal, teachers from the course, the school advisor and Mofet’s national representative – holds a discussion and decides who will be admitted to the Mofet classes.


The following flowchart (Fig. 3) summarizes the various stages in the process of screening students in advance of their acceptance to the Mofet program.


Our experience has shown the following level of achievement by those students who complete the Mofet program.

    • 40% of students completing the Mofet program sit for 5-point matriculation exam in mathematics, physics or chemistry as well as participate in academic courses concurrent with their matriculation studies in high school.
    • 20% of students participate in academic courses while in high school, earning credits toward their academic degrees, concomitant to their matriculation studies in high school.
    • 15% of students complete the 5-point matriculation exam in mathematics, together with an additional 4 points in physics or chemistry.
    • 15% of students complete the 4-point matriculation exam in mathematics, as well as the physics or chemistry exam, enabling their admission to academic institutions.
    • 10% of students complete their matriculation without an emphasis on mathematics and science subjects.
        1. Schneiderman O., Levit E., Marcu L., Zakharova A. From Mediocrity to Excellence: The MOFET Group for the Advancement of Teaching// Proceedings of the Third International Conference on CREATIVITY IN MATHEMATICS EDUCATION & THE EDUCATION OF GIFTED STUDENTS, ICCME & EGS’03, Rousse, 2003. pp 144.
        2. Piaget J. Psychology of Intelligence // Selected psychological works, Moscow: Prosveshenie (in Russian) 1969.
        3. Tzukerman G.A., Masterov B.M. Psychology of Self-development. Moscow: Interpraks (in Russian) 1995.
        4. Davydov V.V. Theory of Developing Education. Moscow: Intor. 1996.
        5. Vigotsky L. Psychology of Children . Collected works, V.4, Moscow: Pedagogy (in Russian) 1984.
        6. Kholodnaya M.A. Psychology of Intelligence: Paradoxes of research . Moscow: Bars (in Russian) 1997.
        7. Kalmykova Z.I.Productive Thinking as a Basis of Ability to Training . Moscow: Pedagogy (in Russian) 1981.
        8. Krutetzky V.A. Psychology of Mathematical Abilities. Moscow: Prosveshenie (in Russian) 1968.
        9. Sukhomlinsky V.A. About Intellectual Education. Kiev: Radyans’ka shkola (in Russian) 1983.
        10. PiagetJ. Les Structures Mathématiques et les Structures Opératoires de l`intelligence: L'enseignement des mathématiques. Paris (in French)1960.

Elena Levit, Ph.D., lecturer & scientist



Larisa Marcu, Pedagogic Coordinator



Orna Schneiderman, Director


Hakfar Hayarok, 47800


Tel: 03-6440493

E-mail: mofet1@bezeqint.net



Mark Saul


The creative teacher can make important contributions on many levels of education. We have gathered here mostly to talk about creativity and the education of the gifted. This paper explores some ways in which we can learn about teaching average students from our work with the gifted. It will also give examples of how we can find out more about working with gifted students by examining our work with average students. Finally, it will give some examples of how we can construct problem sets which start with very simple ideas and build towards rather significant mathematical results.


Mark Saul, Director

National Science Foundation






Miriam Amit,AlexeiBelov

Abstract: Working with young students is both challenging and frustrating. Exposing them to innovative concepts, higher mathematics, complex ideas and sophisticated proofs is challenging, but it can also be frustrating because they lack the requisite knowledge and skills. Usually, innovative ideas in mathematics rely heavily on previous knowledge and on skills such as algebraic techniques, argumentation methods, abstraction and generalizations. We sought to solve this problem in part by creating a special program that circumvents the obstacle of prerequisites and taps the ability of the young pupils. This program was successfully tried out at the Kidmatika Math Club at Ben-Gurion University in Beersheva.

It is well known in psychology that there are logical and geometrical ways of thinking, corresponding to the development of the left and right hemispheres of the brain, respectively. Geometrical thinking involves dealing with both plane or two-dimensional (2D) geometry and with spatial or three-dimensional (3D) geometry. Although there are many similarities between 2D and 3D geometry, there are also substantial differences, not only in subject matter but also in the skills and abilities needed to cope with these two domains. Even students talented for mathematics may lack 3D visual ability, both because 3D geometry is neglected in the schools and because they have had very little experience in handling 3D problems or developing spatial skills.

The purpose of the experiment described below was to combine the presentation of new mathematical ideas with the development of 3D visual ability in young students, thereby overcoming the barrier of prerequisite mathematical knowledge. The topic we chose to present was the mathematical idea of interlocking structures. Suppose we have several convex figures on a plane. It is always possible to find one figure that can be shifted without moving/shifting/touching the others. Hence any arrangement of convex figures can be disassembled one by one. Is this also true for 3D? The natural answer would seem to be Yes, backed up by everyday experience with blocks or spherical figures; however, in 3D space the answer is No! Arrangements of convex bodies have recently been found such that they lock each other and none can be removed. This kind of arrangement is called an interlocking structure. The theory of block arrangements is currently at an early stage of development, but it has already produced unexpected results and found important applications, such as in the construction of large roofs without cement.


The Kidmatika Math Club for young people is an after-school program operating twice a week on the university campus. It has 400 members aged 10-15 who are chosen on the basis of talent and motivation. Not only have they passed a battery of examinations, they have also proved to be persistent and committed to the math club. The guiding principle is a belief in talent cultivated from a young age, accompanied by hard work and constant practice, as in sport or music. The pupils work in small groups guided by 14 highly qualified teachers – mathematicians with special expertise in working with gifted children. These teachers have developed a unique program that combines exposure to subject areas not included in the school curriculum and the development of strategies for problem solving and proofs. The subject areas comprise logic, number theory, combinatorics, 2D and 3D geometry, optimization, non routine problems, algebraic laboratory, algorithms, creative innovation, math and culture, and more... The students participate in all the national competitions and Olympiads and have captured major prizes all over the country. The club was founded and is still directed by the first author, and the two author conduct some of the lessons.


In the experiment reported here, we started with the planar case and asked students to show and prove that in any arrangement of convex figures, any single figure can be shifted away. This is a simple well-known experience, and we were expecting an intuitive, non formal proof. There was a heated discussion with the students, who offered different approaches justified by experiences (we had real figures in the classroom) and drawings. We also discussed with them the problematic transition from an intuitive/feeling approach to formal mathematical expression. At the next stage, we asked students if the same conclusions were true for 3D. The immediate answer was Yes. We asked them to convince us, and of course they found 3D structures capable of assimilating every single component. However, our students knew two things: a) experience is not proof; and b) the very fact that we asked them about 3D indicated a possible trap. After further discussions, they began to rethink the problem. We split the group into teams, provided them with small wooden cubes and boxes full of sand into which the cubes could be placed, and asked them to build a structure from which no figure could be removed. To our considerable surprise, after about 40 minutes two teams succeeded (one completely and the other partially) in creating an interlocking structure.

At the last stage, together with the pupils we developed a semi-formal proof based on the fact that, in an interlocking structure, movement of any body in any direction will be stopped by opposite vectors. We did not use the formal symbols for vectors, confining ourselves to drawings of arrows and to verbal argumentation. Based on the discussion that followed and the group's reactions, we assume that the proof was understood by most of the students. We finished by showing pictures of interlocking tetrahedral structures.


The students in the group ranged in age between 12 and 14, and none had had previous experience in or knowledge of the domain. Yet they performed very well in the session, understood the complicated idea of interlocking structures, and applied this understanding to a concrete model as well as using an abstract approach. We do not claim this experience can be repeated with any class, but it is tangible proof that with the right pedagogy and a carefully orchestrated didactic strategy, even powerful mathematics such as interlocking structures can be grasped and unlocked by simple means.

The whole session was videotaped and documented. Our presentation will include a detailed outline of the experiment, including a video and concrete materials.


Belov Alexei



Dr. Miriam Amit

Head, Center for Science and Technology Education

Institute for Applied Research

Ben-Gurion University of the Negev

Beer Sheva

ISRAEL, 84105



Nobuaki Kawasaki


Bulgarian students have participated in mathematical Olympics every year.So I am interested in the substances of Bulgarian mathematical education. Last summer, I went to Rousse in Bulgaria.At that time, I bought some textbooks inmathematics. This time, I would like to focus on theteaching materialsof secondary education. I tried to find the characteristics of Bulgarian mathematical education by comparing both countries' textbooks in mathematics.


I want to introduce some examples in order to find characteristics of the teaching materials of mathematics inBulgaria and Japan, and to find some differences between the curricula.

When we want to compare curricula of both countries, there are some points of view. This time, I want to research each country'scurriculum of mathematics from two points of view. One point is that what substances are considered during each school year, and the other point is how to teach them.

Chart 1 in the next page has shownthe comparison between Bulgarian curriculum and Japanese one about some kinds of teaching materials of mathematics in secondary education. The grade is from 9th to 10th one.I made Chart 1 with the textbooks of both countries as a reference.

According to the Chart 1, Bulgarian curriculum has put emphasis on algebraand geometry. Especially, equation and inequalitysolving is substantial,and all students study them. On the other hand, Japanese curriculum has put emphasis on analysis, but some students do notstudy them, because Japanese curriculumallows thechoice.There are some remarkable examples in the Chart 1.

Except for the teaching material in the 12th grade, all teaching materials in the Bulgarian row of Chart 1 have been included in the required subject. However, in Japan, only trigonometric ratio has been included in required subject and other teaching materials have been included in selective subjects.

The arrows in the solid line have displayed the comparison of the grade where eachcurriculum of mathematics has adopted teaching materialsof algebra and geometry. In addition, the arrows in the dotted line have displayed the comparison of the grade where each curriculum of mathematics has adopted teaching materials of analysis. The arrows in the solid line haveexpressed that Bulgarian curriculum of mathematics has adopted teaching materials of algebra earlier than Japanese curriculum of mathematics.

The arrows in the dotted line haveexpressed that Japanese curriculum of mathematics has adopted teaching materials of analysis earlier than Bulgariancurriculum of mathematics.


Each country has adopted the same sorts of many teaching materials of mathematics. However, the method of instruction is not the same. I introduce some typical examples in this paper.The examples are equation, trigonometric ratio,differentiation, integration, recurrence formula, and rearrangement of data.


Bulgarian curriculum of mathematics contains complete substances of equation. The reason is as follows:

(i) The Time When Equation is Studied

Bulgarian students study many kinds of equationsearlier than the Japanese students do. The Chart 1has shown that all Bulgarian students study almost all kinds of the equation in 9th or 10th grade. On the other hand, Japanese students study them in 11th or 12th grade, and some students do not study them because of the choice system.

According to the Bulgarian textbooks of mathematics, substances of exponential law and logarithm law are in the 10th grade's textbook, and substances of exponential function and logarithm function are in the textbook of 11th grade. The substances of exponential law and logarithm law are considered as algebra.

On the other hand, in Japanese, there are all these substances in the textbook of 11th grade.

(ii)The Number of Exercises in the Textbooks

In comparison with the Japanese textbook, there are many exercises in the Bulgarian textbooks. I think that Bulgarian students have high ability of calculation. In case of Japan, mathematical thinking or many kinds of problem solving is more important than the ability of calculation. There are not so many exercises in the Japanese textbooks of mathematics in comparison with the Bulgarian ones.

(iii)Difference Method of Solution

In case of Japan, students often use graph for solving equation. But Bulgarian students solve it as problem of algebra, not using graph.

The example is thus equation :

The solution of the Bulgarian students is and , then

As ,

then .

The solution of the Japanese students is:

The graphs of and are given on Fig. 1.

It is possible to calculate the coordinate ofthe intersection in two graphs.

(The calculation is the same as Bulgarian


The value of coordinate is ,then this value is the solution of theequation.The value is not the solution of the equation. According to the graph, this value is the coordinate of the intersection of another two graphs that are and .

If students must verify whether the value of solution is correct, Bulgarian students use the character of root sign, and Japanese students use the character of intersection of two graphs.Of course, students of both countries know these two types of answers.

Iwould like to say that the Bulgarian students are accustomed with algebraic thinking, and Japanese students are accustomed with analytic thinking.


Students of both countries learn Pythagorean Theorem in 10th grade. However, Bulgarian students learn the basic of trigonometric ratio as the application of Pythagorean Theorem. In Bulgarian textbook, Pythagorean Theorem and trigonometric ratio are in the same chapter. Japanese students have learned Pythagorean Theorem without the substances of trigonometric ratio in 10th grade, and they have learned the basic of trigonometric ratio as the application of the characters of triangle in 11th grade.

Bulgarian textbooksof mathematics have an interesting character. The character is that there are many theorems or formulas about the characters of triangle or circle. In Bulgarian textbooksof mathematics about geometry, there are many formulas that Japanese students cannot use as such. I want to introduce two examples. The first example is the Parallelogram Theorem, and the second example is the length of bisector.

Parallelogram Theorem

There is Parallelogram Theorem in Japanese textbook of mathematics. The formula is as follows:


This is Parallelogram Theorem, and students remember the formula as theorem.

However, another expression of this theorem is possible. Bulgarian textbook defines the length of segments as follows (Fig. 2):

AB= , BC= , CA= , AM=

The formula (2) is derived by using trigonometric ratio, and this is another expression of Parallelogram Theorem.If we use the formula(2), we can calculate the length of the median line directly in a short time.


Each formula is usedwhen we need the length of the median line, and the formula (2) is more convenient than the formula (1).Then the formula (2) is more practical than the formula (1).

The Length of Bisector

There is a triangle ABC, and the segment AL is a bisector of (Fig. 3).In Bulgarian textbook, there is the method how we can calculate the length of bisector directly.

Let AB= , AC= , AL= , then the length of AL is as follows:


In case of Japanese, the formula (4) is introduced in

the textbook:

AL2=AB ×AC-BL ×LC (4)

Of course,

Japanese teacher can prove the formula (3) or(4).

However,they do not remember the formula(3)or (4).

Japanese teachers or students do not remember so many formulas as there are in the Bulgarian textbooks of mathematics. As Bulgarian students can use these many formulas, I think that they can solve geometric problemsfaster than Japanese students can.


In Japanese, there are teaching materials of differentiation in the elective subject's textbook of 11th grade. The sort of function is integral one. Other sorts of functions are in the elective subject's textbook of 12th grade.

On the other hand, teaching materials of differentiation are only in the elective subject's textbook of 12th grade in Bulgaria, and the students study differentiation of almost all sorts of functions.The method of curve tracing is one of the applications of differentiation.


In Japanese, the substances of integration are the teaching materials of elective subject. The students can learn them if they elect the subject named "math II" or "math III". However, I could not find these substances in Bulgarian textbooks.There are substances of the integration of integral function in math II, and there are ones of the integration of other functions in math III. (Other functions mean transcendental function, fractional function, irrational function, and so on.) The method of quadrature is one of the applications of integration.

There are substances of quadrature in the Bulgarian textbook of mathematics, but the contents is the method of using the Lemma of CAVALIERI.


There are substances of recurrence formula in each country's textbook of mathematics, but the method of teaching is not same.

In Bulgaria, there are substances of recurrence formulain textbook of mathematics for understanding some features of arithmetic sequence or geometric one. The examples are as follows:

There are these substances in Japanese textbook too. However, in case of Japanese, there is certainly next kind of recurrence formula in the textbook.To find the general term is one of the purposes of considering recurrence formula .

As we can change this formula to , the sequence{ } is a geometric one, and we can find the formula of general term.

In Bulgaria, recurrence formula is the way for finding the characters of arithmetic sequence or geometric one. However, Japanese students learn the substances of recurrence formula to grasp therecursive thinking.


In Bulgaria, the students certainly can study some parts of statistic's field. In the 11th grade, all students study rearrangement of data. The substances are how to make frequency distribution, histogram, and how to calculate the mean value of the data.However, in Japan,substancesof statistics are included in selective subjects. Soonly few students study statistics.


In case of Bulgaria, there are many theorems and their proofs in the textbooks of mathematics, and students use them for solving mathematical problems positively. The textbooks are similar to some technical books of mathematics that are used in Japanese universities. For example, there are many topics of expansion of theorem. Therefore, if the teaching instruction is desirable, students may be able to realize the pleasure of making mathematics. Moreover, if students have experience to use theorems or their expansions, they will be able to solve problems of mathematics in short time. Bulgarian curriculum of mathematics has adopted many substances of algebra, geometry, and statistics.

In case of Japanese curriculum of mathematics, our teachers think that abilities of problem solving are more important than those of calculating are. So there are not so many exercises of calculation but there are many substances of thinking.

If we could mix the mathematical educations of both countries, I think it may be possible to find the method of new mathematical education. Then our teachers should become able to teach or train not only gifted students but also many students desirably.


Nobuaki Kawasaki

Senior High School, Otsuka University of Tsukuba



Oscar Joao Abdounur


In this communication I shall consider educational aspects of the development of ratio and proportion, focusing on the arithmetization undergone by these concepts in the light of the relations between mathematics and music.Since such relations, even if confined to the context of ratio and proportion, are fairlywide-reaching and also that the process ofarithmetization is quite complex, we shall concentrate mainly on the instructional aspects of a structural peculiarity presented in such a fascinating dynamics. This peculiarity is the so-called compounding ratios, a curious feature present in the structure of ratio since the Classical Period whose irregular transformation into the operator multiplication is quite representative of the importance of theoretical music in the arithmetization of ratios. As a consequence we shallalso point out features of the differences between identity and proportion , which are capable of being didactically explored with a mathematic-musical approach.

The reason for choosing music for the present approach is not only historical, but more specifically didactic insofar as the subtle semanticdifferences between compounding and multiplication and also between identity and proportion are clearer if one thinks of ratios as musical intervals when looking at such constructs. Grattan-Guinness argues that the well-known difficulties in teaching fractions can be alleviated by converting the latterintoratios, andthususing a musical approach. These considerations corroborate the need to explore didactically specific contexts in which differences between given constructs manifest themselves moreclearly.

In order to fulfill the aforementioned aim we shall first of all introduce some historical aspects of ratio in mathematical-musical contexts as well as of the corresponding structure in which compounding makes sense, and then follow these with examples of the practice of compounding on the monochord and by the didactic-epistemological aspects that underlie such a practice.

The present musical approach widens our comprehension of ratio and proportion in mathematics not only because of its historical-cultural contextualization and the interdisciplinary aspect which underlies it, but also, and most importantly, because of the role that analogical thought plays in the construction of meaning, in this case, that of ratio and proportion. If we wanted to extend Kieren's argument about rational numbers to ratios, we could claim that to understand the ideas of ratios , one must have adequate experience with their many interpretations. Throughout the history of mathematics and theoretical music, ratio and proportions assumed different meanings with discrete or continuous natures in regard to geometry, music and/or arithmetic. Among such meanings, ratio can be seen as a tool of comparison by means of proportions, a musical interval, a fraction, a number, an invariant with respect to proportion, a common thread between distinct contexts with regard to proportions whereas proportion can be seen as a vehicle to compare ratios, an equality, a relation, a function etc. The aforementioned device not only provides a fertile ground for the understanding of the subtle differences and structural similarities underlying the diversity of interpretations associated with ratio and proportions but also contributes to constructing and to experiencing in a broader way their associated meanings.

In a general sense, discovering common schemes and archetypes is an efficient way of constructing concepts that concern in principle different areas. An analogy or metaphor used in a sensible and discerning way may re-configure a student‘s thought in a problematic situation of learning, enabling a better understanding of matters that escape immediate intuition, or that seem too abstract to him/her, such as the many interpretations associated with ratio and proportions as well as with the wide variety of structures historically associated with them.



University of Sao Paulo,

Rua do Matao, 1010 – Cidade Universitaria – Cep 05508 900 – Sao Paulo – SP –


E-mails: abdounur@mpiwg-berlin.mpg.de




Péter Körtesi


Key words : General education, mathematical research and education, researchers and high school students.

The author has been chairing the Junior Mathematical Congress'96 Miskolc, Hungary, one of the official satellite meetings of the 2nd European Congress of Mathematics, and a chain of similar meetings almost each year.

The meetings were aimed at bringing together the future mathematicians of Europe, especially those aged between 13-19 years. Apart from attending lectures given by invited scholars and meeting famous European mathematicians, the young participants themselves gave talks and exhibited posters.

The activity in between the conferences and congresses in sustained by a smaller group of talented young people, secondary school students of two local grammar schools, the Földes Ferenc Gimn?zium and Herman Ottó Gimn?zium.

During the year the students perfom their activity in a special group, we call it the SELF MADE MATHEMATICS group. They learn a bit of Mathematics out of the normal school material, and they present their „findings” for the whole group in the form of a student presentation. We hope that what they do is to convert the idea of the learning by doing method to the deeper understanding by explaining.

The efficiency of the method is due to the facts:

    • While preparing oneself for a presentation the material must be understood much deeper.
    • One must be prepared to defend the statements.
    • While explaining, the knowledge is restructured in new manner, instead of simple reception (passive), this is the new dimension of transmission (active).
The best papers of the actual year have been published in the online edition of the Hungarian language journal Pi edited in Miskolc, see:



Péter Körtesi

Institute of Mathematics

University of Miskolc

Miskolc, P.O.Box 10., H 3515


Phone: + 36-46-565111-1795

Facsimile: + 36-46-565146

E-mail: matkp@gold.uni-miskolc.hu



Risto Malcevski, Valentina Gogovska

Abstract: During the process of learning mathematics, students are in similar position as scientists. It means that they “discover” mathematical truth alone or with teacher’s help. Because of that scientific methods used in mathematical research are at the same time educational teaching methods and they have significant meaning for development of gifted and talented students. Basically without qualitative knowledge of scientific methods, especially of their use, it’s almost impossible for students to make structural knowledge, which, of course,is not necessary for everyone, but at the same time has special importance for gifted and talented students, starting from theirbeginnings.

In this work short explanation of scientific methods used in mathematics is given and few examples according to which we can introduce some scientific methods on higher level are considered. This is especially important, because individual further research is main tool for development of gifted and talented students.

Key words : Scientific methods, Observation and experiment, Comparison, Analysis and synthesis, Generalization, Systematization and abstraction.


A scientific method represents a way of noticing some fact, which will allow to the examiner to discover basic characteristic of overviewedobjects or phenomena. The most commonly used scientific methods in mathematical research are:

Observation and experiment


Analysis and synthesis

Generalization, systematization and abstraction

Observation is a scientific method which is used according to an a priori developed plan, with the aim to discover, to establish and to investigate some properties of some objects and phenomena or connection between other objects or phenomena. During it the objects and phenomena are commonly investigated in their natural environment.

Experiment is a scientific method of learning about objects and phenomena where the examiner intervenes in their natural condition and development creating artificial conditions, dividing them into parts or combining with other objects and phenomena. Experiment and Observation are closely connected, because during the experiment every object is observed.

Comparison is a thoughtful operation during which a thoughtful discovery of similaritiesor differences between investigatedobjects and phenomena is made. It’s necessaryto respectthe following principles:

comparison should have real meaning, theobjects compared should be connected,

comparison should be realizedaccordingly to a plan,

comparison should be realized entirely.

Analysis decomposes the given object or phenomena intocharacteristic elements, with the main idea to investigate them individually, remembering that they are composed parts. As athoughtful operation analysis starts from consequences and goes on to causes.

Synthesis is the merging of given parts or properties into unique composition. As a thoughtful operation synthesis starts from causes and goes on to consequences.

Analysis and synthesis are the most important psychological characteristics of thinking, because in the process of thinking we first analyse and then synthesizeon the basis of results of analysis.At the end we come to generalization, systematization and abstraction which are the results of analysis and synthesis.We should mention that there is not a strict boundary between analysis and synthesis, because they are strongly connected. Because of that analysis and synthesis like scientific methods are always connected forming unique analytic – synthetic method.

Generalization is a result of joining separate basic properties, which are essential for given class of subjects and phenomena. The generalization allows to work with larger set of objects instead of the given set .

A thinking process reverse to generalization is specialization . In this process we separate some properties from the set of properties of investigatedobjects or phenomena. We can say that specialization is the investigation of a subset N instead of the whole set .

Systematization is a thoughtful operation during which the investigatedobjects are organized into system according to some principle or properties.

Before systematization we apply: analysis, synthesis, generalization and comparison.Their results are used and realized within systematization. Final results are systems of phenomena. The most important type of systematization is classification , which is the constructing of groups of objects, based on similarities and differences between them.

Thinking operation during which we isolate unessential and accent essential properties of given subject or source is abstraction . Abstraction can be sensual or thoughtful. All mathematical ideas are formed with thoughtful abstraction. Abstraction and generalization are strongly connected.

Concretization is a thinking process reverse to abstraction. It discovers the content of scientific abstractions, including concrete facts or relations.



In the process of scientific exploration, but also in the education the scientific methods are connected and because of that their independent learning has a meaning only in the process of theoretical teaching of scientific methods, but not in the practice. From the other side for student the early mastering of them is necessary because without good knowing of scientific methods it is impossible to get structural knowledge which is necessary for successful development of gifted and talented students. Practice shows that realization of mathematics during the lessons and commonly accepted forms of working almost don’t allow accepting scientific methods by the students. It’s a big cause for concern because this question is marginal: nobody takes care of it, neither the creator of educational systems nor the teacher. They use different excuses for their behavior like:

shortage of teaching time,

nonappropriate connection between psychophysical ability of students and abstractness of scientific methods

These aren’t real excuses, because with skilled planning of teaching process we can pass the problem of time shortage,and we don’t accept excuse about nonappropriate connection because we think it doesn’t really exist.It's more about not perceiving enough about the problem and missing of preparation and ability of teachers and program creators. We should mention that acceptance of scientific methods by primary and secondary school students should be achieved only if:

solving of heavier problems can be separated into parts in which we can use different scientific methods, like exercises in geometrical constructions (analysis, construction, proof and discussion)

for specific purpose we introduce specific topics not only allowing the students acquire new knowledge, but also effectively introducingthem to scientific methods without strict definitions.

We really think that this kind of introducing will allow qualitativeapplications of scientific methods by gifted and talented students. We will give some examples of improving the level of mastering scientific methods by the students in high school.

Example 1. Great German mathematician Gauss (1777-1855), being a ten yearold child, quickly solved the following task: Find the sum of all natural numbers from 1 to 100.

After waiting long enough for students’ answer about their idea, we go on following “fertile” idea, and if we don’t get it we tell them that probably Gauss used the following table of two progressions each consisting of numbers: 1, 2, ..., 100 during solving this task:

The sum of the numbers in can be found in two ways. First way consists of finding the sums of the numbers in each column(1+100, 2+99, 3+98, ..., 98+3, 99+2, 100+1) and findingthe sum of these sums. So the result is


Second way of finding the sum of all numbers in is finding the sum in every row:

1+2+3+...+98+99+100 and 100+99+98+...+3+2+1.

Than we add these two sums and get


Finally, from (1) and (2) we get the result:


In the next step we can tell the students the next task to introduce the method of generalization.

Find the formula for the sum .

Of course the students should solve the task individually, but during that it’s useful for teacher to mention that this is a generalization of the idea of Gauss.

Further we give the next task, which is similar to the previous one.

Task 1: Find the sum of squares of first n natural numbers


The way of finding this sum is not obvious, but some students will use Gauss idea and will write

But in this example the sums of numbers in columns of table are different, for example


so students will discover quickly that this way is not appropriate for the task. In this case we get infertile idea, from which students go away quickly. It’s important that students (individually or with teacher’s little help)go to the formula

which can be written as


Cleary, last formula is true for every natural number p, and if we put we get

Students will easily find out further that they should add the equalities and get

from which we get


After some manipulations we get


The action of counting the sum of squares of first n natural numbers is different from the action of counting the sum of first n natural numbers. It’s logical that students will use the previous procedure for finding the sum of cubes of first n natural numbers, which we denote with . Following analogy they will probably write

from this, using analogy, they will get

After adding they will get

from where

. (5)

In the last equality everything is already known, except the sum , which can be found from (5):

. (6)

In the next steps students will try to find the sum of fourth powers of first n natural numbers, then that of the fifth ones, etc. But this is an already known procedure, so it doesn’t add new quality to the learning of scientific methods. We should try to reach a new quality, which can be done by insignificant intervention in which we will introduce the method of specialization.

Let’s set

and we write

Adding these equalities we get


from where . Further, using (3) and (5) we get



In the next step students should notice the connection between the formulas (7)-(9) and the role of Newton’s Binomial formula to reformulate the results:




where , are binomial coefficients. After that students should come to the general identity for :


We should expect that students could prove (10) using analogy.We should expect that they will write down the equalities

........ ...............................................................................

and add them to get (10).

The obtaining of previous equalities allowed us to demonstrate scientific methods: generalization, specialization, abstraction, and introducing the recurrence. For the last one students should notice that we can use (10) to find Skif we know already Sk-1, Sk-2, …, S1, and S0 , or that we can find , consequently one after another, because we know that .

Using of scientific methods in teaching mathematics is a priority. During this teacher should take special care of gifted and talented students.

From the history we know about attempts to discover universal scientific methods connected with the universal scientific procedure of preparation young generation for scientific research. Because this did not happen educational systems are paying more attention to partial implementation of scientific methods. This is a huge mistake for mathematics, especially forgifted and talented students. We could not find any mathematical discovery based on only one mathematical method in the history. The most important thing working with gifted and talented students is using different methods interacting each with other. It’s essential if we consider a problem, which not only uses different mathematical methods but also leads to open question in the science. The next task is such anexample.

Example 2. We will consider Ramsey type colourings.We will use scientific methods to investigate this combinatorial topic.

Step 1. Students are introducedto the subject.

Let be a given set. Every map is calledcolouring of the set in colors, and map is called function of colouring.

For a given function of colouring we define the relation with if and only if . Clearly, is an equivalence and because of that set is the union of disjunctive classes of equivalence. A set , is monochromatic, if is a subset of some class of equivalence, the same as is a constant function.

Ramsey theory is a very important part of combinatorics, which started its development in 1930 with work of English mathematician F.P.Ramsey. The basic aim of it is to establish the conditions under which every “irregular” coloring of some structure (points of the line, surface, elements of a set etc.) provides a “regular” monochromatic substructure only if the given structure is “big” enough. A classical example of Ramsey-type problem is: “Prove that in every colouring of sides and diagonals of a regular hexagonin two colors a monochromatic triangle can be found”.Its solution can be obtained by elementary use of Dirichlet’s principle.

A heavier problem of the same type:”Find the smallest n such that for each colouring of the sides and diagonals of a regularn -gon ink colours a monochromatic triangle can be found” is still unsolved. It is well known that for , and for ,but even for only inequality is known. We must mention that this kind of problems is a relatively simple one in Ramsey theory, which is very complicated and offers numerous unsolved combinatorial problems.

Step 2. Discussion with students about the necessaryprior knowledge.

It’s necessary to learn about Dirichlet’s principle and basic combinatorial principles before we start to solve colouring problems.We give some definitions further.

Let n be a natural number and . Set is finite, if a bijection existsfor some natural n. Then hasn elements and . We will consider finite sets only.

Principle of equality . If a bijection between the sets and exists, then .

Principle of sum.

a) If , then .

b) If is a family of sets, , for which holds if , then


Principle of product. , for .

Dirichlet’s Principle. Let objects, ,be put into boxes. Then at least one box has at least objects in it.

Step3 . Involving into research (starting problems)

It’s useful to give one or two elementary problems which will be generalized further.So the principles of generalization will be learned automatically.

Problem 1. The line is coloured in two colours. Does there exist a segment whose endpoints and midpoint are monochromatic?

Naturally students will try to find the number of possibilities of colouring three points in two colours. They will discover that three points can be coloured in two coloursin different ways, and that among these three points two monochromaticpoints exist. This result implicatesDirichilet’s principle, having nine points among which two are monochromatic.

Further, students will experiment and will notice the “rule of distance” between the points in the individual triples. It’s necessary, with teacher’s help, to conclude that they should take nine triples of points of form (for different ), etc.

After that they can go to the next problem.

Problem 2. A plane is coloured in two colours. Does there exist an isosceles right-angled triangle whosevertices are monochromatic?

The idea of solution is similar to that of the previous problem using also its result,so this problem is appropriate for acquiring analytic-synthetic scientific method. Let’s start with three monochromatic points A, B, C on a line such that AB=BC. Consider isosceles right-angled triangles AMB, BNC, AKC, such that M, N, K are on the same side of the line AC; then discuss the possible colourings of M, N, K. Cleary the answer is positive.

Step 4. Acquiring the method of generalization

Problem3. A plane is coloured in two colours. Does there exist a trianglewith monochromatic vertices similar to the given triangle?

This problem and its solution is a generalization of the previous problem (all isosceles right-angled triangles are similar).

Step 5. Acquiring themethod of specialization

Problem 4. Prove that the results of the problems 1 – 3 still hold if the whole plane is replaced by an appropriate finite set of points (these sets are different for each problem.)

Step 6. Further deverbing of the method of generalization

During solving the next tasks it’s essential to work individually and to discuss the methods used. It means that students should individually recognize and comment their actions. For this it’s importantto know the scientific methods and their differences.

In further we will assume that instead of the whole planeonly points with integer coordinates are coloured.


Problem 5. A plane is coloured in two colours. Prove the existence of an isosceles right-angled triangle with monochromaticvertices.

Problem 6. A plane is coloured in colours. Prove the existence of a rectangle with monochromatic vertices.

Problem 7. A plane is coloured in two colours. Prove the existence of a square with monochromatic vertices.

Problem 8. A plane is coloured in colours and is an arbitrary finite set of integer points. Prove that there exists a monochromatic set similar to the given set .

Problem 9. A natural number is placed in each integer point of the plane (the numbers can be different). Prove that for every there existsa square with sides parallel to the coordinate axis, such that the sum of the numbers inside the square dividesby .

Remark. In this placea question can be set about the connection between the previous problem and the next task “Each set consisting of natural numbers has a subset whose sum of elements dividesby .”


The educators should achieve that the scientific methods are acquired by students in primary and secondary school.They can achieve it solving heavier problems that can be separated into parts to which different scientific methods can be applied(an example:analysis, constructions, proof and discussion in geometrical exercises)or maybe introducing specific topics allowing the students to get new knowledge without strict definitions, but introducing scientific methods. The practical application of this work will show whether this is a successful attempt to stimulate the mathematical creativity of gifted students.

Learning and using the scientific methods develops such qualities of thinking as activeness, depth and width as well as the criticism of thinking.


    1. Ganchev, I.; Portev, L.; Sidorov, ?.et al. : Methodic of Teaching Mathematics, 1chast, Modul, Sofia, 1996
    2. Malceski, R.: Methodic of Mathematics Lessons, Prosvetno delo, Skopje, 2003
    3. Malceski, R.; Gogovska, V.: Using Inequality Between Means for Structural Mathematical Knowledge
    4. Polya, D.: Mathematical Discovery, Nauka, Moskva, 1976
    5. Celakoski, N.: Didactic of Mathematics, Numerus, Skopje, 1993


Risto Malcevski, Prof.

Faculty of Natural Science

Institute of Mathematics-Skopje,


Valentina Gogovska

Faculty of Natural Science

Instituteof Mathematics-Skopje,


E-mail: valet@iunona.pmf.ukim.edu.mk



Sang-Gu LEE

Abstract: The purpose of this paper is to introduce an activity of student who found purely linear algebraic solution of the Blackout puzzle. It shows how we can help and work with gifted students. It deals withalgorithm,mathematical modeling, optimal solution and software.

Key words: Motivation, Blackout puzzle, linear algebra, basis, algorithm, mod 2 arithmetic.


From the conditions of real life problems, gifted students see some aspects that others don’t see. Blackoutgame, which was introduced in the official homepage of popular movie “Beautiful Mind”,is a one-person strategy game that has recently gained popularity as a diversion on handheld computing devices. An animated Macromedia Flash version of the puzzle can be found from the official website for the 2001 movie ‘A Beautiful Mind’.

We will show what was the question and answer. We will introduce student’s answer, which is very elementary and intuitive. We also made a software based on his algorithm. This process only used the basic knowledge of linear algebra and can be extended to the fullsize Go board problem and teach how we work with gifted students.

LAYOUT: L inear Algebraic Solution of Blackout Puzzle

    I. Background of Blackout Puzzle
    II. Main Questions
    III. Our Solution of the Blackout Puzzle
    IV. Conclusion

I. Background of Blackout Puzzle

In my recent linear algebra class we were talking about the movie "A Beautiful Mind", starring Russell Crowe as Nobel Laureate John F. Nash, Jr. (2001), where Nash was playing Go game with his friend. Some of my students told me that they have played "the Blackout puzzle" from the Korean official website of the movie. ( http://www.cjent.co.kr/beutifulmind/ ).

One of my students asked me "Can we find an optimal solution for the game?” and further "Is there any possibility that we can not win the game if the given setting is fixed?"

I gave a chance to think about it for one of my students who liked to do so.

He and I met a couple of times personally and made a Mathematical model of the game, and he brought me the right answer.What he found was that we can always win the game. The model is fully based on the basic knowledge of linear algebra.

We made a search of the game at that time but we did not find any good reference, so we did it in our own way. Later we found the following web site

http://home.sc.rr.com/jacobsfam/jared/blackout.html , so we went further.

(1) Introduction of Blackout puzzle

Blackout is a one-person strategy game that has recently gained popularity as a diversion on handheld computing devices. An animated Macromedia Flash version of the puzzle can be found from the official website for the 2001 movie “A Beautiful Mind”.

(2) How to Play

The Blackout board is a grid of any size. Each square takes on one of two colors. (The diagram above used blue and red.) The player takes a turn by choosing any square. The selected square and all squares that share an edge with it change their colors.

The object of the game is to get all squares on the grid (tile) to be the same color (Black or White).

When you click on a tile the highlighted tile icons will change or " flip" from their current state to the opposite state. Remember, the goal is to change all of the tile icons to black (or white).

(3) How to Solve Any 3×3 Game

The diagram below illustrates the shortest sequence of moves for resolving possible scenarios on a 3x3 board.

II. Main Questions

Q 1. "Is there any possibility that we can not win the game if the given setting is fixed?"

Q 2. "Can we always find an optimal solution for the game?”

Q 3. "Can we make a program to give us an optimal solution?"

III Our Solution of the Blackout puzzle

There are patterns of blackout grid.

Among these 512 patterns, there are patterns such that we can win the game with only one more click as following. (Twice of the following basic 9 patterns as we can change all initial colors.)

We checked several examples and had enough trial and error to convince us to answer the first question with any given initial condition.

(Example) Assume the following initial condition

This setting is not one of the above = 18 patterns, but the following 3 clicks make it all white.

Our first step to find a winning strategy was to recognize these 18 patterns.

Now we try to make a Mathematical Model of this game.

Only actions that we can perform are 9 clicks because we only have 9 stones on the board. We assume "the white stone 1 and black stone 0". Then we can classify effects of each action as an addition of one vector (or 3x3 matrix). Any series of our actions is a linear combination of these. Now we use modular 2 arithmetic to make the Zero vector or all 1's vector (or matrix, resp.) to finish the game

So we now have the above 9 vectors (in fact, twice of them) to consider which will end the game with just one more click.

[Modelling example] We have 5 black(blue) stones and 4 white(red) stones in board as below.

Then the given matrix is

and we can click some of 9 positions to take action on it. This can be represented by

So, our problem is to find some a, b, c, d, e, ¦ , g, h and i such that

We can use any computational tool and obtain

But we only need integer vector x, so


We only need 0 and 1 because clicking 2n + 1 times of one stone is same as clicking once, and clicking’s of one stone is same as doing nothing.

So, our answer is

This shows that if we click on positions (1, 1), (1, 3), (2, 1), (3, 2), we will get all white stones on the board with only 4 clicks.

In the following Fig. 9, the command "(Wizard) " tells us "1 3 4 8" that indicates which 4 stones we have to click to win. The number "4" shows we won with 4 clicks(MOVE).

We can run the program from http://matrix.skku.ac.kr/sglee/blackout_win.exe

This works always. Why does this happen? So our next question is

Q 2. "Can we always find an optimal solution for the game?”

[ Proof ] From the 9 matrices

make nine column vectors with the above, and make a symmetric matrix (because of the symmetry in the board) whose columns are these vectors.

Then we have a linear system of equations to find x.

where is a given (condition) matrix.

Then RREF(A)=I9 and . So the columns (rows) are linearly independent, and the system has a unique solution.

(Furthermore all this process can be done in Modular 2 arithmetic.)

For the example considered earlier we have



given (1,1) (1,2) (1,3) (2,1) (2,2)


(2,3) (3,1) (3,2) (3,3) Goal 1 Goal 2

We want to show that such , exist for any B.

Remember that we can represent 3x3 matrices by vectors and let

Given (1,1) (1,2) (1,3) (2,1)

Recall: Ax = - B is consistent iff rank (A) = rank[A |-B]

Let 0 , j

Case 1. If is a zero matrix, it’s OK because a, b, c, d, e, ¦ , g, h, i (mod 2) give a solution.

Case 2. If is not a zero matrix, then it is clear that


In any case, ( or ) is consistent.

So, for given , x, 0, j, and b,


0 , j ,

where the vector b=[b 1,b2,b3,b4,b5,b6 ,b7,b8,b9] t comes from the given (0,1)-matrix ,

the system Ax + b = 0 (or j ) has a solution.

Then x is obtained as

x = 0 - b (or x = j - b )


Let x' x (mod 2). Then x' is a real optimal winning strategy vector (matrix) which can be deduced from x.

Now entries of x' are all 0 or 1 as is in real game situation and we can always find a (0,1) matrix as a real optimal winning strategy vector(matrix).

With this idea, one of my student made a computer program in C++ based on this algorithm

which tells us an optimal strategy to win.

We can download it and run from http://matrix.skku.ac.kr/sglee/blackout_win.exe .

This software also verified our conjecture, and showed the proof was valid.


This practical approachto a real problem of gifted students and leading teacher gave a stimulating mathematical creativity for both. This process can be adapted to resolve other real world problems with basic mathematical knowledge.


        1. Park, H.-S., Go Game With Heuristic Function , Kyongpook Nat. Univ. Elec. Tech Jour. V15, No.2 pp. 35-43, 1994
        2. Park, J.-B.(2003), Software http://matrix.skku.ac.kr/MT-04/blackout_win.exe
        3. Uhl, J. & Davis, W. , Is the mathematics we do the mathematics we teach?, Contemporary issues in mathematics education , Volume 36, pp. 67-74, Berkeley: MSRI Publications,1999
        4. JAVA program by CJ entertainment Inc. , Movie: The Beautiful Mind, Blackout puzzle (2002), http://www.cjent.co.kr/beautifulmind/

Prof.Sang-Gu LEE, Ph.D.

Sungkyunkwan University

Suwon, South KOREA

Phone: +82-31-290-7025






Viktor Freiman

Abstract: The purpose of this paper isto analyze how the model of a teaching based on challenging situation approach would help to engage young children in a meaningful mathematical activity and thus contribute to the identification and nurturing of mathematical talents in primary school through:
    • the questioning, search for patterns, inquiry about mathematical relationships
    • the elaboration of efficient strategies and methods and creating of new tools of problem solving
    • the discussion and communication: thinking about results and reflection on methods
Key words: Mathematical giftedness, Challenging environment, Challenging situation, Early Grades, Didactical tools, Re-organization and (re-)construction of mathematical knowledge


The biographers of famous mathematicians often refer to the evidence of particular nature of their talent which can be detected already at a very young age. One can ask where this deep insight in mathematics comes from. How can teachers discover their talent and nurture it? And, as a result of this discovery, what kind of classroom environment would be advantageous for these children?

Many educational systems are now implementing new reforms of mathematics curriculum that leave the classroom door open to the innovative methods of teaching that meet the interests of all the students. The question is how we as teachers are prepared to handle this multifaceted educational task. Our particular interest is directed to the identification and fostering mathematically gifted students in the elementary mathematics classroom.

Based on our experience of teaching a challenging mathematics curriculum in Grades K-6 we constructed a developmental ‘recursive chain’ of identification and fostering mathematical giftedness: the challenging situation requires rupture with old knowledge and construction of (new) abilities, thus at the same time revealing the obstacles and giving the teacher an opportunity to address them to all students (not only the gifted ones); reflection on the situations and the shortcomings of thinking about it leads to new questions and indeed creates a new challenging situation.


What tools could help to identify mathematical giftedness in young children? Analysis of various studies in mathematics education shows a large spectrum of theories and practices. Baroody and Ginsburg [3] remark that even among children just beginning school, there is a wide range of individual differences. Kindergarteners and first-graders are far from uniform in their informal mathematical knowledge and readiness to master formal mathematics. With each grade, individual differences increase. Although there seems to be no agreement on terminology nor on procedures of identification of the gifted, all researchersagree that school marks donot reflect mathematical abilities. School success in mathematics does not imply the presence of mathematical ability. Conversely, also,children who donot succeed in school mathematics are not necessarily mathematically unable.

Many researchers indicate that the process of identification of gifted children requires many different steps and techniques including classroom observation, surveys, interviewsand a variety of tests. According to Kulm [12], Young& Tyre [18] and Johnson [9] an appropriate assessment should be aimed at revealing the extent, complexity, and functional characteristics of mathematical thinking rather than focusing simply on final, well formed ability or performance. It worth to find out what children can do and how well and quickly they can learn to do more. Finally, the characteristics that separates a gifted from non-gifted child in mathematics is the quality of the child's thinking.

Unfortunately, as it was mentioned by Greenes [8], the bulk of our mathematics program is devoted to the development of computational skills and we tend to assess students' ability or capability based on successful performance of these computational algorithms (so called "good exercise doers") and have little opportunity to observe students’ high order reasoning skills.

Many authors point at the teacher's role in the process of identification of mathematically able children as a crucial element of the process of identification of very able students. Fore example, Kennard [10] affirms that the provision of challenging material and forms of teacher-pupil interaction is a revealing key to mathematical abilities voting for interactive and continuous model for providing identification through challenge.

An identification of mathematically gifted children is only the first step in a long termwork with them, which is usually followed by various methods of fostering and nurturing.

Such instructional program, according to Sheffield [16], helps students develop their mathematical abilities to the fullest. It may thus allow them to move faster than others in the class to avoid deadly repetition of material that they have already mastered. She opts for a program that ”introduces students to the joys and frustrations of thinking deeply about a wide range of original, open-ended, or complex problems that encourage them to respond creatively in ways that are original, fluent, flexible and elegant [16:46]”

In the Enrichment-Triad Model developed by Renzulli, one can find the following three types of activities which are important for nurturing mathematical talents [14: 218]:

    • General exploratory activities to stimulate interest in specific subject areas
    • Group training activities to enable students to deal more effectively with content through the power of mind using critical thinking, problem solving, reflective thinking, inquiry training, divergent thinking, sensitivity training, awareness development, and creative or productive thinking
    • Individual and small-group investigation of real problems in which giftedness manifests as a result of student's willingness to engage with more complex, self-initiated investigative activities.
As pedagogical task of fostering the development of mathematical thinking, Baroody [2], Ernst [5] and Fishbein [6] stress the use of a problem-solving approach which focuses on the processes of mathematical inquiry: problem solving, reasoning, and communicating approach in which a student plays an active role. Thus, the didactical transition of mathematical process from the application of facts, skills and concepts, to the full range of problem-solving strategies including problem-posing happens when the classroom teaching becomes more open and challenging. Facing a challenging task child might not be able to find solutions spontaneously. Rather, he gets engaged in a constructional process combining various conditions producing a method to work on the problem systematically. This aspect of finding a method consciously is fundamental for the development of mathematical reasoning. A teacher's task becomes to create an environment that would require a mathematical attitude, mathematical concepts, and mathematical solutions.

The literature survey shows a variety of models and methods of identification and fostering mathematical giftedness. In our study, we focus on those which can be applied to the everyday teaching practice. We base our study on the assumption that mathematical giftedness appears already in the early school age. Therefore, we as primary school teachers cannot wait till mathematically gifted children are detected in the middle grades and transferred into special programs. We have to takecare of these children as early as we can and create favourable conditions for their development on the everyday basis. Questions of teaching and learning organization become thus crucial for identification and fostering mathematically gifted children.


Our challenging situation teaching model is a didactical response to the teaching paradoxes described by Brousseau [4], Shchedrovitskii [15] and Sierpinska [17]:

    • Everything the teacher undertakes in order to make the student produce the behaviours that, she expects, tend to deprive this student of the necessary conditions for the understanding and the learning of target notion (Brousseau)
    • When we as educators want our children to master some kind of action, we often tend to teach it directly by giving children tasks which are identical to this action. But classroom practice shows that the children not only do not learn actions that go beyond the tasks, they do not even learn the actions we teach them within the tasks (Shchedrovitskii)
    • In order to access a higher level of knowledge or understanding, a person has to be able to proceed at once with an integration and re-organization (of previous knowledge). But we can not tell the students ‘how to re-organize’ their previous understanding, we can not tell them what to change and how to make shifts in focus or generality because we would have to do this in terms of a knowledge they have not acquire yet (Sierpinska, referring to Piaget and Bachelard)
If the teacher does not pay attention to these paradoxes, children would not be able to show (nor to develop) their abilities to formalize, to generalize, to curtail, to demonstrate flexible thinking, to evaluate critically their thinking strategies. But these abilities form a specific mental structure that Krutetskii [11] calls a ‘mathematical cast of mind’ and their development gives an opportunity to identify and to nurture mathematical giftedness.


Our study of mathematical giftedness [7] shows that, at a very young age, children are eager to learn mathematics, they enjoy it, and teachers should use every opportunity to nurture their fresh minds. Thus, a special environment has to be created in order to maintain their genuine interest. We shall call this environment challenging, as it is composed of a variety of situations that provoke mathematical questioning, investigations, and use of different strategies, reasoning about problems and reasoning about reasoning.

We propose the use of teaching approaches based on challenging situations in order to engage all students into meaningful learning through:

    • Early beginning of work on challenging mathematical tasks : 3-5 year old (fostering precocious mind)
    • Stimulating questioning (fostering critical / reflective mind) :
    • Why?
    • What if not?
    • Is there a different way?
    • Does it always hold? etc.
    • Encouraging search for new original ideas by means of open-ended tasks (fostering creative / investigative mind
    • Promote full and correct explanations (fostering logical / systematic mind)
    • Introduce children to the complexity and variety of mathematical concepts and methods (fostering looking at the world with mathematical eyes)
    • Provide children with tasks that require complex data organisations and reorganisations (fostering selective / reversible / analytical / structural mind ).
Our 8 year-long teaching experiment has been conducted at Académie Marie-Claire, a Canadian private bilingual elementary school with French and English both taught as a first language. The school is located in the west suburban area of Montreal, in Quebec. Along with a strong linguistic program (with a third language, Spanish or Italian), the school insists on offering enriched programs in all subjects including mathematics to all its students independently of their abilities and academic performance. The school thus promotes education as a fundamental value by instilling the will to learn while developing such intellectual aptitudes as being able to analyse and synthesize, critical thinking, and art of learning.

The mathematics curriculum is composed of a solid basic course whose level is almost a year ahead in comparison to the program of the Quebec's Ministry of Education (Programme de formation de l'école québécoise, 2001) and an enrichment (deeper exploration of difficult concepts and topics: logic, fractions, geometry, numbers as well as a strong emphasis on problem solving strategies). The active and intensive use of "Challenging mathematics" ; text-books [13]along with carefully chosen additional materials helps us create a learning environment in which the students participate in decisions about their learning in order to grow and progress at their own pace. Each child competes with himself (herself) and is encouraged to surpass himself (herself). Since the school doesn't do any selection of students for the enriched mathematics courses, all children of Académie Marie-Claire participated in the experiment. With some of them, this author started to work at their age of 3-5, as a computer teacher.

In the challenging situations used in our own teaching, we favour open-ended problems which are situated in a conceptual domain familiar enough to the child who appropriates the situation as his/her own and engages in an interplay of trials and conjectures, examples and counter-examples, organisations and reorganisations [1].

In each situation, we observe various elements of the child's mathematical behaviour:

    • How the child enters into the situation (introductory stage, pre-organisation) and how different ways of presenting the problem affect children's actons;
    • How the child constructs his/her process of problem solving (choice of strategy, use of manipulative, systematic search, autonomy, self-control, mathematical components);
    • How the child acts in case of an error (destroys his/her previous work and starts from scratch or tries to modify/correct certain actions);
    • How the child modifies his/her strategy when the conditions are slightly / completely changed;
    • How the child presents his/her results (orally or in writing, clearly or not, communicating or not with other participants (children or adults), symbolism used by the child, organisation of results (on paper)).
Since a real challenge is possible only when the situation is new for the learner, the challenging situation must contain the rupture with what the student has previously learned, provoking the student to reflect on the insufficiency of the past knowledge and construct new means, new mechanisms of action adapted to the new conditions, activating her full intellectual potential. A challenging situation often presents the child with a problem, which goes above or beyond the average level of difficulty. The child is encouraged to surpass what is normally expected from children of her age, thus demonstrating her precocity, which is a sign of mathematical giftedness.

A challenging situation could also provide the student with an opportunity to face an obstacle of a pure mathematical nature, the so-called epistemological obstacle. In order to overcome it, the student will have to re-organize her mathematical knowledge, create new links, new structures following laws of logical inference.

A challenging situation helps to create a friendly environment in which a child compete with herself sharing her discoveries with other children and learning from others. Thus it allows mathematically gifted children who are not high achievers to participate actively in class and to succeed. Altogether, this approach helps to reveal mathematical giftedness while fostering further progress in children.


In this section, we will analyze several examples of children’s work in a challenging environment. These examples illustrate our approach and show how it enables young children’s genuine mathematical thinking and helps to identify and foster mathematical giftedness.

Construction of new means by gifted students

The meaning of multiplication and division is not fully developed in 8-9 year old children. They constantly meet situations in which they need to evaluate critically their means, adapt them to new situation, or construct the new ones. This process can be seen once children go through the problems with the calendar (How many days there are in 3 weeks, in 3 weeks and 4 days? How many weeks there are in 21 days, in 27 days?).These problems can be solved by simple counting and grouping (using manipulatives or pictures). However, more advanced children would use more sophisticated strategies like addition or multiplication (division would be hardly expected).

Now, suppose, we ask children to calculate a number of weeks in the year (in order, for example, to know the number of issues of a weekly published magazine in one year). Even if they know the fact that a year has 365 (or 366) days and a week has 7 days, they will not be able to divide 365 by 7. Once applying a strategy of drawing and counting, they will face the difficulty to represent this big number. So, they are forced to modify their strategy.

Amelie (8) started searching for a different way of counting. She constructed a chart which represents a year divided into 12 small periods (month) (Fig 1.). She invented symbols representing a week (‘X’) and a day (‘C’). Thus she obtained a structure suitable for the use of grouping (days into weeks) and counting (weeks).

She has thus developed a new mechanism which included old means (counting by 7), old objects (days, weeks, year), old process (grouping), and old products (number of days in a month) as well as a reflective action on the problem as a whole (what does not work in my direct approach), that pushed her towards the creation of new means (symbols and procedures). But beside all this, we can see the influence of the combination of abilities and meta-abilities (to work on a problem and to work on the work), which is an important characteristic of a mathematically gifted child.

Logical thinking in mathematically gifted children

According to our model, mathematical reasoning is an important part of fostering theoretical thinking in all children and especially in those who are identified as mathematically gifted. Since logical thinking is a foundation of mathematical reasoning, we can expect that presenting our students with challenging situations that stimulate the growth of logical thinking, we help them reason at a higher theoretical level. That is why logic is an important part of our challenging elementary curriculum.

On the one hand, our children learn to solve different logical puzzles, use logical operations (negation, implication, class inclusion, etc.), play different strategy games (like chess). On the other hand, they are constantly invited to think logically in various mathematical situations (like working with definitions, looking for logical explanations, proofs, using examples, non-examples and counter examples). We challenge them constantly with 'little questions' provoking logically grounded mental actions. For example, the following problem is very difficult for an 8 year old child:

When it's sunny, Tim always puts on his white hat. When he puts his hat he never puts on his blue shoes. Yesterday, he was wearing his blue shoes during a whole day. What was the weather that day?

It requires a deep understanding of negation and ability to apply it in a complex situation. Sarah's solution (Fig.2) is particular in many ways. First, she uses two key words describing her solution: proof (‘preuve’) and inverse (reasons about the reasoning), develops an efficient strategy (making necessary links between data: sun means wearing hat, hat means no shoes, so sun means no shoes), uses schemas to communicate her thinking, gives a short explanation (shoes means no sun, so it rains (‘alors, c’est inverse’; ‘Reponse: il pleut’).

It comes as no surprise that, facing challenging tasks, gifted children use their abilities to the full in making logical inferences and thus demonstrate their potential of proving. In their 'proofs' they use schemas, symbols (letters), relationships between data and generalisations. Let’s look for example at Alice’s work (10 year old, Fig. 3, Fig. 4) on the following problem:

Alex has no brother and is 2 years younger than Peter. Boys are older than girls. Melissa is not the oldest of the group. The sum of ages of Peter and Elisabeth is the same as the sum of ages of Alex and Melissa. Put the ages of four children in the order, from the youngest to the oldest. Give the age of persons that we know exactly. Find the difference of girls’ ages.

Here is a transcript of child’s work (translated from French):

Alex – 13 year old, Elisabeth - ? Melissa - ? Peter – 2 years older than Alex – 15 year old. If the sum of ages of Peter and Elisabeth = 20*, than the age of Elisabeth = 5 and of Melissa = 7, because the sum of ages of Peter and Elisabeth is the same as the sum of ages of Alex and Melissa and Alex is 13 .

As we see, the child took concrete values of the variable (assuming that the sum of ages of Peter and Elisabeth equals 20). That gave her a possibility to establish an order between all ages. But, the key point of her reasoning is the following footnote:

Here is a transcript of it (translated from French):

* this number can be always different but the difference between two ages (Elisabeth’s and Melissa’s) will be always two and Melissa will be always older.

We can see that this child shows abilities to

    • look for relationships ("Si la somme des ages de Peter et d'Elisabeth = 20, alors l'?ge d'Elisabeth = 5 et celui de Melissa = 7, car la somme de son ?ge et celui d'Alex est égale ? celle de Peter et d'Elisabeth et d'?ge d'Alex est 13 ans."),
    • think about relationships making generalisations (" ce nombre (20) peut toujours changer, mais la différence d'?ge entre celui d'Elisabeth et celui de Mélissa sera toujours 2 et Mélissa sera toujours plus ?gée")
    • appropriate use of prepositional phrases in mathematical explanations (therefore (‘alors’), because (‘car’), always (‘toujours’)
Switching from one representation form to another during investigation process

Our next example presents the work of an 8 year old child on the problem. We started our investigation with grade 3 children with the question of what number of lines could relate 2 points. Than we started to ask the same question about 3 points, 4 points. Then, the students had to find a way to do it for 10 points and then for 101 points. Children had to construct a whole process of investigation working systematically, looking for patterns and making generalizations. So, Matthew started to look geometrically into small number of points. Then he discovered that we have to add natural numbers starting with 1 and finishing one less than our number of points. So he came out with calculation of the some of first 100 numbers. This work he does also systematically (making only two calculation mistakes). He didn’t succeed here to discover a faster way but for the child of 8 is calculation of big numbers is still meaningful and interesting task. But what is particularly interesting in terms of mathematical giftedness is to see this ability to switch from one representation (geometric) to another (arithmetic). We can also see the effect of an open ended problem that gives a chance to a child to conduct her own investigation.

Illustration of Matthew’s work: geometric representation (small numbers, search for patterns)

Illustration of Matthew’s work: arithmetic representation (big numbers, systematic work, perseverance)


A challenging situation approach is a system of teaching based on a challenging curriculum as a whole. It presents thus an efficient didactical tool of constructing a learning environment in which every child would be able to demonstrate her highest level of ability.

Challenging situations cannot be used only on exceptional occasions in a teaching approach. Some of them must, of course, be carefully prepared, but, for the approach to work, it must become a style pervading all teaching all the time at all levels of education. The teacher must be ready to use any opportunity that presents itself in class (e.g. a puzzling question posed by a students, an interesting error or unusual solution) to interrupt the routine and engage in reflective and investigative activities on the spot, or suggest that students think about the problem at home. Thus, in fact, what is needed is not occasional challenging situations, but a ‘challenging learning environment’.

This is why, using a challenging situation model, we are not only able to get gifted children involved in genuine mathematical activity but also help all children to increase their intellectual potential.

A challenging situation approach has another opening for gifted children: they can always go further, go beyond situations, ask new questions, initiate their own investigations, and be more creative in their mathematical work.

This gives us as teachers a chance to understand better what it is that makes mathematical talent appear and grow and thus leads to the creation of more efficient didactical tools that would help to keep their interest in learning mathematics.


[1] Arsac, G., Germain, G., & Mante, M. (1988). Probl?me ouvert et situation-probl?me. Lyon: Université Claude Benaud.

[2] Baroody, A. (1993). Problem solving, reasoning, and communication, K-8: helping children think mathematically. Macmillan Publishing Company.

[3] Baroody, A., & Ginsburg, H. (1990). Children's Learning: A Cognitive View. In: R.Davis, C.Maher, & N. Noddings(Eds.), Constructivist Views on the teaching and Learning of Mathematics (pp 51-64). Reston, Va: NCTM..

[4] Brousseau, G. (1997). Theory of didactical situations in mathematics . Dordrecht: Kluwer Academic Publishers.

[5] Ernst, P.(1998). Recent development in mathematical thinking. In: R. Burden, & M. Williams (Eds.), Thinking through the curriculum (pp.113-134). London, New York: Routledge.

[6] Fishbein, E. (1990). Introduction. In: P.Nesher, & J.Kilpatrick (Eds.), Mathematics and cognition: A research synthesis by the International Group for the Psychology of Mathematics Education.Cambridge: University Press.

[7] Freiman, V. (2003). Identification and Fostering of Mathematically Gifted Children, A Thesis In The Department Of Mathematics and Statistics, Concordia University, Montreal, Canada

[8] Greenes, C. (1981, February). Identifying the Gifted Student in Mathematics. Arithmetic Teacher, 14-17.

[9] Johnson, M. (1983, January). Identifying and Teaching Mathematically Gifted Elementary School Children. Arithmetic Teacher, 25-26; 55-56.

[10] Kennard, R. (1998). Providing for mathematically able children in ordinary classrooms”, Gifted Education International, Vol. 13 , No 1, 28-33.

[11]Krutetskii V.A. (1976). The psychology of mathematical abilities in school children . Chicago: The University of Chicago Press.

[12]Kulm, G. (1990). New Directions for Mathematics Assessment. In: G. Kulm, Assessing higher order thinking in mathematics . Washington, DC: American Association for the Advancement of Science.

[13] Lyons, M., & Lyons, R. (2001-2003). Défi mathématique. Cahiers de l'él?ve. 1-6. Montreal: Cheneli?re McGraw-Hill.

[14]Ridge, L., & Renzulli, J. (1981). Teaching Mathematics to the Talented and Gifted. In: V.Glennon (Ed.), The Mathematical Education of Exceptional Children and Youth, An Interdisciplinary Approach (pp. 191-266). NCTM.

[15] Shchedrovitskii, G. (1968) Pedagogika i logika. Unedited version (in Russian).

[16]Sheffield, L. (1999) Serving the Needs of the Mathematically Promising. In: L. Sheffield (Ed.), Developing mathematically promising students (pp. 43-56). NCTM.

[17] Sierpinska, A.(1994). Understanding in mathematics, London: The Falmer Press.

[18] Young P., & Tyre C.(1992). Gifted or able?: realising children's potential. Open University Press.


Viktor Freiman, M.T.M., Ed.Dr., Professeur agrégé

Département d’enseignement au primaire et de psychologie éducationnelle

Faculté des sciences de l’éducation, Université de Moncton

Pavillon Jeanne-de-Valois, Moncton, NB

Canada E1A 3E9

Phone: 1 506 858 44 37

Fax: 1 506 858 43 17

E-mail: freimanv@umoncton.ca





Ziva Deutsch, Akiva Kadari, Thierry Dana-Picard

Abstract: We present a recreational Mathematics Magazine produced by the gifted students from the Math Department at Michlalah - Jerusalem College. The magazine is published in two versions: printed and on-line (interactive), and is targeted to a broad audience. It also includes components for gifted high-school students.By that way we introduce gifted population to an innovative creative process.

Key words: Creative Process, Creative Environment, Interactivity, Gifted Students.


The need for specific activities in Mathematics for gifted students has been researched in detail. The positive influence of computerised teaching and computerised self-learning has also been investigated. References to such studies are given in the report by Ravaglia[5], [6]. It isof great importance that gifted high-school students be given the opportunity to learn more advanced courses than their peers. In some curricula, specific courses are actually built for this gifted audience, at various levels[5], [6], but the syllabus skeleton usually remains the same, the main difference being the course’s depth and pace.

In this paper, we describe a project which is based on a very different approach.The world of Mathematics is a very broad field. Gifted students are able to discover mathematical regions beyond that which is taught in their usual course of study. So, why not inspire them to create their own mathematical material, either educational or recreational? The result will be recreational pursuit with a clear educational value.

We emphasise the following: not all member students of “Alef Efes“’s board, are gifted teacher trainees. Additionally, the audience does not only consist of endowed high-school students. Nevertheless, working with gifted people produces finer materials which can inspire a more general audience to discover new horizons and new trends in mathematics[8].

“Alef Efes” is an exclusive quarterly mathematical magazine in Hebrew, appearing in two non-identical forms: a printed version for subscribers and a free on-line interactive version. It is prepared by a group of gifted teacher traineesguidance from the departmental staff primarily created for the gifted high-school student. It publishes recreational Mathematics together with classical mathematical topics in a unique way. The magazine consists of original articles on popular subjects, riddles on various levels, mathematical games, aspects of the history of mathematics, and more. Non-mathematical topics which containmathematical features are also included. We describe these features in the next section.


1. The Magazine’s Goals

The audience of “Alef Efes” is comprised of different groups, in varying proportions:

    a. Gifted high-school students and their teachers;
    b. High-school students who are less drawn to math(not those who have a

profound lack of interest for Mathematics);
    c. Undergraduate college students;
    d. Teachers at all levels not necessarily involved in gifted youth programs.

Because of the non-homogeneous audience, there are various levels of difficulty within the activities and the proposed challenges. We will describe this point later.

In particular, high school students are faced with unfamiliar mathematical material. Thanks to “Alef Efes” they have an opportunity to discover mathematical topicswhich are not included in their regular curriculum, and to discover mathematical applications in everyday life. All this is done in an interesting and attractive manner, displaying and adding mathematical flavour to ancient problems. This process develops curiosity for new topics as well.

2. The Magazine’s Double Format

Each of the magazine's dual formats provides a unique advantage. The journal's purpose is to make Mathematics enjoyable and to encourage mathematical education within a wide variety of populations. This goal becomes attainable if we are able to shatter the myths and fears that surround this subject. For that purpose, the graphic features of the printed journal comprise a significant issue, and much attention is devoted to their development.

Theinteractive dynamic on-line version, uses the advantages of the Internet, and allows the visitor to try and solve problems on his own, to check the solutions, or make use of on-line hints. The mathematical games, which appear on the site, take advantage of new methods (such as use of Java, Java script and Flash), allowing interactive educational activitiesand illuminating the underlying mathematical principles in depth. The interactivity is a central feature of the project, in particular to decrease the “mathematical fear level" of a general audience.

The original web site, written in Hebrew contains hundreds of riddles, paradoxes, mathematical games and articles. For the convenience of the readers of this paper, we translated a few examples into English and included them in a mini English version of the site.

3. Content

A recurrent myth claims that a gifted population is homogeneous. This claim is not necessarily true, and even a group of very bright students of the same age often show different interests and abilities in Mathematics. Therefore, " Alef Efes" must contain articles and activities of a wide range of mathematical topics.

Such activities can be compared to enriching activities in class teaching. Nevertheless Alef Efes does not intend to replace the teacher, but rather contributes strongly to self-teaching, within the borderlines of the official syllabi (and often beyond them).

Let us give some examples:

a. Expository Articles

The site contains a rich collection of articles of various subjects:

    • Sundry examples from geometry and their applications such as the 4-colour problem, platonic solids.
    • Number theory discussions as in magic squares (with a reference to Albrecht D ?rer’s Melancholia, Fibonacci sequences, etc).
    • Recursive processes like Hanoi towers, Pascal Triangle etc.

b. Math Riddles

Motivating questions, quizzes and challenges, are proposed at various levels, including those designed for an audience with out high mathematical abilities. The different levels of difficulty are indicated by specific icons, as displayed next to every puzzle both in the printed version and at the site.

Regarding this, [8] quotes [3]: “learning takes place when students’ abilities and interest are stimulated by the appropriate level of challenge”.

Examples of a few riddles appear in the mini site at the following URL [2]:



c. Mathematical Games

Recreational activities are a very important part of a “low-fear” approach to Mathematics. As noted previously, gifted students do not form a homogeneous population, and being gifted does not mean being strongly self-confident. The online games proposed by the web-version of “Alef Efes” contribute to lower the reluctance level before mathematical situation. Of course, the mostly graphical nature of such games contributes to developing geometrical intuition, both two-dimensional and three-dimensional. Critics are sometimes formulated against Mathematical Education’s classical way, such as “Until the age of 6, children have a good three-dimensional perception of the world; thenwe flatten them to plane geometry”. Alef Efes brings its two cents to enhancing a better three-dimensional view.

Two examples of our math games appear in the mini English site at the following URL[2]:


d. Paradoxes

One of the most attractive sections of both the printed and the on-line versions is the paradox section. Well known and less famous paradoxes amaze the readers. The dynamic part of the on-line version enables lively illustrations of the paradoxes. For example look at the so-called Galileo paradoxes in Hebrew[1]:


e. History of Mathematics and Ethnomathematics

Mathematics is a constantly developing field made up and discovered by human beings, therefore the journal emphasizes historical topics dedicated either to biographies of famous mathematicians or to episodes from the history of Mathematics, such as the bridges of Koenigsberg, the life of Archimedes and the famous palimpsest of one of Archimedes’s lost works that was recently found and sold.

Special attention is paid to Ethnomathematics (African, Arabic, Indian etc.) Moreover, the journal includes many articles from the Jewish mathematical sources, and Hebrew manuscripts.

f. Special Topics

Besides the regular topics appearing in every issue of the printed version, every new issue includes updated new items, such as mathematical problems on a chessboard, ancient mathematical problems, the “Olympic Corner”, etc. These are intended to either broaden the mathematical landscapes discovered by the readers or to develop deeper mathematicalinsight. In particular, the challenges appearing in the “Olympic Corner” present activities from other countriestargeted to endowed students. This enables the readers to feel as part of a worldwide community.

4. Who Is Involved in the Edition?

This project enrols gifted students of the Department of Mathematics at Michlalah-Jerusalem College, Israel, in creating, formatting, and editing the material for both the hardcopy and the on-line version of the magazine.

The creative process is planned around a slightly modified version of Renzulli-Reis Triad model [7]. That was initiated through an on-line course.

In that course the students deal with interactive tools, gain experience, they discover and sort unprocessed printed material and material from the internet, or other sources (Type One activities), they develop skills to write original material for the magazine, to translate it into Hebrew and to edit it for the Israeli students.

Subsequently, they organize the output according to the level of mathematical understanding of the targeted audience (Type Two activities), and process it into a finished product appropriate for either the print or for electronic publications. For the level of Type Three activities, the department students are less autonomous; they are directed by decisions of the editorial board of the journal.

We should emphasise the fact that one of the main goals of the "Alef Efes" project is to develop gifted teacher trainee’s creativity, and to encourage autonomous research. These are not frequently developed at the undergraduate level throughout the world, as the regular classroom work is generally repetitive and imitates the educator’s expository work. Additionally the teacher trainees developbroadened graphical features (programming animations, applets, etc.). The importance of animations and applets is obvious in quite every course in Mathematics; for an example see[4].

5. How Is the Work Done?

The students work in groups, both in a synchronous framework and in asynchronous interactions, via a specific web-forum. This allows for a continuous, uninterrupted flow of work throughout the week, with constant feedback and open discussions. Of course, the department's staff is present at each step, but wide autonomy is left to students.


    1. A Broader Audience

As noted by J. Stepanek [8], elitism is not perceived as a relevant issue, and in many countries, classes are highly non-homogeneous, mirroring the general population. Aproject aimed exclusively at a high-ability population is therefore irrelevant.

On that basis, the editorial board of "Alef Efes" designed the publication to address an additional issue. A large part of the high-school students and a certain percentage of their teachers feel, at the very least, uncomfortable when dealing with mathematical topics, whether they come from the official syllabus or appear as a by-product of another activity. In the worst case, they don't deal with such a material.

The presentation of “serious” material in a recreational way, using attractive illustrations and high standard designfor the printed version, and all the graphical updated animations for the on-line version, assist in decreasing the level of fear, and increase curiosity about mathematics.

2. Interactions

The connection with a wide an inhomogeneous audience implies the necessity of a high level of interactivity. The online journal fulfils that need in various ways:

    • A few forums are active in the site and supply constant connection in between the editorial board and the audience. The forums are dedicated to discussions about interesting open problems in Mathematics and reader’s suggestions of new puzzles.
    • “The Problem of the Month” is a very popular section offering a new challenging puzzle at the beginning of every month. The winner receives free subscription for the printed journal.
    • A very active guest book enables the readers to give their opinion and send remarks.
3. Statistics

As mentioned previously the audience of “Alef Efes” comprises varied populations according to mathematicallevel, professional occupation or age group. There is a difference between the printed and the online version of the periodical:

    • The printed version is published in 3000 copies (An impressive number, considering the language and the country size). About 50% of the subscribers are institutional subscribers (schools, libraries, colleges and universities). The other 50% comprise teachers, students and math enthusiasts.
    • The on-line free version is very popular (about30000 visitors per month).The following table displays the audience distribution along the day; two maxima appear, one in the morning corresponding to classroom activities, the other one in the afternoon, showing home activity mainly of young students.
    • The following diagram shows the relative popularity of the various sections of the on-line site:


The double magazine (hardcopy and on-line interactive) is an integral part of the students’ teacher training at the Mathematics Department of Michlalah-Jerusalem College, enabling the student to participate in a form of cooperative learning within a very creative environment, in conjunction with the faculty. A by-product of this cooperation is “learn how I work, it’s more important than what I teach”.

"Alef Efes" is a special experiment in which the undergraduate students are not only exposed to existing materials, but also participate in original creation of new material and most importantly, have fun with Mathematics.


        1. URL: http://alefefes.macam.ac.il
        2. Mini site in English: http://alefefes.macam.ac.il/english
        3. Caine, R.N. &G. Caine , Making Connections : Teaching And The Human Brain, Alexandria, VA, Association for Supervision and Curriculum Development. 1991
        4. Kidron,I. &N. Zehavi, The Role of Animation in Teaching the Limit Concept , The International Journal of Computer Algebra in Mathematics Education , 9 (3), 2003
        5. Ravaglia, R., Suppes, P., Stillinger, C., & Alper, T.M., Computer-Based MathematicsAnd Physics For Gifted Students, Gifted Child Quarterly, 39: 7-13. 1995, Available: http://www-epgy.stanford.edu/research/mset.pdf
        6. Ravaglia, R., Sommer, R., Sanders, M., Oas, G. & DeLeone, C., Computer-Based Mathematics And Physics For Gifted Remote Students,Available: http://www-epgy.stanford.edu/research/mset.pdf
        7. Renzulli, J.S. and Reis , S.M., The Enrichment Triad/Revolving Door Model: A School-Wide Plan For The Development Of Creative Productivity, Systems and Model For Developing Programs For The Gifted And Talented, in J.S. Renzulli (Eds), Mansfield Center, CT, Creative Learning Press, 216-266. 1986
        8. Stepanek, J., Meeting The Needs Of Gifted Students: Differentiating Mathematics And Science Instruction, Northwest Regional Educational Laboratory, 1999Available: http://www.nwrel.org/msec/images/resources/justgood/12.99.pdf
        9. Tyler-Wood, T.,M.V. Perez-Cereijo &T.Holcomb, Technology Skills Among Gifted Students, Journal of Computing in Teacher Education, 18 (2), 57-60.

Ziva Deutsch, Ph.D.

Department of Mathematics

Michlalah - Jerusalem College

Bayit Vegan



e-mail: zivad@macam.ac.il

Akiva Kadari, M.A

Department of Mathematics

Michlalah - Jerusalem College

Bayit Vegan



e-mail: akiva@macam.ac.il

Thierry Dana-Picard*, Ph. D.

Department of Applied Mathematics

Jerusalem College of Technology

Havaad Haleumi Street 21

Jerusalem 91160


e-mail: dana@mail.jct.ac.il



Dimitris V. Papanagiotakis, Panayiotis M. Vlamos


Key words: Problem Database, Problem Solving, Collaboration, Creativity in Problem Solving, Web, Creative Process, Web Site, Step by Step Approach, Solution, MathML, XML, JSP.

Themainaspectofthispaperistopresentan Integrated Web Based Step by Step Mathematical Problem Solving System. The system consists of a Database where the problems and the solutions are stored using HTML and MathML format. The mathematical problems database is accessible to registered members who are interestedin solving the problems. The members can be students from all over the world.

The main objective of the system is the collaboration among gifted students on solving complicated mathematical problems. The key concept of the system is the Step by Step approach to the solution of the problem by each group of students using different paths. Each path may lead to the solution or may not.Another innovation of the system is that each step is recorded by the system so at the end the students can have a full tree structure image of the different paths that lead to the solution.

As a result, of the Step by Step process and the collaboration we have the increase of the students’ creativity and imagination. The system was developed using state of the art web technologies Tomcat Web Server, mySQL, JSP 2.0, HTML, MathML,XML.


Dimitris V. Papanagiotakis

Defacto Research Center for Culture and Innovation


Panayiotis M. Vlamos, Ph.D., Assoc. Prof., President

Defacto Research Center for Culture and Innovation





Donco Dimovski

Abstract: The aim of this short note is to give an overall description of most of the mathematical schools, competitions and publications for primary and secondary school students in Macedonia. The primary school in Macedonia is an 8 year school, and the secondary one is a 4 year school. Usually the students start to attend the primary school when they are six and a half years old.

Key words: Competitions.


There are several different types of mathematical schools organized in Macedonia. The mathematical contests in almost all of them are not included in the curricula, they are organized usually by the Union of Mathematicians of Macedonia (SMM) and some of its sections, and usually the students that participate at these schools are the ones which like mathematics and show big interest in it. Below, some of these schools will be described in more detail.

I.1. Mathematical schools for high school students

The Mathematical school for high school students in Macedonia started in 1976/77 school year. At the beginning it was organized by the Mathematical Institute at the University of Skopje. Later, the organizers of this school were the Institute of Mathematics and Computer Science at the Faculty of Mathematics and Natural Sciences in Skopje, SMM and the section for high schools at the Ministry of Education of Macedonia. Today the organization of this School is solely based at the Institute of Mathematics, at the Faculty of Mathematics and Natural Sciences in Skopje and SMI, but because of the shortage of money its activity is very low.

The aim of this school is to broaden the knowledge of talented high school students in mathematics and other disciplines close to mathematics. With its activity, the school gives the opportunity to high school students to see different aspects of mathematical sciences and helps them in the choice of their professional carrier. Most of the participants of the school in the past years have chosen to study mathematics, informatics and technical sciences, and a lot of them today are mathematicians and technical engineers, but some of them study medicine, law, economics.

In 1967, the school started as a Summer school held in Ohrid, with the topics: Elements of vector algebra, and Elements of number theory. The next 1967/68 year, an Evening school was organized for students in Skopje, working in the Fall and Spring 2 hours per week. Since 1968, the Summer school had two stages: Seminar in June held in Skopje, and Summer school held in Ohrid.

Some of the topics of the school in the years 1967-1971 were the following: Basics of computer sciences (1967); Three classical geometric problems (1968); Programming on IBM-1130 (1968); Methods for solving geometric construction problems (1968); Divisibility of numbers (1968); Approximate calculations (1968); Sets and maps (1969); Elements of probability theory (1969); Inversion (1969); Matrix theory 1970); Fortran (1970); Metric spaces (1971).

From 1972 till 1994 the form of the School was as follows.

A topic and a main lecturer for the school was chosen. The main lecturer prepared written material, separately for all of the three stages. For the first stage classes were formed in high schools in Macedonia, working in the spring, 2 hours per week, using the prepared written material. The number of students attending the first stage ranged from about 350 to 1400. At the end of the first stage all the students that attended the school took the same exam prepared by the main lecturer. About 90 students that have shown the best results on the exam, participated in the second stage, held in Skopje in June. At this stage the students continued to work on the topic, using the written material for this stage, for about 3-4 days (12 hours). Again, at the end the students who attended the second stage took an exam prepared by the main lecturer. About 35 studentswho had shown the best results on this exam, participated at the third stage, the Summer school, usually held in a tourist resort on Ohrid lake. Again, at this stage the students continued to work on the topic, using the written material for this stage, for about 10 to 15 days (30 to 40 hours). At the end of this stage the students took an exam on the material of the three stages. Sometimes the prepared material was published as a book in the edition "Library Mathematical school".

Some of the topics of the school in this period were the following: Post and Turing machines and normal algorithms (1972); Fortran programming (1972); Mathematical logic (1973); Basic of linear programming (1974); Basic of graph theory (1975); Set theory (1976); Algorithms, computers and their applications (1977); Semigroups and finite automata (1978); Project planning (1979); Logic and computers (1980); Probability and statistics (1981); Basic programming (1982); Finite automata (1984); Artificial intelligence (1985); Mathematical Olympiad problems (1987); Computer programming (1988); Inversion (1989); Combinatorics (1991); Dynamical systems (1992); Topology via logic (1993); Difference equations (1994).

I.2. Mathematical schools for primary school students

There are several mathematical schools for primary school students held in Macedonia, and most of them have local character. Two of them will be described in more details.

The Society of Mathematicians of Stip organizes a mathematical school for primary school students (grades 5,6,7 and 8) since 1987. It has two stages: Winter school and Summer school. The winter school is held in January for about 7 days, and the summer school is held in June for about 10 days. The students work on a topic (different topics for different grades) using a written material. The number of students from almost all parts in Macedonia attending these schools ranges from 100 to 350.

The Society of Mathematicians of Skopje organizes a mathematical school for the students in Skopje, grade 7 and 8. This school works during the school year, in classes formed of students from almost all primary schools in Skopje, 2 hours per week. A written material, prepared usually by a group of mathematicians, is given to the students who participate at the school, and they attend lectures given usually by primary school mathematics teachers. Similar schools were organized also in other towns in Macedonia.


The mathematical competitions for high school students of Macedonia have been organized in 1955, 1957. Later, since 1959, such competitions are organized each year. Since 1978, a Regional mathematical competition is held each year at the beginning of February, for 2 and a half hours, each year in a different town in 6 regions of Macedonia. At these competitions students usually work on 4 problems, different problems for different grades, prepared by a national problem committee. Usually about 800 to 1000 students take part at the Regional mathematical competition. About 140 students (about 35 from each grade) with the highest number of points from the Regional competition take part at the Republic mathematical competition, usually held at the end of March, for 3 and a half hours. The Republic competition was held in Skopje till 1979, and ever since it is held in different towns in Macedonia.

A Macedonian Mathematical Olympiad is organized each year since 1994, in the end of April. About 30 to 40 students with highest score from the Republic competition take part at the Olympiad. All of them work on 5 problems for 4 and a half hours.

A school competition for primary school students, grade 8, 7, 6 and 5 (and since last year grade 4), is held each year. Students work on 4 problems, two of them prepared by the National problem committee, and two of them prepared by school teachers of mathematics. After the school competition, the students, grade 8, 7 and 6 (and since last year grade 5), with highest score participate at the Municipal competition, usually held in March. All of the students work on same problems prepared by the National problem committee. A Republic mathematical competition for primary school students, grade 7 and 8 (and since last year grade 6) is held usually in May. About 140 students (about 70 from each grade, and since last year 60 + 60 + 20), with the highest scores from the Municipal competition take part at the Republic mathematical competition, usually held at the beginning of May.

A Macedonian Junior Mathematical Olympiad is organized each year since 1997, in June. About 30 to 40 students with highest score from the Republic competition take part at the Olympiad. All of them work on 5 problems for 4 and a half hours.

During the year several types of preparations for primary and secondary school students are organized in several places in Macedonia.

Since 1995, the European Kangaroo Competition is held in Macedonia, with 7000 to 25000 participants in different years. In cooperation with the Union of Mathematicians of Bulgaria, about 100 to 300 students participate at the tournament Cernorizec Hrabar each year. Also, for some years, students from Macedonia participated at the Tournament of Towns.


Usually, the system of national competition is supported by publishing the appropriate literature for the participants of these competitions. The same is practiced by the SMM, the main organizer of the national mathematical competitions, via the appropriate publishing activities.

The mentioned literature for the students of the primary schools consists of the publication "NUMERUS", published since 1975. It is published in four books per school year. In this publication, except the popular literature for the young readers, there is a material for preparation of the students for the National mathematical competitions. This publication contains also complete information about the National Mathematical competitions in Macedonia and the JBMO (the Junior Balkan Mathematical Olympiad). Also, there is a special edition "Library Numerus", which for the time being consists of 9 books.

Similar situation is with the publishing activities for the high school participants in the National mathematical competitions. Namely, for the needs of these competitions, SMM is publishing the publication "SIGMA", three times per school year, since 1979. This publication contains: papers for extending the knowledge in mathematics of high school students; complete information about the National mathematical competitions in Macedonia, the BMO (the Balkan Mathematical Olympiad), the International Mathematical Olympiad, and the Tournament of Towns. Also, there is a special edition "Library SIGMA"with 10 books.

The Institute of Mathematics at the Faculty of Mathematics and Natural Sciences in Skopje publishes the edition "Library Mathematical School" with 13 books, 4 of which are problem books with problems and solutions of the Republic, Regional, and Yugoslav Federation Mathematical Competitions.

For the purposes of Winter and Summer camps and Mathematical Schools, for primary and secondary school students, several internal working materials are published.


Donco Dimovski, Ph.D., Prof.

St. Cyril and Methodius University

Faculty of Natural Sciences and Mathematics

Institute of Mathematics

Gazibaba b.b. P.O.Box 162, 1000 Skopje

FYR of Macedonia

E-mail: donco@iunona.pmf.ukim.edu.mk

  for_gifted_stud.zip (2,900Mb)
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