ࡱ> j @bjbj >؝؝95> > > > > > > R ***8*,R @- 6777888$,R~ > <88<< > > 77hLLL<r > 7> 7L<LLF> > 7z- `b*WGK:ә0.RJBRtR R > > > > R> Ǘ 89L0::)888 R R '-LdR R ' AFFECTION THE MATHEMATICAL CREATIVITY OF THE STUDENTS WITH COMPLEXES OF EXAMPLES OF GOOD PRACTICE SVETOSLAV JORDANOV BILCHEV Abstract: In the paper several complexes of problems useful for promoting mathematical creativity are given. The all investigated problems are examples of good realized practice along the chain of operations for solving problems as a part of the process of mathematical creativity. Key words: Mathematical Creativity, Creative Process, Solving Problems, Posing Problems, Creative Mathematical Product, Research Works in Mathematics, Motivation, Support. INTRODUCTION When we work on mathematics with students the main problem is how to affect the creativity possibilities in every student. It is not enough only to solve the problem in some way, i.e. to realize the following statement this problem have to be solved by this way. The necessary theory and the solving of the problem should be introduced so as to be clearly seen the following: This is the source of a thought or idea. What are the developmental ways of the thoughts and ideas? How are these roads diverted from the correct direction to the wrong one? Which are the main basis postulates for the ideas and what the main used learning patterns are? How it is possible to complete the ideas. How to find a way for moving the ideas forward. How to enlarge the ideas. How to generalize the ideas. How to find the way of constructing a group of problems, which could be attacked with the same ideas. How to accumulate new and new ideas and methods, which could solve a singular group of problems or groups of problems. How to collect not only interesting and at the same time general problems, but also the ideas and methods for their solution, expansion and generalization. As a whole, I have to say that, every time when I face some kind of a problem in front of my students at the famous Rousse Mathematical Circles or at my University of Rousse I try to show them not only the solution of the problem, but also the whole process through which my or someone elses thought has been gone through. Here in the paper I will try to present several complexes of problems useful for promoting mathematical creativity. Of course, in a paper with limited number of pages this is too difficult but some important moments are possible to be underlined. The given investigated problems are examples of good realized practice along the following chain as a part of the process of mathematical creativity: solving problems! solving problems uniquely ! creating problems ! creating new problems! problems posing! group of connected problems posing! methods for problems posing! creating research works in mathematics Group of Problems 1 It is well-known the following problem: Problem 1.1. In a circle (k) an equilateral triangle  EMBED Equation.3  with side  EMBED Equation.3  is inscribed. Let the point  EMBED Equation.3  is an arbitrary point on the arc  EMBED Equation.3 and  EMBED Equation.3  Prove that : (1)  EMBED Equation.3  . Solution 1. (This solution is an example of a constructive way of solving problems see Fig. 1.) If we put the segment  EMBED Equation.3  on the segment  EMBED Equation.3  so that  EMBED Equation.3 , where the point  EMBED Equation.3  belongs to the segment  EMBED Equation.3 , it is necessary to prove that the last segment  EMBED Equation.3  is equal to  EMBED Equation.3 . So, the triangle  EMBED Equation.3  is an isosceles triangle with an angle of  EMBED Equation.3 , i.e. it is an equilateral triangle with  EMBED Equation.3  Then the triangles  EMBED Equation.3 ,  EMBED Equation.3  are similar with:  EMBED Equation.3  and  EMBED Equation.3 . Hence  EMBED Equation.3  and  EMBED Equation.3 . Here we realized the first step of the given above chain solving problems.  Figure 1 Solution 2. By using of the famous Ptolemys Theorem for the quadrilateral  EMBED Equation.3 we get immediately  EMBED Equation.3  , i.e.  EMBED Equation.3  . Now we realized the second step of the given above chain solving problems uniquely. Problem 1.2. For the given construction in the Problem 1.1 prove that: (2)  EMBED Equation.3  . Solution. By using the well-known method of the areas we get:  EMBED Equation.3  , or  EMBED Equation.3  , from where (2) follows. Problem 1.3. Further prove that: (3)  EMBED Equation.3 . Solution. From (1) we get  EMBED Equation.3  , i.e.  EMBED Equation.3  and (3) follows with the help of (2). Problem 1.4. Prove that: (4)  EMBED Equation.3  . Solution. The identity (4) is a simple consequence from the Cosine Law for the triangle  EMBED Equation.3 , or from (2) and (1). For the last three problems we realized the third step of the given above chain creating problems. Problem 1.5. Prove that: (5)  EMBED Equation.3  . Solution. By squaring (2) we obtain  EMBED Equation.3  and from (1) the identity (5) follows immediately. Problem 1.6. Prove that: (6)  EMBED Equation.3  . Solution. By squaring (3) and using (5) we obtain consequently:  EMBED Equation.3  ,  EMBED Equation.3 , i.e. (6). The last two problems give to us the steps: creating new problems and problems posing. Problem 1.7. Prove that: (7)  EMBED Equation.3  . Problem 1.8. Prove that: (8)  EMBED Equation.3  . With the last two problems we continue the step problems posing and realize in general the step group of connected problems posing. Group of Problems 2 Every one of the given bellow problems (2.1 2.5) consists of equivalent true inequalities for the usual elements  EMBED Equation.3  of any arbitrary triangle. The given sums are cyclic. Problem 2.1. (9)  EMBED Equation.3   EMBED Equation.3  (10)  EMBED Equation.3   EMBED Equation.3  (11)  EMBED Equation.3 (  EMBED Equation.3   EMBED Equation.3  (12)  EMBED Equation.3  (Eulers Inequality). Problem 2.2. (13) EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  (14)  EMBED Equation.3  . Problem 2.3. (15)  EMBED Equation.3   EMBED Equation.3  (16)  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  (17)  EMBED Equation.3 . Problem 2.4. (18)  EMBED Equation.3   EMBED Equation.3  (19)  EMBED Equation.3   EMBED Equation.3  (20)  EMBED Equation.3  . The above proved inequalities are possible to be compared one to the other. So we obtained the following interesting chain of geometric inequalities for any triangle with the usual elements  EMBED Equation.3  . Problem 2.5. Prove that the following chain of triangle inequalities exists: (21)  EMBED Equation.3 ( EMBED Equation.3 ( EMBED Equation.3  (  EMBED Equation.3  . The used above idea is very attractive because of possibility to prove many known inequalities with the presented technique but in some lucky cases it is possible to get new sharp results. Let we see some new results. Problem 2.6. The following true triangle inequalities are equivalent: (22)  EMBED Equation.3   EMBED Equation.3  (23)  EMBED Equation.3   EMBED Equation.3  (24)  EMBED Equation.3 . The equality in (22) holds if  EMBED Equation.3  or  EMBED Equation.3 . Corollary. The inequality (24) belongs to the chain: (25)  EMBED Equation.3  . Problem 2.7. The following true triangle inequalities are equivalent: (26)  EMBED Equation.3   EMBED Equation.3  (27)  EMBED Equation.3   EMBED Equation.3  (12) - Eulers Inequality . Problem 2.8. The following true triangle inequalities are equivalent: (28)  EMBED Equation.3   EMBED Equation.3  (29)  EMBED Equation.3   EMBED Equation.3  (30)  EMBED Equation.3  . Corollary. The last inequality belongs to the chain: (31)  EMBED Equation.3  . Problem 2.9. The following true triangle inequalities are equivalent: (32)  EMBED Equation.3   EMBED Equation.3  (33)  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  . Corollary. The inequality (32) belongs to the following chain of inequalities: (34)  EMBED Equation.3  . Problem 2.10. The following true triangle inequalities are equivalent: (35)  EMBED Equation.3   EMBED Equation.3  (36)  EMBED Equation.3   EMBED Equation.3  (37)  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  (38)  EMBED Equation.3 , because  EMBED Equation.3 ,  EMBED Equation.3 . The irrational inequality (60) is obviously new! Corollary. We can compare (38) to right inequality (34):  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  and then we get the following chain of inequalities: (39)  EMBED Equation.3  . Of course, the all given above Group of Problems 2 with their solutions and their novelty realized the last two steps methods for problems posing and creating research works in mathematics. Really, the above given results are obtained after hard creative research work on the group of problems. Now is just the appropriate moment when the novelty and the innovation of the created problems, methods and research works are necessary to be discussed. Some of the obtained results are new ones but what means new result for different people? In this sense we have three different expressions - new result for the creator, new result as far as the creator knows and new result for the world. The meaning of every one of these expressions is necessary to be clarified but this meaning is obvious for the people working on mathematics. My personal opinion is if the obtained result is really unknown to the creator then the right decision is to write that the result is new as far as the creator knows CONCLUSIONS AND FUTURE WORK Every time when we consider and discus the mathematical creativity of the talented students in mathematics it is absolutely necessary to have in mind ad to use the following very important integrative chain: Person  EMBED Equation.3  Background  EMBED Equation.3  Problems  EMBED Equation.3  Solutions  EMBED Equation.3   EMBED Equation.3  Extra Education  EMBED Equation.3  Methods  EMBED Equation.3  Discovering  EMBED Equation.3  Motivation  EMBED Equation.3   EMBED Equation.3  Support  EMBED Equation.3  Investigations  EMBED Equation.3  Research Works REFERENCES Bilchev S. And D.Kontogiannis (1991), A Short and Elementary Proof of Mushkarov Simeonovs Inequality, Hrvatska Akademija Znanosti I Umjetnosti, Rada 456, Matemati ke znanosti, svezak 10, Zagreb, 33  37. Velikova, E. and M.Georgieva (2001). Diagnostic of Mathematical Abilities. Proceedings of the Union of Scientists  Rousse, Ser.5, Mathematics, Informatics and Physics, Vol. 1, p.114-120. Bilchev S., B.Kuiyumdzhieva, R.Chaparov, M.Kunchev, T.Mitev (2002), All Squares Are Non-Negative, 2002 Write A Problem - Set Challenge, International Competition, Best Practices in Education, N.Y., U.S.A., 1-20. Velikova, E. (2002). Stimulating mathematical creativity in 9th 12th grade students. The Government Specialized Scientific Council, Sofia, Bulgaria. Bilchev S. (2003), This Fairy Land Named Mathland, Proceedings of the Union of Scientists Rousse, Ser.5, Mathematics, Informatics and Physics, Vol. 4, p. 168 180. Bilchev S. (2004), About an Unexpected Transition from Algebra to Geometry, Mathematics Competitions, Journal of the WFNMC, Australian Mathematics Trust, Canberra, Australia, Vol. 17, No. 2, 17 27. Bilchev S., E.Velikova, P.Kenderov, S.Grozdev, G.Makrides (2005), The European Project MATHEU: Indentifying, Motivating and Support of Mathematical Talents in European Schools (Bulgarian and English), , Proceedings of the Scientific Conference of the Union of Scientists, 29-30 October, 2005, Rousse, University of Rousse, Bulgaria, Ser.5, Mathematics, Informatics and Physics, Vol. 41, p. 48 65. Bilchev S. and E.Velikova (2005), About Some Basic Principles of the Extracurricular Work with Talented Students, Proceedings of the 4th Mediterranean Conference on Mathematics Education, January 28 30, 2005, Palermo, Italy, Cyprus Mathematical Society, 553 559. Bilchev S. J. (2005), The Concept of the Triple Ladder for Identification and Motivation of the Talented Students, International Conference on Mathematics Education, 3  5 June 2005, Svishtov, Bulgaria, Sofia, 138 - 145. Bilchev S. (2006), Provoking Curiosity and Creativity of the Gifted Students in Mathematics by Obtaining New Results, University of South Bohemia esk Budjovice, Pedagogical Faculty, Department of Mathematics Report Series, Volume 14, 65 68. Bilchev S. (2008), Multiple Connection Tasks or Methods for Solving Multiple Problems, Proceedings of the International Research Workshop of the Israel Science Foundation, Multiple Solution Connecting Tasks, 20 - 21 - 22 February, 2008, Haifa, Israel, CET - The Center for Educational Technology, Tel Aviv, 13 24. Bilchev S. 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$d\$a$gd%$a$gd$x7$8$H$^`a$gd$ x^`a$gd$x^`a$gdCCCCCCD )+2345BCFƴƩwk[kKhmhWQ5CJaJmH sH hWQhWQ5CJaJmH sH hWQhWQ5CJaJhWQhWQCJaJmH sH hWQhWQCJaJmHsH'hhWQCJ\aJmHnHsHtHhhWQCJaJ"hhWQCJH*\aJnHtHhhWQCJ\aJnHtHU*hhWQ6CJ\aJmHnHsHtH"hhWQ6CJ\aJnHtHsults on High Level, Proceedings of the 5th International Conference on Creativity in Mathematics and the Education of Gifted Students, Projects and Ideas, Editor Roza Leikin, Haifa, Israel, February 24 28, 2008, CET - The Center for Educational Technology, Tel Aviv, 277 288, ISBN 965 354 006 8. ABOUT THE AUTHOR Assoc. Prof. Svetoslav Jordanov Bilchev, Ph.D. Chairman of the Department of Algebra and Geometry Faculty of Natural Sciences and Education University of Rousse 8 Studentska str. 7017 ROUSSE, BULGARIA Cell phones: +359 886 735 536, +359 8 999 555 28 -mails:  HYPERLINK "mailto:slavy@ami.ru.acad.bg" slavy@ami.ru.acad.bg  HYPERLINK "mailto:slavy_bilchev@yahoo.com" slavy_bilchev@yahoo.com  PAGE 170 PAGE 170 PAGE 170 PAGE 170 PAGE 170    Svetoslav Bilchev Affection the Mathematical Creativity of the Students with Complexes of Examples of Good Practice PAGE 206 DG 9: Promoting Creativity for All Students in Mathematics Education, Section 3 PAGE 205 ICME 11, Mexico, 2008 Proceedings of the Discussing Group 9 : Promoting Creativity for All Students in Mathematics Education The 11th International Congress on Mathematical Education Monterrey, Mexico, July 6-13, 2008 PAGE 195 ICME 1168DH~tʿʯʿ|d|Q||$hWQhWQ0JOJQJaJmH sH /j?IhWQhWQOJQJUaJmH sH )jhWQhWQOJQJUaJmH sH  hWQhWQOJQJaJmH sH hWQhWQOJQJaJhWQhWQ5CJaJmH sH hWQhWQCJaJhWQhWQ5CJaJhmhWQ5CJaJhm5CJaJmH sH hmhm5CJaJmH sH tvx 012ҿҲ||||pgch%hfh%aJh%CJOJQJaJhjhUh0JmHnHu h0Jjh0JUhWQhjMCJaJmHsHhWQhWQOJQJaJ$hWQhWQ0JOJQJaJmH sH )jhWQhWQOJQJUaJmH sH /jJhWQhWQOJQJUaJmH sH &012i$a$gd|*$'&#$+D,a$gd $ee]e`ea$gd|*y&#$+D,gd gdF"$a$gd%$a$gdf28<=@MWXY[\fhjxȺگreUh;h%6CJOJQJaJh%6CJOJQJaJ%h%0JCJOJQJaJmHnHu hAoIh%0JCJOJQJaJ)jhAoIh%0JCJOJQJUaJh%h%h%mH sH h%CJOJQJ^JaJ"h%CJOJQJ^JaJmH sH (hwh%CJOJQJ^JaJmH sH  hwh%CJOJQJ^JaJ ʵvhZhSʵQvJ hIih%U h-Ch%h-Ch%56CJaJh-Ch%56CJ aJ h;h%CJOJQJaJh%CJOJQJaJ%h%0JCJOJQJaJmHnHu hAoIh%0JCJOJQJaJ)jhAoIh%0JCJOJQJUaJh%h%6CJOJQJaJh;h%6CJOJQJaJ'h;h%6CJOJQJaJmHsHFd\$gd$a$gd:'&#$+D,gd $ 9r  ]a$gdIi$ 9r  ]a$gd52 , Mexico, 2008 hWQhjMCJaJmHsHhh%< 00&P 1hP:pWQ. 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