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
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?
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 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 else’s 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:
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Affection the Mathematical Creativity of the Students
with Complexes of Examples of Good Practice
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 ABC with side a is
inscribed. Let the point M is an arbitrary point on the arc
AB and
MA  x, MB  y, MC  z. Prove that :
(1)
z  x y .
Solution 1. (This solution is an example of a constructive way of solving
problems – see Fig. 1.) If we put the segment MA  x on the segment MC  z so
that MD  x , where the point D belongs to the segment MC , it is necessary to
prove that the last segment DC is equal to MB  y . So, the triangle AMD is an
isosceles triangle with an angle of 60 o , i.e. it is an equilateral triangle with
AD  x. Then the triangles ADC , AMB are similar with: AC  AB, MA  AD
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and ADC  AMB  120 0 . Hence DM  MB  y and z  x  y .
Here we realized the first step of the given above chain – “solving problems”.
Figure 1
Solution 2. By using of the famous Ptolemy’s Theorem for the quadrilateral
AMBC we get immediately a.x  a. y  a.z , i.e. z  x  y .
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)
xz  yz  xy  a 2 .
Solution. By using the well-known “method of the areas” we get:
FAMC  FBMC  FAMB  FABC , or
1
3 1
3 1
3
3
xz
 yz
 xy
 a2
,
2
2 2
2 2
4
4
from where (2) follows.
Problem 1.3. Further prove that:
(3)
x 2  y 2  z 2  2a 2 .
Solution. From (1) we get x  y  z 2  0 , i.e.
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Affection the Mathematical Creativity of the Students
with Complexes of Examples of Good Practice
x 2  y 2  z 2  2xz  yz  xy   0 and (3) follows with the help of (2).
Problem 1.4. Prove that:
(4)
x 2  xy  y 2  a 2 .
Solution. The identity (4) is a simple consequence from the Cosine Law for the
triangle AMB , 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)
x2 y 2  y 2 z 2  z 2 x2  a4 .
Solution. By squaring (2) we obtain
x 2 y 2  y 2 z 2  z 2 x 2  2 xyz x  y  z   a 4
and from (1) the identity (5) follows immediately.
Problem 1.6. Prove that:
(6)
x 4  y 4  z 4  2a 4 .
Solution. By squaring (3) and using (5) we obtain consequently:


x 4  y 4  z 4  2 x 2 y 2  y 2 z 2  z 2 x 2  4a 4 , x 4  y 4  z 4  2a 4  4a 4 , i.e.
(6).
The last two problems give to us the steps: “creating new problems” and
“problems posing”.
Problem 1.7. Prove that:
(7)
x 4 y 4  y 4 z 4  z 4 x 4  4axyz 2  a8 .
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Problem 1.8. Prove that:
x8  y8  z8  2a8  8(axyz) 2 .
(8)
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 a, b, c, s, F , R, r of any arbitrary
triangle. The given sums are cyclic.
Problem 2.1.
(9)
(b  c)2 (b  c  a)  0

(10)
a2 (b  c  a)  3abc

(11)
a  b  c  a 2  2 a3  3abc

(12) R  2r (Euler’s Inequality).
Problem 2.2.
(13)
(14)

a(b  c)2 (b  c  a)  0 
s2 
2
3
4




a

4
abc

a

2
ab
 
  
a.
r
( r  4 R) 2 .
2R  r
Problem 2.3.
(15)
(a  b)2 (b  c  a)(c  a  b)  0

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Affection the Mathematical Creativity of the Students
with Complexes of Examples of Good Practice
a2 )2  3abc(a  b  c) 
(16) 2 a 2 . ab  (
(17) s 2 
r
(r  4 R) 2 .
Rr
Problem 2.4.
(18)
c(a  b)2 (b  c  a)(c  a  b)  0
(19)
a2 (a3  6abc)  (a  b  c)a4  6abcab
(20)
s2 


r
(r  4 R)(8R  r ) .
2R  r
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 s, R, r .
Problem 2.5. Prove that the following chain of triangle inequalities exists:
(21)
s 2  16 Rr  5r 2 
r
r
( r  4 R) 2 
(r  4R)(8R  r ) 
Rr
2R  r
 3r r  4R  
r
r  4R2 .
2R  r
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)
a  b  2c2 a  b  c  0

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 a2   ab 5 a3  39abc  (24)
(23) 3a  b  c 
s 2  15Rr  3r 2 .
The equality in (22) holds if a  b  2c  b  c  2a  c  a  2b  0 or
a  b  c.
Corollary. The inequality (24) belongs to the chain:
(25)
s 2  16 Rr  5r 2  15 Rr  3r 2 .
Problem 2.7. The following true triangle inequalities are equivalent:
(26)
a  b  2c2 b  c  ac  a  b  0
(27)
 a4  abca  b  c  2 a2b2


(12) - Euler’s Inequality .
Problem 2.8. The following true triangle inequalities are equivalent:
(28)
ca  b  2c2 a  b  c  0
(29)
9a  b  c 
(30)
s2 

a3  13a4  10a2b2  4abca  b  c

9r
r  4 R 2 .
7r  10 R
Corollary. The last inequality belongs to the chain:
(31)
s2 
r
r  4R 2  9r r  4R 2  3r r  4R  .
rR
7r  10 R
Problem 2.9. The following true triangle inequalities are equivalent:
(32)
a  b2 a  b  2c2 a  b  c  0

a5  7abca2  4a2.a3  ab.a3  4abcab 
(33) 3
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2


ab
   2abc. a  5 a. a4 

a.

2


r R  2r  s 2  3r r  4R   0

 s 2  3r r  4R .
(12)
Corollary. The inequality (32) belongs to the following chain of inequalities:
(34)
s 2  16 Rr  5r 2  15 Rr  3r 2  12 Rr  3r 2 .
Problem 2.10. The following true triangle inequalities are equivalent:
(35)
ca  b  2c2 4a  4b  3c  0
(36)
4
(37)
14s 4  4r 49r  16 Rs 2  18r 2 r  4R2  0 

 a4  8a  b  ca3  26 a2b2  2abca  b  c
7s 2  r 49r  16 R   2r 316 R 2  518 Rr  616 r 2
(38)


r
s 2  16 R  49r  2 316 R 2  518 Rr  616 r 2  ,
7



because 7s 2  7 16 Rr  5r 2  r 49r  16 R ,
8R  7r  .
The irrational inequality (60) is obviously new!
Corollary. We can compare (38) to right inequality (34):
r
3r r  4 R   16 R  49r  2 316 R 2  518 Rr  616 r 2 
7

34 R  14r  316 R 2  518 Rr  616 r 2


R  2r r  4R  0
and then we get the following chain of inequalities:
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(39)
r
s 2  3r r  4 R   16 R  49r  2 316 R 2  518 Rr  616 r 2  .
7

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  Background  Problems  Solutions 
 Extra Education  Methods  Discovering  Motivation 
 Support  Investigations  Research Works
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REFERENCES
Bilchev S. And D.Kontogiannis (1991), A Short and Elementary Proof of
Mushkarov – Simeonov’s Inequality, Hrvatska Akademija Znanosti I
Umjetnosti, Rada 456, Matematičke 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
Českė Budějovice, Pedagogical Faculty, Department of Mathematics Report
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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. (2008), Activating the Interest of Gifted Students to Work in
Mathematics by Obtaining New Results 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
Е-mails: [email protected] [email protected]
206 DG 9: Promoting Creativity for All Students in Mathematics Education, Section 3