Download What is Creative Problem Solving in Mathematics?

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
WHAT IS THE CREATIVE PROBLEM SOLVING
IN MATHEMATICS?
KIM SOO HWAN
Abstract: The purpose of this paper is to show you some cases of
creative problem solving by talented students in mathematics, and
analyse the characteristics of their problem solving. To promote
the creative problem solving in mathematics, I recommend some
critical points discovered through my experiences of gifted
education in mathematics for 10 years. It is important to make a
reasonable deduction by such creative hypothesis as “half of the
legs in the problem of chicken and rabbit”, to find an original
pattern by intuition and insight for such open-ended task as
Pascal’s triangle, and to have an experience of persuasive
assertion by mathematical modelling in which geometry problem is
transformed to fraction problem in making a hexagon with pattern
blocks.
Key words: Creative Problem Solving, Creative Hypothesis,
Intuition and Insight, Mathematical Modelling
INTRODUCTION
We have made an effort to realize a gifted education in mathematics, science,
and information technology. The Center for Science Gifted Education (CSGE) of
Chongju National University of Education was established in 1998 with the
financial support of the National Science Foundation and Ministry of Science
and Technology of Korea. We have executed talented education programs for 5th,
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Kim Soo Hwan
6th, 7th, and 8th graders in CSGE since 1998.
We have made a lot of programs for Super Saturday, Summer School, Winter
School, and Mathematics and Science Gifted Camp in Chongju National
University of Education. Each program is suitable for 90 or 180 minutes of class
time. The types of tasks developed can be divided into experimental, group
discussion, open-ended problem solving, and exposition and problem solving
tasks(Kim, 2003). Let me show you some cases of creative problem solving in
mathematics by talented students, and analyse the characteristics of their
problem solving.
CREATIVE HYPOTHESIS
Chicken and Rabbit Problem
Some are chickens and some are rabbits. I counted 50 legs in all.
How many of the animals are chickens and how many are rabbits?
(Kim, 2005)
1. simultaneous equations (usual 8th graders in Korea)
x: number of chickens
y: number of rabbits
x + y = 18 … (a)
2x + 4y = 50 …(b)
2. If they were standing with Half of their Legs, Then ... (17C Korean
mathematicians)
If all of the animals were standing with half of the number of their legs, then the
number of their legs would be 25. The difference of the number between 25 and
18 is 7 which is the number of rabbits because the numbers of chickens and the
legs of chickens are same in that situation. They solved this problem by creative
hypothesis. This creative hypothesis can be identified by the following
equations:
(b)/2 : x + 2y = 25 … (c)
(c) – (a) : y = 7 x = 11
3. If all of them were rabbits, then ... (some 5th graders in CSGE)
If all of the animals were rabbits, then the number of their legs would be 72.
262 DG 9: Promoting Creativity for All Students in Mathematics Education, Section 3
What Is the Creative Problem Solving in Mathematics?
72 – 50 = 22
They are 22 more than 50, because they have regarded 11 chickens as 11 rabbits.
x = 11 y = 7
They solved this problem by the creative hypothesis of ”All Rabbits”. It can be
interpreted by Egyptian rule of false position as well. This creative hypothesis
can be identified by the following equations:
(a)×4 – (b) : 2x = 22
x = 11 y = 7
4. If all of them were chickens, then ... (some 5th graders in CSGE)
If all of the animals were chickens, then the number of their legs would be 36.
50 – 36 = 14
They are 14 less than 50, because they have regarded 7 rabbits as 7 chickens.
y = 7, x = 11
They solved this problem by the creative hypothesis of ”All Chicken”or
Egyptian rule of false position as well. This creative hypothesis can be identified
by the following equations:
(b) – (a)×2 : 2y = 14
x = 11 y = 7
INTUITION AND INSIGHT
Pascal’s Triangle
See the table below and find as many rules or patterns in
columns, rows, and diagonals as you can among the numbers.
(Kim, 2001; Tsubota, 1997)
Pascal’s Triangle
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Kim Soo Hwan
1.
binomial coefficients in the binomial theorem (usual 11th graders in
Korea)
n
 a  b n   Cnr a nbn r
r 0
n
 x  1n   Cnr x n
10  1
r 0
n
n
  Cnr 10n Cn010n  C1n10n 1  ...  Cnn
r 0
2.
1 2 1 = 11² (a 5th grader in CSGE)
A really special gifted student’s response is as follows.
1
= 110
1 1
= 11¹
1 2 1
= 11²
1 3 3 1
= 11³
1 4 6 4 1 = 11⁴
He did not know the binomial theorem, but I thought that he must have been
another young Pascal. It can be interpreted as follows.
1
= 110 = (10+1) 0 =1
1 1
= 11¹ = (10+1)¹ = 10 + 1
1 2 1
= 11² = (10+1)² = 10 ² + 2ⅹ10 + 1
1 3 3 1
1 4 6 4
= 11³ = (10+1)³ = 10 ³ + 3ⅹ10 ² + 3ⅹ10 + 1
1
= 11⁴ = (10+1) 4= 104+ 4ⅹ10 ³ + 6ⅹ10 ² + 4ⅹ10
+1
some students’ another responses in CSGE are as follows.

Numbers in the third column are triangular numbers.

Numbers in the seventh column are all multiple of 7, except for 1.

Every number is the sum of the number just above it and the number to its
left.

The arrangement of numbers in each row is symmetrical.

The sum of the numbers in the (n+1)th row is 2ⁿ

The sums of numbers along diagonals drawn upward from the 1’s in the
left column form the “Fibonacci sequence” which is a sequence of
numbers obtained by summing the preceding two terms, starting from 1.
264 DG 9: Promoting Creativity for All Students in Mathematics Education, Section 3
What Is the Creative Problem Solving in Mathematics?
MATHEMATICAL MODELING
Pattern Block
Build the Yellow Hexagon using th e Pattern Block: 250
pieces of six geometric shapes in six colors -50 green
triangle, 25 orange squares, 50 blue parallelograms, 50
tan rhombuses, 50 red trapezoids, and 25 yellow
hexagons .
Can you show that there exist such 6 ways as figures?
(Kim, 2001)
1.
If a hexagon were 1, then... (some 5th graders in CSGE )
Figure1. Making Hexagons
If a hexagon were 1, then a trapezoid would be 1/2, parallelogram 1/3, a triangle
1/6. But squares and tan rhombuses can’t be transformed by some other fractions
within a hexagon, because their combination of angles and diagonals can’t be
transformed to others. Therefore making a hexagon using some figures in pattern
blocks can be transformed to make 1 using 1, 1/2, 1/3, 1/6.
①
②
③
④
⑤
⑥
⑦
⑧
2.
1
1/2+1/2
1/2+1/6+1/6+1/6
1/2+1/3+1/6
1/6+1/6+1/6+1/6+1/6+1/6+1/6
1/3+1/3+1/3
1/3+1/3+1/6+1/6
1/3+1/6+1/6+1/6+1/6
If a hexagon were 6, then... (some 5th graders in CSGE)
If a hexagon were 6, then a trapezoid would be 3, parallelogram 2, a triangle 1.
But squares and tan rhombuses can’t be transformed by some other numbers
within a hexagon, because their combination of angles and diagonals can’t be
transformed to others. Therefore making a hexagon using some figures in pattern
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Kim Soo Hwan
blocks can be transformed to make 6 using 6, 3, 2, 1.
3.
If the number of figures were 1,2,3,4,5,6, then... (some 5th graders in
CSGE)
① 1 haxagon
② 2 trapezoids
③ 3 parallelogram or 1 trapezoid, 1 parallelogram, 1 triangle
④ 1 trapezoid, 3 triangles or 2 parallelograms, 2 triangles
⑤ 1 parallelogram, 4 triangles
⑥ 6 triangles
CONCLUSIONS AND FUTURE WORK
To promote the creative problem solving in mathematics, I recommend some
critical points discovered through my experiences of gifted education in
mathematics for 10 years. Firstly, it is important to make a reasonable deduction
by such creative hypothesis as 17C Korean mathematicians. Some 5th graders in
CSGE chose such hypothesis as ‘all of animals were rabbits, or chicken’. It is
almost the same strategy as Egyptian rule of false position for an algebraic
problem of direct proportion.
Second, it is important to find an original pattern by intuition and insight for
such open-ended task as Pascal’s triangle. Teacher needs to hear a lot of
students’s findings for challengeable good tasks, because he has two ears and
one mouth. We had better develop a lot of Open-Ended Problems, because they
are good tasks for several levels of students in Gifted Education in Mathematics.
Third, it is important to have an experience of persuasive assertion by such
mathematical modeling as a geometry problem transformed to fraction problem
in making a hexagon with pattern blocks. Analogy is good strategy for
mathematical modeling. For example, as the shortest distance between the points
A and B is AB on the straight line in a Euclidean Plane, so is AB on the great
circle in a spehere. So we take a course of great circle sailing in an airplane tour.
REFERENCES
266 DG 9: Promoting Creativity for All Students in Mathematics Education, Section 3
What Is the Creative Problem Solving in Mathematics?
Kim S. H. (2005). A Case Study for Developing the Mathematical Creativity in
CNUE of Korea. Journal of the Korea Society of Mathematical Education
Series D: Research in Mathematical Education 9(2), 175-182.
Kim S. H. (2003). A Study on Developing the Teacher Education Program for
Mathematical Excellence. Journal of the Korea Society of Mathematical
Education Series D: Research in Mathematical Education 7(4), 235-246.
Kim S. H. (2002). A Case Study on Evaluating the Teaching of Mathematics in
Korea. Journal of the Korea Society of Mathematical Education Series D:
Research in Mathematical Education 6(2), 135-143.
Kim S. H. (2001). A Case Study on Gifted Education in Mathematics. Journal
of the Korea Society of Mathematical Education Series D: Research in
Mathematical Education 5(2), 87–98
Kozo Tsubota (1997). Pascal’s triangle. In: J. P. Becker & Shigeru Shimada
(Eds.), The open-ended approach: a new proposal for teaching
mathematics (pp. 76–80). Reston, VA: National Council of Teachers of
Mathematics, Inc. MATHDI 1998b.01596
ABOUT THE AUTHOR
Kim Soo Hwan, Ph.D.
Department of Mathematics Education
Faculty of Mathematics Education
Cheongju National University of Education
330 Chongnamro, Heungdeok-gu, Cheongju, Chungbuk, 361-712
Korea
Cell phone: +82 16 234 0474
Е-mails: [email protected] [email protected]
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