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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, 261 ICME 11, Mexico, 2008 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 ICME 11, Mexico, 2008 263 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 1n 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 265 ICME 11, Mexico, 2008 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] ICME 11, Mexico, 2008 267