Project-Based Mathematics Learning through LOGO Programming Activities : Korean Experience

1. MTC 2011(AMS) Project-Based Mathematics Learning through LOGO Programming Activities : Korean Experience Hee-chan Lew Korea National University of Education 2011.6.2 Singapore

2. Contents • Background of This Lecture • The Characteristics of LOGO for Project-based mathematics learning • Concepts and Characteristics • Role for mathematics education • Project-based Mathematics Learning through LOGO activities • Overview & Methodology • Tasks & Activities • Results • Conclusion

3. Background

4. Background • LOGO was developed 30 years ago, in 1980 by S. Papert. It is very old software. • But, why today? • It is a good environment suitable for today’s conference theme: Communication, Reasoning & Connections • communication among teachers and students • inductive & deductive reasoning • connection between mathematics and other subjects like art, geometry and algebra, and old action(thinking) and new action(thinking)

5. Background • Why LOGO was chosen as my research topic? • It is a good environment for pursuing mathematics educational objectives Korean Government has emphasized currently. • For example, mathematics education policy issued by Korean MEST, May 17, 2011. • Emphasizing connection between mathematics and other subjects like Science, Technology, Engineering, and Art. (STEAM) • Inducing positive image of students and public toward mathematics by designing more interesting action-based mathematics education

6. Background • Training creative manpower through integrated thinking/reasoning and problem solving ability • Constructing democratic society through reasonable communication in mathematics classrooms • Today, I will introduce LOGO as a good environment suitable for current common concerning of mathematics education of two countries: Koreaand Singapore. • Particularly, I will introduce LOGO activities used for project-based mathematics learning as a good methodology for communication, reasoning & connections

7. Background • Conclusively I believed that LOGO combined with project-based mathematics learning environment can strengthen Communication, Reasoning, Connection and eventually can develop positive attitude of mathematics.

8. LOGO as an environment for project-based mathematics learning

9. Concepts of LOGO • LOGO is a “math-land” for the connection of abstractness and concreteness: LOGO is a good place to make students understand abstract concepts like angle, length, variables, functions through concrete activities and to make these concrete activities be a matrix of further higher abstract concepts. • LOGO is a micro-world to provoke a “mind-storm”: In the micro-world, students can enjoy “thinking” freely. LOGO is a language for designing thinking • To carry out these concepts, LOGO has four kinds of special characteristics.

10. Characteristics of LOGO 1 • First, LOGO is closely related to students' actions: body-syntonic. The movement of a turtle implemented by a LOGO command is matched easily to students' actions in the thinking level. Student’s Drawing Action LOGOCommand

11. Characteristics of LOGO 1 • FD 50 RT 90 • FD 50 RT 90 • FD 50 RT 90 • FD 50 RT 90 • or • Repeat 4 [FD 50 RT 90] • For example, when the above command is given, even students who are less experienced with computer can imagine easily the result is a square with the length of 50. • And, when the following square with the length of 50 is given, students can make an above command for the square easily.

12. Characteristics of LOGO 2 • Second, LOGO is a "procedural" language. That is, once a program is written, LOGO can store it as a "procedure" or a “name” to use it in the future with basic commands for another programming task. • To Triangle • Repeat 3 [FD 100 RT 120] • End • To Square • Repeat 4 [FD 100 RT 90] • End • To House • Square FD 100 RT 30 • Triangle LT 30 BK 100 • End

13. Characteristics of LOGO 2 • Thus, a “structured” programming is possible in LOGO: A whole programming process can be divided into several functional units and each unit can be analyzed independently. This method makes the programming process much easier and makes debugging process simpler. • To flower • Repeat 8 [petal RT 45] • End • To petal • Repeat 2 [repeat 9 [FD 5 RT 10] RT 90] • End

14. Characteristics of LOGO 3 • LOGOis a mathematical language. That is, in the programming process, turtle movement or the shapes turtle makes is determined by the values of the variables. • To Triangle :X • Repeat 3 [FD :X RT 120] • End • Triangle 100 Triangle 80 Triangle 50

15. Characteristics of LOGO 3 • TO POLY :SIDE :MULTI • REPEAT :SIDE [FD 100 RT 360 * :MULTI / :SIDE] • END • POLY 7 2 POLY10 3 POLY 12 5 • More “broad” mathematical concepts of variables and functions can be easily and naturally learned in LOGO environment. Many students think that the domain and co-domain of function should be a real number only.

16. Characteristics of LOGO 4 • LOGO is a "recursive" language. That is, one procedure can be used as a command in itself. Recursion sets up a never-ending process in its character. • To POLY1 • FD 100 RT 72 • POLY1 • End • Continuous Repeat of [FD 100 RT 72] • It is a pentagon!!

17. Characteristics of LOGO 4 • A recursion is particularly able to evoke an excited response because the idea of "going on forever" touches on every child's fantasy and makes children feel like mathematicians. • TO STAR :X :R :N • IF :N = 0 [STOP] • LT 126 • REPEAT 5 [FD :X LT 18 STAR :X * :R :R :N - 1 RT 18 RT 144 FD :X LT 72] • RT 126 • END • STAR 30 0.4 1 STAR 30 0.4 2 STAR 30 0.4 3 STAR 30 0.4 4 STAR 30 0.4 5

18. Role of LOGO for mathematics education • What needs to be noted here is a role of LOGO to make children "reflect on" their own thinking. • Because it is body-syntonic, procedural, mathematical, and recursive, LOGO can give an easy and natural environment that encourages children • to be aware of their actions, • to analyze or criticize them, • to generalize them • to control them • to synthesize some actions performed previously, • In reflecting their own thinking, communication, reasoning and connection can be strengthened.

19. Piaget theory • Reflecting on their own thinking is same as the reflective abstraction clarified by Piaget. • According to Piaget, intellectual development is a continuous process of this reflective abstraction. Thus, what is the most important in mathematics education is how to organize an educational environment to evoke the reflective abstraction. • Papert believes that, like an environment to learn the mother tongue, LOGO is a micro-world which is called as “Math-land” in which children can learn mathematics naturally and spontaneously based on reflecting their own thinking.

20. Polya-style problem solving • Programming in LOGO can be considered as a Polya-style problem solving process itself. It passes steps of understanding, planning, carrying out, and looking back. • LOGO can provide students with a natural environment for improving problem solving ability. • Problem solving strategies like "to subdivide" and "to relate to the already known facts" can be exercised in a natural setting. • Particularly, by reflecting on their own planning process and results, meta-cognition or managerial skills can be fostered.

21. Poincare’s mathematical esthetic sense • In LOGO environment, students' intuitive and creative thinking can be improved by training them to grasp the situations in their own 'eyes'. • As a result, an improvement in attitude toward mathematics and mathematics education can be expected. • According to Poincare, the role of mathematics education is to train students’ mathematical attitude, their own mathematical intuition and their own mathematical esthetic sense.

22. Synthesis Papert = Piaget + Polya + Poinacare LOGO = reflecting on thinking + problem solving + mathematical esthetic sense = communication, reasoning and connection

23. Project-Based Mathematics Learning through LOGO Programming Activities

24. Overview and Methodology • This study was undertaken with three sixth grade students chosen at the Education Center for the Scientifically Gifted of the Seoul National University of Education, and one a fifth grade student chosen from a group of 20 high ranked students in mathematics at the Education Center for the scientifically gifted of the Gang-Nam District Office of Education in Seoul. • The four students made two groups of two students. • The students learned the basic MSWLOGO through various project-based tasks during a total of 12 experimental classes • In the first 2 classes, the students learned the basic commands and defined the procedures to draw several diagrams.

25. Overview and Methodology • In the 3rd class, each group was asked to determine the theme for project which will be done in the 10th to 12th classes and to plan how to design and how to approach to the final product. • In the 3th to 9th classes they cooperated to accomplish various tasks given by teachers and selected their own diagrams required for the project for each group and built up programming of the diagrams. • In the 10th to 12th classes, students integrated or modified several diagrams previously made by each group and made new diagrams if necessary.

26. Overview and Methodology • Final result of the group 1

27. Overview and Methodology • Final result of the group 2

28. Project Learning 1st - 9th Classes

29. 1st class • Introduction to MSWLOGO and 3 Kinds of screen: MSWLogo Screen, Editor, Commander • Basic commands • FD, BK, RT, LT • PU, PD, PE • HT, ST, Home, SHOW Pos • CS, CT • FD 50 PU FD 50 PD FD 50

30. 2nd class • Repeat command • Repeat 4 [FD 80 RT 90] • How to make a command to draw a regular triangle, regular Pentagon … a circle (Turtle Journey Theorem) • Repeat 3 [FD 80 RT 120] • Repeat 5 [FD 80 RT 72] • Repeat360 [FD1 RT 1] • Double repeats: Repeat 18 [Repeat 360 [FD 1 RT 1] RT20]

31. 2nd class • Procedure • Square • Triangle • House • Repeat of House • Repeat 36 [house RT 10] • Procedure with variables • Regular n-polygon • Polygon :n • Polygon :x :n • Command for Coloring Blocks

32. 3rd class • Application of Square Procedure(Basic) • Parallelogram, Rectangle

33. 3rd class • Application of Square Procedure(Advanced) • Make Command: Make “Name X • Label Command: Label “Word or sentence

34. 4th class • Application of Triangle Procedure(Basic) • Regular triangle • Tree • Butterfly

35. 4th class • Application of Triangle Procedure(Advanced) • Big House, Rocket, Daisy, Wheel • Moving Turtle: SETXY 90 30, SH 45, SH 0

36. 5th class • Application of Circle Procedure(Basic) • Flower, Translation of circle, Circles with a same center,

37. 5th class • Application of Circle Procedure(Advanced) • Face, Flower, Snowman, Petal

38. 6th class • Recursive Procedure(Basic) • Spiderweb • Maze • Buildings

39. 6th class • Recursive Procedure(Advanced) • Rotation of Square Maze, • Rotation of Flowers • Rotation of Polygons

40. 7th class • Recursive Procedure(Advanced 2) • Line-circle 1, Line-circle 2, Line-circle 3, Line-circle 4 • Dragon

41. 8th class • Fractal • Koch curves • TO KOCH :N :S • IF :N = 0 [FD :S STOP] • KOCH :N - 1 :S / 3 LT 60 • KOCH :N - 1 :S / 3 RT 120 • KOCH :N - 1 :S / 3 LT 60 • KOCH:N - 1 :S / 3 • END • Snowflake • To Snow :n :s • Repeat 3 [Koch RT 120] • end

42. 8th class • Fractal • Sirpinski Triangle • TO CROSS4 :X :R :N • IF :N = 0 [STOP] • REPEAT 3 [FD :X RT 120 CROSS4 :X * :R :R :N - 1] • End • CROSS4 80 0.5 3 CROSS4 80 0.5 4 CROSS4 80 0.5 5 • Tree

43. 9th class Animation To car pd setpencolor [0 0 0] fd 30 rt 90 fd 30 lt 60 fd 30 rt 60 fd 50 rt 60 fd 30 lt 60 fd 30 rt 90 fd 30 rt 90 fd 140 rt 180 fd 30 lt 90 setfloodcolor [255 125 0] pu fd 10 pd fill pu bk 10 pd repeat 360 [fd 1/5 rt 1] rt 90 fd 5 pu lt 90 fd 5 pd setfloodcolor [0 0 0] fill pu bk 10 pd fill pu fd 5 rt 90 bk 5 pd fd 60 lt 90 repeat 360 [fd 1/5 rt 1] rt 90 fd 5 pu lt 90 fd 5 pd setfloodcolor [0 0 0] fill pu bk 10 pd fill pu fd 5 rt 90 bk 5 bk 90 lt 90 pd rt 90 pu fd 50 lt 90 fd 30 pd fd 20 rt 90 fd 40 rt 90 fd 20 rt 90 fd 40 rt 90 pu rt 45 fd 10 setfloodcolor [255 255 255] pd fill pu bk 10 lt 45 bk 30 lt 90 fd 50 rt 90 pd rt 90 end To del setpencolor [255 255 255] setpensize [300 300] pd fd 3 bk 3 setpensize [1 1] end To movie1 repeat 100 [del pu fd 10 lt 90 car] end

44. 9th class • Animation • to delete • setpencolor [255 255 255] • setpensize [125 125] • fd 1 bk 1 • end • to movie :speed • ht • repeat (abs(int(360 / :speed))) [delete rt :speed pinwheel wait 1] • st • end • to pinwheel • setpencolor[255 0 0] • setpensize [2 2] • setfloodcolor [255 125 0] • wheel • fill • end • to wheel • pu fd 30 pd rt 120 • repeat 6 [fd 30 repeat 3 [lt 120 fd 30 lt 120] rt 60] • lt 120 pu bk 30 pd • End

45. The Results of Projective Learning

46. Strategic thinking • In this study, LOGO is incorporated in the dynamic project-based learning that provides students with opportunities to apply and develop their mathematical knowledge and engage in diverse creative activities through the integration of mathematics and art as a positive way to foster higher levels of thinking for gifted students. • Today, I will focus on what kinds of thinking the mathematically gifted elementary students use to plan, implement and debug in the programming as a problem solving process.

47. Analogy Analogy (6th class) How can we draw these two diagrams ?

48. Analogy Students could analogical thinking based on the procedure to draw the following figure. TO POLYGON :S :A :I IF :S>200[STOP] FD :S RT :A POLYGON :S +:I :A :I END POLYGON 5 90 2 POLYGON 5 91 2 POLYGON 5 89 2

49. Generalization Generalization (5th class) Soohyun proposed to input a variable in the vehicle procedure control the size. They decided to change each number to represent distance to the numbers divided 10 multiplied :X They did not change the number to represent angles They made a procedure as a general formula.

50. 9th class