1. Mathematics as a Creative Art Scott Kim Iolani School February 4, 2008

2. What’s missing ¥ English mechanics = grammar ¥ English creative = writing original work ¥ Math mechanics = algorithms ¥ Math creative = ?

3. Mathematics as a creative art ¥ Creating original, expressive work ¥ Like art or writing ¥ Like math research

4. But how can students create math? ¥ Original creative work ¥ Engaging, meaningful ¥ Every student succeeds

5. Answer: Puzzles ¥ Puzzles are math made fun ¥ Something for every ability ¥ Many opportunities for creativity

6. Math Fairs (mathfair.com) ¥ Students build puzzles ¥ Add themes of their own ¥ Present in fair-like setting

7. Game Clubs (thinkfun.com) ¥ Pack of 6 puzzles, 4 copies each ¥ Students work at stations ¥ Reflect on strategies ¥ Teacher manuals

8. How do you create puzzles? ¥ Where do you get started? ¥ How do you get new ideas? ¥ What makes a good puzzle?

9. I design puzzles

10. Railroad Rush Hour ¥ Published by ThinkFun ¥ Sequel to Rush Hour I designed ¥ Rush Hour Extravaganza is a Game Club pack

11. 1. Compose challenges ¥ Work backwards ¥ Add pieces to board ¥ Compose sequence easy to hard

12. 2. Change presentation ¥ Change size ¥ Change appearance ¥ Change story

13. 3. Vary rules ¥ Vary board size ¥ Vary pieces ¥ Vary goal

14. Summary ¥ Creative math = puzzles ¥ Math Fairs, Game Clubs ¥ 1. Compose challenges ¥ 2. Change presentation ¥ 3. Vary rules

15. Thank You

16. ¥ Exploring Math Through Puzzles (keypress.com) ¥ Brainteasers Page-a-day Calendar (pageaday.com) ¥ Discover Magazine (discovermagazine.com) ¥ Railroad Rush Hour, Sudoku 5x5 (thinkfun.com) ¥ ThinkFun teacher guides (puzzles.com) ¥ Math dance (mathdance.org) ¥ scottkim.com shufflebrain.com

17. Outline What makes a good puzzle? Inventing new puzzles Adapting old puzzles

18. W HAT M AKES A G OOD P UZZLE ? A bit of theory from game design

19. 1. Definition of “Puzzle” A puzzle is fun and has a right answer.

20. 1. Definition of “Puzzle” A puzzle is fun and has a right answer. As opposed to everyday “problems”

21. 1. Definition of “Puzzle” A puzzle is fun and has a right answer. As opposed to everyday “problems” As opposed to a game (no answer) or a toy (no goal)

22. 2. Medium ¥ Spoken ¥ Paper and pencil ¥ Manipulative ¥ Computer

23. 3. Goal ¥ Put together ¥ Take apart ¥ Fill in the blanks ¥ Matching ¥ Get from here to there ¥ Unscramble order ¥ Satisfy conditions

24. 4. Parts

29. 5. Rubric ¥ Attractive (familiar, intriguing) ¥ Simple rules (harmonious, few pieces) ¥ Fun to play (manipulative, unfamiliar) ¥ Good feedback (sense of progress) ¥ Clear goal (pleasing, checkable) ¥ Solvable (deducible, maybe unique)

30. A DAPTING O LD P UZZLES You too can invent puzzles

31. 5 levels of originality 1. Present 2. Adapt 3. Compose 4. Vary 5. Invent

32. 1. PRESENT …an old puzzle

33. Presenting a puzzle ¥ Play lots of puzzles ¥ Choose a puzzle ¥ Present it to someone else ¥ Offer hints as needed

34. 2. REVAMP …an old puzzle in a new way

35. Revamp appearance

36. Revamp theme

37. Revamp context

38. 3. COMPOSE …within an existing form

39. Sudoku ¥ Fill the grid so every row, column and outlined region contains the numbers 1 to 5.

40. Sudoku ¥ Fill in solution ¥ Remove numbers ¥ Solve it ¥ Unique answer?

41. Sudoku — Lessons Learned ¥ Fewer numbers = harder (usually) ¥ May be more than one solution ¥ May be no solution at all ¥ Better if the puzzle has a theme

42. Make a sequence: easy to hard

43. Groups of Levels

44. Levels ¥ Go from easy to hard ¥ Common in computer games ¥ Help player learn the game ¥ Levels work in physical games too

45. Tangrams

46. Logic puzzles

47. Rush Hour

55. 4. VARY …an existing game

56. Start with an existing game

57. Vary difficulty

58. Vary scale

60. Vary size

61. Vary the rules

62. Vary the medium

63. I NVENTING N EW P UZZLES Creative mathematics

64. Getting started ¥ Art — doodle ¥ Writing — what have I experienced? ¥ Machines — what is needed? ¥ Music — what do I care about?

65. Asking the right question 1. What can this do? 2. What’s wrong? 3. What’s the question? 4. How can I generalize? 5. What happens if? 6. How can I make this fun?

66. 1. What can this do?

67. Hint: It’s not the letter L.

68. 1. What can this do? Answer: The letter F.

69. 1. What can this do?

70. 2. What’s wrong? …with Sudoku?

71. 2. What’s wrong? Too abstract…make it physical

72. 2. What’s wrong? Too much time…make it smaller Too repetitive…make shaped regions

73. 2. What’s wrong?

74. 3. What’s the question? ¥ The word TEN is made of 9 sticks. ¥ That’s the answer. What’s question?

75. 3. What’s the question? ¥ Remove six matches and leave ten.

76. 4. How can I generalize? Place 8 queens so none attack each other

77. 4. How can I generalize?

78. ¥ What about 9 queens? ¥ What about other chess pieces? ¥ What about other size boards? ¥ What if queen attacks n others?

79. 4. How can I generalize? ¥ Queens are in pairs ¥ Each pair uses up 3 or more rows/columns ¥ 16 rows/columns ¥ 16/3=5.33 pairs ¥ Therefore, maximum queens=10

80. 4. How can I generalize?

81. 5. What happens if? Roll the ball to the end of the maze.

82. 5. What happens if? What if there were 2 balls instead of 1?

83. 5. What happens if?

84. 6. How can I make it fun? Marcy Cook tiles

85. 6. How can I make it fun? ¥ Attractive (tiles) ¥ Simple rules (place all the tiles) ¥ Fun to play (moving tiles) ¥ Good feedback (use every tile) ¥ Clear goal (use all ten digits) ¥ Solvable (unique solution)

86. Example: Mind Games in Discover ¥ Monthly puzzle for science magazine ¥ One page, three puzzles ¥ About a topic in science or math

87. Creative Process 1. Choose a topic 2. Make it work in print 3. Make a range of difficulties 4. Address a range of thinking styles

88. Topic: Manipulatives

89. 1. Choose a topic Cuisenaire Rods Pattern Blocks Geoboards Topic: Manipulatives

90. Hands-On Math (Dec 2002) 1. Choose a topic 2. Make it work in print Build the figure with the ten rods

91. Hands-On Math (Dec 2002) 1. Choose a topic 2. Make it work in print Rods: 10

92. Hands-On Math (Dec 2002) 1. Choose a topic 2. Make it work in print Rods: 10, 9

93. Hands-On Math (Dec 2002) 1. Choose a topic 2. Make it work in print Rods: 10, 9, 8, 7, 6, 5, 4, 3, 2, 1

94. Hands-On Math (Dec 2002) 1. Choose a topic 2. Make it work in print 3. Make a range of difficulties

95. Hands-On Math (Dec 2002) 1. Choose a topic 2. Make it work in print 3. Make a range of difficulties

96. Hands-On Math (Dec 2002) 1. Choose a topic 2. Make it work in print 3. Make a range of difficulties 4. Address a range of thinking styles Numerical Spatial Logical

97. Hands-On Math (Dec 2002) 1. Choose a topic 2. Make it work in print 3. Make a range of difficulties 4. Address a range of thinking styles

98. Hands-On Math (Dec 2002) 1. Choose a topic 2. Make it work in print 3. Make a range of difficulties 4. Address a range of thinking styles

99. S UMMARY

100. Summary What makes a good puzzle? Definition Medium Goal Parts Rubric Adapting old puzzles 1. Present 2. Revamp 3. Compose 4. Vary 5. Invent Inventing new puzzles What can this do? What’s wrong? What’s the question? How can I generalize? What happens if? How can it be fun?

101. Puzzles = art form “A good puzzle can give you all the pleasures of being duped that a mystery story can. It has surface innocence, surprise, the revelation of a concealed meaning, and the catharsis of solution.” — Stephen Sondheim

102. What’s missing ¥ Goal of math education is literacy ¥ Literacy = grammar + writing ¥ What’s missing: creative math MechanicsCreative Grammar Writing Computing ? English Math