Mathematics as a Creative Art

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Mathematics as a Creative Art

• Scott Kim
• Iolani School February 4, 2008

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Whats missing

• English mechanics grammar
• English creative writing original work
• Math mechanics algorithms
• Math creative ?

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Mathematics as a creative art

• Creating original, expressive work
• Like art or writing
• Like math research

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But how can students create math?

• Original creative work
• Engaging, meaningful
• Every student succeeds

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Answer Puzzles

• Puzzles are math made fun
• Something for every ability
• Many opportunities for creativity

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Math Fairs (mathfair.com)

• Students build puzzles
• Add themes of their own
• Present in fair-like setting

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Game Clubs (thinkfun.com)

• Pack of 6 puzzles, 4 copies each
• Students work at stations
• Reflect on strategies
• Teacher manuals

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How do you create puzzles?

• Where do you get started?
• How do you get new ideas?
• What makes a good puzzle?

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I design puzzles
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Railroad Rush Hour

• Published by ThinkFun
• Sequel to Rush Hour I designed
• Rush Hour Extravaganza is a Game Club pack

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1. Compose challenges

• Work backwards
• Add pieces to board
• Compose sequence easy to hard

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2. Change presentation

• Change size
• Change appearance
• Change story

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3. Vary rules

• Vary board size
• Vary pieces
• Vary goal

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Summary

• Creative math puzzles
• Math Fairs, Game Clubs
• 1. Compose challenges
• 2. Change presentation
• 3. Vary rules

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Thank You
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Thank You

• 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

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Outline

• What makes a good puzzle?

Inventing new puzzles
Adapting old puzzles
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WHAT MAKES A GOOD PUZZLE?

• A bit of theory from game design

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1. Definition of Puzzle

• A puzzle is fun and has a right answer.

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1. Definition of Puzzle

• A puzzle is fun and has a right answer.

As opposed to everyday problems
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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)
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2. Medium

• Spoken
• Paper and pencil
• Manipulative
• Computer

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3. Goal

• Put together
• Take apart
• Fill in the blanks
• Matching
• Get from here to there
• Unscramble order
• Satisfy conditions

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4. Parts
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4. Parts
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4. Parts
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4. Parts
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4. Parts
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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)

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ADAPTING OLD PUZZLES

• You too can invent puzzles

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5 levels of originality

• 1. Present
• 2. Adapt
• 3. Compose
• 4. Vary
• 5. Invent

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1. PRESENT

• an old puzzle

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Presenting a puzzle

• Play lots of puzzles
• Choose a puzzle
• Present it to someone else
• Offer hints as needed

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2. REVAMP

• an old puzzle in a new way

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Revamp appearance
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Revamp theme
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Revamp context
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3. COMPOSE

• within an existing form

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Sudoku

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

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Sudoku

• Fill in solution
• Remove numbers
• Solve it
• Unique answer?

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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

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Make a sequence easy to hard
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Groups of Levels
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Levels

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

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Tangrams
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Logic puzzles
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Rush Hour
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Rush Hour
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Rush Hour
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Rush Hour
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Rush Hour
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Rush Hour
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Rush Hour
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Rush Hour
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4. VARY

• an existing game

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Start with an existing game
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Vary difficulty
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Vary scale
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Vary scale
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Vary size
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Vary the rules
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Vary the medium
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INVENTING NEW PUZZLES

• Creative mathematics

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Getting started

• Art doodle
• Writing what have I experienced?
• Machines what is needed?
• Music what do I care about?

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Asking the right question

• 1. What can this do?
• 2. Whats wrong?
• 3. Whats the question?
• 4. How can I generalize?
• 5. What happens if?
• 6. How can I make this fun?

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1. What can this do?
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1. What can this do?

• Hint Its not the letter L.

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1. What can this do?

• Answer The letter F.

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1. What can this do?
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2. Whats wrong?

• with Sudoku?

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2. Whats wrong?

• Too abstractmake it physical

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2. Whats wrong?

• Too much timemake it smaller
• Too repetitivemake shaped regions

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2. Whats wrong?
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3. Whats the question?

• The word TEN is made of 9 sticks.
• Thats the answer. Whats question?

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3. Whats the question?

• Remove six matches and leave ten.

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4. How can I generalize?

• Place 8 queens so none attack each other

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4. How can I generalize?
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4. How can I generalize?

• What about 9 queens?
• What about other chess pieces?
• What about other size boards?
• What if queen attacks n others?

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4. How can I generalize?

• Queens are in pairs
• Each pair uses up 3 ormore rows/columns
• 16 rows/columns
• 16/35.33 pairs
• Therefore, maximum queens10

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4. How can I generalize?
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5. What happens if?

• Roll the ball to the end of the maze.

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5. What happens if?

• What if there were 2 balls instead of 1?

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5. What happens if?
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6. How can I make it fun?

• Marcy Cook tiles

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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)

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Example Mind Games in Discover

• Monthly puzzle for science magazine
• One page, three puzzles
• About a topic in science or math

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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

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Topic Manipulatives
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Topic Manipulatives

• 1. Choose a topic
• Cuisenaire Rods Pattern Blocks Geoboards

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Hands-On Math (Dec 2002)

• 1. Choose a topic
• 2. Make it work in print
• Build the figure with the ten rods

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Hands-On Math (Dec 2002)

• 1. Choose a topic
• 2. Make it work in print
• Rods 10

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Hands-On Math (Dec 2002)

• 1. Choose a topic
• 2. Make it work in print
• Rods 10, 9

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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

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Hands-On Math (Dec 2002)

• 1. Choose a topic
• 2. Make it work in print
• 3. Make a range of difficulties

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Hands-On Math (Dec 2002)

• 1. Choose a topic
• 2. Make it work in print
• 3. Make a range of difficulties

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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

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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

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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

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SUMMARY
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Summary
Inventing new puzzles What can this do? Whats
wrong? Whats the question? How can I
generalize? What happens if? How can it be fun?

• What makes a good puzzle? 
• Definition
• Medium
• Goal
• Parts
• Rubric

Adapting old puzzles 1. Present 2. Revamp 3.
Compose 4. Vary 5. Invent
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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

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Whats missing

• Goal of math education is literacy
• Literacy grammar writing
• Whats missing creative math

Mechanics
Creative
Grammar
Writing
English
Computing
?
Math