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

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

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

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

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

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

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

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

I design puzzles

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

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

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

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

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

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

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

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

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

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

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)

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

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

4. Parts

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)

A DAPTING O LD P UZZLES You too can invent puzzles

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

1. PRESENT …an old puzzle

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

2. REVAMP …an old puzzle in a new way

Revamp appearance

Revamp theme

Revamp context

3. COMPOSE …within an existing form

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

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

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

Make a sequence: easy to hard

Groups of Levels

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

Tangrams

Logic puzzles

Rush Hour

4. VARY …an existing game

Start with an existing game

Vary difficulty

Vary scale

Vary size

Vary the rules

Vary the medium

I NVENTING N EW P UZZLES Creative mathematics

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

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?

1. What can this do?

Hint: It’s not the letter L.

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

1. What can this do?

2. What’s wrong? …with Sudoku?

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

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

2. What’s wrong?

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

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

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

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?

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

4. How can I generalize?

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

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

5. What happens if?

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

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)

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

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

Topic: Manipulatives

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

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

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

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

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

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

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

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

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

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

S UMMARY

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?

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

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