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 The more you invent, the deeper your thinking.

But how can students create math? Original creative work Engaging, meaningful Every student succeeds Danger: designing puzzles that are just fancy textbook problems

Answer: Puzzles Puzzles are math made fun Something for every ability Many opportunities for creativity Give you some perspective on why puzzles are valuable.

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? Danger: designing puzzles that are just fancy textbook problems

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

Thank You scottkim.com shufflebrain.com 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

What makes a good puzzle? Adapting old puzzles Inventing new puzzles Outline What makes a good puzzle? Adapting old puzzles Inventing new puzzles Today’s talk. Behind the scenes. How In invented them.

WHAT MAKES A GOOD PUZZLE? It helps to have some language for talking about puzzle design. 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” The key difference. Subjective.

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) Versus other play.

2. Medium Spoken Paper and pencil Manipulative Computer Will Shortz radio puzzle, two pencil, body as manipulative, nonphysical computer puzzles.

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

4. Parts Schematically, a puzzle challenges the player to get from a problem to a solution.

4. Parts But of course the path is never simple. Every puzzle requires that the player make choices, some of which lead to dead ends.

4. Parts Puzzles in a game have a larger situation that gives the puzzle meaning. Applying the solution lets you move forward in the game.

4. Parts Good puzzles have require insight. The insight above is to walk around the outside of the maze. Obscure insights, however, feel unfair.

4. Parts Different puzzles emphasize different parts of the journey. Persistence puzzles are a slow steady climb. Aha! Puzzles skip the climb and go straight to the insight. Story puzzles work the setup into the story. Crossword puzzles are full of little insights: each word unlocks more.

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) Attractive (familiar, intriguing) Simple rules (few elements) Fun to play (novel thought pattern) Good feedback (sense of progress) Clear goal (pleasing, checkable) Solvable (deducible, maybe unique)

You too can invent puzzles ADAPTING OLD PUZZLES Like any creative act. How do you write a song? Paint a picture? Some of it is technique. Some of it is art. To do it well takes work. But to get started is open to everyone. You too can invent puzzles

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

1. PRESENT …an old puzzle The most basic. Like writing new words to a familiar song.

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

…an old puzzle in a new way 2. REVAMP …an old puzzle in a new way The most basic. Like writing new words to a familiar song.

Revamp appearance

Revamp theme

Revamp context

…within an existing form 3. COMPOSE …within an existing form The most basic. Like writing new words to a familiar song.

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

Rush Hour

Rush Hour

Rush Hour

Rush Hour

Rush Hour

Rush Hour

Rush Hour

4. VARY …an existing game

Start with an existing game

Vary difficulty

Vary scale

Vary scale

Vary size

Vary the rules

Vary the medium

INVENTING NEW PUZZLES Creative mathematics

Getting started Art — doodle Writing — what have I experienced? Machines — what is needed? Music — what do I care about? Art -- start with technique, or start with inspiration. Clearly making something. Writing -- write about what you know Machines -- what do people need? How can I do it better? Music -- levels of originality. Change the words, to much deeper. Puzzles are linke songs.

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? 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?

Too abstract…make it physical 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.

Place 8 queens so none attack each other 4. How can I generalize? Place 8 queens so none attack each other I got interested in puzzles by reading Martin Gardner’s column in Scientific American.

4. How can I generalize? I got interested in puzzles by reading Martin Gardner’s column in Scientific American.

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.

What if there were 2 balls instead of 1? 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

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

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

SUMMARY

What makes a good puzzle? Adapting old puzzles Inventing new puzzles 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? Today’s talk. Behind the scenes. How In invented them.

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 Mechanics Creative English Grammar Writing Math Computing ?