Mathematics as a Creative Art

Scott KimIolani 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

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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

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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

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

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WHAT MAKES A GOOD PUZZLE?

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

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4. Parts

4. Parts

4. Parts

4. Parts

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)

ADAPTING OLD PUZZLES

You too can invent puzzles

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5 levels of originality

1. Present2. Adapt3. Compose4. Vary5. 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

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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

Rush Hour

Rush Hour

Rush Hour

Rush Hour

Rush Hour

Rush Hour

Rush Hour

4. VARY

…an existing game

Start with an existing game

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Vary difficulty

Vary scale

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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?

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?

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?

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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?

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 ormore 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 topic2. Make it work in print3. 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 topic2. Make it work in print

Build the figure with the ten rods

Hands-On Math (Dec 2002)

1. Choose a topic2. Make it work in print

Rods: 10

Hands-On Math (Dec 2002)

1. Choose a topic2. Make it work in print

Rods: 10, 9

Hands-On Math (Dec 2002)

1. Choose a topic2. Make it work in print

Rods: 10, 9, 8, 7, 6, 5, 4, 3, 2, 1

Hands-On Math (Dec 2002)

1. Choose a topic2. Make it work in print3. Make a range of difficulties

Hands-On Math (Dec 2002)

1. Choose a topic2. Make it work in print3. Make a range of difficulties

Hands-On Math (Dec 2002)

1. Choose a topic2. Make it work in print3. Make a range of difficulties4. Address a range of thinking styles

Numerical Spatial Logical

Hands-On Math (Dec 2002)

1. Choose a topic2. Make it work in print3. Make a range of difficulties

4. Address a range of thinking styles

Hands-On Math (Dec 2002)

1. Choose a topic2. Make it work in print3. Make a range of difficulties

4. Address a range of thinking styles

SUMMARY

Summary

What makes a good puzzle? 

Definition

Medium

Goal

Parts

Rubric

Adapting old

puzzles

1. Present2. Revamp3. Compose4. Vary5. 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?

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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

What’s missing

Goal of math education is literacy

Literacy = grammar + writing

What’s missing: creative math

Mechanics

Creative

Grammar Writing

Computing

?

English

Math