Mathematics as a Creative Art Scott Kim Iolani School • February 4, 2008
Dec 22, 2015
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