Mathematics as a Creative Art Scott Kim Iolani School February 4, 2008.
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Transcript of Mathematics as a Creative Art Scott Kim Iolani School February 4, 2008.
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Mathematics as a Creative Art
Scott KimIolani School • February 4, 2008
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What’s 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. Present2. Adapt3. Compose4. Vary5. 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. What’s wrong?3. What’s 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: It’s 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. What’s wrong?
…with Sudoku?
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2. What’s wrong?
Too abstract…make it physical
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2. What’s wrong?
Too much time…make it smaller
Too repetitive…make shaped regions
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2. What’s wrong?
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3. What’s the question?
The word TEN is made of 9 sticks.
That’s the answer. What’s question?
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3. What’s 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 16/3=5.33 pairs Therefore, maximum queens=10
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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
![Page 87: Mathematics as a Creative Art Scott Kim Iolani School February 4, 2008.](https://reader030.fdocuments.us/reader030/viewer/2022032705/56649d945503460f94a7b7ba/html5/thumbnails/87.jpg)
Creative Process
1. Choose a topic2. Make it work in print3. Make a range of difficulties
4. Address a range of thinking styles
![Page 88: Mathematics as a Creative Art Scott Kim Iolani School February 4, 2008.](https://reader030.fdocuments.us/reader030/viewer/2022032705/56649d945503460f94a7b7ba/html5/thumbnails/88.jpg)
Topic: Manipulatives
![Page 89: Mathematics as a Creative Art Scott Kim Iolani School February 4, 2008.](https://reader030.fdocuments.us/reader030/viewer/2022032705/56649d945503460f94a7b7ba/html5/thumbnails/89.jpg)
1. Choose a topic
Cuisenaire Rods Pattern Blocks Geoboards
Topic: Manipulatives
![Page 90: Mathematics as a Creative Art Scott Kim Iolani School February 4, 2008.](https://reader030.fdocuments.us/reader030/viewer/2022032705/56649d945503460f94a7b7ba/html5/thumbnails/90.jpg)
Hands-On Math (Dec 2002)
1. Choose a topic2. Make it work in print
Build the figure with the ten rods
![Page 91: Mathematics as a Creative Art Scott Kim Iolani School February 4, 2008.](https://reader030.fdocuments.us/reader030/viewer/2022032705/56649d945503460f94a7b7ba/html5/thumbnails/91.jpg)
Hands-On Math (Dec 2002)
1. Choose a topic2. Make it work in print
Rods: 10
![Page 92: Mathematics as a Creative Art Scott Kim Iolani School February 4, 2008.](https://reader030.fdocuments.us/reader030/viewer/2022032705/56649d945503460f94a7b7ba/html5/thumbnails/92.jpg)
Hands-On Math (Dec 2002)
1. Choose a topic2. Make it work in print
Rods: 10, 9
![Page 93: Mathematics as a Creative Art Scott Kim Iolani School February 4, 2008.](https://reader030.fdocuments.us/reader030/viewer/2022032705/56649d945503460f94a7b7ba/html5/thumbnails/93.jpg)
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
![Page 94: Mathematics as a Creative Art Scott Kim Iolani School February 4, 2008.](https://reader030.fdocuments.us/reader030/viewer/2022032705/56649d945503460f94a7b7ba/html5/thumbnails/94.jpg)
Hands-On Math (Dec 2002)
1. Choose a topic2. Make it work in print3. Make a range of difficulties
![Page 95: Mathematics as a Creative Art Scott Kim Iolani School February 4, 2008.](https://reader030.fdocuments.us/reader030/viewer/2022032705/56649d945503460f94a7b7ba/html5/thumbnails/95.jpg)
Hands-On Math (Dec 2002)
1. Choose a topic2. Make it work in print3. Make a range of difficulties
![Page 96: Mathematics as a Creative Art Scott Kim Iolani School February 4, 2008.](https://reader030.fdocuments.us/reader030/viewer/2022032705/56649d945503460f94a7b7ba/html5/thumbnails/96.jpg)
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
![Page 97: Mathematics as a Creative Art Scott Kim Iolani School February 4, 2008.](https://reader030.fdocuments.us/reader030/viewer/2022032705/56649d945503460f94a7b7ba/html5/thumbnails/97.jpg)
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
![Page 98: Mathematics as a Creative Art Scott Kim Iolani School February 4, 2008.](https://reader030.fdocuments.us/reader030/viewer/2022032705/56649d945503460f94a7b7ba/html5/thumbnails/98.jpg)
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
![Page 99: Mathematics as a Creative Art Scott Kim Iolani School February 4, 2008.](https://reader030.fdocuments.us/reader030/viewer/2022032705/56649d945503460f94a7b7ba/html5/thumbnails/99.jpg)
SUMMARY
![Page 100: Mathematics as a Creative Art Scott Kim Iolani School February 4, 2008.](https://reader030.fdocuments.us/reader030/viewer/2022032705/56649d945503460f94a7b7ba/html5/thumbnails/100.jpg)
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?
QuickTime™ and aTIFF (Uncompressed) decompressor
are needed to see this picture.
![Page 101: Mathematics as a Creative Art Scott Kim Iolani School February 4, 2008.](https://reader030.fdocuments.us/reader030/viewer/2022032705/56649d945503460f94a7b7ba/html5/thumbnails/101.jpg)
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
![Page 102: Mathematics as a Creative Art Scott Kim Iolani School February 4, 2008.](https://reader030.fdocuments.us/reader030/viewer/2022032705/56649d945503460f94a7b7ba/html5/thumbnails/102.jpg)
What’s missing
Goal of math education is literacy
Literacy = grammar + writing
What’s missing: creative math
Mechanics
Creative
Grammar Writing
Computing
?
English
Math