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2014Intel ISEF Educator AcademyLos Angeles, CA
Paul J. KarafiolWalter Payton College Prep, ChicagoMay 13, 2014
Math Circles and Math Symposia: Levers to Engage Mathematical Creativity
A full bus holds 80 passengers and uses 0.72 liters of fuel to travel 1 km. How much fuel will it take to take 190 passengers on a trip of 12.2 km?
The Problem
What’s wrong?
• Purely numerical
• No general principles
• Fundamentally uninteresting
It’s about computation, not math.
“Saying that mathematics is about numbers is like saying that
Shakespeare is about words.”
Danny Stoll
Solution 1: Math Circles• Out-of-school enrichment
• Emphasizes learning by problem-solving and exploration
• Requires justification and explanation, not just number answers
• Origin in Eastern Europe; spread to United States via the coasts in 1990’s-2000’s.
Payton Citywide Math Circle• Established 2010
• Meets 14x/year (every 2-3 weeks)
• Serves 160 students in grades 5-12
• Free (donation-supported)
• Staffed by local teachers
Math Circle FinancesPer-session
Nametags $ 30
Presenter stipends $400
Building costs $350
Copying, etc. $ 50
Annual Expenses
Sessions $11,620
Guest Speaker $1,000
Web Hosting $100
Practical Issues
• Selecting students• Self-select• Handwritten “Why I want to do math on Saturdays”
• Attendance and Record-keeping• Custom, student-built software
• Grouping students into classes• Class sequences not topical• By mathematical level + interest, not age
• Administrative/housekeeping• Need 1-3 adults + volunteer students every week
Problems and Solutions
• Discipline and Removing students• Students who miss two classes are automatically removed.
• Conversations with individual students and parents are very effective.
• Need to teach professors classroom management
• No-show teachers• Assigned “assassin” for every week.
• Backup adults present at every session.
• Use ARML Power, Math Circle books for session ideas.
Ongoing Issues
• Teacher burnout• Need to keep recruiting teachers
• Need to rotate teachers through groups
• Fundraising
• Curriculum adjustments
Surprising Discoveries
• The supply of 5th-8th graders is nearly infinite.
• Kids who are signed up don’t always want to be there.
• University professors love doing math circle sessions (but need training).
• Pre-service teachers can lead sessions (and improve their practice).
Other Benefits
• Community of mathematically-minded teachers
• “Spinoff” math circles
• New school programs, e.g., to promote geometry in elementary school.
Key ResourcesSam Vandervelde, Circle in a Box
Zvezdelina Stankowa et al, A Decade of the Berkeley Math Circle
Ok, but Math Research?
Two problems:
1.Non-mathematicians define research as a set of experimental trials.
2.Mathematicians define research as tightly-focused proofs of theorems in peer-reviewed journals.
Neither group judges student mathematical work appropriately.
So what is Math Research?
Careful study that is done to find and report new knowledge about something.
“New knowledge” can be “new to you”, not “new to the world.”
Mathematical research is special:• Describe patterns
• Justify them logically
These are things we want all students to do!
QED: Our Own Forum1. Teach teachers how to develop
mathematical inquiry
2. Solicit projects from students in grades 5-12
3. Recruit judges from local colleges and universities
Results• QED affiliated to Chicago
Science fair as a “regional” fair
• 14+ math projects at City Science Fair
• Students report that math research was the most important experience in math class.
Research Titles
Middle GradesWhich route is most likely to get me to school on time?
How many permutations are there in “X” number of pitches?
Cats, Dogs, and Mice: Input-Output Transformations
Upper GradesThe Secretarial Challenge: A Generalization
Least solutions to Pell’s Equation for Integers Near Perfect Squares
The Math Behind the ENI Puzzle
Game Theory and Tablet Pricing
(Almost) Any Question Can Be a Research Question
For what numbers of passengers are buses more efficient than cars?
In how many orders can a pile of papers be generated, if papers are added to and removed from the top only?
What’s the probability of a .500 batter actually hitting half the time?
When is the square of a sum equal to the sum of the cubes?
Research Topics
Number Theory
Game Theory
Combinatorics
Probability
Geometry & Trigonometry
Graph Theory
Abstract Algebra Applications
Problem Sources
Generalizations & Extensions• Of classroom problems• Of contest problems
(AMC, AIME, Olympiads)
Math Puzzle Books
Martin GardnerIan StewartJournals
Math HorizonsMathematical IntelligencerMathematics Magazine
Nuts and BoltsAnnual budget $3000• Trophies and giveaways (math books)• Keynote speaker stipend• Food for participants in symposium & PD• Website• Building costs
One-year cycle• Spring-summer: train teachers• Fall: recruit teachers, judges, and
projects• Winter: present and compete
Professional Development
Three foci:
1.Make classroom mathematics more exploratory. Teach kids to ask “What if?” and “Why?”
2.Extending classroom practices into research projects
3.Nuts and Bolts of QED
Challenges
• Balancing activities for students with unstructured time to observe projects.
• Finding and training judges• Calibrating judging rubrics• Creating rubrics
• City rubric still doesn’t suit math projects
• Our rubric feels idiosyncratic
Lessons Learned
• Collect senior projects ahead of time.
• Teach students how to present their work.
• Judge projects more than once; have conflict-resolution round.
• Recruit local organizations/companies to bring people, giveaways.
Payoffs
• Increased student involvement in mathematics.
• Increased student involvement in research.
• Increased college/university involvement in developing young mathematicians.
• Changes in teaching practice for mentors.
27
Mentoring
Not like classroom teachingChanges practice
• Away from telling kids what math to do.• Towards asking engaging questions and presenting counterexamples.
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