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.

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Mentoring

Not like classroom teachingChanges practice

• Away from telling kids what math to do.• Towards asking engaging questions and presenting counterexamples.

Thank you!28

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