MAA Intermountain Section Meeting March 24, 2007 PETER E. TRAPA Department of Mathematics University...

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Transcript of MAA Intermountain Section Meeting March 24, 2007 PETER E. TRAPA Department of Mathematics University...

MAA Intermountain Section MeetingMarch 24, 2007

PETER E. TRAPA

Department of MathematicsUniversity of Utah

Organizing a High School Math Circle

University of Utah March 24rd, 2007

What is a Math Circle?Provisional definition:

University of Utah March 24rd, 2007

What is a Math Circle?Provisional definition:

• a program for K-12 students devoted to mathematical training and exploration. Particular emphasis is placed on direct interaction with mathematicians.

University of Utah March 24rd, 2007

What is a Math Circle?Provisional definition:

• a program for K-12 students devoted to mathematical training and exploration. Particular emphasis is placed on direct interaction with mathematicians.

• Typically the methods and materials used are non-standard, at least compared with usual K-12 curriculae.

University of Utah March 24rd, 2007

Examples

Three that I’ll discuss today:

– The Boston Math Circle

– The Berkeley Math Circle

– The Utah Math Circle

University of Utah March 24rd, 2007

Examples

But many others exist, including:

– Davis (UC Davis)– New York (Lehman College, CUNY)– Miami (Florida International U)– Mobile (U of Southern Alabama)– Palo Alto (Castilleja School)– San Diego (UCSD, Art of Problem Solving)– San Franscisco (USF)– San Jose (collaboration with AIM)– St. Louis (Washington U)

University of Utah March 24rd, 2007

Toward a National Network?

MSRI is interested in building andmaintaining a national-level “Math CircleInfrastructure”.

University of Utah March 24rd, 2007

Toward a National Network?

MSRI is interested in buliding andmaintaing a national-level “Math CircleInfrastructure”.

Small “seed grants” from MSRI are now available for starting new Math Circles.

University of Utah March 24rd, 2007

Why all the interest in Math Circles?

University of Utah March 24rd, 2007

Why all the interest in Math Circles?

Typical high school math curriculae are almost exclusively template based.

University of Utah March 24rd, 2007

Why all the interest in Math Circles?

Typical high school math curriculae are almost exclusively template based.

This is extremely effective for some purposes.

University of Utah March 24rd, 2007

Why all the interest in Math Circles?

Typical high school math curriculae are almost exclusively template based.

This is extremely effective for some purposes.

But it’s terrible for others.

University of Utah March 24rd, 2007

Downsides of template-based learning

University of Utah March 24rd, 2007

Downsides of template-based learning

• Encourages very short attention spans.

University of Utah March 24rd, 2007

Downsides of template-based learning

• Encourages very short attention spans.

• Implemented algorithms take precedence over their conceptual formulation.

University of Utah March 24rd, 2007

Downsides of template-based learning

• Encourages very short attention spans.

• Implemented algorithms take precedence over their conceptual formulation.

• Creativity is often limited to the confines of the template.

University of Utah March 24rd, 2007

But this is not what mathematicians do!

University of Utah March 24rd, 2007

But this is not what mathematicians do!

Instead we are trained to think deeply about simple things.

University of Utah March 24rd, 2007

But this is not what mathematicians do!

Instead we are trained to think deeply about simple things.

This idea is completely absent in template-based learning.

University of Utah March 24rd, 2007

A Different Model: The Russian Experience

In Russia, a highly evolved and extremely efficient mathematical training infrastructure exists.

University of Utah March 24rd, 2007

A Different Model: The Russian Experience

In Russia, a highly evolved and extremely efficient mathematical training infrastructure exists.

• Early Identification

• Special Schools

• Math Circles

University of Utah March 24rd, 2007

A Different Model: The Russian Experience

In Russia, a highly evolved and extremely efficient mathematical training infrastructure exists.

• Early Identification

• Special Schools

• Math Circles

The objective is not limited to producing only research mathematicians.

University of Utah March 24rd, 2007

Example (courtesy of Josh Zucker)

University of Utah March 24rd, 2007

Example (courtesy of Josh Zucker)

St. Petersburg’s School 239:

2500 sixth-graders take the district Olympiad.

25 or so are tracked into their own math courseand math circle.

University of Utah March 24rd, 2007

Example: 8th grade algebra class

University of Utah March 24rd, 2007

Example Problems

If it is true that (a2 - 1) x ≥ 3a - 1 for all x, what value(s) of a are possible.

What is an inequality whose solution set will be (-infinity, -2) union [3, infinity). 

Solve |2x - 1| > 3x. 

University of Utah March 24rd, 2007

Example of Math Circle Problems

Vlad draws diagonals of unit squares in an 8x8 grid of squares.  Nikita sees to it that they have no common points (including ends).  What is the maximum number of diagonals that Vlad can draw?

University of Utah March 24rd, 2007

Example of Math Circle Problems

Asterisks are arranged in some squares of an n by n grid.  In each vertical, horizontal, and diagonal (even diagonals of only one square count), the number of asterisks in it is known.  For which n is it possible to determine where the asterisks are?

University of Utah March 24rd, 2007

Can any of this be emulated here?

.

University of Utah March 24rd, 2007

The Boston Math Circle

University of Utah March 24rd, 2007

The Boston Math Circle

• Started in 1994 by math educators Bob and Ellen Kaplan.

University of Utah March 24rd, 2007

The Boston Math Circle

• Started in 1994 by math educators Bob and Ellen Kaplan.

University of Utah March 24rd, 2007

The Boston Math Circle

• Started in 1994 by math educators Bob and Ellen Kaplan.

• No official university affiliation (although resources are provided by Harvard, MIT, Northeastern, Tufts,….)

University of Utah March 24rd, 2007

The Boston Math Circle

• Started in 1994 by math educators Bob and Ellen Kaplan.

• No official university affiliation (althoughresources are provided by Harvard, MIT, Northeastern, Tufts,….)

• Grown into a rather large enterprise

University of Utah March 24rd, 2007

Typical Spring Schedule

• Four pre-algebra courses

– Divided, roughly, into grades 3-4, 5-6

– 10 sessions, one hour each

University of Utah March 24rd, 2007

Typical Spring Schedule

• Four pre-algebra courses

– Divided, roughly, into grades 3-4, 5-6

– 10 sessions, one hour each

• One advanced course

– for high school students

– 10 sessions, three hours each

University of Utah March 24rd, 2007

Approach for younger students

• A “seed” topic is a springboard for open-ended explorations.

University of Utah March 24rd, 2007

Approach for younger students

• A “seed” topic is a springboard for open-ended explorations.

• Sample topics for kids aged 5-10:

– Are there numbers between numbers?– Primes– Infinity

University of Utah March 24rd, 2007

Approach for younger students

• A “seed” topic is a springboard for open-ended explorations.• Sample topics for kids between 5 and 10:

– Are there numbers between numbers?– Primes– Infinity

• Ages 12-14 (some algebra):– complex numbers– Fibonacci sequence– continued fractions

University of Utah March 24rd, 2007

Approach for High School Students

• Almost exclusively devoted to a single topic.

University of Utah March 24rd, 2007

Approach for High School Students

• Almost exclusively devoted to a single topic.• Sample courses:

– Algebraic Geometry – Complex Analysis– Knot Theory – Hyperbolic Geometry – Group Theory – Quantum Mechanics

University of Utah March 24rd, 2007

The Boston Math Circle charges tuition

• Each pre-algebra courses

– $225 tuition

• Each advanced course

– $450 tuition

• About $20/hour per student

University of Utah March 24rd, 2007

To start your own Math Circle…

From the Boston Math Circle webpage:“Want to start your own Math Circle?

Founders Robert and Ellen Kaplan can be

hired as consultants to help get you going.

Feel free to contact Bob and Ellen Kaplan

directly to learn more about what they can do

for you. They are willing to travel to your

locale to help out. Their per-diem fee is

$1000 plus expenses.”

University of Utah March 24rd, 2007

The Berkeley Math Circle

University of Utah March 24rd, 2007

The Berkeley Math Circle

• Founded in 1998 with the support of MSRI (Hugo Rossi, Zvezda Stankova, and others).

University of Utah March 24rd, 2007

The Berkeley Math Circle

• Founded in 1998 with the support of MSRI (Hugo Rossi, Zvezda Stankova, and others).

• Closely followed the Russian-type model.

University of Utah March 24rd, 2007

The Berkeley Math Circle

• Founded in 1998 with the support of MSRI (Hugo Rossi, Zvezda Stankova, and others).

• Closely followed the Russian-type model.

• In particular contests were absolutely central.

University of Utah March 24rd, 2007

The Berkeley Math Circle

• Founded in 1998 with the support of MSRI (Hugo Rossi, Zvezda Stankova, and others).

• Closely followed the Russian-type model.

• In particular contests were absolutely central.

• This led to the creation of the Bay Area Math Olympiad (BAMO).

University of Utah March 24rd, 2007

The Berkeley Math Circle

• Founded in 1998 with the support of MSRI (Hugo Rossi, Zvezda Stankova, and others).

• Closely followed the Russian-type model.

• In particular contests were absolutely central.

• This led to the creation of the Bay Area Math Olympiad (BAMO).

• Draw heavily on faculty from Berkeley, Stanford, UCSF, etc.

University of Utah March 24rd, 2007

The Berkeley Math Circle

• Founded in 1998 with the support of MSRI (Hugo Rossi, Zvezda Stankova, and others).

• Closely followed the Russian-type model.

• In particular contests were absolutely central.

• This led to the creation of the Bay Area Math Olympiad (BAMO).

• Draw heavily on faculty from Berkeley, Stanford, UCSF, etc.

• Free.

University of Utah March 24rd, 2007

BMC: Schedule Excerpt

Ravi Vakil, “The Mathematics of Doodling”

Sam Vandervelde, “Geometric Mad-Libs”

Tom Rike, “Infinite Series”

Zvezda Stankova, “AMC 10/12 Preparation” 

Tom Davis, “BAMO Preparation”

Alexander Givental, “Solving Problems with Vectors” 

Vera Serganova, “Combinatorics and Geometry”

University of Utah March 24rd, 2007

Very Successful!

Although the BAMO program only started in 1999, the 6-member team at the International Math Olympiad (held in Washington, D.C., July 2001) included 3 members from this program:– Gabriel Carrol– Tiankai Liu– Oaz Nir

University of Utah March 24rd, 2007

Footnote: San Francisco Math Circle

Specifically targets students (and teachers) from urban schools who may not have access to after-school enrichment programs

University of Utah March 24rd, 2007

The Utah Math Circle

University of Utah March 24rd, 2007

The Utah Math Circle

Lauched in 2001 as part of Utah’s first VIGRE grant.

Hugo Rossi had returned from MSRI and his experience with the Berkeley Math Circle was instrumental.

University of Utah March 24rd, 2007

So We Started

University of Utah March 24rd, 2007

So We Started

Sent out application materials to a hundred or so high schools statewide.

University of Utah March 24rd, 2007

So We Started

Sent out application materials to a hundred or so high schools statewide.

Received a healthy response.

University of Utah March 24rd, 2007

Feeling in the Dark

Immediately met some challenges:

University of Utah March 24rd, 2007

Feeling in the Dark

Immediately met some challenges:

– Had to get acquainted with high school students.

University of Utah March 24rd, 2007

Feeling in the Dark

Immediately met some challenges:

– Had to get acquainted with high school students.

– Our expertise and resources differed from those in Berkeley.

University of Utah March 24rd, 2007

Feeling in the Dark

Immediately met some challenges:

– Had to get acquainted with high school students.

– Our expertise and resources differed from those in Berkeley.

– Gradually we polished our own model.

University of Utah March 24rd, 2007

The Utah Math Circle Model

University of Utah March 24rd, 2007

The Utah Math Circle Model

• Weekly for two hours

University of Utah March 24rd, 2007

The Utah Math Circle Model

• Weekly for two hours

• Two consecutive weeks devoted to the same topic

University of Utah March 24rd, 2007

The Utah Math Circle Model

• Weekly for two hours

• Two consecutive weeks devoted to the same topic

• Typically led by faculty or graduate students

University of Utah March 24rd, 2007

The Utah Math Circle Model

• Weekly for two hours

• Two consecutive weeks devoted to the same topic

• Typically led by faculty or graduate students

• Monthly contests

University of Utah March 24rd, 2007

The Utah Math Circle Model

• Weekly for two hours

• Two consecutive weeks devoted to the same topic

• Typically led by faculty or graduate students

• Monthly contests

University of Utah March 24rd, 2007

A typical session….

University of Utah March 24rd, 2007

A typical session….

• Some lecturing…

University of Utah March 24rd, 2007

A typical session….

• Some lecturing…

• But mostly problem solving

University of Utah March 24rd, 2007

Some Examples

The 15 puzzle:

University of Utah March 24rd, 2007

Some Examples

Higher dimensional projections:

University of Utah March 24rd, 2007

Some Examples

Higher dimensional projections:

University of Utah March 24rd, 2007

Getting Started

University of Utah March 24rd, 2007

Getting Started

• Targeted Mailing (identification)

University of Utah March 24rd, 2007

Getting Started

• Targeted Mailing (identification)

• Settle on a model, collect materials

University of Utah March 24rd, 2007

Getting Started

• Targeted Mailing (identification)

• Settle on a model, collect materials

• Open enrollment

University of Utah March 24rd, 2007

Getting Started

• Targeted Mailing (identification)

• Settle on a model, collect materials

• Open enrollment

• Seed Money

University of Utah March 24rd, 2007

Getting Started

• Targeted Mailing (identification)

• Settle on a model, collect materials

• Open enrollment

• Seed Money

• Release Time

University of Utah March 24rd, 2007

Have Fun!