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:



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



• 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.


Examples

Three that I’ll discuss today:

– The Boston Math Circle

– The Berkeley Math Circle

– The Utah Math Circle


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)


Toward a National Network?

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


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.


Why all the interest in Math Circles?



Typical high school math curriculae are almost exclusively template based.




This is extremely effective for some purposes.




This is extremely effective for some purposes.

But it’s terrible for others.


Downsides of template-based learning



• Encourages very short attention spans.




• Implemented algorithms take precedence over their conceptual formulation.




• Implemented algorithms take precedence over their conceptual formulation.

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


But this is not what mathematicians do!



Instead we are trained to think deeply about simple things.



Instead we are trained to think deeply about simple things.

This idea is completely absent in template-based learning.


A Different Model: The Russian Experience

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




• Early Identification

• Special Schools

• Math Circles




• Early Identification

• Special Schools

• Math Circles

The objective is not limited to producing only research mathematicians.


Example (courtesy of Josh Zucker)


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.


Example: 8th grade algebra class


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.


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?


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?


Can any of this be emulated here?

.


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,….)




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

• Grown into a rather large enterprise


Typical Spring Schedule

• Four pre-algebra courses

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

– 10 sessions, one hour each


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


Approach for younger students

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



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

• Sample topics for kids aged 5-10:

– Are there numbers between numbers?– Primes– Infinity



• 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


Approach for High School Students

• Almost exclusively devoted to a single topic.


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


The Boston Math Circle charges tuition

• Each pre-algebra courses

– $225 tuition

• Each advanced course

– $450 tuition

• About $20/hour per student


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.”


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.







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

• Free.


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”


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


Footnote: San Francisco Math Circle

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


The Utah Math Circle


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.


So We Started


So We Started

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


So We Started

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

Received a healthy response.


Feeling in the Dark

Immediately met some challenges:


Feeling in the Dark


– Had to get acquainted with high school students.


Feeling in the Dark



– Our expertise and resources differed from those in Berkeley.


Feeling in the Dark



– Our expertise and resources differed from those in Berkeley.

– Gradually we polished our own model.


The Utah Math Circle Model



• Weekly for two hours




• Two consecutive weeks devoted to the same topic





• Typically led by faculty or graduate students





• Typically led by faculty or graduate students

• Monthly contests


A typical session….



• Some lecturing…



• Some lecturing…

• But mostly problem solving


Some Examples

The 15 puzzle:


Some Examples

Higher dimensional projections:


Getting Started


Getting Started

• Targeted Mailing (identification)


Getting Started


• Settle on a model, collect materials


Getting Started



• Open enrollment


Getting Started



• Open enrollment

• Seed Money


Getting Started



• Open enrollment

• Seed Money

• Release Time


Have Fun!

maa intermountain section meeting march 24, 2007 peter e. trapa department of mathematics university...

Documents

utah math circle slide

high school math circle

based learning slide

berkeley math

boston math

new math circles

downsides of template

louis washington u slide