Mathematical Circles Topics

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Here is a link to this document in Portuguese: Portuguese pindex.html.

Mathematical Circles are for students of high school age or younger who want to increase their abilities to reason about mathematical problems. Students who study in circles learn to do mathematical olympiad-style problems (essay or proofs, not quick answer).

The web pages for a couple of the local circles are here:

Bay Area Circles and BAMO (Bay Area Mathematical Olympiad)
San Jose State University Math Circle
UC Berkeley Math Circle

An extensive discussion about how to lead a math circle can be obtained here:

So You're Going to Lead a Math Circle, or (in Postscript). (For some reason, the pdf version sometimes doesn't seem to load properly on an Apple Safari browser, but the Postscript version does.)

Here is the National Association of Math Circles website with links to existing circles, problems, lesson plans, and many other things.

For more lesson plans and for information on Teachers' Circles which are like math circles but aimed at teachers, go to the Math Teachers' Circle Network.

I have been volunteering time at three local circles (in Berkeley, San Jose, and Palo Alto, California) for the past nine years, and have prepared handouts for some of those sessions. Those handouts are almost all available here in both PostScript format (files with a .ps extension) or files in Adobe Acrobat format (files with a .pdf extension).

Free Dynamic Geometry Software

Rubik's Cube and Mathematics

Math for Teachers

There is a series of workshops for teachers, called "teacher's circles", that may be of interest. Here is the website:

http://mathteacherscircle.org

The 1,2,3,4 problem:
plustimesminus.pdf(72 KB)

Exploring Pascal's Triangle:
pascal.pdf(148 KB)

An Introduction to Zome:
ZomeIntro.pdf (504 KB)

Julia Robinson Zome Exercises (used for Julia Robinson Festival at Google and Pixar):
googlezome.pdf (148 KB)

An Unusual Way to Combine Numbers
numbercombine.pdf (216 KB)

A Game That's Not a Game
pilesubdivide.pdf (216 KB)

Mathematical Card Tricks
CardTricks.pdf (216 KB)

Conway's Rational Tangles
tangle.pdf (48 KB)

Huge numbers with short descriptions:
bignumbers.pdf (84 KB)

Geometry and Geography:
geography.pdf (60 KB)

Kenken:
kenken.pdf (92 KB)

Counting and Combinatorics

Pólya's Counting Theory:
polya.ps (248 KB)
polya.pdf (168 KB)

Set Theory, Logic, Cardinal and Ordinal Numbers

Infinity (Cardinal and Ordinal Numbers):
Infinity.ps (168 KB)
Infinity.pdf (158 KB)

Computability and Recursive Functions:
Computability.ps (222 KB)
Computability.pdf (180 KB)

Geometry and Transformations

Zome

Games

Miscellaneous

To view PostScript files, you may download the GhostScript programs.

To view Acrobat files, you can download the Adobe Acrobat Reader.

I prepared a following document for people who are interested in leading math circles. If you wish to make a nice version to hand out to prospective volunteers, here it is in PostScript format: MathCircles.ps, and in Acrobat format: MathCircles.pdf. Here it is in HTML:

How to Lead a Math Circle Session

Here are some differences between a math circle and the usual math club:

• One school teacher usually runs all math club sessions. Leaders of math circles rotate. Circle leaders don't burn out, the kids see different approaches to math, and the leaders only need to prepare a few sessions that can be repeated at multiple circles. Circle leaders include teachers, professors, graduate or even undergraduate students, and other professional mathematicians.
• Circle sessions are focussed on a particular topic. ``Here are a bunch of unrelated old AIME problems.'' is usually not a suitable circle topic.
• There is homework, but exciting and seductive homework.
• Math clubs often prepare the math team for multiple-choice or short-answer competitions, without going through the problems in depth. Math circles prepare students for Olympiad-style problems like those of BAMO, the Bay Area Mathematical Olympiad. Circles teach kids to be mathematicians who solve essay-style problems requiring proof.

Make sure your circle session goes as well as possible:

• Hand out a set of problems a week before your session. Not too many, but seductive. Include an easy one and a challenging one.
• Try not to lecture. Even though introducing new theory and techniques is an integral part of math circles, your sessions should be as interactive as possible. Score yourself: 1 point per minute you talk; 5 points per minute a student talks; 10 points per minute you argue with a student; 50 points per minute the students argue among themselves.
• Divide students into groups of 2-4 to solve problems. Have them present their own solutions.
• Be encouraging, even about wrong answers. Find something positive in any attempt, but don't be satisfied until there is a rigorous solution. Wrap up each problem by reviewing the key steps and techniques used.
• If the kids don't answer your question immediately, don't just tell them the answer -- let them think. If they're still stuck, give hints, not solutions.

For details on Math Circles or on BAMO, see:

``Mathematical Circles (Russian Experience)'', by Fomin, Genkin, and Itenberg, American Mathematical Society, 1993.

or

http://mathcircle.berkeley.edu/bamoinfo.html