Books and Articles

"Hidden Harmonies: The Lives and Times of the Pythagorean Theorem"
by Robert Kaplan and Ellen Kaplan
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Bozo Sapiens: Why to Err is Human
by Michael and Ellen Kaplan

"Out of the Labyrinth: Setting Mathematics Free"
by Robert Kaplan and Ellen Kaplan

"Chances Are . . . : Adventures in Probability"
by Ellen Kaplan and Michael Kaplan

"The Art of the Infinite: The Pleasures of Mathematics"
by Robert and Ellen Kaplan

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Disturbed by the poor quality and low level of math education in the country, three of us began The Math Circle in September 1994...



Our courses are located on the campuses of Harvard and Northeastern Universities. [learn more...]

History of The Math Circle

Disturbed by the poor quality and low level of math education in the country, three of us (Bob and Ellen Kaplan, and our colleague Tomás Guillermo) began The Math Circle in September 1994. We rented space on Saturday mornings in a local church and word of mouth alone brought us 29 students for that first (ten session) semester. Scholarships were offered when needed, and we almost covered our expenses.

The Saturday format we set up then continues still - but now on Sunday mornings to avoid soccer conflicts. The students are divided up into two groups (roughly by age) and now we teach a class from 9:15 to 10am. After a 15 minute break for juice, cookies and conversation, we change places and teach a different group from 10:15 to 11. In the last hour we all listen to an invited speaker or work on a joint problem.

In the second semester of our first year, Northeastern University (through Andrei Zelevinsky) offered us free - and much larger - quarters; and demand for a younger class led to a single session on Thursday afternoons, in a room offered us free by Harvard (thanks to Danny Goroff). Enrollment (with some leaving, more arriving) rose to 34.

We began our second year at Northeastern and Harvard with 38 students (some coming via the article about The Math Circle, which had just appeared in the AMS Notices), so we added a Harvard graduate student to our staff, setting a pattern which has allowed for our expansion from 29 to (in the fall of 2001) 205 students. The Sunday classes have, of necessity, stayed about the same size, but the weekday sessions for 5 to 11 year olds have proliferated: this year there are nine classes meeting at Harvard, on Tuesday and Wednesday late afternoons. We have been asked to give demonstrations of our approach, with the prospect of expanding, to public schools in Cambridge, Brookline and Taunton, and up and down the length of Great Britain.

The courses we have taught in these ten years are many and varied:

Weekday Classes

Weekend Classes

for 5-7 year olds For the Young (9-11, no Algebra):
Are There Numbers Between Numbers?
Sequences and Series
The Euclidean Algorithm
Prime Numbers
Triangular, Square etc. Numbers
Graph Theory Invariants
Linear Functions
Big Numbers
Area, Geometry and Number
Set Theory
Polygon Construction
Map Coloring
The Euclidean Algorithm
Modular Arithmetic
Game Theory
Group Theory
Sequences and Series
Mathematical Games
Solving Equations
Pascal's Triangle and Fractals
Concurrency and Collinearity
Pythagorean Triples
The Intermediate Value Theorem Mathematical Origami
Steiner Points
Complementary Sequences
For 7-9 or 9-11 year olds For the Middle Group (12-14, some Algebra)

Cantorian Set Theory

Fractions and Decimals

Straight-Edge and Compass Constructions

Sequences and Series


Eulerian and Hamiltonian Circuits

The Infinite

Interesting Numbers

Polygon Construction

Prime Numbers

Complex Numbers

Min/Max Problems

Functions and their Graphs




Powers of 2

Weird Fractions

Random Walks

Area, Number and Geometry


Periodic Decimals

Continued Fractions

Propositional Calculus

The Fibonacci Sequence

Solution by Radicals

Polygon Decomposition

What is ii?

Krasnoselsskii's and Brouwer's Theorem

Interesting Points in Triangles

Maxima, Minima and Optima

Angle Trisection

The Golden Mean

Which Numbers are the Sum of Two Squares?

Visual Proofs

Information Theory

The Pythagorean Theorem

Cantorian Set Theory

Conway Games

Pick's Theorem etc.,

Linear Algebra

Mathematical Origami

Integer Triangles

Complementary Sequences

Taxicab Geometry

For the Senior Group (15-17, good Algebra and Geometry)

Sequences and Series

Projective Geometry

Induction and the Pigeonhole Principle

Classification of Surfaces

The Four Color Problem

The Pythagorean Theorem

Number Theory

Proofs and Refutations

Algebraic Geometry

Complex Analysis

Cantorian Set Theory

Number Theory

Knot Theory

Hyperbolic Geometry

Group Theory

Conway's Numbers

Mathematical Logic

Information Theory



Proofs from The Book

Banach Tarski Paradox

Combinatorial Geometry


We are careful to choose topics which are unlikely to be in the school curriculum - we see our role as widening and deepening the river, rather than accelerating its flow between narrow banks. You will notice that some courses appear at several levels: one of the glories of math is its constant upward spiral of sophistication. You may have "had" Group Theory when you were ten, but it is a totally different subject when you meet it again as a senior.

In addition to the three of us, a fair number of Harvard math department graduate students, one or two undergraduate math majors, and a few people from M.I.T. and Tufts have taught a course or two from time to time. Outstanding among these was Mira Bernstein, who taught regularly on Sundays for three years.

We've had great good luck with our guest speakers: among them have been Raoul Bott, Persi Diaconis, Dick Gross, Tom Lehrer, Barry Mazur, the historian of mathematics Joan Richards, Andrei Zelevinsky, Fernando Gouvea, mathematicians from Tufts, M. I. T. and Northeastern, and graduate students from the M. I. T. and Harvard math departments (their talks gave us a good opportunity to audition them as possible teachers); as well as a number of applied mathematicians. In every case the presentation (subjects ranged from perfect shuffles to Stirling Sequences) has been more of a conversation than a lecture, in keeping with our aim of active student involvement.

Where have our students come from? We began with a considerable number from the Russian émigré community, some former students, and then friends brought friends and word began to spread from school to school. Inquiries came in from greater and greater distances; some students rose at dawn to come in from Rhode Island or Maine! Home-schoolers are particularly attracted to the Math Circle - students like the casual intensity, and their parents often feel less confident about teaching math than history or literature.

Our essential admissions requirement is the desire to come: meeting at five in the afternoon or nine on a Sunday morning weeds out the faint-hearted. But of the 29 who began with us in September 1994, 10 were still with us in the spring of 1998, and students we met as nine year olds are now seniors. The majority of those who leave have gone off to college, or their families have moved away. Some couldn't bear the transition to getting up on Sunday mornings; some succumbed to our rivals: soccer games and music lessons. A few found it not to their taste. Of the latter, most had been forced to come by ambitious parents: never a good start.

Our gender balance has varied from 40% to 20% girls. We've had about the same number of girls each year, but increasing numbers of boys. Anecdotal evidence indicates that, sadly, a lot of girls are convinced by the age of ten that they won't like math. Another issue we struggle with is the distribution of bold and timid students in a class: just enough of the former are needed to keep a conversational ferment - not so many as to hijack the discussion. Critical mass is important too: a class with only four students hasn't a sufficient variety of viewpoints; more than 12 gives people too little chance to contribute.

We judge our successes by insights won in the midst of chaos, and by growing confidence and competence. We've had to learn how to raise the level of abstraction carefully: newly-won insights have to be fondled for a while. A language needs to be practiced with declarative sentences before elevating it to poetry.

What seems to be the Math Circle Secret is our always striving for understanding generated by the students' own conjectures and counterexamples, rather than our aiming to cover a certain body of material in a fixed amount of time. Better to find out for yourself how to bisect an angle with Euclidean tools than to be told by someone else why these tools won't suffice for angle-trisection.