Here are some brief descriptions of the Math Circles we have successfully held over the past two years. We hope they help to illuminate what this amazing program is all about.

1/9-2/27 (8 weeks, 12 hours of instruction)

Many social, political, and environmental issues have underlying mathematical concepts that can affect outcomes. For instance, elections can depend on voting methods: the candidate who wins a plurality may not be favored by the majority. In this course, we’ll explore and evaluate many voting systems. We’ll also delve into the mathematics of gerrymandering, student loan debt, environmental disasters, and human trafficking. This course is inspired Emily Riehl’s work on the mathematics of social choice. Underlying mathematical concepts include algebra, statistics, functions, combinatorics, voting theory, logic, and financial mathematics.

Ages 13+

Thursdays, 3:30-5:00pm, Cope House

1/9-2/27 (8 weeks, 12 hours of instruction)

$240

10/24-12/12 6 weeks, no class on Halloween or Thanksgiving, 6 hours of instruction)

Students will invent their own number systems after playfully engaging with various number systems. Explorations will include numerals (Chinese Rod, Navajo, Egyptian, and Roman), Lego function machines, the Caesar Cipher, Exploding Dots, Backgammon, and the four-color theorem (map coloring). This course is inspired by Rochelle Gutierrez’s work on rehumanizing mathematics. Underlying mathematical concepts include place value, the meaning of zero, number operations, cryptology, functions, and graph theory.

Ages 8-10

Thursdays, 3:30-4:30pm, Garden Classroom

10/24-12/12 6 weeks, no class on Halloween or Thanksgiving, 6 hours of instruction)

$120

9/19-10/17 (5 weeks, 5 hours of instruction)

A fractal is a pattern that infinitely repeats itself, growing smaller by a scale factor. In this course, students will create their own fractals, study some famous ones, and attempt to define them with age-appropriate language. Students will test their definitions on some things that might be fractals. We’ll look for them in nature, drawing connections between mathematics and life itself. We’ll take some occasional breaks from fractals to engage in activities such as pen-tracing puzzles, Exploding Dots, Mobius strips, and the unsolved No-Three-in-a-Line problem. This course is inspired by Johnny Houston’s tireless work on the No-Three-in-a-Line problem and especially by Maria Droujkova’s work on making compelling mathematics accessible to young children. Underlying mathematical concepts include multiplication, exponential growth, scale, geometry, infinity, self-similarity, dimension, iteration, place value, parity, discrimination, graph theory, and mathematical communication.

Ages 5-7

Thursdays, 3:30-4:30 pm, Garden Classroom

9/19-10/17 (5 weeks, 5 hours of instruction)

$100

Polyominoes are a hands-on geometry activity that develop students’ thinking about classification, combinatorics, symmetry, and more. We will also study characteristics of functions via the book Funville Adventures (or via extensions of this book if the students have already used it) and function machines in order to develop algebraic reasoning skills.

1/24-3/21 (8 weeks, 75-minute sessions, 10 hours total, no class on 3/7)

In this course, students will explore real mathematics problems from ancient history. These will include Queen Dido problems, Zeno’s Paradox, and ancient inheritance problems. We’ll do the math and put the problems in their historical contexts. We may dabble in a few mythological problems as well. Mathematical concepts will include pre-algebra, algebra, geometry, and some calculus, but pre-requisite knowledge of these topics is not required.

10/25-12/6 (6 weeks, 75-minute sessions, 7.5 hours total, no class on Thanksgiving)

Mathematician Eugenia Cheng describes category theory as “the mathematics of mathematics.” Inspired by Cheng’s book “How to Bake Pi,” we will do activities that use abstract mathematics to see, understand, and generalize the defining structure of things. And by “things” I mean mathematical things, logical things, and social phenomena. Visit her website (eugeniacheng.com) for a preview.

Knights and Liars, open questions, story problems, pattern making and breaking, explorations of infinity, proofs, and more. We will have fun with these classic math circle activities as students develop the mathematical-thinking skills of asking questions, forming conjectures, testing conjectures, and generally seeking the underlying structure of things.

3/22-5/17 (8 weeks, off 3/29 for spring break)

Students will engage in hands-on activities to discover some fundamental principles of geometry. We’ll create and classify the platonic solids as we build with Polydrons. We’ll explore fractals as we attempt to build a 3D Sierpinski Triangle from business cards. We’ll make discoveries about area as we fold paper into squares and triangles. We’ll explore and expand upon Euclidian geometry as we fold more paper. We’ll see math in nature through a look at how ladybugs fold their wings. And in a study of empirical versus logical proofs, we’ll use toilet paper to explore what variables come into play – and how they interact – as we try to figure out the maximum number of times you can fold a piece of paper.

An invariant is a quantity whose value never changes no matter what you do to the operation under consideration. For example, when you shuffle a deck of cards, the number of cards in the deck remains unchanged. Mathematicians consider invariance one of the most important concepts children need to know as they go through their math educations. In this course, we’ll engage in problem solving, games, flowcharts, storytelling, and a hands-on exploration of the Euler Characteristic to search for and understand invariants.

11/2-12/7 (5 weeks, no class on Thanksgiving)

Neuroscience has provided empirical evidence of what we intuitively knew all along: that counting on your fingers enhances learning. The discipline of embodied mathematics employs gesturing and physical interactions with the environment to develop conceptual understanding and to facilitate articulation of mathematical concepts. Year after year, young students come into Math Circle with the idea that mathematics is all about quick computation and nothing else. This course will open students’ minds to the reality that math is about more than numbers and can be explored with more than a computational approach. We’ll use our bodies and surroundings to examine symmetry, 2D and solid geometry, equivalence, measurement, spatial reasoning, and arithmetic computation.

9/7 to 10/26, 2017 (8 weeks, 75-minute sessions, 10 hours total)

What are algorithms and how do they drive our culture? We’ll examine the Google page-rank algorithm, Cathy O’Neill’s National-Book-Award-nominated *Weapons of Math Destruction: How Big Data Increases Inequality and Threatens Democracy*, whether random number generators are really random, the mathematics behind “fake news,” the Euclidian algorithm, and much more. These topics will provide context for a study of the algebra behind algorithms: sequences of instructions that usually involve variables and (algebraic) expressions. These expressions can be organized logically into matrices, programs, flowcharts, etc., to produce solutions to well-defined problems. Then, of course, we will debate the appropriateness of labeling problems “well-defined.” While we’re at it, we’ll delve into some statistics and number theory and then compose some algorithms of our own.

Students should be familiar and comfortable with variables, although the ability to manipulate them is not a prerequisite. While this is a students-only course, interested parents and guardians are invited to participate during the final session.

Beginning with non-mathematical functions, then function machines, and finally – probably – ending with algebraic expressions and graphing on the coordinate plane, we will have fun discovering what a function is and how to express it in various mathematical ways. I say “probably” because Math-Circle is student-directed and the students’ mathematical interests could take them elsewhere once I present the initial premises.

March 2 to April 6*, 2017

Using both unsolved problems in mathematics and the book Avoid Hard Work, we will explore general problem-solving strategies. The goal is for students to move toward an understanding that (1) the pursuit of mathematics is not the same as memorizing a bunch of math facts, and that (2) there are ways of thinking that help to tackle a problem. The students will create their own problems for others to solve.

**The schedule may shift by one week to accommodate spring break.*

January 10 to February 14, 2017

In this course, we will attempt to discern whether math is about absolute truth or relative truth, and along the way will discuss how math is a creative endeavor. Using the book Camp Logic and other sources, we will engage in logical reasoning, induction, and proof to explore ideas including invariants and isomorporhism.

October 31 to December 5, 2016

Our math circle will explore the storied history of Fermat’s Last Theorem and some of the underlying mathematics, such as Pell’s and other Diophantine Equations, and Fermat Proofs for Specific Exponents. We will discuss specific work by mathematician Sophie Germain, as well as the drama involved in Andrew Wiles’ Fermat proof.

According to Wikipedia, “Fermat’s Last Theorem states that no three positive integers a, b, and c satisfy the equation a^{n} + b^{n} = c^{n} for any integer value of n strictly greater than two. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of Arithmetica where he claimed he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by mathematicians. The unsolved problem stimulated the development of algebraic number theory in the 19th century and the proof of the modularity theorem in the 20th century. It is among the most notable theorems in the history of mathematics Prior to its proof, it was in the Guinness Book of World Records as the “most difficult mathematical problem”, one of the reasons being that it has the largest number of unsuccessful proofs.”

September 22 to October 27, 2016

Led by Rachel Steinig, students will make ropes dance via specified moves. Rational Tangles was invented by one of Rodi’s favorite mathematicians, John Horton Conway, a living mathematician whose life we’ll discuss in the course. Rational Tangles is rich in mathematical content, including algebraic thinking, transformations, symmetry, classification, geometric equivalence, the order of operations, and some of the more interesting arithmetic of fractions.

#### Open Questions

Designed for ages 11-14, this Math Circle ran from 4/21 through 5/26, 2016

At an age when some kids feel disenfranchised from mathematics while others feel empowered by it, we will collaboratively attempt to solve currently unsolved (“open”) questions. The students will be essentially working mathematicians, with the stated hope of making some progress toward a solution and the unstated hope of experiencing joy in mathematics.

#### Parity, 3/3-4/14 (6 weeks, no class 3/24)

Designed for ages 7-8, this Math Circle ran from 3/3 through 4/14, 2016

The basic definition of parity is this: a property of a number that describes whether it is even or odd. Sounds simple and obvious, right? But parity has implications that are bigger in mathematics and science: alternating groups, or a way of putting things into 2 distinct groups. We will play games that depend upon this concept as a strategy in hopes of ending the course with a true conceptual understanding (vs. rote memorization) of parity. I suspect that the students will lead this discussion into how parity is related to infinity (is it even or odd, is it even a number?), as most Talking Stick math circles with this age group have done.

#### Cantor, 1/5-2/9 (6 weeks)

Designed for ages 14-18, this Math Circle ran from 1/5 through 2/9, 2016.

We’ll examine the life and work of this revolutionary mathematician once called a “corrupter of youth.” Come and have your teens corrupted with Georg Cantor’s ideas: set theory (a concept that seems fundamental and even obvious today); his most famous proof; and more. Cantor’s life story is sad because of his struggle with mental illness. In discussing his personal story, we’ll question (1) the stereotype that the most successful mathematicians are somehow unbalanced, and (2) the apocryphal “math gene.”

#### Compass Art, 9/17-10/22 (6 weeks)

Designed for ages 9-11, this Math Circle ran from 9/17 through 10/22, 2015

What do Michelangelo, Bernini, Zarah Hussein, feng shui practitioners, mapmakers, architects, astronomers, and mathematicians have in common? They all use compasses to construct and deconstruct circles. We’ll create our own compass art while learning about basic circle geometry and some math history. (Each student should bring a compass, sketch pad, and pencils.)

#### River Crossing Problems

Designed for ages 9-11, this Math Circle ran for 5 weeks from 4/28-5/26, 2015.

This Math Circle focuses on the concept that classical composers incorporated variations on themes in their compositions just as mathematicians create them in their work. Isopmorphic problems appear dissimilar on the surface, but have the same underlying structure. We’ll try out some traditional river-crossing problems, and and attempted to solve them. Then we’ll try some problems that are not about crossing a river, and compared and contrasted them. Finally we’ll try to create our own isomorphs.

#### Chromatic Number of the Plane

Designed for ages 7-8, this Math Circle ran for 6 weeks from 3/3-4/7, 2015.

Students will explore graph-coloring questions and tilings to lead up to an exploration of an open (unanswered) question in mathematics: the Chromatic Number of the Plane (aka The Hadwiger-Nelson Problem). But my real agenda here, as it is in just about every math circle, is to move children toward abstraction. We’ll start out by using manipulatives and then hopefully wean from those to explore the difference between objects and symbols and, more generally, the difference between things (the concrete) and ideas (the abstract).

#### Escher and Tesselations

Designed for ages 11 and up, this Math Circle ran for 6 weeks from 1/6-2/10, 2015.

In this math-meets-art circle, students will experiment with the four types of symmetry in a plane to create their own tessellations (tilings). We’ll look at the work of MC Escher and that of the mathematician whose work inspired Escher, George Polya. We’ll draw and draw and draw. We’ll also attempt to determine which regular polygons can tessellate a plane, and then verify our answer with proof.

#### Infinity

Designed for ages 5-6, this Math Circle rang for 5 weeks from 11/11-12/9, 2014.

We’ll explore the idea of infinity using drama (puppets!) and embodied mathematics. The kids will use their imaginations and physical movements to play with patterns that have limits and patterns that don’t. And we’ll work on verbalizing our mathematical ideas as we try to figure out what patterns exactly are.

#### Martin Gardner

Designed for ages 9-11, this Math Circle ran for 6 weeks from 9/23-10/28, 2014.

Before there was Vi Hart, there was Martin Gardner. Celebrate the Martin Gardner Centennial with an exploration of Recreational Mathematics. For 25 years, Gardner wrote the Mathematical Games column in Scientific American, and became legendary for his unconventional approach to mathematics. In this circle, we will explore his life, his influence, and of course, his mathematical puzzles. The goal of this math circle is the same as the goal for all of them: to develop mathematical thinking. Recreational mathematics is yet another avenue for seeking patterns when none are obvious, and for seeking ways to crush seemingly obvious patterns that aren’t really patterns at all.