# Math Circles

Approx. ages TBA | Spring 2024: I am overwhelmed and excited by my list of courses that I’ve always wanted to offer. I will offer one or two of these in spring 2024 during the day at Lovett Park in April-May.
Approx. Ages 11-16 | Spring 2024: How many different ways are there to do something? What is an efficient way to count groupings of things? How do you decide which things to count and which to exclude? What is the optimal structure for a group of objects? These and related questions constitute the mathematical field of combinatorics.
Approx. Ages 11-16 | Winter 2024: In this course, we will collaboratively work on problems from Sergey Dorichenko’s book A Moscow Math Circle. The goals of the course (and book) are (1) to interest students in mathematics, (2) to show that mathematics is beautiful, (3) to teach them to reason, (4) to distinguish a solution from a nonsolution, and (5) to show students that they are able problem solvers.
Approx. Ages 7-17 | Fall 2023: In mathematics, there is a rich history of doing math by correspondence: mathematicians corresponding with other mathematicians, students corresponding with teachers, family members corresponding with other family members.
Approx. Ages 7-9 | Fall 2023: Every activity in this course will somehow relate to the number two. Maybe the problem will have two solutions. Or be a function with two conditions. Or involve parity (even/odd). Or probability via the pigeonhole principle. Or the binary system. Or exponential versus linear growth.
Approx. ages 10-14 | Fall 2023: We will do the group theory activity “A Fox and a Sock” to build up algebraic intuition of concepts such as simplification, isomorphism, inverses, and commutativity.
Approximate ages 7-17 | Spring 2023: In this 2-session mini-course, we will set up a bunch of games that have mathematical strategies and move among them at will as we engage in math conversations about them.
Approximate Ages 9-17 | Spring 2023: We’ll get our hands and minds to work together as we explore map coloring, Mondrian Art Puzzles, and more
Ages 9-12 | Fall 2022: In this course, we will do mathematical activities that Rodi learned about at the G4G conference and Celebration of Mind events
Ages 5-7 | Fall 2022: We’ll use sidewalk chalk to play with math problems that no one knows the answer to. We’ll use puppets to explore formal logic. And we’ll use storytelling to try to make sense of both.
Ages 7-9 | Fall 2022: In math, axioms are statements or rules that are accepted as true without having to prove them. But if you don’t have to prove them, how can you accept them? We’ll explore this and other interesting questions through the lens of set theory, an area of mathematics often used to make sense of axioms.
Ages 10-12 | Postponed: How many different ways are there to do something? What is an efficient way to count groupings of things? How do you decide which things to count and which to exclude? What is the optimal structure for a group of objects? These and related questions constitute the mathematical field of combinatorics.
Ages 13-18 | Winter 2023: 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.
Ages 10-12 | Winter 2023: 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.
Ages 9-13 | Spring 2022: It’s springtime, let’s follow our annual Math Circle tradition and just play with math picnic-style. We’ll explore math that is accessible, joyful, useful, and beautiful. We’ll play the game Mathematical Zendo and do joyful activities from the Natural Math books Camp Logic and Playing with […]
Ages 6-8 | Spring 2022: Fun and meaningful problems. Cooperation. Beauty. Delightful Discovery. It’s springtime, let’s follow our annual Math Circle tradition and just play with math picnic-style. We’ll do joyful activities from the Natural Math books Avoid Hard Work, Socks Are Like Pants Cats Are Like Dogs, and Bright […]
Approx. ages 11-14 | Winter 2022 | In this course, we’ll explore concepts such as quantifying uncertainty, probability, cognitive science andgame theory with real-life scenarios: how insurance works, auction theory, Nash equilibrium, stablematching, probability paradoxes (i.e. the two envelope problem), Bayes Theorem, and more topicsrelated to economics. We’ll take intermissions […]
Approx. ages 5-7 | Fall 2021 | In this course, we will play games that reveal the underlying mathematical structures of things, thataddress some philosophical/historical/cultural aspects of mathematics, and that foster development offocusing skills helpful in mathematical exploration. In August 2021 Math Circle leaders from all over gottogether at MathFest […]
Approx. Ages 7-10 | Fall 2021 | We’ll explore the differences between axioms, conjectures, theorems, and proofs. (An axiom is a self-evident truth, something that we assume to be true without having to prove it. You can’t prove somethingfrom nothing so we have to start somewhere.) We’ll delve into set […]
Age: 9-11 “Students who have just learned to count can play with the Cookie Monster Problem, yet there is sufficient depth to engage graduate students,” says mathematician Gabriella Pinter in Inspiring Mathematics. Participants “experiment and practice unstructured problem solving, make up and test conjectures, and develop different strategies for the […]
Age: 7-9 “Toilet paper turns into a flexible number line,” according to mathematicians Dave Auckley and Phil Yasskin in Inspiring Mathematics. “In addition to the usual number line explorations, one is able to rip off pieces, fold, and rearrange.” Participants will explore equivalent differences, which can be explored and discovered […]
Age: 11-13 Students will use bean bag tossing to explore modular arithmetic. Writes mathematician Amanda Serenevy in Inspiring Mathematics, modular arithmetic “results from arranging numbers in circles instead of on a number line. Students make and prove conjectures about patterns relating to factors and remainders… students will gain a deeper […]
Age: 13+ A state machine is an abstract set of instructions that can describe the behavior of a common object or situation. When written correctly, a computer can be programmed to follow these instructions precisely. Students will engage in various activities to learn the basics, and then will present to […]
Age: 5-7 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. […]
Age: 8-10 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 […]
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 […]
Age: 9-11 In this course, students will explore three underlying themes: math can be used to model real life; mathematical structures underlie much in life; and the study of mathematics meets many human emotional needs. Planned activities will include the M&M Death and Immigration Problem, Conway’s game of Life, the […]
Ages: 8-9 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 […]
Age: 5-7 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.
Age: 10-14 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 […]
Age: 10-13 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 […]
Age: 8-10 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 […]
Age: 13-17 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 […]
Age: 5-7 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 […]
Age: 8-10 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 […]
Age: 10-14 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 […]
Ages: 12-14 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, […]
Ages: 14-18 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 […]
Ages: 6-7 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) […]
Ages: 10-13 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 […]
Ages: 9-11 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, […]
Ages: 14+ 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 […]
Ages: 7-8 In math, parity is the quality of a number that describes whether it’s even or odd. The parity of 3, for instance, is odd. Why study it? It has relevance for computer science, physics, and math topics such as algebra, group theory, analysis, and combinatorial game theory. For […]
Ages: 11-14 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 […]
Ages: 9-11 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 […]
Ages: 5-6 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 […]
Ages: 11-13 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 […]
Ages: 7-8 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 […]
Ages: 9-11 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. Isomorphic problems appear dissimilar on the surface, but have the same underlying structure. We’ll try out some traditional river-crossing problems, and attempted to […]
Ages: 8-9 You’ve read about the Egyptian God Horus and the Greek God Narcissus in Rick Riordan’s books. What is mathematical significance of the Eye of Horus? What is the smallest Narcissistic Number? Was Thalia named after Thales? We’ll discuss these concepts, and more, in this Math Circle that explores […]
Ages: 13+ Blaise Pascal said “All men’s miseries derive from not being able to sit in a quiet room alone.” We will explore the implications of this quote as far as mathematics is concerned. We’ll also examine, from a 100% secular perspective, the mathematics of Pascal’s famous wager, which uses […]
Ages: 11+ What does your sacred symbol look like? Join Gina Gruenberg and Rodi as we combine the math of sacred geometry with the ancient art of Henna, also known as Mehndi. A temporary way of decorating the body (or other materials), Henna offers an opportunity to study geometric shapes, […]
Age: 5 Students will be using their bodies, imaginations, and minds to explore mathematical thinking. Children will be using clock (modular) arithmetic to learn to look for patterns. We will do some jumping and digging activities to begin to think abstractly (negative numbers). We’ll talk about what math is. And […]
Ages: 6-7 This number theory course will explore triangular and square numbers, multiplication, and counting strategies. We’ll move back and forth from the visual to the abstract (and discuss what those terms mean), debate whether math is invented or discovered, and have a lot of fun with Function Machines.
Ages: 11-14 Mathematics, like most subjects, cannot really be segmented into discreet topics. Everything overlaps. When students ask what Fermi Problems are, I say vaguely, “data.” These problems often involve geometry, arithmetic calculations, optimization, logic, statistics, and maybe most importantly, question asking. We’ll tackle some of them while we talk […]
Ages: 9-10 Students will be exploring Number Theory in our November-December Math Circle. We’ll start out with place value, and will probably end up getting into different bases, such as binary. We’ll also likely explore infinity: What is it? How big is it? Is there only one type of infinity? […]
Age: 8 What is true, what is false, and how can you prove it? What are assumptions? If you can deduce the rule for a function, can you find its inverse function? Does every function have an inverse? We will use story problems and function machines to explore these important […]
Ages: 6-7 participants will “accompany” 2 fictional children who venture into an unknown kingdom only to encounter a game-loving elf who won’t let them leave until they beat him at a game. The games require students to exercise the logical thinking that underlies mathematical thinking. We will also engage in […]
Ages: 13-18 How do you know for sure that the commutative property works? In other words, can you prove that ab=ba? And can we really trust the Pythagorean Theorem? And how many prime numbers are there anyway? Is it okay to take things on faith in mathematics? We will do […]
Ages: 5-7 Which mathematician is considered The Male Prophet of Feminism? Come to the math circle to learn about him and his wife, and also to collectively create a story about a zoo that will require us to explore the mathematics of voting theory.
Ages: 10-12 Is it possible to construct a spiral using only straight lines? And by the way, what exactly is a straight line? In this class we will explore how to construct polygons with a compass and straightedge, experiment with polydrons, and create Baravelle Spirals. Depending upon the students’ interests […]
Ages: 11-13 Fractals, the Math Doodling of Vi Hart, tesselations, and geometry proofs
Ages: 8-10 Polyhedra construction, the platonic solids and the Euler Characteristic (with a possible exploration of the famous Konisburg Bridge problem)
Ages: 6-8 Introduction to the Fibonacci series, triangular numbers, and negative numbers, using a narrative story woven throughout

### Math Buddies

Math Buddies is a girls-only, invitation-only, yearlong Math Circle. It began as a student-created year-long program for 10-year-old homeschoolers in 2014. The students’ goal was to make math fun again. Rodi’s goal was to do math, of course, but also to talk about and foster skill-building with the highly emotional component of mathematics. The mathematical content was originally based upon Natural Math’s Multiplication Explorers Course. Over a year, Math Buddies grew in scope to include student-requested topics and also games such as Go and a weekly student-designed logic game called Math Squares Math Buddies grew in size to include younger and older siblings and parents. The participants bonded over math and decided to rename what was originally called “Girls Math Circle” Math Buddies. Contact us if you are interested in reviving this program for your group of (approximately) 10-year-old girls.

##### What is a Math Circle?

In a Math Circle, a leader presents a vague but interesting math problem. The participants ask clarifying questions, pose conjectures, invent methods to test conjectures, justify or reject conjectures, again and again, to the eventual ends of generalizing, abstraction,understanding truth, solving problems, and proof. The problems are low-floor, high-ceiling so that students can enter at and progress to varying levels of concepts rooted in advanced mathematics. The leader acts more like a secretary, in a supporting role, as the participants work collaboratively to solve the problem.

##### The Math Renaissance Math Circle (formerly Talking Stick Learning Center Math Circle)
Locations

OUTDOOR LOCATION

The in-person courses will be outdoors, following all CDC-recommended precautions. The outdoor location will be Lovett Park in Mt. Airy in Northwest Philadelphia. We hope to schedule when Lovett Library is open, but the librarians (Miss Marsha and Miss Dana) report that they don’t yet know what their hours will be in the fall. Right now the library is open on Monday and Wednesday afternoons, but will hopefully open more frequently when the academic year begins. Lovett Park is a city park open all day regardless of library hours. We will meet on the library’s covered, open porch during the rain. There is an Acme supermarket across the street with a public restroom if the library is closed during Math Circle.

INDOOR LOCATION

We are not scheduling any indoor courses unless a participating family requests it, organizes it, follows age-appropriate CDC recommendations, and hosts it in their home. Please contact us if you are interested in doing this.

Mission Statement
• to expose participants to the richness of mathematics content
• to foster deep conceptual understanding of mathematical topics
• to help participants realize that mathematics isn’t defined by arithmetic or performance
• to allow participants to experience the creativity of mathematics
• to explore mathematics as a collaborative group, vs. competitive group or individual, pursuit
• to invent and/or discover mathematics in a non-coercive environment
• to help participants find joy and beauty in mathematics
• to facilitate the understanding that mathematics is for everyone (in other words, to “rehumanize mathematics”)
• to relieve math trauma for students who have experienced it
• to inform adults about mathematics pedagogy so that they can increase their children’s, families’, and students’ math enjoyment and success in general

NOTE: While the MRMC is not a way for students to “get ahead,” all of our activities can be connected to at least one (and usually multiple) math practice Common Core State Standard.

History and Land Acknowledgement

Land acknowledgement: We acknowledge that Math Renaissance works on the unceded Indigenous territory of the Lenni Lenape people, who were and continue to be active stewards of this land. We affirm Indigenous sovereignty and with this Land Acknowledgement pledge our intention to do work that not only does no further harm to Indigenous communities but also benefits Indigenous students.

The Math Renaissance Math Circle has been serving students since 2011. Until then, Philadelphia was the only major US city without a student Math Circle. Our program was founded as an in-person program hosted by Talking Stick Learning Center (TSLC) to provide programming to both homeschoolers and school students. We switched to a virtual program in the spring of 2020. The Math Renaissance Math Circle will resume some in-person courses for the 2021-2022 academic year.

Over the years, we have offered 46 courses: each with 6-13 students enrolled; each course meeting weekly for 6-8 weeks; each session lasting 60-90 minutes. These 46 courses have covered 45 topics – our goal is to never repeat a topic in order to keep the material fresh and exciting for facilitators. We did repeat our Compass Art course at the request of parents; we discovered that it was still a different course because the group dynamic and student math backgrounds were different the second time around. This was no surprise, since the program is student-driven.

In May 2021, TSLC reorganized into a smaller program that currently is not hosting supplemental programs. TSLC has become Harmony Learning Community. The name change reflects not only the reorganization, but also an awakening to the disservice/perniciousness of cultural appropriation. The Talking Stick Math Circle followed Harmony’s lead in renaming our program.  Read here for Harmony’s Founding Director Katie O’Connor’s illuminating answer to the question “Why did you change your name?”

Logistics

Math Circle Tuition – Tuition is \$25 for a 60-minute session, prorated accordingly.