Blog
(Spring 2026) This is a story about how a planned course changed course based upon student curiosity and excitement. What began as a course called “Problem Solving with Alice” ended up primarily questioning whether there is a relationship between counterexamples and proof. Here’s the original course […]
This quarter has been a whirlwind of advocacy, leadership, and "swimming against the current" to create spaces where math feels human.
Our organization, the Philadelphia Area Joyful Math Alliance, in partnership with some of our favorite organizations and sponsors, will be presenting an online screening of the popular feature documentary Counted Out. We’ll also provide a post-screening get together and discussion with Nancy Blachman, Executive Producer of the film. Counted Out investigates […]
The Phila. Area Joyful Math Collaborative started as an idea in 2022 and a research project from 2022-24 and has quickly grown into a focused effort to bring inspiring mathematical experiences to a broader community DECEMBER 2024 Anne Paoletti Bayna joined Rodi Steinig (Founding Co-Director) as […]
On Wednesday, June 7, I will give a participatory presentation at Fundapromat’s “Math Webinar in English” invited-speaker series. My virtual talk on “The Emotional Experience of Math” is at 6pm EDT, 5pm Panama time. Fundapromat is one of the world’s most influential Spanish-language mathematics outreach organizations. […]
(11/30/2022 ) Ugh, pride. I was so filled with it after that first session of this course, when I first brought the ropes. On That Day, I had instructed “Stand in the red circle if you like dogs and blue if you like cats.” L put […]
(November 22, 2022) We did a few rounds of “regular function machines,” in which students put in numbers and I tell them the outputs and the students have to figure out the rule. The rules students figured out were f(x) = x+7, f(x) = x-12 (this […]
MU PUZZLE (November 16, 2022) For about 20 minutes we had been working on the famous puzzle from the book Gödel, Escher, Bach when Z asked, “What’s the point of this?” (besides the obvious problem-solving). I explained that I wanted to talk about axioms. “What are […]
(November 9, 2022) “Could you empty out that backpack for me?” I asked Z and V, students who had arrived a few minutes early. It was our first session of the course “Axiomatic Set Theory” for students ages 7-9. They pulled out a bag of two […]
The Learn’d Astronomer (9/14/2022) We began our exploration today with a reading of the Walt Whitman poem, When I Heard the Learn’d Astronomer. This poem has a huge connection to mathematics, but I resisted the urge to tell who wrote it (our famous local poet) or […]
MATH Statements and Domains (2/22/22) We began our session finishing up our function machine from last week: the input is a statement and, as the students figured out last week, the output is that statement’s truth value (“true” or “false”). But what the students still wanted […]
THE BUTTERED TOAST PROBLEM (2/15/2022) After last week, I realized that I have gotten away from an old, important practice of mine in facilitating Math Circles: listing assumptions, questions, and conjectures on the board/slide. My goal for today was to recover this. Since O was absent […]
OVERVIEW (February 9, 2022) In this course so far, we spent weeks 1 and 2 of this working The Pirate Problem to introduce the ideas (a) that math can describe human behavior, (b) that backwards induction can be a useful problem-solving method, (c) that the math […]
MATH THE PIRATE PROBLEM, CONT’D In the third week of working this problem (1/25/22), O asked “Can we just look up the solution?” and N commented again about how horrible these pirates are, but then the students solved it quickly with backwards induction. They determined not […]
(NOTES: I will be posting reports on the Nontraditional Games and the Axioms of Mathematics sessions soon – thank you for your patience. Students are referenced below by the last or penultimate letter of their first names.) (Jan 11 and 19, 2022) There are 5 pirates: […]
MATH Zeno’s Achilles Paradox (October 27, 2021) Draw 2 long lines. On each of them label one end START and the other end FINISH. What are the fastest and slowest runners in the world? Done. Cheetah and snail, same responses as the School House Lane group. […]
MATH Map Coloring (October 19, 2021) I drew a few shapes on ground. Me: Finish creating a map of your country, with regions. There’s only one restriction: each region needs to be enclosed. S: “Are we going to name the regions?” Me: After class, if you […]
School House Lane, 10/11/2021 MATH: Criss Cross We took a break from axioms to play the game Criss Cross today. You draw 7 dots/vertices inside a triangle. On your turn, you connect two of them with an edge/line segment. The last person who is able connect […]
I don’t want to “present” things; I want to “ask” things. I want to just use words without defining them. To just expect that if students don’t understand, they will ask and I will answer, hopefully by throwing a question right back at them. With this […]
Going into the second week of our Axioms of Mathematics courses, my plan with the School House Lane group is to linger, to savor the beauty of the math in Hofstadter’s book, Gödel, Escher, Bach, doing everything in the book that can be adapted to this […]
Both current Math Circle courses are exploring the Axioms of Mathematics: School House Lane, virtual, ages 9-11, a year-long course, and Lovett, in-person, ages 8-10, a 7-week course. The first session for each course started the same way: Rodi (me) announces I am lying (the Epiminedes […]
“Aw, are we talking about human trafficking again? I don’t want to feel sad,” said F at the beginning of the session following the human-trafficking discussion. I was grateful to F since I wasn’t sure whether to continue that topic or move to something else. Thanks to his comment, I knew what to do.
Middle-school Math Circle students use graph theory to examine the problem of human trafficking.
(February 5, 2020) “Suppose there was an election for Official Ice Cream Flavor of the USA, and the choices were ravioli, mint chocolate chip, butter pecan, mango, vanilla, and chocolate. Which voting methods could be used?” So began our class, which was filled with debate, questions, […]
(January 30, 2020) Last week I made up a silly example of laws about dairy-product usage and decided today that I’m just going to run with it. “Suppose that your state is going to vote for a leader and that all citizens and candidates only care […]
(January 23, 2020) Is it possible to teach students about gerrymandering and the Four-Color Theorem by asking questions and not lecturing students about anything?* My plan was to ask the students just four questions: 1) What’s the difference between the House and the Senate? 2) If […]
(1/16/2020) Our overarching Math Circle goal is for students to invent and discover math for themselves. In this course, the plan/hope is that I’ll ask a bunch of questions and the students will invent their own voting methods before discovering what voting methods people have created […]
(November 1, 2018) “That’s a pig!” said F, as we all walked over the picnic tables to start our Math Circle outside today. “No, Penelope is not a pig. She is a pig puppet. There’s a big difference,” I replied as we sat down. This seemingly […]
(October 25, 2018) An important concept in category theory, indeed in mathematics itself, is deciding which attributes to ignore when you conduct mathematics. Context matters. When you’re signing a contract to buy a car, you may not care what color the contract is printed on. But […]
(September 20 and 27, 2018) On a particular island, every inhabitant (puppet) is either a knight, who always tells the truth, or a liar, who always lies. Which puppet is a liar? Which one a knight? You can either listen to their statements, or ask them […]
MATH CIRCLE COURSE DESCRIPTIONS 2018-19 Unofficial schedule Classic Math Circle Problems Dates: Thursdays, 3:30-4:30pm, 9/20-10/18 (5 weeks) Suggested Ages: 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 […]
POLYDRONS (April 5 -19, 2018) In the past, I’ve often made the mistake of getting out “manipulatives”* to help students discover a certain mathematical concept only to find that the students wanted to engage in open-ended exploration. They weren’t interested in my agenda. So, for this […]
(3/8/2018) During our course on Invariants, the eight-year-olds spent most of the time exploring the Euler Characteristic (click here for details). This report is essentially a list of other activities we did to start or finish our sessions. An invariant is something that never changes. Piagetian […]
(Jan. 25 – March 8, 2018) The five students (plus one occasional visitor) in our math circle spent six weeks doing like mathematicians do – savoring a math problem, learning it in depth before any attempts to solve it. I felt like Andrew Wiles in his […]
(October 19 and 26, 2017) “Today we’re going to do something we’ve never done in a math circle here. We’re going to take a quiz.” Stricken looks on faces. “We’ll do it together, it’s just 2 questions, and I think it will be fun.” Relief. “I’m […]
(October 12, 2017) “Everybody get a partner. Take a piece of paper. Take a pencil. Number your paper 1-30. You’re all going to flip a coin 30 times. In each partnership, decide which person will flip a real coin and which person will flip an imaginary […]
(October 5, 2017) We continued work on the Stable Marriage Problem today. After a discussion of the field of mathematics known as game theory, I put on the board the Jane Austen example that Emily Rhiel used in her Numberphile video. Six students used this example […]
(October 2, 2017) I discovered this problem and was so excited – my students would love it, it directly tied in to the topic of our course, and it was a good example of a mathematical algorithm that all of the students would have sufficient math […]
(September 14, 2017) This week we talked about the Google PageRank algorithm. I facilitated it in the same way that Dr. Emile Davie Lawrence did at the recent National Math Festival in DC. I was fortunate to be in the audience during her presentation, which was […]
(September 7, 2017) Today was the first session of our teen course “Our Algorithmic Culture.” My goals for today were to stimulate people’s imagination and curiosity, give students a sense of what an algorithm is, and provide some baseline cases and examples to keep returning to. […]
Here are our unofficial course schedules for the upcoming year. Once they become official (after some possible extremely minor tweaking), registration info will be posted on the Talking Stick website. Our Algorithmic Culture Dates: Thursdays, 3:30-4:45pm, 9/7-10/26 (8 weeks, 75-minute sessions, 10 hours total) Suggested […]
(May 16 and 23, 2017) I thought that this course was about functions. We did a great exploration of them for the first four weeks. But in the last two weeks, something magical happened. Our math circle transcended a single topic to arrive at an exploration […]
(April 18 and 25, 2017) We spent most of our third session creating mapping diagrams, which are a visual way to present and evaluate functions. The students were surprised, after several weeks of numerical function machines, that functions do not have to involve words. They can […]
(April 18, 2017) The goal of today’s session was to get everyone familiar with what typical functions look like. Over the weeks we’ll be ramping this up. Today, though, was the first session of a six-week “Picnie Math” course on functions designed for students approximately age […]
(March 30 and April 6, 2017) I’ve had the song “Pushin’ Too Hard” in my head for the past two weeks. Math Circle is to blame. (Or to credit – I like that song!) The students seem to want to really challenge themselves. Especially since they’ve […]
(March 23, 2017) “What’s that,” asked L, the first student to arrive for math circle. It was a balance scale. I showed him how to calibrate it, then, as the others trickled in, showed everyone how to use it to weigh things. Once our group was […]
(March 9, 2017) “The math circle teacher is here!” announced S to the other students who were playing outside, as I walked onto the Talking Stick grounds. I was early, but before I even had my bag open and the whiteboards set up in the classroom, […]
March 2, 2017 As I prepare for today’s math circle, I realize that I am in a privileged position of utmost responsibility: only one person in this group of six has ever been to a math circle before. For years, the majority of students in our […]
WHAT IS THE MOST IMPORTANT NUMBER IN THE WORLD? The students came up with a list that includes just about everything that mathematicians say is important. (Non-mathematicians have a completely different list, though!) What would a mathematician say is the most important type of number in […]
(October 31 and November 7, 2016) “What are the greatest human accomplishments in history?” I asked at session one of our teen math circle. The students generated a list on the board. “What made them great accomplishments?” I asked. The students gave reasons for the greatness […]
WEEK THREE Voting on How to Proceed with Conjectures: R wasn’t here last week, so the students did a brief recap for her of what we did last time, which was testing their conjectures for which mathematical operation the move “Rotate” represented. “I’ve been thinking about […]
So we’re two weeks into the Rational Tangles math circle. Rational Tangles is an activity within the mathematical realm of knot theory where the students tangle and untangle ropes to uncover mathematical properties. With students this age (12-13), topics such as negative numbers, geometry (rotations, reflections, transformations), […]
Talking Stick Learning Center will offer 5 New Math Circle Sessions This Year Math Circle is a supplementary program at Talking Stick, led by Mt. Airy math educator Rodi Steinig. Math Circles are a form of education enrichment and outreach that bring mathematicians into direct contact with students. […]
(May 26, 2016) Split into 3 teams. Each team gets 5 dice. Roll them all, look at them, but don’t let the other team see them. The first team makes a 2-digit bid. The first digit of the bid is your prediction of the total number […]
(May 19, 2016) This is a graph-theory story best told with pictures, so feel free to go straight to the photo gallery below. But I’ll try to explain it in the words I used with the students: Draw a tree. Any Tree. Sequentially number both ends […]
(May 12, 2016) We spent almost the whole hour examining the “total stopping time” of sequences of Hailstone Numbers, those formed by the Collatz Conjecture. (Pick a number. If it’s even divide it by 2, if it’s odd triple it and add 1.) Is there a […]
(May 5, 2016) We only had 4 kids today due to some illness going around. I had a lot more to say about the Collatz Conjecture from last week, and the kids wanted to talk about it too. But shouldn’t we wait until next week, when […]
(April 21 and 28) So far, the questions I’m giving the kids to work on in Math Circle are leading them to ask very interesting questions on their own. The bullet points below are all questions and conjectures posited by the students, not me. The problems […]
(April 14, 2016) The kids finally solved The Very Clever Prince problem. My helper J and I put a list on the board entitles “What We Know.” (We had never done this before.) One student posited a conjecture which made someone else think of something. That […]
(April 7, 2016) Remember my report on last week’s session, my thoughts about the nexus of pedagogy, kid behavior, and older kids helping? This week put all those conjectures to a real test. We began with 11-year-old J facilitating a problem-solving discussion on our ongoing logic […]
(March 31, 2016) Some older kids helped to lead the circle. At the beginning, R (age 16) sat down on the floor with them to play nim, which the kids know how to play. She asked some open-ended questions: “How do you want to play?” “How […]
(March 24, 2016) The kids got right to the business at hand, no introductions necessary: NIM. They chose teams. The experimented with strategies. They changed their teams (multiple times). After about six games, some strategy emerged. “We should have taken two,” said S at one point. […]
(March 17, 2016) Rachel co-led this circle of 8-year-olds, and with her it began. NIM She threw 4 piles of stones onto the ground and said, “We’ll take turns removing stones from piles until there’s just one left,” she explained. “I’m playing against the team of […]
(February 9, 2016) Our last session.Sigh. TRANFINITE ALGEBRA When I had asked the students two weeks ago what topics they wanted to be sure to hit in this course, F had responded “something with more answers than questions, something definite.” I came prepared with a list […]
(February 2, 2016) My recap of our session is going to be hard to write about for 2 reasons: (1) I’m writing it almost a month later, and (2) we used a lot of jargon and symbols that parents may not understand. But, I remember the […]
People frequently ask me for recommendations on developing their own pedagogical skills and their children’s/students’ mathematical thinking skills. There are a number of exciting opportunities coming up in the next few days, the next week, and next month: FUN MATH FESTIVAL 2/14 Yulia Shpilman would like […]
(January 26, 2016) “COUNTING” THE RATIONALS We began where we left off last week: how can one list the set of rational (fractions and integers) numbers in a one-to-one correspondence with the naturals (counting) numbers? I gave the students some time to work on it, and […]
(January 19, 2016) “I realized as soon as we left class last week that there was a big mistake in our Venn Diagram of number types,” I said to start our session today. “Huge?” asked someone. “Enormous,” I teased. VENN DIAGRAM REPAIR I felt like […]
(January 12, 2016) We had a small group of 3 teens today, so to start the session on an energetic note, I shamelessly presented some controversial material. I showed the students E.T. Bells’ 1937 book Men of Mathematics, a book many consider an accessible classic of […]
(January 5, 2016) “Are there more books or bookcases in this room?” This question thus began our first math circle of the new year. “Do you mean bookshelves or bookcases, not that it would make a difference?” asked W. Everyone agreed immediately that there are more […]
(October 15, 2015) I showed the students a bunch of pictures of basic mandalas.1 People were intrigued, but alas not all intrigued by the same style. SECTIONING CIRCLES EUCLID’S WAY One thing that all of these mandalas had in common, though, is that they start […]
(October 1 and 8, 2015) I’m combining 2 sessions into one report, mathematical conversation topics grouped by how they went over with the students. LAME Compasses: Some students are struggling with their (and my) compasses. Parts get lost. Positions slip. The leads get lost. No one […]
(September 24, 2015) I decided to do something today that I rarely do: to demonstrate to the students how to do something. Last week the kids learned a whole bunch of geometry, but the group did not feel cohesive. Everyone was pursuing different leads and goals, […]
(September 17, 2015) Our new math circle course on compass art for 9-11 year olds began with a switcheroo activity: I gave everyone a printout of an image, and about every 30 seconds said “switch!” I asked them to pass their page clockwise one person. The […]
(May 24, 2015) It was a melancholy day – the last math circle of the year, and also (for those who attend the Day Program) the last day of Talking Stick. As students slowly trickled in, we warmed up by attempting two of Martin Gardner’s line […]
(May 19, 2015) Only R and J were there; the rest of the kids were out sick. We didn’t want to continue with the Dark Bridge/Unicorn problem without the others, so we tackled the famed river crossing problem Missionaries and Cannibals: In the missionaries and cannibals […]
Due to popular demand, we are offering courses on Tuesday and Thursday afternoons this year, for ages 5-18. Expect an announcement any day now for registration. Here’s the course info: COMPASS ART What do Michelangelo, Bernini, Zarah Hussein, feng shui practitioners, mapmakers, architects, astronomers, and mathematicians […]
(April 28, May 5, and May 12, 2015) We’ve had 3 sessions so far, and I see 3 big themes developing in this 5-session course for 9-11 year olds: Everyone thinks that “Everyone Else in the Room is Better at Math than Me.” Not everyone realizes […]
(April 14, 2015) In case you haven’t been following along, a bunch of 6-8 year olds have been tackling the Hadwiger-Nelson Problem (to determine the chromatic number of the plane) with the support of a narrative tale about aliens on a planet called Botso. This week […]
FUNCTION MACHINES (March 31, 2015) The first student to present his Function Machine to the group was N.* He drew the machine, then called on students whose hands were up to suggest “in” numbers. He stated the out number for each, as follows: IN OUT […]
(March 24, 2015) “This is the last function machine I will ever lead in this course,” I announced at the start of class. “After today, you will take turns leading them.” My function machine of the day produced 7 from 3, 17 from 8, 6259 from […]
CHROMATIC NUMBER, 2, ACTING IT OUT (March 17, 2015) “What do you remember of our problem from last week?” I asked. The students called out details until the problem was pieced together. THE PROBLEM Here it is in everyday language: What’s the fewest number of colors […]
(March 10, 2015) As the kids entered the room, they saw on the table were some images of life pods on Mars and… “Mars!” exclaimed someone. Immediately their curiosity was piqued. MARS ONE I told the kids about the Mars One project – a private company […]
(March 13, 2015) As many of you know, I am co-writing a book about math education with Rachel. The working title of the book is How (NOT) to Ruin Math. It will be published by Delta Stream Media, the publishing arm of Natural Math LLC. We […]
(February 24, 2015) Our last session. One parent asked – as everyone was walking out at the end – “Did you come to a conclusion in your math circle?” The answer was a resounding no. This course was more of an exploration. We often begin a […]
ESCHER #5: All About Assumptions (February 10, 2015) I brought in some soccer balls to continue our discussion about which regular polygons can be tessellated. The kids discovered that the balls were a pattern of both pentagons and hexagons. The question became “Why?” Conjectures: Roundness? Size? […]
ESCHER #4: Can Any Regular Polygon be Tesselated? (February 3, 2015) The story of today’s session can be told by a list of questions the participants asked in response to one question I had written on the board at the beginning of class: Can any regular […]
(January 20, 2013) “What’s this for?” asked the kids as I handed everyone a piece of triangular graph paper. I explained that you can make fancy tessellations on this kind of paper. “What’s wrong with squares?” asked A, referring to traditional graph paper. I then handed […]
(January 13, 2015) For the most part, I’m going to let the pictures do the talking this week. They tell the mathematical story of our session via the students’ work and question. We continue to wrangle over the definition/types of symmetry. I posed 2 new questions […]
(January 6, 2015) How do you define symmetry? This question became the crux of our first Math Circle on Escher. Our 11-13 year-olds debated this question for well over an hour. It all started when I passed a bunch of images around: >beehives and other natural […]
TAKEAWAY GAME (December 9, 2014) We began our final session in this course with a few rounds of The Takeaway Game. My goal was to focus attention for deep mathematical thinking, and to plant seeds of a solution strategy for The Very Clever Prince. (We never […]
(December 2, 2014) You may recall that last week everyone realized that “snails” were too big to use as pawns in this game – the piece could be on the finish line and not on the finish line at the same time – a contradiction. So […]
(November 25, 2014) Rather than my conventional in-depth play-by-play, this is a quick report on the highlights of our last session. With the Thanksgiving holiday, 2 kids were absent, so our group size was 5. MATH RED LIGHT GREEN LIGHT, CONT’D This week, we switched […]
I LOVE YOU AS MUCH, CONT’D (November 18, 2014) Right away I got out the book “I Love You as Much,” which we were using as an infinity discussion prompt last week. One of my two assistants, J, read it aloud to the group. When we […]
(November 11, 2014) For the past 2 weeks, I’ve been trying to figure out just how to facilitate the topic of infinity for 5-6 year olds in a way that kids could make some discoveries without me feeding them any facts. I had a bunch of […]
LIFE (October 28, 2014) The kids began by explaining our continuing dilemma to K (who was absent last week): How should we proceed in the attempt to see if any set-ups in Conway’s Game of Life lead to a pattern of continued growth? On the one […]
LIFE (October 21, 2014) Last week’s boardwork was on display before the students arrived. M, who was absent last week, was early for class today. She saw the board and immediately asked, “Why does it say ‘die?’” “Ask them,” I told her, indicating J and L, […]
(October 14, 2014) “I’d like to show you my creatures,” I announced as everyone arrived. I already had everyone’s rapt attention – the word “creatures” will do that. I opened a box that seemingly contained a Go board and Go stones. “That’s a Go board,” protested […]
OLD BUSINESS (October 7, 2014) Since I had forgotten my Gardner books last week, this week I began class with reading aloud his versions of the problems. “His wording is confusing,” commented one of the kids. (To me, his wording is remarkably clear; I suspect that […]
BRONX vs. BROOKLYN (UPTOWN VS. DOWNTOWN SUBWAYS) (9/30/2014) We talked briefly about this problem from last week. It turns out that everyone did remember and agree on its solution. I told the kids what I had been thinking about during the week – that if there […]
In our first Math Circle of the year, we ushered in Gardner’s centennial with a look at some problems from his classic book My Best Mathematical and Logic Puzzles. The problems seem almost whimsical because of how compelling they are. They are, in fact, quite serious; […]
Talking Stick Math Circle Course Offerings, 2014-15 MARTIN GARDNER Approximate Recommended Ages 9-11 6 weeks, 9/23-10/28, 2014 4-5pm Tuesdays 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 […]
(May 20, 2014) Rachel Steinig co-led the session with me today, so I’ll let her tell the story of what happened: Today at Math Circle we did a lot of math (no kidding). WE started out reading a puzzle from the book by Smiullyan. I had […]
THE CRIMINALS OF THE WEEK (May 13, 2014) “It’s time talk about the criminals,” I announced to an exuberant group of kids who were not quite ready to settle into math circle. The word criminals got their attention. We tackled Smullyan’s Inspector Craig mystery puzzle #74. […]
(May 6, 2014) Today I have a quickie report and lots of photos. We started with Smuyllan’s Inspector Craig puzzle #72. Today our variables A, B, and C represented Able, Beatrice, and Cable. I asked the kids if they realized that these are different people from […]
(April 22, 2014) Today was the first of a six-week math circle for ages-6-7. We began with a crime-solving logic puzzle involving 3 characters, Abigail, Bartram, and Carle. This puzzle was Smullyan’s puzzle #71 of his “From the Files of Inspector Craig.”1 The four kids […]
(April 8, 2014) We started out playing a few logic games – “Picking Fruit” and “Wearing Hats.” The students had fun using process of elimination to deduce what couldn’t be seen. The students announced one conjecture, then would immediately change their minds, then revert, etc. […]
COUNTING STRATEGIES (March 25, 2014) We started out on the floor counting pennies. “I want to buy a piece of candy – a Swedish fish. It costs 10 cents. Do I have enough?” The pennies were arranged in a particular way.1 The students counted, with […]
Mathematical Thinking with Five-Year-Olds, 1.0 (March 18, 2014) When I arrived for today’s session, some of the 5-year-olds were already there, bounding with excitement having spied on their older siblings in Math Circle over the past few years. Now it was their turn, and they were […]
(February 25, 2014) Kids arrived to see some yarn and 3 different-sized cans on the table: salt, coffee, and espresso. I asked the kids to point out the key parts of a circle on the cans – the circumference, the area, the (invisible) center, and the […]
(February 18, 2014) I received two interesting Math Circle emails this week: “It sounds like you all had a delightful and spiraling investigation and the participants are really owning the learning… Thanks for packing so much into math circle!” This came from a parent and professional […]
Making Henna, Talking about Petes The story of last week’s Math Circles is best told in photos, over 50 of which you can see here. But here are a few details. In order to make up a snow day we had 2 sessions. On the first […]
Bouncing between Rationalism and Empiricism (February 4, 2014) I felt a smidge of trepidation coming to Math Circle today: Gina was not able to make it with the art/henna component, and I know that at least a few of the kids signed up more for the […]
Mehndi and Circles NOTE: This report was written by Gina Gruenberg this week, with a few mathematical notes from Rodi at the end. (January 29, 2014) Yesterday was finally the second meeting of the “ Nexus of Sacred Geometry and Henna”. I was a bit concerned […]
What is Your Sacred Symbol? (Math Circle January 14, 2014) Students saw two items on the board when class began today: “Name of this course: The Nexus of Sacred Geometry and Henna Question: What is your sacred symbol?” Our visiting instructor, artist/acupuncturist Gina Gruenberg, and I […]
Newcomb’s Problem (Math Circle Teens 4) NOVEMBER 26, 2013 You have a choice of boxes. Box A is transparent and contains $1,000. Box B is opaque, and contains either $0 or $1,000,000. You may take just the opaque box (Box B), thereby “one-boxing,” or you […]
Insuring your Soul (Math Circle Teens 3) NOVEMBER 19, 2013 Imagine you are an adult professional gambler and have the opportunity to play the following game: “Pay $10 to roll a 5-sided die. You win nothing if you roll a 1, 2, or 3; you get […]
Insuring your Bed November 12, 2013 “What is your most valuable possession?” asked Raissa Schickel, our guest instructor. “Probably my bed,” responded a student. Raissa went on to use “bed insurance” as an example as she explained the job of an actuary. That job is […]
A Mathematician’s Task (TEEN CIRCLE 1) NOVEMBER 5, 2013 Suppose you have an opportunity to play a game that costs $1 to play. You have a 50% chance of winning. If you do win, you get $3, but if you lose, you get nothing. Should […]
Pumpkin Numbers and Math Circle Names (Eye of Horus 6) OCTOBER 22, 2013 This week, we finalized our Math Circle names, explored the number theory behind some interesting numbers, did some mathematical thinking, discussed some important ideas, and then presented our own Pumpkin Numbers. NUMBER […]
A Matter of Perspective (Eye of Horus 5) (October 15, 2013) Students arrived to see 7 arrays of blocks on the table, with the following numbers of blocks in the piles respectively: 1, 2, 4, 8, 16, 32, and 64. Well, that was the numbers from […]
A Breakthrough, and Inventing our own Math (Eye of Horus 4) (October 8, 2013) As students arrived, they saw three cubes of different sizes positioned, left to right, on the table with a “do-not-touch-my-number” sign. By the time everyone was there, S and V had concluded […]
Function Machines, Etc. (Eye of Horus 3) (October 1, 2013) In order to imbue our Math Circle with some levity, and to get kids thinking deeply about arithmetic operations, we played Function Machines.* On the board, I had drawn a blob-like shape, of which everyone had […]
MATH CIRCLE: Questioning Everything (Eye of Horus 2) EXCITEMENT AND ENTHUSIASM (September 29, 2013) Many students arrived feeling very enthusiastic about today’s session. Several parents had emailed me about how excited they and their kids were after the last (first) one. K and her mom told […]
(May 21, 2013) There’s more than one way to prove the vertical angle theorem: verbally, numerically, algebraically (with various approaches), intuitively, visually. Our Math Circle participants debated and attempted them all. When the theorem was finally proven algebraically, everyone smiled. It was a satisfying proof. And […]
THEOREM: A cat has nine tails. PROOF: 1. No cat has 8 tails. 2. One cat has one more tail than no cats. Therefore, a cat has nine tails.1 (May 14, 2013) We began today’s Math Circle debating the merits of the […]
(May 7, 2013) It is said that Pythagoras promoted the belief that every number can be expressed as a ratio of whole numbers. This idea was still a bit confusing to our Math Circle participants, who tried to brainstorm some number that couldn’t be expressed this […]
(April 30, 2013) As I entered the building, L, one of my Math Circle lurkers, asked excitedly, “Are you studying The Pythagorean theorem?” I never erase the board work at the end of class so that L (and other building users) can study and think about […]
[juicebox gallery_id=”25″] (April 23, 2013) Before continuing our TV problem, the students recapped last session for A, who had been absent last week. This week’s problem solving once again presented rich opportunities for delving deeply into arithmetic; it has not been the algebra and geometry that […]
PROOFS #1: Aspect Ratios, the Golden Ratio, and Z’s TV (April 16, 2013) Your cousin has just gotten a new job as an announcer on the Golf Channel, and is so excited for you to watch every broadcast. While you may enjoy playing golf, you don’t […]
Logic for the Very Young (February 26 and March 5, 2013) “One afternoon, 2 children wander into a Kingdom unknown to them ….” So began our two-week Math Circle for children aged 6-7. This story framed an exploration of logic games, questions, and strategies. I had […]
The Jabberwock and the Converser “’Twas brillig, and the slithy toves Did gyre and gimble in the wabe” So begins Dodgson/Carroll’s poem Jabberwocky. I read it aloud without introduction as the students colored their own Jabberwock puppets. The coloring focused students’ attention, as they were, as […]
Upcoming Mini-Course: Logic Games In our 2-week Math Circle for 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 […]
The Kids Take Over (February 5, 2013) What would you do if you were settled around the table with your students, about to engage in a civilized discussion about math history, when suddenly the students started chanting in loud rhythmic unison “Puppets! Puppets! Puppets!”? That’s exactly […]
Noise, Normals, and Negators (January 29, 2013) The kids bounded in, filled with curiosity about whether the new puppets, Wags and Rooney, were knights or liars. Each child also wanted to hold a puppet. We had 4 puppets and 8 kids. Easy to manage, I thought: […]
(January 15, 2013) As students trickled in, the kids immediately resumed debate over one of last week’s question, “What happens if an irresistible cannonball hits an immovable post?”1 The new kids, H and L, were now clued in, but the question was not resolved. Once everyone […]
January 15, 2013 We lounged around the room, talking about math. M and I sat on the floor, slouched against the wall while E lay on the floor. J did the occasional unobtrusive headstand while D, V, and C sat in chairs. I never moved from […]
The Monty Hall Problem You are a contestant on a game show. The host shows you 3 doors. He tells you that the prize behind one door is $1,000,000 and behind each of the other doors is a goat. He instructs you to choose a door; […]
Signaling Problem Solution via Proof by Contradiction December 11, 2012: We began our final session of our Signalling Problem Math Circle with a few rounds of Exploding Dots, this time in binary. The large number of explosions in binary (base two) compared to decimal (base ten) […]
Aliens, Number Systems, and Contemplative Mathematics “Some alien spies have been helping our captain. It turns out that you were right: there are enemies hiding on the field behind the house. The captain wants to know how many. The alien photographer comes down and takes a […]
Navajo Code Talkers, Plato’s Cave, and Dots Since the students have been creating their own codes in an attempt to solve our signaling problem, we began class 3 with a lively discussion of the history and cryptology of the Navajo Code Talkers of WWII. Almost everything […]
Plato, Codes, and Exploding Dots “I must add how charming the science of arithmetic is and in how many ways it is a subtle and useful tool to achieve our purposes, if pursued in the spirit of a philosopher, and not of a shopkeeper!’” This was […]
Clarifications, Flaws, Assumptions, and Codes November 6, 2012: “You and your parents are in your coastal house at a time of great danger. A person who can end the danger is in a ship. At some point in the night, the ship’s captain hops into a […]
Hearing What We Want to Hear October 23, 2012: After 3 weeks of work, we finished Bertrand’s Paradox with a discussion of why it’s a paradox. (You can get different correct answers, both theoretically and experimentally, depending upon how you define the term “random.”) This led […]
It’s Good to be Perplexed in Math N came up to the board a number of times as we continued last week’s probability problem.* D had planned to start class with a new approach, but didn’t because his ideas had changed over the week. N first […]
Answers Usually Boil Down to How You Defined the Terms in Your Question October 2, 2012: “Death is a constant.” Or at least so claimed most of our group, after R had contributed “12 inches per foot” and G “60 seconds per minute” to our list […]
This Week Our Group Invents Parameters. SEPTEMBER 25, 2012: Here’s what I expected to happen in Math Circle today: The students would walk in, discuss A’s discovery of an important calculation error last week, and then continue the Popcorn Problem. I had placed unpopped popcorn kernels […]
Geometry, arithmetic, optimization, logic, and question asking. SEPTEMBER 18, 2012: “What would it take to fill this room with popcorn?” “a big popcorn machine” “the machine running a long time” “a lot of patience” “corn, heat, oil” “a small factory” “a lot of people” At this […]
Should we require kids to produce math facts on demand? What does it mean that something is “true,” “valid,” or “proven?” (Hint: they do not mean the same thing.) And upon what fundamental system is arithmetic based? (I’ll just tell you: place value.) These are the […]
“The idea of a Math Circle is to not tell anyone anything.” So began the Math Circle Summer Teacher Training Institute, led by Bob and Ellen Kaplan. Bob continued: “To discover and construct mathematics for yourself is to make it yours. You can use it again […]
My instincts told me that this group was not going to get The Solution to this problem in the time allotted. Is that okay? Falling off your chair, Pokemon figures, a shoe, 1 strand of hair, reading, anger, candy, God: this was the list of […]
Mathematics does not simply mean arithmetic. Everyone noticed the drawing of the birthday cake on the board as they entered today. As kids were arriving, we played “Mathematician Concentration” to focus our attention, but attention was split between the game and the birthday. “Whose birthday […]
Everything has something to do with math. “What do pick-up sticks have to do with math?” asked one of the kids. We were playing this attention-focusing game as people arrived for Math Circle this week. “Everything has something to do with math,” I posited. “What […]
No one had any clue what the rule might be. We started last week’s Math Circle with 2 rounds of “Tens Concentration.” This arithmetic game is also a mindfulness practice, focusing and sharpening attention. You could have heard a pin drop as the kids tried […]
This question was first asked nearly 800 years ago. “In a land far away, King Fudge and Queen Ramona, guided by the wisdom of Red Bunny, peacefully ruled their people.” So began the first session of our spring Math Circle for young children. With the […]
Kids were experiencing math collaboratively without my help. In DC this past weekend, I gave a taste of our Math Circle to some of the 350 children who participated in the 2012 Circles on the Road Workshop. While our Circle informed that workshop, that workshop […]
The idea that math is an art and not a science is not a new idea. Today we examined the life of mathematician/writer Sonya Kovalevsky. We first talked about some of the childhood experiences that shape her life, then took a break to watch Vi […]
What are the numbers in the Fibonacci sequence? What’s the difference between a sequence and a series? Is it still the Fibonacci sequence if you start with a different first number (or two)? What phenomenon in nature prompted Fibonacci to study these numbers? How is the […]
The arbitrariness of variable names. When cake is 20, pie is 5. When cake is 100, pie is 85. So pie = cake – 15, or p = c -15. The students named these variables, figured out the rule, and created the equation after we […]
Kids inventing their own algebra. How do you represent an infinite sequence or series in mathematical symbols? Our group attempted to answer this question in preparation for viewing Vi Hart’s video “Infinty Elephants.” So again we began with function machines. Once the students deduced three […]
Experience the joy of discovery. • the sense of humor to say “I want to whisper the function in your ear because I’m too lazy to do the math.” • the honesty to say “I want to whisper the function in your ear because I […]
Developing collaborative problem-solving through the joyful art of mathematics. Since Euler codified the concept of functions, we started with a function machine. It was challenging to get past the design phase to play with the math, but after applying a chimney to a fish, we […]
Do you feel like you learned some real mathematics here today? Before showing Vi Hart’s “Binary Trees” short film, we did a few function machines as we waited for everyone to arrive. (This session was class 1 of a new session for middle-school kids.) The […]
People working together to solve a problem. This photo tells half the story of today’s Math Circle: people working together to solve a problem. The question was whether the famous “Gas Water Electricity” problem is solvable or impossible. Some students first tried to find loopholes […]
It’s fine to not solve a problem. “I’ve got it!” said X, J, M, N, A, G, and D. Almost everyone came up to the board (excitedly and without invitation) to demonstrate a solution to the famous Konigsberg bridge problem. “Uh, wait… I just had […]
After the total silence that accompanies very hard thinking, the kids were thinking aloud and producing conjectures. After we played with Polydrons and Schläfli symbols* for about 10 minutes, we created a function machine. The students dictated the machine’s parts (various polygons, a cupcake, and […]
Our Math Circle is about playing with, conversing about, and collaborating on math. “We got those for Christmas,” announced G as she arrived and saw the Polydrons scattered across the table. “They were the only educational thing we got.” A brief flash of worry entered my […]
Geometry is not just the basis of language, but is a language in itself. “Oh no, now she’s going to ask us the definition of the word flat,” said Z, after the whole group had agreed that a plane can be defined as a “never-ending […]
The kind of real mathematical thinking that a math circle encourages. Photos of a Julia Set crop circle and a Cissbury Ring crop circle whetted our appetites for circles this week. Everyone had many questions about how the crop circles got there, particularly how they […]
People enjoy math from many different pathways. “I worked on my Flower of Life at home,” said Z as the students entered the room. As she showed me her work, the others began to draw compass designs long before our circle officially began. I unwrapped […]
Some things in math we choose to believe even though we can’t know for sure. Michaelangelo, Bernini, Lloyd Wright (or, the students wondered, was it simply Wright?), Zarah Hussein, Native American geometry designs, mapmakers art, the feng shui compass, and a photo of a piece of […]
All we know for sure is that math is art. I had hoped that we could quickly focus attention by starting with a game of “How Are They Different?” I had thought that we could eliminate the distraction of inequitable turn-taking by saying, “We’ll go from […]
Another mathematical attempt to save the unicorn. The green tape on the rug hinted at a quadrilateral shape, but was actually composed of 4 line segments that did not meet at corners. Is it safe to say that the lines would definitely meet and form corners […]
The 60 seconds of disorder in out classroom reveals something about how people enjoy math: different people enjoy it in different ways. We all have what the Kaplan’s refer to as “the architectural instinct” – a natural desire to pursue revelations about structure. But some of us seek structure numerically, others geometrically, others logically, and so on.
In our group, different styles are emerging, and it will be fun to see if and how they change over time.
I next asked whether math questions always have just one answer, and discovered that not everyone in the group realized exactly what “math” is. The group decided that this stepstool question is identifiable as math because it deals with numbers. We will surely expand this discussion in the future. This would have been a nice place to discuss what numbers are, but alas, we only have 55 minutes each week.
The idea that math questions don’t always have a single answer felt like a turning point in our circle, which has been characterized by frequent musings about what “the answer” could be.
We had a few visitors at Math Circle today, so we started with a collective recap of our voting scenario from last week. Kids immediately started suggesting still more voting/counting methods. I reminded them that the question is whether there is a system for voting that is fair. I briefly told them about Borda Counts, which count votes in such a way.
Some kids were getting distractingly physical with some of the math manipulatives on the table, so we engaged in an attention- focusing activity: the Bobble-Head doll.
We began this week’s math circle sitting in an ogre - not a circle, not an oval, but an ogre, as suggested by the kids. This used the math skill of mentally shifting from the concrete to the abstract.
Once situated, we played a version of the shell game aimed at increasing our attention spans and developing the Jeffersonian hand required for extended mathematical inquiry. Then we moved to our task at hand: to create a scenario that would involve land plots and voting so that we can explore the topic of our semester, “maps, voting, and counting.”
“There is a unicorn dying at the end of a bridge. He has 17 minutes left to live, unless 4 people can join hands around him and recite a magical spell. There are 4 people on the other side of the bridge, but it is very dark, they have one flashlight, and the bridge can only hold 2 people at a time. Ginny can cross in 1 minute, Ron in 2, Fred in 5, and Percy in 10. Can they save the unicorn?”
So began our younger Math Circle today...