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 Paradox). In both courses, we then looked at Escher’s Drawing Hands, discussed some other things, and ended with Function Machines.

### School House Lane (SHL), 9/21/2021

#### MATH

WHAT MATHEMATICIANS DO

Before announcing that *I am lying*, I said *Welcome to Math Circle, where we think like mathematicians. What does a mathematician do?* Five students didn’t know, and one posited that mathematicians engage in problem solving. We then discussed how mathematicians talk about things together, and that problem solving isn’t about applying a rule, but trying to solve a problem without knowing the rule. In other words, trying to discover the rule. (There are 60+ types of math, many of which are not about numbers, and mathematicians can be wrong 90% of the time.) Half of the students in this group have not done Math Circles before, so Melissa, the Director of SHL, suggested that I give them an idea of what to expect. Good idea!

I AM LYING

Students responded to *I am lying* with questions about whether that is true. The main discussion, which emerged entirely from student comments and questions, was about big issues in formal logic: what is a statement, the truth value (a term we did not use) of statements, and much more.

MATHEMATICAL THINKING

Today’s most-uttered phrase was “What do you mean by.…?” The students and I all asked this about terms like “it,” “figure it out,” “statement,” and “prove.” Not that Math Circle is about the Common Core Standards, but “attending to precision” is huge in this group of skills.

Since we were in a virtual classroom, students did a lot of signalling their conjectures and agreement/disagreement with their hands: thumbs up, thumbs down, the wishy-washy wiggly hands. I could actually see students changing their minds as their hand signals changed. It was also exciting to see students sometimes very hesitantly giving a thumbs up or down, and when I asked why the hesitation, students would cast doubt upon an entire conjecture, changing others’ minds.

ESCHER

The students’ discussion of this built on the above, with people wondering what was going on and whether it was possible.

FUNCTION MACHINES

The way function machines work is that students draw a machine that takes numbers in, does something to them, and spits out a number. The students’ job is to figure out what the machine is doing. Today I challenged students to figure out the rule x+5 and its inverse (not using those terms).

#### PEDAGOGY

One of the student goals for MC is to normalize struggle (to a reasonable point). I find that I am a student too, learning through struggle as I facilitate MC sessions. Today’s session was a great teacher for me. I said a few things and asked students whether it was a statement. Then the group took over. I struggled with how to ask students to do this. Do you want to give a statement? Implies that the answer is yes. Finally we settled into me asking students “Do you want to put one to the test?” I don’t want to ask leading questions; Math Circle pedagogy is divergent, not convergent, so the Socratic Method I use in some other types of classes doesn’t work here.

I also struggled in my mind with whether I wanted to give the mathematical definition of the word “statement.” It got to the point where one student asked “If something is false can it be a statement?” How to answer if I’m trying to just be a secretary (or a “sherpa” as Bob Kaplan of the Global Math Circle calls it)? I decided to give two more samples for the class to evaluate and then tell them the definition.

One more thing: I’m so happy that we can linger and savor everything since this is a year-long course!

#### TO-DO LIST FOR NEXT TIME

Next week please bring a pair of scissors, a roll of tape, a piece of paper, and a writing instrument.

### Lovett, 9/22/2021

#### MATH

REFINING CONJECTURES

“Quick, it’s a spotted lanternfly – kill it!” said a student before I could tell them that I am lying. The work to eliminate this one harmful invasive insect generated a student discussion on whether there is math in lanternflies. Once the students concluded yes, I let them know that I’m considering running a course on math in nature in the spring. One student mentioned the Fibonacci pattern as evidence that there is math in all nature. “There’s no such separate thing as nature,” said another student, “since everything originates in nature.” Is there math in chairs? “Yes, chairs come from nature.” Another student said “There’s math in everything touched by humans.” Is there math in things human don’t touch? Another: “Math is everywhere you look.” Close your eyes. “There’s math in everything and everywhere.” I loved how the students used observation and questions and discussion to engage in the mathematical thinking skill of refining conjectures.

EPIMENIDES PARADOX

“I am lying,” I stated to the class. The students immediately discussed this among themselves and agreed that this is some sort of “infinite regress” or “infinite loop” or just “it’s infinite.” (We’re covering material in Douglas Hofstadter’s *Gödel, Escher, Bach*, in which he calls the Epiminedes Paradox an example of a “Strange Loop,” but we’ll stick with the students’ terminology here.)

I gave some history of that problem, involving Epiminedes statements “All Cretans are liars” and “Zeus is mortal.” Our students agreed that didn’t do a good job convincing the citizens of Crete that Zeus is immortal with this argument. [SPOILER ALERT – students in the SHL group have not heard about Epiminedes yet – please don’t tell them]

ESCHER

The students’ discussion of this built on the above, with people immediately latching on to the “infinite regress/loop.” “I detect a theme here,” said one.

BARBER PARADOX

[SPOILER ALERT – again don’t mention this problem to the SHL group.] *The barber is the “one who shaves all those, and those only, who do not shave themselves”. The question is, does the barber shave himself?*

Students questioned terms, looked for loopholes, questioned the question itself (could there be 2 barbers, could a shave be done with scissors and not be considered a shave, does “the barber” mean just one, what do you mean by “themselves,” what do you mean by “shave,” could the barber be female and not need a shave, does the barber with a beard even have to get a shave….?”). Another infinite regress/loop!

RIDDLES

One student remembered a riddle about a barber and presented it to the class. Another presented a locked room puzzle. Then another wanted to present a riddle. I asked if it was related to the math we’re discussing? Since it was not, I said let’s do it at the end. I think we may have already developed a custom for our group.

FUNCTION MACHINES

Everyone in this group had done function machines before, so we had a bit of fun figuring out x+5 and its inverse. One student (and this may have been in the School House Lane group) proposed putting into the machine “sheep.” The student was doing so facetiously, and all were surprised when I said that you can put “sheep” into some function machines, and we’ll do some like that, but that the domain of this machine is numbers only.

I asked whether running a number through the machine both forward and backwards and ending up back where you start, in mathematical notation g(f(x)) = x, whether you’re doing an infinite regress/loop? I expected confusion and discussion and debate, but every student immediately said no. Why not? Because with (“cancelling” or “going backwards”) inverse functions you have the option to stop, they’re just cancelling each other out, you can choose when to be done, but with the loop you have to go on forever. Then students talked about whether a shredder has an inverse function (all agreed no!)

We were out of time, so I suggested thinking about something at home: are there any numerical functions that can’t be undone? (Preliminary conjecture – there are some that involve zero.) Students didn’t want to leave (“Can’t we just keep going since we’re all here?”) but sadly I had to go.

#### PEDAGOGY

I told students up front that I am going to try to tell them nothing in this course, just ask questions. They held me to this throughout the session until we had to refine my goal to now be tell them nothing except necessary background (i.e. I am lying) and math history vignettes. I hope I can manage this!

#### TO-DO LIST FOR NEXT TIME

Next week please bring a pair of scissors, a roll of tape, a piece of paper, and a writing instrument. Also bring a blanket or towel to sit on in the grass.

## 3 Responses

What an exciting and scary exploration. Leaving space for uncertainty and struggle is so challenging. And to keep track of all the students’ curiosity is amazing. Thank you for this blog!

I love that you use the word “scary” in your comment. I’ve heard it said that math is the MOST emotional of all the academic disciplines for the reasons you describe.

The beautiful thing here (and in many math circles) is the allowance of time, and a relaxed atmosphere. On a journey to explore and discover, the students are not forced and pressured by a deadline or specific content. You have created a beautiful space in which your students can pause and wander and think about math.