The post Opening the Bag appeared first on Math Renaissance.

]]>(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 to initiate a discussion about what it’s about. But artifacts matter. So I did show the book I was reading it from: Martin Gardner’s Best Remembered Poems. I told of Gardner’s relevance to our course. Anecdotes matter too. R remembered an anecdote from a prior course, the story of his famous April Fool’s joke. Students were also interested in the story of why I was given the book, by whom, and the cards that I received in the book. The human connection to mathematics was a big theme of all our explorations today, and always.

I then read the poem, let it sit, then jumped into the math.

I told the class briefly about the bag, which had come from the 14^{th }biennual in-person Gathering 4 Gardner (G4G14). I explained that at this celebration, everyone who wanted to participate in a Gift Exchange could, and I had. I hadn’t explored any of the items in the bag. I was waiting to share it with our Math Circle participants.

Me: “Go ahead, look in the bag. Pick out anything that interests you.”

Z chose the little bag with construction pieces labeled Zometool. Z looked at the instructions and asked “What is a 3D cube?” (I added this question to running list that I had started of assumptions, questions, and conjectures.)

Me: Is there a difference between a 3D cube and a cube? Do you know what 3D means?

Z – sorta

Me: Do you know what dimensions are?

Z – not sure

Me: a bunch of hard-to-understand info

W: 3D means it’s not flat!

Me – so can a cube be flat?

Students – flat means like a pancake. Maybe cubes could be flat

I talked about nets, which are 2D figures that can be folded into 3D shapes without gaps or overlap. This was very hard for students to imagine, so I plan to bring some physical nets in for the students to explore another week.

This set of Zometools was a carefully curated kit of pieces with instructions for creating specific geometric objects. Z had started following the instructions but then creativity took over and they created their own mathematical object.

I told the students that I was wondering who at G4G contributed the Zometools. We couldn’t find a person’s name in the package. One of my goals in this course is that students realize that there are mathematicians alive and working now and doing interesting things. Real things. Another of my many goals is for students in this course to see math in action, to present applied mathematics, not just all theoretical work like Walt Whitman’s Learn’d Astronomer did.

The students asked “What did you make for the Gift Exchange?” I told of (1) the G4G Gift Exchange book, (2) that I had written an article for it about a prior Math Circle from years ago, and (3) how I presented some of this writing to the hundreds of people gathered together back in March at G4G 14.

Cube

R pulled out of the bag a small 3D-printed cube (contributed to the Gift Exchange by puzzle designer Oskar van Deventer). R had a lot of questions about it.

R: What are you supposed to do with it?

Someone: Do you think you’re supposed to take it apart?

R: The instructions say disassemble it. But HOW do you dissemble it?

The pieces were somehow linked together. After some struggle, R had disassembled it.

R: I noticed that almost all the pieces are identical except for one – why? And what do I do next?

Me: I don’t know. Do you think that maybe we are supposed to put it back together into its original form, or maybe create something new?

P: Create something new!

At this point, R passed the cube to P, who created something new but not interconnected.

Me: Do you think the creator intends us to build something interconnected or connected?

(Here, I wish I had asked “Do you think it matters whether we do or think about what the creator intends?” That is the real question, IMHO.)

Accidentally but fortunately, P took this item home. Maybe she’ll explore it more. If not, at some point before the course ends, after everyone who is curious gets to play with it, I may show the website directions on exactly what the puzzle is and how to solve it. But for now, I want questioning, exploration, and discovery to be the focus.

R next chose an item labelled “Celt Decision Maker.”

R: How does it work?

W: Is it like those things you can make with paper and two pencils to get answers from ghosts? (Everyone got interested and wondered the same thing.)

Me (not wanting to break the spirit of questioning or to reveal too much): Chances are not since a mathematician made this.

R explored it more, testing it’s motion, figuring out how to use it. Others joined it. R asked me to give it a question about a decision I’m facing.

Me: Should I go home and make my lunch todaCely or buy something prepared?

R: You have to ask a yes-no question!

Me: Should I buy something prepared for lunch today?

(The Celt Decision Maker answered in the affirmative -yum!)

Me: What’s the math question here? What’s interesting about how this thing works? This thing, by the way, is generically called a razorback. Celt Decision Maker is it’s brand name.

More exploration.

R: It controls itself when you spin it in one direction!

Others rushed over to see what R meant.

Me: Do you want to come up with some ideas about why it works that way?

Students explored, but so far no conjectures about that. They were more interested in the question of who created this, since there is an interesting name on the package.

W: Who is Sirius Enigma? It sounds like the name of a plant.

After discussion about whether this might be a pseudonym, students turned their attention to another gift.

W choose this gift, and wondered

- Is it made from 1 piece of paper?
- How do you make it? (W took it apart to see how it was made – like the math strategy of working backwards, starting at the endpoint and going back to the beginning)
- What are we supposed to do with this object (wear it? Create one of our own?) Can you wear it? Is it a pin (jewelry)?

W was worried about damaging this seemingly fragile object. The pin fell apart and got lost for a bit. I explained that I had made a conscious decision that these things can get ruined, rained on, blown away in the wind, etc. No worries. Exploration is more important than preservation!

W noticed that there is a website for this object, and that maybe we can look at it when I bring my laptop.

Me: you can learn almost anything about geometry from folding paper (origami).

The students said they knew this. What they didn’t know or notice is what this object is called and the significance of both words in its name. People did wonder who Akio is and whether this came from Japan. I told an anecdote about G4G and the huge Zoom screen allowing people from very far away to present.

W then noticed that “G4G14” was backwards on the item. Was this intentional? If so, why? How could we find out?

Me: Remember that the creators of these gifts made over 200 of them. If you had to make so many gifts, might you make a mistake?

The students doubted that it is a mistake. Maybe it wasn’t. But one of my goals here is to humanize mathematics, so a mathematician making a mistake is a useful thing to consider.

The gift that P chose had the exact same name and instructions: Build 14 Bridges. She, and everyone, wondered

- What do the instructions mean?!
- Do you build them all at once, or one at a time? Do you build them on the platform provided or elsewhere?

P also pointed out that one of the pieces was broken.

Me: That must have gotten broken in my suitcase. I had to get this entire bag of gifts into my suitcase for the flight home.

Students: Wait, where was this thing?

Me: Atlanta.

The students’ eyes opened wide. They were impressed!

Students pulled out a few other items but decided not to explore them for various reasons. One needed us to look something up online and I didn’t bring my laptop. One, students thought, required us to tie-dye something and “that would be messy.” One gift seemed to want us to have a 3D printer, thought students. Some were things students weren’t in the mood to do, so those went back in the bag for another day or another student. “That’s okay,” I said. “We’re mathematicians. We’ll pursue what interests us!”

I ended today’s session with one of Martin Gardner’s “Sneaky Arithmetic” puzzles. I won’t write it here since I suggested, as did Gardner, that now that student’s figured out the answer, that they challenge you (their families) with the problem. “But our parents are right there!” protested R, pointing at a few parents nearby. I said I don’t think they heard. So ask your students for the one about the seventeen sheep.

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]]>(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 to know is “What is the definition of a statement?”

Me: Try putting something in and I’ll tell you whether it’s in the domain of he function. I’ll start by putting in the sentence *“The sky is red.”*

Students: “False!”

Me: So it’s a statement.

N: *“The FBI killed JFK.”*

Me: Not a statement, therefore not in the domain, therefore, we can’t put it in the machine.

K: *“K boycotted N’s server.”*

Me: Not a statement, therefore not in the domain, therefore, we can’t put it in the machine.

Students: *“O likes monkeys.”* (Background: O has been experimenting with monkey-themed Zoom backgrounds the last few weeks.)

Me: Wait a minute, I think I might be doing the machine wrong. Maybe N’s and K’s sentences are statements. I could be wrong. O, do you like monkeys?

O: Yes.

Students: “True!” This statement went into the machine.

Me: K, did you boycott N’s server?

K: Yes.

Me: I was wrong. It is a statement and goes into the machine.

I had forgotten earlier that we could discern a statement’s truth value by asking for verification. By now students were conjecturing that something is a statement if you can tell whether it is true or false. In other words, if you can discern its truth value. So I presented another sentence.

Me:* “There’s a million dollars buried in my backyard.”*

The students immediately started mentioning ways to test this sentence. “We could ask you.” “We could come over and dig up your yard.” Everyone agreed that this sentence could be tested, and its truth value ascertained, so it is a statement. It went into the machine. We revisited K’s sentence “The FBI killed JFK.” Students debated whether the truth value could be ascertained definitively. They decided that it could not be, so this sentence remained a sentence, not a statement, and it didn’t go into the machine.

Me: What about this one: *I am a liar.*

The students debated it. K suggested that it is vague by using the pronoun “I” and changed it to *“Rodi is a liar.”* The students decided that if I (Rodi) say “Rodi is a liar,” the truth value cannot be ascertained so this sentence was excluded from the domain of the function.

Me: What about this one: *I’ll give you a million dollars if you can make yourself believe that the sky is red.*

The students debated how to discern the truth value of this one. They asked

- Do you have a million dollars?
- If you had a million dollars would you actually give it to me or are you lying? This is probably not a statement since “I’ll give you” can’t be tested with certainty.
- Is it possible to convince myself of this?

The conversation moved into the concept of rationality, something we’ve talked about almost every week. Three of the four students thought they could convince themselves of the sky being red if they were not rational. No one felt that the task could be done by a rational person.

I presented this famous problem by sharing a great video by Julia Galef. I told the students that Galef is a philosopher famous for being a skeptic and for studying rationality. We pondered this for a moment, then went straight to the video, which I stopped every minute or so for discussion.

Galef in video (me paraphrasing): You enter a tent at a carnival and are shown two boxes. One is clear and contains $1,000. The other is opaque and contains either nothing or $1,000,000.

Me (interrupting, stopped video): You can choose one or the other. Which do you choose?

O and K: “Of course you take the box with $1,000 since it’s certain you’ll get something.”

Z: “I would take the opaque box. $1,000 isn’t that much money so it might be worth the risk to maybe get a million.”

N: “I would take neither. I have a terror of gambling.”

Me: Good point. You are right that gambling is something to be afraid of. It is addictive. You should not do it. But is it gambling if you don’t have to pay to do the game? Let’s assume here that it’s not gambling since you’re not risking anything. The game is free. N was okay with this assumption.

Galef in video (me paraphrasing): You can choose just the opaque box or you could choose both.

Me: I was wrong! You can take one or both, not one or the other. Even more interesting! Does this change anything?

I didn’t even need to ask the question “which would you take.” Students immediately started asking their own questions.

K: Do you know the odds? Can you trust this person? How many times can you choose a box? And is it realistic that someone who works in a circus tent would have a million dollars to give?

Me: There’s another version of this same mathematical problem where the person offering you the money is an alien who just landed in front of you, not a circus-tent worker.

Z: This question seems really simple. What else is going on? (Z and the other students know that Math Circle problems usually run deep.)

Galef in video (me paraphrasing): As you walked into the tent, there was a perfect predictor who could tell in advance with 100% certainty which box you would pick. If the predictor predicts that you would take just the opaque box, the circus tent woman puts $1,000,000 in that box, but if the predictor predicts that you would take both, the circus tent woman puts nothing in the opaque box.

Again, I didn’t even need to ask the question “which would you take.” Students immediately started asking their own questions.

K (and others): When does it predict? Does it predict before or after you hear the offer?

No one was sure, so we replayed that part of the video to recall that the perfect predictor was at the entrance and scanned you as you walked in, before you heard the offer.

Z and K: What is the prediction based upon? How you act or how you think?

No one was sure, so we replayed that part of the video to recall that the perfect predictor bases the prediction on your “psychology.”

K: Is the outcome being skewed somehow because you know the judgment? Am I being too nitpicky with these questions?

Me: You are NOT being too nitpicky. You are thinking like a mathematician. A mathematician’s job is to question and doubt every word and assumption in a problem.

More discussion. Confusion. Interest. Enjoyment. Frustration.

A: Is this something a mathematician would wear? I dressed up for math circle in what I think a mathematican would wear. Take a look.

Me: Great question. We’ll come back to that when we finish talking about Newcomb’s Problem. I’ll show you some pictures of mathematicians and what they wear.

K: This is frying my brain.

Me: It’s supposed to!

At this point I was mixed up about a detail of the problem, but fortunately Z helped us remember the basic premises of the problem.

**Spoiler**

O had asked at one point whether you are supposed to base your own box choice on your character, your personality, etc., or on the rules – that everyone who had chosen just one box had gotten the million dollars. We played more of the video, where Galef explains why some people consider this problem a paradox. That in Decision Theory, there are two types of thinking – “causal” versus “evidential,” and that normally both should dictate the same outcome/choice, but in this problem they don’t. (No wonder it fries your brain!) Students also saw in this video that one of the critiques of this problem is that rationality is punished. (Galef gives the example of “I’ll give you a million dollars if you can make yourself believe that the sky is red” – I was trying to foreshadow this in our earlier work on the function machine about statements.)

“I said it seemed really simple at first until I heard it’s a paradox,” said Z.

“I don’t know why you would (or wouldn’t?) pick both boxes,” asked O. K explained the paradox. I was happy to say nothing, thrilled that one of my favorite Math Circle things just happened – students answering each other’s questions and even directing questions at each other not me.

**What DO mathematicians wear?**

N was wearing khaki pants, a white button-up shirt, and a belt – definitely pretty dressed up for a middle-school student doing a virtual session from home! I showed the students my mathematician cards – we looked at them for the wide variety of attire. Some were even wearing t-shirts. We agreed that all of us, even O in the Reese’s Peanut Butter Cup sweatshirt, were dressed like mathematicians.

Me: We’ve been taking about pirates, prisoners, aliens, and circus-tent workers. But also climate talks and social experiments. What is the difference between theoretical and applied math? Z posited that applied math is about real life. I agreed, and said that next week we’re going to apply game theory to a real-life scenario from science, kidney exchanges.

It turns out that not all of the students had heard of organ transplants, organ donors, and organ recipients. We spent a brief time talking about how this works for kidneys, and then I tried to elicit from the group a list of the decision factors that influence whether recipients go ahead with the operation when a kidney becomes available, and whether donors go ahead when a kidney need comes on their radar. The students immediately said that there would be biological matching criteria, but I had to lean heavily on leading questions to elicit the other factors.

Me: “If you needed a kidney and were offered one from a 90-year-old man, would you take it?”

Me: “If you needed a kidney and I said to you that your doctor is on vacation and I’m filling in and I just graduated from medical school and I’m so excited for this opportunity to do this surgery because a kidney just became available today, would you take it?

Me: If you are healthy and I said to you that “my pen-pal in Sweden needs a kidney, can you donate” would you say yes? That last question elicited some interesting conversation about risk. Students asked about gender as a factor, and I wasn’t sure – I told them that the background articles I read (linked below) seemed to indicate that gender used to be a biological match factor but might no longer be. Students were surprised to find out that research shows that race affects the decision to have the transplant because there are different health outcomes for different races. This topic did not capture student excitement/interest the way all of the other topics/problems we’ve done have, so I may decide to present the same problem next week but in a different context. The problem is called the Stable Matching Problem, and it has many possible contexts.

I didn’t like how heavily I had to rely on the Socratic Method to elicit the decision factors for the kidney exchange problem. I wouldn’t have minded had the point of the class been to teach some science, some new info. But my main point here is to coach mathematical thinking. When possible, I prefer to use a pure inquiry-based learning (IBL) approach. The main difference to me is that in pure IBL, we are not trying to direct students to a specific, pre-determined answer. I do sometimes end up using both the Socratic Method and IBL, a sort-of hybrid approach. But I’m thinking that next week, I’ll change the context of this problem to something that the students are already familiar with, and maybe even care about, so that their intellectual energy can be more focused on the mathematical thinking.

I love that I made a mistake with the function machine; I literally did the math wrong! And then with Newcomb’s Problem, I asked the question incorrectly. What an opportunity! Since making mistakes is so necessary in math, I hope that students will feel more willing/confident to take the necessary risks when they see me mess up. I used to think I had to be all-knowing; I was afraid of not knowing the answer. Now it’s the opposite. I feel grateful when I make a mistake that the students see. This brings to mind one of my favorite literary characters, Louse Penny’s Armand Gamache, who is to known to give the advice that the four most important sentences on the way to wisdom are

- “I don’t know.”
- I need help.
- I’m sorry.
- I was wrong.

This article on the Wisdom of Gamache refers to applying these concepts to police work, but I think they apply to teaching as well.

As I was struggling to keep track of the details of Newcomb’s Problem in class, I told students that in my preparation this week I had done an internet search to see if anyone had ever used this problem in a Math Circle before and had written about it. What turned up was my own blog! It turns out that I had done (and written about) this problem nearly a decade ago. I don’t remember it at all!

It helped me so much to have a beginner’s mind, to not let my past experience create any self-fulfilling prophecies. Fortunately, this group of students did the problem very differently than the prior group.

Here are links to that prior session a decade ago and an interesting article about beginner’s mind:

Newcomb’s Problem (Math Circle Teens 4)

How a Beginner’s Mind Can Improve Your Teaching and Coaching

I do carry around with me a set of Mathematician Cards for moments just like today. And I have them on my computer for virtual classes. I was grateful for this today. These cards help me to attempt to “rehumanize mathematics,” to quote Rochelle Gutiérrez, who is pictured in the cards. Her talk, “Rehumanizing Mathematics: a Vision for the Future” has been greatly inspiring me for the past four years:

**Background**

Newcomb’s Problem: The full title of Galef’s video is “Newcomb’s Problem and the Tragedy of Rationality.” I highly recommend it:

We only watched about half of it. I think these students would enjoy finishing it on their own, especially about 7 minutes in when Galef presents the problem “Parfit’s Hitchhiker,” which is about rationality and what it means (it’s connected to “the sky is red” example). If you don’t have time to watch a video, here’s a very brief write up of Newcomb’s Problem from a site I like. (This is a version with the alien, not the circus-tent worker.)

N’s comments got me curious, so I later looked up the legal definition of gambling. Gambling “means the staking or risking by any person of something of value upon the outcome of a contest of others, a sporting event, or a game subject to chance, upon an agreement or understanding that the person or another person will receive something of value in the event of a certain outcome.” So our in-class assumption that if nothing is risked it’s not gambling holds.

I ran this course and wrote the majority of this report back in February. In August, I had the opportunity to talk to some mathematicians and Math Circle leaders at the conference MathFest. Several people insisted that “it is impossible to teach probability without talking about gambling.” Others, myself included, argued that it’s really important to respect students’ and families’ values on this and find a way to teach probability without examples that model gambling. I think I was successful with this goal today, but not with the Buttered Toast Problem the past two weeks. Of course, my students, who developed an understanding of some of the basics of economics, would probably argue that you are gambling in Newcomb’s Problem if the “opportunity cost” was your time and your time was “of value.”

I used the articles below to learn about what the decision factors are. I love science, the scientific method, experimental design, results analysis, and meta-studies. But just because I find these topics fascinating doesn’t mean that you or my students do or will. Today was a good reminder of that for me.

- Determinants of kidney transplant candidates’ decision to accept organ donor intervention transplants and participate in post-transplant research: A conjoint analysis
- Analysis of Factors Influencing Kidney Function of Recipients After Renal Transplantation in Southwestern China: A Retrospective Study
- What are the most important donor and recipient factors affecting the outcome of related and unrelated allogeneic transplantation?

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]]>The post Toast, Coins, & Mud appeared first on Math Renaissance.

]]>(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 last week, I had the other students explain this problem to him as I acted as secretary on the slides. (We hadn’t finished solving the problem last week, but the students did finish writing it.) Also, I wanted to get O’s input on the problem before getting into answering it.

*The problem: You’re locked up in the basement of Leshy’s cabin (but not in a bad way). You have 5 arcade tickets that you can cash in for prizes or cash once you get out. You may play a game that costs 1 ticket to play. If you win, you earn 3 tickets. If you lose, you get nothing. You have a 50% chance of winning. Should you play?*

Me: Do you have any questions, O?

O: Yes! I want to know what the game is.

Me: We never stated that last week. I had something in mind last week, but I never mentioned it. I didn’t think it would work with the group’s stated assumption that the problem takes place in a basement.

Students: What were you thinking?

Me: Don’t you prefer to choose an assumption that makes sense about what the game is? (Last week, the students stated assumptions, thereby creating some of the premises of the game.)

Students: No. (Huh! Usually the students want to write the problem. I like to think that they didn’t want to write the problem – specifically the context – this time because they wanted to get on with the math, but I’m not sure. I wish I had asked.)

Me: Okay. I was thinking that the game is this: you stand on a high platform (as high as a bungee jump) and drop a piece of buttered toast off it. If it lands butter-side-up you win, and butter-side-down you lose. Assume you have a 50-50 chance of either side. (I expected arguments about this probability and had even read up on it, but no one questioned it.)

K: Let’s just use that. (Everyone agreed, and I edited the problem to state that Leshy’s basement has a very high ceiling.)

Students were still using their gut feelings (math intuition plus emotion) to answer the question. I asked “What would the outcome be if you played the game one time? Two times? Three times?”

I took periodic polls to see if the group had come to a consensus about the answer to the problem (should you play?). Eventually they all said yes, after simulating playing 5 times. Some of them doubted that they would play if they had only one chance to play. I chose not to encourage more work on this problem because the students really, really wanted to flip their coins, which I had asked them to bring. I did ask them to name the problem – something that normally students are excited to do, but not this time. K suggested calling it “The Buttered Toast Problem” and everyone concurred.

Before starting the coin tossing, I wanted to talk about it for a moment, but the students just wanted to start flipping. Finally I was able to get in my questions:

*What is the probability that you get tails if you flip once?**If I just flipped a coin 10 times and got tails each time, what’s the probability of getting tails on the 11*^{th}flip?

The students said that the probability of tails is 50% if some very specific requirements are met (see below) – I also introduced the idea of a “fair coin.” At first the students said there were pretty low chances of getting tails on the 11^{th} toss after 10 tails, but then realized it would be 50% because of the probability of independent events. Now time to toss!

I asked them to first flip their coin one time and record (on a Google sheet) how many tails they got. Then to do it twice, then three times, etc. “What does tails mean?” asked Z. The others explained. Then the flipping and recording began. “What number should we go up to?” asked O. I suggested 12. After everyone had done trials up to 12 flips, we looked at the data.

The students did notice that with the smaller number of tosses, there was a lot of variability in the percent of tails results, but that it got closer to 50% with more tosses (the Law of Large Numbers).

“What I’m interested in,” said K, “is what the total is.” I totaled up the column with the sum formula so that we could immediately see that 312 tosses produced 150 tails – 48%. We were all impressed with the math.

N also pointed out how impressive spreadsheets are at doing calculations – almost magical! Some of the students had never seen spreadsheets in action before. At one point when the students were doing the experiment and recording their raw data, N had typed a number over the percent formula in the row and said “I broke the spreadsheet!” I fixed it in less than a second, again exciting to students.

I asked for predictions about how many tails you would get if you tossed the coin a very large number of times. Students suggested the thought experiment of tossing it 1,000 times, 50,000,000 times, and 2,020 times. They predicted 50% tails each time based upon the Law of Large Numbers.

Me: How does this relate to the Buttered Toast Problem?

Students: The more you do it, the more stable the results become.

K: There’s a name or rule for that.

Me: Yes, the Law of Large Numbers.

Me: Would you eat mud for a million dollars?

Students: How much mud? Would it kill you?

Me: A pie pan full of mud, and it would not kill you.

Everyone but N said yes. N said he would not eat mud for any amount of money.

Me: Would you eat it for a *billion* dollars?

Everyone but N said yes. N said he would not eat mud for any amount of money.

Me: Would you eat it for a *trillion* dollars?

Everyone but N said yes. N said he would not eat mud for any amount of money.

Me: What if you already had a billion dollars. Would you eat the mud for a million dollars?

K: That depends. Do I have a source of income?

Me: Yes.

K: What is my salary?

Me: A million a day.

Now everyone said no, they would not eat the mud.

Me: But a few minutes ago you said you would eat it for a million!?

The students explained why they wouldn’t, which I told them is called the “diminishing marginal utility of money” in economics. (In other words, context matters – a key idea in the mathematical discipline of Category Theory, which I did not talk about.)

Me: How is the Buttered Toast Problem different from the Pirate Problem, the Prisoner’s Dilemma, and the Triple Dominance Game?

Students: Random chance. It’s about what you would do, not about figuring out what some made-up people would do

Me: In Decision Theory, one actor is making a choice. In Game Theory, multiple actors whose choices affect others.

I used James Tanton’s material in his book about function theory to present functions without numbers, which the class had requested last week. I presented some function machines (input and outputs) and the students, as usual, were supposed to figure out the rule. We ended up delving into formal logic as well as function theory.

My original goal in doing functions without words was to get into the concept of the actual definition of a function beyond the idea that it’s a rule. That there is only one output for each input. That they can be mapped. That there is a notation system. That some functions are not well-defined. Again, time to let go of my expectations! I feel so privileged to be able to teach in a way that allows me to follow the students’ curiosity for some very deep learning.

Buttered Toast: My original goal for the Buttered Toast Problem was for the students to lead themselves to discovering the mathematical (probability) concept of expected value. These students are so interested in the application of math to the behavioral sciences that I thought we could progress into a lot of different things with expected value. Didn’t happen. (Had we delved into “expected value,” the answer to “should you play?” would be mathematically yes.) At first, I couldn’t figure out why these students weren’t excited by this problem; I had done a similar problem with older students – ages 13-15 – and they had come up with this concept and solution.

These students are younger, ages 11-13, and haven’t had the same math experience, much less work with percents and probabilities compared to that older group from the past. OTOH, this current group has been doing game theory for six weeks now, which the other group had never seen, so of course the math will go differently! (These students, unlike the older ones a few years ago, are using the language of game/decision theory, i.e. decision factors, rational actors, etc., naturally and automatically. Note to self: I need to let go of my expectations more.

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]]>The post Tragedy of the Commons, Elinor Ostrom, and the Triple Dominance Game appeared first on Math Renaissance.

]]>(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 discipline of game-theory exists, and (d) that there’s a Thing called Behavioral Economics that explores whether humans behave rationally. In class 3, we explored The Prisoner’s Dilemma to discover that if humans behave “rationally,” we may choose outcomes that benefit the individual at the expense of the group. In class 4, we applied The Prisoner’s Dilemma to climate talks to (1) see how people apply mathematics to attempt to solve real-world problems and (2) debate the appropriateness of this math model.

One member of our group, O, had been a bit discouraged about humanity after last week’s discussion, so I chose some topics for today that offer hope. Sadly, O was absent today, but I decided to do these topics today anyway to keep the narrative arc of our course going.

Today, in class 5, we continued examining situations like the Prisoner’s Dilemma and Climate Talks, often called “The Tragedy of the Commons.” I delved into some math history, explaining the work of Elinor Ostrom, who won the Nobel Prize. We did a simulation of the Triple Dominance Game, which uses math to try to understand some of the psychology of human decision-making. Then we moved into another area that uses math to measure uncertainty and attempts to use math to predict human behavior: probability. Here’s how that all went:

“The Tragedy of the Commons” is the conjecture that when there are collective “common pool resources” that can be depleted by individual users, humans may act like the prisoners in The Prisoner’s Dilemma; in other words the “dominant strategy” might/will be chosen at the expense of an outcome that is good for everybody and that would leave the individuals better off than the dominant strategy does. Yikes.

So the core question is this: Are common pool resources inevitably destroyed?

I gave our group some examples to lubricate their thinking: grazing lands, fisheries, and irrigation systems. The students came up with more, including

- Peace (international relations)
- Ocean
- Oil (fossil fuels)
- Land
- Cimate
- The Earth (as evidenced in the movie Don’t Look Up)
- Characters (in the context of gaming, where people might choose fun characters over wise ones because of the short-term benefit of enjoyment)

I mentioned the work of ecologist Garrett Hardin, who first coined the phrase “tragedy of the commons.” To put things into historical context, I also showed an image of him from the Southern Poverty Law Center site, to show that he is not only famous as an ecologist but also infamous as a white-nationalist extremist. I asked, “If someone behaves badly does it mean their ideas are bad?” I also asked, as we’ve been asking over the past three weeks, does The Prisoner’s Dilemma or The Tragedy of the Commons give us reason to lose hope? Or, using more mathematical language, can the conjecture of The Tragedy of the Commons be proven or disproven?

I told students that Hardin saw only two solutions to avoiding the destruction of common pool resources. I asked the students for conjectures on what these solutions might be. The students came up with one of Hardin’s solutions: strict government regulation. Hardin’s other solution was privatization, which I briefly explained as ownership, free markets, capitalism. “But why,” asked Z, “would that solve the problem?”

“Maybe it’s a long-term solution, that would address things over a very long period of time,” replied K.

“The East India Tea Company!” argued N. The students were driving the conversation into politics and economics beyond the scope of Math Circle, so I put a world map up on the screen.

I said that Elinor Ostrom did research on this very question: Is the Tragedy of the Commons prophetic? I asked the students to mark the places where Ostrom did this research. I explained that Ostrom discovered that as long as the decision makers are physically close to the resource, the resource is not destroyed. She disproved Hardin’s conjecture! Hope! Not just a win for the good guys, but hope for the planet.

I then showed a slide with an image of Elinor Ostrim on the left and a picture of the Nobel Prize on the right. I asked, “Why do you think I’m showing you this?” The students excitedly responded that she won the Nobel Prize.

*Here’s a game you play with another. You can choose how it’s scored:*

*You get 550; the other person gets 300.**You get 500; the other person gets 100.**You get 500; the other person gets 500.*

*Which would you choose?*

Immediately the students got into a brisk debate about what people in general might choose and what their choices reveal about them. I told them nothing about anything. The consensus conjectures was that A will be the most common choice and C the next.” I gave them a link to a Google form to fill out as many times as they could.

They could fill it out however they thought people might answer and could use pseudonyms for each form. Then I set the timer and said “Go.” Here were the results:

After seeing the results, participants asked each other whether they were trying to use their responses to “throw the poll” to obtain a particular result. It turns out they were not doing this much. They were happily surprised to learn that their results mirror what studies have found, that the majority of people in our culture are probably “cooperators.” Our group likened cooperators to Communists, Bengali people, and/or South Indian people in general. These students were also relieved to learn that their simulation resulted in a larger proportion of “competitors” than studies show. Our group identified competitors as “mean” and/or “cruel.” Our group likened “individualistic” responders to “Americans,” “capitalists,” and “Mark Zuckerberg.” Our group also brought up that there could be a fourth option: getting less if you win. (Turns out that theorists identify this group as well: “altruistic.”) The students had a lively discussion about this game, which encompasses topics including decision theory versus game theory, social psychology, data collection, graphing, positing conjectures, and that most-important skill needed for mathemathics: communication. How can you share your ideas if you aren’t able to express them, right?

FYI, I learned about the Triple Dominance Game in this article in The Shuttle (scroll to page 9).

Now we had just a few minutes left.

*Here’s another game you could play. It costs one ticket to play. If you win, you get 3 tickets. If you lose, nothing happens. You have a 50% chance of winning. What do you think the question is?*

The students immediately knew that the question is “Should you play?” They looked at this I the way economists would:

- “How many tickets do you have?”
- “Are there lots of other fun games with better payoffs?” (In other words, what is the opportunity cost?)
- “Or are you stuck in a cabin in the woods somewhere?”
- “Is money your only decision factor?”

For each question, I answered “I don’t know. You tell me.” I suggested answering the questions in a way that would make the problem the most interesting mathematically. The students then stated the following assumptions:

- You have 5 tickets.
- The tickets are each worth $1.
- You are locked up in the basement of someone’s cabin. In fact, you are locked up in the Leshy’s cabin. Leshy doesn’t want to hurt you; you are in no danger. It’s just for fun. (I had to ask. N explained that the Leshy is the forest spirit in Slavic mythology.)

With only a few minutes left in class, the students decided unanimously, without a mathematical explanation, that you should play the game. (“Why not, there’s nothing else to do!”) To be continued next week.

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]]>The post Using Math to Model Climate Talks appeared first on Math Renaissance.

]]>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 just the Nash equilibrium solution, but also which way each pirate would vote. Sadly, I forgot to take a screenshot of the student work, but we made a quick summary in week four (2/1) to recap for K, who was absent when we finished.

We moved on to the most famous of game theory problems:

*You and your partner-in-crime get caught. The police separate you and offer you each the same deal:*

*If you both stay silent, you each get 1 year in jail.**If one of you confesses but the other stays silent, the one who confesses goes free and the other (silent) one gets 4 years.**If you both confess, you both serve 2 years.*

*What should you do?*

The students struggled with this using a decision matrix for a while, until Z said “I’m having trouble understanding who you’re referring to.” (I was saying “you” and other pronouns and it was confusing!) The students then suggested naming the criminals; N proposed calling them Gallavan and Rubbert, which made things much easier to keep track of.

**SPOILER**: After a lot of work, the students determined that the dominant strategy for both Gallavan and Rubbert is to confess, that confessing is always better, that the outcome that’s best for the individuals is not the outcome that’s best for the group (which would be both staying silent.)

*Can we do something to insure the outcome that’s best for the group?* I asked.

“I don’t think so,” said O. Everyone discussed and then agreed with O, for now.

Sadly I (again!) forgot to take a screenshot before the 40-minute-Zoom break so I lost the students’ work. We quickly reconstructed a summary after the break, but this image understates the hard work that went into solving the problem.

At the start of the session 4, we watched a quick video of the problem to refresh memories and to introduce the problem to K, who was absent last week. K commented that “these rules make absolutely no sense in the real world!” (Another chance for me to make the point that math problems don’t have to be consistent with the real world; they only need to be consistent with themselves.)

*Would it change things if the prisoners knew the payoff schedule in advance*? I asked. Would both actors both still be best off by confessing? At first the students all said yes, but after more discussion, they thought that the answer comes down to whether you can trust your partner.

“Then don’t have a partner in the first place! I wouldn’t!” said N. I mentioned that changing the premises of the problem, as A also suggested doing in the Pirate Problem, can sometimes be a very useful math strategy.

*There are two countries in the world: North and South. The world needs to prevent the world temperature from increasing by more then 2 degrees Celsius or life as we know it will be drastically harmed. Is climate mitigation good for everyone? *(“Yes”)* Mitigation will cost 6 in every region. If only one region does it, it will generate a profit of 4 in both regions. If both regions do it, there’s a profit of 8 in both regions. What do you think the question is?*

“Should you do it?” said several students.

“Of course you should do it!” responded the other students immediately.

“Do the countries trust each other? What are the decision factors?” asked Z.

*Let’s first assume countries don’t trust each other, but then talk about how it would play out if they do. Assume that profit is the only decision factor.*

**SPOILER**: The group used a decision matrix to determine that the dominant strategy for each country is to pollute not mitigate. In other words, just as in the Prisoner’s Dilemma, looking out for one’s own best interests only is better mathematically than cooperation, even though cooperation gives a better outcome for the group as a whole. The students decided that you get a different answer depending on whether we are using the conventional game theory assumption of rational actors. “It comes down to trust,” they said.

*Can countries trust each other, and if so, under what circumstances?* I asked.

The students, some of whom are more knowledgeable about current events and history than I am, pondered how this problem might play out differently if the countries “North” and “South” represented the US, China, Russia, Korea, Bosnia, and others. I found it interesting that the students related Climate Talks to The Pirate Problem more than to The Prisoner’s Dilemma, the traditional model in the field of International Relations (IR).

“This is all politics!” said both N and K.

*This is International Relations, where people in this field really do use math*! I explained.

Now I was finally ready to ask the question that I had been excited for weeks to ask (four weeks of work to build up to this):

*Is The Prisoner’s Dilemma a good model for climate talks?*

I explained to students that some people who discuss the field of IR think it is and that things are pretty hopeless. Some people are unsure. Some people think it’s a bad model. *What do you think?*

The students had a huge discussion about this, with their many perspectives summarized below.

- It’s a good model because
- things do feel pretty hopeless

- It’s hard to tell because
- what it if leads to war?

- What if countries can punish or threaten each other?

- What if the threat of war is the only deterrent?

- What if sanctions could work?

- What if binding agreements could work?

- What if countries do look at the long term, instead of the short-term reelection of their leaders? (Students pointed this out as an issue in The Pirate Problem too.)

- It’s a bad model because
- in reality, not all countries are equal (in their power/influence and in their contributions to global warming).

- In reality, “not all leaders are scumbags” – this is what K really said when I wrote the bullet point “ethical leaders” on the slide. My in-class paraphrases in the interest of speed and brevity lost something in translation.

- In reality, there are other decision factors; climate change is more complex than The Prisoner’s Dilemma.

- In reality, people (governments and companies) realize that mitigation is a public good

- In reality, it might not be so expensive to mitigate.

- In reality, members of the public can/do put pressure on their leaders.

- In reality, decision makers “are not robots” (i.e. they are not rational actors like game theory assumes).

In the end, K, Z, and N came down on the side of The Prisoner’s Dilemma being a bad model for Climate Talks. O came down on the side of hopelessness. I didn’t want to leave students depressed from this topic. So I stressed that my belief is that there is hope.

Z concluded that “It’s cool that people try to apply math to real-world problems.”

Then I moved on to a more whimsical problem for some emotional relief, one of my old favorites.

*“You are exploring a land populated by hydrophobic vicious animals. You are safely wading in a one-foot deep stream when you come to a fork in it. Each branch leads to a different pond. Each pond has a helicopter on the other side of it that can transport you to safety. A sign at the fork tells you that the pond on one side has an average depth of 5 feet, and the other is 7. Oh, and did I mention that you can’t swim?”*

I presented the problem and said nothing. The students initially agreed that the answer is 5. But then someone eventually questions the word “average.” Students debated and calculated and asked questions such as Z’s: “How tall are you?” They came to the realization that distribution/spread actually matters more than average/central tendency. O concluded that if you’re guessing, you might as well assume that this is a typical pond that is more shallow around the edges. The others agreed.

In week 3 (1/25), O presented a function machine where the students had to guess the rule y=x/3. This led to a quick discussion of square roots, and that “divided by 3” not the same as square root

In week 4 (2/1) N asked to limit the domain to less than 100 but the students wanted me to present a rule for them to guess. First I gave a rule where the students derived a conditional function (“the next even integer if it’s odd, plus two if its even.”) This conditional rule does work. I challenged them to think about after class whether they can simplify, in other words, think of a way to express this rule as a single, not conditional, function? (Note from future me: we never came back to this; the unconditional rule I was thinking is that the output is the next larger even integer.)

With 2 minutes left, I gave another function in which some students defined the function as y=x-x and others as y=0x. I mentioned that different equations can give the same results, that even if the output is the same every time it is still a function.

N brought a nice math quote to class today: “Every triangle is a love triangle when you love triangles” (Pythagoras)

The students have been enjoying The Pirate Problem on such a deep level that I’m sticking with game theory to build up to problems about using game theory to model real-life situations. The particular students are very interested in philosophical themes – cooperation, rational actors (is there such thing and how do they act), trust, and the validity (or not) of mathematical models.

NOTE: This reports includes references to human trafficking, Chernobyl, hypothetical violent actions, guns, and student reactions to these topics. Feel free to skip it and jump down to the next topic.

During Session 1 of this course, N was appalled by one of the Decision Factors of the Pirate Problem: that *If the majority vote no, the proposer is thrown overboard and the next most senior proposes a distribution. * I had given the students a trigger warning but the shock value was still there. I didn’t anticipate that. There has been ongoing discussion in the Math Circle community on how problems involving life and death are highly motivating but … are they actually appropriate for students?

During Session 4, when I showed the video of The Prisoner’s Dilemma, which was essentially a slide show of drawings, the first image generated a reaction from N: “Guns!” (The image was a drawing of a pirate holding a gun.) I didn’t anticipate that.

I also didn’t anticipate O’s reaction to the seeming hopelessness of climate change; nor did I anticipate the class discussing the possible need for force and threat to keep countries in line

A few years ago, I taught a Math Circle course on “The Mathematics of Social Change.” In this course, we discussed how math can be used to reduce human trafficking. I fortunately did anticipate student reaction to this and was able to inform and educate families in advance and also invite parents to sit in on class. I also polled parents in advance on whether to explore this topic. All said yes. In a course on the history of Fermat’s Last Theorem, I decided to not include anecdotes that involved violence. Planning helped a lot. I have used my judgment about what age groups to mention things to.

Unfortunately, I haven’t always anticipated which topics may generate a reaction, and these students are not adults! (In the current group, they are 11-13. They were a bit older in the Social Change course.) Sometimes the students, not me, bring up potentially-upsetting topics (like Chernobyl, or anecdotes involving violent behavior, which I immediately quashed). Sometimes something flies under my radar, like the gun in the Pirate video – I knew better because in a past course there was some family preference that I not teach the problem “The Truel” which involves guns. But I just didn’t notice, and that will happen sometimes.

Moving forward, I think my plan will be to always try to remember that these are children! I’ll continue to give trigger warnings and ask parents up front via email. Ideally, I want to have a list of things that I should warn people in my Math Circle community about: crime, guns, climate change, human slavery, war, international relations, etc ….. I invite members of this Math Circle community to put in the comments your suggestion of topics and age groups for this list.

On the other issue: how to handle things students bring things up. I plan to continue to generally shut it down right away, saying just the mention of that thing could make someone the class feel bad. Nuclear war was very relevant to our discussion today, however, so didn’t shut it down, but again wish I had anticipated and warned people up front.

Just a quick note about things I’ve been learning about this specific platform of Zoom: one student got locked out after the break because he came back a bit late and I was sharing my screen, so didn’t see the “admit” button so easily. Also, ugh, twice I lost the work when I had to exit and reenter because of the 40-minute rule. (In the past, I was using a different version of Zoom without this limit.) Here’s my new system: when we are about to break, have the students remind me to take a screen shot! (Future me reports that this has worked extremely well.)

Function Machines: On student asked “can you give us a hint?” I said to ask the others, not me. The others did not want a hint. They said that if they didn’t solve it after a few more minutes, then a hint would be okay. I was so close to giving the hint and thereby lessening their joy of discovery. So glad I didn’t!

**Differing Perspectives on Applying the Prisoner’s Dilemma to Climate Change Talks**

- The Prisoner’s Dilemma – Explained by Climate Change (This is Astrid Lensink speaking in Dutch with subtitles, and is easy-to-undertand and great! It’s presentation is exactly what I copied with the students.)
- Climate Change: The Prisoner’s Dilemma
- Why Climate Change is No Prisoner’s Dilemma
- Prisoner’s Dilemma and the Environment

One little detail: with my 20/20 hindsight, I would rewrite this problem to say “revenue” instead of “profit” (my economic training showing here – but the students didn’t question it).

**Game Theory**

I’m using these elements in a game – the actors, the rules, the decision factors, equilibrium, and the problem.

The Pirate Problem – Famous Game Theory Puzzle

The Prisoner’s Dilemma – the Most Famous Problem in Game Theory

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]]>The post THE PIRATE PROBLEM (Applied Math 1+2) appeared first on Math Renaissance.

]]>(Jan 11 and 19, 2022) *There are 5 pirates: A,B,C,D, and E. They plunder a treasure of 100 gold coins. You are the captain, A. How do you propose splitting the treasure?*

*Here are the Pirate Rules for Splitting Treasure:*

*The most senior pirate proposes the split.**All pirates get a vote on the proposal.**If the majority vote yes, the proposal is accepted.**If the majority vote no, the proposer is thrown overboard and the next most senior proposes a distribution.**If there is a tie, the proposer casts the deciding vote.*

*What do you think the math question is here?* I asked.

“It’s obvious,” said the students. “How do you divide the money?”

*Yes. If you are A, what distribution should you propose? *I clarified.

“It’s obvious,” said N (and others). “Everyone gets 20.”

*But these are pirates*, I countered. *What do you think they care about?*

“They want the most money. The captain would like it all if he could.”

*What else do you think they value?*

From this discussion, the first two of the problem’s decision factors emerged. Then the third, and finally the fourth.

*Each pirate wants to survive.**Each pirate wants to maximize their share of the treasure.**Pirates want to see someone thrown overboard.**Pirates don’t trust each other.*

(I loved that the students were able to figure these out without me telling them. Those of you who’ve been reading these reports this academic year know that one of my personal goals in improving my pedagogy is to become even more inquiry-oriented. Hence, not telling the students the question or the decision factors.)

**MAKING THE PROBLEM OUR OWN**

I’ve heard versions of this problem where the “strongest” pirate proposes the distribution, or the “most senior.” Our group wanted to refer to the proposer as “the captain.” If he/she gets thrown overboard, the next one in line becomes captain.

The students named the pirates Albert, Biraj, Callie, Daniel, and Esther. By the end of the first session, however, most students were referring to them as ABCDE. The math quickly became compelling enough to let go of some of the narrative.

**RATIONAL DECISION MAKERS**

In game theory, the actors are rational decision makers. We had a lot of discussion about what “rational” means. The students understood it as all actors anticipating every move like a computer playing game of chess. Each actor knows everyone else’s strategies/priorities/decision factors. The students corrected each other throughout the sessions with the reminder “These pirates are robots!”

I repeatedly had trouble understanding K’s question about whether the pirates care about the future. Finally, in the second week of K clarifying this question, it dawned on me that the question is whether future treasures beyond this one are decision factors. They are not.

The students spent time both weeks discussing how the decision factor of mistrust affects the solution to the problem. I clarified that pirates will not form alliances with each other.

This problem gets pretty confusing and frustrating when you try to wrap your mind around how each pirate would vote on any distribution of five amounts, especially when you must meet all four of the decision factors. If the captain would propose that she gets 60, wouldn’t she then want to try for 61? Or 62? And so on. That’s what happened in our group – people disagreed, couldn’t justify their answers with definitive reasoning, and proposed solutions that go beyond the premises, especially the requirement of rational decision making. (A normal reaction to this problem.)

O proposed changing the problem: “Lets make one of the decision factors be who do you want to see die?” A few others liked this idea.

*This is a famous problem, and many people have created variations over the years, but let’s stick to the problem as written for now, *I replied. *Your idea would make a really interesting problem.*

“It does seem like there are a lot of politics in this problem,” observed K.

**GAME THEORY**

*Do you know what area of mathematics this problem is?* I asked the group.

“No.”

*Do you think it’s algebra, geometry, number theory, something else?*

“Something else.”

*Do you agree that this is math?* (I asked because often in Math Circles, students ask when we are going to start doing actual mathematics after we’ve been doing math for a long time.)

“Yes.” Everyone agreed. Phew.

*This is game theory. Have you ever heard of it?*

“Oh yes, I love game theory!” said O.

*Game theory involves not just math, but psychology, economics, computer science, politics, and more*, I said.

“That’s not the game theory that I was thinking of,” explained O. The students then educated me that there’s an online show/game called Game Theory that’s a lot of fun.

Students continued to propose distributions and debate them. There were soon many conjectures. I was sensing a bit of frustration. I said

*By looking at this board, I can tell we’ve made progress. What do you see that indicates progress?*

“There are a lot of mathematical-looking things on there that are not nonsense,” posted K.

*You are right, although that’s not what I was thinking. I think another sign that we made progress is that we crossed off a conjecture. In math, we learn so much from rejected conjectures.* (Unfortunately I didn’t get a screen shot of this point in the work.)

“Is there a definite answer to this problem,” asked O, “or is the point of it the processing?”

*There is definitely an answer, although there are a lot of Math Circle problems where the processing is the point or where it may be an unsolved problem in mathematics. I think it’s time we try another approach to the problem.*

“Why do you think that?” asked Z and K.

*Everything people are saying has had the word “maybe” in it for a long time.* *And you’re starting to wonder if there even is an answer.* Students found this a reasonable reason to try things another way. O proposed that each person play a pirate and answer for that character. Students tried this for a bit, but ran into the same problems.

**SPOILER ALERT**

Stop reading here if you’d like to try to figure out the problem for yourself.

**BACKWARDS INDUCTION**

*Let’s see what would happen if Esther were the only pirate. What would the distribution be according to the above rules and decision factors?*

“E would get 100,” everyone agreed quickly.

*What if there were only 2 pirates, Daniel and Esther?*

This took a bit more work, but finally all agreed that D would get it all and E would get none, a 100-0 distribution.

*What if there were 3 pirates, Callie, Daniel, and Esther?*

Students proposed various distributions and then were able to reject almost all using the decision factors. Everyone agreed that the solution – the Nash equilibrium where no matter what an individual does, one can’t get a better outcome for oneself – was the C gets 99 and either D or E get 1. But which one of them gets 1 and which gets 0? Would E ever vote for getting 0? Would E always vote yes? Would D always vote no? This was the meat of the mathematical discussion. Students disagreed. Students argued. Students posited conjectures. Students defended. At one point, Z said “I’ve changed my mind.” Then we ran out of time.

At the start of class two, I put last week’s work on the board (Zoom whiteboard). I assumed that the students would continue where we left off last time. But they wanted to go back to their original method with new conjectures for 5 pirates. They ended up in same rabbit hole: if the captain can get some, could the captain get more?

I suggested we use backward induction again. Some students went along, others stuck with the original method. Students were working both methods simultaneously on the board. Lots of disagreement. What should we do? Z felt strongly that we should stick with backwards induction. (I loved that students were using this term naturally at this point.) This was met with resistance. N was adamant that the money be distributed evenly. K adamantly stated this week and last that “pirates are not communists!” I promised N that our exploration of game theory will move into problems where cooperation is a thing.

After a lot of frustration, everyone agreed to stick with backwards induction. They worked through the possible scenarios with the decision factors and after a lot of work, concluded that with 3 pirates, the Nash equilibrium distribution for 3 pirates is 99-0-1.

Similar discussions went on when discussing the four-pirates scenario: Biraj, Callie, Daniel, and Esther. But by now, students had the hang of backwards induction, had a strong understanding of the hierarchical nature of the decision factors, and had confidence that there is a logical solution to the problem. Frustration had shifted to excitement. So they pretty quickly arrived at the conclusion that the Nash equilibrium distribution for 4 pirates is 99-0-1-0.

The students had worked hard for an hour and 25 minutes. We had 5 minutes left in class but N had to leave to get to another class. The students agreed to table finishing this problem until next week when everyone will be together. We spent the last 5 minutes guessing the rules for function machines. Today’s rules were y=4x (way too easy) and y=x^2+1 (a good challenge). We collectively decided to end every session with this activity, but with interested students presenting the functions.

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]]>(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. (After a brief debate about G’s suggestion that we use birds such as the peregrine falcon that “can fly faster than land animals can run.” Then slugs: A mentioned that “every animal has a role; scientists just found a way to use slug slime in medicine.” R claimed that spotted lanternflies, which seem to come up in every session, do not have a beneficial role. But then A reminded everyone that spotted lanternflies are invasive here but beneficial in their native habitat.) Back to the problem…

The problem: if the snail gets a certain distance head start, and the cheetah has to wait until the snail covers that distance to then cover the same distance, then the snail gets another head start as soon as the cheetah reaches first head-start distance, who will win? The problem is originally known as Zeno’s Achilles Paradox – you can see it in this video. We started to work the steps of the problem.

“What’s the point of this problem, anyway?”

*An ancient person named Zeno did this problem to prove that motion is impossible.*

G: “Actually, there’s also no such thing as touching. Nothing can really touch each other.” He started stroking a tree. “I’m not really touching this tree, since there are electrons racing around the protons of each atom.”

*True. You’re using science to look at the underlying structure of things. Mathematicians and scientists have a similar goal: to figure out the underlying structure of things. That’s what Zeno was trying to do with this problem with the cheetah and the snail.* (At this point, A also chimed in with an example of how science looks at the underlying structure of things.)

The students intensely debated the parameters of the problem:

- How long should the race be? Should it be measured in units of time or distance? (Many opinions, finally settled on 100 since half the class agreed)
- What units should we use? (Miles? Kilometers? Kilo-miles?) There was no consensus or majority or even plurality for this decision. I ended up using the term “units” throughout the class to set an example of the important math skill of generalizing.
- How big of a head start should the snail get?
- How far would the cheetah reasonably move after the snail got the head start?
- Is it realistic that a snail could cover such a long distance? Or that a cheetah would be racing a snail?
- How should we mark on the line where each animal is at each point? Draw the animal with chalk? Use a rock or shell? Both? (You can see in the photo that the decision was “both.”)

As the students worked the problem, it was becoming apparent to most that the cheetah might not win. This defies common sense. Several of the students, especially R, started looking for loopholes that would resolve the cognitive dissonance that this problem often produces. He said something like “Wouldn’t the cheetah have an advantage since he would be resting while the snail was doing the head starts, so the snail would be getting tireder?” This gave me another opportunity to revisit the idea of formal systems – we have to stick to Zeno’s rules in this formal system that Zeno designed. We have to stay in what Hofstadter calls M-mode (machine). We can’t use I-mode (intelligent) to jump out of the formal system. In M-mode, the snail doesn’t tire sooner than the cheetah. We talked about it as a “thought experiment.”

S: “Are we supposed to be rooting for the snail?”

*Rooting isn’t the point. The point is that this problem is supposed to convince us that motion is impossible.*

Everyone: “This problem is not convincing us!”

*That’s okay. Zeno had three other problems he used with this one to convince us, and people have been studying them for thousands of years. We might not get convinced today.*

Students: “Can you show us those other three problems?”

*If we have time.* (I was doubtful)

After working on it a bit more, we left it unresolved as A, G, and S thought that there’s no way cheetah can win working within the system. R asked how much time we had left. Not much. He wanted to keep looking for loopholes that would allow the cheetah to win, but also agreed with the others that it would be nice to spend our last few minutes on another activity.

*Let’s do a function machine.*

R asked to draw the T-chart and G asked to draw the machine.

A: “What’s the difference between the T-chart and the machine?”

*The T-chart is your organizational device, the machine is what you literally put the items in the domain into. For this machine, the domain is animals, so don’t draw anything that would hurt them*.

A helped G draw the machine (notice the “in” and “out” locations), then students started suggesting animals to put into it.

“Put in narwhal.” Out comes 0.

“Put in snake.” Out comes 0.

“Put in kangaroo.” Out comes 4.

“Put in butterfly.” Out comes 6.

“Really?”

After students tried about six different inputs, A figured out the rule, but did not tell the others – instead they told the others what came out from each additional input. (If you’ve been reading along over the weeks, you may realize that this is the same rule that we worked on in the School House Lane group.)

Most of the discussion of this machine was about its domain (the rule about what you’re allowed to put into a machine). I repeatedly had to make the domain more clearly defined because at first, G kept putting in animals that I had never heard of.

*Let me redefine the domain: the domain is not just animals, but animals that Rodi has heard of.*

“Put in a tardigrade,” said G.

*I’ve heard of that but know nothing about it.*

Both G and A started telling me about it, but not what I needed to know to be sure of what number comes out of the machine.

*Let me redefine the domain: the domain is not just animals that Rodi has heard of, but animals that Rodi knows something about.*

“But you do know something about it; we just told you,” said the students. They were correct.

*Let me redefine the domain: the domain is not just animals that Rodi knows something about, but animals that Rodi knew something about before arriving at Math Circle today.*

Getting that precise seemed to resolve the issue (for now, dun dun dun duuuuun – imagine suspenseful music – foreshadowing of next week’s report!)*.*

S was very focused on conjectures; she came up with many that involved the spelling, which were not the rule I was thinking. We talked about how wrong conjectures still help with reasoning.

A: “I could suggest an animal to put in that would make it easier to figure out the rule, but that might spoil people’s fun.”

*I agree – why don’t you wait until next week to do that if everyone else agrees.* (They all did.)

Spoiler alert: I will tell you all in the next report what the rule is.

At the end of the session, I gave students printouts of the map they created last week, as they had requested. (But I had cut them out into rectangles instead of circles as the students had asked – sorry!) Everyone wanted 2-3 copies to work with at home. During class G drew borders on his more clearly with sharpies, since the printouts were a bit hard to read. You may want to try this at home. Please email me if you’d like me to send an electronic file with the map.

I didn’t use the word paradox when I named and described the history of this problem. I wanted to avoid the spoiler that the formal name of the problem implies. I just called it things like Zeno’s Motion-Is-Impossible Problem, Zeno’s Race Problem, etc.

Our discussion about domains is somewhat analogous to a process whereby I tell you that the domain of my function is “numbers.” You might say “Put in 2.” I’d say “Out comes 0.5” Then you say “Put in 11” I’d say “Out comes 0.09.” After a few more examples, you might say “Put in 0” and I’d say “That breaks the machine.” Or, I might refine the domain and say “No, the domain is actually all numbers but 0.” If you say “Put in pi,” I’d also say “that breaks the machine because I’m using the calculator on my phone to save time and it doesn’t let me input pi.” So we need to refine the domain to all non-zero rational numbers. And so on. Maybe you’ve guessed by now that my function is . In this process, we’re developing skill in the 5^{th} grade Common Core Standard for algebraic thinking of “analyze patterns and relationships” and “attend to precision.”

While the students worked Zeno’s problem, they explained everything to each other – why conjectures wouldn’t work, what is the definition of something, what makes sense as the next step, etc. Usually at least one person can answer anyone else’s conjecture/question. They continue to draw everything – nothing on the sidewalk chalk “board” from me. At this point in the course, the group is coalescing into a good workplace team (as I think it’s called in the field of project management). All they need me for really is providing the interesting math problems to explore and to watch the time, or today, hand out rocks since people’s papers were blowing away. Or sometimes come up with systems for decision making/consensus, or noticing if someone is waiting a long time to talk. Otherwise, I just sit quietly.

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]]>(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 want. Let’s take a day off from talking about axioms and strange loops, and celeberate Martin Gardner’s Birthday.*

A: “Who’s Martin Gardner?”

I told a bit of his history – his love of puzzles, magic, jokes – how he is considered by many the founder of the field Recreational Mathematics, the G4G Gathering, the G4G Celebration of Mind events. Then the students took turns adding regions to the map. “It looks like a bird.” “We should call it Bird Country.” “No, it looks like a ….” Soon each student on their turn was adding a region to make the map look like a certain thing, but each student wanted it to look like something different. Finally it was done. For now.

Me: *One math problem Martin Gardner had a part in the story of is map coloring. Can one of you color in a region?* (Done.) *Now someone else color another region. See if you can color the whole map with just 2 colors. One rule: no two adjacent regions can share a color. In other words, the same color can touch a vertex but not an edge/border. Can you do it with only 2 colors?*

Student conjecture: “No, unless we change the map.” They posited a 3-color conjecture (that they could color the map in 3 colors only, following the rule).

Students took turns coloring, all standing around the map (while social distancing), with me pretty much saying my favorite thing: nothing. Students were coaching each other on where to color and which color to use. They quickly rejected their 3-color conjecture and posited a 4-color conjecture. A, R, and G added some additional regions to continue making the map look like “a bird” … or “someone doing magic with a cupcake on top of their head” … or “a top hat on top.” S asked questions about the mathematics and the history.

I told some history (maybe playing a bit fast and loose with it). I said that mathematicians were convinced for many years that any map could be colored (according to these mathematical map-coloring rules) with 6 colors. Then someone came along and said something like “All of those maps you needed 6 colors for, I can do in 5!” People were convinced, and the 6-color conjecture became the 5-color conjecture. Then someone came along and said something like “All of those maps you needed 5 colors for, I can do in 4!” People were convinced, and the 5-color conjecture became the 4-color conjecture. Then came along Martin Gardner, who published the McGregor map in his column in Scientific American, saying something like “You were wrong, people, there is a map that needs 5 colors.”

S: Do you have a picture of the map that MG drew?

Me: *I do!* I handed out copies of the McGregor map.

Students looked at the map, came up with their own conjectures, and asked “But was he right?”

Me: *I’m not going to say, since you may want to take these home and have some fun trying to figure it out yourself. I will tell you one thing* (imagine suspenseful music here): *He published this map on April Fool’s Day.*

Students: “Oh!” “Maybe he was lying!”

Me: *Yes, maybe he was pulling an Epiminedes on everyone! *(The students by now knew that Epiminedes was known for using lies to make points.)

G: “So many math names end in ‘edes!” There’s Archimedes, and others.”

I got up on my metaphorical soapbox and explained *that it’s just the Western European mathematicians who have names that may sound similar that have gotten the most attention. There have been mathematicians from Asia, Africa, and other places doing amazing work for several thousand years. But in our culture, we haven’t been taught as much about them.* I promised to tell of some of these mathematicians at some point. Back to map coloring…

I left the McGregor map question without an answer. Some students wanted to work on it at home. I did explain that this was the first famous math problem “solved” by a computer that was convincing to humans.

Me: Can you change your own map so that it would would need 5 colors to color it?

Students: Yes! (They drew a line segment splitting one of the regions into two.)

Me: Could you color this revised map with 4 colors if you had used a different coloring strategy from the start?

Students: Yes.

I encouraged students to try making maps at home that would require more than 4 colors.

[SPOILER: I called this problem the “Four-Color Conjecture.” I did not call it the Four-Color Theorem, which it is now called, since for a conjecture to become a theorem, as these students know, it has to be proved. I didn’t want them to know it was proven so that they could work on it more themselves at home if desired.]

By the end of class, students were using math vocabulary including theorem, conjecture, edge, and vertex with ease. The students asked me to go home and print out copies a photo of their map and bring it back next time. They requested that I cut the map out in a circle and not a rectangle so that everyone could choose their own orientation for the map. There was no consensus on which part was the top. As they left, students discussed making a coloring-book with map-coloring challenges and also coming back next week to work together on naming the regions on the map they created today.

If a meta-analysis is an analysis of analyses, and metacognition is cognition about cognition (in other words, thinking about thinking), then meta-pedagogy must be pedagogy about pedagogy. At the start of class today, S asked “Why are you asking us to draw the regions instead of you drawing the regions yourself?” Other students asked similar questions about why I was facilitating Math Circle in the manner that I was that day. I explained the pedagogy of our Math Circle, which is basically that the students are the ones who are supposed to be doing all the thinking, and I’m only supposed to be asking questions or providing interesting math problems. I also explained the benefits of doing it this way.

Blogging about blogging: this paragraph is for those of you who also write blogs or reports about your work with students. What practices aid in your being a reliable historian? Each time I lead a Math Circle, I do a brain dump that same day with details that catch my fancy and that I want to share with parents. Then I go back into the document later and fill in the important context. I did a brain dump on October 19 and have been thinking about meta-pedagogy a lot since then. On November 13 I sat down to fill in the context. I opened up the document and saw that under “meta-pedagogy” I had written “see note in pants pocket.” Ugh. Of course those pants have been through the wash a few times in the past 3 weeks. How do you quickly collect your anecdotes and quotes if you write later? I’d like to know.

Why do I write these blogs/reports?

- (THE MOST IMPORTANT REASON BY FAR) …to show parents/guardians what we’re doing in class so you hopefully agree that you’re getting your money’s worth from the tuition you pay and will therefore enroll your children in the course. I need students with whom to share this joy!
- …give you (parents/guardians/students) info about the content so you can do more at home, if desired
- …give me a reflective process to enhance my teaching/facilitation
- …to entertain myself. Writing is a creative pursuit. It’s fun. It’s gets me into the zone, or “flow,” as coined by the famous psychologist Mihaly Csikszentmihalyi, who sadly left this world last week. Joyfully engaging in mathematical explorations is another way to experience flow. See the diagram of his flow model below to see why.
- …to spread this pedagogy to other educators and their students

After we talked about the math names that end in ‘edes, G asked what Archimedes was famous for in mathematics. At the time, I had forgotten, but promised to refresh my memory and report back next week. I never did remember to do this. Here are a few Archimedes items I’ve discussed with students in Math Circles over the years:

- Did Archimedes really coin the phrase “Eureka!”?
- There are multiple versions of the story of his death. How can we determine which, if any, are accurate?
- Archimedes developed a novel approach to the determination of a more precise valuation of pi.
- Archimedes discovered the relationship between force and distance and how it affects leverage.

He did a lot more. If you type “Archimedes math contributions” into a search engine, you’ll find a lot of material, including some animated videos.

Many thanks to A’s grownup B for the great map photo, and to R’s grownup N for action shots over the weeks.

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]]>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 two vertices wins. Is there a winning strategy?*

I did a demonstration game with the students, and fortuitously got an unexpected result. I didn’t understand why at first, but then realized I had forgotten to tell the class one of the rules. *What rule do you think I forgot?* (I just LOVE that I got to ask that question!) The students figured it out – the line segments are not allowed to cross. I love that this happened organically because sometimes in Math Circles we leaders posit an intentionally vague question with the goal that students ask clarifying questions. But this time I didn’t even plan it!

Melissa paired students into breakout rooms to play Criss-Cross on Jamboards. I asked them to keep track of number of vertices, which player went first, and who won for each round. I bounced around from room to room to check in and offer support.

Even before combining results, the students had an initial conjecture. “Seems like the first player wins if there’s an odd number of vertices, second if even.” Then students combined results. Almost all the results supported the conjecture.

But S&S got a different result…

*What could explain this?* Dead air.

I rephrased the question: *If you yourself had played every single one of these games and gotten these results, what would you be asking yourself?* Now students had some ideas. It’s easier to come up with a question than a conjecture. Perhaps more accurately, it takes more courage to voice a conjecture than to ask a quetion. I missed an opportunity on my part to say this to the students. Drat.

“Maybe S&S have a move the rest of us didn’t see.”

*Should we take a look?* Together we tried to recreate their game. S and S coached the rest of us on their moves. But we were out of time. We decided to continue this exploration next week.

To learn more about this game, its underlying structure, and an introduction to graph theory, take a look here: https://www.mathteacherscircle.org/session/game-criss-cross/

*Next week, we’re going to start Math Journaling. Melissa will provide paper (drawing paper and graph paper). Bring your own pen/pencil and something for drawing in multiple colors.*

I told students this at the start of class, after Melissa and I had brainstormed about how to engage students who want to draw and at the same time keep the Zoom screen easier to follow for students who get distracted. (Take a look at the image in the left-hand column of this blog post to see an example of a busy Zoom screen:)

A confession: I’ve led weekly math circles all year long for over a decade and have never played a competitive game with students. I had a Good Reason. I just can’t remember it. Was it that I thought games would somehow “cheapen” mathematics, or reinforce the idea of math as a tool, or what? I don’t know. This past summer, I was toying with idea of introducing some games, when knew I’d be leading a year-long course. Historian/artist Nell Painter, in her memoir Old in Art School, talks about fellow scholars who run the risk of losing something important by clinging to the gravitas of academic culture. This quote was the final nudge I needed to start playing some games:

“They identified with my break for freedom but feared their academic or lawyerly selves that had already quashed their inner Beyonce, that the wet blanket of professionalism had smothered their flame. … Yes, yes, I loved all the steps entailed in scholarship, but I reached for more, to take other steps, additional steps, call them side steps, for freedom from evidence-based knowledge of things I could know for sure, things that stood for much smaller as well as larger things, beyond and around the truths of the archive. Fiction. Visual fictions. I wanted to make art. Seriously. And to make serious art unfettered from the mandate that I address larger truths.” (page 17)

After reading this, I said this to myself: *In Math Circles, do we always have to be fettered to the mandate that we address larger truths? Or can we sometimes just have fun? In other words, can we just do graph theory? Graph theory, of course, is actually devoted to uncovering larger truths, but maybe sometimes we can delegate the processing of larger truths to our subconscious and just make art, play games, or whatever.*

So that’s what I had to say to myself to stop taking myself so seriously and free myself to just play Criss-Cross. But I have no plans to do anything more competitive than Criss-Cross.

I’m posting this weeks later, at a time when the School House Lane Math Circle has gone on hiatus indefinitely. For those of you who want to do math journaling at home, I recommend that you follow the work of Denise Gaskins. For those of you who want to further explore the axioms of mathematics, I’d recommend you try the MU Puzzle. You also may want to try playing Criss-Cross at home. You can follow this blog to read about other activites related to Axioms that the Lovett group is doing. Feel free to contact me for direction on recreating those activities at home.

This week I “strewed” a bunch of things on the ground to see what, if anything, would catch students’ interest. (Actually, since I have a broken foot, I had the students doing the strewing without telling them what/why.) I began class by asking *What should we talk about?*

“What?” Confused looks.

*I mean, what should we do today?*

“Huh?”

*There’s all this stuff that you all helped me put out.*

“Let’s see what these papers are,” said S

*Good idea – why don’t you pick a pile of your choice and see what it says.* (Session proceeded by each student taking a turn choosing something on the ground to explore.)

S picked up a pile of papers, which contained the rules in words to last week’s student-created formal system. S transcribed the words into symbols on a piece of paper

.

The students starting talking to each other.

“Should we keep or get rid of Rule 2 and Rule 4?” they wondered. These rules were not needed to solve the student-designed “Pink to Purple Puzzle. The students’ conversation moved into puzzle design:

- Should the puzzle be easy or challenging?
- Should we give hints?
- Should we make it more complex by “having codes for the colors, and challenge people to use a decoder to figure out what the colors are?”
- If we give hints, should we “make it a lift-the flap book?” Or some other type of hidden hints? (I loved that me being quiet resulted in the students making book-publishing plans.)
- Should we give a hint that “you don’t have to use all the rules?”
- Should we give a hint that “some rules are fake?”
- If we give hints, since not everyone will want to use them, should we “write them upside down or hide it on the back?”
- Should we “add a rule or two” whereby “if you get a certain thing you can unlock a hint?”

I finally interjected with a question: Are you still going to name the system itself, like how in Gödel, Escher, Bach Hofstadter designed his “MU” Puzzle within the framework of what he called the “MIU System?”

Huge discussion! The students produced many more name ideas, in addition to the first five that they came up with last week:

- “The Not-Boring System”
- “The Green-Pink-Purple System”
- “The Not-Boring Colors System”
- “The Not Boring Green Pink Purple System”
- “The Not Boring Pink-Into-Purple System”

At this point the students decided that they should use their first initials within the phrase “not-boring:”

- “The Not-Boring AGSR System”
- “The Not-Boring RSAG System”
- “The Not-Boring SARG System”
- “The Not-Boring GRAS System”
- “The Not-Boring RGSA System”
- “The Not-Boring ASGR System”
- “The Not-Boring SRAG System”
- “The Not-Boring AGRS System”
- “The Not-Boring SRGA System”
- “The Not-Boring SAGR System”

“Wait, how many combinations are there?” wondered the students at this point. Conjectures: 8, 16, 24. I asked, *How could you figure out how many combinations there are? *They ignored me, and kept trying to convince each other that their conjecture was right, without explanation or proof. I asked again. They ignored me again. I let it go. (See the “pedagogy” section below if you’re curious about all the thoughts my brain was yelling at me at this point.) Their attention moved from conjecturing about combinatorics to coming up with more possible names:

- “The Not-Boring 7
*xxx*System” - “The Not-Boring 3GRAS System”
- “The Not-Boring 7SRGA System”
- “The Not-Boring (insert all students’ first or full names here) System”
- “The Not-Boring (insert every person’s favorite color here) System”

Each name had a good explanation. Students proposed voting. “Can I vote for more than one?” someone asked.

“How much time do we have left?” asked R, who has evolved into the time-manager of the group. Once R asks, the group somehow arrives at a consensus about whether to continue with the current problem, or get in one other short thing before we have to leave. Today, we tabled the name-choosing so that someone else could choose from the pile of strewn papers.

The next pile of papers was images of tattoos, and the next was a game of Odd One Out. This was all about Epiminedes, whom we’ve discussed before. Lots of laughter, which ended with G role-playing being an oracle.

Strewing is a pedagogical method whereby the facilitator places seemingly random things around the space in seemingly random placements, and then switches up the things and the placements after a certain amount of time. The goal is that students will pick up what they notice get curious about it. Here are the items I strewed today:

- the rules of the MU puzzle in words (paper)
- the rules of the MU puzzle in symbols (paper)
- the rules of the Pink-to-Purple puzzle in words (paper)
- 4 images of ancient people (paper)
- An image of a tattoo-covered body (paper)
- A long line drawn in chalk with “start” on one end and “finish on the “other” (sidewalk chalk)
- A T-chart with a column labelled “in” and a column labelled “out” (sidewalk chalk)

*Combinatorics! Oh boy oh boy oh boy! I’ve always wanted to talk about combinatorics in Math Circle and now here’s my chance! The students have discovered a Need for The Topic. Teachable Moment! Every teacher’s dream! I know lots of ways to present it, and I love the topic myself, and they may learn about this in middle school or high school so now they’ll Get Ahead, and maybe it will be on their SAT, and how cool is it that I’m showing it to them when they’re only 9 years old, I’m such a pedagogical rock star that I led them to this point, and … wait. Drat. They’re not interested. They don’t want to know the underlying structure of how combinatorics work. They don’t have a real Need for The Topic. I didn’t lead them to this point. They just want to generate a list. Sigh. Okay. I’ll keep my mouth shut. Back to respecting the students’ curiosity. I can do this.*

In other words, my Ego told my brain to jump into the proverbial “Sage on the Stage” role. Fortunately, my Inner Wise Teacher noticed and kept me in my lane as the proverbial “Guide on the Side.” Actually, Guide on the Side is an overstatement of my role if I’m facilitating in the way I want to be doing it.

For details on the Odd One Out game and info about Epiminedes’ tattoos, see the left column in last week’s blog and Wikipedia. (The activity played out similarly in the Lovett group this week as it did in the SHL group last week.)

For details, see the right column in last week’s blog.

“I wish we could do Math Circle every day.”

“Yeah. Or at least 4 days a week.”

“That’s too much. Maybe twice a week at most.”

“I wish this could go on forever.”

In every session the students lament that this is only a 7-week course. They ask for more. I do plan to offer a 6-week course in the spring, probably from mid-April to late-May. The topic will probably be Set Theory, specifically as it related to the axioms of mathematics. This current course will not be a prerequisite. Anyone in the age group (TBD) is welcome to sign up. I’ll be posting an interest form on the website within the next month to collect scheduling preferences. In the meantime, let me know if anyone is interested in a course that meets twice a week instead of once.

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]]>Students looked at 4 images of artist’s renditions of figures from ancient Greece.* *Which one is the Odd One Out?*

(Before reading what the students said, which one do you think?)

“The one on the bottom left, with the eyes closed.”

“The one on the top left, with the different color.”

“The one in the middle since it’s a painting, not a sculpture.”

“The one on the right since the ears are showing.” (I explained that I really enjoy playing Odd One Out this way, trying to make a case for each item in the group being the Odd One.) Today, I wanted to focus on the bottom left: *Why would an artist depict someone with their eyes closed?*

We had fun talking about Epiminedes’s history – *Why are his eyes closed in the art? Why did he have tattoos? Was he a God? How old do you think the stories say he was? What’s the difference between an oracle and a shaman? Did he really sleep for 57 years or was he a hermit?*

*I later briefly mentioned who the others were – Zeus, Pythagorus, and the Oracle of Delphi.

*Who or what is the fastest runner in the world?*

A cheetah.

*Who or what is the slowest runner in the world?*

A snail or a sloth.

*Let’s model a race between a snail and a sloth so I can prove that motion is impossible. How long should the racetrack be?*

700 miles

*Let’s give the snail a head start to be fair. How much of head start is reasonable?*

300 miles

*What is the snail doing after running its 300 miles when the cheetah is running its 300 miles?*

Running

*How far?*

2 miles

*So who is winning so far?*“The snail.”

“This is hard.”

*This is definitely hard. That’s why people have been doing this problem for over 2,000 years.*

The students followed these rules (that the snail gets a head start, the cheetah starts when the snail gets to that distance, but that the snail is running while the cheetah is) and did another few examples. The snail was at 358 when the cheetah was at 350. *Will the snail ever catch up?*

Discussion: A reality check on the part of the students revealed that the cheetah will win, look at the biology and physics of how animals and racing work. But Hofstadter refers to this scenario as “can’t catch up.” It depends whether you frame it in reality or within a formal system, where these are the rules and they have to be followed. You can’t come out of the system or change the rules to account for science. Students did end up agreeing that if stick with this system, the cheetah can’t catch up.

I mentioned that the ancient Greek philosopher Zeno proposed this problem (with several others) to demonstrate that motion is impossible. Instead of a cheetah, Xeno suggested Achilles. Instead of a snail, Zeno suggested a tortoise. But no one in our group was convinced that motion is impossible so we’ll need to do Zeno’s other 3 famous paradoxes to see if the students are convinced.

(youtube: Smart by Design)

This time the domain is not numbers, it’s animals. Which animals do you want to put in?

IN: horse – OUT: 4

IN: racoon – OUT: 4

(and many more)

The big skill needed here: strategically choosing an input that may result in a different output. It went on for a while.

“Are you not going to tell us the answer?”

No, but Melissa put it in the group’s Google classroom for students to ponder.

You can see from the above images that some students are doodling on the screen, enough that a student who gets distracted by a lot of visual stimulation could struggle. Melissa helped me to come up with some strategies to still enable doodling but keep it either off-screen or much smaller. I’ll be experimenting over the upcoming weeks to find the sweet spot. (Some ideas: Lhianna Boditoro is teaching me how to use Miro boards. Denise Gaskin’s upcoming book on Math Journals is also inspiring.)

Why did I play Odd One Out? The math in that game is compelling, but I also wanted to discuss some interesting Epiminedes history by asking questions and generating conjectures from the students. Conjecturing is a huge skill in math that many students are reluctant to do.

Why didn’t I just use the original problem with Achilles for this paradox of Zeno’s? I hoped that students would feel more ownership in the problem if they had co-written it.

Here’s a fun video depicting the race between Achilles and the tortoise.

*Last week we explored Hofstadter’s Formal System the MIU System and the MU Puzzle. Today, let’s develop our own System and Puzzle. What would we need to do to start?*

At first some students wanted to work more on the MU puzzle after working on it at home. S had come up with an idea of a way to prove that it’s impossible with the given rules, and asked whether it’s okay within a Formal System to add a rule. (No.) G wanted to revise a rule to make the puzzle solvable. *No, we’re going to stick to M-Mode.*

What should we do first? Everyone strongly agreed that to make our new system and puzzle that we start with the “objective,” in other words, create the puzzle before making the rules. “This reminds me of a metaphor,” said G, “You can’t make something from nothing.” This prompted a big discussion about how this may apply to painting: you have to choose your tools (Brush? Canvas? Paint? Hands?), which are the axioms, but you don’t have to choose the topic/subject of the art. So maybe choosing the tools is a formal system but choosing the topic is not since you can change your mind easily. But wait, you can change your mind about the tools partway through (i.e. what if Jackson Pollock didn’t like those thrown splotches and decided to pick up a brush?) so maybe even that’s not a formal system. Back to M-mode versus I-mode again.

S pointed out that the puzzle might be too easy if we choose it before writing the rules, so the group came up with some goals for the puzzle:

- Challenging to others
- Possible to solve
- Not boring

Students proposed some different puzzles within the MIU System. I said *let’s create our own system*, so students proposed various things to be the “alphabet” of the system: letter, numbers, colors, shapes, trucks, symbols, emojis. Since we were limited in our work tools to sidewalk chalk and cement, everyone agreed colors would be the best alphabet at the moment. They chose the alphabet to contain pink, purple, orange, and green. The puzzle would be to transform pink into purple.

The biggest discussion of the day was what axiom to start with. Most students wanted to start with the axiom that you possess a string of one green, that it’s fine not to start with the thing that’s the initial position of the puzzle. So we started with green.

Then students started proposing rules that would make the transformation from pink to purple (after starting with green) possible. “The Pink-to-Purple Puzzle.” A and S took turns as scribes, writing down the rules with symbols:

**Rule #1: If you have one green, you can double it.**

**Rule #2: If you have one orange, you can double it.**

**Rule #3: If you have 2 greens, you can transform them into one pin**k. (At this point in the discussion, everyone agreed that the next rule ought to get purple into a string somehow.)

**Rule #4: If you have 2 oranges, you can transform the first one into a purple.** (“That ruins my plan;” said G, when S put this rule on the list. Discussion: one challenge of collaboration, having to adjust.)

**Rule #5: If you have any string that ends with pink, you can add one orange at the end.** Discussion: How to notate “any string?” Mathematicians like to cross off instead of erase so you can see rejected work again later if you need it. *Can this rule be applied if the string is just one pink?* The students said yes since that technically ends with pink. Two students were standing up on their chairs bouncing with hands up, so excited to contribute suggestions for rules at this point. I couldn’t remember whose turn it was to talk, so … pick a number …

**Rule #6: If you have pink then orange, you can transform the pink to purple and add an orange on the end.**

(At this point, it was 10:57 and class ends at 11:00. No one could believe how time had flown. So close to a solution. What to do? Most agreed to make up one additional rule that would solve the puzzle.)

**Rule #7: If you have the string purple orange orange, you can drop the two oranges.**

Then, students verbally derived purple from pink by applying the rules in this order: 1 -> 3 -> 5 -> 6 -> 7.

Hesitation….

“Why do we need all these rules?”

The students crossed off rules 2 and 4, but then reconsidered – they could help achieve the goal of challenge to others. (Missed opportunity here on my part to talk about elegance in math.)

*What’s the name of your system?*

- “The Not-Boring System”
- “The Green-Pink-Purple System”
- “The Not-Boring Colors System”
- “The Not Boring Green-Pink-Purple System”
- “The Not Boring Pink-Into-Purple System”

Students lingered after class, looking at the work, some copying down. I sent them home with two questions mathematicians would ask

- How many solutions are there?
- Can you do better – optimize (i.e. do it in fewer steps)?

TERMS: I am so excited that the students are using the word “axioms” in conversation! My main goal for today was to use these terms/phrases naturally in conversation, without stating them with definitions:

- Axiom
- Theorem
- Requirement of Formality
- Decision Procedure
- Produce/Derive
- Rules of Production
- Lengthening/Shortening Rules
- Guaranteed to terminate

I even wrote them in my notebook in advance so I could score myself later. I ended up using about half, and one unplanned term: transformation. Now onto using the word “theorem!”

CONSENSUS VERSUS MAJORITY: Three people wanted our alphabet to consist of 4 colors, one wanted 3. Three people wanted an initial axiom of green, one wanted pink. Three people wanted to finish the list of rules and quickly explain verbally how they could be used to solve the puzzle, one wanted to finish the problem next week by formally deriving theorems/strings. And so on. Throughout the session, there was not consensus. My attempts to facilitate consensus might have worked, but it would have taken the whole session since students were very attached to their ideas. I ended up using majority to move forward with the math, but want to move toward consensus going forward.

INDIVIDUAL VERSUS COLLABORATIVE: Each student had their own idea of the exact progression of rules and what they should be, but since we are collaborating, I asked that each person contribute one rule at a time. This made this much more challenging for students.

BITING MY TONGUE: Early in the session, I tried so hard to ask leading questions to get the students to consider making the rules before the puzzle. (They really wanted to make the puzzle before the rules so that they could guarantee that there would be a solution.) Then I bit my tongue, remembering that I want to students to come up with this idea based upon their own work. They later on did briefly bring up how it might have been different had they started with the rules.

AXIOMS: The after-class discussion with a few people who lingered revolved around what exactly is an axiom. (Something we take for granted as true without proving it – Hofstadter calls it something we’re given for free.) This conversation led to the question of why aren’t things like this often taught in school to students or teachers? Which led to me recommending A Mathematician’s Lament, by Paul Lockhart.

MU Puzzle – we will work on this again, as students are excited!

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