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]]>Ugh. Pride. Here’s what happened when the students got out the ropes today:

“Stand in the yellow if you have a head and stand in the green if you are human,” I commanded. With a bunch of trial and error, the students came up with two concentric circles, the outer (yellow) circle for having a head and the inner (green) circle for being human. (Nefarious laugh here, my plan was working: the students under my brilliant guidance, had invented a way to model subsets using Venn Diagrams!)

But N was standing here with one foot in the green and one foot in the yellow outside of the green.

“N, do you not have a head?” I asked.

“Of course I do,” he said, “but I identify more with being human than with having a head so I put one foot in each.” I realize now that this was such solid reasoning, that N had introduced another variable into the mix, identity. A variable that can be evaluated with inequalities and that can be modelled with rope diagrams. In the first week, when students tried to introduce another variable (preference), I had shut it down. But did I learn my lesson? No. Pride again. What do you think I did?

I tried to talk him out of this. (Ugh!) Today pride got in the way of my teaching again as I forced the math of subsets when curiosity was pointing students in another direction. The students were still having fun inventing and discovering math, but I could have done better by them in our context. It’s not like they were going to take a test on Venn Diagrams tomorrow!

Let me back up a bit. It was a rainy day. We’ve been meeting outside and have a covered porch at the library/park, but it was still damp and raw for our fourth session. Only four students showed up. They worked on the MU puzzle for a while – a continuation from a previous session. To use a chess metaphor, the group made some solid progress on the opening. Stuck, they switched (without my input) to working backwards, examining the final desired position to come up with an endgame plan. But no matter how well their ideas were building off of each other’s, a middlegame did not emerge. I assured students that this puzzle is solvable, that it’s famous, and that if they want to work on it at home that they could ask their grownups to Google it if needed.

At this frustrating point, someone asked, “Is this all we’re doing today?” I said no, that I have a list of activities longer than we could ever accomplish in an hour. I shared my list with them. “I was thinking of doing a game that I’m trying to invent called perhaps *Axiom or Theorem*. I might be a good game or it might be a terrible game. We’ll only know by trying.” The students were so intrigued because of (a) game design and (b) said Q, “Wait a minute, you never said the word theorem before. What the heck is that?”

We never got to the game *Axiom or Theorem* this week, but I promised that we’ll start with it next week. I hope/think I have their curiosity built up enough that the game will be engaging and that they’ll leave the course with a basic idea of what an axiom is versus theorem, and more importantly, the role of proof.

I shared a few other brief activities from my long list of rainy-day ideas. First, I did another of Zvonkin’s Magic Tricks – they were yelling at me (in a good, nice way). My tricks are So Bad! These tricks are Not Magic! These tricks are just math! I promised a “better” magic trick this coming week. My heard was bursting with joy to be scolded about this trick.

Then I had each student take 3 slips of paper from my bag. One paper had a common version of the Epiminedes Paradox (“I am a liar”) printed on it. One had an image of M.C. Escher’s Drawing Hands. And a third had Russell’s Barber Paradox (“Who shaves the barber?”) After a lot of debate about what each of these things even mean, I announced that the real math question of the day is “What do all three of these things have in common?” (This is a take-home question!)

We finished off with a round of the game Odd One Out with these images, somehow related to these slips of paper:

Pride. I think it’s a common thing that can happen in a classroom – that things can go too well, or we can have a self-serving agenda. Why was I personally so susceptible to pride? Maybe I wanted the Applause. Maybe to please the parents/grownups. Maybe I got a dopamine hit from doing these Venn Diagrams, as Ralph Ryback explains in Psychology Today. Who knows how I let pride steer me away from my real agenda of letting students invent and discover math for themselves? It’s complicated. I see that pride is right next to hope on Plutchik’s Wheel of Emotions. I’m hoping to enhance my emotional literacy by looking into this more!

As some of you know, I’ve been co-editing a special issue of the Journal of Math Circles, honoring two of my mentors, Ellen and the late Bob Kaplan. This work has given me the privilege of reading other people’s writings about Ellen and Bob and their pedagogy – which is to never tell anything and let kids invent/discover math for themselves. This work is giving me the gift of a beginner’s mind. A reminder. A big aha moment. “What am I doing?!” I said to myself. Bob and Ellen helped me realize now that I had gotten off the path.

(I HOPE)

I called this course Axiomatic Set Theory because a few years ago I read an article online about how a lot of fundamental axiomatics of mathematics can be explained to children (and all of us) using set theory. It was so wonderful that I decided to someday learn this and run a course on it. I scheduled it, but now haven’t been able to find that article or anything like it. So I’m hoping that our concurrent explorations of axioms (mostly from Hofstadter’s book *Gödel, Escher, Bach)* and of Venn Diagrams will naturally converge, that students will discover a golden thread or hidden harmony.

I’m hoping that in next week’s final session, I can actually allow students the freedom of being creative, of inventing what they want. And then stating the axioms of their own formal system. Their own formal system that’s even better than Venn Diagrams! Or is this my pride speaking, inviting you to the Grand Finale of this course?

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]]>I didn’t mention algebra, variables, or “f of x” notation but conceptually the students were doing it. This point here was just to make sure that everyone knew how basic function machines work. Then we moved to Legos.

The Legos were in little bags. A student would build an “input” Lego structure with the bricks from their bag, then I would reach into a box and create an “output” structure following some rule. Again, the students had to guess the rule.

I said, “But before we start, let me show you a few of my mathematician cards so you know the history of these problems. Dr. Brandy Weigers invented Lego function machines.” I showed a card with Dr Brandy on it. I had a paper printed out from Dr Brandy with the instructions for doing Lego function machines. I referred to it as “Dr Brandy’s mathematical secrets,” so of course a few of the students tried to playfully sneak a peek.

I showed another mathematician card: “Dr. Eugenia Cheng wrote about how you can talk about the axioms of mathematics with Legos.”

“How do you do that?!” asked Z. D remembered that axioms are the answer to the question “why” when the reason is “Because I told you so.” I tried to explain it as well as Cheng does in her modern classic, *How to Bake Pi*, but struggled. Why are Legos plastic? Why do they attach the way they do? The answers to those questions are axioms. But, on the other hand, you can prove the answer to this question: Can you build a stack of 6 Legos that will stick together? So that’s not an axiom. U built a model to demonstrate. Then U said, “Can we get to the game now?” A few others were still interested in the idea of axiom, so I tried to multi-task and talk while doing the Lego rules, but that was hard. Creating the outputs required a LOT of concentration.

Some of the rules, thanks to Dr. Brandy’s mathematical secrets were f(x) = x in terms of shape and color, f(x) = 2x shape and color, replace a certain color, count the studs/pips, and x+5 in terms of number of studs. Sadly, we didn’t have enough time to get to the really exciting part of the game, composite functions, in which you start with a certain struggle and then run it through multiple functions.

But this activity did help me in my own struggle with how to define axioms for students this young: the students like this definition of an axiom: “because I said so.” (In our final sessions I hope that students understand that axioms are also generally accepted and also sorta obvious. I’m speaking to 7-9 year olds here. I am not presenting a rigid definition of axioms here, more of a contrast to theorems, which are things you can prove.) In the first session, we had worked on Zvonkin’s proofs, so the students easily understood the contrast between axioms and things that are proven.

Can the students develop a lasting understanding of axioms versus theorems in a five-week course? We shall see.

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]]>(November 16, 2022) For about 20 minutes we had been working on the famous puzzle from the book *Gödel, Escher, Bach *when Z asked, “What’s the point of this?” (besides the obvious problem-solving). I explained that I wanted to talk about axioms. “What are axioms?” the students asked, and I gave a very superficial explanation. I mentioned formal systems, showed the book, and said that axioms are “because I said so” answers to the question “why.” I’m very fortunate to be able to leverage the assets of the students: Z and D are very interested in mathematical philosophy and pedagogy so I didn’t even have to introduce the idea of axioms in any kind of pedantic way!

U: My brain hurts!

Me: Your brain is growing!

Students: Did you bring the ropes again today?

Me: Yes.

Students: Good!

I heard one of the students mention fried chicken.

Me: Stand in the red circle if you like fried chicken and the blue if you like fried, uh, ah…. I’m trying to think of something fried that not all of you would have tried before.

D: Fried pickles!

Me: Yes! Stand in the blue if you like fried pickles.

This opened up a whole wonderful can of worms.

Students: I don’t know/I don’t like either/I know my answer about one food but not the other.

Fortunately, they noticed that there was now a green rope in the bag. They debated extensively on whether to make an “I don’t know” circle with the green. If so, where to put it? Outside the red and the blue? Overlapping one or the other or both?

U: My brain hurts!

Me: That means it’s growing!

Z: Can I do some?

Me: Yes!

Once again, the students’ curiosity and excitement spared me from having to give direct instruction or information. I had arrived today with a plan to let the students take turns being the “Caller.” Z took over as Caller. Gave various prompts regarding holidays. “If you celebrate this holiday go in red and that holiday go in blue.” At first it was straightforward – red, blue, intersection, or universal set without being in red/blue. Other students took turns as Caller too. I asked for another turn.

Me: This is a yes/no. Stand in blue if you celebrate Bastille Day and red if you don’t celebrate Bastille Day. No one had heard of Bastille Day. What is it? (I answered.) Does it count if your parent is French and probably observes it somehow?

Students: What do you mean by “celebrate?”

Spoken like true mathematicians! They discussed what it means to “celebrate” and came up with a more specific definition. They debated how the ropes should be rearranged and moved them. Everyone took turns as Caller, trying to think of unfamiliar holidays.

U: I’d like to give a riddle. Can I do that now?

Me: I think that’s fine as long as you can turn it into a Venn Diagram activity.

Somehow this was possible. Go into the red if you know the answer and the blue if you don’t. Then someone standing in the red gave the answer to the riddle. Everyone was satisfied. (And I must admit that I was self-satisfied that I had started using the term Venn Diagram without ever having told the students what it was.)

V showed the group the solution to the Make 37 Puzzle, which students had worked on last week. V had figured it out at home with Cuisinaire Rods. V also gave an explanation that proved the answer. Most (or all – ?) of the others understood and appreciated this.

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]]>They pulled out a bag of two large coiled ropes. “Do you want to untangle them and set them in the grass in large circles?”

“Yes!”

Now everyone had arrived and was curious about what we would do with these rope circles. “Let’s make one circle represent yes and one represent no.”

“The red should be no,” said N. Everyone agreed.

“Okay. So stand in the blue circle if you agree with the statement and the red if you don’t. Here we go. First statement: I LIKE DOGS.”

Most students ran into one circle or the other. But two stood outside. They said, “What if the answer is yes and no, like sometimes I like dogs and sometimes I don’t?”

“Hmmmm…. I need to make a better statement. Here’s one: I LIKE TO EAT PIZZA FOR BREAKFAST.” This one worked a bit better – only one student remained outside of the circles. And we had an interesting short discussion about the distinction between actually HAVING pizza for breakfast and LIKING pizza for breakfast.

“I’M TALLER THAN A T-REX.” Most found the “no” circle but several didn’t know. “How would you know?” they asked.

“I CAN SWIM.” Bouncing back and forth – does doggie paddling count?

“I CAN DOGGIE PADDLE.” Better.

“I CAN SPEAK ANOTHER LANGUAGE.” Not so good – what if I’m starting to study another language?

“I LIKE WHAT A WHALE’S SKIN FEELS LIKE.” This was interesting. Two students went into the “yes” circle and the other four stayed outside the circles. Yes, the students in the “yes” circle had felt a whale’s skin! The others realized that if they don’t know, don’t automatically say no.

“I AM TIRED.” Lots of running from circle to circle, indecision, questions, confusion.

“Boy, I did not design this game very well. It’s not working. Hmmmmmmm…. I have a better idea! Let’s change what the circles represent! Maybe Yes and No are bad categories. New instruction: GO IN THE RED CIRCLE IF YOU LIKE SOCCER AND THE BLUE IF YOU LIKE TENNIS.”

“What if I like both?” said L.

“My dad likes tennis but I never tried it – I guess I’ll choose tennis,” said Z.

Amidst lots of laughter and hard thinking that was going on since we began the game, I announced excitedly, “Okay, I’ve finally figured it out! I finally have the statement that will make the circles work the way they should: RED IF YOU LIKE DOGS AND BLUE IF YOU LIKE CATS!”

“Wait,” said D. “We need to move the circles closer so that people who like both can have one foot in each.” The students moved the circles closer. Some straddled.

“But what if I like both and like dogs more than cats? I think I’ll go in red,” said L.

“Hmmmm,” I said, “would it be better if I give a few statements that might not involve liking something more?” (In other words, does degree matter in Venn Diagrams? The students were divided on how to design this system.)

“Yes!”

“RED IF YOU ARE OLDER THAN 2 AND BLUE IF YOU ARE OLDER THAN 4.” Debate here. Are we both? Is it possible to be older than 4 and not older than 2?

Okay, hmmmm, try this one: RED IF YOU LIKE TO EAT MAC N’ CHEESE AND BLUE IF YOU LIKE TO EAT SAND.” At first this seemed pretty straightforward. Five people went into red and one person went into blue. (N confirmed, after questioning, that yes, he does like to eat sand and does not like mac n’cheese.) Students questioned the problem again. But what if you like both? What if you like one more than the other?

“Ugh. Maybe this statement will be easier then: RED IF YOU HAVE A HEAD AND BLUE IF YOU HAVE FEET.”

The students now came up with the idea to move the ropes again to form a region of overlap (the intersection of the sets in a Venn Diagram, in formal set theory language). Students convinced each other to stand in the intersection, although one student went in grudgingly (wondering about alternative definitions of “have,” “head,” “you,” and “feet,” spoken like a true mathematician), after stating “I don’t like committing.” Peer pressure is alive and well in mathematics. It takes courage to go against the group.

(wikipedia.org) |

RED IF YOU LIKE HAVING A HEAD AND BLUE IF YOU LIKE HAVING FEET.” This generated a brief discussion on the pros and cons of having heads and feet. (I hadn’t realized there were any cons! I always do learn a lot from the participants, who have the benefit of youth and therefore less biased assumptions about how the world works.)

Here’s another one: RED IF YOU LIKE GETTING STUNG BY A BEE AND BLUE IF YOU LIKE GETTING STUNG BY A WASP.” Young people started running into circles but then all abruptly slammed on their brakes and looked at each other. “What if I don’t like either?” They slowly stepped outside of all regions. They realized that there can be elements of the universal set (students in this class) who are not elements of either designated set (like bee stings or like wasp stings).

“Excuse me, Rodi, are we going to do any other games today?” asked someone. They were having fun, but also were wondering if there’s more fun on the agenda after 30 minutes of Venn Diagrams. I told them that I had a lot of other activities that we could move to. The students decided to try some other things, but that next week continue this to devise a definitive system that includes a separate set for the answer “I don’t know.”

My goal for this course is that every activity we do contains elements of set theory, axioms of mathematics, and/or, of course, proof. We may these things together by the last session. Or we may not, depending upon student interest.

I asked students to prove or disprove the following statements (written by Zvonkin):

“I see with my ears and hear with my eyes.”

“The sun is closer to the earth than are the clouds.”

I laid down a line of colored blocks. What’s the next one? Everyone said “brown.” Nope, that’s not it. Then I put down more. Eventually the students got it, but the big underlying question is how can you prove it’s a pattern? TBD.

Students attempted the activity “Make 37.” There are a bunch of numbers in bags. Can you take numbers out of each bag so that you can make 37?

I instructed V, D, and L on how to draw it on the ground in sidewalk chalk, and they all got to work. Some did mental math. Some started adding with chalk. U realized that there were too many of some numbers in the bags so started crossing off the extras. (In other words, since 5×9=45, you need fewer than five 9s.)

At one point, I reminded students that the question is “Can it be done,” not “How do you do it?”

“That changes everything,” said U, who had a conjecture.

We were running out of time, so I handed out printouts of the problem to continue at home for those who were interested.

“Excuse me, Rodi,” said V. “Why does this paper have a bag of fives but you told L to draw nines? There are no fives here.”

“Oh no,” I groaned, “I told the problem wrong! You are right, the bags are supposed to be full of 1s, 3s, 5s, and 7s, NOT 1s, 3s, 7s, and 9s! Do you want to try it again with the right numbers? A few did, but most were happy continuing with the numbers they started with.

As we finished up and cleaned up, I asked students whether zero is a number and to prove or disprove it. Also, I showed a mathematical “magic” trick with my fingers. V and N were still trying to Make 37 as we were leaving.

This Venn diagram activity evolved from initial activities suggested in a Natural Math book (I can’t remember which one) and also Zvonkin’s Math from Three to Seven.

My apparent feeblemindedness today produced so much joy in students! I witnessed this “feeblemindedness pedagogy” in the work of the late Bob Kaplan. Bob gave me the confidence to transfer this approach that I formerly used in teaching preschool years ago to mathematics. Specifically the confidence to not let the trope of gravitas conventionally connected to mathematics hold us back from true exploration, discovery, invention, and joy

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]]>(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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]]>(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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