Statements and Domains

(2/22/22) We began our session finishing up our function machine from last week: the input is a statement and, as the students figured out last week, the output is that statement’s truth value (“true” or “false”). But what the students still wanted 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.

The Sky is Red

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.

Newcomb’s Problem

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.


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.

Theoretical Versus Applied Math

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.

The Math of 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.


Socratic Method versus IBL

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.

My mistake

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.

Beginner’s Mind

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

Mathematician Cards

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:


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.”

Kidney Exchange Decision Factors

 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.

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