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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]]>The post Axioms 6 appeared first on Math Renaissance.

]]>(October 27, 2021) *Draw 2 long lines. On each of them label one end START and the other end FINISH. What are the fastest and slowest runners in the world?*

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

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

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

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

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

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

The students intensely debated the parameters of the problem:

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

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

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

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

Everyone: “This problem is not convincing us!”

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

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

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

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

*Let’s do a function machine.*

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

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

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

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

“Put in narwhal.” Out comes 0.

“Put in snake.” Out comes 0.

“Put in kangaroo.” Out comes 4.

“Put in butterfly.” Out comes 6.

“Really?”

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

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

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

“Put in a tardigrade,” said G.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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]]>The post Happy Birthday, Martin Gardner appeared first on Math Renaissance.

]]>(October 19, 2021) I drew a few shapes on ground.

Me: *Finish creating a map of your country, with regions. There’s only one restriction: each region needs to be enclosed.*

S: “Are we going to name the regions?”

Me: *After class, if you want. Let’s take a day off from talking about axioms and strange loops, and celeberate Martin Gardner’s Birthday.*

A: “Who’s Martin Gardner?”

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

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

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

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

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

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

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

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

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

Students: “Oh!” “Maybe he was lying!”

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

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

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

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

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

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

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

Students: Yes.

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

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

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

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

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

Why do I write these blogs/reports?

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

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

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

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

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

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]]>We took a break from axioms to play the game Criss Cross today. *You draw 7 dots/vertices inside a triangle. On your turn, you connect two of them with an edge/line segment. The last person who is able connect two vertices wins. Is there a winning strategy?*

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

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

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

But S&S got a different result…

*What could explain this?* Dead air.

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

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

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

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

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

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

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

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

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

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

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

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

“What?” Confused looks.

*I mean, what should we do today?*

“Huh?”

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

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

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

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

.

The students starting talking to each other.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

“Yeah. Or at least 4 days a week.”

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

“I wish this could go on forever.”

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

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

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

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

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

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

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

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

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

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

A cheetah.

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

A snail or a sloth.

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

700 miles

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

300 miles

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

Running

*How far?*

2 miles

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

“This is hard.”

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

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

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

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

(youtube: Smart by Design)

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

IN: horse – OUT: 4

IN: racoon – OUT: 4

(and many more)

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

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

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

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

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

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

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

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

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

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

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

- Challenging to others
- Possible to solve
- Not boring

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

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

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

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

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

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

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

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

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

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

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

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

Hesitation….

“Why do we need all these rules?”

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

*What’s the name of your system?*

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

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

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

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

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

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

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

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

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

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

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

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]]>Here’s what I told the class first:

*Last week, I said “I’m going to give you a math statement: I am lying.” Was that a statement?*

So much dead air. *We’ll come back to that*, I said.

Instead, I drew students’ attention to Epimenides. I gave a one-sentence bio and then Melissa chimed in to ask students to put him in historical context, as the students have been talking about historical eras at SHL outside of Math Circle. I told the class that *Epimenides told his fellow Cretan citizens that “all Cretans are liars” in order to convince them that Zeus was (is?) immortal. What would Epiminedes need to say next in order to be convincing?*

The students pretty quickly came up with a solution, which is what Epimenides is famous for saying, “Zeus is mortal.” (The discussion reminded me of last week’s session, when the students taught me about Opposite Day.)

I played a recording of “The Song that Never Ends.” Discussion: *Does it end? If so, when/how does it end? How is this song different from me singing “I love my dogs I love my dogs I love my dogs” over and over?* The students pointed out that “The Song that Never Ends” has something in it that “flows.” I tasked students with precisely describing what makes it flow.

The students made their own Moebius strips. I asked whether this activity has anything in common with anything we’ve talked about in the course so far? I was certain that they’d say The Song that Never Ends. But no. Instead, it reminded them of Epimenides because of how it loops. I love that students rarely say what I expect them to. I also am getting excited that they are connecting these seemingly-random activities by describing them as loops (what Hofstadter calls “strange loops”).

The rule from the Function Machine of the Day was this: . After students deduced this, they worked on figuring out its inverse. The first conjecture was . We tested it by plugging in x=7 and found that it didn’t work. (I emphasized that plugging in numbers is such a useful strategy in math.) I gave them a hint: *when you do inverses, the order of operations is at play, so you may need to reverse the order*. (Honestly, I just wanted a chance to say the phrase “order of operations” out loud – in my experience, students of this age group perk up when they hear that, unlike older or younger students.) Anyway, this hint helped, and the students came up with the correct inverse, , or in Function Machine language, . (FYI I do plan to gradually transition into algebraic notation as the year goes on.)

When I facilitate in-person Math Circles, I just nod toward whoever’s turn it is to talk, so I don’t influence what they are about to say. That’s harder on Zoom. In this group, I’m developing a practice of saying to students *What are you thinking?* Instead of something like *What are your thoughts about X*, or *Do you have a conjecture about Y*, etc. The contributions to discussion from every participant in this Math Circle are so valuable that I find myself consciously trying to avoid contaminating their thinking with questions that might unduly influence. My challenge now is to see whether *What are you thinking* is unduly influential. Hmmmm….

Hofstadter didn’t actually talk about The Song that Never Ends in his book. Or at least not yet – I’m not done with the book yet; but it’s not in the index. I came across the connection of this song to strange loops in the James Propp’s article “Breaking logic with self-referential sentences” in his blog Mathematical Enchantments. Adults who are following along may enjoy this article, in which Propp connects the work of Hofstadter and Gödel to a lot of interesting mathematical paradoxes and popular songs. FYI, if you read the article, in Math Circle, I am never going to use Curry’s Paradox to prove whether Santa Claus exists. Feel free to use that one with your children at home!

You may be wondering about all the doodling on the screen while we were playing Function Machines. I generally draw a line on the screen to separate my territory from that of the students. In my territory I put the t-charts with the numbers and in the students’ they draw the machine before we start experimenting with numbers. But the drawing keeps on going while we’re conjecturing. I posit that the cognitive benefits of doodling enhance the problem-solving.

*Let’s start a new custom: every time a bus or truck goes by and we can’t hear each other, let’s take a breath*, I said at the start of class on a trafficky morning. “I know some meditation breaths!” said A. *Mathematicians like to take deep breaths. Why do you think that is?* “Frustration! Math can be frustrating,” said the students. *It can be helpful to build up frustration tolerance so that you can explore some things in math.* (I replied) *Here’s something that may or may not be frustrating: the MU Puzzle.*

[SPOILER: If you’re from the SHL group, students haven’t seen this yet – don’t mention it to them – thank you!] *We’re going to talk about formal systems. What is a system?* Discussion: what they are, problems in how they work, etc. *What do you think a formal system is?* More discussion: concluding with the definition of a formal system. *We’re going to talk about the MIU system. All that’s in there are the 3 letters MIU. You can make strings out of them.* Discussion: what are strings in math? *Here’s the puzzle: If you start with the sting MI can you produce the string MU from it if you follow the rules? What do you need to know?*

At this point, I really thought students would say that we need to know the rules, and then I’d present the rules to them one by one so that students could play with each and come up with conjectures. But no.

“We need to know the alphabet!” said R. And we were off!

The students posited conjecture after conjecture about what was needed until they realized they needed to know the rules. But they didn’t ask me what the rules are. They asked questions about the rules. (So exciting for me!) “Is there a rule where you can remove an I? … add a U? …delete something? …copy something? ….switch the order of something?” I had a yes/no answer for each question: for instance, *Rule #1 states that if you possess a string whose last letter is I, you can add on a U at the end. *As the rules emerged, I added them to our running list on the ground in sidewalk chalk.

Once students had arrived at all 4 rules, they got to work experimenting. For a long time. The students eventually entered “If-Only” mode (“if only you could triple a letter,” etc…). This mode reinforced the definition of a formal system, that you have rules and you have to follow them. Then students started working backwards (I affirmed that WB is a big strategy in math) – “If we could get MUUU, we could use Rule #4.” S was playing with Rule #2, trying to find a string with a number of elements after the M that’s divisible by both 3 and 4. I showed them a decision-tree approach that someone had tried. They engaged in other problem-solving methods until they started wondering whether the answer is that “it’s impossible.” “Why in the world did Hofstadter make a formal system like this?” someone asked. I don’t want to put any spoilers about the puzzle here, so I’ll skip to a later discussion.

*How would a computer tackle this problem*? I asked. Lots of ideas. Then G announced excitedly “A bee just landed on my thumb!” We talked about how bees like to land on us and see if we are flowers and when they find that we’re not, they fly away. The talk went from how computers handle problem-solving to how bees handle problem-solving to how humans (the students) handle problem-solving (like the MU Puzzle). If a computer uses what Hofstadter calls “Mechanical-Mode,” what would you call what humans (and maybe bees?) use? “Biological Mode,” said one student. “Loophole Mode,” said another. I told students that Hofstadter calls this “Intelligent Mode,” that trying to jump out of the system, as the students wanted to, is a fundamental characteristic of Intelligent Mode. I then showed them the subtitle of the book: “A metaphorical fugue on minds and machines in the spirit of Lewis Carroll.”

*If something seems to go on forever, looping around without an answer, how can you know whether there’s eventually a solution*? After students talking about possible tests, I mentioned that formal systems must have a test so that we know we won’t be waiting around forever for our answer. I did eventually cave, at students’ insistence, and let them know what the final result of the MU Puzzle is. I challenged students to think about at home, if they want, how to prove the answer to the MU puzzle, or come up with a test for it.

The noisy trucks and the bee: Sometimes the most interesting discussions come from the side topics . I’m really interested in, and respond to, those moments when students sense something beyond the overt mathematics, that seem tangential but actually hit on the underlying structure of things, or hit on

a useful way to think that structure.

The MU Puzzle: Those of you who know me may be wondering “Why did she cave? That’s not like Rodi to give away a solution.” My aim in this course is to talk about theorems versus axioms (which is what we were doing without using those terms yet) and to discuss M-Mode and I-Mode. Since we only have 7 weeks, I don’t want to lose a week or two on coming up with a proof for a result. I think several students may be interested in doing this at home, so I didn’t reveal how to prove it. Wikipedia shows the proof. If you haven’t played with the puzzle, I recommend you don’t read below the puzzle itself in the article because it gives away the answer too fast. Or better yet, read about it in Hofstadter’s book (pages 33-41) to enjoy the commentary.

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]]>Both current Math Circle courses are exploring the Axioms of Mathematics:

- School House Lane, virtual, ages 9-11, a year-long course, and
- Lovett, in-person, ages 8-10, a 7-week course.

The first session for each course started the same way: Rodi (me) announces *I am lying* (the Epiminedes Paradox). In both courses, we then looked at Escher’s Drawing Hands, discussed some other things, and ended with Function Machines.

WHAT MATHEMATICIANS DO

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

I AM LYING

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

MATHEMATICAL THINKING

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

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

ESCHER

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

FUNCTION MACHINES

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

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

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

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

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

REFINING CONJECTURES

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

EPIMENIDES PARADOX

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

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

ESCHER

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

BARBER PARADOX

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

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

RIDDLES

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

FUNCTION MACHINES

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

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

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

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

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

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]]>The post Human Trafficking, COVID, and Graph Theory appeared first on Math Renaissance.

]]>**Isomorphic problems**

“Aw, are we talking about human trafficking again? I don’t want to feel sad,” said F at the beginning of the session following the human-trafficking discussion. I was grateful to F since I wasn’t sure whether to continue that topic or move to something else. Thanks to his comment, I knew what to do.

The planned follow-up activity to the previous week’s was to use US highway maps and graph theory to analyze travel between states. * I asked the group how we could talk about the same mathematical content in a context other than slavery. In other words, how to change the problem without changing the math. The students rose to the occasion with multiple ideas – a car with money falling out of it, something about solar energy, something else about a bus. They wrote these new problems and then we then shifted gears to other topics for the rest of the day.

I’ve been thinking about F’s question ever since he asked back in February.

**COVID**

Once the pandemic hit and my daughter J’s school district switched to virtual learning, several of her teachers assigned projects related to COVID: things like the math and science of disease spread, etc. I was concerned because she and many students experienced increased depression related to the pandemic (isolation, fear, etc – nothing you haven’t heard about or experienced, I’m guessing). Some students I knew became sad and anxious when tackling these assignments. I talked to my friend R, a therapist, about this. R felt very strongly that schools should be a sanctuary away from immersion into pandemic studies, since students were living it at home. She did not think schools should do lessons centered around COVID. R, of course, was looking at this from a mental-health perspective: school as a place of safety when some students were saying that pandemic-related lessons scared them.

OTOH, some mathematicians I collaborate with were advocating strongly for presenting to students the mathematics of epidemiology. Many of their students and students’ families did not have access to accurate information about how COVID spread, how virulence works, the benefits of testing, etc. Here was an excellent chance for natural learning about math in a relevant context. And for mathematics to be a vehicle to get important health info home to families.

I came up with a grand plan to ask a variety of therapists, teachers, mathematicians, and parents for their thoughts and then to write about it here. Of course I got caught up in the change in lifestyle the pandemic has wrought, so never followed through. J’s school is no longer studying pandemic-related topics, although some schools and colleges are. This morning I asked her to apply her 20/20 hindsight to the matter: she said “it was a good idea (to study this topic) in the beginning of the pandemic but not now,” that the pros outweighed the cons initially only.

Regarding COVID: I still wonder what the right thing to do is/was in terms of weighing the emotional versus the intellectual/practical impacts.** Regarding human trafficking: can a certain context diminish a student’s enjoyment of mathematics, even make them not want to come to Math Circle? Regarding both: by exploring these topics with students, are we saving lives, which probably trumps all? I would like to see some discussion, analysis from psychology, and educational research on these questions. Please point me toward it if you know of it. And let me know what your thoughts are.

**Graph Theory**

In her email about the math of human trafficking, B reported that she was “struggling to apply math concepts to this topic.” She read in my blog***

*“Suppose law enforcement has enough employees to focus on just four cities in the US. How should they choose which ones? M suggested (and the others agreed) that we can choose the cities with the most lines, in other words, the vertices with the most edges. In other words, we could calculate the degree of each vertex on the graph.” *

Her question was “Would you mind explaining why each vertex’s degree on the graph can represent the city with the most lines?” I realized others reading this might wonder the same thing. Here is my reply:

*Please let me know if I am misunderstanding your question and answering something else! I suspect that your confusion may come from the terminology that I am using because in this course we were applying the math topic of graph theory, not geometry. There is some overlap in terms between these fields. Are you familiar with graph theory? A good quick intro to it is here: https://www.mathsisfun.com/activity/seven-bridges-konigsberg.html*

*Also, if you haven’t already, take a look at the photo on the blog post under the heading “Results and Reaction.” Take a look near the center of the image of the boardwork where it says “NY” with the number 6 in a circle. NY indicates the location of New York City. It is, in graph-theory language, a vertex on the graph. There are 6 lines coming out from the NY vertex. Each of these are called edges. These 6 edges/lines indicate flights. Can you see the line connecting NY to Kingston? This line/edge means that one can fly from NY to Kingston on one of the airlines we investigated. Kingston is another vertex (point). The “degree” of the vertex (point) means how many lines (edges) originate there. NY has 6 edges, or a degree of 6, because we drew 6 lines representing flights from NY.*

Thank you for reading and thinking about these things! I’m wishing all of you good health in the new year.

Rodi

NOTES * This planned follow-up activity is described in depth on pages 87-88 of Mathematics for Social Justice: Resources for the College Classroom.

** I also wonder about the specifics of studying this topic in a public school where students can’t opt out. In my Math Circle on human trafficking, I was able to give trigger warnings to students and their grownups so people could opt out, request that I not cover this topic, or process it at home outside of the sessions.

*** The original blog post is here: https://talkingsticklearningcenter.org/reducing-human-trafficking-through-math/. I had introduced students to graph theory in week one of this course, which I blogged about here: https://talkingsticklearningcenter.org/intro-to-voting-theory/ (scroll down to “Getting away from Numbers”).

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]]>The post Reducing Human Trafficking through Math appeared first on Math Renaissance.

]]>On the day of the class, not everyone had arrived by class time, so I began with some other topics. “But I thought we were talking about human trafficking today,” questioned F. We soon turned out attention to that issue.

“What is that?!” asked A about the box that said “Flash Gordon” on the table. After a quick discussion about who Flash Gordon is, I explained that this Flash is a bobblehead doll, and that we were going to use his bobbling to help us focus. “Why?” wondered the students. I explained that a focusing activity can help us bring our best to conversations about emotional topics. We then proceeded to watch Flash bobble until he stopped. He. Didn’t. Ever. Stop. So we began our conversation quietly.

I gave some background statistics on human trafficking and students contributed what they know about it. I then passed around a picture of former victim and present author Zana Muhsen, explaining to students that in math and science we can get so focused on the interesting questions about a problem that we can forget that we are talking about human beings. We kept the photo of Zana in the center of the table for the whole session.

**PUTTING MATH TO WORK**

“Math can be a tool to lessen human suffering,” I said to introduce the math problem:

*“How can law enforcement know where to focus their
efforts to reduce or stop human trafficking in the US?”*

(This lesson was first used by and published by Dr. Julie Beier of Earlham College; I did things just a little bit differently. In our course, we did not do a deep study of human trafficking before getting into the math. We did not preface the presentation of the problem with an in-depth study of graph theory; we did do the famous Konigsburg Bridge graph-theory problem several weeks before. Also, I did not instruct the students to “determine the major entry points using airline maps;” instead I hoped that the students would come up with this strategy themselves, which they mostly did.)

The students had a lot of ideas for law enforcement and the discussion moved into how the victims, who are mainly from Central America, the Caribbean, and Asia, get to the US. Once students realized to target cities with a lot of flights from those regions, I passed out airline maps. Students examined these maps (with help from parents) to create a master map. This map was essentially a graph theory graph with cities/regions as vertices and airline paths as edges. I used these mathematical terms for the rest of class.

“Suppose law enforcement has enough employees to focus on just 4 cities in the US. How should they choose which ones?” M suggested (and the others agreed) that we can choose the cities with the most lines, in other words, the vertices with the most edges. In other words, we could calculate the degree of each vertex on the graph.

**THE RELIABILITY OF MATH MODELLING?**

Before getting to work on that calculation, students had a lot to say about the lack of accuracy on our graph. For one thing, we had only examined flights from 2 US, 2 Asian airlines, 1 Mexican, 1 Latin American, and 1 Caribbean airlines. “How many airlines are there in the US, in Asia, etc.?” wondered the students. Parents, our in-class researchers, looked this up. Not surprisingly, we had a dramatic underreporting of airlines and cities. Also, the printouts of the maps were a bit hard to read, so we couldn’t even include every flight from our 7 airlines.

“Would it be okay for law enforcement to take action based upon our data and analysis?” Absolutely not, said the students. “Before we do our mathematical analysis, then,” I explained, “we have to state and write down our assumptions.” Students stated that the graph is “not thorough,” “a dramatization,” “an oversimplification,” and based upon “incomplete data.” We were all emphatic that any conclusions were drew were just an exercise, that real-life math modelling required more thorough and more accurate data.

**RESULTS AND REACTION**

Finally, students were ready to calculate degree and choose the four cities to target. Three cities exceeded the others in degree: Anchorage, San Francisco, and New York City. But four cities tied for fourth place: Seattle, San Francisco, Orlando, and Dallas. “How should we choose if we are not going to complete the data with more flight maps?” Students suggested various methods. M suggested we choose Seattle based upon some other data she had seen, so we did.

“Let’s suppose that you work for law enforcement in one of these cities. How could you use math to further focus your investigation beyond the airports?” I asked. Students started to discuss this a bit until a student asked “What time is it?”

It was 4:51. We had been at it since 3:30. We usually start cleaning up at 4:55. Everyone was surprised by how late it was. There was no time for further discussion or another activity.

A said, “I guess time flies when you’re having fun,” but he said it in a wistful voice. The students, who usually exit class in upbeat moods, were a bit subdued. I could see that they would probably need time to process the emotional content of this lesson. I acknowledged this and told them (and the parents present) to please talk about this more at home.

I did want to end class on an upbeat note, so I told students that one activity I want to do next week is to try out another voting method, and that I want to have the puppets vote for a place to go on vacation. This (puppets!) perked people up quickly – specifically thinking about what kind of places that puppets might want to go on vacation. (So far, the students’ list includes Sesame Street, F’s house, Disneyworld, washing machine and then on top of a laundry basket, Turkey, toy shop, and Stonehenge.) F asked if he could bring someone from home (not a puppet but similar) in next week for the voting, and I said, “the more, the merrier.” I encourage all students to bring in more voters, especially since we’ve been talking the whole course about voter turnout issues!

I haven’t yet decided whether we should continue with this topic. There is a lot of deeper mathematics to explore, but OTOH we only have 2 sessions left and one student already missed this lesson. We may want to explore other topics and finish off our exploration of voting theory. Parents, let me know if you or your students have strong feelings either way.

I am filled with gratitude for the parents who helped out in class and the parents who were not able to come but trusted me to cover the topic appropriately with your children. I am grateful to have had the chance to do it.

**OTHER TOPICS**

As we were waiting for everyone to arrive, we talked about voting-theory math in this week’s news. We looked at articles about

- How some pundits find that plurality voting in the Democratic primary problematic
- How close to representing the country demographically Pennsylvania is
- A study done on why so many people don’t vote (This prompted a student-led collaboration on figuring out what percent of the US population doesn’t vote. The article stated that about 100 million people sit out elections. Students decided to look up total population, child population, felon population, and were incredulous to figure out what a large portion of the population qualified to vote the non-voters comprise.)

We also did more work on the 4-color theorem. Students tried to color M’s map that previously seemed to require 5 colors. This time they did it quickly in 4.

We discussed whether there is an optimal strategy to do the work on this problem. (M started with one color and colored every region possible with that color before switching to a different color. F started with one region and colored every adjacent region before moving to a different region.) I asked again whether any maps require more than 5 colors, or could these 4-color maps be done in 3. “Are you going to tell us the answer?” asked F. (I didn’t promise, but I probably will at the last session. I don’t usually do this, but mathematicians required a computer to solve this one.)

Rodi

*For anyone who didn’t see it, here is my pre-class email to parents:

*As
you know from reading the course description of our current course, one of our
topics is human trafficking. Tomorrow we will begin our mathematical
exploration of this topic. The math involves graph theory, mathematical
modelling, and data analysis. To give the mathematics some context, we will
talk for a short bit about what human trafficking is.*

*I
am using a module that is used in a course at Earlham College, described in the
book “Mathematics for Social Justice: Resources for the College
Classroom.” I will not be doing the background/contextual studies that a
college class would. In the book, Dr. Julie Beier of Earlham suggests
supporting students for the emotional content by “setting classroom
guidelines for discussion, practice with less intense topics, and starting with
silence to encourage students to bring their best self to a conversation.”
She also recommends that instructors “explain the purpose of silence”
and to encourage students “to make a list of their reactions.” We
have already been doing the first two suggestions and plan to do the others as
well. I do plan to start with a totally secular focusing activity – probably a
bobblehead doll. I would like to invite any of you to sit in for the topic
introduction, and to email or call me with any questions or concerns up front.*

*The
background that I will introduce is from the UN Office of Drugs and Crime
(UNODC):*

*only 63% of 155 countries providing data to the UN have “passed laws against the trafficking of people”**“approximately 79% of human trafficking is for sexual exploitation”**“approximately 79% of all victims are female”**“forced labor accounts for about 18% of the reported trafficking**“the percentage of humans trafficked that are children is 20% globally, although in some parts of the world it is as high as 100%”*

*There
is not much more known or studied about this problem, and beyond these
statistics and information about the logistics of how the transport of victims
works, I do not plan to delve into background. Students will, of course, have
questions. (Ellen, you will of course use your judgement about which questions
to research in class and which to defer to outside study.)*

*I
STRONGLY encourage you to talk with your students in advance about this if you
suspect your student will have an emotional reaction to hearing this
information for the first time.*

*Also,
feel free to respond to this email (reply to all, please) or call/text me with
any other questions or concerns.*

*Best
regards,*

*Rodi*

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