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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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]]>The post Axioms 2 appeared first on Math Renaissance.

]]>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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]]>The post Same Topic, Different Directions appeared first on Math Renaissance.

]]>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”).

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

]]>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*

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

]]>The post Data Interpretation and Analysis appeared first on Math Renaissance.

]]>*“Suppose the country votes state by state on different
days; we are in Hawaii; and actual state preferences are ravioli 1%, mint
chocolate chip 12%, butter pecan 12%, mango 40%, vanilla 20%, and chocolate
15%. What voting method would each flavor lobby want?”*

*“Suppose there was a caucus with low turnout, a pushy
person there advocating for butter pecan, and a required tasting session first.
What would happen and what would be the implications?”*

I asked these questions to facilitate evaluation of the Iowa Caucus, which took place 2 days before our session. Students brought up important issues, explored possible outcomes, and moved the conversation in some unexpected, interesting directions such as the population of Hawaii, mosquito control, and how percents work. They also came up with some novel voting methods.

**PERCENTS WITH MOSQUITOES**

A had a lot of questions about how percents and percent change are calculated. In response to his question “Is it possible to have percents that are over 100%?” M came up with some examples involving mosquitoes. For instance, “The mosquito population increased by 200 percent.” “What does it mean to have minus 200 percent?” asked someone. We made up some examples with mosquito population changing over time to illustrate the calculation. Then F asked, “Why are there only 10 mosquitoes in the whole world?”

“Because this is math!” I replied. “We can make up anything we want within a system as long as we follow a consistent set of rules within that system.” We also talked about, as we have before, how the math itself can become clearer with easier examples.

Students also pointed out that mosquito growth isn’t linear, it’s probably exponential. Again, though, we can imagine it to work how we want it to work since this is math and our goal is precision and logic.

**THE IOWA CAUCUS**

I then handed out photos and the detailed results of the Iowa Caucus at various points in time. “What do you notice?” Students analyzed how the results varied depending upon what percent of the results had been counted. They also noted how the first alignment begets the second alignment begets delegates.

We looked at a map of results by county. “Would it be fairer if the delegates were elected proportionally to county instead of proportional to population?” All of the students were adamant that this was a terribly unfair idea. My helper Ellen and I told them that it has been and is done this way in some circumstances in the USA. Could the students guess where? J knew that the electoral college had some geographic issues built into it, but everyone was pretty skeptical of the concept of how representation is done in the Senate. Another issue that came up is how Iowa can influence the entire electorate while not necessarily representing the entire electorate in terms of various demographics.

**THE WRONG KONIGSBURG BRIDGE PROBLEM**

I accidentally presented an oversimplified version this famous problem. When I posed the question, I forgot that you have to start and end on the same side. Students made fast work of it, testing it first using an actual map and then using a graph theory map. They definitely preferred exploring the problem using graph theory, coming up with the conjectures that it is impossible to cross all the bridges once with 7 bridges in this position but possible with 5, 6, and 8. This being math, the most important questions were “Why?” and “Can you prove it?” I asked “Is it possible with 117 bridges.” The working conjectures at this point are yes if you move the bridges so that they are all parallel and no if you keep them where they are because there is an odd number. To be continued next week, with the correct problem.

**TO EXTEND AT HOME FOR THOSE WHO DESIRE:**

- Why do some of the Andrew Yang signs at the Iowa Caucus say “MATH” instead of “YANG?” (The students noticed this but we didn’t get a chance to explore.)
- Explore percent change

See you next week!

Rodi

The post Data Interpretation and Analysis appeared first on Math Renaissance.

]]>The post Caucusing and the Four-Color Theorem appeared first on Math Renaissance.

]]>*“Suppose that your state is going to vote for a leader
and that all citizens and candidates only care about one issue: the law that
you can keep dairy products in your refrigerator for no more than 21 days. You
hold a caucus. What happens when the following four candidates run and the
puppets vote?*

*DAIRY-LOBBY CANDIDATE (dark blue): The soy milk industry is damaging the dairy milk industry. Our state’s workers feed their families with money earned from working with cows. I will work to make soy milk illegal, and if not illegal, pass a law against calling it “milk.”**COW-OWNER CANDIDATE (light blue): I have some cows but also have child who is lactose intolerant and needs an alternative to dairy products. I support the applying the 21-day rule to both dairy and non-dairy milk products.**HEALTH-PERSPECTIVE CANDIDATE (green): Since expired dairy milk and expired soy milk have different health consequences, the 21-day law should only apply to dairy milk. I will pass a law exempting soy milk.**NO-NANNY-STATE CANDIDATE (pink): What is the government doing in our refrigerators? The 21-day milk law should be struck down. The government has no business making and enforcing laws about how we use products.”*

First the students reacted to these candidates. Disbelief that someone would propose criminalizing the word milk applied to non-dairy beverages. Confusion about what soy milk is. Uncertainty about the word “exempt.” Laughter…

We had the puppets (so that we don’t get into discussions of student preferences) caucus for these candidates. We explored all sorts of scenarios of first and second alignment, based upon explanations from the Des Moines Register.*

After experiencing a caucus first-hand, I asked the big question: I told the students the 5 changes that Iowa was instituting this year and asked the students for conjectures on the reasons for each change.

**MAP COLORING**

Can you make a map that requires 5 colors to color? At various points students thought they did. We talked about how a mathematician’s job is to make patterns and break patterns. I walked around trying to break patterns (in other words burst people’s bubbles of hope that they had succeeded in this mission). In most cases I was able to point out a way to use fewer colors.

But there were 2 maps where I couldn’t do this quickly or easily.

I promised students that next week, they (not I) will try to reduce these student-created 5-color maps to 4—or-fewer color maps.

ANTI-PLURALITY VOTING

We played out and analyzed Wikipedia’s Tennessee capital example of anti-plurality voting. Students discussed and realized about how this method can lead to results where a more middle-of-the-road/bland/non-extreme candidate is more likely to win. “Is that a problem?” I asked. Students were split. My helper Ellen then read from an article on presidential candidate Andrew Yang’s website about voting methods.

QUESTIONS/UNEXPECTED TOPICS

Our mathematical explorations led students to pose some new questions. (Some of you may be familiar with Rochelle Gutiérrez’s work on rehumanizing mathematics. Giving students opportunities to follow their own curiosity and to connect math to other disciplines are two of the ways we can rehumanize mathematics.) Our students wondered and investigated/discussed

- HISTORY: Which candidate is a Freemason?
- ETYMOLOGY: Why is milk called milk?
- CURRENT EVENTS: Are Republicans caucusing in Iowa too, or is it just the Democrats?
- POLITICAL SCIENCE: If the Senate convicts the president and he cannot run for a second term, would the Republican party be able to field a candidate at this point, and if so, whom?

Rodi

‘* Our discussions were based upon good background info from the Des Moines Register and NPR.

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]]>The post Gerrymandering via IBL appeared first on Math Renaissance.

]]>1) What’s the difference between the House and the Senate?

2) If your state has a population of 50 people, 30 affiliated with the blue party and 20 affiliated with the red party, and you’re electing 5 representatives to the House, what’s the most fair way to draw district lines?

3) Could the red party win a majority of your state’s seats in the House?

4) What’s the minimum number of colors needed to color the Pennsylvania Legislative Map?

My hope was that when the students started discussing and asking questions, that my answers provided any direction they need.

**VOTING ON MILK**

When I posed the first question above, it turned out that the students weren’t totally sure what the House and the Senate are. One student tentatively posited that they are part of the government.

“What is the government for?” I asked, as I wanted to give our lesson some context. The students knew that governments engage in foreign policy and make laws. So we had two branches of the U.S. government covered. I wanted students to know that there is another branch, but I didn’t want to just come right out and tell them about the Judicial branch. I didn’t have a question planned, so I just blurted out the first thing that jumped into my mind:

“Suppose the Legislative Branch passed a law stipulating that no citizen is allowed to keep milk in your refrigerator for longer than 21 days. What are some possible problems with that law?” Students came up with ideas, but I was hoping to get them to think of the challenge of interpretation. (An important skill in mathematics is defining your terms, so I was hoping students might ask “How does the government define milk? How do they define refrigerator?” Those are the questions a mathematician would ask. The students did not ask these questions, so I asked “What if some people drank soy milk and wanted to keep it for longer? Would that be okay?”

“That would definitely be the problem in my house!” announced M. Then followed discussion that generated the idea of courts and the Judicial branch and also the House versus the Senate. (I almost wrote here, “We were finally ready to get into some math,” but we were already into some math with the issues of definitions and precision.)

**DOES PROPORTIONAL MEAN FAIR?**

I drew on the board 5 rows of 10 dots: the top two rows red and the bottom three blue. “Suppose your job is to decide where the boundaries of your 5 districts will be. How would you draw the lines?”

F immediately saw a flaw in this model. “Why do they live in nice little rows?” I acknowledged that, as we sometimes do in mathematics, we’re oversimplifying to make a point and real life is messier.

No one had an idea how to draw the lines. “If we imagine each row is a street, what would the election results be if each street was a separate district?” Students calculated: 3 blue representatives and 2 red.

“Are those reasonable places to draw the lines?” The students said yes. I told them of the belief that legislative districts be “compact.” “What does compact mean?” Students talked about it for about 30 seconds and were done. I must tell you that I spent a very long time before class wrestling with and researching the idea of what compact might mean. It seems that no one can come up with a good definition of this term. I find that incredibly interesting, probably the most interesting concept in this entire discipline. But the students did not find this interesting at all, so in the spirit of IBL, I moved on.

What the students really wanted to talk about was whether this district division was fair. I told them that the conventional wisdom is that proportional representation is fair. M brought up the point that if the red party only has 40% of the representatives, it will always be outvoted and its interests will never become law. At this point, we had moved into the context of our milk-law discussion from earlier. The students wondered: if 40% of the citizens want to legally keep soy milk for more than 21 days, but 60% of the citizens want soy milk to get discarded at the same time as the dairy milk, how is that fair for soy-milk drinkers? In other words, how is proportional representation fair if 40% of the people don’t get their legislative desires satisfied? (This is something I had never thought about. I am so happy to be exposed to a new idea from students!)

At this point, A took the marker from my hand and demonstrated on the board what might be more fair: 5 people from the blue party switch their allegiance to the red party. Of course, A explained, we would have to know what the positions of each party are so that these 5 people really want to switch. At this point, we’d have 50% from each party. “If your district is entitled to 5 representatives,” I countered, “then what would the election results be?”

Students realized here that things are more complicated than they seem on the surface. They came up with a few ideas of their own:

- Instead of having a national government with a Congress comprised of people from each state, let states govern themselves and just have one nationwide annual meeting.
- Let’s have neutral representatives! Why do they have to be connected to one specific party?

**MAKING LEGISLATIVE DISTRICTS
MORE COMPACT**

“Suppose people on one end of the street didn’t even know their neighbors ten houses down and decided that the districts would be more compact if grouped vertically, with their neighbors behind them. Would the election results change?” Students calculated and immediately agreed that this districting is more compact but less proportional and less fair, as the blue party gets all of the seats. By optimizing one variable, we weakened another.

“When are we getting to gerrymandering?” asked F presciently.

**RIGGING THE SYSTEM**

“Can we draw the lines so that red wins?” I asked. M had seen this example before, so we already knew the answer. So I changed the question: “How can we draw the lines so that red will win?” I gave each student a paper with a 5×10 grid of dots, a red pencil, a blue pencil, and a regular pencil. I instructed them to circle the dots in the top two rows red and the bottom three rows blue. “Use the regular pencil to draw your district lines.” The students got to work.

“Is this legal?!” asked someone incredulously.

“I can’t do it,” said F.

“Imagine it’s your job. Your boss has hired you to redistrict in favor of a certain party. If you can’t, you’ll get fired.” I hoped drama component would stimulate creativity. It didn’t.

“Good. I wouldn’t want that job,” said F adamantly. As the students worked, they were still holding on to some shock about the legality of this.

Soon, M had figured out a way. “Are your districts compact?” I asked. They were. “What happens sometimes is that districts get gerrymandered into shapes that are not compact, that are the opposite of compact. Can you do it that way?” No one was able to get five non-compact districts of the same population with red winning, but M came close. “Can you look at that weird-shaped district and see an animal shape?” She saw a rhinoceros-shaped district.

I passed around the legislative map carved out by Massachusetts Governor Elbridge Gerry in 1812 and asked students why they think it’s called gerrymandering. They figured it out. We also discussed the pronunciation of the word and name. The students helped to correct my pronunciation for the rest of the class.

“Are there any examples of this
happening in our times, or did it just happen in history?” asked someone. I passed
around some maps of current gerrymandered districts (Maryland’s 3^{rd }district,
North Carolina’s 12^{th}, Florida’s 5^{th}, and Texas’s 35^{th}).
Then one of the most famous gerrymandered district, our local former 7^{th}
District of PA.

**GERRYMANDERING IN PENNSYLVANIA**

I then gave students a handout comparing PA’s congressional districts from 1992 versus 2011. After students discussed differences, I gave them a handout showing how the districts changed from 2011 to 2018 after the PA Supreme Court declared gerrymandering illegal in our state. The students examined these maps and asked a lot of questions about the legality of gerrymandering. My helper Ellen did some in-class research and answered student questions about this.

**MAP COLORING**

I then gave everyone a map of the current congressional districts in PA. “What’s the minimum number of colors you need to color this map?”

“Can we try?” asked someone.

“First, in the spirit of mathematics, make a conjecture about how many colors it will take,” I responded. Conjectures ranged from 3 to 18. They then tried it. F realized it would be quicker to test conjectures using hashmarks to symbolize colors. After a short time of asking clarifying questions, making mistakes, and testing ideas, the consensus was that you need four colors to color the PA Legislative Map. I asked 2 follow-up questions:

- Is it possible to color this map with 3 colors if you try harder?
- Are there any maps that require more than 4 colors?

M had seen a map of the U.S. colored in four colors, so we knew that was possible. I added these questions to today’s running list of questions on the board.

We were almost out of time when the students asked to return to our graph theory problem from last week.

**GAS WATER ELECTRICITY PROBLEM**

I asked whether anyone had thought about this problem since last week. A few had. No one had come up with a solution. “If you were allowed to change the rules to make it easier, what would you change?” The students agreed that allowing one pair of lines to cross would make a solution possible. As we were discussing this, F jumped up to the board with an idea.

“Never mind,” he said, partway through, and sat down.

**FOLLOW UP AT HOME**

- Come up with an actual mathematical definition for “compact” (if that interests you!).
- Do some research on different perspectives on whether proportional voting systems are fair.
- Find out whether gerrymandering happens outside of the U.S.
- Ask your student what PA’s former
7
^{th}district resembles. (Some call it Goofy Kicking Donald Duck, which I did not mention in class.)

**GRATITUDE**

This class would not have been possible without my helper Ellen or content suggestions from Haley Horton. Before leading this session, I posted to the Facebook group 1001 Circles a request for help connecting gerrymandering to the Four-Color Theorem. Haley came through, with the idea of using our local legislative map and also content from this Washington Post video on gerrymandering. The video is less than 3 minutes and covers a lot of what we did in class today.

Rodi

*This plan was different from a pure “Ask Don’t Tell” approach because I was prepared to give students information in answer to their questions. We don’t have enough time in our 90-minute sessions to allow for students to research all answers. So I planned to answer (with Ellen’s help) at least some student questions beyond the scope of mathematics. In a pure IBL (inquiry-based learning) class, students would do their own research and possibly go off on a tangent. I used a guided-inquiry approach here in the interest of time and also because I wanted the focus to stay on mathematics.

The post Gerrymandering via IBL appeared first on Math Renaissance.

]]>The post Intro to Voting Theory appeared first on Math Renaissance.

]]>**Pet of the World**

“What would happen if you got to vote for Pet of the World, you had to choose between Dog and Cat, that there were exactly 100 people in the world, and you were the only people who showed up to vote?” After I posed this question, the students voted and Dog won with 75% of the vote.

“Why did Dog win?” I asked.

“Dog had more votes!”

“So whoever has the biggest number of votes wins, right?”

“Yes!”

“What would happen if all of those people who didn’t vote actually wanted Cat to be named Pet of the World?” Now the game was afoot, the game being to shake up our assumptions about math. This question incited a big discussion about fairness and problems with voting systems and methods, so we began a list on the board and added to it every time someone raised an issue.

**Plurality versus Majority**

Of course students wanted to talk about the current US election. One goal for this course is to let students experience Emily Rhiel’s wonderful presentation “The Mathematics of Social Choice.” I’m using her presentation as an outline, turning her statements into questions, and letting the group take it from there. (In other words, we’re taking an inquiry-based approach.) *

“Can you name every candidate who was running for the democratic nomination this past summer?” The students brainstormed, with help from me and Ellen (my helper). We got a full list on the board and then I asked students to cross off a few so that we had only 20 to make the calculations easier. **

“What would happen if all 100 people in the country voted and Cory*** got 6 votes, Seth got 4, and everyone else got 5 apiece?” Great hesitation from the students now, not that they didn’t know the answer, but the results were disturbing. Then an even more disturbing question: “What if Wayne dropped out and all 5 of his supporters preferred Seth to Cory?”****

After some discussion, the students suggested aggregating votes to fewer candidates. They dictated who should drop out and to whom those votes should go. Each time one candidate dropped out, I asked, “Now does any one person have enough support to be sure of winning?” The answer was no every time. Things got really interesting when the students manipulated the voting so that Pete had 34, Bernie had 41, and Elizabeth had 25. Students soon stated that a single candidate had to have one more than half of the votes to be sure that the real preference won. We discussed plurality versus majority.

“Is majority the best and fairest method?”

“It’s a no-brainer,” said A. Everyone agreed. After a little more discussion, J pointed out that it depends upon how many candidates there are. Majority is fine for a two-candidate election, but harder to achieve with more than two. M brought up another issue: gerrymandering. Then F brought up yet another issue: the electoral college. And the students already knew about strategic voting. By now our list of issues/problems with voting systems was getting longer.

**Student list (so far) of Problems with Voting Methods**

- People not voting
- Most preferred candidate not winning
- Electoral College (is this a problem or a solution?)
- Gerrymandering
- Miscounts
- Majority could produce a bad leader
- Strategic voting (is this a problem or a solution?)
- Uninformed voters
- Too many candidates (is this a problem?)

We didn’t have one single discussion of problems during our 90 minutes. Instead, this seemed to be the theme that every discussion throughout the session circled back to. The “problem” discussion generated a new list: “ideas.”

**Student Ideas**

We kept a running list on the board of ideas as they occurred:

- A point system whereby voters list multiple preferences and candidates get different numbers of points. (This is what I mean by students inventing and discovering math for themselves! Mathematicians have developed ideas like point systems for hundreds of years. We’ll talk more about that method during the course.)
- Ruler is appointed by the predecessor but can be deposed by a majority of citizen votes. (This idea prompted a discussion on whether we were assuming a democracy in this course, and whether this system is democratic. Again, prescience here with the idea – there is a precedent for voters stating whom they disapprove of – we’ll get to that soon.)
- Add in an element of chance to the election. (I will research this!)
- Write-in candidates. (Once again, students had not previously heard of something that is done in real life but thought of it themselves.)

**Vote for Two Method**

“What democrats are left in the election as of today?” We got the list of 12 on the board. “Let’s suppose that there are only 6 left. Of these 6, suppose every voter is allowed to vote for two. What would happen?” The students winnowed the list down to 6 candidates. I pulled a puppet out of my bag and said to M, “If this puppet was allowed one vote, whom would she vote for? And if she had a second choice, what would that choice be?” I circled around the room several times getting students to say whom each of the 26 puppets would vote for in a vote-for-one and a vote-for two election. We then compared the results of the two methods. The students agreed that different methods can produce different results and that different methods might be better for different situations. The students had no conjectures *yet* about what those situations might be.

**Other questions**

As we worked, students asked other questions. Ellen acted as our Math Circle’s real-time fact-checker. So some of these questions got answered on the spot:

- When is the last date a candidate can join an election?
- How does the Iowa Caucus work?
- Can people write in anyone?
- Is there an age minimum for write-ins?
- What if nobody votes?

**Getting away from numbers**

At this point, students’ brains were tired. I abandoned my plan to introduce yet another voting scenario and instead mentioned a topic in which politicians are involved: utilities. I posed the famous “Gas Water Electricity” Problem.

“Three houses in a row need to be connected to three utilities (gas, water, and electricity), the sources of which are also in a row. Can it be done with no lines crossing?”

M immediately noted that this problem is similar to a problem we did with names a year or two ago. I explained that it’s similar since it’s in the field of graph theory. As soon as I said “graph theory,” one student reacted excitedly with “yes!” The students spent the rest of the session with paper and pencils trying to solve this problem.

**Follow-up at home**

There is no homework in Math Circle. But for those of you who have asked, here are some ways to extend what we did here today. If you chose to do that, try having the students create the scenarios.

- Explore methods for how to tally up large sets of data (i.e. if you had 100 people voting for Dog versus Cat, how to count quickly?).
- Make a fraction from a ratio (i.e. if 65 voted for Dog and 35 for cat, what fraction of people voted for Dog?).
- Convert from a fraction to percent.
- Research how utilities are related to politics.
- Explore the name problem that M mentioned. It’s an unsolved problem posed in 1977 by Krishnamoorthy and Deo. Visit https://mathpickle.com/unsolved-k-12/ and click on the video at the top right in the array of videos.
- Work on the Gas Water Electricity problem.

A big goal for this course is to expose the students to as many voting methods as possible and for them to evaluate them at the end to see whether any truly is a no-brainer.

Rodi

‘* I plan to do this with the guideline that we only talk about the mathematics of voting theory and not about policy or candidate preferences. Using puppets helps to keep the conversation non-partisan.

**One student asked “Why just talk about democrats?” I explained that it makes the math more interesting since there more candidates.

*** The students helped to formulate the questions. This writing seems to insinuate that I had a script of questions and was just reading them. I didn’t. I only came in with a printout of the slides from Dr. Rhiel’s presentation. The students not only asked a lot of questions, but they also came up with the names and numbers when our candidates dropped out or their supporters switched preferences.

**** In the first situation, a candidate won with only 6% of the votes. In the second situation, the last-place candidate became the first-place candidate when one person dropped out.

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]]>“No, Penelope is not a pig. She is a pig *puppet*. There’s a big difference,” I replied as we sat down. This seemingly inane comment of mine captured everyone’s attention.

** **

**PIG-PUPPET YEARS**

“You know how it is said that a year for humans is like 7 for dogs and 5 for cats?” I asked? Everyone nodded.

“There are also fox years,” added A.

“Yes, and there are pig-puppet years. But they work in the opposite direction as dog years and cat years. For every 10 years that a human ages, a pig puppet only ages 1 year. So even though I got Penelope almost 30 years ago, her age is really 3.”* We discussed this concept for a moment, and then I announced, “I’m having some trouble with Penelope that you can probably help me with. I’m trying to teach her about numbers, but listen to how she responds to my lessons:”

Me to Penelope the pig puppet: “If Grandma gives you five cookies and Grandpa gives you five cookies, how many cookies will you have?”

Penelope to me: “None, because I’ll eat them all!”**

** **

**DOES CONTEXT MATTER?**

Penelope’s statement set off a huge mathematical conversation. Students had questions and comments about what numbers are, how to explain them, what counting really means, the difference between numbers and numerals, and how numbers first came into being. We talked about all of these things, and then returned to the original scenario. The students tried and tried to teach Penelope the math problem above by changing the context:

Student: “What do you get when you combine five blobs and five blobs?”

Penelope: “One, since blobs squish together when you combine them!”

Student: “What do you get when you combine five pieces of titanium and five pieces of titanium?”

Penelope: “One, since titanium is a metal and metals melt at high temperatures.”

Student: “What do you get when you combine five wooden blocks and five wooden blocks?”

Penelope: “Zero, because I like to play with matches!”

Student: “What do you get when you combine the numeral five and the numeral five?”

Penelope: “Fifty-five, since the fives are right next to each other now!”

Turns out that no matter what context the students came up with (probably 20 examples in all), Penelope had a way to make the problem not work. No matter what, five things plus five things didn’t equal ten.

A: “How can Penelope know so much about other things and not know anything about math?”

Me: “She’s a science prodigy.”

F: “But aren’t math and science related?”

Me: “Yes, but that’s another thing that’s special about pig puppets. We can take some creative liberties.”

By this time, the puppet Penelope had somehow moved from my hand to F’s hand, and the students had taken over both roles – coming up with new contexts and finding ways to contradict the hoped-for result. No one was able to come up with a context that Penelope (in most cases actually M) couldn’t knock down. I was just enjoying the show.

** **

**STRIPPING AWAY CONTEXT**

“What’s the difference between the problem *five plus five equals ten* and the problem *five things plus five things equals ten things*? I asked. The students’ thinking even further intensified. They posited conjectures, debated them, rejected them until S*** said “My brain hurts!” The others agreed.

“What’s the difference between numbers and things?” I asked more directly.

“Well, numbers are something that we made up to talk about things,” answered A.***

“Do numbers exist as things in the natural world?” I asked.

“Yes,” said about half the students.

“No,” said the other half at the same time.

They all looked at each other. Then those that said Yes changed their answers to No.

“Are they ideas?” I asked.

“Yes!” everyone agreed. We talked about ideas versus things. How mathematicians use the word abstract to describe ideas that can then be applied to multiple scenarios.

“Like abstract art,” said S excitedly. Then she quickly reversed herself: “Actually, no, since abstract art is a thing.”

“My brain really hurts now,” said A.

“Do cookies behave logically?” I asked?

“No. People eat them!”

“So would mathematicians rather study things that behave logically or things that do not?” The students all agreed that “logical things” is the answer. I explained that mathematicians like to strip away the context to get at the underlying abstract structure of things. This can reveal similarities, I continued, like in that problem we did last week with the symmetries and arrangements.

But is it always mathematically sound to strip away all context? If a problem is totally abstract, will you arrive at a useful answer?

**RECONSIDERING CONTEXT**

I presented the students a paraphrase of a problem from Eugenia Cheng’s book __How to Bake ____π__:****

*You run a company that takes people on tours. You’re organizing a trip for 100 people.* *You’re renting minibuses and want to maximize your profit. Each minibus holds 15 people. How many do you have to rent?*

The students started out by trying numbers: 10 busses – too many. 9 busses – still too many. Then S suggested dividing 100 by 15, yielding 6.6̅. After some discussion, they concluded that we need to rent 7 busses, so 100 ÷ 15 = 7.

“That’s it? That’s the problem?” said S, a bit disappointed. She was happy to hear that no, that’s just the first part. The problem continues:

*Now you’re shipping some chocolates to a friend. You pre-paid for a stamp that covers the cost of mailing 100 ounces. Each chocolate weighs 15 ounces.* *How many pieces can you send to your friend without paying extra for shipping?*

Immediately the students saw that it’s the same calculation but a different interpretation of 6. 6̅. They all were talking but not so much to each other or me. More like each was thinking aloud, simultaneously. S persevered the longest and gave a solid explanation of why in this case, 100 ÷ 15 = 6.

“So here’s the real problem,” I said to the students. *“Why is the answer 6 when you’re talking about chocolates and stamps but the answer 7 when you’re talking about people and busses?”*

“Context matters,” they all agreed. I quoted Cheng to them: “Be careful not to throw away too much… Category theory brings context to the forefront.”

“What would be the answer,” I asked, “to a person who looked at this problem purely abstractly, with all of the context stripped away?”

“6. 6̅” they all agreed. They definitely were grasping abstraction versus reality and some key points about context. But it was time for a break. People’s brains had started hurting 20 minutes ago.

** **

**FUNCTION MACHINES AND CAKE CUTTING**

I gave everyone an apparent brain break by doing a function machine with them. (The students provide a number that goes “in,” I tell them what number comes “out,” and their job is to discern the rule.)

“Now try this one: If you slice a cake, what’s the function for the maximum number of pieces you can get with a certain number of cuts?” (I also worded it in the language of circles at F’s request: “What is the function for the maximum number of regions you can create with a certain number of chords in a circle?”)

I started sketching this on the board with the students’ verbal instructions (“2 pieces from 1 cut, 4 pieces from 2 cuts,” etc.). But almost immediately, the students were all at the board figuring it out for themselves. Once again, I sat back and enjoyed. Most of the students quickly got 7 pieces from 3 cuts.

“I got 10 pieces from 4 cuts,” announced M.

“Can you get more?” I asked. She tried, without success, and then went on to test 5 cuts and 6 cuts. By then, at least three students had diagrams with 10 pieces from 4 cuts. “You can get more,” I promised. “There’s something that all of you are doing that you could change to get more.” They kept working.

“Can you get more than 7 pieces from 3 cuts?” backtracked S.

“No one ever has,” I said.

“But just because no one ever has, does that mean it can’t be done?” she asked. “Has anyone demonstrated that it definitely can’t be done?”

“That is one of the key questions in mathematics,” I said, so excited by this question. A huge goal of our math circle is to teach kids to be doubters. “In math, it’s not enough that no one has ever done something. There has to be a proof that it can’t (or can) be done for us to believe anything. And yes, there is a proof that you cannot do more than 7.”

“I got 11!” announced A, who had been fervently trying to beat the class record of 10 from 4 cuts.

“Now you’ve reached the number that has been proven to be the maximum.”

My intended point of this activity had been to look for a pattern/function/rule to determine the number of slices. We had a nice sequence of numbers (2,4,7,11), but no one was interested in pattern-seeking. They just wanted to keep testing. So I played the big-number card: “How many pieces could you get from 500 cuts?”

“We would need bigger whiteboards and more markers,” said someone, defeating my attempt to redirect the approach.

I played the ridiculous-number card: “If you had to determine how many pieces you could get from 5,000 cuts, would you rather have a bigger whiteboard or know the rule?” Someone grudgingly said that the rule would be better in that case, but it didn’t detract from anyone’s enthusiasm for drawing.

We were out of time. Had we gotten to the point where students showed interest in determining a rule, I would have burst their bubble anyway with some talk about how patterns don’t mean rules without a proof. (Again, training doubters.) So we ended on a high note with me connecting this activity to the idea of abstraction.

**A FEW OTHER THINGS**

Early in the session, the students brought up the golden goose problem from last week. (*Would you rather have golden eggs, a goose that makes golden eggs, a machine that makes those geese, or a machine that makes those machines?*) Some students had talked about it at home and were reconsidering their answers from last time. We talked about the levels of knowledge you needed for each item in this hierarchy, and how that ties to mathematics. The students wanted to explore whether the answer to the question would be different if we removed the goose as a possibility. I explained that “What would happen if we changed the question a bit?” is exactly something mathematicians ask all the time. This came up later in the class, when S was doing the cake-cutting problem without realizing that the cuts had to be chords, not just random line segments. She made a quick shift from initial disappointment that she had misunderstood the problem to excitement to hear that this might a new way to do this problem that people hadn’t worked on before.

Rodi

*I’m relaying this anecdote so that interested parents can have a jumping-off point to talk more about ratios and proportions at home.

** Eugenia Cheng, __How to Bake ____π__, p19. We also dramatized with Penelope the pig puppet Cheng’s examples of the difference between having memorized the sequence of counting numbers and actually understanding what they mean. (p20) Cheng discusses the cake-cutting problem on pages 33-34. You can find an algebraic explanation of the problem on Wolfram MathWorld and many other places. It’s a classic problem.

***We have two students whose names begin with S, and I’m using S for both. Ditto for A.

****Cheng, p21

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