(April 28, May 5, and May 12, 2015) We’ve had 3 sessions so far, and I see 3 big themes developing in this 5-session course for 9-11 year olds:
- Everyone thinks that “Everyone Else in the Room is Better at Math than Me.”
- Not everyone realizes what math is really about.
- There’s culture/folklore in mathematics that the kids are starting to get a glimpse of.
I’m using a platform of the Dark Bridge problem and of Function Machines as the anchoring point for discussion. Here are the specifics.
SESSION 1, INTRODUCTION
Oh, the glory of spring! We’re back outside, in the garden, under the trees. After we gathered in a circle (yes, and actual circle for Math Circle) on blankets under a tree, I told the kids that we are revisiting Math Circle history in this course, in three ways.
“Are we going to do the unicorn bridge problem? That’s my favorite math problem of all time!” said M, who had done this problem in Math Circle nearly 4 years ago. (Two others in this group, J and L, were present back then too.)
“Yes, that’s one of the three,” I replied.
“Martin Gardner?” asked someone hopefully (L or A), remembering our fall course.
“Yes, we will do a problem that he published,” I said, also giving a three-sentence bio on Gardner to R, who had not been in that particular circle. “Any other guesses about what we’ll be doing?” No one had a clue, so I told them. “We’re going to go back, for J and R, 2 weeks, and for M and L, a few years, to revisit function machines.”
“Does 2 weeks ago really count as history?” asked A, skeptically. No one was really sure.
A had never seen a function machine, so R and M explained what one is, then the kids designed one: a hockey skate with bunny ears and a hand (see photo gallery!). The kids started putting in numbers. In went 88, out came 178; in 1 out 4; in 11 out 24; in 16 out 34. I expected these four ordered pairs to generate some conjectures about the rule, but I was wrong. The next “in” number suggested was infinity. Much to the kids’ confusion, the “out” number was infinity. (An aside here about infinity might have clarified things.) Then in went 7, out came 16 – a pair of better-behaved numbers. But then in went 0 and out came 2. (Another badly-behaved number, disturbing or even dislodging any conjectures that may have been brewing in people’s minds.) After that, the kids suggested 5 well-behaved “in” numbers (3, 2, 22, 10, 4), conjectures began to flow, and the group solved it.
“Are we going to do the unicorn problem soon?” asked M excitedly.
DARK BRIDGE PROBLEM
“There is a unicorn dying at the end of a bridge. He has 17 minutes left to live, unless 4 people can join hands around him and recite a magical spell. There are 4 people on the other side of the bridge, but it is very dark, they have one flashlight, and the bridge can only hold 2 people at a time. Ginny can cross in 1 minute, Ron in 2, Fred in 5, and Percy in 10. Can they save the unicorn?”
Immediately the kids got to work. Well, maybe not immediately immediately. First they had to debate who these people might be. Obviously, said some, they’re characters from Harry Potter. No, said another, Percy is Percy Jackson (of children’s literature fame). Then why would there be a Ginny if it’s that Percy? They could be a combination of characters from multiple books, posited someone. Definitely not, said another. I interjected here to say that this problem is an old problem from math folklore.
“It could have been created before Harry Potter was written then,” said A to the others.
“Maybe it’s about a different group of people with those same names,” I suggested.
“Yeah, but what are the chances of that?” asked the kids.
“I think there’s definitely a chance. I think those are pretty common names in England. We could do a math circle just on that question alone some day,” I said, making a mental note to possibly return to this question some other time.
“Did you just make up those names and slap them on this problem?” he asked me. Yep, I freely admitted. But still, the literary/dramatic/metaphoric connection held as kids worked the problem.
They posited and tried numerous conjectures. Every conjecture failed. Determination was faltering. “Cmon, guys,” said M encouragingly, “we solved this problem when we were like six!” I reminded the kids who had been in that math circle years ago that it took a larger collective of kids six weeks to solve this. Also, when they were younger, it took four weeks for the group to understand that if a slow and fast person walk together, the slow person determines the rate for the couple. This time, it took about 3 minutes for the students to come to that conclusion. This conversation gave them a bit more hope.
Another thing that happened much sooner this time around is the dropping of the assumption that the fastest person has to go first, and that the fastest person has to go multiple times. (Interesting, since my meta-assumption has always been that the younger you are, the looser your attachment to assumptions. Hmmm…. )
Soon, this group was attacking this problem like a combinatorics problem – trying to exhaust every possibility of combinations/orders (see photos). But they didn’t have a systematic way to list all the possible arrangements, so they got tired. They decided to do what mathematicians do and take a break.
“Here’s another one,” I said.
FOLKLORE – WOLF/GOAT/CABBAGE
Next I presented this one: A man has to cross a stream in a boat that can hold himself and only one other object. He needs to transport a wolf (or a lion, or a jackal), a goat (or a sheep), and a cabbage (or bundle of hay, or pumpkin). He must be sure that when he is out in the boat the wolf does not eat the goat and the goat does not eat the cabbage.
“Ooh, are we also going to do that one about the adults and the children of different weights, or that one about…” asked A, clearly familiar with this genre of math problem. His question led into a discussion of the folkloric nature of mathematics. Most kids don’t know about math’s cultural aspects, or if they do, they buy into stereotypes of apocryphal “math people.”
Then the kids spent some time solving this classic problem from math folklore.
HAILSTONE FUNCTION/COLLATZ CONJECTURE
In week 2 I presented a function machine rule called (by some) the Hailstone Function. Every conjecture about the rule that someone posited led to another conjecture by someone else. The conjectures were building upon each other, getting closer and closer to the solution. Finally, one student pointed out that even numbers were all halved. A few conjectures later, M realized that the odd numbers were being tripled then one added.
“M solved it!” said someone. (More on that comment later…)
Then we took it a step further. We used each out number as the next in number (a special kind of function called a sequence). The kids noticed quickly that each sequence seemed to end with 4 then 2 then 1. Aha!...
“Do they always end in 4 2 1?”
“This is an unanswered question in mathematics,” I said. “This question – whether the Hailstone Sequence always ends in 4 2 1 is called the Collatz Conjecture. No one knows. People are trying to figure it out. So far, every number tried has ended this way.” Eyes opened wide, as they always do when encountering open yet comprehensible math questions. Our group tried a bunch more numbers, but got weary from the arithmetic involved. The mental math involved was a bit beyond the group, and the time/labor involved in the arithmetic operations were overwhelming. We talked about using computers for math – for calculations, and whether they could be used for proof.
Then we ended session 2 acting out more scenarios in the Dark Bridge Problem. (When you look at the photos, those kids lying down are playing the dying unicorn.) The kids acted it out in session 3 too, but in that session, I gave them total control over casting (which they had been begging for). We did not, therefore, get to very much problem solving on this particular question. Next time, I’ll see what happens when I ask them use paper and pencils to attack the problem. Most of session 3 was devoted to discussions about math philosophy and pedagogy anyway.
ASSUMPTIONS/CONJECTURES/MYTHS/THE ELEPHANT IN THE ROOM
When the kids came back for week 3, there were these 4 phrases on the boards:
- The Elephant in the Room
We had a lively discussion about what each of these things means. I told them that I my conjecture is that the elephant in our room is the assumption (common at this age) that everyone in the room is better at math than me. Most agreed, although some of the kids wanted me to qualify my statement with terms like “sometimes” or “usually”
Then I told them my conjectures about myths in mathematics education:
- Myth: Math = competition
- Myth: Math = arithmetic/computation/calculation only
Our discussion got everyone to eventually realize that last time, it wasn’t really true that “M solved it.” Really everyone did, but no one realized that without me explicitly explaining it. No one had realized how collaborative their work had been.
The second myth was a bit easier to bust since these kids, all math circle veterans, understood with not much discussion that what makes math math is the idea of proof. We got into this when I asked
“What made our exploration of the Hailstorm Sequence/Collatz Conjecture math?” The kids said that it wasn’t really about performing the arithmetic operations. (In their hearts, though, I know they hold some insecurities about this.) Their big question was this:
“What’s the difference between a mathematician and Siri?” You can imagine what a fun discussion that was. What things in math can Siri not do?
MISSIONARIES AND CANNIBALS
We spent our last few minutes discussing this problem:
In the missionaries and cannibals problem, three missionaries and three cannibals must cross a river using a boat which can carry at most two people, under the constraint that, for both banks, if there are missionaries present on the bank, they cannot be outnumbered by cannibals (if they were, the cannibals would eat the missionaries). The boat cannot cross the river by itself with no people on board. And, in some variations, one of the cannibals has only one arm and cannot row.
A few conjectures were produced, then rejected. I feel like there might be something culturally insensitive about this problem, but couldn’t wrap my mind around what to say about it, so I said nothing. (Advice?)