(Jan. 25 – March 8, 2018) The five students (plus one occasional visitor) in our math circle spent six weeks doing like mathematicians do – savoring a math problem, learning it in depth before any attempts to solve it. I felt like Andrew Wiles in his decades-long work on Fermat’s Last Theorem.
BEFORE THE COURSE: THINKING ABOUT IT
I didn’t want to spoon feed the math in worksheet form where I tip my hat to what’s cool about the Euler Characteristic. I spent a long time developing an approach that I hoped would allow students to make some deductions but not be led too much. (See references at the end for my inspirations.) My big question was how much was enough leading but not too much?
WEEK 1: PRESENTING THE PROBLEM
The goal was for students to know what the question is. I spent so much time sent on setting up a dramatic narrative because this is a hard problem for 8-year-olds. It’s especially hard because I was hoping that they would come up with the idea that there is a pattern. I did not want to end up telling them that there’s a pattern.
Here’s the setup:
I need people to play some roles – a farmer, a horse, a carpenter, a secretary, and an accountant. The farmer wants to build pastures for her horse so that there’s a different crop in each for the horse to graze on. Horse, what do you want to eat? Farmer, can you draw some dots to show where you want the fenceposts to be? Carpenter, can you connect the dots with lines to indicate the fences? The rules are that the fences can’t cross and every post has to be connected to every possible other post. Horse, can you count the pastures? (Fun debate here about whether outside the fences counts as a pasture/region.) Secretary, can you keep track on the board everything that we are counting? Farmer, how many dots did you draw? Carpenter, how many fences did you install? Accountant, what do you get when you add the number of dots to the number of regions?
The carpenter’s bid depends up on the numbers of dots, lines, and regions. The farmer will hire the carpenter to do the work if the sum of dots and regions is equal to the number of lines. The farmer and horse really want this thing built so the horse can eat that pizza! Will this thing get built?
Not everyone understood the math. They did get the general gist that the mathematical requirements were not met to get the fence built. “Let’s change it up!” They tried, but the counting got really tedious and confusing.
WEEK 2 – UNDERSTANDING THE PROBLEM CONCEPTUALLY
Since I didn’t think the students really understood the problem last week (as mathematicians often don’t at first), we delved into some background. I asked the students how electricians, tile-installers, painters, and carpenters decide how much to charge for a job (“bidding”). What happens if the bid is too high? Too low? How much would you charge to paint the room we’re sitting in right now? The purpose of this discussion was to demonstrate the ideas of formulas/algorithms/rules for bidding on jobs, since our carpenter is putting in a bid to build the fence.
Also, since the diagram the students constructed was pretty complex, I handed out paper and asked them each to draw their own sample pasture, with “any number of dots.” I hoped that if each student had their own example that they created themselves, that they’d understand the problem better. I also hoped that each would create a less-complex example and therefore would have a better shot at coming up with an answer.
Turns out most of the students had a hard time drawing it and sticking to the rules (no lines crossing, connect everywhere possible). Kids did 19, 20, 25 – covered their pages with dots. I thought to myself that I should repeat this in week 3 with an assistant helping the kids draw. I also thought to myself that I could make a handout with our diagram from the whiteboard and dashed lines so that the students could change it. (Alas, I never did either of these things – the assistant or the handout.)
So no progress on the problem this week. (Just like what happens to mathematicians!)
WEEK 3 – STARTING TO LOOK FOR PATTERNS
At this point I started to worry about time. I was starting to get nervous won’t have time to connect it to course topic invariants. Unlike Andrew Wiles, we didn’t have a lifetime to make progress on the problem. Only 3 more sessions after today. So I led more than I had originally wanted to. Used the strategy of starting small and gradually building. Kids wanted to jump ahead to larger numbers but I reigned them in a bit. I neglected to tell them that we were using the strategy of starting small – a lost teachable moment. Oh well, can’t get them all, I had to remind myself later when I was beating myself up mentally a bit about this.
Week 4 – ASKING QUESTIONS
I wasn’t sure whether all kids are following the record keeping on the board; we needed to make it more clear. I insisted to trying to do this investigation systematically – increasing by 1 the number of points in each trial – but they still wanted to skip 6. Even when they noticed the gap, they didn’t suggest to try 6. I insisted only because we had only 2 sessions left. (Had we more time, I would’ve just let them skip 6.)
C asked what if you position the dots a different way?
S asked why are all the results odd except when there are 7 dots?
A asked why are they always going up by 2 except…
Someone asked can we do curved lines?
Someone else asked can we ever get a different result?
The students were excited, curious, asking many questions about the problem. Moreover, they were no longer talking about it in context of farmer/carpenter problem. They were saying “dots/lines/regions” not “posts/fences/pastures.” These 8-year-olds had transcended the material world to the abstract! (After class, I asked myself, “Why are we still calling points dots?!” I set an intention to shift terminology to the more accurate term points, which are different from dots. I never explained that difference but did make the shift.)
WEEK 5 – DOES THE PATTERN HOLD FOR ALL CASES?
I brought out the students’ original diagram from week 1, the one with 13 points. The students knew exactly how to assess it now. I asked for a conjecture ahead of time: Do you think you’ll end up with points + regions exceeding the number of lines by 2? Most did. Then they counted and discovered that they still got the same result. So is this an invariant? The consensus was… maybe. Most students said we’d need to try more cases, and C argued vigorously for the need to try different arrangements of dots for the completed trials. One student said we’d need to have a proof. (Most students didn’t know what proof meant, so we didn’t get into because of time. Had we more time, we certainly would have.) So some students worked on trying examples with larger numbers of points while C attacked rearranging the points for several cases (4 points, 5 points, and 6 points).
With about 15 minutes left, we shifted gears to a new problem, “Cross-Country Race,” which I’ll explain in a different report – click here for that one.
WEEK 6 – STUDENT OWNERSHIP
I had another activity for today that we all started with (in honor of the approaching Pi Day). During the Pi Day activity, students were anxious about returning to our prior problems. Of the only four students in attendance that day, two were desperate (yes, desperate, I mean it!) to get back to what we were calling at that point “The Horse and Carpenter Problem.” The other two were tired of that problem and really wanted to explore the new problem that we started last week. We didn’t have time for both. It seemed that no matter what we chose, half the class would be disappointed.
“You don’t need me for either of those problems. You own them now. How about you two tackle one and you other two tackle the other?”
They looked at me in seeming shock. “We can’t do them without you!” someone exclaimed.
“Yes, you can. You own these problems!” I handed out markers and that was that. They really didn’t need me. I answered a question here and there, checked their progress when they wanted to show me, and that ended our course.
I promised to publish these pictures so that the students can continue to work on the problems at home. Like real mathematicians often find (and we discussed), six sessions just isn’t enough time to tackle really interesting mathematical problems.
Joel David Hamkins. Math for Eight-year-olds: Graph Theory for Kids http://jdh.hamkins.org/math-for-eight-year-olds/
Harvey Mudd College Math Department. Mudd Math Fun Facts: Euler Characteristic https://www.math.hmc.edu/funfacts/ffiles/10001.4-7.shtml
Simon Singh. Fermat’s Enigma: The Epic Quest to Solve the World’s Greatest Mathematical Problem (book)
Owlcation. Some Practical Applications of Mathematics in Everyday Life https://owlcation.com/stem/Some-Practical-Applications-of-Mathematics-in-Our-Everyday-Life