(April 5 -19, 2018) In the past, I’ve often made the mistake of getting out “manipulatives”* to help students discover a certain mathematical concept only to find that the students wanted to engage in open-ended exploration. They weren’t interested in my agenda. So, for this course, I put the Polydrons on the table with no guidelines for two weeks. The students just played with them as we worked on other mathematical questions.
Finally, in week three, I said “This week we are only getting out the Polydrons that are regular polygons. Can you sort them so we can put away the irregular Polydrons?” The students quickly learned what regular polygons are. Then I said, “Let’s make some Platonic solids!” What are they, the students wondered. “There are only two rules: they are constructed from regular polygons and all vertices are the same.” The students spent some time asking questions and understanding these rules, playing with the Polydrons with this goal in mind. “Now we can get to the question,” I announced.
“We haven’t even gotten to the question yet?!” exclaimed the students.
“Yep! The question is this: how many different Platonic solids are there?” After some time, the students had discovered three of them (actually four, but they don’t know yet that they discovered a fourth).
THE HANDSHAKE PROBLEM
Since the students in this course ranged far in age (10-14) and didn’t all know each other, in week 1 I gave a classic math problem that easily generates interaction among students:
“If everyone in a room shakes hands with everyone else, how many handshakes will there be?”
The students reasoned that we have 8 people, so it’s 8 times 8. Wait a minute, do we shake our own hand? No. So we each shake 7 hands, 8 times 7=56. So 56 shakes. Done. Confident they had solved it after 3 minutes. “Are you sure?”
“We have to be sure! Let’s try it out!” declared F. They realized soon that shakes were being double counted. 56 divided by 2=28. So 28 shakes. Done, confident they had solved it after 3 more minutes. I insisted they finish gathering evidence (by completing their experiment). They did get 28 after coming up with way to keep track. Confident they had solved it. (F and Z asked clarifying questions – i.e. what if you do two-handed handshakes? etc)
The following week I asked them to generalize their process, which they did. They even they came up with an algebraic formula for it (with a bit of help from me). “How can you be sure that because this works for 8 people, it would work for all numbers of people?” This introduced doubt big time. That’s great news as far as I’m concerned. I am coaching them to doubt conclusions arrived at through induction. I want to move into proof so that they know beyond a doubt that their formula will work for any number of people. In the spirit of true mathematicians, they're asking does is work for multi-digit numbers of people, etc etc etc.
I also challenged the students to explain how this problem relates to the Platonic Solids. No conjectures yet.
FOILED BY MY EXPECTATIONS AGAIN
The handshake problem did turn out to be a great icebreaker. Actually, the students came up with an icebreaker: Go around the table, say your name, one thing you like to do, and name your favorite Youtuber. (Turns out that two of the students “knew” each other from playing Minecraft online, and loved meeting in person.) “Funny you should mention your favorite Youtuber,” I said, since mine is Vi Hart and I brought in one of her videos to show you today. I showed them one of my favorites: Binary Trees.
My mind was aglow with how the students were going to watch this video, become enraptured by the Sierpinski triangles, and demand time to doodle these on their own. Ha! That didn’t happen at all. I was operating under the false assumption that because something happened once before (7 years ago in a math circle) that it would happen again. No one was interested. Even when I told them that you can make a 3D Sierpinski triangle (a Platonic solid!) out of recycled business cards. “Sounds like a lot of work,” several of them muttered. Foiled by my expectations again. Will I ever learn? OTOH many years ago I tried the Platonic solids with Polydrons activity in a course and those students had no interest in that. These kids now are very interested. It’s actually quite wonderful that the same activities turn out differently each time when you let them.
THE BEAUTY/GLORY OF FUNCTION MACHINES
On the first day, S (an experienced math circle participant) asked, “Are we going to do function machines in this course?” I hadn’t planned on it but decided to throw it into the mix as a crowd pleaser. The math that has come out of this so far has been unexpected and delightful.
For those of you unfamiliar with function machines, you play by saying a number that goes in to the machine and the person operating the machine tells you what comes out. Your job is to guess the rule from a series of ordered pairs (in and out numbers).
When J presented a machine, her rule brought up a discussion of negative numbers once it became apparent that when the opposite (negative) of a number went in, the same number came out as it did from the original. So what kind of function would generate the same output from its negative? Turns out that squaring a number does this. What is squaring? What happens when you multiply two negatives? And many more questions… The math behind this that I didn’t mention (and wish I had) is that her function, (x^2 + 50)/2 is an even function. In mathematical symbols,
f(x) = -f(x).
I do allow the presenting student to use a calculator. This saves time, keeps everyone interested, and opens up its own Pandora’s Box. When S presented a machine, the out numbers didn’t seem to make sense to him. Everyone waited patiently as he input the numbers into different calculators and got different results. (One of the many things I love about this group is their patience.) I mentioned that not all calculators follow the order of operations. This led to a discussion about what the order of operations is. Z broke the ice for this discussion with her comment “The order of operations can be confusing.” We also talked a bit about the necessity of knowing how to use parentheses on calculators.
“How would you get the in number for these two functions from the out number? I asked. This led to a discussion about inverse functions. I gave two analogies students are often taught for these – (1) undressing and (2) peeling corn. The students seemed to find the undressing analogy (you take of your shoes before you take off your socks even though you put them on in the opposite order) more accessible. F pointed out a flaw in the corn analogy, that there are obvious things smaller and underneath the kernels with corn.
This exploration of function machines looks likes it’s going to converge with the Platonic solids, as both can be looked at through the lens of symmetry. More on that next time.
Also, next time, I’ll tell you more about some of the other things we’ve been talking about – some logic questions, a paper-folding problem, and more.
*Manipulatives are physical objects used as teaching tools. In mathematics, they offer concrete experiences with abstract concepts.