Kids inventing their own algebra.

How do you represent an infinite sequence or series in mathematical symbols? Our group attempted to answer this question in preparation for viewing Vi Hart’s video “Infinty Elephants.” So again we began with function machines. Once the students deduced three relationships between input and output numbers, I told them that my hand was too tired to write out the rules in words. In response, they collaboratively constructed equations using the words “in” and “out” for the variables:

out =in²

out=2ⁱⁿ

out=(1/2)ⁱⁿ

These came easily once we reviewed the definition of an exponent. For the next function, the students quickly supplied the equation out=in times 2.

“But can we get rid of that word times?” I pleaded.

Kids first suggested in x 2, and then remembered that we can assume multiplication when writing equations, and simply write in2. “There’s a convention in math, though, about the order …” I began. Immediately, L raised his hand and suggested 2in, instead of in2. “So, in math, the convention is to write the thing that does not vary to the left of the thing that does vary. By the way, in math, there’s a name for the thing that varies.”

“A variable!” chimed in the few kids who are already doing some algebra.

“So,” queried M, “why don’t you just write i instead of in?” I did.

“But that’s the square root of negative one,” replied L. Most faces looked seriously perplexed, as they had never heard of such thing. (This will be covered in a future math circle, but not this year.) It was not the time to explain imaginary numbers and therefore lose the momentum of this moment, as the kids were now inventing their own algebra.

“In math, i does stand for something that does not vary,” I explained. “But, there is a convention for using one particular letter to mean the number that goes in to a function machine.” I was surprised by all the guesses at this point: is it v (for “variable,” suggested A)? is it a (“no, that stands for a known quantity,” says R)? is it g (says one of the kids whose name starts with that letter)? I had to give a hint to elicit x. I’ve used x habitually for so many decades that it sometimes surprises me that we aren’t born automatically knowing this. It took more guesses to come up with the letter that represents the output of a function: y. Then the kids rewrote the above equations using variables:

y=x²

y=2^x

y=(1/2)^x

y=2x

“What kind of math does it feel like you’re doing?” I asked. When they responded with algebra, I let them know that they had been doing algebra all along, just not symbolizing it with single letters. It’s important that the introduction of the use of variables be accompanied by the user’s need for variables. In this case, my hand was “tired” from writing all those words, and we have a tiny blackboard and need to optimally utilize the board space. For kids who are already using algebra, it’s important to occasionally revisit the need for (and benefits of) variables to avoid falling into the trap of the cookbook approach to math.

Speaking of kids inventing their own algebra, the next functions were sequences (Fibonacci, for one) and series. By the time the kids had written a few of these as equations, they had collaboratively come up with their own symbols to represent a prior term in a sequence (subscript “b”) and the term before that (subscript “e”). The kids discarded their first attempts for the “2 before” subscript when they realized (on their own) that “b2,” “2b,” and “bb” could all be misconstrued as multiplication instructions. The collaboration here was impressive considering that this Math Circle has 13 participants, not a number that facilitates easy conversation. In a group this size, I have to constantly check myself against the didactic instinct, so that the kids are truly (to loosely quote Maria Droujkova’s slogan) making math their own.

That pesky didactic instinct almost led me to introduce the math conventions of using n, n-1, n-2, and sigma to symbolize sequences and series, but somehow I resisted. This week, since this topic was so new to most of the kids, the main point was to grasp the idea of using symbols. I may show them these conventions once it’s obvious that the concepts are solid in most minds, or when someone asks, whichever comes first.

Anyway, we were now ready for the video. No one wanted to see it twice this week because everyone wanted to draw immediately after viewing it. So drawing commenced (infinity elephants and Appolonian Gaskets), as did quiet conversations with neighbors about the math:

“Make them sort of ovalish.”

“This is confusing.”

“These are fwomps – genetically engineered elephants – I can’t draw elephants.”

“One fish, two fish, red fish, blue fish.”

At that point I (foolishly) pounced on a “teachable moment”: I shared an anecdote about a mathematician’s likening the Appolonian Gasket to Dr. Suess’s Cat in the Hat. Conversation stopped. It was awkward for a moment. I quickly stopped talking and moved around the table and asked quiet individual questions to restart child-led group conversation: “Do you think that’s easier or harder to do with a compass?” “Do you think you could see the repetition more clearly if you colored your fractal?”

At this point, S picked up the conversational ball with “What if I make circles within a circle?” and conversation was restored.

N stood up and exclaimed, “I just got it! If I keep filling in circles, it goes on for infinity.” A few minutes later she showed how she “did triangles in a circle” instead of the Appolonian circles within a triangle. People experimented with all kinds of fractals and visual representations of infinite series until time was up.

Our Math Circle had reached a crossroads at that point. Should we continue with a new Vi Hart video each week? The videos are exciting, and filled with curiosity-provoking rich mathematical content. On the other hand, we could spend a lot more time on the material from two videos we have seen so far. In order for the math content to sink in, we will move in direction of contemplative mathematics (i.e. slow down). I was curious, however, whether the kids felt the same way, so I asked. They too generally feel the need to explore more deeply what we’re doing. So this coming week, we will be doing more drawing, more functions, more fractals, and some mathematical history. See you then!

— Rodi