Today's class began with an activity counting manipulatives. Yellow and orange blocks were used to represent a fictional currency that a certain "Matt" had accumulated while at summer camp. The exercise was for us to work with our elbow partner to count how many blocks we had been given in a plastic baggie. We quickly divided the allotment into groups based on size, or denominations, and then went about counting the number in each group. It was fairly straightforward counting our sets of ones, tens, and hundreds. We had 272 blocks total.
After tallying up our sets, I thought we might want to further divide them into colours. We had been given no information about the meaning, or lack thereof, of colour to the exercise, but we thought it might become important. Did different colours have different values? Did they come from different activities? Were some blocks simply left out in the sun too long and they all should have been orange? The role colour played was never discussed, but I like to image that one paranoid camper kept them closely guarded in their pockets through rock climbing, kayaking, and even nervous moments around the campfire. But I digress.
The consensus in our class was that everyone had taken a similar approach to the counting problem, counting largest to smallest. When asked to add two groups together, students took one of two approaches: adding the new blocks to the initial group from largest to smallest, or by regrouping all of the blocks and counting all over again. Here we saw a clear demonstration of how two students might approach a problem differently. We were then asked to add two numbers the way we have always been taught, have always practiced, and have always thought of doing so. Ones, tens, hundreds.... Smallest to largest? A clear contradiction from the method we had all used in our camp economics class.
This new method made sense from an approach using manipulatives--I would never count up all of the money in my piggy bank by starting with the pennies. Pennies don't even exist anymore! So why is this different from what we all know and practice? Because we have been taught an arithmetic tool for evaluating such expressions. But it is no way for students to learn how to add. I know, use, and love this tool, but I certainly don't add smallest to largest place values when doing mental math!
So what do we do with these tools? We ignore them. They're great when you don't have reams and reams of paper for writing longhand, but the purpose here is to understand the principles of what we are doing. Any other time we can use a calculator to save paper.
Reading parts of the curriculum, one thing in particular piqued my interest: that all children are capable of doing math. Just as children read and write, math is its own language and follows a set of rules that are, unlike many constructs of the English language, quite logical. We then talked about brain plasticity, a concept that I am quite familiar with from courses in psychology. It was an idea that I had never considered before. That one's ability to do math, much like speaking a foreign language or playing a musical instrument, requires practice to maintain and improve upon. Unfortunately, most people don't exercise those synaptic pathways enough, and worse yet, are often discouraged to. Math is "geeky" and people "hate" math. But math plays a role in everything we do. What is time? What is money? What is a kilogram? These are questions that math is important for, and the kinds of questions that I absolutely love to think about.
The "Handshake Problem" was then posed. If you have n number of people in a room, how many handshakes must occur for each person to shake hands with every other individual only once? I quickly wrote on a whiteboard that it was equal to (n-1)!, and started drawing tree diagrams. But when we began talking as a class I realized I had been impulsive and mistaken. My grade 12 data management teacher would have cringed! It is actually 5C2. Woops. But we all learn from our mistakes, right?