We played a game in grade 4 called Math Castle, in which alien ships would shoot lasers at your castle if you weren't fast enough at solving increasingly difficult math problems that would float up in the clouds. I wasn't very good at the game. I was slow and didn't know my times tables past ten. I felt embarrassed and discouraged, and my confidence took a blow. It took another in grade seven when variables were introduced. What was x? What can we do with it? Simplification didn't seem all that... simple.
I worked hard at math though. Math was supposed to be my thing and I sort of felt that if I couldn't do math that I shouldn't be in a gifted program. I spent hours studying math at home, worked on algebra on weekends, and those courses I fast-tracked in high school? I spent hours each night slowly working my way ahead. I didn't have a gift, I just worked at it. And when I started university and had a little more freedom, I stopped working so hard. And I sank.
I excelled at math within the traditional confines. I did my worksheets, learned my tools, memorized my terms, and practiced using identities. But my exceptionality wasn't in math per se, it was in problem solving. I think sometimes that this, combined with my dedication, was what made me a strong math student. Math was a puzzle for me to solve, and for the most part, I loved it.
[Note: My IEP also said I had no social skills, so that's where I found all the time to work on math]
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I saw my abstract problem solving come into play today in class. Perhaps I spent too much time watching Dan Meyer talks on youtube this morning when I couldn't sleep, but I found myself looking at our math activity and thinking "there has to be a better way!" This was in regard to the horse racing activity in which we had to add all of the numbers on the cards to see how far our horse had gone. Ten three digit numbers, no calculator, and little patience. My problem solving nature kicked in and assessed the situation. The numbers differed by regular intervals, with no gaps, so one could "add" them by finding the average value and multiplying it out. But I didn't bother calculating the average longhand, I listed the numbers from highest to lowest and paired them off, highest-lowest, next highest-next lowest, etc. This works in from the outside towards the middle of all of the values, and you are left with 147 and 157. The average of these two will then be the average for all of the numbers: 152. No long calculations. And if there had been an odd number of values, the average would have simply been the middle value, or the last one remaining when pairing them off.
I was pretty proud of myself for this lazy approach. Not because it was easier or better, but because I had set out to find another way of approaching it, and it worked. These are the sorts of discoveries we want our students to make. I love challenging myself with little things like this. Borrowing a term from Dan Meyer, you could say that I find these patterns perplexing and want to know more, want to find ways of breaking them down, and want to find models that can be used to represent them.
I want to inspire students to find these connections and learn math from the ground up. Tools aren't always helpful---someone else made them up! I borrowed my neighbour's wheelbarrow and it was rusty and hard to push, and when I wanted to give it back he gave me a hard time because he didn't want to go back out and open his garage. This is our traditional education---or just a really bad metaphor, I'm not sure. But it's someone else's tool that they found useful and are trying to pass off on you, even though sometimes it's more trouble than it was worth.
I used my knowledge of math and love of science to go on and study chemistry. It was in a graduate level course on electrochemical systems that I encountered the Butler-Volmer equation. This thing is gross, but it didn't just come out of nowhere. No one handed this formula down to Butler and Volmer, they studied what perplexed them, and came up with it. There are many other formulas that branch off of this equation, and I would consider them tools because most have certain stipulations in order to use them properly. You have to know how to apply them. And I think I've come full circle now because the tools we spout to students are the same. You have to understand them, and you won't unless you start from the ground up. Let students be perplexed and discover strategies that work for them, even if they're longer. There's nothing wrong with using a little more paper than the person next to you in class.
