Math Blind Spots

As a Math tutor, you'd expect that I'd be pretty good at Math. And, in fact, in primary and secondary school, I was. I routinely sported Math marks in the 90+% stratosphere, sometimes finishing first in my grade and pretty much always in the Top 3.

When I got to university, however, I discovered Math of a tougher sort. My first year Calculus mark at the University of Waterloo, for example, was a very humbling 58%. (In Algebra there, at least I managed something like a 75%.) That Calculus result, more than anything before or since, convinced me that I was no Math genius. Compared to the general population of Canada, I'm quite good at Math... but once you narrow the sample set down to just those who can actually handle the post-secondary version of the subject, I'm probably average, at best.

With the virtue of hindsight all these years later, I can see where some of my struggles hailed from. I didn't study like I should have; I didn't form alliances with fellow students who might have been strong in the areas in which I was weak; and I wasn't humble enough. At 19 years of age, I entered university convinced that I was a top Math student (thanks to ridiculous Grade 13 marks) and it took me most of that first year to work that undeserved hubris out of my system. I like to think that, were I able to go back in time and take that Calculus course over again, I'd do at least somewhat better today. But I may once again be fooling myself.

Regardless, I've also come to another revelation on this subject: I, like virtually all of my tutoring students, have the odd "blind spot" when it comes to Math. Some kids just don't "get" fractions, for example. And so the operations of adding, subtracting, multiplying, dividing, and reducing or reformatting fractions typically involve the student guessing at strategies until they get the right answer. Others didn't learn how negative numbers work when the topic was introduced in or around Grade 7, and so their Algebra in the later grades always falls apart if they can't manipulate the terms sufficiently to eliminate any negatives.

In reading up on one of the Grade 12 Math courses, I realized that logarithms were and are one of my blind spots. Unlike most high school Mathematical concepts, log functions don't come to me naturally. In fact, when faced with a question involving log (or ln when dealing with base e) I have to stop and remind myself what a logarithmic function really is. Fortunately, I'm wired to react that way in such a situation, because I can usually then carry on with the problem once I've re-established my frame of reference. (For anyone who cares, the log of X base B is that number Y such that B raised to the power of Y is equal to X. Got that? In other words, log of 100 base 10 is 2, because 10 raised to the power of 2 is 100.)

For many students, though, they just completely hit a wall when they're not sure how something works. I've tried to teach a few of them mitigation strategies, like the following:

"Suppose you don't remember what to do when multiplying with exponents. You know there's some rule about what to do with the exponents if the bases are the same, but you can't remember if you're supposed to add them, multiply them, or what. So when you get to that point, do a simple example, where you can actually work out the numbers to see what the right answer is. For example, try 2 squared times 2 cubed. You know that 2 squared is 4, and that 2 cubed is 8, and you know that 4 x 8 is 32. So now all you have to do is figure out what power of 2 equals 32. Once you see that it's 2 to the power of 5, then you'll know to add the exponents when multiplying the bases."

The kids who have the potential to get good at Math usually embrace that kind of approach... eventually. Getting them there, though, can be half the battle. Which is one of the reasons I so liked that quote from the Freakonomics site yesterday: it speaks to the value of being willing to work through a tough problem, instead of just giving up on it when the answer isn't immediately obvious.