I’ve been thinking a lot about teaching math lately, as I prepare for my entrance into grad school. I’ve tutored a lot of calculus and a little bit of lower-level math, and in a year I’ll be teaching classes of my own. I’ve worked with a number of students, and had a few friends, who fall into the (all too common) afraid-of-math category, and on a broader level I worry about the state of mathematics education. So many students leave high school afraid of math, unable to do basic algebra or even simple addition without reaching for a calculator, and I feel that’s wrong. With the exception of those few people with severe learning disabilities of one kind or another, it seems to me that everyone should be able to learn math, at least as well as they learn reading and writing. I’d like to think everyone can get a kick out of discovering patterns, and solving tricky puzzles in an elegant way. Maybe that’s too idealistic, I don’t know. I do know that more experienced minds than me have studied these problems (John Allen Paulos comes to mind), but I want to do what I can to combat them as well. Part of this means being the best teacher I can, and helping instill a sense that math is interesting, and doable.
To that end I’d like to reminisce on some exceptional experiences, good and bad, that I’ve had learning and teaching math. I’ve been lucky enough to have some very good teachers throughout my education, and my parents took plenty of opportunities to introduce interesting concepts. I’ve also witnessed some singularly terrible math teachers. In the next few posts I’ll describe some of each.
Logarithms on a Pizza Box:
One of my favorite stories to tell is how my parents taught me logarithms on a pizza box. One night when I was in maybe sixth grade, we were eating pizza for dinner and I decided to share an interesting fact I had picked up somewhere: “Did you know that as the pizza cools, the temperature drops in half every ten minutes?” Leaving aside the fact that this is an oversimplification (it’s the difference between the temperature of the pizza and the air that decays in this way, and I’ve no idea if the 10 minute figure is correct), this little tidbit sparked a dinner-long discussion about logarithms and exponential decay. My parents got out a marker and drew axes for time and temperature on the pizza box, then plotted the temperature at ten-minute intervals. When the points were connected I could see that they didn’t form a straight line, but rather a curve that got closer and closer to 0 and never quite touched.
My parents then explained that when graphed on a special kind of paper, the curve would become a straight line. My dad got out a piece of log paper, with equally-spaced vertical lines and strangely spaced horizontal lines, where every group of ten was spread out at the bottom, but bunched up at the top of the interval. My parents explained that on a normal axis, each line represented one unit more than the one below. But on a logarithmic scale, each line (actually, each group of ten lines) represented ten times as much as the previous one. When we plotted our pizza temperature vs. time graph on the log paper, sure enough, we got a straight line! On the other hand, a graph that would have been a line on the original axes now became a curve that looked very similar to the original temperature graph.
My dad also pulled out some log-log paper, where both axes have a logarithmic scale. (I must be the only person of my generation who’s ever seen real log paper–who needs it when we can plot things so easily on computers these days?) I didn’t understand why you would need logs on both axes–it wouldn’t straighten out the temperature graph, and lines on normal axes would still be straight on these (my parents weren’t about to try to explain exponents and properties of logarithms in a mathematically rigorous way to me at age 11!) Nevertheless, I thought it was interesting, and the odd-looking distribution of the lines had a kind of nice flow to it.
Now I don’t expect every parent to teach their kids about advanced math concepts at the dinner table. Most adults have never learned even this much math, or learned it but forgot it as soon as the final exam was over. I was extremely lucky to have parents with both the capability and the interest in math to teach me these kinds of things just for fun. (It is fun–how wonderfully geeky is it to draw mathematical diagrams on a pizza box? I wish I’d saved it.) And I don’t expect every kid to be capable of understanding this while in elementary school; I was pretty exceptional. But mathematics doesn’t have to be hard, and it doesn’t have to be boring–I firmly believe every high school student is capable of understanding the concepts my parents taught me that night. More than that, I believe every student is capable of having fun playing with ideas, discovering patterns and trying to figure out why they work.
Anyway, this remains one of my favorite experiences with learning mathematics. What about you? What are your some of your best and worst stories about learning (or teaching) math?