Sunday, January 25, 2015

The "How" Is Rooted In The "Why"

Here's a great post explaining, with a robot analogy--and who doesn't love robots?!--why it's important in math not just to memorize things, but to understand why the rules you're memorizing make sense:
Every rule – even the craziest, most arbitrary mandate – has a reason rooted in this essential purpose...

And so it is in math class. If you understand slope not as “that list of steps I’m supposed to follow” but as “a rate of change,” things start making more sense. (Why is it the ratio of the coefficients? Because, look what happens when x increases by 1!)

You get to work a lot less, and think a lot more.

Now, conceptual understanding alone isn’t enough, any more than procedures alone are enough. You must connect the two, tracing how the rules emerge from the concepts. Only then can you learn to apply procedures flexibly, and to anticipate exceptions. Only then will you get the pat on the back that every robot craves.


maxutils said...

This is one of the reasons why common core bugs me sooooo much. The premise seems to be on teaching the 'why,' which I believe we both agree is a good thing … but a) before Common Core, any good teacher was already doing this -- slope is a good example, but I would go further … introduce delta x/delta y terminology. It gives the same understanding, and allows for a more smooth transition to calculus. It also eliminates that stupid slope formula, with it's overly complicated way of doing things … and which textbooks can't even agree with the order (since it doesn't matter). Try telling that to your child, who has been told to always use the formula, but can't tell which point should go first; and b) Common Core, at least for math, from what I've seen -- attempts to promote understanding through even more complicated disconnected algorithms. As to the arts? If you're teaching memorization as opposed, primarily to analysis, you really aren't doing your job.

PeggyU said...

It was a very good article. And I have always found that understanding the principles allows you to derive what you need, rather than to lug around a whole bag of formulas. I try to pass that on to the students I work with as well. Still, as with arithmetic, if you understand it and have internalized those formulas it frees your mind and gives you time for other tasks.