Favorite Comments of '12: Michael Paul Goldenberg

On Letter from Huck Finn: Thinking about Inverting, Multiplying, Understanding, and Hanging

Michael Paul Goldenberg said...

You have decided that practice MUST precede understanding. That's certainly your right as a learner, but not as a teacher. You're imposing your prejudices and learning style on everyone you try to instruct. But you will (predictably) fail with many of those students. A few months down the road, many of those kids you're so sure "got it" will not recall how to divide fractions. Some will swear no one ever taught it to them. Unless, of course, the derivation, the WHY, has sunk in and made sense to them of nonsense. In that case, they'll very GLADLY return to that derivation and figure out the 'short way' again for themselves. If you've given them nothing to ground the concept, however, they'll be hopelessly lost. And in the inner-city schools where I've worked for two decades, that's been what mathematics has been for most kids most of the time: meaningless rules to be memorized and repeated without sense. And it fails them, time and time again. Do two negatives make a negative (addition) or maybe a positive, maybe a negative (subtraction) or a positive (multiplication and division)? Who knows? Certainly not most of these kids, because there's no conceptual ground upon which they can stand.

You really need to rethink your biases, though I know you won't (you've made that abundantly clear in post after post, anonymous or attributed). You're SURE you're right. Maybe that's what it means to be left-brained, in which case I am thankful that I am whole-brained. I have successfully completed graduate degrees in literature,educational psychology, and mathematics education. And I continue to learn in as many areas as I can, in whatever ways I can, without needing to insist that there's no place for rote, but merely that it isn't the best option for many sorts of learning. Would that you could see past your nose.

Katharine Beals said...

"rote is a waste of our time (speaking as a learner) when it comes to things that can be UNDERSTOOD."

Wasting time is very much relevant to the tradeoff between rote processes and conscious understanding (and not just when it comes to muscle training).

The acoustics of speech can be understood, involving lots of interesting patterns. Why not consciously analyze the speech signal we receive when people speak to us rather than doing it by rote?

Why not conduct etymological deductions rather than rote memory lookups when listening or reading?

Why not do conscious applications of the parabolic trajectory law every time we try to catch a ball?

Why not translate all arithmetic problems into their set theoretic underpinnings?

Alternatively, one could handle these things as the Quadratic Formula used to be handled in traditional math. Have the students derive it a few times on their own (no Reform Math algebra book I've seen has students do this very important activity, preferring formula memorizations and guess & check plug-ins) and then move on.

One might also explore the Strawman-free middle ground between "practice MUST precede understanding" and "understanding MUST precede practice," and make decisions based on feedback from students and student performance. And, doing our best not to impose our "prejudices and learning styles" on "everyone we try to instruct", keeping our minds open to what might be holding them back versus helping them forward.