Clearly, I’ve been introspective about both the specifics of my teaching and the general properties of higher education. I was feeling like I had found some closure internally, when I sat down last night to read a bit of Seth Godin‘s Linchpin. Two nights ago, I had read a few pages on education, and I was so struck by them that I set my bookmark back intentionally to read them again. I don’t remember ever doing that before, for what it’s worth.
From page 39:
We’ve been taught to be a replaceable cog in a giant machine.
We’ve been taught to consume as a shortcut to happiness.
We’ve been taught not to care about our job or our customers.
And we’ve been taught to fit in.
None of these things helps you get what you deserve.
That’s reasonable, and it echoes what I believe about striving for excellence, about “good enough” simply not being good enough. He goes on to say [p44]:
Teaching people to produce innovative work, off-the-chart insights, and yes, art is time-consuming and unpredictable. Drill and practice and fear, on the other hand, are powerful tools for teaching facts and figures and obedience.
Again, I agree. I read this paragraph over and over again, and it is sobering. He continues [p47]:
What should they teach in school? Only two things:
- Solve interesting problems.
Right-on. However, then he extrapolates:
The idea of [computing a hypoteneuse] by rote, of relentlessly driving the method home, is a total waste of time.
Now, I’m not so sure that Godin and I are talking about the same thing. While it’s true that most people do not need to memorize the Pythagorean theorem in order to succeed in every day life, this ignores one of the most important aspects of a liberal education: that it trains the mind to think deeply about problems. I came across a compelling argument recently (though I cannot remember where) that said that the point of K-12 math is not to teach mathematical skills, but to hone the learner’s ability to mentally manipulate complex and abstract concepts.
Taking the Pythagorean theorem as an example, it may be true that there are diminishing returns in “relentless” repetition, but on the other hand, it takes 10,000 hours to be an expert. If we want students to be able to even conceive of higher mathematics (i.e. to gain any insight beyond novice or advanced beginners), would they be able to do this without practice? I’m not so sure.
I can’t help but think about Math and Computer Science together here. My Calculus professors in college were traditional mathematicians who made us do volumes of exercises, the same exercises the students before had done. I suppose I learned some Calculus at the time, although now I can only remember the very basics of differentiation and integration, and I certainly don’t remember “series” except by name. Honestly, I don’t really remember clearly what they are for.
My Computer Science professors were, by and large, mathematicians also. (It was a Mathematics and Computer Science Department while I was there.) Our assignments were very similar: build up something non-novel that is a little step above what we built before. Thinking about data structures, I remember binary search trees since we used them again and again as examples, and they are conceptually simple. I remember the names of red-black trees and AVL trees, but I could never build one now without either searching the Web or essentially re-inventing it. That is to say, I don’t have any working knowledge of either advanced data structure aside from knowing that they’re balanced trees.
Also, I am OK with this. I brought up a similar point in a curriculum committee meeting once, and some of my colleagues seemed to think I was nuts, that I didn’t mind not remembering “fundamental” Computer Science ideas. However, it hasn’t stopped me from solving interesting problems, as Seth Godin puts it.
The question remaining for the scientifically inclined is: would I still be able to solve interesting problems without the physical changes in my brain brought about by years of mathematics practice? Several studies in CS education point to time-on-task as a major factor in success, suggesting that motivation works only in that it encourages fruitful practice.
I have been enjoying Linchpin, but this discussion of education is not as black-and-white as the author portrays it, not according to recent research on the science of teaching and learning. Adopting Dreyfus’ terms again, it’s one thing for an expert to spend his/her time solving interesting problems, but novices still need rules and scaffolding to move up the skill acquisition ladder.