More On Framing
But, to my mind, mathematics education serves to change students' minds, not pander to their ignorance or even to their common sense in order to justify its own existence. In other words, mathematics learning should be seen as an imposition of alien ideas that serve as powerful tools for understanding the world, not, instead, as an extension and/or validation of informal, S1 thinking.
The broader, related argument—that education is a cultural product, not a therapy--is likely more familiar to readers. Ravitch said it this way:
There are certain fundamental ideas, events, and principles that educated people know; they don't know them by osmosis. They know them because they have been educated. They are not educated by happenstance but because adults have designed a curriculum to teach them "the best that has been thought and said in the world."
Certainly my own experience is that conceptual understanding in mathematics comes only after a considerable amount of procedural practice (much of which therefore is of necessity carried out without understanding). How many of us professional mathematicians aced our high school or college calculus exams but only understood what a derivative is after we had our Ph.D.s and found ourselves teaching the stuff?
In fact, I can't imagine how one could possibly understand what calculus is and how and why it works without first using its rules and methods to solve a lot of problems. Likewise for most other areas of mathematics. In fact, the only parts of mathematics that I find sufficiently close to the physical and social world our brains developed to handle that there are innate meanings we could tap into, are positive integer addition and subtraction for fairly small numbers, and perhaps also some fairly simple cases of division for small positive integers.
Labels: education
Comments:
I think it's true for ALL disciplines - for language and literature, for science, and arts... "Alien ideas" should come first, understanding and applications - second.
I hate it when poetry is taught as "express yourself child, who careas about classics". If you don't know ANY good, clssic, eternal poetry you can call your feces "a poetic expression".
Unfortunately, schools obviously supposed to "entertain and motivate", not make the students practice and memorize.
So Devlin is the guy that I hear from time to time on NPR. And I'll admit that I'm coming into this a little biased. I don't really like the guy but I can't say why.
I think Devlin is going beyond simply saying that learning goes from the informal to the formal. I don't know who is saying otherwise. The constructivists, from what I gather, are not arguing for formality in K12, they are arguing for "conceptual" understanding, which isn't necessarily the same thing.
On a different note, Devlin annoys me because to me he represents yet another mathematician, like Morris Kline, who after having had the benefit of years of formal study of mathematics will then console us, the intellectual "proletariat", that we don't need anything that has earned a mathematician the academic respect he deserves.
And who can argue with "needs" based education? We don't NEED concepts and proofs for basic skills in math. We don't NEED Latin, Greek, or philosophy either in education, but isn't knowing "the whys" part of what it means to be educated? It's not all needs-based, is it?
It is clear what Devlin's priorities are, "progress into the sciences and engineering"
and
we owe it to our students to prepare them well for life in the highly technological world they will live in.
Maybe he says this because this is what most people want to hear. "Not knowing the concepts is okay. Don't worry, only a really smart guy like me with a PhD and teaching experience can REALLY understand the concepts." Why doesn't he just come out and say that pure math isn't an important field?
He doesn't seem serious when he says that later on the concepts will be learned. Lots of people say this sort of thing and then they whisk you off to the next course in the sequence which just inundates you with more algorithms.
I'm going to have to go join Monty Python's Black Knight on the bridge and bite someone's leg off.
Despite my non sequiter ranting that was a good post, Mr. Person.
I think Devlin is going beyond simply saying that learning goes from the informal to the formal. I don't know who is saying otherwise. The constructivists, from what I gather, are not arguing for formality in K12, they are arguing for "conceptual" understanding, which isn't necessarily the same thing.
Just so we're clear, I wrote "from the formal to the informal," not the other way around, so I'm not sure how to interpret the rest of your paragraph.
You're right, though, that conceptual understanding isn't the same thing as formality in mathematics education.
As I mentioned, I think Devlin's argument is narrower but related to the point I try to make about framing. But I would disagree that he completely discards conceptual understanding as valuable:
Of course, there is plenty of evidence to show that mastery of skills without understanding is shallow, brittle, and subject to rapid decay. Understanding mathematical concepts is crucially important to mastering math. The question is: What does it take to achieve the necessary conceptual understanding, and when can it be acquired?
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