the lion’s share of the difficulties isn’t the silliness of the problems, no matter how silly they are – it’s the illogical, poorly defined, and trite mental worlds that most folks live in! _That’s_ our challenge!
One of the most useful pieces of educational research comes from the folks at the University of Wisconsin. Cognitively Guided Instruction (CGI, about which I have written a couple of times) demonstrated quite definitively that young children think quite a bit about mathematics-especially early number concepts, and addition and subtraction. What they don’t do is think like adults. This they need to be taught. But it’s not “procedure first then word problems”. It’s “procedure is an abstraction that follows from students’ informal ideas that come from their interaction with the every day world”.
I tell the kids pure math is strength training for the brain. Just because you never do bicep curls while in a “real sports” situation, it doesn’t mean that doing those curls is irrelevant to your sporting ability.
This is such a tempting analogy but we really don’t have a shred of evidence that it’s so. What I learned from reading Thorndike, the godfather of transfer, is that practice on certain skills makes one better at those certain skills and on nothing else. We have never been able to demonstrate that learning thing A makes you better at thing B. Even so-called “near transfer” is unsubstantiated. What waldo points to is “generalized transfer”-a harder nut to crack and completely unproven to exist.
Now there are lots of other reasons to study pure mathematics. And I have no objection to the idea at all. Mathematics, like poetry, is a beautiful achievement of human intellect for instance. But the cans of peaches problem is not one from pure mathematics. And it’s not from applied mathematics either. It is a word problem. A clever little puzzle, to be sure, but not a useful representation of the subject.
Mark Schwartzkopf writes:
I’m not sure what could be meant by “thinking mathematically”. Translating words into algebraic expressions, without the need to understand the situation fully IS thinking mathematically.
The emphasis was intended to be on thinking, not on mathematically. Recall the Johnson quote was, “students having difficulty can learn basic procedures even if they are unable to reason out a problem.” I was objecting to the author’s satisfaction with getting students to apply procedures without reasoning (thinking).
And I’m not so sure I agree that “translating words into algebraic expressions…IS thinking mathematically.” Especially when the words are written with this translation being the only purpose. If “of” always means “multiply”, am I really thinking mathematically when I rewrite “1/2 of 3/4” as “1/2 x 3/4”?
Before 1500 or so, the science of math was developing at a snails pace. It was extremely hard to think about. So much so that people would have to travel to other countries in order to learn the arcane skills of multiplication and division. At this point, math and algebra texts were pretty much exclusively word problems; algebraic notion had not been invented yet. As the mathematical community began to develop the means of translating word problems into algebraic notation, math became way easier, and began to develop at a faster and faster rate.
It is absolutely the case that algorithms and memorized procedures free up the human mind to tend to other, more important matters. I have no issue with algebraic symbolism, nor with its use in K-12 classrooms. My beef is with curriculum that offers students little of intellectual value and little in the way of honesty about the actual uses of the subject.