Dan Meyer linked to and quoted from my screed on the end of word problems the other day. This led to some robust discussion on his blog, which I now take the opportunity to reply to.

jg writes:

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”.

waldo writes:

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”?

Schwartzkopf continues:

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.

Thanks for the link on CGI. I’ve been looking for something like that for a while, and never had a name to put with it.

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One of the curricula I have read (its name rhymes with Balugal Rydell) has, at the start of every lesson, a “Why?” statement as a hook to students. To use your phrase, it offers little in the way of honesty about the actual uses of the subject. I work with complex numbers so I can “solve problems involving fractals” like deciding whether or not 1+i is in the Mandelbrot set. Really, that’s why complex numbers are taught? Or I learn inverse functions so I can convert temperature from Fahrenheit to Celsius. I learn to multiply and divide rational expressions so I can “compare the efficiencies of two designs”. I graph ellipses so I can “model an elliptical region” after a lightning strike. Really? Is that why kids take math, so they can be better at identifying where lightning struck?

Why not tell kids the real story: they’re learning math so they can learn mathematical thinking skills, like generalization, structure, argument, modeling. Yes, we’re preparing them for college and careers, but the preparation for those is far more about their thinking and learning skills and far less about whether they know the rule for log A/B and even less about whether they can “model the wind speed of a tornado” (via a provided formula).

Do kids really get grabbed by these “Why” statements? I always thought if a student asked me “Why are we learning this?” it was code for “I am bored and your teaching sucks”.

Quick question about language and translation: why do you think there is a big difference in the number of kids who correctly answer the question “What’s 3 less than 20″ and the number of kids who correctly answer “What’s 3 less than x”? I think there’s more to it than language and vocabulary.

Thanks for the discussion!

Chris,

Hi. I met you this summer at the CMP conference in Michigan.

I am wondering if there is a way to electronically alter the CMP2 assessments. There are a few problems in each CMP book that I would like to change a little. When I open them from the TeacherStation they seem to be read only documents. So, I don’t know if this is possible??

Scott Swita

Neillsville, Wisconsin

I sent you a note by email.

Thanks for stopping by the blog and let me know if I can be helpful with anything else.