Thursday, February 17, 2011

The Birds and the Bees and Constructivist Math

Years ago, parents could buy little picture books to read to their kids when the tykes asked "Where do babies come from?". Designed for thoroughly embarrassed parents (isn't that all of us?), the books went through the conceptual habits of most of the animal kingdom (insects, fish, birds ...) before arriving at humans. This way, blushing Dad had time to work up to the really difficult stuff, and if he was lucky, the kid might have fallen asleep!

My college biology teacher told us that some genius did a follow-up study. It was very simple: he went around to kids whose parents had dutifully read them the book, and asked the kids "where do babies come from?" He discovered that the kids, understandably, had gotten the different species all mixed up. A typical answer:
"Daddy holds the babies in his mouth until it's time to go to the hospital."

Constructivist math takes a similar approach, with similar consequences. When the kid asks "how can I multiply two big numbers together?", constructivist math behaves like the blushing Dad, loosening his shirt collar with one finger while replying: "Here's an ancient lattice method, and over here is a partial products method, and oh, look! let's measure our armspan!" while never getting around to the method that everyone actually uses.

Since the kids don't really understand any of the methods, they tend to combine them, using steps 1 and 2 of one method, followed by 4 and 5 of another method, and coming up with the wrong answer after a great deal of effort and frustration. I say, cut to the chase! Just teach them the standard algorithm.


  1. FedUpMom -- I'm loving these posts on constructivist math, even if I have my own doubts about math instruction that are different from yours. But I do wish you could get some pro-Constructivist-math people to engage in this conversation. I'd really like to hear their responses to the points you're making, and their defenses of programs like Everyday Math.

  2. Hilarious.

    I agree that it would be nice to hear from pro-constructivists. I don't think that EM is all bad, and it is definitely better than Investigations, which my older child started out with.

    My experience is that the most proconstrucutivists end up either bashing "the old way", or end up sounding like an infomercial for the program....but in the end I was still befuddled about why the pro-constructivist way was superior.

  3. @Chris, if you have different doubts, feel free to voice them. That's what we're here for!

  4. Oh, I guess I've been working my way around to it. It isn't really something I've given a whole lot of thought to. Mostly I just wonder about how much difference all these different curricular choices really make in terms of how capable people are at math when they're adults. My sense is that most adults actually retain very little of the math they're taught, no matter what math program they had in K-12, so part of me wonders whether it's worth all the hours we put the kids through to learn it. In some ways, that makes me agree with you -- because I think: just make sure they know the basic algorithms for the fundamentals; the rest is all wishful thinking.

    Of course, I realize that there are a number of shorter-term reasons why you'd want your kids to know more advanced math, even if they're destined to ultimately forget it all: so they can keep up in their high school math courses, so they can score well on the SAT, etc. But if the wide majority of American adults get along just fine without knowing the quadratic equation or what sines and cosines do, then why do we make math skills such a major sorting device for college admissions, etc.?

    Don't get me wrong, I think it would be great if we had a more math-literate population (especially if it sharpened people's ability to evaluate public policy claims). But it just seems like the prospect of achieving anything more than de minimus gains is pretty slim. I mean, measured by the adult population's actual math skills, the history of math curricula is just the history of one failure after another, isn't it? So I just wonder what the point is of piling hour after hour of math on the kids when we pretty much know in advance that nine-tenths of it isn't going to stick.

    As for the stuff that we could realistically hope might stick, it seems like we could get that stuff across more quickly and less painfully if we just waited until the kids were a little older. (I tend to think that the experiment discussed here is onto something.) Again, why does a fifth-grader need to know how to divide by fractions?

    Another way of putting it is: I think the reason Americans don't know math isn't because of the math curriculum they had in elementary school. It's because they (to some degree accurately) perceive that they don't need to know it -- that they can get along just fine in life with very few math skills. (It's the same reason Americans don't speak foreign languages.) We can wish they felt differently, but I don't see how choosing one curriculum over another is going to have any effect on that perception.

    It's as if we insisted that all schoolkids become high-level bodybuilders -- because, after all, it's good for you! -- but then, as soon as we left school, we all sunk back into our couch-potato, weight-gaining ways. We'd just be using the kids to act out some fantasy about ourselves. When I see my six-year-old sitting through hour-long math lessons five days a week, I get that same feeling.

    Of course, there will always be people who choose to learn more math anyway (wouldn't you have been one of them?) and those are the people who are more likely to retain what they learn. But teaching everyone huge quantities of math just because some people will want or need to know it makes no more sense than teaching everyone advanced biochemistry just because we need some people to become doctors. I'd say let people self-select for most of it.

    Of course, I do agree that schools should choose math curricula that don't make the kids miserable, so I sympathize with a lot of your criticisms just for that reason.

    Sorry to ramble on. Again, I'm still figuring out my thoughts on this topic and am just kind of thinking out loud.

  5. "Of course, there will always be people who choose to learn more math anyway (wouldn't you have been one of them?) and those are the people who are more likely to retain what they learn."

    Chris—My problem with constructivist math is that I believe it turns these sorts of people off math as well (perhaps especially)—which means in the future there will be fewer students able to handle the higher level math needed in university and in certain sectors of the economy.

    In other words, for me the key issue is not whether a math program promotes the retention of mathematical facts or concepts, but rather whether or not it stimulates or enables further learning. I think constructivist math programs, by continually frustrating kids, and by denying them basic competence, make it difficult for students to know whether or not they like math or have any real interest in it.

    That said, I have engaged with pro-constructivist math people (after I wrote my math post, for instance), and I believe their concerns are actually quite similar to mine. They think constructivist pedagogy is the way to stimulate enduring interest in math. I think we disagree about means more than ends. Though there are some constructivists who believe that the end—as well as the beginning—of math instruction must always be real world applications. I disagree with this, since to me math is a language, and any pleasure I once derived from it came from this aspect of it.

    (My apologies in advance for any incoherence—I'm half asleep as I write this!)

  6. northTOmom -- That sounds exactly right. I think the first task of any math curriculum should be to make sure that it's not creating mathphobes. The second task should be, as you say, to stimulate interest in math and facilitate further learning. But both of those goals seem inconsistent with the drive to achieve certain milestones by certain grades (and to meet one-size-fits-all standardized testing benchmarks), especially if we're talking about the early grades.

  7. I think the big problem with constructivist math is that kids never even get the core math skills. Chris, you're right that most people probably don't remember more advanced math and get along just fine. I personally don't remember anything about calculus or trig and barely any geometry. But I can add, subtract, multiply and divide integers, decimals and fractions and do most algebra with no problem. I don't see how kids in constructivist programs ever get that because the curriculum spends so much time beating around the bush.

  8. northTOMom, well said. I agree that one goal of all teaching should be to encourage future learning -- a teacher whose students never want to see the subject again has failed.

    @Chris, I understand what you're saying about the benchmarks, and they can certainly be overdone, too rigid, or developmentally inappropriate. But what I'm seeing at Natural Friends is that without clear goals, they just drift around and don't accomplish anything in particular. It's boring for the kids too.

    @northTOMom, I'm going to excerpt your comment for a stand-alone post -- I presume that's OK --

  9. @Matthew, I agree with you too. Kids need their basic skills. Even if neither of my daughters goes into a math-heavy field, they should at least be capable of managing their own finances. How can you do that without a solid understanding of fractions, decimals, and percentages?

    The makers of Trailblazers will say things like, "we teach deep conceptual understanding instead of arithmetic", but I don't believe it. You can't have deep mathematical understanding without the ability to work with numbers. That's not how it works.