Thursday, June 25, 2009

How Children Learn Mathematics, by Richard W. Copeland

As I read this book I kept thinking, "My daughter would love this guy." Why? Because the entire book can be boiled down to the argument, "Kids can't do math, so let's not make them yet." Copeland says, "The child must first develop to the stage of conservation of number, usually sic and a half to seven years of age, before he can be 'taught' number" (21). He says they can't learn measurement until they can understand continuity. (Note to Copeland: I just took a real analysis class and I still don't understand continuity.) He says they shouldn't learn addition until they are eight, and then they should only learn addition facts they teach themselves. He says some kids won't be ready to learn addition until junior high. I'll pause here for all my Asian readers to finish laughing and wipe the tears from their eyes.

Initially I thought, "This is the kind of book I would have loved when I was in school," but then I realized that, actually, it wasn't. My problem in school was being bored out of my mind, and spending the first three years of school doing counting worksheets, as Copeland suggests, would have done me in. The frightening thing is that the ages Copeland suggests are averages, meaning he thinks there are some children who, even at seven, can't learn how to count. (At least there were when he wrote this book in 1970; these days every child is above average.)

This is one of the (many) books I checked out of the library while I was waiting for the two kids I tutor to not show up. (They don't show up more than they do, actually.) I wanted a book more like What's Math Got to Do With It? by Jo Boaler, which I read last year (in preparation for becoming my daughter's primary math teacher) and really enjoyed. Instead, with Copeland's book I got 300 dense pages of him verbally fellating Jean Piaget. Page 10 is a reproduction of a letter from Piaget to Copeland. The rest of the book presents the findings of Piaget as conclusive. "Here's a proposition, and if you think it sounds wacky, rest assured that Piaget agrees."

The problem with citing Piaget as the final authority is that, I think, the guy is dead wrong about a lot of stuff. His argument for why it's pointless to teach counting before the age of seven is that some younger kids make crazy counting errors that show they don't really know what counting means. When they are seven, they don't make those mistakes anymore. A reasonable person would say it's because they've mastered the skill; Piaget and Copeland say it means they shouldn't even have been taught the skill yet.

One of the great things about these outrageous claims was that I had two kids at home to test them on. Every time Copeland would tell me a kid can't do something until he's eight or nine, I'd ask my five- and six-year-olds and usually get the right answer. For instance, my five year old thought I was a fool for asking, "If there are three dogs, two cats, and a mouse, are there more dogs or more animals?" Copeland says kids' brains can't hold onto the idea of three dogs while simultaneously thinking of the dogs as part of the animals group. My son just thought I was an idiot who didn't know that dogs were animals.

Sure, two kids is a pretty small sample, and since I'm a modern parent, it's a given that my children are exceptionally brilliant, but nearly every time my kids answered Copeland's brain-busters with no problems. Copeland suggests showing kids a collection of blue circles, blue squares, and red squares and asking, "Are all the circles blue?" Says Copeland, "A four- or five-year-old when asked 'Are all the circles blue?' responds that they are not because 'there is a blue square.' He cannot make the distinction between 'are all the circles blue' and 'are all the blue ones circles?' He is unable to establish a logical class and will usually be unable to do so until nine or ten years old" (39). I swear to you, I am not making this crap up. In practice my five-year-old said, "Yes, I mean no," because he immediately realized he'd misheard the question, and my six-year-old just laughed at the question, refusing to take it seriously (as I was tempted to do with most of this book).

I wanted to use the Binet Simon test on my kids, but of the five statements given in the book, four deal with homicide, manslaughter, or suicide. This is supposed to be a test for nine- to 12-year-olds. For the suicide question, "The poor response on this item seemed to be due mainly to the children's refusing to admit the premise, 'If I ever kill myself'" (134). I think I'd prefer to live in a world where children refuse the premise of suicide. I cannot believe that people who spend their careers studying children couldn't come up with more age-appropriate questions.

Continually Copeland claims that children who make mistakes are showing they don't understand the underlying concept, and they should not have been taught the operation yet. Thus a kid who looks at a small pile of four things and a spread-out pile of four things and says the spread-out pile has more things doesn't understand number and can't count. I think this is more the result of learning. Take an adult and show him a five-by-five square, then tell him to add a one-square border around the outside.

How much of the entire area is black? I think most adults would intuitively say most of it, but the answer is just barely more than half (25/49). Does this mean that adults don't know how to measure? Math is a tool we use where intuition might lead us astray. Kids learning math are learning to distrust what their intuition tells them. We shouldn't be surprised that there is some trial-and-error involved, and we definitely shouldn't tell the kids, "We'll try again in a few years."

I also wonder if the problem is language-based. It might not be that the kid doesn't know how to count, but that the kid is learning the nuanced meanings of "more" and "less." A spread-out set takes up more area visually, and until the kid learns that this is unrelated to the discrete number of set elements, it makes sense to think of the bigger set as "more." After all, what is the largest nation on Earth, Russia or China?

Copeland gets incredibly worried about needless distinctions, like whether two sets of three things is the math problem 3 x 2 or the problem 2 x 3. (And don't ask, "Does it matter?" You better believe it matters to Copeland!) Later he makes an argument for children not understanding the concept of area measurement until they can invent the square root themselves (or at least realize that something like the square root should exist). "He then knows he needs the square root of 18 as the measure of each side of the new square. He has to approximate the square root of 18 as somewhat more than 4" (237). Approximate? Come on, Richard. Why don't we just expect him to invent infinite series and get the real answer?

The problem with this way of thinking is that it completely discounts everything that came before it. The great thing about Newton and Leibniz was that they were smart enough to invent Calculus. I'm not smart enough to invent it, but I can understand it when someone else teaches it. If we just wait for me to come up with a way to find the area under a curve, thus ensuring that I really understand the concept and I'm not just parroting something out of a text, we'll be waiting a long time.

This book made me happier than ever that we home school. All it takes is one crack-pot with a clever new idea to ruin an entire year of my kids' educations. "I know, we won't teach letters until the kids develop their own writing system, because sometimes a kid writes C where he should have written S." Piaget must have been studying Swiss trailer trash, Copeland was too enthralled with Piaget to see any problems with his conclusions, and American math education is in the crapper precisely because of books like this.

Rating: 1.5 out of 7 giant inflatable monkeys.

1 comment:

  1. What does this book say about adults who can't answer whether there are more dogs or animals? Maybe some of the cats ran away since the last roll call... who knows? I think this concept sounds awesome... just wait for my kids to spontaneously come up with complex math concepts on their own. Totally makes sense. On the other hand, I'm still waiting for psychologists to come up with some relevant child-related concepts on their own... This could take a while.

    I don't think I would like the book, but I enjoyed the review! So not only did you save me from a bad book, but entertained me in the process. Nice.

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