October 4, 2011
“I’m going to pick on one little corner of school, namely mathematics. I am a mathematician, so it is easy for me to talk about math, but what I have said applies to everything. Now in mathematics the dominating feature is the content. I recently did a little research on what discussion there is in the world about real change in the content of school mathematics.
For example, should we teach fractions? Some people say yes, you need to know that a half is two quarters. OK, well this is stuff that I’ve noticed that my six-year old grandson knows. He likes playing in the kitchen and he has learned a few things about halves and quarters and thirds. I have no doubt that is very good stuff. But that is only a tiny sliver of what they do in school. There is all this other stuff, like knowing how to add and divide fractions. What reason could there possibly be for teaching this?
Now, I am not advocating the abolition of fractions. I am drawing attention to the fact that here is a human activity on which billions of dollars and countless hours are spent. Incredible psychological harm is done to people who don’t succeed in doing it and therefore classify themselves and are classified by others as stupid or incapable or whatever else they are classified as. And there is no discussion whatsoever about why this is a good thing to do.
Sometimes people have proposed abolishing teaching fractions and there has been some discussion at those times. For example, there was a big movement in the late ’30s to abolish fractions on the grounds that people had done statistical studies of how much people used the fraction stuff they learned in school. And they came up with the obvious result that you all know, that nobody ever uses them. Not even mathematicians. Nobody ever uses things like dividing one fraction by another, so it was said, “Let’s throw it out of school.” So the other people said, “Well, people do use it occasionally.” You can find an occasional use here and there. But that’s not the basis on which we should settle this debate. You can’t judge whether the knowledge is good on the basis of whether you actually use it, because knowledge can serve all sorts of other purposes, and this discussion seems to have quelled that movement. Everybody wanted it quelled anyway because it is just too hard to contemplate the idea of eliminating these things from the curriculum.
Now this other thing that learning factions is supposed to do for you is not very clear. It’s never been spelled out. It’s rather like the old argument that learning Latin was good for developing the mind. And you might say that learning fractions is good for developing the mind. Some people say that learning fractions is good because you need it to do more advanced mathematics. Well, why do you need it to do more advanced mathematics?
OK, you can find an advanced mathematics book that will use an example that presupposes that the reader knows about fractions. Of course, why shouldn’t the writer of the book use that example since everybody has been through this experience? But that doesn’t prove that you needed that experience. If you take it all together, my personal view is that this is just harmful stuff to teach. In any case, there is no rational discussion about why it should be taught. So there is room for making theories about why it is taught, and I think there are a couple of these theories.
One theory was that manipulating fractions was actually closer to what people needed back before there were calculators. So a lot of school math was useful once upon a time, but we now have calculators and so we don’t need it. But people say that surely we don’t want to be dependent on the calculator. To which I say, “Look at this thing, these eyeglasses, that make a dramatic difference to my life and the life of everybody who reads or looks at any tiny detail. Once upon a time we would have been crippled, severely handicapped. Now we’ve got these and we don’t need to go through all that suffering. So we are dependent on this little thing.
Well, so what? There is nothing wrong with being dependent on a little thing that everybody can have lots of. It doesn’t even cost much. So, that is no argument. But I think historically that was a factor. I think the other important factor was for various reasons people thought we ought to teach something called mathematics because since the days of the Greeks mathematics was ensconced as one of the major elements of knowledge. In fact, I don’t know how many people know this, but if you want to know where the word mathematics comes from, the stem, math, comes from Greek mathein which is the word to learn. In fact, all the words in math in ancient Greek didn’t mean what we mean by them, they meant learning. And somehow in the course of the intervening centuries, my sort of intellectual ancestors, talking now as a mathematician, somehow managed to con the world into thinking that the only good learning was this kind of learning. And so the word slipped over with hardly a trace of its original meaning.
Well, there are some traces, like the word polymath. A polymath is not a person who knows a lot of math. It is a person who has learned a lot of different things. But that word has been thoroughly appropriated by mathematics and so by definition knowledge includes doing some mathematics. Actually, I would agree, except that I don’t think that working with fractions is really mathematics. And I do think that if we think about what mathematics means to me as a mathematician, it’s got nothing to do with things like those formal operations that you do with fractions and it’s got absolutely nothing to do with the way you do it in school. And so if we are going to prescribe mathematics for children we need to do something very different.
Now this something very different isn’t very hard to imagine, although it will need a lot of work to develop. And that is one reason why it isn’t done. It is not hard to imagine in the context of modern technology. We have developed lots of examples to show how with computers there can be a radically different relationship between children and learning – learning all sorts of things, including mathematics.”
Papert, S. (1997). Looking at School Through School-Colored Spectacles. Logo Exchange, Winter 1997.
A version of this article was published in Logo Exchange in the winter of 1997. It was adapted from a talk delivered by Seymour Papert at the MIT Media Lab, June 4, 1996, at an event sponsored by The American Prospect Magazine.