Aharoni discovered exactly what I did, when I started getting involved in my son's education; having a Ph.D. in math didn't mean I knew a darn thing about how to teach it to elementary school kids: I didn't. Not elementary school mathematics. It was different from any sort of teaching I'd done before. I was totally nonplussed, and Aharoni expresses it for both of us very well:

But what surprised me most was that I learned mathematics. Actually, a lot of it. This would not be the case had I gone to teach in a high school. The mathematical concepts there are known to a professional mathematician. In elementary school, it's the teaching of the most basic principles that counts; the nature of numbers, the meaning of the arithmetical operations, the principles of the decimal system. About these, it is rare for a mathematician to stop and think.

He addresses, in his article, the fact that the most common operations have multiple meanings. We adults get so used to moving between these meanings that we conflate them all in our minds, and when asked about the difference, can't even recall that there is a difference. Here's an example from Aharoni's article:

I was experienced enough to know that such confusion almost always originates from having skipped a stage. In this case the missing stage was the understanding that subtraction has more than one meaning. There is the meaning of diminution, where objects are removed: I had 5 balloons, 2 of them burst, how many do I have left? But there is also the meaning of comparison of quantities, where nothing disappears: There are 5 children in a group, 2 of them are boys. How many are girls? Or perhaps: How many more green apples than red apples are there? In these cases, too, the exercise is one of subtraction, but the meaning is different.

The various meanings of subtraction are an example of a fine point that has to be taught explicitly. Skipping this stage will result in later difficulties with word problems.

Another example of this conflation happened when Catherine asked me, and a couple of other math types, what the meaning of 'partitive' vs. 'quotitive' division was. She'd come across the concept in Liping Ma's book, which claimed that Chinese teachers could generate word problems involving fractions that were of either type, whereas most American teachers couldn't.

Well, mathematically there's no difference; division is division. Partitive vs. quotitive is just a pedagogical distinction that a teacher needs to know, in order to be sure that she can generate word problems that cover the whole set of possible problems, so that the kids will understand that the division operation is the same for all.

I didn't see the distinction at first; neither did any of the other mathematicians Catherine asked. We didn't need to in order to do our own work; but teaching elementary math is a whole different ballgame.

And here's an insight that I just love:

When I started teaching in elementary school, I was convinced that precise formulations and the explicit naming of principles was a matter for grownups. Children should learn things on an intuitive level, I thought. One of the greatest surprises that awaited me was to realize how wrong I was about that. Children need precise formulations. Such formulations consolidate their knowledge of the present layer and make it a safer basis on which higher layers can be built. Moreover, children love "adult" formulations and notations, and are proud of being able to use them. First-grade children who learn the notation "1/2" are happy to discover the notation for "1/3" by themselves.

It makes complete sense to me: grade schoolers realize that knowledge and skills are power, and grownups have the knowledge and skills. They don't want to learn dumbed-down, intuitive formulations of problems; they want to do what YOU do. It reminds me of how my Dad used to leave papers around with derivatives and integrals on them; I thought they looked like the coolest thing, and I wanted to grow up and do what he did.

More to come from both Catherine and me.


Aharoni article, part 1
Aharoni article, part 2: America's 'new math' goes to Israel
Aharoni on the fifth operation of arithmetic
Ron Aharoni on teaching fractions & forming units
What I Learned In Elementary School by Ron Aharoni (AMERICAN EDUCATOR)

-- CarolynJohnston - 08 Oct 2005

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