This is interesting. First of all, Richard is correct that "Teaching the kids how a slide rule works requires they have a basic knowledge of how one uses logarithms," and that's rough with an eight year old.

But, just because a kid doesn't understand why a slide rule works doesn't mean that you can't teach him a rote procedure for using it to, say, divide two numbers. An eight year old can learn that if he does "this, this, this, and this", the answer will show up right here. That's fundamentally no different than teaching him a rote procedure for doing the same thing on a calculator. He can learn to do "this, this, this, and this" and expect the answer to show up on the display, even though he doesn't have a clue about how the calculator works internally.

Now, whether either of those two rote procedures is fundamentally any different than the one for doing long division on paper is probably open for debate. When you're writing down the numbers and doing a bunch of trial divisions and multiplications and subtractions yourself, you're definitely a lot more involved than when you're just punching keys on a calculator or lining up marks on a slide rule. I would guess, however, that the value of that involvement varies from one kid to the next. Some kids probably understand completetly what's going on, and are better off for having gone through the exercise. I'd venture that others, however, get so lost in the mechanics that it has no more real meaning to them than pushing those buttons on that calculator.

In any case, Erik has put his finger on the real problem, and that's how to give kids a gut feel for what it all means, so that they just know which arithmetic operation should apply to a given "word problem", and so that they just know when an answer they they have obtained looks wrong, regardless of whether they got it from a slide rule, or a calculator, or by scratching some numbers on a piece of paper.

How do you fix it so the McTeen behind the counter doesn't give you $14.28 in change for something that you've just paid for with a five dollar bill? In other words, how to you teach numeracy? Does practice with the mechanics of arithmetic help do that? (My guess: probably not directly, and probably not equally with all kids.) What would be some other approaches towards giving kids that hard-to-define "gut feel"?

-- Russ