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What an excellent story! No doubt there are millions of things like this involved in every concept in mathematics, most of them hidden, and most of them picked up automatically--who knows how--by normal students at some point in their education.

Since I was in grade school I've had a vision of how math education should really go, using an interactive system. Why do we still have textbooks? Textbooks take the student through at a certain pace, a frozen pace, which has been determined by the author at publishing time. After that it is set in stone. The problem is that each student has his or her own pace, necessarily different from the author's pace. Given an individual student, some parts of the book will be boring because the pace is too slow, and some will be excruciating because the pace is too fast. Because there is a hidden conceptual gap of some sort. My vision is a system which automatically checks the student's pace, notices when the student is bored and moves to a higher level of abstraction at that point, or notices when the student is lost and starts to fill in the gap at that point. Much like the high-priced tutor Catherine was discussing a while ago.

My theory is that conceptual gaps almost always occur because of the absence in the educational process of sufficiently concrete examples. One man's concrete example is another man's abstraction, and much of mathematics can be characterized as a sort of process of making abstractions concrete, but I strongly believe that the student must move from the familiar concrete to the unfamiliar abstraction.

The problem is not simply one for elementary school students. The problem occurs at all levels of mathematics, from fractions to schemes to categories. I'll cite one example, which happened to me last night. I was reading some probability theory and they used Lagrange multipliers to get some results. I'm ashamed to say it, but despite having taught it years ago, I had completely forgotten about Lagrange multipliers and couldn't even remember what they were for. I opened up a calculus textbook; the relevant section begins with a most abstract theorem about the topic, so abstract that I couldn't quite follow what they were trying to say. I turned the page and found a simple example, which clarified all. The theorem could be easily derived from the example, but not conversely. I'm sure this textbook is more or less a disaster for the average undergraduate who hasn't yet learned to look for the examples first and skip the theory till they need it. So, I had a double conceptual gap to fill, one within my knowledge of calculus and another within the specific meaning of Lagrange multipliers. Once I had seen the specific example, a function constrained to a circle, I had a sudden epiphany and instantly realized without any conscious thought exactly how this was being used by the probabilists and why.

Why do textbook writers write so poorly? I hypothesize that it is because they frequently aren't really writing for the students reading the books; it's because they're subconsciously writing for their colleagues. They want to impress their colleagues with how smart they are. Research mathematical papers are often, or usually, written the same way.

The disease is not unique to mathematicians; plenty of middle-management businessmen push products that are duds with the consumer because they couldn't care less about their market and only care about impressing their bosses. Likewise, many engineering products are created which are never bought because the designer wanted to impress the other engineers.

-- WichitaBoy - 16 Jul 2005

Even though the boy was able to intuit the solution with smaller numbers, I believe his real problem was conceptualizing the whole in this fraction problem (Joe walked 288 feet to the end of the pier and back. How long was the pier?)

I would have liked to try the following approach: Draw a line segment representing the length of the pier. Imagine walking from point A to B and draw the same line segment below the first. Then walk back from B to A and add the line segment to the first segment to form a straight line. This would visualize the whole (288 feet). The length of the pier should then immediately become apparent.

-- KtmGuest - 18 Jul 2005

Thank you, Ktmguest. I'm thinking of what you are saying, but using popsicle sticks instead of your two lines. Easy for him to have moved the second stick to add it to the first.

Too bad, I can't go back two years and try it. I'll keep your idea in mind though, for the next student who has difficultly with this problem. Thanks.

-- CarolynMorgan - 18 Jul 2005

That's an incredible story; it put a smile on my face (plus I teared up a little).

When I finally got around to showing Christopher the movie STAND AND DELIVER, I had tears streaming down my cheeks at the end, and Christopher kept staring over at me.

Finally he said, "You're crying."

-- CatherineJohnson - 18 Jul 2005

ktm Guest

I think you're talking about the same thing Carolyn brought up way back at the beginning of Kitchen Table Math.

I'm going to see if I can find that comment using Carolyn's cool new Comments search engine.

Wow.

I did it.

It wasn't as easy this time; I did 2 or 3 searches before I realized I should look for the word 'manipulatives.'

That took me to the manipulatives Topic thread, and from there it was easy.

QuickThoughtsAboutFractionManipulatives

if you want to talk about improper fractions -- fractions greater than one -- with your kid, then the pie-shaped manipulatives add potential for confusion because you can't make a single connected object that represents a quantity greater than one. If you want to represent, for example, 3/2 with pie manipulatives, then you'll have one whole circle and a half circle. You can tell a kid that that represents a single object, the quantity 3/2, all you like; but to him it will look like two objects. Fractions are confusing enough without that.

Conversely, you can make a single line of tiles that is as long as you like.

-- CatherineJohnson - 18 Jul 2005

That's what you were talking about, right?

-- CatherineJohnson - 18 Jul 2005