metacognition and math

Steve H raised a question about overconfidence the other day (in this case, college kids assuming they know stuff they don't), which coincided with my having discovered the concept of metacognition in math ed.

I haven't had time to write anything about it yet, but metacognition is a hot topic in radical constructivism and non-radical constructivism, and it's a terrifically useful concept for me, too.

Metacognition at its simplist means knowing what you don't know.

That's probably not how most researchers would define it, but it's how I define it, and how I have defined it for a number of years now. I think the ability to know what I don't know is one of my most important skills as a journalist.

Math has me stumped. I have just about zero metacognition when it comes to math. I don't know what I don't know; I barely know what I do know.

No, that's not it; I do know what I know, I just don't know if any of it's right. In other words, I may know what I know, but should I know it?

Or should I forget it right this minute, because it's stupid, illogical, and wrong?

I don't know.

Here's Bernie Johnston:

I don't think you can ever "learn math". There's just way way too much of it. As a matter of fact, I'm certain even a professional mathematician can't learn the names of all the things that are being produced today within mathematics, let alone understand them.

The real question is: when do you know you've learned a piece of mathematics? For example, when do you know you've understood dividing fractions? As a matter of fact, you may never have "completely" understood it. That's because a successive generation may come along with a new idea which sees dividing fractions from a completely different point of view.

[For example, although the ancient Pythagoreans knew that the square root of 2 is irrational, and suspected that pi was irrational, they had no way of knowing that the former is algebraic (which means "not too irrational"), while the latter is not. They couldn't even have formulated the terms.]

I guess all that we can really aim for then is having the experience of understanding a single mathematical idea using a single point of view. At some point the additional insight gained from seeking different points of view is no longer worth the additional effort.

Mathematics is a journey, not a destination. (What can I say? I had to say it. I'm a child of the Sixties.) I guess the only sensible guideposts we have are reports from those who have already travelled down that particular path.