More on Understanding Math

I was sitting with a small group of math teachers at a meeting and I asked about their methods of "Testing for Understanding." It seems that for many (most?) the answer was a combination of "They can do the homework." or "They can pass the tests." Am I just looking for too much?

Anyway, while doing some serious research (playing around on google search) I came across a page called Understanding Mathematics, a study guide by Peter Alfeld.
He writes," You understand a piece of mathematics if you can do all of the following:

Explain mathematical concepts and facts in terms of simpler concepts and facts.

Easily make logical connections between different facts and concepts.

Recognize the connection when you encounter something new (inside or outside of mathematics) that's close to the mathematics you understand.

Identify the principles in the given piece of mathematics that make everything work. (i.e., you can see past the clutter.)

By contrast, understanding mathematics does not mean to memorize Recipes, Formulas, Definitions, or Theorems. "

He then goes on to give a few examples (powers, logarithms, quadratics) which he uses to amplify his explanation.

I think the student who can explain why 8^-4/3 = 1/16 without too much armwaving probably has a pretty deep understanding of the exponentiation process.

I'm not as sure about his quadratic solutions model. He talks about the quadratic formula and then states, "Forget about the formula, it's an example of clutter. (To this day I cannot remember it.) But since you understand these matters you know how the formula was derived."

I'm not sure I believe him. Ok, I know that sounds kind of harsh, but if you derive the quadratic formula a few times it sort of sits there looking at you all the way through, doesn't it... I see it and know what it is I'm working toward.

I agree that one ought to be able to derive the quadratic formula by completing the square, or at least solve quadratics without the formula, but then I remember a quote I have from Euler in my notes on "Twenty Ways to Solve a Quadratic Equation": Even a great mathematician like Euler, after deriving the formula, suggests “it will be proper to commit it to memory”. from An Introduction to the Elements of Algebra . Euler understood math, and when crunch time comes, you gotta' really sell me to make me disagree with him.


I don't have a wide range of clear delineators of understanding, but a kid who looks at a quadratic with a positive leading coefficient and a negative constant term and can't tell me how many real solutions it has does NOT understand solving quadratics, no matter what he has memorized. And if I write some ugly equation on the board the kid has never seen before and ask, "What happens to the graph of this if I replace all the x's with x-2 and he says it moves to the right (or left, I'm easy) I think they understand something big. If I tell him that (2,3) and (3,5) are on the graph of f(x) and ask him what f^-1(3) is and they have no clue, I think they have missed something big.

Here is a recent example, we were talking about H1N1 and working with the logistic curve and I was explaining that the maximum rate of growth occurred at half the maximum value, such as here.
I suggested that from this point they should be able to see that the ab^-t = 1 and that would lead them to see that b^t=a. But for many (most) of them sorting out that the exponential part = 1 was too many steps at once... in fact for a large number, it just did not make sense that 1+ab^-t=100/50 (seeing past the clutter?). Whether you think of this as a division property or the means and extremes property of proportions, it seems to me that if you don't see this "clumping" (I think that was Polya's term) you can't be very flexible in math patterns.

Ok, so send me your gut tests... If they can do THIS, they mostly "get it", or, conversely, if they can't do THIS, then they definitely don't "get it".