> I have nothing special but I do believe the heart of the problem is in > your very premise, that manipulation facility is devoid of, or at > least separate from, understanding.
Below is an instructive experiment, Wayne, that you can carry out in the privacy of your own classroom--where you can identify the students and the instructor involved. Do this in a first semester calculus section, right after the standard discussion of natural logarithms. (The standard discussion, mind you--not a discussion that's been fortified with this experiment in mind.)
Right after the natural logarithm discussion, we should be confident that our students are as understanding about logarithms as they're likely to get for a while--and as understanding as they've ever been. And first semester calculus students should certainly understand straight lines and how to write equations for them. After the natural log discussion, they've *recently* had to demonstrate facility with manipulating both.
What do they really understand?
Here's the experiment: On the first exam after the natural logs discussion, put the following problem:
A scientist has generated a collection of data points of the form (x_k, y_k), where k varies from 1 to 20. She plotted the points (x_k, ln y_k) on a graph, and found that the points all lay very close to a straight line. Upon seeing this, she concluded that there were constants a and c such that the law y = c a^x governed the phenomenon she was studying. Explain her reasoning. How should she go about finding the constants a and c? (Approximately, of course.)
No matter how well your students manipulate logarithms and straight lines, I think you will find that very few of them understand the two well enough to do much with this rather easy problem.
If you're unsatisfied with the result, go over the problem in class after they flub it on the test. Pass out a hard copy of your best presentation of the solution. Put it on your web site, so they can't complain that they lost their copy. Tell them how important you think it is. And then put it on the final exam--but have her plot the points (ln x_k, ln y_k) and conclude that the law has the form y = c x^a.
Betcha the results won't be very different unless you cheated and told them what was gonna be on the final. Might not be very different even if you told them.
--Lou Talman Department of Mathematical & Computer Sciences Metropolitan State College of Denver