I've done this one several times, always with the same results: rectangles chosen randomly do a better job of estimating the mean population area than the "guess method" and the "judgment method".
But I have found I need to organize things quite carefully, in advance. For example, if anyone had told me four years ago that you have to _explain_ carefully to students how to use a random-number table, I would have scoffed. But I was wrong. It needs to be explained and rehearsed in advance. (Using the TI-83 to generate the random numbers probably would be easier; I'll try that next time.)
I use an overhead copy of the page of rectangles, and supply a sheet with prompts on which their results are recorded. I also arrange it so I can check their use of a random number table. Even then, I get strange results at times, like the student who guessed that the average area on the page was 20, despite the fact that the largest area on the page is 18.
This is a second opportunity to say to students that "You can't expect yourself to behave randomly!" Of course, it also supplies an opportunity to talk about selection bias.
============================================== Bruce King Department of Mathematics and Computer Science Western Connecticut State University 181 White Street Danbury, CT 06810 (firstname.lastname@example.org)