Here is a reply concerning mathematical procedures from Kimberly Vincent:
---------- Forwarded message ---------- Date: Mon, 31 Mar 1997 13:35:27 PST8PDT From: Kimberly Vincent <email@example.com> To: Ronald A Ward <firstname.lastname@example.org> Subject: Re: Eval.Stds.XI [3/28/97] (fwd)
Ron's latest post on the Standards brings up something taht is near and dear to my research. Ron wrote the following:
> > > 67. The NCTM Standards have often been criticised as if they placed little > emphasis on procedures. But I note that one of the bulleted items says > that students should be able to "reliably and efficiently execute > procedures." And, in fact, this Standard goes even further. It > demands a knowledge of procedures that requires much more: when to apply > procedures, why they work, how to verify that they give correct answers, > understanding concepts that underlie a procedure, as well as the logic > that justifies it. And assessment "must emphasize all the aspects of > procedural knowledge." That's pretty demanding, don't you think? Have > the critics been wrong? >
I do think the critics have been wrong. I feel that the thrust of what has been said in the Standards is that we value how each individual student solves a problem. Rather than requiring all students to use the same algorith or procedure for finding a solution, we allow each of them to use the alogrithms or procedures that they understand the best.
There are times it will be approriate for all students to use the same algorithm. When this is the case, it is not too much to ask that the students understand how the process works, why and how to verify their answers. I have been using this approach in a college algebra course (not a prerequisite for calculus). In this class the students have found they are less frustrated when they rely on understanding rather than memorization and they have more avenues for verifying their work is correct, expecially at test time. So by understanding why and what they are doing they gain confidence and gain a firmer ground to stand on. Unfortuanley there are always a few students who feel more comfortable just memroizing the process for it is less work and they are comfortable mimicing the process. However, for the majority of my students I have found that they can live up to the expectations and that they are better mathematics problems solvers because they understand the process, how, when and why it works. This gives them a wider range of situations to apply the process to. Kimberly Vincent Department of Mathematics University of Idaho Moscow, ID 83844-1103