Got cut off at 4 this morning in some sort of an attempt to add a possible point of focus for some of the recent discussion. I don't know if it gets anyone anyehre, but it was sort of helping me evaluate some of the ideas presented and the questions being asked. You can flame me if you don't like it. Message follows.
mike To: nctml Subject: Dead Brains/Batteries
The discussion that has been going for a while now about algorithms/under- standing/basic skills etc. has brought up some really good questions. I don't really see "two sides" or any fundamental confliict, but rather a great deal of thinking/questioning about what we want students to be able to do. As Che-Tien has asked, "what are the goals? what is the optimum path to those goals?" (or something approximately like that). I think this is the essential question. It seems that if we could answer this question, the answers to the following questions would, well, follow.
(Paraphrasing (these are all recent, I can supply authors if you want, but since they are my paraphrases, I thought the original authors might not agree with my take on them) How are you going to be able to solve a life-endangering problem that occurs in a location that has no computational facilities?
Does/should anyone teach log tables any more, or rationalizing the denominator?
Is the technology (tv, video games, etc) that (it is supposed) has contributed to passiveness in kids today also capable of creating mathematical passiveness if the same kids have unlimited access to computational technology?
Suppose that computers could produce perfectly spelled and grammatically correct papers from our (possibly grammatically lacking) dictation. Whould we quit teaching writing and just teach reading? Are there certain skills that we want to be sure that we keep teaching? (end praphrasing)
I have the inescapable feeling that there is a profound truth that we are digging all around here that would essentially answer these questions in a perfectly obvious way (no, I'm not claiming to have discovered it or anything. I'm not a smart man....). I know that this is only an approximation to something that would be really useful, but it may be a passable first step.
I think that what we really (should) want is for our kids to learn how to think. In the end, it doesn't matter if they started with a calculator and had some brain-stimulating experiences or if they got things kind of figured out and then were able to accelerate their understanding with appropriate use of technology. The question is not about whether or not technology is a good or bad thing, as we all know, but how it's used. And it would be surprising to me if we could answer that question--"How should technology be used?"--even if we had all year with unlimited experimental resources to work on it. The most probable "short" answer to that question is likely to be "In a variety of very different and perhaps even apparently conflicting ways." It is not clear that two seemingly very different approaches are necessarily rankable as better or worse than one another. The question to ask is, "Does this method encourage/teach/force/help/require students to think?" Are they learning problem solving skills? Are they learning to ask questions, to question the answers? Are they learning how to create a set of criteria to distinguish "better" answers from others, and to allow for the possibility of equivalence or indistinguishability?
As far as I am concerned, if my kids come out of high school knowing a lot about graph theory or abstract algebra and have only a rudimentary knowledge of addition and multiplication algorithms, I will be more than pleased. At some point, if they want to, they can develop numerical skills. if iF _IF_ they know how to think. How to break a problem into component parts. How to relate it to other things they know. When to walk away from it and give it some time to gel in their minds. How to get "the answer," and then keep looking at it until they can see another way to get "the answer" or another way of thinking aobut the answer they already have. How to guess what it might be, test that guess, etc. Who to talk to about it, what books to look in. Whatever, as long as it uses their brains.
Now, supposing that any of those ideas have any value, what are the implications for, say, the drudgery/practice questions? The appropriate time and way to use technology? The place of algorithm? If we assess each question in terms of "which way would maximize the amount/quality of thinking required or produced or encouraged?", we may come up with some answers that make sense. But it will require thought...How do you know whether more practice is just going to bore the student and cause a loss of interest or whether it is necessary exercise for upcoming challenges?
Having run into mathematics and the teaching thereof as a layman, I haven't ever really read the "constructivist" and "behaviorist" positions, but from the few references I've heard, it seems to me that we want something from both sides. We would like to have students donstruct their own knowledge, but being able to do that requires certain behavioral skills. I would be completely at a loss as to tell you whether I value my "brain habits" more than the knowledge I have obtained/constructed with their help. It is through certain behavior (clear thinking, taking things one step at a time (perhaps the most difficult thing to do in any type of thinking situation is to make sure you haven't leaped to a conclusion you can't support), questioning the answer, etc) that I am able to construct knowledge.
Does this put me into either camp? (I need to know so I can buy a bumpersticker supporting whiichever side I'm on...) ( <== a joke )
Personally, I would like to see the average high school graduate have a lot better idea of basic logic--how to recognize the logical fallacies that the press, politicians, and advertising executives (did I just repeat myself there?) are smothering public information media with. Well designed courses (not just in math, but in writing classes and history classes and you name it) could accomplish this. I am a lot more concerned that my kids grow up knowing how to think about history than knowing a lot of facts that are available on CD-ROM.