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Math Forum » Discussions » Education » math-teach

Topic: Focus
Replies: 14   Last Post: Jul 30, 1995 11:07 AM

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Michael South

Posts: 16
Registered: 12/6/04
Focus
Posted: Jun 16, 1995 1:03 PM
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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.

let the flames begin...


mike






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