In article <[EMAIL PROTECTED]>,
Richard Ulrich <[EMAIL PROTECTED]> wrote:
>I have kept all of Herman's post, and put in my commentary.
>- I hope that I am not bothering many readers of sci.stat.edu
>by expanding in this direction. "Consciousness" is something
>that interests me enough that I don't want to drop the dialog yet,
>or shut out other possible contributors.
>On 29 Apr 2004 09:48:08 -0500, [EMAIL PROTECTED] (Herman
>Rubin) wrote:
>> In article <[EMAIL PROTECTED]>,
>> Richard Ulrich <[EMAIL PROTECTED]> wrote:
>> > - warning - Another digression (being bright, and being conscious).
>> This is very definitely the wrong word; see below.
>> >On 27 Apr 2004 11:12:04 -0500, [EMAIL PROTECTED] (Herman
>> >Rubin) wrote:
>> >> In article <[EMAIL PROTECTED]>,
>> >> Art Kendall <[EMAIL PROTECTED]> wrote:
>> >> >part 2
>> >> >One thing that is being done very frequently today is to have children
>> >> >teach each other some of the time. Recall Seneca's "docens discimus",
>> >> >"in teaching, we learn". Trying to find different ways to communicate
>> >> >the same concept to people broadens and deepens our understanding.
>> >> >In addition, my recall of my grammar school education is based on my
>> >> >perception at the time when my mind was much less developed.
>HR >
>> >> This may be the case for adults, but not necessarily for
>> >> children. When my son was 6, he understood, and could do,
>> >> algebra and logic, but he could not explain anything.
>> >> One has to learn a lot to explain something which is, to
>> >> him, completely obvious, to someone who does not see it.
>RU >
>> >Oh! Now we introduce 'consciousness'.
>HR >
>> I said nothing about "consciousness".
>Sorry - Donald did not see the connection, either, in his post.
>I jumped ahead, I guess, figuring mathematicians might already
>share my language and conclusions about this. The phrase,
>"completely obvious", seemed to encode my own early experiences
>of the unconscious solution of easy math problems.
>On consideration of these negative reactions, I remember that I have
>now read numerous philosophical discussions of consciousness
>and intelligence. The terminology is a bit abstracted from the
>commonplace; or perhaps it is more exact to say, there are some
>widely shared conclusions about consciousness that are not
>yet commonplace.
>For instance: Expert performances are mostly not 'conscious' but
>employ well-learned unconscious routines under slight guidance.
>(The caterpillar walked just fine until he tried to figure out how.)
However, it is quite possible for a person to know how, and
be aware of this. When someone carries out routine calculations,
or spells words, or uses mathematical concepts, while the use
may be "automatic", this does not mean that it is not also
conscious. Even if the person cannot describe it, it can be
very conscious.
>I've been encouraged to think the ideas are spreading. Last
>year, there was a major league ballplayer whose problems with
>overthrowing first were blamed on being overly-conscious --
>that was in the sports pages.
>HR>
>> Ramanujan, who produces hundreds of results in analytic
>> number theory and related parts of analysis, in fact only
>> published little of the ideas behind them. Most of the
>> proofs were published by others; I believe they are still
>> going through the "lost notebook", sent by his widow to
>> Hardy on his death, and relatively recently found. As to
>> how he got the results, he attributed them to a particular
>> Hindu goddess.
> - or, in the vocabulary, "unconscious processes." That is
>what is going on, when you know the result, and then have
>to figure out afterwards how you got there.
>HR >
>> The human mind is quite capable of dealing with concepts
>> which are, as far as we can tell, purely abstract, and even
>> of communicating them by describing their formal properties.
>> As to how one uses them, or decides which ones to use, this
>> is much harder to explain.
>Mathematicians do know that conscious thinking can be
>non-verbal. Much of it is spatial (spacial?), but I do not
>identify that with "purely abstract".
Nor do I identify spatial with purely abstract. In fact,
mathematicians are more likely to compute (in the extended
sense) to see what is happening than to use pictures.
It is the
purely abstract part which is the more important, and which
should be started very far back, even in kindergarten or at
the latest with beginning reading.
In addition to the problem
>of figuring "how you got there", mathematicians sometimes
>need to figure "how to put it into words" or other symbols.
The use of symbols (variables) with arbitrary meaning,
but with rigid syntax, is the linguistic idea needed to
allow mathematics to be done easily, and is THE most
important idea for non-mathematicians to use mathematics.
This one abstract tool is what is used to convert "real"
problems to mathematical problems, provided one knows the
mathematical concepts involved; as one bright child stated,
"Algebra is like cheating. It makes word problems trivial."
The same holds when more mathematics is used; the user of
mathematics and statistics needs to formulate first, not
even to have any idea of how to solve.
>RU >
>> >I have previously assumed that consciousness was a good thing
>> >for science and math, and underlies future learning. So, I would
>> >have expect that your well-advanced son could learn by teaching.
>HR >
>> He was fully conscious about what he knew. It is quite possible
>I don't see that you have any reason to assume that he was
>"fully conscious" in my terminology; if he is good at it, that
>would arrive some time *after* he could solve the easy ones.
Concepts are not "partially learned". One develops practice
with using them, but this is different from understanding.
>> that he might have been able to present the material as he
>> learned it, although I doubt that most can present the courses
> - and, as you have indicated again and again, most students do
>not *learn* much in their courses. I know that I improved
>some of my math understanding by teaching others.
One should learn the concepts; details can come later.
This is the approach opposite to what we are not taking.
Learning the details first interferes with learning
concepts, and even good mathematicians have trouble
with this.
And GOOD students learn much in their courses, especially
if the courses are mainly conceptual. Methodology courses
do not help in understanding, no matter how well they are
learned. Memorization and routine are the educational bane,
and this is now demanded by the "students" who have come out
of elementary and high school.
Now,
>maybe this is a bad approach for some subtle reason, but you
>have said nothing to show that the approach is bad, have you?
The approach is not automatically bad; it is rarely that
good, and it certainly interferes with the student moving
ahead in the subject. For those who think the current
amount taught is what should be done, this might not matter,
but it is a driving force in the dumbing down of our youth,
and of dulling or even debilitating their brains. When one
understands a concept, move on. Anyone who keeps a child
back should have to pay heavily for the attempt.
>> they took, but this would not help someone to solve a problem
>> in algebra as he did.
>RU >
>> >I thought that the arguments for 'natural ease' were confined to
>> >production in certain of the arts, as performed by very young
>> >people. Yes, adults want to get back to naturalness and
>> >unconsciousness, but we do that most fruitfully after training --
>> >which (I think) is characterized by being conscious.
Again, one can do things "naturally" and be quite conscious
about what one is doing. This is the case even if the act
occurs too fast for the consciousness to be in control. Those
who consider statistics as a collection of algorithms might
not be able to see this, but in that case, they have been very
badly mistaught.
>> >I am curious, do folks here assume the same, or otherwise?
>HR >
>> Most of those reading this newsgroup are familiar with the
>> Neyman-Pearson Lemma. I have yet to see a textbook provide
>> other than a formal proof, and low-level ones just a statement.
>> It is not difficult to present enough to make it "intuitively
>> obvious" to someone who can understand high-school level
>> discrete probability.
This last position is what teachers should always keep in mind.
It is not enough to produce a formula, or a proof. One should
try to understand.
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
[EMAIL PROTECTED] Phone: (765)494-6054 FAX: (765)494-0558
.
.
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