In article <[EMAIL PROTECTED]>, R. Martin <[EMAIL PROTECTED]> wrote:
>Herman Rubin wrote:
>> In article <[EMAIL PROTECTED]>, R. Martin <[EMAIL PROTECTED]> wrote:
>> >Richard Ulrich wrote:
>> >> On Sun, 16 May 2004 13:25:07 GMT, Art Kendall <[EMAIL PROTECTED]>
>> >> wrote:
..................
>> >> Further: I have to wonder, Are today's kids really smarter in ways
>> >> that matter? I suspect that they are. Now, how do you measure it?
>> >Anecdotal evidence, which may or may not be applicable to the
>> >question, and comes from a non-random sample and has other known
>> >uncontrolled variables: I know that some of my younger relatives
>> >know more about some academic subjects than I did at their ages.
>> >I had what I feel was a decent education for the time (1960s and
>> >1970s), and in certain subjects I'm sure I had an outstanding
>> >education.
>> The educational system has made SOME improvements since
>> Sputnik, but also some declines.
>> My younger relatives have been often two (sometimes
>> >more) years ahead of where I was at their ages. Different school
>> >systems, yes, different times, definitely, and my relatives probably
>> >don't represent anything like a cross section of the population.
>> >At young ages I attributed it to "Sesame Street", etc. When they're
>> >doing things like full blown calculus in high school, I'm not sure
>> >what the sources of the differences are.
>I have not followed this thread with perfect attention, so forgive
>me if I misunderstand any of the points you're making.
>> Are you sure they are ahead?
>I'm reasonably sure they are in many respects.
>> They may be in memorization
>> of facts and methods of computation, but understanding in
>> the schools has not improved much.
>Certainly a 20 month old being able to line up 20 toys and
>count them does not imply the child has Fermat's facility for
>number theory, as it certainly is largely memorization, but IIRC
>I was learning to count on my fingers when I was in kindergarden
The process of counting (not memorizing the names, but
realizing that it is a process) is an important concept
of the integers; it is not the only one. This is not
what is stressed, and may not even be taught. It is
mistakenly assumed that someone who can do arithmetic
understands the concepts.
>(OK, so I'm not Gauss :-), although I did figure out division
>on my own a few years later), so I still think the achievement
>level of younger people says something worth noting.
Did I ever say otherwise? But can one expect them to
learn the precise concepts by themselves? This is
VERY unlikely; however, someone can be taught concepts
fairly easily if there is not too much interference by
having "learned" how to "do it" without any idea of what
it means. From Aristotle's writings, he clearly had a
totally mistaken idea of lines and "real numbers".
Is it
>"intelligence"?
If a 20 month old can do that, this is strong evidence
of giftedness right there, and that child should be
considered for much faster and more rigorous education.
If it isn't, it is at least a building block
>that is needed for the quantitative reasoning portion of
>intelligence.
I agree. The "new math" was introduced because the
teaching of arithmetic manipulations did not seem to
enhance quantitative reasoning to any major extent,
if at all.
>> As for "full blown
>> calculus", knowing all the methods of computing derivatives
>> and antiderivatives in closed form provides NO understanding
>> of what any of it means. I would not trust the high school
>> teachers on this point. The old "Euclid" course should be
>> a requirement, but many schools do not even offer it as an
>> option.
>> This continues at the college level. Those taking methods
>> courses in statistics may end up unable to understand the
>> concepts of probability and of decision making under
>> uncertainty which they might well have been able to master
>> in primary school. Knowing how to compute is the least
>> important thing; it is knowing how to speak and communicate
>> in the formal language which is most important.
>No doubt your concerns have some validity, but there are levels
>of understanding, especially in math, and I'm not willing to say
>that only the highest level of understanding is the one that
>exclusively counts in the IQ or education debate. If a high school
>students can recognize the calculus to apply to a problem and apply
>it correctly, that still allows them to solve problems I couldn't in
>the same grade, even if my understanding of fundamental mathematical
>concepts was better (which it may or may not have been).
There are two parts to this. The first part is using the
concepts of limit and derivative to apply to a problem.
The second part is being able to calculate the solution.
Few calculus courses even teach the first part at a D-
level; if the student cannot get it from the textbook,
which also does not teach the concept, it is not learned,
and the only part learned is the catalog of computational
procedures. It is important for the economist, psychologist,
biologist, chemist, physicist, engineer, etc., to have the
concepts; it is far less, if at all, important for them to
be able to compute answers to problems which have a closed
form solution, but that is what they learn.
A person
>can make intelligent use of, say, calculus without being able to
>derive the properties of the real number system from first principles,
>formally or informally.
Derive, no. Understand them well enough to use them, yes.
Unless the person is really bright, in which case ability
to easily follow the derivation from first principles would
be present, they will not be understood, even informally,
unless specifically taught. Many of the teachers of
mathematics in the schools have no understanding of them.
If a person can do this at the level I
>attained as, say, a freshman in college when he/she is a junior in
>high school, then IMO that person is, for practical purposes, two
>years smarter than I was. Again, if this isn't germane to the
>argument your making, feel free to ignore my ramblings.
But if a person can do the calculations without understanding,
that is not the case. And this is what is happening.
--
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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