In article <[EMAIL PROTECTED]>,
Peter Flom <[EMAIL PROTECTED]> wrote:
>Herman Rubin wrote (heavy snipping, since this is getting out of hand in
>length
>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 don't understand, then, what you mean by 'conscious'.
><<<
>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.
>I can agree with this, mostly, but when should the details come?
>e.g., the concept of 'derivative', is, in one sense, very very easy:
>The slope of a curve. I think students could learn this idea very
>fast.
You are oversimplifying; what is the slope of a curve? I
have seen ONE precise direct explanation of the slope of a
curve without already having derivative, and this was not
in any text. It was in the R. L. Moore film. I believe
that derivative is the easier concept, but this requires
having the concept of limit. Limit should already be there
when infinite decimals are discussed. Euclid's students
knew the concept of limit for points, and to some extent
for numbers. They did know about approximants to square
roots of rationals.
But on a more important note, one does NOT learn a
concept from merely getting a formal definition or
characterization; it needs to be developed before the
concept is learned. Concepts are also not extensional,
but intentional; there are at least two distinct key
concepts about the positive or non-negative integers, and
the cardinal and ordinal concepts are distinct, and both
should be taught. One can question as to whether the
others are mere consequences of these, or are themselves
basic concepts.
>I agree that it is important to get this idea across before teaching
>methods of differentiation. But WHEN should the details come?
As above, a concept is not learned by merely hearing the
words, or even being able to repeat them. One needs to
look at examples AFTER the definition, and to see proofs.
A concept needs to be internalized. There ARE cases where
the concept may remain unclear, even if the topic can be
used. I am not convinced that I know the concept of
"analytic function", although I can certainly use them
and their properties.
After
>learning the idea of derivative on day 1,
See the above. But I will consider "day" to be "appropriate
period of time", and comment accordingly.
should students learn
>integration on day 2,
They should have ALREADY learned integration, and very
definitely NOT as the inverse of differentiation. The
Greeks definitely had the notion of the "Riemann" integral,
but the more basic idea goes back to very ancient times.
The earliest use of computing an integral I am aware of is
the determination of the total of a merchant's bill. All
other computations of integrals of real-valued functions
come from minor extensions of this, and the use of limits.
The full notion comes from extending to other types of
measure and function values, again using limits. I did
not have as full an understanding of integration as I have
until I got away from usual "definitions".
partial derivatives on day 3, and multiple
>integration on day 4, and THEN go back to learn details?
There is a limit to how much has to be done together.
These can easily be postponed. The ideas have to be
developed.
>And when should they learn about proofs? After all, concepts don't
>need to be proven; theorems need to be proven.
They should learn about proofs in primary school. Now for
those who will not be mathematicians, or at least research
scientists, being able to prove theorems is not overly
important; it is the concepts, the language, and the
understanding of formulation which is basic. The engineer
does not need to know how to solve the differential
equations he uses, but he needs to know what the equations
mean, and what the solution means.
>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.
>I agree that this is bad; I disagree that it is new. It's been done
>for at LEAST 35 years (since I was in elementary school that long ago);
>anectodal reports are that it's been around much much longer.
I can personally attest to 70 years.
>I had a 4th grade math teacher who refused to teach division to anyone
>who could not do a sheet of multiplication problems without error. And
>she was an old woman when she taught me; I assume she had been using
>similar methods earlier in her career.
"Mastery learning" is not always desirable. But one must
understand the subject to know what is NEEDED to do what,
and very definitely perfection is not what is needed.
>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.
>Again, I am confused as to just what you mean by 'conscious'.
>To my way of thinking, making something conscious is part of the nature
>of true understanding. If I read a theorem which I find intuitively
>obvious, I understand it at one level. But if I have to formally prove
>it, or even if I follow a formal proof in a book, I understand at
>another level. I think the main difference here is whether I am
>conscious of WHY the theorem is true, and how I KNOW that it is true.
One can easily be able to produce many proofs of a theorem,
and still not understand the underlying concepts. One can
also understand the concepts, even the concepts underlying
the proof, and still not be able to carry out the details.
There are many theorems which I know how to prove, but I
still do not know WHY they should be true. The bounds on
the convergence to the Central Limit Theorem is one of
these, and the fact that a counter with "dead time" can
be considerably better at producing random bits than one
with no dead time at all.
>Of course, it is hard to define 'conscious' in a precise way, and this
>may not lead anywhere (but it is interesting, nonetheless)
--
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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