Re: Not Just the US With Education Problems

On Jun 9, 9:54 pm, John McKendry <jlastn...@xxxxxxxxxxxxxxx> wrote:
On Mon, 09 Jun 2008 03:08:39 -0700, tgdenning wrote:
On Jun 8, 11:43 pm, John McKendry <jlastn...@xxxxxxxxxxxxxxx> wrote:


....


 The broader argument here seems to be about what counts as monkey
work, and I have to vote with Tim Norfolk overall. I agree that
calculating the sine of 39 degrees is monkey work, but writing out an
expression for the sine of 67.5 degrees is not. Anyone who wants to do
any level of creative work involving math should recognize the latter
as a couple of applications of basic trig identities to the sine of 45
degrees (or better, 135 degrees). I don't care if he/she looks in a
book for the trig identities; the important thing is to know they
exist. You will never learn the trig identities from a calculator. If
all you care about is the numeric answer, it doesn't matter, but if all
you care about is the numeric answer, you'll never learn what it is to
do math, either.

There's a recurrent fallacy here---maybe several different ones
depending on the writer, although they speak to the same point.

The question that has to be answered is "what is the goal?", and it has
to be answered without emotive language but with clarity and precision.
*My* goal is to have a population that has a positive attitude towards
science, and uses reason and scientific reasoning to arrive at
decisions. I would suggest that once you set up an educational system to
do that, people will flow into the appropriate professional disciplines,
without mandates, based on their own inclinations and perhaps economics.

 I think we've identified the difference, then. My goal is to have a
population that understands what mathematics is. I think such a population
will also have a positive attitude towards science and so forth, just
like your imaginary population, but my imaginary population will have
the additional advantage of being able to do creative things with math.


Ok, but first let's be clear about what population we are talking
about. I'm talking about the entire population of the US, not the
population of math grad students. I would be pretty confident in
saying that at no time in the past, despite the teaching of four or
five or six ways of 'solving' quadratic equations (metaphor alert),
has any but a tiny percentage of that population been able to do
'creative' things with math. (Probably a very consistent percentage.)
What we have mostly are people who can't do *non*-creative things that
rely on very basic principles, and so they are misinformed about and
alienated from math and science, and subject to manipulation by
businesses and politicians. But you seem to be wiling to discuss some
specifics below, so let's see if that will clarify further.



I don't see that learning to perform algorithms that can be done by a
machine advances my goal. And I don't mean just computation, but
manipulation of symbolic forms. But what you and various others tend to
do is *define* the goal as learning to perform those algorithms. And
then some even set up the rather looney strawman that says attempts to
teach that de-emphasize those algorithms have 'failed' because they
don't produce people who *can* do the algorithms.

So yes, if you don't learn to rearrange symbols according to some rules,
you will not learn to rearrange symbols according to some rules. But if
you have a tool that will do that for you, the symbols will still be
rearranged. Tell me why that is a Bad Thing, *without begging the
question*.


 Math is not rearranging symbols according to rules. That's a logician's
canard; no practicing mathematician believes it. If you think that,
I have to wonder whether you have ever had the experience of understanding
a theorem. Mathematics happens when you finally see why it is that the
antiderivative is the integral, or why the area under the hyperbola 1/x
between x=1 and x=2 is the same as the area between x=2 and x=4, or
how to get the trig identities by multiplying rotation matrices.


My concept of the term "understanding" is as follows:

The first step is the ability to describe something in words. But
since such a description may be memorized, we test for understanding
by requiring the ability to articulate the concept in different ways.
Further, we test for the ability to articulate how the concept might
apply in different situations.

I would also observe that one improves these abilities over time, by
'practicing one's craft'; I do not suggest at all that it isn't
necessary to 'do problems' in order to achieve understanding.

The question at hand has to do with *what* concepts we would like the
general population (excluding math grad students but including large
parts of the science and engineering community as well as 'voters') to
understand, and *what problems they should be doing* to achieve that
understanding.

I see it as a rather stark choice, between (again metaphor alert)
sitting in a classroom doing 5 different ways to solve quadratic
equations, and sitting in a classroom working on examples (using
mathematical tools on computers of course) that demonstrate that
correlation does not imply causality. But maybe that's another
logician's canard, even though I don't consider myself a logician ;-).

Anyway, this always gets too long, so my question is: Why *is* it that
the antiderivative is the integral? How should we understand it,
outside the context of rearranging symbols?

 Getting the symbols rearranged correctly is the booby prize if it
doesn't help you get the big general concepts. It seems like you
agree with the first part of that statement, but I don't see any
evidence that you understand the second part.

No, that's exactly my point. I'm saying that we waste enormous amounts
of time having students *not* get any concept at all, because the very
large majority of their time is spent rearranging symbols, and so
that's what they understand math to be.

-tg




It looks as if you
think that getting the symbols rearranged correctly is all there is
to mathematics.

 Maybe I'm reading you wrong; I confess I haven't read everything
you've written about this. But everything I *have* read from you says
that all you want is the number or the expression at the end of the
calculation.

John


.

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