Re: contribution of mathematics...

["Donald E. Simanek" <dsimanek@eagle.lhup.edu>]

Jerome's very comprehensive analysis of the problems of math preparation
is the best I've seen in a long while, and addresses the real issues
which, I think, many teachers ignore or dismiss. I'll take off on just a
few points.

On Mon, 27 Apr 1998, Jerome Epstein wrote:

> The antithesis between physics concepts and mathematics is indeed a
> straw man as others have pointed out. The real questions are these:

In my view concepts and math complement and strengthen each other. The
problem is that students come out of math courses with no grasp of math
concepts, even if they have acquired some math "skills" (blind
number-crunching and plug-and-solve). 

> (calculus often masks real lack of understanding by getting answers too
> easily) 

A very good point. This is why calculus was developed and why it is used
in physics: because things go *easier* with calculus. It can shield us
from a lot of mathematical drudgery, but it can hide concepts if it is
done blindly. If calculus is learned properly, it adds a rich collection
of powerful concepts which can strengthen physics concepts.

Any mathematics (even algebra) can mask lack of understanding if used
blindly. But to avoid math in physics means that every new physical
situation requires the student to re-invent the wheel, using tools
inadequate to the task. The student is denied the sense of unity and
relatedness of one thing to another which mathematics can provide.

I think that many high school teachers advocate the minimal math physics
courses because the teachers themselves never achieved a strong conceptual
and technical proficiency in math, and never appreciated physics on that
level. They were products of their own mis-education and now are
perpetrating it to another generation. Thus they are poorly equipped to
discern what is an appropriate level of math in the physics courses, and
how best to present it to students. They fall back on one of two methods:
(1) strip most math from the course, or (2) encourage blind application of
equations, using plug-and-chug and number crunching. Either way the
students fall back on rote memorization of facts and procedures to get
through, learning very little physics in the process.

Another tactic teachers in all disciplines use to ensure that their course
"looks good" is to give exams that demand so little understanding that
even the half-awake students pass. I've even seen "conceptual" exams, with
essay questions (such exams always "look good"), in which answers were
scored "right" if they had a few key words strung together, even if the
logical connections were totally confused or outright wrong. I've even
seen partial credit points awarded on problems if the student merely wrote
down an applicable equation, but did nothing with it. The student may have
written down five equations, only one of which was applicable, and still
got half credit. Teachers, look honestly at your own scoring practices. Is
this any way to encourage "thinking" and correct use of concepts? I think
this is an example of what Jerome calls the fraud of education. 

I've often raised this embarrassing question to teachers. Suppose you had
a student with eidetic imagery, who could easily memorize anything: facts,
equations, and procedures, but had no conceptual understanding. What score
could that student get on your exams? If such a student could score a
passing grade on your exams, then something is wrong with your exams. 
Those exams must be based on recall of information and application of rote
procedures. If we really think that concepts are more than facts or
procedures, and that conceptual understanding is what we are after, then
the student who mearly memorizes facts and procedures shouldn't pass. 

Then a follow-up. Suppose you had a student who could memorize all the
verbal discussion of the text, all the textbook and class examples and the
analogies and slogans which are the meat of conceptual physics courses,
but simply couldn't do even simple math correctly. Let's say this student
could memorize the worked examples, and could plug new numbers into
examples and do the arithmetic, but couldn't understand how to do algebra. 
Could this student pass your exams? If so, something is wrong with your
exams. If we really think that learning physics generates the ability to
correctly deal with *new* physical situations as they arise, then we
shouldn't reward the student who can't.

I recall a doctor who complained about lack of understanding shown by his
assistants fresh out of college, saying "They can *only* do what they've
been taught."

Jerome, have you ever administered your math diagnostic exams to teachers
of the sciences and math? What other studies have been done to document
the math competence of teachers in disciplines which use math? That might
be the *real* root of our present problems.

Also, Jerome, can you give us a few concrete examples of questions which
can distinguish conceptual grasp of math vs. blind rote application of
math? We need more examples of this distinction for physics, also.

	-- Donald

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Dr. Donald E. Simanek                            Office: 717-893-2079
Professor of Physics                                FAX: 717-893-2048
Lock Haven University, Lock Haven, PA. 17745
dsimanek@eagle.lhup.edu                 http://www.lhup.edu/~dsimanek
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