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Re: [Phys-l] What comes first, the equation or the explanation?

I like to raise the question of "the unreasonable effectiveness of mathematics" in math methods courses, and in more advanced courses wherein one can show how ideas are so usefully generalized. Even an advanced course benefits greatly from progressing from counting things, through proposing operational inverses, closure under various operations, and so on, and then up through the necessities for imaginary and complex numbers, and into the dizzying heights of exterior forms, coordinate transformations (and why coordinates are so useful much of the time, and it's so useful to do away with them other times), and the like.


Though students very often don't have time (what an unfortunate phrase, and I feel a bit sick even thinking it!) to think about such out-of-the-usual-courseware questions, I do like to raise the issue from the viewpoint of Wigner's famous essay. ("The Unreasonable Effectiveness of Mathematics in the
Natural Sciences," in Communications in Pure and Applied Mathematics,
vol. 13, No. I (February 1960). New York: John Wiley & Sons, Inc.
Copyright © 1960 by John Wiley & Sons, Inc.)
   Students who are inclined to a bit of deep thinking seem to get the entire gist from the title of the essay itself (one which is, to my mind, perhaps one of the most provocative ever), and several have evinced quite some pleasure at tackling the question on their own. It's one of those issues which seems to stick fast in a lot of people's brains, once it's formalized in such a way that it seems approachable. 







________________________________
From: Anthony Lapinski <Anthony_Lapinski@pds.org>
To: phys-l@carnot.physics.buffalo.edu
Sent: Thursday, December 22, 2011 8:28 AM
Subject: Re: [Phys-l] What comes first, the equation or the explanation?

I read a quote somewhere (about calculus?), semi-related to this
discussion:

Is mathematics invented or discovered?



Forum for Physics Educators <phys-l@carnot.physics.buffalo.edu> writes:
That is certainly not how the original science was done, which leads to
the question of whether the student learning new science should be
treated like the scientist learning new science.  That is slightly
different philosophy of learning.

joe

On Dec 21, 2011, at 8:44 PM, ludwik kowalski wrote:

On the other hand, as John D. emphasized several years ago--I am 
nearly certain it was him--one proceeds from what is known to what is 
unknown. This implies linearity; topics are not chosen randomly. The 
sequence of topics is essentially the same in all textbooks.

The way of teaching evolves; it reflects experience of generations of 
teachers. Back and forth changes, in the forms of interacting with 
students--qualitative, quantitative, labs, homework, lectures, 
demonstrations, etc.--were, and still are, very common. Teachers try 
to accomplish two things, make sure that interactions are as effective 
as possible, and that learning is a pleasure-generating activity. That 
was my "philosophy of teaching."

Ludwik
=====================================

On Dec 21, 2011, at 5:29 PM, John Denker wrote:

On 12/21/2011 03:05 PM, Peter Schoch asked:
"Professor, when we ask you a question, do you see the formulas first
and then create the explanation or get the explanation first and then
see the related formulas?"

It's a very good question, but IMHO the answer is "neither of
the above".

The human mind does a *lot* of parallel processing.  There are
something like 10^11 neurons and 10^14 or 10^15 synapses, all
working at once.  The explanation(s) and the equation(s) get
recalled all at once, in parallel.

When speaking (or writing), we need to present things serially,
not in parallel, which is a royal pain, because it (a) requires
the speaker to serialize a set of ideas that is highly inter-
connected, with no natural order, and (b) requires the hearer
to de-serialize the ideas, i.e. to reconnect them.

Again:  Serial communication gives a /misleading/ impression of
the structure of the ideas, and a /misleading/ impression of how
thinking in general (and recall in particular) get done in the
brain.

So: to answer the original question:  When somebody asks a
question, I get to choose, based on what I think will be
most helpful.  I have both the equations and the pictures
available, and I can talk about them in whatever order I
choose.

When learning (which includes book-learning as well as original
cutting-edge research) the conceptual understanding comes before
*and* after the equations, iteratively.  Some level of understanding
is needed in order to know what calculations to do.  The results
of the calculation gives a better conceptual understanding, which
motivates more and better calculations, and so on, iteratively.

When talking about this, I do the itsy-bitsy-spider thing with my
hands:
  http://www.youtube.com/watch?v=eiWbjoOOly4&t=0m55s
_______________________________________________
Forum for Physics Educators
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=======================================================================
Ludwik Kowalski, whose profile is at:

http://pages.csam.montclair.edu/~kowalski/my_profile.html






_______________________________________________
Forum for Physics Educators
Phys-l@carnot.physics.buffalo.edu
https://carnot.physics.buffalo.edu/mailman/listinfo/phys-l

Joseph J. Bellina, Jr. Ph.D.
Retired Professor of Physics
Co-Director
Northern Indiana Science, Mathematics, and Engineering Collaborative
574-276-8294
inquirybellina@comcast.net




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Forum for Physics Educators
Phys-l@carnot.physics.buffalo.edu
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_______________________________________________
Forum for Physics Educators
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