Re: Simple explanations. Was: what are the labs for?

["Dewey Dykstra, Jr." <dykstrad@bsumail.idbsu.edu>]

Donald E. Simanek wrote:

> On Tue, 10 Feb 1998, Dewey Dykstra, Jr. wrote:
>
> > What about good "explanations" which do not involve formal mathematics?
> > What about mechanical models?  Sure lots of models can be described with
> > formal mathematics, but can't models be thought of as physical or
> > mechanical ones by someone without having to resort to formal mathematics?
> ...
> > But, what about mechanical models?  Don't they "work".
>
> I suppose it depends. Perhaps you can suggest some particular ones you
> think work very well, and we can discuss them.

I guess this point it is important not to get too distracted by the details
of any particular model because such a distraction allows us to avoid the
original issue.  But, one such model that comes to mind is one my students
routinely construct of how a converging lens "assembles" a real image from
many pin hole images which it receives, one at each point on the face of
the lens.  This is in the context of thinking of light as rays.  By
mechanical model I do not feel restricted to a physical object, but to
something that in principle a physical representation of the model could be
assembled from physical objects; in this case lines on paper or strings or
threads in the air.  One might argue that the ray model is mathematical,
but I think here the issue is formal mathematics and after all a common
claim is that _everything_ can be represented mathematically.  At any rate
a remarkable amount of what can be seen from on focus to very far from good
focus can be explained via the model my students build and _they_ are not
using any formal mathematics.

> Any model is an analogy,
> and all anaolgies break down at some point. The beauty of mathematical
> models is that when they break down, you can fix them, or invent new
> mathematics for the purpose.

Sure and the model I describe above is subject to the same features.

> The problem with mechanical models is that you first must understand
> mechanics. The problem with visual models is that some things can't be
> visualized. Too often we use them as crutches to avoid sweating the
> details. At some point we must "throw away our crutches" to achieve fully
> adequate and powerful descriptions of nature.

I certainly am not trying to say that some alternative can be the
be-all-end-all tool for model building.  But, it is the case that in any
model I understand or know about, regardless of the amount of mathematics
involved, _meaning_ lies as a foundation for that model and _meaning_ is
not mathematics.  Starting with the _meanings_ students come to us with,
they can be induced to construct new ones, often involving the sort of
mental models I've described above.  With this newly constructed meaning
and models they are able to give explanations and make predictions which
students who only know _our_ definitions of terms (real, virtual,
converging, concave, focal point, etc.) and the thin lens equations don't
even begin to consider.

It strikes me that our profession (anyone who teaches physics topics in
grades 1 - 16) makes a big deal out of the mathematics, but frequently in
"giving the mathematics" gives only a ghost and certainly little
understanding to most of the students we teach.  One might take issue with
my inclusion of such a wide range of teachers, but "they" teach way more
students that "we" do.  If we don't take responsibility, who will?  Have
those to whom we have left the responsibility so far done as good a job as
we could if we put our minds to it?

I think Bill Beatty is trying.  Thanks Bill.

Dewey



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Dewey I. Dykstra, Jr.                   Phone: (208)385-3105
Professor of Physics                    Dept:  (208)385-3775
Department of Physics/MCF421/418        Fax: (208)385-4330
Boise State University                  dykstrad@bsumail.idbsu.edu
1910 University Drive                   Boise Highlanders
Boise, ID 83725-1570                       novice piper

"Physical concepts are the free creations of the human mind and
are not, however it may seem, uniquely determined by the external
world."--A. Einstein in The Evolution of Physics with L. Infeld,
1938.
"Every [person's] world picture is and always remains a construct
of [their] mind and cannot be proved to have any other existence."
--E. Schrodinger in Mind and Matter, 1958.
"Don't mistake your watermelon for the universe."  --K. Amdahl in
There Are No Electrons, 1991.
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