[Phys-l] Math Teaching (Was: Equations)

["John Clement" <clement@hal-pc.org>]

The teaching of HS math relies heavily on teaching by rote rules of
operations and how to solve equations.  The math teachers think they are
teaching students how to think, but this is not confirmed by results from
Piagetian pre and posttesting.

One of the greatest weaknesses is that they are trying to push abstract
material onto students who are at a low thinking level, and as a result can
not handle it.  Of course the same thing is happening when physics is taught
in the traditional fashion.

I am always asking students whether they learned to write equations.
Apparently this is not done in math.  So if they can't write equations why
should they be expected to learn how to solve equations?  One is seldom
given an equation to just solve.  You have to write the equation that
represents the physical situation.

Once students can handle abstract equations, then the very abstract notions
involved in causality may be introduced.  Until that point, I believe that
one must present many concepts in a causal fashion.  This is based on the
notion that students must make connections using the current level of
reasoning that they are capable of.  Even many intro. physics students can
not think at a high enough level for the traditional abstract method of
teaching physics.

One proof of this last assertion can be found in the studies which showed
that elementary teachers taught by PER methods can out perform engineers
taught traditionally in problem solving.  The one thing that has not yet
been demonstrated by PER methods is a high gain on a Piagetian test of
scientific thinking skills.

I find some of the bickering over whether to use causality in teaching a bit
silly when physicists can not even agree whether NTN2 is a causal
relationship.  While there is no proof that explicitly invoking causality
improves student understanding, the PER methods do tend to promote
understanding in terms of a causal relationship.  I would recommend that
anyone who wishes to comment on pedagogy should read the many fine PER and
JRST papers that are available on the subject.

John M. Clement
Houston, TX

>  	Good question, because it focuses on a major weakness in the
> teaching of math in this country.
>  	What is algebra?  It is a formal, systematic method of reasoning.
> You start with a relationship (F=ma, or its differential, vector
> equivalent) and deduce consequences of that relationship.  Using algebra,
> and obeying its rules, guarantees that the result is logically equivalent
> to the starting point.  In this way you discover that a huge class of
> physical phenomena is consistent with that relationship.  It is not that
> nature is obeying the rules of algebra, it is that your conclusions are
> correctly obtained if you correctly apply the rules of algebra.
>  	Sometimes, of course, you reason correctly and obtain a result
> that is different from the physical result.  When that exciting event
> occurs you are guaranteed that your starting principle was incorrect.
>  	High school algebra teachers, many of them, have never learned
> that algebra is nothing but a system for reasoning.