Re: [Phys-l] Equations

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

While the comments about causality and equivalence are certainly true, the
pedagogical analysis is off base.

Students at the lower level and beginning students will be completely
overloaded by considerations of causality.  Students must start from what
they already know and then work towards higher levels, not the other way
around.  This has been proven by experiments by Lawson, Karplus, Renner...
Students must first see force as causative and acceleration as a result.
This gives them a familiar context to begin with.

Also this is in line with the way you do an experiment to demonstrate
constant acceleration.  Students first see an experiment with constant
force, and measure the acceleration.  So a is the dependent variable and F
the independent.  The a=F/m is precisely the way many of the reformed
curricula formulate NTN2, and they demonstrably get better understanding of
physics concepts.

Until students are completely comfortable with the idea of variables, the
idea of an equation as just being a relationship is way too abstract.  They
first have to see an equation as a formula.  Once students have gotten to
this level, which is probably the theoretical level as defined by Anton
Lawson, they can be exposed to the idea of causality.

You do not have to teach a wrong concept about the mathematics, students
naturally start with a wrong concept.  This is actually part of the natural
development of a more productive concept.  The recent Physics Teacher has a
very interesting article which bears on this point.  Redish has an article
where he shows that even when students are using unproductive ideas, they
can actually be making progress toward more expert problem solving.  Part of
the difficulty is that when you propose this abstract concept, you are not
aware that students have not internalized the many concepts that are
required to come to this concept.

John M. Clement
Houston, TX
> > ....  I
> > always write it a = F/m and I tell students that writing it this way
> > ought to help them grasp it better than writing F = ma.
>
> This is unhelpful for several reasons.
>
> First, it needlessly blurs the distinction between equality and causality,
> i.e. between equation and causation.
>   -- Equality is (by definition!) an equivalence relation.  As such, it is
>    reflexive, symmetric, and transitive.  Reasonable definitions and
> discussion
>    can be found at
>      http://en.wikipedia.org/wiki/Equivalence_relation
>    In particular, "symmetric" means that if A = B, then B = A.
>   -- Causality is (by any halfway reasonable definition) not symmetric.  A
> ==> B
>    does *not* imply that B ==> A.
>
> Equations is an important idea.  Cause-and-effect is an important idea.
> These are not the same idea!  Absolutely, positively not the same.
>
> It is a tremendous disservice to the students to teach them to confuse
> causation using equation.  They will develop wrong ideas either about
> causality or about equality.