Re: [Phys-l] Equations

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

> >  I also avoid implications of causation.  However, my
> > reasons are not only semantic but also pedagogical.
>
> Excellent.  That's an important and helpful point.
>
> [more good stuff snipped]
>
> > So, yes, students naturally start out with strong conceptions of
> > cause-effect.  It just isn't clear to me that we want to reinforce such
> > conceptions.
>
> Another most excellent point.

Yes, but why is it harmful?  I don't think there is any documented evidence
that it is harmful.

 
> Applying this analogy to our main subject:  Teaching students the
> "advantages" of a=F/m is bad pedagogy.  It is letting the inmates
> run the asylum.  It reinforces misconceptions about the nature of
> equality and the nature of causality.
>
> Sure, in the short run it is _easier_ for all concerned to accommodate
> misconceptions rather than confronting misconceptions.  But in the
> long run there is a terrible price to pay.

I don't think anyone has advocated teaching that a=F/m has any advantages
over F=ma, so this is a red herring.  What has been said is that a=F/m is
more understandable, and establishes a relationship that students can begin
to make sense of.  There have also been comments that it seems to work
better.

As to causality, there are really two definitions.  There is the
mathematical one where the cause must precede the result, then there is  the
commonly used definition.  The commonly used one presumes there are agents
and actions and that an agent causes an action.  There is also bound up in
this definition the idea of intention.  Virtually all students operate with
the common definition.  When you propose equations which violate their
understanding, they will treat it as memorizable things which are not
understandable.  Also, when the students think at a low level, they can not
really understand the mathematical definition of causality.

Now what is the advantage of a=F/m?  It basically makes the relationship
more concrete, so it is a very good way of introducing the law.  Indeed when
one models the law you measure the a and vary either the F or the m.  Then
you find that a is proportional to F and inversely proportional to m hence
a=F/m.  This makes it an experimental problem which students can readily
model in the lab.  Notice that the independent variables are F and m.  Now
most students will not tend to think of m as being a cause, but they will
most certainly think of F as being a cause.

It is very difficult to make up a sentence in which F is not the cause, and
almost impossible to make up one where a is the cause.  An example is the
questions "How do you change velocity?".  the answer is "You push the
object."  So immediately F because causative and a is the reaction.  Whether
we know that this is not true mathematically is totally irrelevant to the
way the language and our thinking tend to work.

In actuality there is some causation here.  The intention to change the
velocity preceded the change.  The initial impression of the force came
before the change.  The continuation of the impressed force is certainly
simultaneous with the desired action.

Is there any evidence for this point of view of pedagogy?  Yes, there is
some support.  The equation i=V/R has been traditionally taught and
understanding has usually been low.  However in the Real Time Physics labs
they found that students gain better understanding if they can see the time
dependence of the current.  They clearly see that the current takes time to
rise, which gives them a causal handle on the effect.  It makes the
situation more concrete, and causal.

Why is the concreteness of the initial exposure important?  Well the
research of Karplus, Renner, Lawson, Shayer, Adey,... have shown that
students must proceed from the concrete to the abstract for better
understanding.  Even more important is the finding that thinking ability
rises when students are exposed to concrete -> abstract sequence.

How important is the specific language you use?  There is little evidence to
show that large gains result from very careful language.  In reality
students do not notice the carefulness until they begin to understand the
concepts.  Most of the studies that I have seen have shown very little gain
from the language used, but there are many that show large gain from
following a closely tailored concrete -> abstract sequence.  There is one
that shows high gains by varying the language.  Lawson essentially gave two
groups of students different readings and got much higher gain from the
experimental group.  But the experimental group got the same text, just
rearranged to follow the learning cycle.  I have looked at studies of
refutational text, and the gain is statistically significant, but low.  I
would point out that Hewett uses refutational text extensively.  Notice,
that I am not absolving us of responsibility for using exact language.

One person proposed that Arons was the guide to good practice.  I would tend
to agree with that up to a point.  Much of his book preceded the latest
research.  His instincts were good, but there are a number of specific
points that are probably not optimal.  He also made some big goofs that were
not obvious at the time.  He proposed that students came into some subjects
tabla rasa.  Now we know that is never true.

So in the end, I propose that insisting on introducing F=ma as NTN2 is not
optimal.  Whether of not you say that force causes acceleration, it is
implied in the minds of many students.  If you try to deny this idea, the
lower students will probably never have a good handle on NTN2, and will in
the end memorize it as a piece of random mathematics.

Is there research to support this particular point of view?  Well this is
how it is done in Modeling, which results in high student gains.  There is
no specific research on this one particular point.

My position is that students have to go through various stages or construct
various models of reality before they come to the most productive one.  The
idea that F causes a is one stage that they will go through, and it is
probably a necessary stage.  So if you actively try to deny this stage, they
will be stuck at the memorize and search for equations stage.  A correlary
is that helping them get to this stage is productive.  But then you can help
them go beyond it.  So just as physics proceeds by building a model which
approximates the data at hand, but later changes the model, so students must
do the same thing.  They can not end up with the correct model all at once.
They need to arrive at it in stages.

Is it really necessary for most physics students to develop an understanding
of mathematical causality?  This is probably only necessary for students who
are going on to higher education in math and science.  There is a point at
which they need to acquire a proficiency in manipulating equations and in
making reverse connections.  The reverse connection is that when you see a
change in velocity you know there was a net force.  This particular
understanding is an example of operational reasoning.  This happens
automatically in formal operational reasoners.

There is a real difficulty with the reverse reasoning, which is also present
in F=ma or even F_net = ma.  Students tend to think that F_net is a real
force, rather than the sum of the forces.  The same happens when you use the
label centripetal force.  They tend to think that circular motion implies
there is a force which is "the" centripetal force.  So one must ask them
what is the centripetal that causes this force.

Finally what works pedagogically is often not what you think works.
Pedagogy is in the end an experimental science, not a piece of mathematical
logic.  It involves equal knowledge of psychology and the subject matter.
The sorts of dialogs that students must engage in are often disgustingly
inaccurate according to out understanding, but it is necessary for them to
make sense of physics.

John M. Clement
Houston, TX