Re: [Phys-L] proportional reasoning, scaling laws, et cetera

["Robert Cohen" <Robert.Cohen@po-box.esu.edu>]

Jeffrey Schnick wrote:
> On a related note, what do you do to convince a person in a lasting
> manner that the reciprocal of (1/x + 1/y) is not, in general, x+y?

Rauber, Joel wrote:
> x = x_0 + v_0 t + 1/2 a t^2
> is one I think is OK

Philip Keller wrote:
> I've never had a student gush over the algebra that leads to delta X =

> Vo t + 1/2 a t^2
>
> But I have seen many happy faces when we explain that the Vo t is a 
> rectangle and the 1/2 a t^2 is a triangle, and the formula gives the 
> area of the trapezoid on the V vs t graph.

I believe the three notes above all reveal a common problem.

I don't have an answer to Jeffrey's question.  You can try having
students try x=y=2.  They can then see that it doesn't work.  However,
knowing that their "trick" doesn't work does NOT help them solve the
problem.  All it does is tell them that their "trick" doesn't work.
They will still search for an alternate "trick".  Then, they'll likely
forget the alternate trick and end up using the original trick that
doesn't work.

This is because they only know how to use tricks.

I've gone to professional development sessions for K-8 math teachers and
it is easy to see that they are also looking for "tricks".  The clearest
example of this is with dividing fractions.  They teach that one should
"invert and multiply".  This allows students to get the right answer
without understanding what they are doing (i.e., without understanding
proportions and ratios) -- as long as they can remember the "invert and
multiply" trick.  Some teachers are surprised to learn that there are
"other" tricks for dividing fractions, like getting a common denominator
and then cancelling the denominators.  However, they see no use for
those "other" tricks -- why memorize more than one trick when one trick
alone will work?

Joel writes that x = x_0 + v_0 t + 1/2 a t^2 is OK.  I don't think so,
because to students it does not represent any physics.  Instead, it is
an equation that gives them the right answer.  It is no different than
the "invert and multiply" trick -- this equation is simply a "trick"
that gives them the position.  They likely don't even know why it works
or what it has to do with acceleration. [I'm assuming an algebra based
class with students who have poor proportional reasoning skills -- in a
calculus class, the physics is represented by the definitions of
position and velocity, which they can then integrate assuming a constant
acceleration]

You have to keep in mind that a lot of students are looking for ways to
do physics without actually thinking about the physics.  Finding an
equation that "works" is what they are looking for, in the same way that
they are looking for a rule that works to solve the reciprocal of (1/x +
1/y) problem.

It seems to me that these are the same type of people who "want" the x =
x_0 + v_0 t + 1/2 a t^2 equation.  Yes, this is a useful equation if you
are faced with such a problem repeatedly.  However, once they finish
their course with you, what use is that equation?

Finally, I want to comment on Phillip's point that students don't gush
over the algebraic derivation but are happy to see how one can find the
displacement via the "area under a curve".  

The reason they don't gush over the algebraic derivation is because the
derivation has no value to them -- all they need is the result, not the
derivation.  In fact, I think every derivation one does in class is
simply ignored by the students we are talking about (problems with
proportional reasoning, scaling laws, et cetera).

As for the graphing, I admit this likely gives *some* students an
insight into the "physics" they did not have previously.  However, my
guess is that for the students we are talking about, this will simply
give them another "trick" for solving a problem without really
understanding the physics (e.g., what this has to do with acceleration).
After all, displacement is not DEFINED as the area under a velocity vs.
time curve, is it?

Instead of using the graphing or the equation, why not have students
predict the displacement by using the idea that the average velocity
should equal the displacement over time?  Yes, it is more steps than
simply using x = x_0 + v_0 t + 1/2 a t^2 equation (after all, they have
to know what is meant by an average and how an acceleration changes the
velocity over time), and yes students with poor proportional reasoning
skills will struggle mightedly.  But isn't it better than giving them a
"trick" that allows them to solve the problem without really
understanding it?

Robert A. Cohen, Department of Physics, East Stroudsburg University 
570.422.3428   rcohen@esu.edu    http://www.esu.edu/~bbq