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Re: [Phys-l] Equations



Again there are two questions.
1. Is there a cause and effect relationship
2. Should students be led to think there is a cause and effect relationship

The first question is directed at someone who has a mature understanding of
physics and is probably a theoretical thinker.

The second question is pedagogical, and requires an understanding of how
students form ideas. Notice the premise that they form the ideas rather
than we give them the ideas. There has already been testimony that students
tend to immediately understand that i=V/r but not V=ir. There is also the
research of Thornton, Sokoloff, and Laws where they found that when students
look at the current vs t and observe that the current rises with time rather
than instantaneously with the voltage application that they do better on the
evaluations. The evaluations test fairly basic understanding of the
macroscopic aspects of Ohm's law and not the microscopic view.

Then how you present it should depend on how it is modeled. If students are
given resistances with various values, and them measure current as voltage
is varied the model is i=V/r. This is because the voltage was the
independent variable and current the dependent. Of course this immediately
sets up the idea of causality in student minds. If they are given just one
resistance and graph V vs I, then r is the slope of the line and the
equation is V=ir.

Now the idea of causality is not a problem here, because one can easily
demonstrate the current rises gradually. The fact that the equation does
not imply causality is also not a problem. Students should be led to the
idea that physics is a matter of building mental models that reproduce what
we know about the world. These models are never absolute, but can be
modified as needed to explain more complex or novel situations. Indeed most
equations students see never imply causality, even when it is there in a
mathematical sense. So you must first get students to understand a model
using the ideas that they currently have. Then that model can be modified
to fit new facts. At the initial model building causality of the lack of it
can be a serious distracter which interferes with model building.

Books that just give the equation without the model justifying it, are
seriously deficient because they deny the students a mental model to build
understanding. As to which model works better, there is no firm evidence.
There are research based curricula which do it either way. But if it is
just presented as an equation, I would bet that i=V/r is the better
formulation because it implies a model. This formulation may also work
better if NTN2 was introduced as a=F/m. Indeed the best model may be one
which is dependent on the type of student you are interacting with.
Elementary school teachers my respond to a different model than budding
physicists. Engineers, chemists, or biologists may be different from
physicists.

Incidentally the problem of students thinking N3L implies causality is
partly because it is usually stated as being action and reaction. If it is
introduced before the other laws as interaction, the pedagogical problem is
partially solved. When introduced this way, the words action and reaction
need not be used. The other difficulty with these words is that action and
reaction are always associated in students' minds with what happens rather
than with the abstract idea of forces.

John M. Clement
Houston, TX


Here is another example about how I think students view equations.
Consider the equation typically referred to as Ohm's Law. Should we
write it as R = V/I, V = IR, or I = V/R? Note that some textbooks
introduce R = V/I as the definition of resistance.

However, when students first encounter this, what experiences, if any,
have they had? They know that batteries have a particular voltage and
they most likely have been taught to think of this voltage as a type of
pressure. They also know that wires and other electrical devices have
resistance to the flow of current. Therefore, if we present the
equation I = V/R it immediately makes sense to them. They view the
voltage as causing the current, and they view R as restricting the
current, so voltage in the numerator and resistance in the denominator
makes intuitive sense to them.

Therefore I first introduce it as I = V/R so they can see the logic of
it.