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Re: [Phys-l] What are your answers for this teacher?



Simple algebraic manipulation is truly understandable if you have
proportional reasoning. According to Shayer and Adey, 3 variable equations
are not understandable unless you have proportional reasoning. Students can
get by learning the rules, but deeper understanding will be lacking.
Basically the equations covered by the triangle rule are 3 variable
equations. Of course it does not work with 4 variable equations. It does
not work or if they are given a=F/m.

So getting students to think at higher levels is the ultimate key. The
triangle rule is just a notch below in desirability getting them to memorize
the algebraic manipulations to isolate a variable. Both can be just
mindless manipulation.

My favorite example of how lack of proportional reasoning makes physics hard
is the hi/ho=di/do equation for magnificaiton in optics. The students are
just looking at similar triangles and using proportions. But I can remember
how there were students who seriously could not remember this equation so
they memorized hihodido as the jingle for the equation. I can remember how
I thought that was stupid. Now I know that the students who did this lacked
proportional reasoning, so this was their only way to remember such
equations.

Algebra to most of the students is just a memorized set of rules, some of
which make absolutely no sense. Similarly division by fractions is a
memorized rule, so even college students when asked to divide 1/2 by 2 will
often get 1 as the answer. I have seen this too many times to consider it a
random error. They don't visualize than think about what is going on. But
once you have the higer reasoning skills such as proportional, two variable,
and compensation reasoning algebra is often vary obvious.

So again, what you do to keep your job may be very different from what you
do to really help the students. If you don't have the time to improve the
thinking, you may have to resort to cheating by pumping the students up with
rules that they will temporarily use. My reaction to the triangle rule is
that if the student can explain why it works, I will let them use it. But
any trick which is inexplicable, may not be used.

John M. Clement
Houston, TX


So are you saying that one cannot help students understand
this simple
algebraic process in, say, half a year's time, so they can do
well on
an exam? While proportional reasoning might take longer than
this, the
process underlying the triangle does not.