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Re: [Phys-L] [Norton AntiSpam]Re: Conceptual Physics Course



While the check your work suggestion is very good, the additional proofs
won't work. The suggestion of a proof is exactly the problem I mentioned.
Advanced thinkers come up with connections and sequences that have no
meaning to lower level thinkers. Anyone who has looked at the work of
Redish would be aware that students tend to think of proofs as merely giving
them permission to use the equation.

If they don't understand the basic algebra the idea of an operator such as
// is at a level even way too far above them. This is an experimental
problem. You have to try various strategies and see which ones actually
work. But I would say that anyone with this misconception is missing some
basic concepts. Again, give the Lawson test and note the correlations.
There is no substitute for learning the level your students are at, and in
the end some cognitive enhancement might be the needed method.

Everyone should read "An Inquiry into Science Education, Where the rubber
meets the road" by Steinberg.

John M. Clement
Houston, TX



On 05/17/2012 08:22 AM, 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?

Well, when I was in kindergarten, the emphasized the idea of
"Check Your Work".

In this case, even the simplest check would tell the tale.
I'm pretty sure the reciprocal of (1/x + 1/y) is not *ever*
equal to x+y, not for any real-valued x and y, except in
the trivial case where they are both zero.

Apparently the authors of certain widely-used introductory physics
texts have never heard of "Check Your Work". I say that because
of the astonishing amount of bogus physics in those books ... stuff
that would not withstand even a moment's scrutiny.

Still, we can hold students to a higher standard. We can remind
them to "Check Your Work".


=================

Here's another suggestion, more focused on this particular
case: Tell them to choose y=1 or some other positive constant,
and then plot z = 1/(1/x + 1/y) as a function of x. Also show
z = x and z = x+y on the same plot.

It should be clear that x+y is always larger than x, while
1/(1/x + 1/y) is always smaller (for positive real-valued
x and y). This is something well worth knowing, in the
context of resistors, which presumably have positive
real-valued resistances.

====

As a related theoretical exercise: Define the mathematical
operator "‖" (pronounced "parallel") according to

1
x‖y := ------------
1/x + 1/y

Then ask students: prove or disprove:
a) is ‖ commutative?
b) is ‖ associative?
c) does multiplication distribute over ‖?
d) does ‖ distribute over addition?
e) does addition distribute over ‖?
f) based on the above, would you say that ‖ has the same
operator structure as addition, or multiplication, or
subtraction, or what?

In real-world electrical engineering, the concept of x‖y is
well known and quite useful.

Also this is an example of the spiral approach to learning.
We are spiraling back to the axioms of arithmetic, reinforcing
them and extending them ... making *connections* between old
ideas and new.
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