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Re: [Phys-L] Conceptual Physics Course

John,

I've interspersed some comments below but first: Thank you so much for a useful and thoughtful answer. Your response shows that:
1. You read the original question carefully so you knew it was about x+y vs. 1/(1/x+1/y) rather than x+y vs. 1/x+1/y.
2. You comprehended the question so you knew it was about convincing people in a lasting manner rather than providing a proof.
3. You gave some thoughtful, thought-provoking, and potentially useful suggestions.
I am truly grateful.


-----Original Message-----
From: phys-l-bounces@mail.phys-l.org [mailto:phys-l-bounces@mail.phys-
l.org] On Behalf Of John Denker
Sent: Thursday, May 17, 2012 4:09 PM
To: Phys-L@Phys-L.org
Subject: Re: [Phys-L] Conceptual Physics Course

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".


This is huge (and the humor in including a typo in the statement is not lost on me). I have included the idea of getting students to check their work in the current goal list. I already had it down as a point of emphasis for the calculus-based introductory course I will also be teaching next fall. I think I have to do more than encourage it, show how useful it is, and set an example. My plan is to assign some questions/problems in which checking whether an answer is reasonable is the only thing that the problem solver needs to do (see: "Find-the-Flaw Problems, The Physics Teacher -- May 2011 -- Volume 49, Issue 5, pp. 277"), and to include such questions/problems on tests. Another option is to require a check to solved (by the student) problems to be worked out and shown, again, on tests. An earlier suggestion you made about creating the final exam now, comes to mind.


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".


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



I think the following suggestions all come under the heading of make a big deal of it and more importantly establish as many connections (as you mention at the bottom of your message) as you can even if you have to add things to connect to, to the tapestry. (I'm using the metaphor used by Feynman of knowledge being a tapestry in which every piece of knowledge is connected locally to everything piece of related knowledge and globally to every piece of knowledge period.)

Your suggestions bring to mind a general solution for a whole class of issues of which the one under consideration is just an example. The solution is to ask the students themselves to come up with the connections (and to facilitate their doing so, ideally, with the need for one to facilitate being less and less as time goes by).


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.


This (just above) suggests a laboratory exercise in which an ohmmeter and a bunch of resistors are used as an analog computer to investigate the properties of the binary operator "||" that you discuss below.

====

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?


Hmmm. Perhaps it would be better to start with a unary "flip" operator where flip applied to x is 1/x.
flip x = 1/x
Then ask, whether flip(x+y) is equal to flip x + flip y.
Then ask about the inverse operator to the flip operator.
etc.


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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