Re: [Phys-L] checking the work for minus signs ... et cetera

[John Denker <jsd@av8n.com>]

On 05/11/2015 12:06 PM, Bill Nettles wrote:

> When I find someone who is a beginning teacher of algebra in
> HS/MS/JHS I tell them "It's a minus sign." I then explain they should
> teach their students to examine their own work for sign errors when
> they can't get a problem correct.  After a semester, the teacher will
> tell me I gave them good advice.

OK, let's take the next step down that road:  As 
the calculations get more complicated, there are 
more things that can go wrong.
 -- missing minus signs
 -- missing factors of 2
 -- missing factors of π
 -- bad unit conversions
 -- et cetera

They're all trivial ... but any one of them will
ruin the calculation.  In a multi-step calculation,
the chance of getting the correct final answer is 
an exponentially-decreasing function of the number
of steps ... so it is essential that each step be 
super-reliable.

Most physicists are very, very good at keeping
track of all the minus signs and factors of two
... but some are not.  I am definitely in the
latter category.  I take some comfort in the fact
that the smartest guy I know has an even worse
case of this disease.

Constructive suggestion:  There exist computer
algebra systems!  These are great for checking
the work.  It is essential to realize that the
computer will *not* do the work for you.  It has
no insight.  Also, it doesn't know the goal, and
there is no good way to tell it the goal.  However,
if you do the calculation, you can type in your
equations at each step along the way, and the 
computer will instantly catch math errors.  It 
won't catch conceptual errors, but it is great
for keeping track of units, minus signs, factors 
of 2, factors of π, et cetera.

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

Tangentially related idea:  Checking the work is
a Big Deal.

The obvious way to check the work is to redo the
calculation, checking and re-checking each step
along the way.

The less-obvious way is to do a completely different
calculation that gets the same answer.  This has a
tremendous advantage, because it can catch conceptual
errors, not just arithmetic errors.  (The downside
is that if there is an arithmetic error, it might
not tell you exactly where to find it.)

Simple example:  Check by doing the inverse problem.
Given the task 10 - 7 = ____, the answer plus 7 had 
better equal 10.

Intermediate example:  If you use trig functions to
find the sides of a triangle, the answer had better
uphold the Pythagorean theorem and other trig identities.
Similarly, any physics answer had better uphold the
various conservation laws.

Fancy example:  Given the task of estimating (closed
book) how much Mississippi River water flows past 
New Orleans in a year, there are at least two
completely different ways of approaching the problem,
leading to completely independent estimates.

For students who have never seen the problem before,
finding one solution is hard ... and finding two
independent solutions is even harder, much worse
than twice as hard.