[Phys-l] crutches versus shoes

[John Denker <jsd@av8n.com>]

First: A parable:  Once upon a time there was a student
with a broken leg.  We gave the student a crutch
 -- *after* we took direct action to treat the underlying
  problem (i.e. after setting the break and applying a cast)
 -- *after* we carefully briefed the student on the correct
  usage and limitations of the crutch.

On the other hand, crutches can actually /cause/ secondary
injuries, especially if overused or abused.  And even in
the best of cases the crutch gets in the way, obstructing
development of normal performance.

So ... there are upsides and downsides to crutches.  We
should not over-react to the upsides or the downsides.

If you see somebody using a crutch that is not really needed, 
it is a good idea to wean them off the crutch, sooner rather 
than later.

It must be emphasized that we are dealing with shades of gray.

To put things in perspective, consider shoes for example.  Like 
anything else, shoes can be used properly or improperly:
 -- You probably don't want to wear shoes while playing water polo.
 -- Industrial safety shoes are different from track shoes. 
 -- et cetera
Still, all in all, shoes are considered normal attire.  Shoes
are not in the same category as crutches.

As another example, consider a bicycle.  A good bicycle costs
more than a good pair of shoes.  It takes some skill to ride a 
bicycle at all.  A bicycle used improperly can be very dangerous.
On the other hand, the upside is tremendous.  The bicycle offers
many advantages to a normal person (not just an injured person).
Bicycling skills will not interfere with development of normal
skills in other areas.  Bicycles are definitely not in the same
category as crutches.

On 04/09/2011 08:19 AM, Bernard Cleyet wrote:
> I have had many students approach me this year showing me the
> triangle and claiming "here's an easier way to solve the problems!"

> Do any of you use this method with your students? Have you found it
> useful? Do you have any tips to make it more meaningful and
> universal? Do you see it as a hindrance?

The triangle is a crutch.  It may be possible to find some case
where it is beneficial to some handicapped student, but it is
not helpful to normal students.  It interferes with development 
of normal skills.

Just so everybody knows what we are talking about, here is the
infamous "density triangle"

                         / \
                        /   \
                       /  M  \
                      /-------\                    [1]
                     /  D | V  \
                    /_____|_____\

where M = mass, D = density, and V = volume.

My point is that there is nothing you can do with this triangle
that you cannot do better (and just as easily) by erasing the
triangle and writing a proper equation:

                          M 
                       -------  = 1                [2]
                        D   V

There is no upside to scheme [1], and one downside is that it does 
not generalize to the case where there a multiple variables on each
side of the equation ... in contrast to equation [2], which does
generalize.  For example:

                        P V
                     --------- = 1                 [3]
                       N  kT

Another downside is that scheme [1] does not generalize to any
relationship other than simple proportionality ... whereas equation 
[2] does generalize.  For example:
                         y
                     ---------- = 1                [4]
                       mx + b

Yet another downside to scheme [1] is that it operates strictly at 
the rote level, whereas in equation [2] the key operation (cross
multiplication) can easily be explained in terms of the axioms 
of arithmetic.

Bottom line:  the "triangle" is a crutch of the worst sort.  For
normal students, the downsides far outweigh the upsides.

> Do you have any tips to make it more meaningful and
> universal? 

Replace every instance of the triangle [1] with the corresponding 
equation [2].