Re: [Phys-L] playing for keeps

[John Denker <jsd@av8n.com>]

On 06/29/2013 02:17 PM, Bruce Sherwood wrote:

> An example of confusion is the frequently seen A = 2pi*R^2.

This provides a delightful illustration of a couple of points 
I've been trying to make.
  -- Check your work.
  -- Look for connections between things.

1a) Write down the formula for the circumference of a circle.
    My guess is C = 2 π r

1b) Write down the formula for the area of a circle.
    My guess is A = π r^2

1c) Check the work!
    Look for connections!
    The area of an annulus had better be 2 π r dr, and 
    if you integrate that you get π r^2.
    Conversely if you differentiate the area you'd better
    get dA = 2 π r dr

Similarly ....

2a) Write down the formula for the area of a sphere.
    My guess is A = 4 π r^2.
    In other words, there are 4 π steradians total.

2b) Write down the formula for the volume of a sphere.
    My guess is V = 4/3 π r^3

2c) Check the work!
    Look for connections!
    There's a connection between the area and the volume,
    and you can use that to check the work.

==========

There are two choices:  Use it or lose it.

A) If you check things like this every time,
  a) you are more likely to get the right answer to today's problem.
  b) in the long run, it reinforces the knowledge of the geometrical 
   formulas, and
  c) it also reinforces the rules for integrating and differentiating.

B) If you don't form the habit of checking things, you might get
 the wrong answer to today's problem, and in the longer run
 everything evaporates ... circumference, area, volume, integral
 and derivative ... everything.

This started out as a discussion of "indicators of quality
teaching".  As I see it, a good education is mostly about
forming good habits, such as the habit of checking the work,
and the habit of looking for connections.  This is important
in the short run, in the long run, and in every other way.

The FCI was designed with certain goals in mind.  Its goals
are not the same as my goals.  It does not try to measure
the things I care about.

On 06/29/2013 05:11 PM, Larry Smith wrote:

> ... spiral ....

> So, obviously physics students should practice what they've learned
> to retain it too, but they've gone on to the next chapter and have to
> learn a LOT of stuff in one year.

I see no point in "learning" stuff that will not be retained.
If the point is not learning but "exposure" of a kind that
is supposed to make it easier to actually learn something
during later turns of the spiral, somebody should please
please please come up with a test that measures that.  In
the absence of such a measurement, we are all groping in the
dark.

I'm willing to believe that some subconscious subliminal
"groundwork / exposure" learning is retained, but if so, 
it seems to be exceedingly inefficient, unsystematic, and 
irreproducible.

In particular, it's rare to see a textbook that even pretends
to do this.  When was the last time you saw a textbook say
"we don't expect you to understand this now, but you will
understand it later, when we revisit the subject two years
from now."

I am skeptical of this "groundwork / exposure" argument, but 
I am willing to listen.  I would like to hear the argument
made in much more detail.  For example, pick a widely-used
textbook and identify some section that you expect will not
be remembered, but should be covered anyway.  Explain why
that makes sense.

I am particularly skeptical that it pays to teach a detailed
picture that will not be remembered, as opposed to a less-
detailed picture that will be remembered.  I know there are
enormous pressures in the direction of making education
a mile wide and an inch deep, but I say these pressures
must be resisted.  The time would be much better spent
covering a small number of important principles, covering
them and connecting them and reinforcing them so that
they stand a chance of actually being remembered.

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

On 06/29/2013 03:30 PM, Ludwik Kowalski wrote:

> 2*Pi*r^1 length ( 1 dimensional object)      [1]
> Pi*r^2 area ( 2 dimensional object)          [2]
> (4/3)*Pi*r^3 volume (3 dimensional)          [3]

That's not the logical progression.

It would be much more logical to look at the *interior* of
the generic N-dimensional object, i.e. everything within
a distance r of the center in N dimensions.  This category
of object is called a /hypersphere/.  (Line [1] above does
not fit the pattern, because it is the boundary of a 2D
object, not the interior of a 1D object.)

In 1 dimension, the set of all things within a distance r of
the center is an interval of length 2 r.

In two dimensions, we have a disk (the interior of a circle)
with area π r^2

In three dimensions, we have a ball (the interior of a sphere)
of volume 4/3 π r^3.

In four dimensions, we have a hyperball with content 1/2 π^2 r^4.

In d dimensions, the general expression is

      π^(d/2)
    ---------- r^d
      (d/2)!

Note that it picks up a new factor of π every /second/ dimension.
Note that (0.5)! is √π and all the other half-integer factorials
can be found using the usual recurrence relation.