Re: serializing the facts

[John Denker <jsd@MONMOUTH.COM>]

At 09:06 AM 8/22/00 -0500, Glenn A. Carlson wrote:

> As I said, "there is a great flexibility in specific topics to cover
> within each level."

Understood.

> while one can
> simultaneously work on different parts of the walls, one does not
> build the roof before the walls, and one does not put in the
> foundation after the roof.

Partly we agree;  partly this a question of emphasis:
  * is the glass 10% full, or
  * is the glass 90% empty?

We agree that some natural ordering relationships exist.  I find them to be:
 -- helpful when they exist
 -- relatively few
 -- relatively obvious
 -- not deserving of emphasis.

I find it much more illuminating to emphasize the hard work that is
required when there is a lack of a natural ordering;  this can be due to a
lack of pairwise ordering requirements, or (more commonly) due to
conflicting pairwise ordering requirements.

As an example, conflicting requirements manifest themselves in the endless
"physics first" discussion on this list.  The discussion is guaranteed to
be endless, because there cannot possibly be a good answer.  There are good
reasons to put physics before X, Y, and Z.  There are also good reasons to
put physics after X, Y, and Z.

Another type of conflict is the arch-builder's dilemma:  You know that the
finished arch is stable.  You know that the higher blocks rest on the lower
blocks.  But each block is also held in place by higher blocks in a
nontrivial way.  In many cases, if you just add a block in the "natural
order" it will fall off before you can get any additional blocks in
place.  Real arch-builders use a lot of falsework.
  http://www.m-w.com/cgi-bin/dictionary?va=falsework

Now imagine a very high-dimensional generalization of the arch.  Each
"block" of learning is slippery;  it needs to be held in place by others
(above, below, north, south, east, and west, plus numerous hyper-dimensions).

The teaching process is often like building a hyper-arch.  William James
and others have discussed how the persistence (not to mention the
usefulness) of a memory depends on the number of connections it makes with
other memories.
  http://mfp.es.emory.edu/tt12.html

The architecture of science is vastly more complicated than the
architecture of a gothic cathedral.  Conflicts are everywhere:
  -- Experiment motivates theory, but theory gives meaning to
     experiment... So which comes first?
  -- Differentiation sheds light on integration, and vice versa...
     So which comes first?
  -- Does ma cause F, or does F/m cause a?  Which comes first?
  -- Do you start with a survey of the entire field, then add rigor
     in a later phase... or are you rigorous starting from day 1?
     How much falsework is necessary for student A?  For student B?

Foundation, then walls, then roof....  That's fine as far as it goes, but
it's far from being the whole story.