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[Phys-L] a matrix of goals



In the recent thread on "proportional reasoning",
on 05/21/2012 03:09 AM, Anthony Lapinski cited ten "major" goals
motion ... projectile motion ... forces ... circular motion; gravitation ... energy
momentum ... heat ... electricity ... sound ... optics

The remarkable thing is that "proportional reasoning" is not on
the list ... even though it was the nominal subject of the thread.

Here is a partially-baked idea that helps me organize some of the
various good ideas that have been discussed over the last couple
of weeks: I suggest that we should think in terms of a /matrix/
in which the ten topics mentioned above are the columns, whereas
proportional reasoning is one of the rows.

*) For example, we encounter proportional reasoning in the "motion"
column, because force is proportional to accelerations (and vice
versa).

*) We again encounter proportional reasoning in the "electricity"
column, since voltage is proportional to current, at constant
resistance.

Actually I would prefer to call this row "scaling" rather than
"proportional reasoning" because scaling is the more general term:

*) Current scales /inversely/ as resistance, at constant voltage.

*) The period of a pendulum is not related to its length by a
proportionality relationship, but still there is an elegant and
useful scaling relationship.

Scaling is a super-important part of physics, and has been since
Day One of modern science.
http://www.av8n.com/physics/scaling.htm
I reckon proportionality could be a subtopic, organized under the
scaling topic.



So .... The question arises, what are some of the other rows?

I suggest that /conservation/ makes a good row. We encounter this
in the energy column, in the momentum column, in the circular motion
column, in the electricity column, and again (with a twist) in the
heat column.

Also /modeling/ might make a nice row. By that I mean simply
making quantitative models and checking them against real data.
This is pretty much what science is all about, and has been since
Day One of modern science. We get to make a check-mark at the
intersection of this row with each and every column.

Also /metacognition/ might make a nice row. This makes contact with
another of the recent threads. Feynman famously used almost every
physics topic as a pretext to say something about metacognition:
"Here's a good way to think about this ...." Why should we not do
the same?

Also simple /special functions/ such as sine, cosine, exponential,
and logarithm.

Also simple /functions of more than one variable/. such as PV=NkT or
resistors in parallel. Note that some of the parallel resistor behavior
can be understood in terms of scaling, but there is also some non-scaling
behavior.

Just getting students to the point where they don't panic when they
see a function of more than one variable counts as some sort of
accomplishment.

And then there are even deeper principles, such as /check your work/.
Just saying /check your work/ is already slightly useful, and it
becomes more useful if is spelled out, including
*) Check the dimensions. More generally, check the scaling.
*) Check the typical cases, then check the corner cases.
*) Structure your calculations /from the beginning/ so as to
facilitate later checking. In particular, if you want to
substitute equation "1" into equation "2", don't do it in
place. Copy the equation before doing the substitution.
This makes more work during the first pass, but it makes
less work during subsequent checking passes, and less work
overall. Paper is cheap.
*) Ditto for software and for physical constructions.
Design for testability.
*) When doing multi-step calculations, be tidy. Even with
something as simple as long division, be fastidious about
keeping the columns lined up. Graph paper is cheap.
*) et cetera.

Another row might be the principle of /mathematical induction/
and the related notions of /iteration/ and /recursion/ aka
"reducing it to the problem already solved."

For example, in chess, in the endgame, there is a technique for
winning given a king and a rook against a king. Somebody showed
it to me once. I don't remember the details, but I could figure
it out in less time than it takes to talk about it, because I
remember the general idea.

As a more physics-relevant example, once upon a time somebody
spent about five seconds telling me how an R/2R ladder works.
If I ever forgot, I could figure it out again in about a
femtosecond, because I remember about Thévenin equivalents
and I remember about recursion.

Then there is always the general principle of /figure it out/.
There are a lot of things you don't know explicitly off the
top of your head, but which you could figure out if you thought
about it for a while.

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


Einstein famously said "An education is what remains after you
have forgotten everything you learned in school."

It might be better to say that the stuff you actually learned
was different from the stuff you thought you were learning.

The rows are not more important than the columns; indeed the
rows can't exist without at least some columns. However,
IMHO the rows need more emphasis than they usually get nowadays
... explicit systematic emphasis.

It is typical to teach things in column-major order. This implies
a spiral approach to the row-topics. Every time you go down a new
column you revisit all the rows. That's not terrible, but it's
not sufficient. We cannot expect students to learn about the
rows "by osmosis".

In Manhattan, you need the streets as well as the avenues. It
strikes me as odd that the typical course-catalog describes the
columns but all-too-often says nothing about the rows. Maybe
they think this stuff "goes without saying" but in my experience
this stuff does *not* go without saying.

When designing the course, it might be good to schedule some
"row time" to go along with the "column time".

I had the sort of education that most people literally cannot
imagine ... and the professors were not shy about talking about
the rows:
-- Figure it out.
-- The same equations have the same solutions.
-- Lookie here, we have conservation of energy and conservation
of momentum, which from a spacetime point of view are the same
thing, and there's also angular momentum and charge and baryon
number and various lepton numbers ... all of which are best
understood in terms of the continuity of world-lines.
-- There are ladders in the game of go, and ladders in electronics,
and iteration and recursion in computer programs. You can prove
stability using loop invariants which are sorta like Lyapunov
functions .......
-- Paper is cheap.
-- Check the typical cases, then check the corner cases.
-- Design your work from the get-go with checkability in mind.
-- Never put yourself in a situation where one mistake is
catastrophic.
-- The most important step toward eliminating mistakes is to
remember that mistakes are possible.
++ etc. etc. etc.

This stuff is cloyingly obvious to us, but it isn't obvious to
ordinary students.

The longer-term goal is to make this a metacognition issue unto
itself. That is, we want the students to realize that the rows
exist. We want them to internalize the idea that they need to
explore the rows, routinely, even when they haven't been explicitly
instructed to do so. They should be able to name the rows and
discuss the rows.

=====

The idea of a matrix is a simplification. It talks about connections
in two dimensions, when in fact the structure has tens or hundreds
of dimensions. Still, talking about row-to-row connections in two
dimensions is waaay better than not talking about them at all.