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[Phys-L] reasoning versus concepts and/or problem-solving



Hi --

Here's my take on the issue of /reasoning/ ± concepts
± problem-solving. Some constructive suggestions appear
at the end.

By way of analogy, consider Legos, or full-sized
masonry bricks. Any individual brick is not
particularly useful or interesting. Even when you
have a huge pile of bricks, in their disconnected
state they are not very exciting. They become
interesting and useful only when you put them
together, using your imagination, creativity, and
skill.

So it is with physics principles. When principles
per_se are the focus, they are hard to learn and
easy to forget, for the simple reason that they are
boring and useless. They become interesting and
useful only if-and-when they are used to solve important
problems. This generally requires weaving concepts
together into an elaborate /fabric of reasoning/.

Therefore the only thing that makes sense to me is
an integrated approach: reasoning based on concepts
and problem-solving together:
-- Solving made-up problems without reasoning is pointless.
-- Rote-memorizing concepts without reasoning is pointless.
++ Reasoning requires (among other things) both concepts
and problem-solving skills.

Textbook-style equation-hunting is an example (but not
the only example) of problem-solving with little or no
reasoning. Equation-hunting often works all-too-easily
for end-of-chapter exercises. In the real world, it
sometimes works; indeed some of the most famous results
in the history of physics involved an element of equation-
hunting. However, real-world equation-hunting is rare,
because it is orders of magnitude harder than textbook-
style equation-hunting. The list of real-world equations
is so huge that hunting through it is horribly inefficient
and error-prone.

To say the same thing another way: The problem with
mindless equation-hunting isn't the equation-hunting;
it is the mindlessness. The problem with textbook-style
equation-hunting is the ridiculously short list of
candidate equations.

I mention this because it is an article of faith in
the PER community that:
improving conceptual understanding is the most natural place to
start.

I don't know whether to laugh or cry when I see
statements like that. It's not the right answer.
The opposite answer is also wrong. Splitting the
difference is also wrong, because it's not even the
right question. The important thing is /reasoning/.
To support reasoning, it is necessary (but not
sufficient!) to have both conceptual knowledge and
problem-solving skills. If you don't have reasoning,
it doesn't matter whether you emphasize principles
or practice or both or neither.

As a secondary supporting point: Asking about
where to start is also the wrong question. The
only thing that has ever made sense to me is the
spiral approach. After a few turns around the
spiral, nobody knows or cares where it started.
We need an integrated approach, where principles
and problem-solving are tightly coupled, and
neither gets priority over /reasoning/.

The best way to get rid of shallow problem-solving is
to get rid of shallow-problem solving. That is, get
rid of the shallow problems. If the problem really
is shallow, anyone with common sense will solve it
using shallow methods.

Students initially imagine that rote memorization is
easy and reasoning is hard ... but they are mistaken.
It would be more accurate to say that students initially
find memorization to be familiar and reasoning to be
unfamiliar, because that's how they've been trained.
However, a major goal of the course should be to
change that, by giving them experience and expertise
in the art of reasoning.

In the textbook business, there is an ancient (but
not venerable) practice of including "end-of-chapter"
exercises that tightly focus on the material presented
in the chapter. I understand why they do that, but it
strikes me as penny-wise and pound-foolish. It makes
it easy to measure what you've just done ... but hard
to measure what you /should have been doing/. Because
of the tight focus, the questions don't exercise,
encourage, or measure reasoning.

Designing exercises to resist equation-hunting is
necessary but not sufficient; the goal is to promote
reasoning. Designing exercises that require reasoning
is not easy, but it can be done. Here are some examples:
*) One of the first exercises in Chapter 1 of the MTW
_Gravitation_ book asks simply, "Estimate the height of
the tides." The non-savvy student will complain that
there is nothing in Chapter 1 that explains how to do
that. The retort is: "The question did not restrict
you to using the methods of Chapter 1 only ... it asked
you to estimate the height of the tides. Without looking
at the book at all, you should be able to do that, using
high-school physics. So stop equation-hunting and start
reasoning. In this course -- as in real life -- the
rule is to solve the problem using /everything you know/.
In the real world, when your boss or your customer comes
to you with a question, they generally do not provide a
short list of techniques that are guaranteed to solve
the problem."

*) Also: Robert Cohen has a number of questions where equation-
hunting doesn't work. For example, in the chapter on F=ma:

A 1000-kg elevator is initially at rest. The elevator is hanging
from a single cable. If the tension in the cable becomes 8000 N
and maintains that tension for 3 seconds, how fast is the elevator
moving at the end of the 3 seconds?

This violates the unwritten (and very unwise) rule that anything
not mentioned in the question is negligible. The non-savvy
student will neglect gravity, hunt up the F=ma equation, and get
the wrong answer. Equation-hunting cannot be used here, but not
because it is forbidden by policy or by artificial restrictions
à la Heller&Heller. Instead, it cannot be used for the best of
reasons, for natural, real-world, physical reasons. It must not
be used, because it gets the wrong answer.

Again the guiding principle is simple: The course should reflect
the real world to the extent possible. In the real world, when
your boss and/or your customer come to you with a question, you
absolutely must not assume that everything not mentioned in the
question is negligible.

I understand why teachers need /some/ exercises that tightly
focus on the material of the chapter. However, that's only a
small part of a balanced diet. IMHO the other 90% of the effort
should be devoted to /reasoning/ which requires /integrating/
the new information with everything else the student knows. I
realize that the integrative questions are hard to grade, because
when things go wrong it's not immediately obvious what went wrong
or how to fix it ... but still, this is what the job is. Just
because it's hard doesn't mean it's OK to skip it.

Constructive suggestion, for the many people on this list who
write their own handouts and books: Exercises should not be
keyed to a particular chapter -- especially not keyed to a
particular section -- because that invites and rewards
equation-hunting. By "not keyed" I mean two things:

a) At a deep level: Every exercise (with rare exceptions)
should combine ideas from multiple sections and multiple
chapters, along with real-world outside-the-book ideas.

b) At the level of appearances: In rare cases where an
exercise focuses on ideas from a particular section, this
should /not/ be readily apparent to the student, so far
as possible.

Specifically: Assign each exercise an ID number that doesn't
mean anything. Then typeset them in more-or-less random order,
not corresponding in any obvious way to chapters or sections.
Instead, have an index (preferably an online search app) that
implements a one-way mapping from topics to ID numbers.
Rationale:
++ Teachers and diligent students need to be able to find
exercises that apply to a particular chapter or section.
-- OTOH they don't need the reverse mapping, which just invites
equation hunting.
-- Assuming that each exercise combines ideas from multiple
sections, arranging them by chapter and verse is impossible
anyway.

The /author/ needs the reverse mapping, for checking that
each section has adequate coverage, but this ought not be
published. The mappings can be implemented using a /tag/
system (rather like the github tracker). Each question gets
multiple tags, indicating which sections and which ideas it
exercises.

Troublemakers can uncover the reverse mapping if they work
hard enough; it's not cryptologically secure. However, we
don't need to make this easy. We need to make it obvious
to students that using the reverse mapping is cheating.

There needs to be a way to search for exercises that combine
ideas from a certain chapter and /earlier/ chapters only.
Otherwise searches will turn up exercises that drag in ideas
that haven't been introduced yet.

A system where exercises are given multiple tags (rather than
being keyed to a particular section) has numerous advantages.
++ At the deepest level, it brings about a salutary change
in how one thinks about the design of each exercise. There
should not be a one-to-one correspondence between sections
and exercises, but rather a multidimensional matrix.
++ If an idea was introduced in chapter 1 and never used again,
that's strong evidence that it was a worthless idea. There
should be little need for a late-term "review" of earlier
ideas; if the ideas were any good, they should have cropped
up naturally, again and again. As I like to say: Utility
is the best mnemonic.
-- OTOH if/when some review is needed, the teacher should be
able to throw in a review question without making it obvious
exactly what section is being reviewed. This is the natural,
appropriate way of raising the cost of equation-hunting.