| Chronology | Current Month | Current Thread | Current Date |
| [Year List] [Month List (current year)] | [Date Index] [Thread Index] | [Thread Prev] [Thread Next] | [Date Prev] [Date Next] |
-----Original Message-----
From: Phys-l [mailto:phys-l-bounces@www.phys-l.org] On Behalf Of Lulai, Paul
Sent: Monday, September 28, 2015 2:39 PM
To: Phys-L@phys-l.org
Subject: Re: [Phys-L] reasoning versus concepts and/or problem-solving
I have quickly read through this note. I like some of the points laid out. I am
unfamiliar with the Robert Cohen source mentioned in the note.
What is the title of the Robert Cohen source?
Thanks for your help.
Paul.
Paul Lulai
Physics Teacher
St Anthony Village Senior High
St Anthony Village MN 55418
On Mon, Sep 28, 2015 at 8:17 AM, John Denker <jsd@av8n.com> wrote:
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.
_______________________________________________
Forum for Physics Educators
Phys-l@www.phys-l.org
http://www.phys-l.org/mailman/listinfo/phys-l
Forum for Physics Educators
Phys-l@www.phys-l.org
http://www.phys-l.org/mailman/listinfo/phys-l