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Re: Pedagogy of misconceptions.



Chris Horton wrote:

But is it good for the students to be presented in print with misconceptions
and mental wrong turns to struggle with unless they are clearly identified
as such?

That sounds like a really terrible idea to me.

0) The following remarks apply mainly to introductory
students. See below for extensions to other situations.

1) When I mention a misconception, it is always labelled
as such. There is always a discussion of how to recognize
it as a misconception, or (better) how to recognize its
_precursors_ so that you don't even come _close_ to falling
victim to such a misconception. The misconception is
contrasted with the corresponding correct conception.

2) I don't mention misconceptions unless there is a good
reason to do so. Good reasons usually involve the "expectation
value" i.e. the magnitude of the mistake weighted by the
likelihood of the mistake. If a misconception is truly
common, I'll discuss it. If a misconception is life-threatening,
I'll discuss it.

What I won't do is spend time conjuring up a misconception
that the student otherwise wouldn't have ever had, just so
we can discuss it.

3) Here's my rationale: The number of misconceptions is
essentially infinite. By way of analogy, consider how
many different ways there are to put random spots of
paint on a canvas ... then ask what fraction of those
represent beautiful (or even recognizable) paintings.
It's less than 0.00000001%. If I'm teaching an art
course, I don't want to spend 99.99999999% of my time
discussing "snow crash" patterns.

Even the set of somewhat-plausible misconceptions is so
vast that exploring it is usually a waste of time. Every
true statement is surrounded by dozens of almost-true
statements.

4) I particularly detest the relativity books that
wallow in contrived "paradoxes". They go to great
lengths to teach the student the wrong way to think
about things, just so they can state the "paradoxes".
The time would be much better spend teaching the
student the _right_ way to think about things. A
well-trained student can hardly even express the
"twin paradox" without giggling. It's like asking
whether 2 plus 2 makes 13.

We "could" teach introductory Newtonian mechanics
the same way, by contriving mechanical "paradoxes".
(I know some pretty sneaky ones.) In the immortal
words of Richard Nixon, we could do that, but it
would be wrong.

5) I don't want to overstate the anti-misconception
viewpoint. There is a book called _Counterexamples
in Topology_ which is actually quite nice, contrary
to what I would have expected. The trick is that
the authors have exercised excellent taste as well
as judgement in selecting _interesting_ topics, i.e
things that are counterexamples to highly plausible
and/or interesting conjectures.

6) The rules are different when teaching grandmasters.
It is fine for us, as professionals, to discuss
misconceptions among ourselves, especially when they
are genuine wild-type misconceptions that have been
misconceived by real students. (I'm still allergic
to contrived misconceptions.)

the proof of the pudding is in the eating.

Yeah. It sure would be nice to have hard data.