The following topic has come across my desk four times in the last week,
in various guises:
Question: Which is more likely to fall on Friday the 13th:
Christmas, or Easter?
It is impressive how many people will simply answer the question, and
even defend their answer if challenged.
Suggestion: Whenever you see a question, start by asking yourself how
badly ill-posed it is. Maybe it has multiple inequivalent right answers
... or maybe it has no right answers. (The formal terminology for this
is /underspecified/ versus /overspecified/.)
In a textbook, almost all of the questions are well-posed. In real life,
almost none of them are. (There may be some jobs where the questions
are well-posed, but those jobs tend to involve asking, "Would you like
fries with that?")
This is one of the many ways in which the classroom environment differs
from the real-life environment. This is a problem, because supposedly
one of the major purposes of the educational system as a whole is to
prepare people to do well in the real world. Teaching them to rely on
tricks that only work in ivory-tower busywork situations defeats the
purpose of the whole system.
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In the context of sending a scientific paper out for review, Hans Bethe
once said:
The reviewer is always right.
That's because the reviewer represents the readership.
If the reviewer is too damn stupid to understand what I've
written, the general readership will be too damn stupid also.
I agree with that in spirit, although I think Bethe's first sentence is
overstated; the second and third sentences provide crucial clarification.
So here's my restatement of the idea:
Everything the reviewer says is important,
even if it's wrong ... especially if it's wrong.
It tells me I have to rewrite the paper
to make it more clear and convincing.
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All this applies to everyday teaching, because teachers get 100 ill-posed
questions per day. It applies to everyday life as well.
One standard good practice (among others) is to figure out where the
questioner is coming from, so you can rephrase the question and then
/answer the question that should have been asked/.
This is easier said than done. The teacher may need to dig down deep,
through multiple paleontological layers, to find something that the
student and teacher have in common, i.e. something they can agree on
and use as a basis for moving forward.
This is one of the ways I distinguish great teachers from not-so-great
teachers: Given an off-the-wall question, how fast can they figure out
where the student is coming from.
Often this involves asking a flurry of questions to figure out the
student's background and the context the student has in mind.
The concept of /bisconception/ is important to my thinking. In a great
many cases, including the most persistent and pernicious cases, the
student has an idea that actually makes perfect sense in some context
-- just not in physics context.
misconception = wrong idea
bisconception = two ideas (sensitive to context)
The way forward is to teach the student that context matters. A lap in
the swimming pool takes you to the far end, whereas a lap on the race
track brings you all the way back to the starting point. Context matters.
The homespun definition of «force» is not wrong, but it is very very
different from the physics force. Neither one is wrong in its proper
context. The only real mistake is not recognizing the distinction, i.e.
using one idea in a context where the other is required.
In contrast: Simply contradicting or ridiculing the "wrong" half
of a bisconception is a losing pedagogical strategy. The student
is likely to become defensive and/or angry, and simply decide not
to trust the teacher. This is the well-documented "backfire" effect.
The student may temporarily pretend to accept the new idea, but
then revert to the old idea at the first opportunity.
This requires a delicate tightrope act. The teacher must be respectful
of the non-physics notions, even while working to shift attention to
the physics notions.
This is one of the strange things about the FCI. People often wonder
how FCI scores can possibly be so low. The answer is complicated, but
part of the answer is that the FCI sets up situations that invite the
students to use homespun non-physics notions of force, and then penalizes
them for doing so.
I consider this to be deceptive. It's unfair to students and teachers
alike. Still, even so, we can learn from it. Like Bethe reading his
reviews, we ought to take it seriously even if we know it's wrong and
unfair. It tells us there's a problem, even if it doesn't tell us
much about the nature of the problem or how to fix it.
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This is related to the psychological principle that old ideas never
really disappear from the mind. The best you can hope for is to
build new structures on top of the old.
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Oddly enough, people learn more about this in English class than they
do in physics, logic, or education classes. Topics include satire,
metaphor, roman-à-clef, allusion, and allegory, where the words say
one thing but mean another.
I am astonished at how little attention this gets in education circles.
You would think it would be front-and-center: What do you do when a
student asks an off-the-wall question?
I tried googling for this, with disappointing results.
Here are a few hints:
*) First teach them how to handle the issue, then give them experience
doing so.
There is an elaborate mathematical formalism for dealing with ill-posed
questions. Most of it is overkill for present purposes, but even so,
everybody should know the basic ideas such as well-posed, solution set,
underconstrained, and overconstrained.
*) Students need a steady diet of ill-posed questions, so that they
become accustomed to asking, at every turn, how badly ill-posed the
question is.
*) Among other things, students should not assume that every variable
mentioned in the question is relevant, and every variable not explicitly
mentioned is irrelevant.
*) If students can answer the question by equation-hunting, there is
nothing wrong with the students and nothing wrong with the equations,
but there is something desperately wrong with the question.
*) Students (and especially teachers!) should learn to treat every
question as an ESP exam. You have to read the mind of the person
who posed the question, to figure out what they intended. Sometimes
the words are unclear, and sometimes they are outright deceptive.
*) In the real world there are politicians, lawyers, and merchants
who understand all this very clearly. They specialize in tricking
people into making bad decisions.
*) Please don't tell me it is important to strip the exercises down
to the bare essentials, so that students can focus on the basic
physics ideas. I'm not buying it. Learning to think is infinitely
more important than learning any particular physics factoid.
I am quite serious about this: If the students cannot handle a
question with extraneous information, forget about the physics and
deal with the fundamental reasoning issue first. It's by far the
higher priority.
*) Please don't tell me it is important to strip the exercises down
in order to «compel» the students to solve things a particular way,
as e.g. Heller&Heller recommend. I'm not buying it. Having multiple
ways of solving the problem makes it possible to CHECK THE WORK.
This is central to any notion of critical reasoning.