Principles of Teaching and Learning
John Denker
* Contents
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1 Overview
- 2 Dealing with Misconceptions
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2.1 Emphasize Correct Conceptions
- 2.2 Confronting Misconceptions
- 2.3 Outright Ambiguity
- 2.4 Insufficient Specificity
- 2.5 Recurring Misconceptions
- 3 Terminology
- 4 A Few Prevalent Misconceptions
- 5 Next Generation Science Standards
- 6 References
1 Overview
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Emphasize learning as opposed to mere teaching.
It doesn’t directly matter what you teach; it matters what the
students learn.
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Emphasize personal responsibility.
Learning is the students’ responsibility; you cannot do it for them.
Your job is to inspire them and help them to fulfill their
responsibilities.
Responsibility can be taught, bit by bit, over the long haul. The
topic of responsibility should be addressed directly. It should be
discussed early and often. There should be clear rewards for
responsible behavior.
I tell students that more than 90% of what they need to know, they
need to learn on their own. All I can do is help get them started in
the right direction.
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It helps to know where the students are coming from.
A useful proverb is:
Learning proceeds
from the known
to the unknown.
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To say the same thing the other way, you cannot explain idea X in
terms of idea A unless the students already understand idea
A.
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Accentuate the positive, and emphasize constructive
suggestions. In particular, start by teaching the correct ideas, as
opposed to starting by confronting misconceptions.
For details on this, see section 2.
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Students can’t learn everything at once, but
they have to start somewhere.
That is: If the long-term goal is for the students to have
a comprehensive, detailed, sophisticated, rigorous understanding
of the subject, they can’t learn that on the first day.
Typically the students start by learning some examples, analogies, and
approximations. This should include explicit disclaimers emphasizing
that the examples are not entirely representative and the
approximations are not exactly accurate. Next the students should
learn some math and some theoretical principles, which they can apply
to the examples. Also they should always make every effort to see the
connections between the new information and everything else they know.
Then they should spiral back and learn a wider range of examples,
better approximations, more math, more theory, and more connections.
And so forth, iteratively.
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Explicitly teach students how to learn, and how to
remember what they have learned.
There are a number of standard techniques to help with memory, some of
which date back 2500 years.
Memory should not be thought of as a substitute for critical thinking,
or vice versa. Indeed, memory itself is a thought process.
Being able to recall relevant information is a necessary ingredient
for critical thinking.
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If you want to improve your memory, it is far from
sufficient to think “harder” about something at the time it must be
recalled. Instead, one must make the effort to form useful memories
at the time the memory is laid down, days or months or years before it
is needed. It takes time and effort to lay down such memories. As
mentioned in item 5, thinking about the connections
(aka associations) between ideas is important. More than 100 years
ago, in reference 1, William James wrote this about each
remembered idea:
Each of the associates is a hook to which it hangs,
a means to fish it up when sunk below the surface. Together they
form a network of attachments by which it is woven into the entire
tissue of our thought. The ‘secret of a good memory’ is thus the
secret of forming diverse and multiple associations with every
fact we care to retain.
Thinking about the connections between a newly-learned idea and older
ideas is first of all a check on the correctness and consistency of
the ideas, and secondly serves to reinforce the memory of both the new
and old ideas.
To say the same thing another way: A rote memory can be recalled in
one way, so technically it counts as a memory, but it is not a very
useful memory. In contrast, a well-constructed memory can be recalled
in 100 different ways, which makes it 100 times more useful.
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There is a fine line between an approximation and
a misconception.
In teaching, as in every other facet of life, approximations are
necessary. On the other hand, any approximation can be abused.
This topic needs to be taught – directly, emphatically, early, and
often. Students need to learn to think clearly about approximations
in general, not just this-or-that approximation du jour.
Things they need to know include:
- Not all approximations are created equal. There are good
approximations, mediocre approximations, and bad approximations. It
takes skill and judgment to know the difference between a good
approximation and a bad approximation in any given situation.
- Whenever possible, it is desirable to have a controlled
approximation. For example, it is of some limited value to say π
is approximately 3.1, but it is incomparably more valuable to say that
π is greater than 3.1 and less than 3.2. The latter is a
controlled approximation, which means you know how accurate is. You
know the limits of validity of the approximation.
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Preservice and novice teachers beware: the space
of misconceptions is larger than you can possibly imagine. Just
because you were never confused about this-or-that doesn’t mean your
students won’t be confused about it. That’s one of the N reasons
why you give tests: to find out what you didn’t cover sufficiently
well.
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Don’t become part of the problem. Don’t teach
misconceptions, and don’t teach nonsense.
That advice is harder to follow than it might seem, because we are
constantly besieged by misconceptions and nonsense. This includes the
misconceptions we grew up with. The misconceptions and nonsense are
particularly pernicious when they appear in textbooks and in
state-mandated “standards” and tests.
Almost any topic can be taught in N different ways, most of which
will cause trouble later, and only a few of which form a good
foundation for further work.
Just because it was badly taught to you doesn’t dictate that it
must be badly taught by you.
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If a student seems confused or hesitant, don’t be shy about
asking the student what the problem is. Sometimes they don’t know
what the problem is ... but sometimes they do.
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Learn from other teachers. Whenever
possible, drop in on other teachers, to see how they handle things.
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Learn from your students. I have a long
list of nifty ideas I learned from my students.
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Do not tolerate cheating.
It is sometimes claimed that in the long run, the cheaters harm only
themselves, but it is not true. Sometimes jobs, scholarships, and
even the privilege of staying in school are awarded partly on the
basis of grades, and unfair grades can cause serious harm to innocent
students and third parties.
In particular, you must not tolerate any situation where students who
are not predisposed to cheat feel obliged to cheat just to keep up.
I don’t want to live in a culture where cheating is considered normal.
School should not train people to think that cheating is normal.
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Do not tolerate plagiarism, since it is a form of
cheating.
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More generally, do not tolerate anything that gives an
unfair advantage to some students over others.
For example, you should assume that some students will have access to
the questions and answers to tests given in previous years, filed away
in fraternities, private homes, et cetera. I mention this because
even if such files are not against the rules, they are unfair, because
not all students have equal access to the files. Making a rule
against such files is not helpful, because it is more-or-less
unenforceable.
Therefore: do not re-use previous years’ questions, unless you have
made sure that all students have equal access to the questions and
answers.
Constructive suggestion: If you want to re-use any questions from
previous years, you can level the playing field by making previous
years’ questions and answers available, on the web or otherwise.
By way of example, consider the tests given by the Federal
Aviation Administration. For each test, there is a pool of
approximately 1000 possible questions, of which a few dozen appear on
any given instance of the test. The questions are a matter of public
record, but the large size of the pool discourages rote memorization,
since most people find it easier to to learn the underlying principles
than to memorize specific answers to specific questions.
If you want to re-use a question from a previous year, another way to
encourage understanding (as opposed to mere rote regurgitation) is
to rejigger the question so that even though the idea is the same,
the answer is not literally the same.
2 Dealing with Misconceptions
2.1 Emphasize Correct Conceptions
Whenever the topic of misconceptions comes up,
there is an important contrast that needs to be made:
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Explaining the correct conceptions should always come first.
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Confronting misconceptions can come later, if at all.
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“The light shines in the darkness, and the darkness cannot
overcome it.”
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“You cannot beat something with nothing.”
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By way of analogy: If you have a healthy lawn, you won’t have
much trouble with weeds. They can’t compete.
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If the lawn is weak and
sparse, due to insufficient water and fertilizer or otherwise, weeds
will invade.
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In an unhealthy lawn, if you just cut off the top of a
weed, it will grow back. Even if you manage to kill the weed, it is
likely to be replaced by some other weed. See
figure 1.
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Correct ideas need to be linked to each other, and supported
by evidence.
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Misconceptions do not exist in a vacuum; they are
supported by their own evidence. If you simply contradict a
misconception, it will grow back, sooner or later, probably sooner.
Furthermore, often an imperfect notion
contains a germ of truth, so if you flatly contradict the
whole notion you’re not even correct.
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On this web site, there are hundreds of documents that try to
explain correct ideas.
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There is only one part of one document –
section 4 – that has focuses on misconceptions,
and it should not be the starting point.
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If you are tempted to search the PER literature for a list of
misconceptions, my advice is: Don’t go there! Reasons for not
going there include:
- The literature will tell you that students have more
misconceptions than you could possibly imagine ... but if you’ve
made it this far, you already know that.
- There are so many misconceptions that it would be
impossible to list them all, let alone analyze them.
- It is OK for teachers to talk amongst themselves about
this-or-that misconception, but talking about it in front of students
is at least as likely to reinforce the misconception as to dispel it.
- This brings us back to the main point: To a first approximation,
with isolated exceptions, the best policy is to teach the correct
concepts and move on.
Figure 1 is one way of visualizing the situation.
If students move randomly away from any
given bad idea (shown in red), they are more likely to settle onto a
new bad idea than onto a good idea (shown in blue). You need
to attract them to the good idea, not merely push them away
from this-or-that bad idea.
Figure 1: Lead Toward Good, Not Merely From Bad
We now turn to figure 2, which is another way of
visualizing the situation. Imagine a very narrow path through a vast
swamp. The path is safe, and everywhere else is unsafe. If you pick
a random point in the swamp, there is a 99.99999% chance you don’t
want to visit that place, or even talk about that place.
One valid path from A to B is shown in black. (There may be more
than one valid path, but even taken together, the valid paths are a
subset of measure zero in the overall space.)
Figure 2: Stay On (or Near) the Valid Path
On rare occasions, as you lead students along the path, you might
want to point out something nasty that is just next to the path, so
they can recognize it and avoid it (as indicated by the yellow places
in the diagram). Still, even so, it is usually easier to recognize
the path than to recognize the nasties. This is the Anna Karenina
principle: All happy students are alike, but every unhappy student is
unhappy in his own way.
In a one-on-one teaching situation, you can sometimes afford to deal
with misconceptions as they come up. However, this is tricky and
hard to plan, because students often come up with weird
misconceptions that you never dreamed of. Meanwhile, in a classroom
situation, things are even worse, because each student is going to
have a different set of misconceptions.
- The misconceptions most worth worrying about are the ones that
are exceptionally prevalent and exceptionally pernicious. The
classic example concerns the washed-out bridge. As discussed in
section 2.2, it is well worth confronting misconceptions of
this kind.
Section 4 contains a list of misconceptions that
seem particularly prevalent. Please keep in mind that compiling
and/or studying such lists is usually not a good use of resources.
- I’m not so much worried about the misconceptions that the
students bring to class as the misconceptions that the teacher and
the textbook author bring to class. When a misconception is passed
on from teacher to student, it can cause tremendous difficulties for
the students in later courses, and in later life.
Remember that it is proverbially difficult to unlearn something.
Some textbooks contain large numbers of misconceptions. For an
example – not even the worst example – see reference 2.
- Reading the PER literature is definitely a source of
misconceptions. I don’t mean you will get a tidy list of avoidable
misconceptions labeled as such; I mean that after reading the
literature, if you believe what it says, you will suffer from more
misconceptions than you started with.
For example, the book by Arons, Teaching Introductory Physics
is — unintentionally — an extensive compendium of bad
pedagogy and wrong physics. For a detailed review,
including a list of some of the misconceptions propagated
and/or introduced by this book, see reference 3.
2.2 Confronting Misconceptions
In many cases, when confronting a misconception, the first step is to
recognize that you are dealing with a notion that probably contains a
germ of truth. Indeed, the most dangerous ideas are the ones that are
usually mostly true, but then betray you at some critical moment.
The classic example concerns a washed-out bridge. Most people take it
for granted that it is OK to drive across the bridge, and this is
usually true. However, if the bridge has been washed out this becomes
a misconception, and could have fatal consequences. Therefore it is
worth putting up some “Bridge Out” signs and barriers, and possibly even
some flashing lights.
As always, it is better to make constructive suggestions, as opposed
to merely pointing out a problem. In this case, it would be silly to
put up a “Bridge Out” sign just at the edge of the washout, because
by the time drivers could see the sign it would be too late to do
anything about it. Instead the proper procedure is to go back to the
previous intersection and block off the approaches to the bridge.
This includes putting up signs directing traffic to a suitable detour.
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There are very few life-critical misconceptions that show up
in the introductory-level physics classroom. In the case of
misconceptions that are not acutely dangerous, you have the option of
confronting them or not.
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By way of contrast, in the research lab you
might have high-power invisible lasers, high voltages, toxic
chemicals, et cetera. Safety-related misconceptions must be
confronted.
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As an example of a classroom misconception that might be worth
confronting, consider the first law of motion. Practically everybody
starts out with the Aristotelian notion that objects at rest
tend to remain at rest, and objects in motion tend to come to rest.
This directly conflicts with the Newtonian principle that objects in
motion tend to remain in motion.
As is so often the case, we are dealing with a notion that contains a
germ of truth: In situations where friction is overwhelmingly
important and taken for granted, objects do tend to come to
rest. That’s fine. It is OK for students to retain that idea,
provided they learn to distinguish it from situations where
friction is not so important. In physics we start by considering
situations where friction is completely negligible. Later we consider
cases where there is a moderate amount of friction, but even then we
do not take friction for granted, but instead account for it as one of
the forces that change the state of motion.
Again, we want to do this in a constructive way. That requires:
- First, explaining the new concept and supporting it with
reasonable amounts of evidence
- Secondly, contrasting the new concept with the old concept.
- Only then does it make sense to explain what part of the
old concept is considered a misconception.
It must be emphasized that it is pointless (or worse) to contradict
the old idea before the new idea has been presented. It is not
helpful to push students away from a bad idea unless/until they have a
good idea to latch onto.
The same principle applies to everyone you deal with, not just
students. It applies at every age, from infancy on up. For example,
if a young child is banging a Wedgwood teacup against the tile floor,
it is better to give the kid something else to play with, rather than
simply taking the teacup away. A small plastic bottle with a few
dried beans inside makes a much better toy, from everyone’s point of
view.
2.3 Outright Ambiguity
It is quite common to find the same term being used to describe two or
more ideas. This is a perennial source of misconceptions. For
example:
2.4 Insufficient Specificity
Consider the assertion that “cows are brown”. Is that a
misconception? I don’t know, because I can’t figure out what is being
asserted. Possibilities include:
- Some cows are brown.
- All cows are brown.
Statement (A) is entirely correct, whereas statement (B) is
entirely incorrect.
Many ideas that are a good approximation in one context are a bad
approximation in other contexts. The goal is to formulate a
more-specific version of the idea, containing enough provisos so that
you know which is which.
Here is another example that touches on the notion of specificity:
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| In Euclidean geometry, |
| a2 + b2 | | = | | c2 | |
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| and c is the hypotenuse |
| (2)
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If you are going to teach people that a2+b2=c2, you have an
obligation to tell them that the result is valid for Euclidean right
triangles only. This is important, because there are lots of
triangles in this world that are not right triangles ... and lots that
are not Euclidean.1
2.5 Recurring Misconceptions
While some misconceptions are only lightly held, others are quite
deeply held, based on the student’s lifetime of experience (in school
and otherwise). As mentioned in reference 5, when you
confront a deeply-held misconception, students may become wary,
defensive, or even angry. It is likely that the students will pretend
to discard the misconception, but then re-adopt it at the first
opportunity.
If a misconception keeps coming back, there are several possible
explanations, and correspondingly several ways of dealing with the
situation. These include:
- It may be that the misconception is supported by a great deal
of bogus data and fallacious reasoning. In such a case, rather
than directly attacking the symptom, it may be helpful to go
upstream a step or two, and pull the supports out from under the
misconception. That is, figure out what sort of mistaken evidence
is supporting the misconception, and then explain why that evidence
is invalid.
- It may be that the real problem is that the correct conception
is not sufficiently supported. In such a case, it is best to focus
on the correct conception, tying it to additional data and
additional reasoning.
- It may be a case of outright ambiguity, as discussed in
section 2.3. In such a case, it may be possible to
disambiguate the terminology by adding adjects, or it may be easier
to switch to completely different terminology for one or more of the
affected concepts.
- It may be a case of insufficient specificity, as discussed in
section 2.4. In such a case, you need to explain why
the notion is correct in some contexts but not in others, and
explain how to tell the difference. A change in terminology might
help (but might not be sufficient by itself).
- Consider the first law of motion: an object in motion tends to
remain in motion. Contrast that with the widely-held opinion that
objects in motion tend to come to rest. This directly contradicts the
first law of motion. It is a notorious misconception, widely and
deeply held.
There is however another way of looking at this
situation.
- If you are a flagellate bacterium, then you live in a world
with a verrrry low Reynolds number. Friction is dominant, and
inertia is an utterly negligible correction.
- If you are an aircraft, your Reynolds number is much higher.
Inertia is dominant, and friction is a relatively minor correction
term.
In the introductory physics class, we choose to start from the
low-friction case. Students’ intuition about the high-friction
case is not wrong; it’s just incompatible with our chosen starting
point.
A direct attack on the idea that objects in motion tend to come to
rest will never be successful, because the idea has too much
supporting evidence. The best you can hope for is to place
limits on the validity of the idea, to restrict it to tiny
objects moving slowly through a sticky medium.
- I take a similarly tolerant attitude toward scalar acceleration:
It’s not crazy wrong; it’s just ambiguous, as mentioned in
section 2.3.
- The same attitude works for notions of “heat content” aka
“caloric”. Such ideas are not crazy wrong, and indeed it is easy
to find supporting data. However, such ideas apply only cramped
thermodynamics, i.e. to situations so heavily restrictricted that it
is impossible to build any kind of heat engine. Such ideas greatly
interfere with any attempt to understand uncramped thermodynamics.
There is a pedagogical / psychological dimension to this. There is a
mountain of evidence suggesting that established ideas are virtually
never truly unlearned, not on any pedagogically relevant timescale
anyway. Instead the best you can hope for is to hide the bad ideas
behind a wall of better ideas, so that in any given context the right
idea is more likely to be recalled. The wrong (or merely
inapplicable!) ideas are still there; they just won’t be the first
things that come to mind. So, rather than figure 1,
the picture is more like figure 3.
By telling students their ideas are not crazy wrong – just restricted – they are less likely to get defensive. It enhances
the teacher’s credibility. It gives students a framework that
accounts for all the data. This upholds one of the core
principles of critical reasoning: account for all the
data.
3 Terminology
Some people use the word “misconception” in very narrow ways, or
avoid it altogether. One teacher sent me a list of thirty different
terms intended to describe different types of misconceptions,
preconceptions, and related ideas.
I am not interested in such fine distinctions. I use words like
“idea” and “notion” in a broad sense, including ideas that are
completely correct, completely incorrect, and everything in between.
Almost all ideas are imperfect in some way. I use the term
“misconception” to apply to whatever part of the idea is incorrect.
4 A Few Prevalent Misconceptions
The following list is restricted to misconceptions that afflict
professionals in the field. (There is of course a far wider class of
misconceptions that afflict naïve students.)
Some related issues of weird terminology are discussed in
reference 6.
Keep in mind that you should always start by emphasizing correct
conceptions, as discussed in section 2.1. To say the same thing
the other way: creating and/or studying lists of misconceptions is
usually not a good idea, and should never be a starting point.
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Far and away the biggest problem is an overall lack
of critical thinking skills. See reference 7.
This includes, far too often, accepting a “rule” without
differentiating between a “rule of thumb” and a “rule in all
generality”.
This includes learning a “rule” without reconciling it with other
experimental and theoretical things they know.
This includes learning the headline of a rule without learning the
provisos, without learning the limitations on the range of validity
of the rule.
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Chronic and pervasive inability to tell the
difference between a good approximation and a bad approximation ...
and even unawareness that this is even an issue.
See reference 8.
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Multiple misconceptions about scientific
methods. For example, fixating during the planning stage on a
single hypothesis. Common sense and basic scientific principles
demand considering all the plausible hypotheses. Indeed this
is required for safety if nothing else. See
reference 8.
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Multiple misconceptions about “significant
figures” and/or how to handle uncertainties. See
reference 9. See also item 6.
It is easy to find examples of professors being completely
confident about the wrong answer.
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That includes pervasive misunderstanding of what
“error” means in the context of “error analysis”. See
reference 9.
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Innumerable misconceptions about probability.
For example, suppose I toss a coin 100 times. On every “heads”,
I take one step to the north. On every “tails”, I take one step
to the south. After the 100th step, how far away am I, on average,
from where I started? (Most kids – and more than a few teachers –
say “zero” ... which is not the right answer.)
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Widespread misconceptions about the fundamental
principles of quantum mechanics. The fact is that even in fully
quantum mechanical systems, not everything is quantized,
not all waves are quantized, not all states are discrete,
et cetera. See reference 10.
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Innumerable misconceptions and/or archaic conceptions about
special relativity.
This includes velocity-dependent mass, rulers that can’t be
trusted, clocks that can’t be trusted, et cetera. It is a
misconception to think those are a good idea (even if they are not
provably wrong). Certainly they must be unlearned as a prerequisite
to any modern (post-1908) understanding of special relativity,
spacetime geometry, and 4-vectors ... not to mention general
relativity. See reference 11.
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Misunderstanding of the famous equation E=mc2. Hint:
this E is the rest energy. If the mass is moving, we need a more
complicated formula. Mass is the invariant norm of the [energy,
momentum] 4-vector. See reference 11.
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Misunderstanding of general relativity, especially as to what
is curved, and in what direction it is curved. A marble rolling
around inside a bowl does not illustrate general relativity. See
reference 12.
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The idea that «Kirchoff’s laws are fundamental,
and can be directly derived from Maxwell’s equations».
A previous version of the wikipedia article said exactly that
(before I changed it). In fact, for AC circuits, both of Kirchhoff’s
laws are flatly contradicted by the Maxwell equations.
Also note that both of Kirchhoff’s laws are routinely violated in
practice. There are tremendous misconceptions about this.
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Assuming that every voltage must be a
potential, and every electric field must be the gradient of some
potential. This assumption is embodied in Kirchhoff’s laws. We
know it can’t be true when there are time-varying magnetic fields
running around.
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Misconception that thermal energy
(whatever that means) is random kinetic energy to the
exclusion of potential energy. See also item 19. For
details, see reference 13.
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There are some who try to define energy as “capacity to
do work”. This formulation is fairly common in nonscientific
books. It appeals to those who know nothing about
thermodynamics. See reference 13.
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Entropy as “disorder”.
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Entropy as “spreading of energy”.
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Misconception that
expansion of the universe correlates with thermodynamic
irreversibility i.e. entropy production
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Innumerable other misconceptions regarding entropy. See
reference 13.
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Trying to think of thermodynamics in terms of a
“heat content” or “thermal energy” state function. See
reference 13.
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Writing d(something) for ungrady one-forms, e.g. dQ =
T dS. (This is somewhat related to item 19.) See
reference 14.
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Terminology and thought patterns that confuse
“heat” with “enthalpy”, e.g. tables of the “heat of reaction”.
Counterexample: reversible reactions such as electrochemical cells.
Similarly, misconception that heat is conserved (e.g. in the
typical statement of Hess’s law). See reference 13.
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Holy wars about the various definitions of “heat”. Example:
adding heat to something “surely” raises its temperature. See reference 13.
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Confusion about the relationship between energy and
temperature, e.g. Define temperature as the measure of thermal
energy
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Confusion about the meaning of “intensive”
versus “extensive”. (Possibly confused with intrinsic versus
extrinsic?)
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Conflict over whether the rate constant
for the reaction x A → y B + z C
should depend on the stoichiometric coefficient x. In particular,
deciding to scale the rate constant by a factor of x without
regard to the order of the reaction. I claim that for an Mth
order reaction, the normalization factor is x(M−1). I claim
you want to define the rate constant per unit of “→” not
per unit of [A].
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Misconception that electrons must “jump” from
stationary state to stationary state. To say the same thing in NMR
terminology, the misconception is that π/2 or π/10 tipping
pulses are impossible.
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Misconception that electrons “like” to pair up,
like Siegfried and Roy. In fact, physics says they hate each other.
Spectroscopy (as summarized by Hund’s rule) says that in the ground
state, they pair up only as a last resort.
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Misconception that breaking a chemical bond
releases energy, the way that breaking an eggshell releases what’s
inside.
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Misconception that during changes of
state the temperature remains constant.
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Multiple inconsistent definitions of “molecule”.
– molecules = “stable particles of matter” is a non-starter,
because water molecules are not stable in aqueous solution.
– molecules are “covalently bonded” is a non-starter, because
many things that ought to be considered molecules are not covalently
bonded.
– molecules obey the “law of definite proportions” is a
non-starter, because many macromolecules do not uphold it.
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Uncertainty about the definition of “compound”.
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Related minor point: There seem to be widespread misconceptions
about the definition of “dimer”.
Also note that the IUPAC definition of polymer is very broad,
and does not parallel the definition of dimer.
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The alleged dichotomy between “ionic
compounds” and “molecular compounds”.
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The whole notion of “physical change” as
distinct from “chemical change” is disconnected from reality. There
are multiple inconsistent definitions of the terms. All the usual
definitions conflict with the usual examples. See
reference 15 .
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In most chemistry texts, there is some kind of a
flowchart that uses various crude criteria to classify substances as
elements, compounds or mixtures. This incorrectly implies that it
is impossible for any substance to simultaneously be an element and
a mixture. In particular, it gets you into trouble with an element
that is a mixture of isotopes. The right way to think about this
is in terms of equivalence classes. Things that are “pure” w.r.t
one property may not be “pure” w.r.t to another.
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Misconception that “Like Dissolves Like”.
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Le Chatelier’s principle is highly problematic.
LeChatelier in his lifetime gave two inconsistent statements of the
“principle”. One is trivially tautological, and the other is
false. See reference 16.
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Widespread deep-seated misconceptions about osmosis and
osmotic pressure. This includes using glycerin or other
hygroscopic substances as pedagogical examples of osmosis. See
reference 17.
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Way too much emphasis on the Bohr model of the atom,
i.e. electrons following Keplerian orbits within atoms.
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There are profound problems with Lewis dot diagrams
in general, and the idea of filled Lewis octets in molecules in
particular. These ideas are are fundamentally inconsistent with all
the spectroscopic data ... and other data. The paramagnetism of
O2 makes for a nice, graphic, in-class demonstration. See
reference 18.
It’s particularly comical when they arrange Lewis dots (falsely
representing electrons) onto little Keplerian circles (falsely
representing orbitals) to make molecules
(falsely suggesting that there are filled “Lewis octets”
in molecules).
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Talking about “position” of the electron within a
“p orbital”. (These are incompatible variables, incompatible in
the Heisenberg sense.)
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Uncertainty about “orbitals”. Does the
term refer to wavefunctions describing actual electrons,
or does it refer to basis wavefunctions that are purely
mathematical?
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The whole idea of oxidation numbers in redox
reactions is grossly abused. When balancing redox reactions, it is
simpler and better to just use conservation of charge, directly.
See reference 19.
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Arons asserts there are two kinds of electrical
charge. He says the two-fluid model is right, and the
one-component model is wrong. See reference 20
for a refutation.
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Arons also suggests teaching students the
difference between “passive forces” and “active forces”.
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Questions about where the lanthanoids and
actinoids belong in the periodic table. There should not be any
questions. These are basically elementary fencepost errors.
See reference 21.
Also, a related bundle of misconceptions revolve around the
ill-conceived notion of p-block, d-block, and f-block elements
... and the corresponding “block structured” periodic table. See
reference 21.
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Numerous misconceptions concerning absolute zero,
degeneracy, zero-point motion, et cetera.
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Teaching Boyle’s law, Charles’s law, and Gay-Lussac’s law
separately, on the theory that in each case “the” variables not
mentioned are held constant. This is an OTBE fallacy. For example,
T1/P1=T2/P2 is not valid if we hold N and S constant.
See reference 17.
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Allegedly “Temperature is not a state function.”
(I’m not kidding. A professor vehemently asserted that.)
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Wild misconceptions about the shape of the H2O
molecule. In fact, contrary to what several chemistry professors
have said:
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The H2O molecule has a “bent” shape,
more-or-less U-shaped. The stick-figure that approximately
represents the bonds between nuclei is V-shaped, and the
electron-density is U-shaped. See e.g. reference 22.
- The electron density in water does not exhibit tetrahedral
geometry.
- The electron density in water does not exhibit tetrahedral
symmetry.
- The electron density in water does not exhibit tetrahedral shape.
Fussing with the terminology will not change these facts.
See reference 22.
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Attempts to build a classical ball-and-stick model
of the isolated ammonia molecule.
Fact: The isolated gas-phase ammonia molecule in its ground state
has D3h symmetry. See reference 23.
(It has a much lower symmetry, C3v, in liquid
ammonia or in aqueous solution.)
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You can allegedly determine the temperature of a
flame by looking at its color. (Not just a black body, but a
flame!)
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Confusion regarding negative temperature coefficient
versus negative activation energy. These are not the same.
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Persistent failure to understand (even after being told)
that dimensional analysis
can sometimes give the wrong answer, both false positives
and false negatives. That is, arrested development at the
level of dimensional analysis when a scaling analysis is
called for. See reference 24 and
reference 25.
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Teaching the “density triangle” as described at
e.g. reference 26.
This approach has many weaknesses, but we should not overreact. You
can solve the problem by converting this to an equation M/(D V) = 1
and then forgetting the triangle.
If you don’t convert, and stick with the “cover up” rule, it is not
just opaque to the underlying math, it is actively misleading. The
problem becomes obvious if/when the underlying equation has more than
one variable in the numerator.
The triangle is a crutch. Normal students should not need any such
crutch.
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Allegedly, the term “algorithmic” is synonymous with rote,
i.e. turning the crank without thinking. This is crazy wrong.
Algorithms are good for you. Do not confuse the presence of one thing
with the absence of another.
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I once heard a professor talking about a «adiabatic
calorimeter .... “Adiabatic” means no heat.» In other words, we’re
talking about a no-heat heat-capacity experiment. This is what comes
from restricting the definition of “heat” (“flow across a
boundary”) without checking the consequences for consistency.
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True or False? – There is no such thing as centrifugal
force. See reference 27.
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True or False? – As part of the recovery from a severe
spiral dive, it is important to roll the wings level and then pull
back on the yoke.
Hint: John-John Kennedy probably didn’t know the right answer to this one.
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True or False? – The airplane’s stability depends on the fact that the tail is
producing a downward force.
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True or False? – In a Cessna 172, starting from normal flight, if you increase the
throttle setting (without moving any of the other controls) the
airplane will speed up.
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True or False? – If two parcels of air flow past a wing, they move from the front to
the back in essentially equal amounts of time, even if one passes
above and the other passes below the wing.
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True or False? – To work properly, an airplane wing must be curved on top and
relatively flat on the bottom.
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True or False? – Blowing a jet of air across the top of a piece of paper is a good
way to demonstrate the principle that “faster-moving air has lower
pressure”.
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True or False? –
Bernoulli’s principle is only valid for incompressible fluids,
which means it cannot be trusted for something as obviously
compressible as air.
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True or False? –
As suggested by the saying “power plus attitude equals
performance”, if you put the airplane into a particular attitude with
a particular power setting, the airplane will give you the
corresponding performance (airspeed, rate of climb, et cetera) and if
you maintain this attitude and power setting you will continue to get
that performance.
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True or False? –
To perform an ordinary steady roll to the right, the upgoing wing must
produce a greater amount of lift (compared to the other wing), and therefore
a greater amount of drag, which is why you need to apply steady right rudder
during the roll.
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True or False? – During flight at very low airspeeds, some
sections of the wing are unstalled, while other sections are stalled
and contributing practically nothing to the lift.
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True or False? – During a normal steady climb, lift is necessarily greater than
weight.
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True or False? – During a steady, coordinated turn to the
left, dihedral creates a tendency for the airplane to roll back toward
level, and you generally need to apply steady left aileron to overcome
this.
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True or False? – P-factor (i.e. asymmetric disk loading)
explains why, early in the takeoff roll in a Cessna 172, you must
apply right rudder to keep it going straight.
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True or False? – On approach, you should never retract the flaps to correct for
undershooting, since that will suddenly decrease the lift and cause
the airplane to sink even more rapidly.
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True or False? – When properly performing turns on a pylon in the presence of
wind, the airplane will remain at the pivotal altitude, and the
pattern will be shifted somewhat downwind relative to where it would
be in no-wind conditions.
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True or False? – To model the earth’s magnetic field, take a globe and skewer it
with an ordinary bar magnet, putting the bar’s “N” pole in northern
Canada, and its “S” pole in Antarctica.
I emphasize that the foregoing list is restricted to misconceptions
that afflict professionals in the field. There is of course a far
wider class of misconceptions that afflict naïve students.
I have seen collections of student misconceptions, but they all seem
so incomplete as to be virtually useless. Furthermore some of them
tend to replace old misconceptions with new ones; see e.g.
reference 3.
5 Next Generation Science Standards
A great many misconceptions are being spread by the NGSS,
as discussed in reference 28.
6 References
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William James, Talks to Teachers On Psychology; and
to Students on Some of Life’s Ideals (1899).
http://books.google.com/books?id=XYSsCLlF_mkCprintsec=frontcover
Chapter XII deals specifically with memory.
http://ebooks.adelaide.edu.au/j/james/william/talks/chapter12.html
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John Denker,
“Review of Hewitt, Conceptual Physics
http://www.av8n.com/physics/hewitt.htm
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John Denker,
“Review of Arons, Teaching Introductory Physics”
http://www.av8n.com/physics/arons-1996.htm
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John Denker,
“Definition of Weight, Gravitational Force, Gravity, g,
Latitude, et cetera”
http://www.av8n.com/physics/weight.htm
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Christopher Horton,
“Student Alternative Conceptions in Chemistry”
http://www.daisley.net/hellevator/misconceptions/misconceptions.pdf
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John Denker,
“Weird Terminology”
http://www.av8n.com/physics/weird-terminology.htm
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John Denker,
“Teaching (and Learning) Thinking Skills”
http://www.av8n.com/physics/thinking.htm
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John Denker,
“Scientific Methods”
http://www.av8n.com/physics/scientific-methods.htm
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John Denker,
“Measurements and Uncertainties versus Significant Digits”
http://www.av8n.com/physics/uncertainty.htm
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John Denker,
“Coherent States”
http://www.av8n.com/physics/glauber-states.htm
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John Denker,
“Welcome to Spacetime”
http://www.av8n.com/physics/spacetime-welcome.htm
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John Denker,
“Tabletop Geodesics, General Relativity, and Embedding Diagrams”
http://www.av8n.com/physics/geodesics.htm
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John Denker,
“Modern Thermodynamics”
http://www.av8n.com/physics/thermo-laws.htm
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John Denker,
“Thermodynamics and Differential Forms”
http://www.av8n.com/physics/thermo-forms.htm
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John Denker,
“Chemical versus Physical Change?”
http://www.av8n.com/physics/chemical-physical.htm
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John Denker,
“Spontaneity, Reversibility, and Equilibrium”
http://www.av8n.com/physics/spontaneous.htm
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John Denker,
“Gas Laws”
http://www.av8n.com/physics/gas-laws.htm
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John Denker,
“How to Draw Molecules ... Just Like Lewis Dot Diagrams,
Only Easier & Better”
http://www.av8n.com/physics/draw-molecules.htm
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John Denker,
“Balancing Reaction Equations w.r.t Charge and Atoms”
http://www.av8n.com/physics/balance-charge-atom.htm
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John Denker,
“One Kind of Charge”
http://www.av8n.com/physics/one-kind-of-charge.htm
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John Denker,
“Periodic Table of the Elements – Cylinder with Bulges”
http://www.av8n.com/physics/periodic-table.htm
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Martin Chaplin,
“Water Molecule Structure”
http://www.lsbu.ac.uk/water/molecule.html
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Feynman, Leighton, and Sands
The Feynman Lectures on Physics volume III chapter 9
(“The Ammonia Maser”).
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John Denker,
“Dimensional Analysis”
./dimensional-analysis.htm
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John Denker,
“Scaling Laws”
http://www.av8n.com/physics/scaling.htm
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Christine Weber
“Ultimate Review Page”
http://www.weberscience.org/ES_review.htm
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John Denker,
“Motion in a Rotating Frame”
http://www.av8n.com/physics/rotating-frame.htm
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John Denker,
“Next Generation Science Standards”
http://www.av8n.com/physics/ngss.htm
