Copyright © 2010 jsd
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.
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.
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.
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.
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.
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:
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.
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.
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.
A word of advice:
I was explicitly told that by a mentor, back when I was getting into the teaching business.
This applies to pretty much everything, not just lectures.
Ironically, sometimes students learn a tremendous amount from a situation where the teacher makes a mistake, catches the mistake, and deals with the consequences. That’s important ... and very tricky to do properly.
As usual, all the extremes are wrong. Yes, it is possible for the teacher to be too slick, too over-prepared ... but it is of course possible, easier, and far more common to be under-prepared, which not good either.
Again, this applies to pretty much everything, not just lectures.
It is also super-dependent on where the audience is coming from. The level of slickness that is appropriate for an invited talk at the annual American Physical Society meeting is wildly different from what’s appropriate in September in the introductory physics course.
For starters, consider the ultra-basic advice:
|The introductory students “should” have been exposed to this advice every year for the last 10 years ... but they probably haven’t. What’s worse, they probably haven’t been exposed to anything complicated enough to make scrupulous checking worth the trouble. Therefore it is important for them to see you carry out the checks.||In contrast, the APS audience already knows the importance of checking, and assumes you checked the results 17 different ways at home. Therefore they don’t need to see the details.|
As a related point, the introductory course isn’t just about physics factoids; it’s also about problem-solving techniques. So it’s important for students to see how it works when you make a mistake, catch the mistake, fix the mistake, and proceed from there.
True story: I learned some of this by watching Richard Feynman. One day he was going over the midterm in an undergraduate course. In the middle of solving one of the problems, he mentioned that a lot of students had made a certain mistake. He spent some time explaining how to recognize that mistake, and (!) a whole family of similar mistakes, so as to make it easy to stay on the strait-and- narrow path in the future.
Think about it: Feynman never made that mistake in his life, not recently, not when he was an undergraduate, not ever. He was way too smart for that. So how did he know that the students were vulnerable? It wasn’t from grading the papers; he had TAs to do the grading. Here’s how it really worked: He talked to the graders, looked over a sample of the graded papers, and then talked to some smallish percentage of the students. (This is what he didn’t do enough of when presenting his famous lectures in 1961/62 and 62/63. Evidently he learned from the experience.)
So, ironically, students got to see him deal with a mistake, not because he was under-prepared, but because he was more prepared than most of them could imagine. He was of course prepared to discuss the right answer, but he was also prepared to say useful things about the wrong answers. He put serious time and effort into it, far more than the minimum required.
On the other hand, it would have been even better to teach students how to stay on the strait-and-narrow path before the midterm. Making mistakes is part of the game, but it is better done on homework rather than on the midterm. So there is room for improvement.
Again, I am not arguing in favor of being under-prepared. For most teachers, being under-prepared is not an option; they know the material backwards and forwards and they will never be able to unlearn it. The point is, being under-prepared forces you to go slowly ... but the skilled teacher can go slowly without being under-prepared. Being under-prepared forces you to show the intermediate steps, show the checks, and recover from mistakes ... but the skilled teacher does all that without being under-prepared.
It’s a bit ironic, but this is one of the ways in which a video is better than a live lecture: the student can pause and/or backspace the video. So being “too slick” is slightly less of a problem on video than it is in person. Different students need different pauses ... different amounts in different places. Books can also be paused. Web documents with lots of hyperlinks are even better.
Although finding and fixing mistakes is part of the game, mistakes should not be the first thing students see. They should see the right approach at least 10 times before they start encountering mutations. Otherwise they pick up misconceptions that are super-hard to unlearn, as discussed in section 3. So this is yet another reason why being under-prepared is not advantageous; you want to control how and when the mistakes are introduced. You can even telegraph the mistake in advance: “Let’s redo the calculation and see what would have happened if we didn’t keep enough guard digits....”
The idea of learning from mistakes applies to students also, not just to teachers. Often they learn more from lessons where they screwed up and then had to deal with the consequences.
At the next level, at some point not too near the beginning of the course, students need to learn how to handle ill-posed questions, where there is no answer that fits all the facts, as discussed in reference 2. Such questions abound in the real world. The under-prepared teacher will generate plenty of these by accident, which is not good. The skillful teacher will control the introduction of such questions, after explaining how to deal with them.
More than once upon a time, there was a young student who kept getting bad grades on the assignments, because he didn’t show the work.
Student: The answers are correct, aren’t they?
Teacher: Yes, but you have to show the intermediate steps.
Student: There aren’t any intermediate steps. I just figured it out.
Teacher: But you have to show the intermediate steps.
Teacher: It’s the rules. Don’t blame me; I don’t make the rules.
Teachers complain bitterly when students blindly follow some procedure without understanding what they’re doing. However, they shouldn’t be surprised. All too often, the entirely foreseeable effect of the lesson is to encourage conformity and obedience, while discouraging reasoning.
Every member of the school board emphasizes that they want to encourage critical reasoning. In our example, the student wasn’t asking for a rule; he was asking for a reason.
The following should be treated as advice, not as arbitrary rules:
In contrast, when the student is given a non-complex non-teamwork situation, it doesn’t make sense to convert this advice into a set of arbitrary and unhelpful rules.
Education is supposed to help the student. The teacher’s job is to make sure it really is helpful, and then to make sure the student recognizes it as helpful.
In this particular example, good ways to proceed include:
Whenever the topic of misconceptions comes up, there is an important contrast that needs to be made:
|Explaining the correct conceptions should always come first.||Confronting misconceptions can come later, if at all.|
|“The light shines in the darkness, and the darkness cannot overcome it.”||“You cannot beat something with nothing.”|
|By way of analogy: If you have a healthy lawn, you won’t have much trouble with weeds. They can’t compete.||If the lawn is weak and sparse, due to insufficient water and fertilizer or otherwise, weeds will invade.|
|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.|
|Correct ideas need to be linked to each other, and supported by evidence.||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.|
|On this web site, there are hundreds of documents that try to explain correct ideas.||On this site, there is only one part of one document – section 5 – that has focuses on misconceptions, and it should not be the starting point.|
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:
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.
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.)
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 more detail: Every student who understands the material understands it in more-or-less the same way, but every student who is confused is confused 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.
Section 5 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.
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 3.
On top of that, even though statistics will tell you what’s common and what’s not, that is nowhere near sufficient; it takes serious judgment to decide what’s important and what’s not.
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 4.
A great many misconceptions are created by the way the subject is taught. We call these didactogenic misconceptions. For example, ask yourself which will reach the ground first: a ball thrown downwards or an identical ball thrown horizontally from the same height? According to reference 5, only students who had taken physics got this wrong.
Many didactogenic misconceptions are preventable. So, one might naïvely think the solution would be to put a big sign at the back of the classroom where the teacher can see it: DON’T TEACH WRONG STUFF.
That would be consistent with the recommendation of wise guys of every era, from Hippocrates to Henny Youngman.
However, for reasons discussed above, that’s not the whole story. We need to take a more nuanced view. I say there are roughly three categories of misconceptions:
(Tangential remark: This trichotomy appies to misconceptions in general, but for now let’s keep focusing on didactogenic misconceptions in particular.)
To a highly nontrivial degree, teaching involves managing whatever misconceptions arise ... without entirely preventing them. Sometimes this involves making a mess and then spiraling back to clean up the mess before too much damage has been done. Cleaning up messes is expensive, but a strict no-mess policy would be impossible.
Clarification: When I say teaching involves managing misconceptions, that means (with rare exceptions) encouraging correct ideas rather than directly confronting incorrect ideas. As discussed in section 3.1, the maxim should be: “The light shines in the darkness, and the darkness cannot overcome it.” Similarly, another maxim is: “A healthy lawn crowds out weeds.” There is a lot of evidence that talking about a misconception is more likely to reinforce it than to dispel it. In addition, there is the Anna Karenina principle: given 30 students, they will have 75 different misconceptions, and it is simply not possible to directly address them all. To clean up the mess, you do not need to understand every detail of the mess.
Also: The whole idea of “didactogenic misconceptions” is a bit of a chimera.
Constructive suggestion: If you’re writing a textbook, please footnote this kind of stuff. Say at least something about the range of validity. For example, if you write S = log W, please add a footnote saying something like “this is valid in a microcanonical situation only. See page 456 for a more robust expression.”
This idea applies to the definition of “entropy”, the definition of “atom”, the definition of “energy”, and a thousand other things.
Rationale: A textbook is not an ideal reference book (nor vice versa). The organization is completely different. The spiral approach is appropriate for a textbook but not for a reference book. However, people use their textbooks as references nevertheless. I get really tired of arguing with people who say the entropy «MUST» be equal to log W because it says so right there on page 123 of the textbook. Ditto for the definition of “atom” and “energy” and a thousand other things. If we could make students aware of the more robust discussion on page 456, everybody would be better off. They don’t need to hear the whole story on the first trip around the spiral, but they need to know there is more to the story.
Another constructive suggestion, from the keen-grasp-of-the-obvious department: Don’t teach wrong stuff. Don’t teach stuff that is 100 years out of date for no reason. Don’t create big messes for no reason. Cleaning up misconceptions is very expensive. Just because some misconceptions is inevitable does not give you a license to create misconceptions with reckless abandon.
There is waaaay too much of this nowadays. When I look through a quote «modern» quote «research based» textbook from a «Big Name» publisher, I see hundreds of easy-to-fix conceptual errors, without even looking very hard. (This is not counting typos, and not counting multiple occurrences of the same error.)
Consider the contrast:
|Sometimes an idea is just completely wrong. Such ideas are relatively easy to deal with.||However, in many cases, you are dealing with a notion that contains a germ of truth. It is important to recognize this. The most dangerous ideas are the ones that are usually mostly true, but then betray you at some critical moment.|
One classic example of a misconception that is “almost” true 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, usually not a misconception at all. 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.
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:
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.
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:
It would be a fundamental mistake to assume that there is only one definition of acceleration. (In contrast, neither definition is fundamentally wrong.)
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:
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:
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
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 8, 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:
There is however another way of looking at this situation.
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.
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.
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.
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 9.
Keep in mind that you should always start by emphasizing correct conceptions, as discussed in section 3.1. 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.
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.
It is easy to find examples of professors being completely confident about the wrong answer.
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 14.
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.
– 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.
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).
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 24.
An example involves 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 a notorious OTBE fallacy. For example, T1/P1=T2/P2 is not valid if we hold N and S constant. See reference 20.
Fussing with the terminology will not change these facts. See reference 25.
Fact: The isolated gas-phase ammonia molecule in its ground state has D3h symmetry. See reference 26. (It has a much lower symmetry, C3v, in liquid ammonia or in aqueous solution.)
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.
Hint: John-John Kennedy probably didn’t know the right answer to this one.
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 4.
A great many misconceptions are being spread by the NGSS, as discussed in reference 31.
Copyright © 2010 jsd