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For example, if you're teaching integration of $\int \frac{dx}{1+x^2}$, would you mention the common wrong answer of $\ln\left(1+x^2\right)+C$?

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For myself, I very rarely mention common mistakes since I'd feel I'm possibly causing the very problem I'd seek to avoid. But if I do mention issues, I will emphatically state before and after that it's the wrong approach (and often, I'll just write it on at the edge of the board and then promptly erase it so it's not written down by students)

One of the reason why I avoid warning of common mistakes is because I can imagine how a student could see the intuitive appeal in the answer, even though the logic is flawed.

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    $\begingroup$ You make sure to put giant red X through the untruth once you're done introducing it. Also, take joy in teasing the students who still make the mistake even though you've warned about it. For example, I tell students, if the eigenvector equations force the eigenvector to be zero then your "eigenvalue" is wrong. Go back, fix it. Still, so many just punt and put down the zero vector as an eigenvector. Anyway, don't erase, cross out so the kids lagging 5 minutes behind also have visual record of it. $\endgroup$
    – James S. Cook
    Jul 21, 2020 at 5:21
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    $\begingroup$ Your question is more general, but I see two specific characteristics of this example that are important. (1) This mistake won't show up if you've pounded it into your students that they should always check an indefinite integral by differentiating (and assuming they understand the chain rule well enough to get the derivative right). (2) The mistake is a symptom of students' desire for integration to be a mechanical and rule-based process like differentiation. $\endgroup$
    – user507
    Jul 21, 2020 at 14:36
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    $\begingroup$ +1 For the question. I think this is a great thing to be aware of, and likewise, I try to avoid the "common error" warnings until they actually show up at least twice. I've seen this issue mentioned in passing in an article, but wish there was more prominent research findings on it. So we could spotlight to other educators this common error. $\endgroup$
    – Daniel R. Collins
    Jul 22, 2020 at 22:42
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    $\begingroup$ It's math - if the answer is wrong, demonstrate that it is wrong using mathematics. You're not teaching a set of facts, you're teaching a method that allows students to determine the correct answer for themselves. Show them how that works. $\endgroup$
    – J...
    Jul 23, 2020 at 17:53
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    $\begingroup$ @LSpice: In a class session (roughly). $\endgroup$
    – Daniel R. Collins
    Jul 23, 2020 at 23:39

6 Answers 6

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This is a 100% subjective opinion, but it is based on teaching in various venues for close to 20 years (although none of that teaching was pure math). Also, my college calculus courses are close to 30 years behind me, so please excuse me if my examples aren't directly related.

IMHO, one of the biggest mistakes in teaching is failure to compare and contrast items that are similar in some ways but different in others. I mean, that's where students make the most mistakes, right? It's failure to pick up on fine distinctions.

I agree with the other replies that say "Yes, you should point out this common error." But I would go further than that. Don't present it as the "wrong answer." Present it as the "right answer to a different question." Then go into the details of why it's a different question and how students can tell the difference.

As an alternative to that direct approach, homework assignments that mix and match the two types of questions can give students the "ah ha!" moment to see the difference. That's how I learned it.

March 2023: Based on comments, my attempt at describing "the right answer to a different question" wasn't successful. Maybe I was trying to be too poetic. All I meant by that is that this hypothetical student correctly recalled the formula $\int\frac{1}{x}dx = \ln|x| + c$. The problem is that she is applying it in the wrong situation.

My own calculus is rusty, so I hope I'm getting these details right and using appropriate terminology. But she hasn't learned that x in this situation can't be any expression. It must be the variable of integration (as shown in dx). It can't be a polynomial in x and it certainly can't be a composite function of x. Those situations require different methods. And it's comparing and contrasting those methods, explicitly calling out when they apply and when they don't apply, that helps students. Sure, the requirements for using $\int\frac{1}{x}dx = \ln|x| + c$ were probably covered when the formula was first introduced. But because homework problems for that night were all meant to practice that same situation, she never had to practice determining if a problem met the requirements or not. Also, it's not until after other methods (appropriate for polynomials and composite functions) are covered that a student will be in a position to fully appreciate a comparison and contrast between the different methods.

Some students will realize their thinking is muddled. They will go back to their notes, textbook, or office hours and dig until they understand the boundaries between concepts with fine distinctions. Other students either never realize their thinking is muddled, or they don't know what to do about it. That's where helping them through a comparison and contract can be very helpful.

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    $\begingroup$ "Mixing and matching the two types of commonly-confused questions" is often called "interleaving" and it is an excellent way to promote learning among students. $\endgroup$
    – Opal E
    Jul 21, 2020 at 21:38
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    $\begingroup$ I don't have formal maths teaching training, but have done a fair amount of small size teaching (2:1 student:me ratio). I really can't see how Don't present it as the "wrong answer." Present it as the "right answer _to a different question_." applies here. Sure basically every statement is the right answer to some question. If someone wrote $\int x^2 dx = 2x$ then you can go "ah ha, no, that is differentiation!". But in the example that the OP gave the common mistake answer is just the integral of a completely different function. Sure, they both have $(1 + x^2)^{-1}$ in the denominator... $\endgroup$
    – Sam OT
    Jul 22, 2020 at 10:29
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    $\begingroup$ but they really shouldn't be though of as that similar. One could -- and should -- say something like "ah, up to the additive constant, $\int (1 + x)^{-1} dx = \log(1 + x)$ but $\int (1 + x^2)^{-1} dx \ne \log(1 + x^2)$. Why is this?" \\ So instead of saying "the answer you gave is the right answer to a different [related] question", you say "the method you applied is correct for a different [related] question. Why doesn't it work here?" The difference between what I am saying vs what you said is suttle but, I feel, crucial. ... $\endgroup$
    – Sam OT
    Jul 22, 2020 at 10:29
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    $\begingroup$ Perhaps your suggestion is valid in other scenarios of which I am not familiar -- and you have an enormous amount of experience compared with my very minimal experience! -- but I have to respectfully say that I strongly disagree that this would be effective in the scenario described by the OP -- and in maths in general, which is very black-and-white in these ways $\endgroup$
    – Sam OT
    Jul 22, 2020 at 10:30
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    $\begingroup$ Building on "right answer to a different question" you can explain that it's easier to verify if an integral is correct than to derive it, and show that the "wrong" answer is the integral of 2x/(1+x2) (right?). $\endgroup$
    – Hans Olsson
    Jul 22, 2020 at 15:37
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Here's another approach when there is a common pitfall that you wish the students to avoid. After teaching the correct reasoning: present the error to the class and ask a student to identify, explain, and correct the mistake. Being able to correct others' mistakes shows a high-level of understanding, and students who make that same mistake might be able to realize and fix it after seeing how.

Then, the students are forming an understanding of why a mistake is wrong, rather than thinking that it is actually correct.

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  • $\begingroup$ Are they now scared of making mistakes because they'll have to do public speaking? $\endgroup$
    – user253751
    Jul 22, 2020 at 12:03
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    $\begingroup$ @user253751 I don't think Eliza is suggesting that you allow a student to make the mistake and then challenge them to correct it. Rather, the suggestion is that the teacher "make the mistake" and then ask someone in the class to identify/explain/correct it. $\endgroup$
    – Doug Deden
    Jul 22, 2020 at 19:58
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    $\begingroup$ At university, I found that others asking me for help greatly improved my work because they essentially found and explained the difficult parts of the questions. It's much easier finding fault in the work of others! I like the technique that simulates that approach (the student doesn't feel stupid for the initial wrong answer). $\endgroup$
    – Philip Oakley
    Aug 5, 2020 at 9:54
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I don't think you have anything to be afraid of here, unless you spend all of your time playing what-if games with common pitfalls, or peppering the board with false statements and not clarifying them as such. As a class conversation about mistakes-to-avoid or a way to encourage students to always check their answers, I think bringing these up can be really helpful.

One of the reason why I avoid warning of common mistakes is because I can imagine how a student could see the intuitive appeal in the answer, even though the logic is flawed.

These are common errors, whether you introduce them or not. You want to teach your students to apply some slow thinking to check an answer or analyze an argument, and having some examples ready that test their ability to do this on the fly could benefit them.

For my part, I always end a conversation of this type by writing "NO" or "FALSE" or "DON'T DO THIS" above the common pitfall we were discussing, so as to not leave something misleading on the board.

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I constantly present wrong answers, identified as such, in my class. My strategy for dealing with exactly the issue you present is that wrong statements, and only (intentionally) wrong statements, appear in red. I hope that this is a convenient and reliable visual cue. Students hate it, though.

One approach that has given me reasonable success, and that students, while not praising, at least don't complain about, is this: when I know students will make a common mistake, I present them with an exam question saying:

Your friend thinks that $$\int \frac{\mathrm dx}{1 + x^2} \qquad\text{equals}\qquad \ln(1 + x^2).$$

(a) What mistake(s) did your friend make?

(b) What is the correct solution?

Examples of this sort have to be chosen with some care—it is not always reasonable for all errors to expect the student to be able to answer (a) beyond "it's wrong"—but I think that this particular one fits well.

(This is quite similar to @ElizaWilson's suggestion. It avoids possible issues with fear of public speaking, but maybe just substitutes another kind of pressure.)

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    $\begingroup$ Watch out for excluding color-blind students. It's find to use color but also include another visual cue, such as underlining, the word "no", etc. $\endgroup$
    – JoelFan
    Jul 24, 2020 at 18:08
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    $\begingroup$ "Spot the error" questions used to be one of my exam go-tos. I still think that they are invaluable, and I use them in homework a lot. Signal to the student that something is wrong, then have them figure out what. These kinds of questions can be rewritten as T/F questions (with an "explain" component) as time goes on, with the eventual goal of completely removing the scaffold. $\endgroup$
    – Xander Henderson ♦
    Jul 24, 2020 at 18:38
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    $\begingroup$ @JoelFan, great point! Another totally unexpected point that one student (not in my class) brought up to me was his issue with synesthesia, which he said made it difficult for him to concentrate on lectures in anything but black, since the board colours for, e.g., numbers conflicted with 'his' colours. $\endgroup$
    – LSpice
    Jul 24, 2020 at 19:31
  • $\begingroup$ It's even easier if you actually have that friend ask you about their problem, as you get the mutual interaction. From the other side (the friend), it's the "Rubber Duck" technique. In a sense it's a critique of the 'own work' philosophy promoted in parts of academia (Isn't collaboration is the new norm?) $\endgroup$
    – Philip Oakley
    Aug 5, 2020 at 9:59
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You could perhaps use examples and non-examples.

This is less about explicitly telling students what the common mistakes are, but more about where the theorem or result applies and where it doesn't.

At high school, this might be showing students what is meant by angles in the same segment and what is not meant by angles in the same segment, for example.

In the case of your example, it might be about giving examples of what looks like $\frac{kf'(x)}{f(x)}$ and what does not. Discussion can then be focused about why thinking it would be a log integral is wrong and that students should consider other techniques.

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  • $\begingroup$ It's not $\frac {k f'(x)}{f(x)}$, it's $\frac {du}{u}$. $\endgroup$
    – Acccumulation
    Jul 26, 2020 at 5:14
  • $\begingroup$ If an integrand looks like $\frac{k f'(x)}{f(x)}$, i.e. the numerator being a multiple of the derivative of the denominator, then the integral will be $k \ln|f(x)|$. Of course your example is only a subset of the integrands I am referring to, with $f(x) = u$ $\endgroup$
    – PhysicsMathsLove
    Jul 26, 2020 at 14:30
  • $\begingroup$ My point is that speaking of $f'(x)$ isn't good pedagogy. We take the integral of $f(u)du$, not $f'(x)$. $\endgroup$
    – Acccumulation
    Jul 26, 2020 at 21:37
  • $\begingroup$ I don't know see any bad pedagogy; one can indeed take the integral of $f'(x) dx$. Have you seen the formula for integration by parts: en.wikipedia.org/wiki/Integration_by_parts $\endgroup$
    – PhysicsMathsLove
    Jul 27, 2020 at 12:03
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Whether you mention the common mistake or not, it is likely to be made by some student. I think the best approach is to mention this mistake but you need to be very clear about why it is wrong, and show the thought process that arrives you at the correct answer. Many educators concern about the consequences of introducing the idea of a wrong answer because it may concern students. But in my experience when I was a student, often in my study I would come across these wrong answers and it would have been helpful to be able to talk myself out of them being right, by applying the same thought process that my teacher provided me with.

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