# How can teachers warn students about common mistakes without causing the student to make the mistake?

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

--

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.

• 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. – James S. Cook Jul 21 '20 at 5:21
• 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. – Ben Crowell Jul 21 '20 at 14:36
• +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. – Daniel R. Collins Jul 22 '20 at 22:42
• 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. – J... Jul 23 '20 at 17:53
• @LSpice: In a class session (roughly). – Daniel R. Collins Jul 23 '20 at 23:39

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.

• "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. – Opal E Jul 21 '20 at 21:38
• 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... – Sam OT Jul 22 '20 at 10:29
• 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. ... – Sam OT Jul 22 '20 at 10:29
• 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 – Sam OT Jul 22 '20 at 10:30
• 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?). – Hans Olsson Jul 22 '20 at 15:37

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.

• Are they now scared of making mistakes because they'll have to do public speaking? – user253751 Jul 22 '20 at 12:03
• @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. – Doug Deden Jul 22 '20 at 19:58
• 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). – Philip Oakley Aug 5 '20 at 9:54

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.

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.)

• 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. – JoelFan Jul 24 '20 at 18:08
• "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. – Xander Henderson Jul 24 '20 at 18:38
• @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. – LSpice Jul 24 '20 at 19:31
• 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?) – Philip Oakley Aug 5 '20 at 9:59

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.

• It's not $\frac {k f'(x)}{f(x)}$, it's $\frac {du}{u}$. – Acccumulation Jul 26 '20 at 5:14
• 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 $f(x) = u$ – PhysicsMathsLove Jul 26 '20 at 14:30
• My point is that speaking of $f'(x)$ isn't good pedagogy. We take the integral of $f(u)du$, not $f'(x)$. – Acccumulation Jul 26 '20 at 21:37
• 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 – PhysicsMathsLove Jul 27 '20 at 12:03

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.

This question is just old wine in new bottles. The situation referred to is true across the board in cognition. The mechanics of the situation is that mls always trumps gls, ‘mls’ standing for the ‘momentary life situation’, and ‘gls’ standing for the ‘general life situation’, these terms having been introduced by Kurt Lewin, the founder of Social Psychology. Therefore anyone who gives any thought to this situation realizes, at least intuitively, the problem with foregrounding an error. As is often the case, humor can make the issue very clear, as in the well-know joke/insult: “Your house burn down and all your family perish in the fire, God forbid.” In Language Arts, when it is felt necessary to cite an erroneous item, it is customary to precede it by an asterisk. There is another phenomenon at play here: any foregrounded condition is taken as necessary. (That is why definitions, even in a rigorous discipline like Mathematics, are given simply in ‘if’ form (as opposed to ‘if, and only if,’ form) – the ‘only if’ (necessary) condition is universally understood by the mere fact of foregrounding.) Thus we come to an error inadvertently introduced by mathematics teachers, namely, the belief that the logarithm (function) doesn’t exist. By heavily foregrounding the fact that n is not equal to -1 in the well-known integration formula, students internalize that n not being equal to -1 is a necessary condition, and therefore the integral of f(x) = x raised to the -1 power ‘does not exist’, all the more so because this integral is typically not dealt with until a significant amount of time later in the course. All this is due to operating at the object-level. There is not problem if, as in discussions like this, you clearly move to the meta level – but moving to the meta level on the fly in a classroom presentation and quickly transitioning back to the object level can be difficult. Meta probably has to be done in discussion mode, not merely lecture mode.

• What are you talking about? I can't figure out what your answer to the question is. – Chris Cunningham Jul 22 '20 at 0:07
• If this is trolling. My hat is off to you good sir. – James S. Cook Jul 22 '20 at 0:30
• This actually seems like an answer with helpful ideas in it, but it's written in a way that makes it hard to understand and makes it easy to mistake for trolling or babble. I'd suggest editing to break this up into several paragraphs -- people don't like to read walls of text. Mark up math using mathjax, and write out "the well-known integration formula." Then, finally, you might want to reread what you wrote and put yourself in the shoes of someone else trying to understand your point. – Ben Crowell Jul 22 '20 at 1:00
• From what I could understand, using my integral as an example, "common mistakes" develop from not understanding the logic behind the rules. To quote the answer, "By heavily foregrounding the fact that n is not equal to -1 in the well-known integration formula, students internalize that n not being equal to -1 is a necessary condition" Thus, in theory, a student who knows the rules for integration laws wouldn't make such a mistake since they wouldn't misapply them. (But wow, either my English skills need work, or I should read a bit more Victorian literature) – user13544 Jul 22 '20 at 6:18
• @EulerSpoiler If you aren't interested in tailoring your answers to a slightly more general audience, then StackExchange sites aren't a good fit. StackExchange is designed for people unfamiliar with your insights into General Systems Theory to find these questions and answers years later and maybe get a piece of your insight. If you are interested enough in this process to post here, then you should also be interested in splitting your contributions into paragraphs, giving "dumbed-down" summaries as invitations to interested readers, and engaging in constructive discussion. – Chris Cunningham Jul 27 '20 at 18:58