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

Question

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.

Answer 1

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.

Answer 2

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.

Answer 3

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.

Answer 4

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.)

Answer 5

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.

Answer 6

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.