How to handle the situation where a student insists I am wrong during the class?

I had one very vocal student in my Calculus recitation last year. Sometimes she would point out if I made a mistake in the lecture.

However, sometimes she would insist that I had made a mistake, even if I concluded it was NOT a mistake after careful consideration. She would continue to contradict me in front of the class, which was disruptive.

(I am a graduate student, so I probably seem less authoritative.) I want her to keep thinking critically and carefully during lecture. But disrupting the lecture excessively is a problem. How would you handle this situation?

Here is one example. I stated that the cotangent function is decreasing where it's defined, and gave some explanation based on the sine and cosine functions. She insisted cotangent was increasing. The encounter was rather short and I moved on quickly, but she stopped coming to recitation soon after. I think she was offended.

• I've met people who did such just for vandalism. – vonbrand Apr 16 '14 at 21:59
• How do you react when she is right? (This is the behavioral standard you can hold her to.) – user11235 Apr 20 '14 at 13:42
• @SueVanHattum, here is one example. I stated that the cotangent function is decreasing where it's defined, and gave some explanation based on the sine and cosine functions. She insisted cotangent was increasing. The encounter was rather short and I moved on quickly, but she stopped coming to recitation soon after. I think she was offended. – kathleen May 7 '14 at 23:59
• Strange behavior, that. People often look offended when they are embarrassed, so I'd chalk it up to a bad day for her, and not worry about it. – Sue VanHattum May 13 '14 at 15:51
• Side comment: cotangent is not decreasing on its set of definition. It is only decreasing on each interval contained in this set. This is related to common erroneous conceptions of students, and care here is valuable. That said, cotangent is not increasing on any interval, so this does not make her right. – Benoît Kloeckner Dec 5 '18 at 11:13

It depends on how much time I can afford to spend on the problem. I check, as you did, whether I see a mistake. If not, I try to explain the math in a different way: a different conceptual approach, which is good for everyone anyway; perhaps replace the variables with numerical values, if appropriate; or replace a general function $f$ by a specific function. If there is time and the student seems fixed in her idea, I might ask her for an example. Often the student is assuming something extra or false, and her example might reveal it. Also, I can check the example and confirm again that there is no mistake. Finally, if there isn't time to work it out, I ask the student to see me after class or in my office hours. ("Maybe we can work this out after class. Are you free?")

Of course, it's very helpful to remain calm, receptive to the criticism, and seem happy to look into it. Controlling the tension in my voice and the look on my face took me a few years to learn. I now tend to seem excited by investigating mistakes. In part, I know I'm doing at least one of two good things: clarifying the lesson or dispelling someone's misconception.

• It doesn't look like anyone has mentioned it, but the first step might be to ask, "Show me where I made a mistake." If they can point to a line on the board, at least you now know what to discuss. If they can't, they just know that the answer is "wrong", then the next question is how do they know it's wrong? Did they memorize the "correct" answer from a book or web page, or is it just a feeling? If it's a website or book, you again have a point of reference, probably to be used out of class. If it's just a feeling, office hours or after-class discussion. – Wayne Apr 16 '14 at 22:26

I agree with @kathleen, that being a young (and female) instructor can lead some students to be less respectful. In this case, we can use that to our advantage...

Long ago, when I was less secure as a teacher, I would be so embarrassed and bothered by making a mistake that it would decrease my ability to teach well for the rest of that class period. I knew that I couldn't be sure of never making mistakes, so early on in my teaching career I came up with a way to make those mistakes a positive part of classroom culture.

At the beginning of the semester, I explain that I will make mistakes from time to time, and I want students to catch them, so they do not stay on the board, and get written into students' notes. Every mistake a student catches is a 'donut point'. When the class has caught me 30 times over the course of the semester, I will bring in donuts. I tell them that there is a second reason for this system. Students too often believe anything a math teacher says, and that's not a good way to learn math. They should be questioning (in their minds) everything I say. Students like this system. Some try too hard to catch me, and I joke about them being awfully hungry.

Now, when someone thinks there is a mistake, and I show there wasn't, they are usually apologetic. I say that's no problem, because they have been brave enough to trust their own reasoning, which is a good way to approach math. If the person were to continue to insist (this hasn't happened to me, at least not in many years), I'd ask the class how many followed my reasoning. If even 2 or 3 people admit to not following my reasoning (there are more not admitting it), I would find other ways to explain. If this student is the only one, I'd ask them to see me in my office for help.

• This is such a wonderful answer. It's so much more than merely a tool to decrease class interruptions, it's a masterclass in pedagogy and maybe even a life philosophy. – Connor McCormick Oct 21 at 17:51
• Why thank you. I needed that just now. Interestingly, it is much harder to keep the donut points system going when I can't offer them real donuts due to covid. But I do keep it going, and record the points on our canvas home page. – Sue VanHattum Oct 22 at 10:12
• <3 I mean, it’s not even the donuts per se, it’s the willingness to be wrong – Connor McCormick Oct 22 at 11:02
• I've always had the impression that mathematicians were better at that than other people. But maybe I'm wrong about that! ;> – Sue VanHattum Oct 22 at 19:18

I have had this problem before with students who always think they're right. If a student continues to insist you made a mistake, when you know that you haven't, then tell the student to hold the thought and ask them to discuss it with you after class. Once the class is over, write the original problem on the board and ask the student to solve it. If their answer is correct, congratulate them on their thinking. If their answer is incorrect, show them why they are incorrect.

Secondly, make sure you plan your recitation and do the homework. I had a student that kept contradicting me one day in class, even after I showed him where his thinking was flawed. I told him to wait after class. It turned out that he had used the Chegg website to get his answer, and when we went to the website to look at it together, the answer was incorrect.

Put the onus back on them to prove to you they have a correct answer. This is very common in College Calculus, especially from students who took AP Calculus in High School.

• Very good advice. My offspring tended to be vehement (but not vociferous) about her solutions, and luckily was completely OK with being asked to take the conflict offline. – Carl Witthoft Apr 16 '14 at 14:24

Taking some extra care to verify the details of a confusing bit of material (preferably with class involvement, intuition-building examples, etc.) is usually an excellent use of class time. Offering to talk about the issue further after class or during office hours is also a reasonable tactic when time is short or when dealing with a particularly insistent student.

However, I think there are also times when it is appropriate to take a firmer hand. A balance must be struck between nurturing critical thinking and being fair to the rest of the class—the loudest student should not get to dictate the course of the lecture. It's one thing if a good portion of the class is confused and one student is simply voicing their collective concerns. But it also happens that one student decides to treat the class as if it's their own personal tutoring session. In this case, it's appropriate to tell them to stop: "I understand you disagree, and like I said I'll be happy to assuage your concerns after class, but right now I need to finish this chapter, so please hold your comments."

In an ideal world, perhaps, classes have no official "end" time, the students are there to learn, and they simply stay until they are satisfied with their own understanding. But unfortunately, that's not the way most university systems work, and the types of classes graduate students are likely to be TAing tend to come with very tight time constraints. Something has to give.

Your student reminds me of me in my first algebra class, in 8th grade. I insisted that my answer was a 'better' solution to a homework problem. The teacher was an ex-Marine, who took the time to step outside the room with me and say something to the effect of "I understand you disagree with me about this problem, but I'm responsible for teaching this class, and I can't have you undermining my authority in the classroom, nor can we hold up everybody else's learning. I'm happy to talk about it with you, outside of class." It was the most respectful treatment I'd ever received from a teacher, and it meant a lot to me that he was willing to speak frankly to me this way. A math classroom isn't just about being right or wrong, it's a social activity.

Projecting wildly from my own life, the student might be insecure, have been rewarded excessively for being 'smart' or 'right', and be a bit behind the curve in social skills. Frankly pointing out the non-technical aspects of the situation can be a big help to a student like that ;-)

• Nah. Math is one of the few things that is not about authority but about correct or not, and often there are several ways of being correct. This ex-Marine should have been rather shaky in his math if he went the authority route. He is a typical military type. Not impressed. – Rusty Core Dec 5 '18 at 4:28
• @RustyCore I agree that mathematics is not about authority, but classroom teaching often requires that the teacher maintain that they are the authority in the room, which I believe was what the teacher in this example was expressing. A high school/middle school teacher should never engage in an argument with a student in front of the classroom, as a matter of maintaining classroom management. – Opal E Dec 5 '18 at 17:46

I stated that the cotangent function is decreasing where it's defined, and gave some explanation based on the sine and cosine functions. She insisted cotangent was increasing.

How about this: "Hmm...I'm pretty sure my argument was right, but maybe I'm just having a brain fade this morning, so let's check. Those of you who have a calculator handy, could you pull it out and calculate cotangent of 0.1 and cotangent of 0.2, so we can check?"

Advantages of this approach:

(1) It demonstrates the correct real-world technique for checking your own work, which is to find some independent way of checking it, rather than just going over your own steps and looking for a mistake. Recapping your own work usually doesn't work because you're mentally locked in to your own steps.

(2) If the class sees two different ways of establishing the same fact, it should erase any doubt in their mind as to whether you were right.

(3) It models civil behavior toward other people when you are disagreeing over an objective fact. You are humble enough to admit that you could be wrong, and you don't dismiss the other person's objection based on status or bluster. You approach it as a situation where the fact is what is to be discussed, and if there is a mistake, what should be criticized is the mistake, not the person.

(4) Admitting the possibility of your own error has the paradoxical effect of increasing your authority as an expert; denying that you could have made an error is behavior that people will easily recognize as a sign of insecurity. The expert is not in doubt about their knowledge or self-worth, so they approach the situation in a relaxed way.

• "Those of you who have a calculator handy, could you pull it out and calculate cotangent of 0.1 and cotangent of 0.2, so we can check?" - calculators, really? Cotangent is adjacent to opposite, going counterclockwise in the first quadrant the opposite increases while the adjacent decreases, here you are, five seconds to dispel the wrong statement. – Rusty Core Dec 5 '18 at 4:34
• @RustyCore That is fantastic, but it requires your students to understand something that is somewhat abstract. Many students will grok the intuition, but are not strong enough to really internalize the idea. Giving them an alternative demonstration to help convince them is good. The fact that it isn't mathematically rigorous is not as important---it is about having independent methods for checking results. – Xander Henderson Dec 5 '18 at 4:39
• @XanderHenderson Two years later, I am amazed that your comment got six likes and my got none. My son is in high school now, they gave him a big-ass graphing calculator for solving simple linear equations (despite that he is supposed to have 10th grade Math, huh?) Really, I think you and Ben are so wrong in your mechanistic, calculator-based approach. My son would reach the damn calculator at every simple case, I took it away, but now because he has remote Covid-safe classes, he uses web tools like Desmos. Pitiful. – Rusty Core Oct 19 at 20:43
• @RustyCore I take offense at your accusation that my approach is "mechanistic" and "calculator-based". My point was that students and instructors should have multiple ways of verifying results. In the example given here, the student did not believe the algebraic and/or geometric arguments. So, rather than repeating yourself, give them a visual---have them look at the graph. Then go back to the algebra and see how it matches the intuition given by the calculator. There is a difference between relying on a calculator, and using a calculator to help build intuition and understanding. – Xander Henderson Oct 19 at 21:31

Humbly accept that you could be wrong, and be open to feedback. Turn it into a teaching moment when possible by inviting the student to come up and show their solution. If they are wrong, don't tell them why, but instead ask they class if they see any problem with the approach. It will not only get the class involved, but it will humble the student from trying to show off their knowledge. If you are the one that is wrong, simply admit it, explain the error in your thinking, thank the student and move on. This way it moves away from an "I'm right and you are wrong" argument, but a way for other students to recognize where mistakes are possible and to learn from it.

If the student is consistently disruptive, speak to them personally and ask them not to be disruptive, but to please bring their concerns to you privately. Obviously, you want the class to know when/if you are making mistakes, and you should humbly tell the class when the student does reveal an error in your logic so they can learn from it. But you don't want the disruptions to become a distraction from the materials you need to get through.

Look at it this way: would you rather focus be on proving the student wrong and defending your reputation, or would you rather focus on helping the class to learn, regardless of who is making mistakes? We all make mistakes. Help your students to learn from them.

It is not good enough for you to "think carefully" and conclude and state that you did not make a mistake.

You need to prove to yourself and the class that you are correct.

If you cannot do this on the spot show a bit of humility and tell the class that you think you are correct and you will confirm via email ASAP after the lecture.

• This is the situation: A student thinks the teacher makes a mistake, the teacher says she didn't, the student insists that she did. I am saying it is not good enough for the teacher to just have a situation where she "concluded it was NOT a mistake after careful consideration". She needs to prove she was correct. – JP McCarthy Apr 15 '14 at 11:42
• If the student is not being satisfied then I am not surprised that she is making herself loud: it should be black and white and this isn't a student saying she is not understanding, this is a student claiming that the teacher is wrong. In mathematics we should be able to clear this up. – JP McCarthy Apr 15 '14 at 12:08
• Surely the common case is that the student states there is a flaw in the proof and the teacher states there is not. Since you cannot practically provide all proofs in fully written-out formulae of predicate logic, there isn't really any difference between "thinking carefully" that you haven't made a mistake, and believing that you have proved you're correct. One student cannot keep holding up the class by rejecting proofs that the others all accept, or else you'll be lucky to have proved that 1+1=2 by the end of an hour. – Steve Jessop Apr 15 '14 at 15:07
• ... which might be reasonable progress in a set theory class, not so much in calculus :-) – Steve Jessop Apr 15 '14 at 15:08
• @staticx The benefit is that if the student is correct then she can teach everyone something. (And if the student is incorrect, then she can learn something, but I agree that it would be selfish of her to interrupt the class too often in this case.) – Trevor Wilson Apr 17 '14 at 0:31

This is a tricky one, since there's a combination of two things:

1. The student is probably genuinely interested in the math and wants to grapple with the material.

2. The student is disrupting the classroom, and interrupting the lesson for other students.

On the one hand, you want to foster the student's interest and enjoyment of mathematics. On the other, you can't allow a single student to disrupt the lecture repeatedly. I think as others have suggested, a somewhat delicate approach is needed.

I recommend that the next time the student does something along these lines, don't react angrily or allow them to fluster you, but instead in a very calm way slowly walk through the derivation and show why the student is incorrect. If it can be done with elementary methods, use them. But if not, don't necessarily shy away. The goal is simply to get to the truth. The student needs to recognize two things: (1) you, the instructor, are an expert on the material, (2) they, the student, still have a lot to learn. Being slow and methodical will hopefully make the student realize just how disruptive they're being, and responding calmly will help defuse any desire they have to elicit a reaction from you.

Having been an instructor for many freshman non-major math courses, in my experience many of the students are still very much like high-schoolers, unfortunately. But they're at an age where they want to be seen as more mature. If you keep the level of classroom maturity high, they'll respond well. But if you get dragged into a conflict with a student, it will only get worse.

Lastly, I think it's important to remember that this is only one student among many. If they're innocently asking many questions out of curiosity, it's definitely not malicious and shouldn't be treated as such. But if you get the impression that they're trying to disrupt the class for their own enjoyment, that's a serious problem, since they're detracting from the learning experience of the entire class, and I would have zero patience for students like that. If you get the impression it's the latter, I wouldn't feel badly about reporting the student to the professor. Everyone else (or, more realistically, their parents) is paying good money for the opportunity to learn in that class.

This happened to me this year briefly. "I have literally a degree in this. I have done it professionally now, for something like 5 years. When I say it's this way, it's absolutely this way." Shut them up immediately.

Edit: Since more details were requested, I will follow this up. You have three problems:

• They don't respect you

• They are disrupting your class

• And they are being rude TO you.

The most immediate issue is the classroom disruption. It is evident from your comment that you are not in charge of the situation here. This may be an unusual situation, or it may be that you were never in charge and this student is making it substantially worse. In either case, the correct thing to do is to move on without them. Put them into a position where they have to accept your decision to move on, or brazenly violate your control of the class. State you are moving on, AND THEN DO IT. You are giving them permission to disrupt your class by doing anything else. Don't consent to it. The other students in the class are well aware the student is being a jerk - the social pressure exerted on them when it is apparent you know what you are doing and they don't is enormous, so if they insist on doing the fight publicly, you can win it. They WILL shut up.

Next up, there is the issue that they don't respect you/your abilities. You can address this directly as I did in my pithy answer, or you can do it indirectly. This student is evidently confident about their math abilities - you are a grad student. You want to see them realize how out of their depth they are? Put some real analysis on the board. Or better yet, some complex analysis. Nothing like a zeta function to terrify a student. If they are the keen and excited student you describe them as, this is the perfect way to deal with the issue. (Edit: to be clear, I suggest you do this after class, in private)

Last, if all this fails, you have the issue that they are being rude to you, and apparently your efforts to control them in class or gain their respect have failed. At this point you need to deal with this directly through the university bureaucracy. This is the worst case, but it will deal with the problem. You have a student who is refusing to let you teach, and you have every right to have them removed. This isn't elementary school, they don't have a legal right to this education, if they don't like it they can get out. Just the threat of this is sufficient in the once in a VERY rare while it is needed. I've TAd for a decade now, and I've only had to threaten this once, and certainly I've never needed to carry it out.

• Since you asked, I edited it to include a few more details. – moose May 5 '14 at 0:29
• One huge problem with this approach is that if you're in the classroom for long enough, it's guaranteed that you will make a mistake now and then. If this is your automatic approach to dealing with a student who thinks you've made a mistake, then pretty soon your students will have seen a case where you try to use authority to deal with your own mistake. – Ben Crowell Dec 5 '18 at 2:38