I made a small mistake while teaching a very simple concept. It was mainly a numerical mistake, and the essential parts of the concept were well illustrated. I want to admit my mistake and even though most students will receive it well, I fear to lose my authority with some students, mainly the ones that think that they could teach the class.
As a student, I actually learned a lot from watching how professors handled mistakes. In a lecture, if you didn't intentionally make a mistake, this is an unscripted moment, and therefore a glimpse into how an experienced person deals with something unexpected. I came to respect professors who were able to make a mistake and fix it; it taught me how to think, and how to be honest.
By the same token, I lost respect for professors who were more concerned with the appearance of authority than just admitting to and fixing the mistake.
Addendum After reading some of the other responses here, I feel I should add that I'm a white male from the US, and didn't consider other cultures or underrepresented groups when writing this answer. I certainly see how perceived authority among students could become more of an issue if the cultural norm is that a teacher shouldn't be challenged, or if the teacher is working to overcome implicit bias. I still feel, on balance, that admitting to and fixing mistakes confidently is intellectually honest and will engender more confidence and trust than ignoring mistakes, and actually provides an opportunity for a positive learning experience. However, I am not an expert on these issues, so I wanted to highlight that as a caveat.
I think the genuinely legitimate authority of "the teacher" is that they have far more experience than their students. Not that they are perfect, etc. Yes, experience does tend to diminish mistakes, but does not eliminate them. In fact, part of "the lesson" can/should be about how to cope with inevitable errors! :) (as opposed to pretending that no errors will be made...) In particular, the teacher should not pretend to take up a "position" that requires "defense". :)
I make mistakes in class - not often, but more often than I'd like. I note them and correct them as soon as I see them (or a student points them out to me).
Sometimes I'm asked if I do that on purpose to see who in the class is awake. I reassure the class that there's no need since I err often enough by accident.
You may annoy the few troublemakers who think they can teach the class better than you, but you will earn the respect of the true learners.
Whoa, there is actually a lot to unpack in a post that short.
Presumably you're teaching in a university setting. What "authority figure" are you even talking about? If students look up to you, great! But it is not even remotely a part of your job to make sure you're "above" them in some imaginary hierarchy; your job is to teach them. Trying to enforce it in some way is usually unproductive both for the teaching and your "authority".
mainly the ones that think that they could teach the class
Do you actually feel contested? Why? If they want to teach the class, they are free to apply to the department asking to do so. If they don't, but feel like you're doing a poor job - why don't they serve as TAs helping other students? There are plenty of ways to handle this tension if it exists, and productive ones don't involve wrestling with the troublesome students.
Finally, when correcting mistakes, consider how impactful they were. If it is something silly like 2+2=5, either don't bother or mention it at the start of the next class briefly asking students to go over their notes since they didn't catch it earlier. If it is more impactful, dedicate some time to it. You have still delivered value to them, so chin up and try to see the situation through their eyes. No one wants to spend ten minutes on someone addressing insecurities more so than they do the actual content of the lesson. Stick to the point, and keep educating them: respect is earned by doing your job well. Your job is teaching, not some sort of power play.
Back when I was in high school I remember situations where the teacher makes an obvious mistake - obvious enough that even students who haven't mastered the topic can spot it, such as a simple arithmetic mistake - and doesn't admit it. Some students joked about it privately afterwards, but the underlying reason for the jokes wasn't that the students think they understand the material better than the teacher. It was because suddenly everything the teacher says is suspect.
For example, suppose the teacher says X. The students think about a homework problem unsuccessfully, and guess that maybe X is wrong. But the teacher insists X is right. Can you trust that X is right if the teacher never admits they are wrong? Even worse: at this point, the teacher is definitely not an authority figure.
You're better off biting the bullet and admitting the error, especially since it was mainly a numerical mistake and the essential parts of the concept were well illustrated.
Frankly, you would lose respect from those capable students far more quickly by not admitting mistakes or worse still trying to cover them up. This is because they probably already know your mistakes, so their respect or disrespect would be based not on your mistake but rather on your honesty regarding mistakes. Students can often tell if you know you have made a mistake but try escaping without admitting.
The best policy as a teacher is to ensure the students know that you are human, just like they are, and can make mistakes. Let them know, also, that the best learners, and most knowledgeable people, are those who never close their minds but continue to learn. Those who think they know everything, besides being wrong, are destined to remain ignorant of many things they might have otherwise learned had they been willing to make corrections to their own preconceptions and mistakes.
Respect is earned, not commanded; unless you are teaching at the lower elementary school level (up through about the fourth grade). While you can always receive a certain level of respect by imposing it on the students, they will not have actual respect for you unless you have earned it.
In Asian societies, students would never dare to correct their teacher because it is taboo to cause one's superior to "lose face." But this, while maintaining a certain form of "respect," loses the advantage of teaching independent thinking. Students merely copy their teachers, generally finding little advantage to recognizing a teacher's potential errors in the absence of an acceptable way to disclose them. When an error is noticed, the student just stays quiet about it.
In Western societies the pendulum is at the other extreme: respect for teachers is secondary to one's own "right" to independent thinking--and correcting a teacher can become a matter of one's personal pride. I recall the story of a European boy whose family had moved to America. He was in primary school but was far advanced (a genius) in math. His father saw fit to enroll the nine-year-old in a math class at the local university. One day, the mathematics professor incorrectly worked a problem on the board, and the child corrected the professor--undoubtedly in the less mature manner of a child. The professor took umbrage at being corrected by one so young, and got very angry. The father ended up defending his son to the professor, and the professor had no defense because he had made a mistake. (Mathematics is somewhat more black-and-white than some subjects might be.) The professor had only made things worse for himself by attempting to hold himself up as infallible.
When an Asian student of mine corrected me in class one day, I was so surprised that I praised him! It is great for the learners if they are alert enough to catch a teacher's mistakes, and I preferred to have it corrected than to have inadvertently taught something which might be incorrect. I always invited my students to correct me--though it was often the case that I had, instead, opportunity to explain why their offered correction was mistaken. I would appreciate them for trying, and it was a teaching moment from which all of the class could benefit.
In Asian societies teachers are not supposed to appear fallible. They are not expected to admit any mistake. They are expected to maintain their superiority always. But, in my experience, the principle of earning respect in place of commanding it actually works very well with Asians, and being humble enough to accept one's own mistakes helps to earn their respect.
In the end, if your authority is based on the students' respect of you, you will maintain your authority best by putting the students' learning ahead of your own pride.
You can flip it round, and use it as a teaching moment, which also takes the wind out of the sails for anyone who might act up.....
Instead of focusing on your mistake, focus on their learning. I'd tackle it like this:
Before we carry on, I want to teach you one of the most important lessons in (subject).
How many of you, feel you've got a good grip on the material so far?
So, how many of you spotted the mistake I made last time/last week?
(pause, some hands go up, some people look down)
At this point, you can either ask someone to explain what it was, or tell them yourself, followed by the punchline, where you ask why people didn't say something. They probably won't speak easily. It's an awkward question but a really good one. If needed, pick one of your bright sparks who says they saw the error, and ask directly, why they didn't say something. It's a very valid, legitimate question.
Do this in a cordial, supportive way. So they don't feel picked out for bad reasons.
Then, draw the lesson from it. People in authority will make mistakes at times, whether in academia or commerce or other areas. And often, nobody will say a thing. This will happen periodically throughout their entire lives. "I want you to think about that, and decide, what you feel should happen, in situations like this." Or whatever else you feel is the lesson you want as a take-away.
In this way, you flip it round. You maintain authority and initiative. The people most likely to feel they know it all, are in the position of "well, why didn't you say something?", which deflates them and reduces scope for issues.
And most importantly, everyone learns a really important point, in the best way possible - seeing it taught, by their teacher.
I do a lot of live programming demos in my lectures (I'm about to do another one in about 10 mins※ ;o). I make mistakes each time. It's no big deal, and it is why compilers give error messages. It gives me a chance to show how things are done that can't be adequately explained by conventional lectures, such as how to go about programming in such a way that you minimize errors and ensure that the ones you do make are found and corrected quickly and easily.
I had the same approach with my chalk and talk lectures back when I taught maths.
If you are not making mistakes as a learner you are too far inside your comfort zone to learn efficiently. As a teacher it possibly means you are too far ahead of the students to find their difficulties easy to anticipate.
In short, don't worry about mistakes, and try not to worry about authority. You have demonstrated that you have a good grasp of the subject by the fact you have been appointed to teach it.
Having said which, as a middle aged, white male marsupial, it is easy for me not to worry about authority, which is something that society needs to be working on. I suspect it is a lot less easy not to worry about authority if you don't automatically get the respect that your ability deserves because of qualities that have nothing to do with your ability to teach the material.
※ I provided several "learning experiences" during the demonstration ;o)
You've received a lot of very good answers already. I'll just add that one more possible way to maintain authority and still correct the mistake is to make a joke about it. Which one depends on your sense of humor. When asked by a student why I wrote something that is wrong or misplaced or noticing that myself, I just say "because after 50 I have a brain-mouth-hand coordination problem, so occasionally I think one thing, say another, and write/do a third one". I usually add that "unfortunately, some younger people suffer from this disease too". It may look somewhat lame to you (or it may not), but usually it causes some laugh and smiles and if you can manage to make your audience smile at your will, you may be sure that you are still in full control. I also tell my students in the beginning of the class that "people usually look most stupid when they are afraid to appear stupid". The same applies to losing authority: one loses it most exactly when one acts of the fear to lose it.
Just my two cents.
You really should admit to the mistake. As others have pointed out, the absolute worst thing for a teacher's reputation is to make mistakes that students can see, and not admit them; or even worse that that, to argue with, deflect, or browbeat the students who point them out.
That said: Time is of the essence. The quicker you can fix the error, the better. Within seconds, by yourself, is ideal. If a student points it out in a few minutes, or anytime in the one class session, then that's perfectly fine; usually easy to go back and fix it.
But if it's outside the class session then the half-life on how useful this is starts to tick down. The particular numbers may not matter a whole lot if the essential concept or process was shown correctly. It becomes more of a burden if you need to recreate all of the work (board work lost, etc.) from scratch. If days have gone by and no student noticed, I'd be prone to let it slide for time-efficiency purposes; if a student does point it out or it's really crushing my conscience, then I've written up a handout document showing the correct solution. (Although Krantz in How to Teach Mathematics argues against even that, specifically.)
When I do make a mistake in class, and a student catches it instead of me, then my standard framing response is like this:
Thank you so much! I usually make about one mistake a day, and there it is. That's why we need to do the hard stuff in teams, and I'm depending on you to watch what I'm doing for mistakes like that. I'm so glad we're doing this together!
One of the best solutions I’ve seen used by a math teacher of mine is to reward the students for catching your mistakes.
My teacher set this up by making it so that if a student pointed out a mistake during class, she added a point to the running tally. When she reached ten mistakes caught by the class, she bought treats for the entire class.
Without needing to expend money, this could also be done by giving extra extension days to the class, or any other reward.
This completely changed the dynamic of the class, which did a few things:
- It made it so that students were proud to share their expertise and were happy to understand the material and point out mistakes. The mistakes made the student who pointed them out look good, rather than the teacher look bad, because the students were focused on the reward.
- It provided a clear mechanism for the teacher to fix mistakes and the students to correct mistakes. If the teacher corrected a mistake before the students noticed, she was saving herself from giving up a point to the class, and her doing that re-engaged the class to pay more close attention. If a mistake was made, a student catching it was a success for the class, rather than a failure of the teacher, and so students were quick to point out mathematical errors and it wasn’t awkward or degrading to do so.
- Writing math could be stressful for the professor at times because she was worried about giving up too many points. I believe this was due to setting the point requirement too low.
- Finding a suitable reward mechanism can be tough, especially since the most motivating rewards are often those which require extra money or time spent to set up.
- It can be awkward or difficult to set this up in the middle of a semester since it introduces a whole new mechanism into lecture, so I recommend setting it up at the start of the class. However, if needed, this can be done later. In fact, in the OP’s original case, it could be introduced with “here’s a free point because I made this mistake - see if you can catch the other ones!”
If a student made such a mistake in a graded piece of work then you would probably take points off for it. That makes this a good teaching opportunity to show why one should always check one's work, because anyone can make some minor mistake. Depending on how much time you have for that topic you can go into greater detail about how you could have checked your work (estimated the answer and compared, using your answer in the original equation and verifying, etc.). Or if you don't have much time, you can just mention it in passing: "Minor correction for yesterday's problem, here is the correct result. This is an example of why you should always check your work. See xyz resource for examples of how to check your work."
It's important to explain that one actually needs to check, and ideally give strategies for checking, because otherwise students tend to conflate "being good at math" with "never making arithmetic mistakes"...if you can separate those two in their minds you will both preserve your dignity and show them that even learners can produce routinely correct work, since catching and fixing errors comes from diligence rather than some stroke of genius.
Just so it's said, authority is a function of attitude, not a function of action. One person can do everything perfectly and still be unable to command authority; another can goof left and right and maintain it. The key is to be calm and assertive. Don't question yourself or your capabilities, and students will naturally fall into line.
With respect to the specific problem, my approach has always been to turn it into part of the lesson. Walk into class the next day and lead with: "How many of you noticed the mistake I made in last lecture?" Get a show of hands so you have an idea of how many people actually did notice (which won't be as many as you think); explain and correct the mistake; move on. There's no need for you to 'admit' you goofed — treat that as a mere obvious fact — and no need to explain anything beyond the basic issue and correction. You on't have to 'make it up' to the students, you just have to fix it. Hold in your mind the fact that you are an expert on the material and the goof is inconsequential; the force of your own self-assertion will move people past it.
Yeah, I know it's a lot harder in practice than it sounds, but it does work.
I would say you are making too much of this. Students do not expect perfection from their teachers. In the first few minutes of each class I always run over the most important things to remember from the last lecture, clarify things I think I didn't teach well, and present errata. In no way has this ever caused me to lose face.
If you seem embarrassed when discussing the mistake, perhaps the students will think less of you. But if you just present it as a correction or clarification to the last lecture, if anything it will improve you in the eyes of your students. You set the tone in the class. Mistakes on your part or theirs are great teaching moments, not a source of shame. If your students are not respecting you much, your tone and confidence may be areas in which you can improve. Confident people are comfortable talking about mistakes they have made.
Being accessible and teaching at the appropriate level are important considerations in students' evaluation of their teachers. Perfection or seeming to know everything is not.
I absolutely agree with the idea that it's a better lesson to admit mistakes, show humility and adaptability, as that's teaching the students many of the skills they'll likely need in complex maths and sciences. Many could use help learning how to spot errors or thinking outside the box (such as you would show by revisiting it, since it's going off script some)
That said... if teaching a course with a fairly limited timeframe, I very much encourage you to carefully consider what your personality is handling a revisit in class, how to handle it best timewise, and perhaps even consider other ways. What I mean by that is that I can remember quite a few classes where a teacher\professor made such a mistake... and then we ended up wasting a lot of time going back over it, getting lost in additional mistakes, etc.
I know when I taught, I might be a little more nervous going over such mistakes, and that might open me up to more tangents or minor questions spiraling out of control on the topic. That's kind of the thing, it may indeed be a bit of a shakeup in the environment in the class, and that can indeed lead to make some students feel freer to be challenging and disruptive going forward, depending on the course level and such. It is indeed a chink in the armor. It's good to show you're real and human. But do keep wary that some students may take it as an opportunity to become too comfortable and derail your direction some in the future. Or invite many more questions of "are you sure you aren't making a mistake here as well", I got X, which in some situations can derail other students from understanding if they become too pervasive, rather than keeping quiet and reviewing their work, and usually finding they made an error. You have to feel you can maintain authority in that sense. You may well have to be stern at some later point if such questions or disruptions kept coming. And so if your concern is you may not have the confidence and force to handle that, you may want to avoid making a big deal of it.
In addition, in courses where time is at a premium... you have to balance that aspect. At the least, if it's a long concept that brought you to the mistake, you may want to have the bulk of the notes rewritten up on the board before class so you can quickly remind them where you went, they can refresh through the notes for a few moments, and then you can point out the issue, and hopefully move on in pretty short time. Or it may be better if it's going to be real time consuming and complex to do like a handout (students probably won't mind that, notes they don't have to write!). You can point to that, or hope they'll look at it in their own time (very much dependent on the course level\passion for the subject again).
Perhaps offer to talk about it more in office hours (or whatever concept you use for outside of class help) if anyone is one or two struggle with the idea, and the others have no issue.
Or if it's more minor, you can mention in it brief passing to start the class, just that you made a minor [numeric/conversion/equation/whatever] error...
Or indeed just skip it entirely if it's just not notable enough to be worth the trouble. Humility and teaching the students how to find mistakes is great, but at the same time, there's no need to try to reflect perfection.
But don't worry about the students who "think they can teach the class". Maybe they can, maybe they can't, but you're paid to, and your duty is to the students who are trying to learn. There are certainly courses where I've had to make that clear to difficult students. You can't let the troublemakers shake you or overwhelm your thought processes. If you're afraid of those students, that they may be show you up or such, it shows. I'm guessing you know your stuff, you've got nothing to worry about from them, just focus on doing what your students as a whole need, and tend to forget those who choose to cause more trouble than trying to learn anything in the course. They aren't your central concern, even if they're trying to make you such. If they feel too smart for the course, they either shouldn't be taking it, or they can up the inability to test out of it with administration or whatever. But it's not your issue. They can doodle or read the paper or something else that's nondistracting. But the world doesn't revolve around them, and you shouldn't stress over such ones that think they can teach the course.