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I don't know whether you guys have made a similar experience but it just happened to me: I made a very stupid mistake in front of the class. I can't really tell you how it happened and I feel too ashamed to even tell you what it was (and it doesn't really matter) but anyway

my questions are:

How should one best handle the situation in the following cases:

  1. you realize it yourself more or less immediately
  2. you realize it only much later because something (e.g. a proof) is not working properly
  3. you don't realize it at all but a student tells you in front of the class (this happenend to me)

So my questions really aim at best practices of experienced educators.

EDIT: Seeing all the great answers and the upvotes I am really relieved that it is not only me - Thank you all for these great tips! :-)

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Hm, I'm just wondering whether the mistake I made some time ago - I somehow screwed the formula for $(a+b)^2$ - was more stupid than yours or not. Don't be ashamed, it's normal to make (even stupid) mistakes. (Unless you're an android or something, that is.) –  mbork Apr 9 at 17:51
    
@mbork: Ok, but don't tell anybody ;-) I didn't realize immediately that $\lim_{x \to \infty} 2^\frac{1}{x}$ of course equals $1$, I was somehow distracted, I don't know... –  vonjd Apr 9 at 19:00
    
I had one that would leave the mistake there and hope the class caught him, resulting in "Oh, so you are paying attention!" - Unfortunately, it doesn't work if no one notices... –  Izkata Apr 10 at 16:24
    
@Izkata mine simply complained that we are not paying attention in such cases (although the mistake was unintentional... probably). –  Maciej Piechotka Apr 10 at 16:48
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Basically, don't pretend to infallibility or authority-of-knowing. Only that, inarguably, you have greater experience, and are a fallible human being. This eliminates many of the formal role-acting issues that we could imagine arise. –  paul garrett Apr 11 at 23:47

9 Answers 9

up vote 26 down vote accepted

Anecdote: I know a very experienced and well-liked professor (emeritus, now) of mathematics who would send a small square of chocolate via snail mail to any student who pointed out a mistake. The professor brought in typed up versions of what he was going to cover each class meeting, and the mistake could be anything said aloud, written on the board, or typed up in his notes. (Related: Donald Knuth's policy.)

Responses:

How should one best handle the situation in the following cases:

$1.$ you realize it yourself more or less immediately

Either correct it immediately or pause and tell the class that you made a mistake. Ask everyone to spend a minute looking for it. (Of course, this will only make sense in certain scenarios; for example, it probably would not make sense in writing down the givens before proving a theorem, since students don't know what result you are aiming for.)

$2.$ you realize it only much later because something (e.g. a proof) is not working properly

If it's straightforward to go through and correct it (e.g., a minus sign is missing in several places) then either follow the advice of $1$ above or draw a hard line on the board, remark what the error is, and write what's correct. If you aren't even sure what the correct conclusion should be by then, then admit the mistake, note that everyone makes them (it would be nice for students to get 100s on all tests, but unlikely) and abandon ship to focus on the next result.

In any event, type up a clean version of the result and its proof to hand out (and possibly discuss) during the next class period.

$3.$ you don't realize it at all but a student tells you in front of the class (this happened to me)

Thank the student, and ask him or her (if possible) to explain why it is an error to the rest of the class. Ask if another student can suggest a way to fix it. Then decide whether going back over what has been written would be a good use of class time, or whether you should ask that they accept the conclusion on faith for the time being. (You might even put such a decision to a majority vote.)

Once again, in any case you should type up a clean version of the related mathematics to distribute (and possibly discuss) during the next class period.

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An anecdote of my own: I had a high school (grade 11) geography teacher who told the class that if he said something wrong and we could prove it, he'd substitute a low grade for a 100% at the end of the semester. (I had to prove that steel wasn't an element and iron wasn't made from steel and other elements) I prefer the above method from Benjamin because it involves the entire class, rather than a single student learning from the teacher's mistake –  David Wilkins Apr 9 at 17:16

In my view, mathematics education is not about the authority of the teacher, it's about developing the students as their own mathematical authorities. Ultimately, the authority in mathematics is that any one of us can use what we have learned, even via different methods, to verify what other people have done.

In mathematics classrooms, we do not learn only mathematical rules and structure. We learn to think mathematically. And this involves learning something about our identity relative to "mathematics." Valuable things for your students to learn:

  • Every one of them can wield mathematical authority; that's the power of learning mathematics.
  • In mathematics (and in science, and in life) we have some responsibility to trustworthiness, which means an attitude of eagerness to be challenged.
  • Justification is not an inconvenient response to a challenge, it is a vital mathematical practice
  • We should avoid anything that teaches the student that making a mistake is something to be ashamed of; when students learn that, they begin to attribute their own mistakes to "not being good at math." They start to believe that someone good at math would not have made that mistake.

A teacher who is relying on cultivating the air of an infallible mathematical authority will have a hard time conveying important aspects of mathematics to the students. But worse, that concern seems to put one's own personal emotional well-being ahead of the students' learning. If this façade helps you avoid challenges from the students, is it for their good, or is it ego-driven? It's a question educators should at least consider. Who benefits?

If students learn from you that even teachers make mistakes, but yet they persevere in correcting them, they may see themselves as possibly reaching your level, with some work. Isn't that, ultimately, our goal as mathematics educators?


EDIT:

I thought I would offer one peek into the implications of the development identity in mathematics classrooms, since I am interested broadly in STEM education and how people come to see themselves as entering STEM fields. Jo Boaler has done research on the implications of different models of teaching in math classrooms, and one perspective she has used to compare traditional and more heavily discussion-based classrooms is to look at the developing identities of the students. One brief excerpt from one of Prof. Boaler's papers illustrates some of what students say when they begin to think mathematics is not for them:

The disaffected students we interviewed were being turned away from mathematics because of pedagogical practices that are unrelated to the nature of mathematics (Burton, 1999a, b). Most of the students who told us about their rejection of mathematics in the 4 didactic classrooms – 9 girls and 5 boys, all successful mathematics students – had decided to leave the discipline because they wanted to pursue subjects that offered opportunities for expression, interpretation and human agency. In contrast, those students who remained motivated and interested in the traditional classes were those who seemed happy to ‘receive’ knowledge and to be relinquished of the requirement to think deeply:

J: I always like subjects where there is a definite right or wrong answer. That’s why I’m not a very inclined or good English student. Because I don’t really think about how or why something is the way it is. I just like math because it is or it isn’t. (Jerry, Lemon school)

In other words, there was a sort of filtering process going on in heavily didactic classrooms, teaching students to select themselves out of mathematics if they were interested in having control of their own choices (human agency), self-expression, and interpretation. The students who stayed interested were ones who felt relieved at the lack of complexity of "one right answer."

By contrast, the students in a more discussion-oriented calculus classroom did not see mathematics learning as conflicting with their developing identities as expressive, interpretive people with personal agency. Part of the difference between these two types of classroom is in how authority is seen.

It's more than just how mistakes are handled, of course. I offered this example to show that there is much more going on in math classrooms, and part of it is very important to how students determine what they are going to put their effort into.

Something to consider!

Works Cited

Boaler, J. (2002). The development of disciplinary relationships: knowledge, practice and identity in mathematics classrooms. For the Learning of Mathematics, 22(1), 42–47.

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Well, at least in cases 1 and 3, I usually say (in a serious tone): "Well, I did it on purpose, to see if anybody is actually paying attention to what I'm saying". It usually takes the students about 3 times (sometimes over a couple of weeks) until they realize it's a joke. Then, I usually explain to them that it's normal, and in the process of learning one is actually expected to make mistakes - this is just natural. It is probably a good idea to proceed then to explaining that the difference between a student and a teacher is not that the latter does not make mistakes, but that the teacher (as a more mature human being, at least in theory;)): (i) makes them less often, (ii) can more easily see whether something is correct or not, and (iii) does not build his/her self-esteem on whether s/he makes mistakes. (That said, I don't try to claim that I always live up to the ideal described here; in fact, I'm still far from it...)

In general, I find that making a bit fun of it is a good way to cope with the natural stress of such a situation. A few other running gags I employ are:

  • "Well, you see, I'm an idiot, but don't tell that to my dean, 'cause he's going to fire me."

  • "I'm lucky I already passed this course."

I think that (besides being funny, though your mileage may vary on that) it's quite formative for students to see that their teacher: (i) also is a human being and makes mistakes, (ii) does not have a problem with admitting it, and (iii) does not treat himself too seriously.

Of course, it's not that easy, and requires some effort to develop such an attitude (and even then, it does not work all the time, obviously). But my experience tells me that especially the trait described in (iii) is very useful here (and also in general).

That said, some time ago I did an extremely stupid mistake of making a bad drawing (it was an elementary geometry problem) and discarding a correct answer because it didn't fit to my bad drawing. Even though I was very embarrassed (so it must have been a really stupid mistake), since it was an introductory course for freshmen, I decided that it's too good an opportunity to waste it, and I did a mini-lecture on how bad drawings can hinder our reasoning. So I think that both a student's mistake and a teacher's one may make a good opportunity to teach something.

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I like your use of humor. Lightheartedness lowers the stakes a bit. I think it contributes to an environment where the students might be more expressive, and willing to make mistakes that they can learn from. –  JPBurke Apr 10 at 13:37

Making a mistake in maths is a positively good thing to do. There are lots of students these days that seem to think maths is something they can never be good at and that if you are good at maths it means you are some sort of wierdo with a brain that functions completely differently to theirs. However this isn't true and it can give the students confidence in their own ability to see that you make mistakes as well.

I also teach programming and quite often I use lecture time to demonstrate how to go about writing and debugging programs by writing a non-trivial bit of code live during the lecture. Of course I make mistakes, and frequently the student spots them before I do. One of the benefits of this is that I can show them how to go about writing programs in a way that limits your opportunity for mistakes and that makes fixing them as easy as possible. The same is true for maths, the fact that you make mistakes is a good way of teaching the importance of checking your maths as yo go, and if there is time to see if you can get the same result by a different method. If we never made mistakes, this sort of thing wouldn't be "good practice".

So for the three cases:

  1. you realize it yourself more or less immediately

ask the students if they have spotted it - good opportunity for involving the students in the class

  1. you realize it only much later because something (e.g. a proof) is not working properly

use it as an opportunity for demonstrating how to go about finding out what the problem is, and how to check as you go along.

  1. you don't realize it at all but a student tells you in front of the class (this happenend to me)

commend the student, point out that we all make mistakes when doing maths and that this is to be expected and nothing to be embarased about. If maths was easy there wouldn't be any fun in it, and the fact maths is hard means you are entitled to enjoy your successes.

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Welcome to the site! I absolutely love the comparison to programming. No one writes [code / elementary algebra] that works the first time unless they work in a certain slow, methodical way and [include comments / show work]. –  Chris Cunningham Apr 9 at 14:44
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Thank you, I especially liked: "If maths was easy there wouldn't be any fun in it, and the fact maths is hard means you are entitled to enjoy your successes." :-) –  vonjd Apr 9 at 15:33
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I agree with @ChrisCunningham: comparison to programming is great. And that reminded me of the famous Knuthian "Beware of bugs in this code: I didn't test it, I only proved it correct." –  mbork Apr 9 at 18:00
    
@mbork, I like the Knuth quote. My definition of a trivial program is one where you know it to be bug-free ;o) –  Dikran Marsupial Apr 9 at 18:10

It happened to me and I think to most of us. The most important thing is that this should not happen too often, then you can handle it. If this happen too often, then you lose credibility.

But if you react the correct way, there is no problem if this happens once. I think it is important to acknowledge your mistake and make a correction as soon as possible. Students are no fools, and we would like to teach them to think critically. The whole scientific world relies on acknowledging our mistakes and searching for the truth. We are no gods, and if this happens only once then this can be even beneficial.

If you recognize the mistake immediately, then it is the easiest: you just correct yourself. If a student recognizes your mistake shortly after you made it, thank her and correct the mistake. Write out the correct part in detail.

If you recognize an earlier mistake, then the same applies, but, if it is important, you should even hand out notes on the corrected statement. And if it was a grave mistake long undetected, then I would (silently) avoid that part in the exam not to get in argument in students.

But the most important thing is, as I said, that you should not let this happen too often.

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I'm not so sure about >And if it was a grave mistake long undetected, then I would (silently) avoid that part in the exam not to get in argument in students. –  Benjamin Dickman Apr 9 at 8:58
    
@BenjaminDickman: me neither. Fortunately, I never had this, my mistakes were usually detected during the same class, and were never grave. –  András Bátkai Apr 9 at 9:54

Making a mistake is OK! In fact, I think your students would react positively if you explain that you made a mistake, and challenge them to find it.

I've been praised for the way I handle my own mistakes in class.

If you're especially afraid of making mistakes, try implementing mistakes on a more regular basis, on purpose, and still challenge the class to find it. That way, when you make a mistake for real, and a student notices it, you can just say "good job" after having them elaborate on why it's wrong. It'll give you time to find out if it's a true mistake, or if the student is mistaken.

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@Aleksander: what part did I handle wrong? I would be happy to learn, I am also unsure about this problem. –  András Bátkai Apr 9 at 9:56
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@AndrásBátkai The part where you said "And if it was a grave mistake long undetected, then I would (silently) avoid that part in the exam not to get in argument in students." It seems to me like you're withholding knowledge from your students by just avoiding it. If you were more open about it, they'd be better equipped to build on the knowledge. One of the largest problems for students, is that they try to acquire knowledge that requires previous knowledge. If they've been taught wrong, then that basis is full of holes, making it virtually impossible for them to learn higher levels. –  Aleksander Apr 9 at 10:00

You just tell the truth and say something like, "Oh, I made a mistake", and correct it by saying something like "it should be ...". You are the instructor and you are required to deliver the truth. If students dont do something correctly, you correct them. I you dont do something correctly, you correct yourself. If the students correct you, you accept it. No matter how ridiculous the mistake, my students are completely accepting of it. Each time, the event passes as if nothing happened. Its no big deal. the whole point is to deliver the correct information. The truth is more important than any ego.

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There is absolutely nothing wrong with making a mistake in front of the class. I even claim that it may be helpful for the students to see you make mistakes. Think about it, we work from examples that we have planned in advance. The students don't see us struggle with problems, we make it look easy on the board. Then they go home to work on homework and don't understand why they struggle with it.

There is also a psychological aspect to this as well. Students tend to judge themselves (and their mathematical ability) on whether or not they can get the right answer. Furthermore, they also need something to compare their mathematical ability against, and usually they will compare themselves against the instructor. I tell my students the first day of class, that I am human, I don't know everything there is to know about mathematics, (or whatever course I am currently teaching) and I will make mistakes on the board. I also tell them that I will sometimes make intentional mistakes to see if they can catch them. The main thing is not to make a big deal of it. I made a mistake, now the learning can start.

The jist is that students need to see that we make mistakes also, they need to know that we struggle with our level mathematical work just like they struggle with their homework.

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Case 1: If it is a minor mistake (e.g. wrong sign) point it out, correct it and go on. If the mistake is a grave one, stop and explain what went wrong (maybe even ask students to spot the mistake themselves)

Case 2: Depends on how much later. If it is still in the current class, correct it immediately (see case 1). If it was in the latest class mention and correct the error at the beginning of the next class. If the mistake was made far earlier try to hand out corrected papers and/or take some time at the next class to address it, especially if it isn't something as simple as a wrong sign.

Case 3: Congratulate the student who found it, proceed like case 1.

Reason: Everybody can and does make mistakes from time to time, but it is handled best if you can spot and correct the mistake instantly before the students have time to internalize (memorize) it. Better yet, explain how to avoid that kind of mistake - especially if your students can't find the mistake on their own.


I personally can relate to the third case, but as the student who spotted the mistake. My teachers handled it quite well, correct the error and proceed with the lesson as planned. They didn't take it personal, they even encouraged others to find their mistakes. As a student this was kind of a relief: one was sure to get a "right" result and one could see that the teacher isn't perfect either, which gave some students a morale boost.

Generally I think it's a good idea to explain how to spot and avoid most common mistakes, since this is the problem most students have when studying on their own. Most teachers don't plan to do this at all because normally their prepared classes work out nicely so there aren't many possibilities for mistakes to happen.

P.S.: English isn't my native language, feel free to correct me if something isn't clear/easy to understand ;)

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