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From time-to-time a question of the form "My professor said $X$ is true, but my book says that $X$ is false. Is my professor wrong?" is asked. For example, I've seen questions similar to:

Question 1. My instructor said in class that every continuous function is differentiable, but I just read in my book that the function $f(x)=|x|$ is continuous but not differentiable. How is this possible?

I'm surprised to see comments and answers stating "Your professor is wrong". While it's true that if the professor made this statement, then he/she is wrong, I'm not comfortable with automatically accepting this assumption.

It seems very unlikely that a calculus instructor would make such a statement. It seems overwhelmingly likely that the OP in this example is incorrectly recalling something the professor said. I try to communicate this in my comments and answers with statements like "It seems unlikely your professor made this statement, as the example you give is a famous continuous function that is not differentiable. Are you sure your professor said this?"

While healthy skepticism is important, I've noticed that many (especially younger) students tend to search for excuses for their misunderstanding of a topic rather than admit their misunderstanding and address it. Therefore I think it's important to communicate this (without discouraging students from questioning statements they think are false).

I'm curious if others think this is as important as I do and how others deal with questions of this form.

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    $\begingroup$ More generally, this is one of the major difficulties of teaching in public. Users are often embarrassed to admit their mistakes in public, so they may try all kinds of ploys to avoid doing that, which can lead to numerous problems. Good teachers can be role models here, by behaving nobly when they make mistakes (as we all do). $\endgroup$ – Bill Dubuque May 20 '14 at 21:14
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    $\begingroup$ More generally, it becomes increasingly more difficult for people in a "make or break" culture admit in their mistakes. This causes the mind to rationalize, and sometimes to create false memories. I have learned not to trust my memory when it comes to these things, and I am right to do so. Many people are ashamed of not knowing, and ashamed of not understanding. Perhaps it is our fault in not over stressing to students that they should be proud of these facts, rather than shameful. Knowing what you don't know is a big step forward. But maybe this is an inevitable result of modern culture. $\endgroup$ – Asaf Karagila May 20 '14 at 21:24
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    $\begingroup$ I had some (very few) professors who did make lots of mistakes including the wrong definition of a vector space and repeatedly claiming to prove $A\Rightarrow B$ when they were proving $B\Rightarrow A$. Their problem was that they did not care enough to make sure that everything was right and they did not want to admit mistakes. Many other students would never believe me that the professor was wrong because "I am so bad at math that I am sure that the professor knows better.". So, the whole premise of this question has HUGE cultural assumptions that are not true everywhere (or always). $\endgroup$ – Phira May 21 '14 at 9:20
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    $\begingroup$ So, while I agree that the student might very well have misheard the professor, I disagree strongly with saying "it is unlikely that your professor said this". Why weigh in at all at this issue? Just say that "It is not true that all continuous functions are differentiable. The mentioned function is indeed a counter-example.". Why defend a professor if you simply do not know whether he deserves your blind solidarity purely based on status? $\endgroup$ – Phira May 21 '14 at 9:26
  • $\begingroup$ @Phira I don't understand your argument. I'm making the claim that in my example it is more likely that the student is misrepresenting his or her professor's statement than it is that the professor actually said the false statement. Obviously professors make errors all the time and I'm not claiming otherwise. I'm simply pointing out that many students are stuck in a paradigm of "if I don't get it, then it's my professor's fault". As Asaf points out, these students will, either consciously or unconsciously, fabricate memories to justify this paradigm. $\endgroup$ – Brian Fitzpatrick May 21 '14 at 9:32
  • $\begingroup$ I also don't view the statement I quote as a defense of the professor nor do I view it as an expression of "blind solidarity purely based on status". I merely want to alert the student that his or her record of what happened in class may be off. If a student is aware of this possibility, then I think he or she is likely to be more careful when note-taking in the future. $\endgroup$ – Brian Fitzpatrick May 21 '14 at 9:40
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    $\begingroup$ @BrianFitzpatrick My argument is that in my country of origin, most students are stuck in the paradigm "I am really bad at math and the professor knows better", and my point is that you should try to imagine yourself in the situation of a student who indeed has a professor who makes this kind of mistakes, who comes here for clarification and who is told by professors that they assume that his professor did not make this mistake because he is a professor. The message is clearly that you value rank over mathematical content even though that is not your intended message. $\endgroup$ – Phira May 21 '14 at 9:41
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    $\begingroup$ @BrianFitzpatrick I get that you do not view your statement as defense. Because you do not for a moment put yourself in the situation of a student who actually speaks the truth about the professor. If you had been a colleague of my former professor, you would have enabled his behaviour by not believing my account. $\endgroup$ – Phira May 21 '14 at 9:43
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    $\begingroup$ @Phira In my question I say verbatim "Therefore I think it's important to communicate this (without discouraging students from questioning statements they think are false)." Your issue must be with my particular phrasing. I asked this question in the first place to solicit suggestions on how to communicate what I'm trying to communicate without discouraging students from questioning statements they think are false. $\endgroup$ – Brian Fitzpatrick May 21 '14 at 9:55
  • $\begingroup$ @Phira You insist that we're not sympathizing with students with bad professors, but my sympathy is implicit in the fact that I asked this question in the first place. My question is: "how can we alert students that they may be blaming others for their misunderstanding without discouraging them from questioning authority when appropriate?" $\endgroup$ – Brian Fitzpatrick May 21 '14 at 19:24
  • $\begingroup$ @Phira You've made it clear that you disagree with how I've phrased my responses, but it seems your own sympathy for students in bad situations is not allowing you to acknowledge that the issues I'm pointing out do, in fact, exist. Why can't we have a discussion that addresses both issues like I intended in the first place? $\endgroup$ – Brian Fitzpatrick May 21 '14 at 19:25
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    $\begingroup$ @BrianFitzpatrick No, there is no comment of mine where I insinuate that students never mishear their professors. Your question, however, speaks of "overwhelming likelihood" which I can very strongly disagree with without denying existence. $\endgroup$ – Phira May 21 '14 at 21:10
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    $\begingroup$ @Phira If I replaced the phrases "very unlikely" and "overwhelmingly likely" both with "possible" would you then be interested in productive discussion? You're splitting hairs at this point. I think I've posed an interesting question that is far less controversial than you're trying to make it out to be. $\endgroup$ – Brian Fitzpatrick May 21 '14 at 21:48
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    $\begingroup$ Perhaps the example was unfortunately chosen. It is indeed highly unlikely that a professor (or TA, what have you) would make such an egregious blunder. Aren't most alleged professorial mistakes more credible, like this one? $\endgroup$ – Daniel Fischer May 21 '14 at 23:13
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    $\begingroup$ But to weigh in actually on the discussion for what it's worth, I don't think addressing the professor issue at all in either direction would be very beneficial. If the assumption is wrong, someone is wrong, but once the right answer comes out in the end, should be the emphasis. I mean if it's the student you'd hope they'd notice a trend, or in comparison with others that their notes were different. And if it's the prof, however unlikely, surely it's a matter for the people actually involved. To sum up, and give my opinion properly, encourage dropping the habit of mentioning anyone $\endgroup$ – snulty May 22 '14 at 0:53
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What do people think about this answer format?

The statement, as it is written here, is [generally] false. [Here is a counterexample.] [Here is a more accurate statement.] Since I wasn't at the lecture, it's hard for me to know if this is what your professor actually said or if there was some mistake in your [notes/understanding/transcription/memory].

If you can meet with your professor, for example at office hours, then you might consider politely bringing it up with them. There is a chance that their wording was confusing or wrong because they were trying to emphasize a subtlety, and you might learn something important from the misunderstanding.

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In response to the question

My instructor said in class that every continuous function is differentiable, but I just read in my book that the function $f(x)=|x|$ is continuous but not differentiable. How is this possible?

I would not say

Your professor is wrong.

Neither would I say

It seems unlikely your professor made this statement.... Are you sure your professor said this?

The question "how is this possible [that my instructor disagrees with my book]?" is not a math question, so I would just ignore it. It would be more useful to the site (and probably to the OP as well) to instead answer the questions "is every continuous function differentiable?", "is the absolute value function continuous?", and "is the absolute value function differentiable?"

If the student insists on an answer to the question "how is it possible that my instructor disagrees with the book?", then I would vote to close the question as off-topic. I don't think it should be our goal on Math.SE to teach the OP about the relative probabilities of error of textbooks, professors, and students, even if we think we know more about these probabilities than he or she does.

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    $\begingroup$ To me this just seems to be taking the question too literally. $\endgroup$ – Jack M May 24 '14 at 21:45
  • $\begingroup$ @JackM Brian's question, or the hypothetical question in Brian's question? $\endgroup$ – Trevor Wilson May 24 '14 at 22:19
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    $\begingroup$ The hypothetical question. The underlying question is something like "what's going on? Which one is correct?", and in particular: "Is this theorem true or false?". $\endgroup$ – Jack M May 25 '14 at 5:21
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    $\begingroup$ @JackM Then I don't understand your objection. My answer was to ignore the literal question "how is this possible" and answer the "underlying" math question, just as you seem to suggest. I don't see how consciously ignoring the literal meaning of Question 1 counts as taking Question 1 too literally. $\endgroup$ – Trevor Wilson May 28 '14 at 21:23
  • $\begingroup$ Addressing the underlying math question also makes the question clearer for future readers, and thus more likely to be found by search and to be useful. $\endgroup$ – Jeffrey Bosboom Aug 10 '15 at 23:05
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For what it's worth, students asking things on MSE also get things wrong or leave out critical information when simply transcribing from a textbook; see Reflection in a plane. which happened just now. I cannot recall answering any "my professor is wrong" questions within, say, the past year; those who do so may decide how gentle to be.

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  • $\begingroup$ I'd say this should've been a comment, although I can understand why you didn't want to bury it under all the arguing up there. $\endgroup$ – Ilmari Karonen Jun 2 '14 at 13:11

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