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This is a typical problem in my undergraduate Calculus I class. Many of these freshmen come in having made an A in their High-School Calculus class, and believe they know everything they need to about Calculus. They love to try to challenge my explanations or they always have a "better way" and they feel the need to share it with the rest of the class interrupting my lecture. Most of the time I can talk to the student after class and explain to them that the way I teach will set them up for success in Calculus II and III, but I have had one student in particular who refuses to admit that he can learn anything from me.

I asked him if he would like to me to assign him a more challenging homework set instead of the current class set, but then he accused me of trying to fail him (he currently has an A in the class). I also suggested that he switch to another class, but mine is the only one that fits in his schedule. What techniques can I use help this student understand there is always something that you can learn even if you know it all?

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    $\begingroup$ Is attendance of your lecture compulsory? How much time can you set aside to discuss these 'better ways'? Is there the potential from learning why these ways may not be better? If the students have explanations, is there the room and time to ask the students for explanations first, and then add relevant facts yourself? $\endgroup$ – Roland Mar 26 '14 at 13:18
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    $\begingroup$ No, I do not require them to attend my lecture, however I do give "clicker quizzes" in class that count as 10% of their grade and they do not know when I will give them. Believe me I would love to tell him not to come to class then. $\endgroup$ – Todd Thomas Mar 26 '14 at 13:22
  • $\begingroup$ @Todd I'm curious about your comment. Could you elaborate? I would interpret your clicker policy as a way to require students to attend your lecture, but it sounds like you do not think of it that way. $\endgroup$ – Chris Cunningham Mar 26 '14 at 14:44
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    $\begingroup$ I try to ask 5 basic questions after each topic I teach to make sure that students have understood what I have just taught. If I see that 10% or more of the class got the wrong answer, then I will revisit the material right then. This counts as participation points in their overall grade, but I don't use it take attendance, I use it to make sure a majority of the class has a basic understanding of the material. $\endgroup$ – Todd Thomas Mar 26 '14 at 15:45
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    $\begingroup$ While I like the idea of class participation, I think it's overrated and it has set up your current lose-lose situation. A student you wish would skip class (because of his attitude), and who wants to skip class (because he thinks he's so smart) can't skip class because it would mean he doesn't get an A. Besides, "class participation", in general, rewards one personality/learning style and penalizes others. $\endgroup$ – Wayne Apr 16 '14 at 22:33
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You should avoid a "I am smarter than you are" war with the student, the teacher must be above that. (You're already avoiding it, I am saying so for the benefit of the Internet.) In Japan you could just smile mysteriously, but I suspect we're talking about the Western world.

The first and most important thing to resolve is this: Is your student right? If his or her suggestions are actually improvements over what you do, or if they can lead to improvements, then you should suspend your ego and use them. And give the student proper credit, and figure out how not to inflate his ego beyond the size of the solar system. Teaching is tough.

Assuming your student is actually suggesting irrelevant modifications of teaching material, the nastiest thing you can do to your student is to relativize his little world:

  1. There may not be the best way to explain or teach any given topic in math. What is best depends on previous experience, background knowledge, personal taste etc. You can tell the student that he is correct to think that for him another way is best, although he should not presume that everyone else has exactly the same background as he does.

  2. There is nothing wrong with knowing how to do a thing in two different ways. The student already knows one way to attack the topic. You're offering another for him to compare. He should therefore listen, and in the end think not just in terms of "absolutely better than" but also in terms of "what are the pros and cons of each way". (See first point.)

Regarding interruption of your classroom, you can use your authority to tell the student that such interruptions should be replaced by off-line conversations.

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    $\begingroup$ It also helps to give an example (where possible) of how having two different methods in your bag is essential or how their way of doing things might not work in a specific situation. $\endgroup$ – Alexander Vlasev Mar 27 '14 at 8:57
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Does this student really already know everything? I ask because of an experience I had many years ago, teaching a "semi-honors" second-semester calculus class in the first (i.e., fall) semester. So the students were there because they had already done some calculus in high school and had done well on a placement test. The first topic of the semester was integration, so I talked about the general notion of integral, applications to specific sorts of problems, and techniques of integration. As far as techniques of integration were concerned, these students could integrate messy functions at least as well as I can; that and analogous computational skills were what got them through their high school calculus courses and the placement exam. But the first exam I gave them included questions that required knowledge of what an integral actually is. Many of them had not learned that in high school and had ignored my attempts to explain it (attempts that they viewed as unnecessary prose surrounding the "real math" of computing integrals). The grades on that first exam were quite a shock to these students, and most of them soon learned that the non-computational prose is also math.

If your students are at all like this, I recommend showing them (preferably in a milder context than an exam) that they still have some things to learn from you.

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Telling them "this is the way to II and III and ..." is doomed to failure, they know their way is right (it got them straight As before, and they are so smart that they made it to college!). Ask them to discuss offline/during office hours, go over their technique and yours. Let them compare. Where relevant, point out what/how yours is more general (i.e., presumably applicable for next courses). Sometimes the difference is just in untenable basic assumptions (i.e., all functions are obviously continuous/differentiable/...).

Most of the time this is due to an ego the size of the Earth's orbit, coupled with a sense of inadequacy they won't acknowledge to themselves (being intimidated by their new environment, where they aren't by far the smartest kid of the block, as they are accustomed from school back home).

Handling those people is difficult.

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they feel the need to share it with the rest of the class interrupting my lecture.

Is the problem (1) that they're interrupting, (2) that their interruptions are excessive or not of interest to the rest of the class, (3) that their comments are incorrect, or (4) they the way they do it is rude? I wouldn't consider #1 a problem at all; there is little excuse in 2014 for making students sit through a noninteractive presentation that they could just as easily have watched on youtube.

Some suggestions:

Avoid reactions that might give the impression to other students that their participation would also be unwelcome.

Don't shut down these students in a way that they will interpret as an attempt to embarrass them in front of their peers, because teenagers are extremely sensitive to this kind of thing, and even if you're right, it can be damaging to the student-teacher relationship. However, if the problem is rudeness or excessiveness, address that. "I think the issue you've raised is an interesting one, and I can tell that you're not yet satisfied with my answer. However, we've spent about 10 minutes on this topic. Class time is limited, and we need to be sensitive to the fact these other folks in the room may not be as interested in this question as we are. Let's continue this discussion after class or in my office hours."

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One way of handling this situation that I haven't seen above is an examination of their knowledge of the subject. The first day of class, review your syllabus and typical basic class procedures, and give an optional examination that could be a mock final for the class, if they get (whatever grade) they get an A in the class. Students are free to leave at this point if they don't wish to take the exam. This does a couple of things, the people that leave this examination are people who you will not have this problem with. (because if they really thought they knew it, they would have stayed). It also lets you pick out which students (if any) in the class might actually know the material, and saves them waisted time in a class they understand. The kids who take this exam and don't pass, will get their examination back the second class and take a private blow to their ego that doesn't offend them, or make them think they made a stupid mistake.

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