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How should instructors deal with the fact that the students who know the least are precisely the ones who think they know everything?

NOTE: This question is not and never was intended to be primarily about mathematics instruction, but was intended to be equally applicable to all subjects to which it can be applied (which probably means all subjects). It has been edited by seemingly many people. I hope I've now expunged the most offensive edits from it. The examples below are about mathematics for the obvious reason. END OF SPECIAL NOTE

Someone takes high-school math courses in which all that is required of them is that they learn algorithms for solving assigned problems, and they think that learning those is what learning math consists of (in other words, they learned no math) and since they always worked hard and got straight "A+"s, they think they're really good at math. Then they say they're entitled to take a course requiring only learning technical skills and algorithms and not requiring them to understand anything, and they think it's outrageous that an instructor would think there's something more that should be required of them. You can't get more ignorant of mathematics than to think that learning procedures to follow mechanically in solving assigned problems is what learning mathematics consists of (or at least you probably can't get more ignorant than that and still get admitted to a university), nor more arrogant than to think that an instructor needs instruction from people who think that.

High-school courses in which students effectively learn (without being told so) that learning mathematics consists of learning procedures to apply mechanically to solve assigned problems are the result of official decrees that everybody must learn mathematics. So ignorance of mathematics results from decrees intended to have the opposite effect.

In one case a student wrote on a test something like "Are you kidding? u-substitution? We shouldn't see that until next semester!" An entirely inexperienced person would have laughed out loud at a student mistakenly thinking under the circumstances that they were being asked to use the technique informally called "u-substitution", and moreover the class had been told in advance that that particular problem would be there. And sometimes they're not angry (as in the example in my previous comment) but they neglect a topic of which they are completely ignorant because they think they already know everything about it.

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    So is the question "How to deal with the fact that..." or "How to deal with the students who..."? I voted to reopen anyway... – fedja Sep 23 '16 at 4:55
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    There is an entire site on the SE network dedicated to math educators. I think this question is much more relevant there. – eykanal Sep 23 '16 at 12:15
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    Set them tasks from the Math Olympics. As extras. Give brownie points for solving them. That should show the good ones what they need to stretch towards, and the others will find themselves behind and feel the limits of their "competence". – Captain Emacs Sep 23 '16 at 13:40
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    I think this question has nine down-votes and nine up-votes so far. The large number of down-votes without attempts to say what anyone objects to about the question makes me suspicious. Will admitting to suspicions incite more down-votes? – Michael Hardy Sep 23 '16 at 16:24
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    I downvoted this question originally because it was extremely ambiguous - it didn't say what actual student behavior you were trying to fix. "Dealing with the arrogance of ignorance" can mean a lot of different things. I tried to fix the title to better reflect your specific question, hopefully that will help with the votes. (If you keep the edit I'll retract my own downvote..) – ff524 Sep 23 '16 at 18:44
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I think this is an important issue, and wrestling with it has informed a lot of what I do as a community college lecturer in mathematics. Helping students make the transition from the BS they get in many K-12 programs, and getting to share real math with them for the first time, is a great challenge and responsibility.

What gets me a surprising amount of traction is to address this explicitly, as the first thing on my syllabus. Top goal: "Read and write math properly with variables." I verbally quiz them on this on day two. I touch back on it almost every day. I explicate how I'll be grading for this on tests, and show grading examples from old tests. Why? The professional writing (a) provides a shared language, (b) allows them to read any math book, (c) serves as an explanation to other students and colleagues, and (d) makes it easy to find and fix errors and disputes.

Example interaction from yesterday (day 9 of the fall semester): One student has garbled the writing of a polynomial multiplication; I point this out, and she does the "But I got the right answer" bit. I ask, "But what's the number one goal for the course?". She says, "I don't know" (which even she can tell is not a good response), and almost all the rest of the class calls out, "Reading and writing math properly".

So this makes the expectation very clear, and by talking about it as item #1, I get the majority of the class on my side, and the community standard (peer pressure) works greatly in my favor. There aren't many silver bullets in teaching, but I'm delighted at how effectively this one works for me.

(P.S.: While the above is math-specific, I think the basic idea of setting a reading/writing/justifying "top goal" can work in many classes. E.g., in my C++ programming course I start with a quote from Bjarne Stroustrup, "Design and programming are human activities; forget that and all is lost", and then likewise emphasize making one's code readable to other programmers via a common style. As Ken Bain writes in What the Best College Teachers Do, "Finally, the best educators often teach students how to read the materials..." [Ch. 4]).

  • Telling them the right bottom-line answer in every case at the outset might help deal with those who think that getting that is the whole point. – Michael Hardy Sep 23 '16 at 16:18
  • @MichaelHardy: I do a little bit of that; I hand out practice tests that are presented that way. – Daniel R. Collins Sep 23 '16 at 17:28
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You're going to not come over as very likable if you're the one who always corrects the same student. But you can put this task onto other students. For example, if problem student A (again) provides his or her opinion on a topic, ask the rest of the class what they think. The other students will do your task in correcting A. Of course, you ought to also ask the class about opinions brought forward by others, which is generally good class design anyway.

  • What is typical is that several students are like this and sometimes it's entirely private. In one case a student wrote on a test something like "Are you kidding? u-substitution? We shouldn't see that until next semester!" An entirely inexperienced person would have laughed out loud at a student mistakenly thinking under the circumstances that they were being asked to use the technique informally called "u-substitution", and moreover the class had been told in advance that that particular problem would be there. – Michael Hardy Sep 22 '16 at 21:36
  • . . . . and sometimes they're not angry (as in the example in my previous comment) but they neglect a topic of which they are completely ignorant because they think they already know everything about it. – Michael Hardy Sep 22 '16 at 21:42
  • I don't seem to encounter these kinds of students, so don't quite know what to suggest. – Wolfgang Bangerth Sep 23 '16 at 4:33
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    @MichaelHardy I don't understand the u-substitution comment - Is this a problem that explicitly asks a student to use substitution to integrate something when it was also explicitly mentioned that problems requiring the use of that technique would not be on the test? Or is it a problem that is "usually" solved with substitution that can also be solved with a different method and the student just assumed that he/she was supposed to use substitution? – Kevin L Sep 23 '16 at 13:46
  • @KevinL : No integrals were involved. Nor derivatives. – Michael Hardy Sep 23 '16 at 16:19
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Some students can be extremely proud (or arrogant), especially if they have been straight A students their whole lives. You may be better off explaining to them that there is a huge gap between the math they have learned in K-12 and the point where college starts off. Effectively suggesting that there are things they need to learn in addition to what they already know instead of implying that they didn't learn anything.

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    Those who think they know everything "know" that any such messages are intended for their classmates and not for them. – Michael Hardy Sep 24 '16 at 1:09
  • There shouldn't be such a huge gap, and probably isn't among students are are good at the subject. – Michael Hardy Sep 29 '16 at 22:32
  • "Effectively suggesting that there are things they need to learn in addition to what they already know instead of implying that they didn't learn anything." A problem with this is that some students, and some professors, try to be deliberately stupid in some circumstances because they know that in some circumstances that is a good strategy for getting ahead in the system, as opposed to actually learning. The kind who think they know everything are usually the kind who do that. – Michael Hardy Oct 22 '16 at 17:46

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