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When I want to point out to my students that getting the right result is not enough, I like to show them the example:

$$\frac{16}{64} = \frac{1\hskip-.1cm- \hskip-.4cm{6}}{-\hskip-.2cm{6}\,4} = \frac14.$$

This example works well in some of my classes where the error is just the right level of "stupid": The students feel absolutely safe from making this error, and they also do not feel that this is a "kindergarten" example too far removed from their level of mathematics.

But I would like to have some more examples like this for students with less and with more knowledge about mathematics.

What are some good simple preferrably amusing examples that getting the right result is not enough?

user11235
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  • I will admit that I was a loss to find good tags that I would use to search for this question. Also, mathjax does not work. – user11235 Mar 13 '14 at 21:22
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    I have posted a feature request for MathJax on meta. – Jim Belk Mar 13 '14 at 21:32
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    Please take a moment to refine your question based on this advice. I think the better question here is to ask not for examples, but concrete techniques to use in this situation. – Jon Ericson Mar 13 '14 at 22:27
  • @JonEricson My question is really not about techniques. In a way, presenting an easier example with a funny error is the technique, but one needs example to use the techinique. – user11235 Mar 13 '14 at 23:41
  • It seems like this thread on MSE should be helpful. – Jared Mar 14 '14 at 00:20
  • @GammaFunction Why? – Geoff Pointer Mar 14 '14 at 00:31
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    I notice that the close votes are not for "duplicate", but for "opinion-based" and "off-topic". Since this is exactly the type of question I have committed to this site for, I think that it would be a very good idea to discuss this on meta. – user11235 Mar 14 '14 at 09:16
  • I think very few of the issues mentioned at @JonEricson's link are applicable to this question. I don't see a problem with this question, but it seems to have been pretty well answered already by the MSE thread. – Mike Shulman Mar 14 '14 at 22:39
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    @JonEricson If the suggested example given by the OP is not concrete then what can be? Lists of concrete examples for mathematics teachers are not shopping lists. They are very useful resources and the more the merrier. Teachers have repeat lessons, new examples help them stay interested in their work. I can imagine looking up this question at some future date after it's been given a chance to grow and getting some very good ideas to use in lessons. And, a correct answer coincidentally given by false mathematics, is not an opinion based matter. – Geoff Pointer Mar 14 '14 at 22:53
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    @user11235 I do agree that the question could be better framed. For example: What ways are there to emphasise to students the importance of understanding processes as opposed to just getting the right answer? The difficulty of course is you're not supposed to ask what is the best way or what are good ways because of the opinion based bogey man, but every answer to this question will be an opinion about how to go about this and so what? That's what this site is supposed to be about, Mathematics Education. The content may be concrete, but the delivery is very opinion based. – Geoff Pointer Mar 14 '14 at 23:11
  • This meta answer lays out my reason for putting this question on hold. – Jon Ericson Mar 15 '14 at 00:04
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    $\dfrac{x^2-y^2}{x-y}=\dfrac{x-y}{-}=x+y,\checkmark$ – حكيم الفيلسوف الضائع Mar 16 '14 at 14:44
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    Check out Barbeau's "Mathematical Fallacies, Flaws, and Flimflam" (MAA, 2000) for lots of hairraising (or just subtly wrong) derivations giving correct answers. – vonbrand Mar 17 '14 at 17:17
  • @user11235 I'm still not sure I understand why you haven't tried to edit your question based on advice given. You haven't been asked to get rid of it entirely just to clarify it's scope. Among other refining points that have been mentioned, most important to me as a teacher is that I'd be more willing to scan through a long list if I knew it was at least relevant to the level I'm teaching. – Geoff Pointer Mar 17 '14 at 23:17
  • @GeoffPointer The advice was to ask for "techniques" which is a completely different question. – user11235 Mar 18 '14 at 00:21
  • @user11235 Okay, so if there is a ground swell that lists of resources should be allowed and you make it clear that your question is one of these then the criticism aimed at you should be phrased in terms of wording it as an example of a good list question. I'm not sure what they're after. Perhaps "Give outlines of presentations suitable for emphasising the importance of understanding process over just getting the right answer - aimed at beginning undergraduates." I think some are just anti lists period, I would hate to think you'd get shut down by non educational professionals. – Geoff Pointer Mar 18 '14 at 02:14
  • I tell my students (in the frame of computer programs, but it is the same here, essentially) that "the right result" isn't enough, what is needed is to be able to convince someone that it is really correct. – vonbrand Apr 12 '14 at 03:02

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