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There are a number of skills needed in maths (I'm teaching undergraduate pure maths) that are not really topics on their own, such as interpreting a definition, taking negation, or giving counter-examples. Part of me thinks that it's useful, while covering the official content of a course, to also bring in these ideas, so that the students get some practice. On the other hand, part of me thinks this risks confusing the students, by bringing in more than one idea that they are not comfortable with at once. Is there any evidence for which, if either, is better?

Ideally I'd like research papers, but I'm also open to personal experience. I'm also more interested in the case where there are time constraints (ie there is a quantity of material that must be covered that fills the available time).

(A connected question: what about bringing in ideas from earlier courses that students are expected to remember but don't?)

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  • $\begingroup$ For the related question, if they are expected to remember something, then you can use it. I would just mention the topic prior in class as something to "recall" or "remember" and perhaps give a reference. $\endgroup$
    – Chris C
    Feb 16 '15 at 17:24
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    $\begingroup$ For the main question, my first thought is that teaching heuristics/strategies explicitly doesn't necessarily help. Students, faced with problems for which they have the background skills, may still be unable to draw on them. The paper I have in mind is: Schoenfeld, A. H. (1979). Explicit heuristic training as a variable in problem-solving performance. Journal for Research in Mathematics Education, 173-187. Link. $\endgroup$
    – Benjamin Dickman
    Feb 16 '15 at 18:54
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    $\begingroup$ @BenjaminDickman Did you possibly mean to say 'teaching... implicitly'? The conclusion of the paper seems to be that teaching heuristics explicitly does help more than teaching them implicitly, but there's an additional problem of getting the students to use what they've been taught. $\endgroup$
    – Jessica B
    Feb 17 '15 at 12:05
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    $\begingroup$ No, I meant what you/I wrote: Teaching these skills explicitly can still lead to scenarios in which students have them but fail to draw from them. I don't think the article answers your main question (hence the comment rather than an answer) but you might use it as a starting reference and see who else cited it thereafter... (it is over 35 years old!). $\endgroup$
    – Benjamin Dickman
    Feb 17 '15 at 22:00
  • $\begingroup$ To your connected question- Absolutely review skills from previous courses. In high school teaching we typically spent the first 5-10 minutes of class reviewing foundation skills for upcoming topics. The students knew there would be a transition to possibly unrelated skills preventing confusion. $\endgroup$
    – nickalh
    May 9 '15 at 8:19
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My experience is that teaching "tools" (the things you mention in this case) beforehand/separate from their applications is counterproductive. When the tools are needed, they are already forgotten, on not really assimilated because the student didn't see the point at the time.

I try to integrate discussion of the needed skills into the class, whenever appropriate. Don't separate them unless necessary (yes, that's your call), don't make a fuss of them, but show how to apply them by example whenever relevant. Perhaps you can summarize a the end what was learned, separately for "technical stuff" (theorem so and so, this and that solution method) and "tools" (dissected a definition, checked how/when it applies). The second part will probably repeat often.

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