Can someone please point to research papers that analyze different ways of expressing informal proofs from an educational point of view? I am particularly interested in proofs by induction but I would like to read about different ways of expressing logical inferences in a way that students understand them better.

For example, a commonly used one is the calculational style. Are there any others? How do they compare to this one?

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    $\begingroup$ Generally my sense is that students simply haven't had logic. In particular for induction, the inductive step is proving an implication $P \rightarrow Q$, and if they've never practiced that and are jumping into it in the midst of an induction discussion, that's a lot of moving parts to pick up on at once. So a modification to the search might be "does teaching logic improve success in later proofs courses?" $\endgroup$ Commented Feb 19, 2016 at 5:31
  • $\begingroup$ Along the lines of DRC's comment, of the basic methods of proof, induction is probably the most difficult. If you haven't mastered other, simpler methods, induction will be somewhat hit-and-miss. You might Google "software to teach methods of proof." $\endgroup$ Commented Mar 21, 2016 at 13:46
  • $\begingroup$ I don't know about so-called learning styles as avenue of research, especially for something as abstract as proofs by induction. I have seen some postings here attempting to debunk the whole notion saying that there is really no science behind it. Look into it before putting a whole lot of time into it. $\endgroup$ Commented Mar 21, 2016 at 13:55
  • $\begingroup$ See matheducators.stackexchange.com/questions/746/… $\endgroup$ Commented Mar 21, 2016 at 14:05
  • $\begingroup$ Probably not what you're looking for, but I like The Grammar According to West. $\endgroup$ Commented May 19, 2016 at 3:38

1 Answer 1


All "real" proof are (more or less) informal. You should prove things more or less the same way as a professional would, but for the sake of newbies (and not constrained by procrustean page limits, as in formal papers) you should spell more steps out, and perhaps add some further comments and motivation for the steps.

You don't just want to show the proof, you want to show the scaffolding used to construct it, teach techniques to find proofs: work from both ends, hoping they'll meet in the middle; work through some examples; try to find counterexamples (not being able to find them might hint at why it is true).

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    $\begingroup$ Do you have references to literature? $\endgroup$
    – Tommi
    Commented Feb 19, 2016 at 7:02
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    $\begingroup$ @TommiBrander, Pólya's "How to prove it" comes to mind. There are lots of books on proofs, like Richard Hammack's "Book of Proof" (2nd edition, 2013) $\endgroup$
    – vonbrand
    Commented Feb 22, 2016 at 14:16
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    $\begingroup$ Just for clarification, Pólya's book is "How to Solve It", whereas "How to Prove It" is by Velleman. $\endgroup$
    – J W
    Commented Mar 20, 2016 at 7:47
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    $\begingroup$ @JW, you are right. Brain slip. $\endgroup$
    – vonbrand
    Commented Mar 20, 2016 at 13:12
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    $\begingroup$ I agree with the answer of @vonbrand. I would just like to add that in the traditional way that mathematical proofs are written sometimes adding more detail may cloud the big picture and make the proof confusing. In this regard, here, Leslie Lamport describes a way to write mathematical structured proofs that don't suffer from this problem. I like them and I think it's worth taking a look. $\endgroup$ Commented Dec 10, 2020 at 0:16

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