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If this is the wrong forum for this post I apologize but I'm not sure of another well-suited medium for this question (and any reference to one is appreciated).

I am wondering if any research in mathematics education has been done about the value of seeing proofs in a course.

On the one hand, seeing proofs done by mathematicians seems very valuable. However this takes up time and perhaps is better discovered on one's own than seen.

On the other hand, suppose that you just discussed statements instead of their proofs and actually used the statements to prove new results. This also has merit and is more likely what we want students to do within the scope of the course and to some extent as a mathematician.

My question is:

"Has research been done in say a proofs class where one class sees proofs of certain statements and (possibly some/fewer) applications and another class only sees the statements and their applications?"

Does anyone have actual experience with this question?

The ideal answer here would be a paper reference to this topic but my searches of this have yielded no fruit.

Thanks in advance.

EDIT: Ideally a first proof course would be the topic of this question (so say first or second year of undergraduate mathematics) but any experiences would be interesting. Maybe even a second course (but something close to elementary definitions like an Elementary Number Theory course let's say where a student would have already taken a proofs course but has no other knowledge of the subject)

EDIT 2: Thanks for all the posts thus far! The ideal situation would be a study where say a class does the proof of Fermat's Little Theorem (for example; maybe more than one example) in class and maybe one instance of its use and another class does the statement of Fermat's Little Theorem and does some number of instances of its use. Then on say a final exam both classes get tested on:

  1. Questions using Fermat's Little Theorem
  2. Questions that require some sort of proof technique similar to that of Fermat's Little Theorem (maybe like a similar binomial theorem + induction type question).

and see how students differ in these respects. I suspect there's more factors involved than just seeing the proofs and that the likely situation is that this factor alone is negligible but I'm still curious. Thanks for the current references and I'll be sure to check them out!

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migrated from mathoverflow.net Nov 30 '15 at 14:50

This question came from our site for professional mathematicians.

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    $\begingroup$ Could you specify what level of class you are talking about? A high school geometry class, a freshman-level calculus class, and an upper-undergraduate level linear algebra class are all settings where I can imagine that an instructor might choose to include or omit proofs to varying degrees, and I would not expect the effects to be the same in all of those settings. $\endgroup$ – mweiss Nov 30 '15 at 16:03
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    $\begingroup$ It also would be helpful to know what kind of effects you are interested in. Presumably students' ability to write proof would not be developed well by a class in which they don't see proofs. But on the other hand, just seeing proofs (without getting practice at writing them) probably doesn't help students very much either. $\endgroup$ – mweiss Nov 30 '15 at 16:05
  • $\begingroup$ Are you talking about writing proofs on the board versus reading proofs in the text? Are you implicitly assuming that students will never read the text? The evidence is that lecturing in general simply isn't an effective mode of instruction pnas.org/content/early/2014/05/08/1319030111 . Therefore it seems almost guaranteed that presenting proofs as part of a lecture is not an effective mode of instruction. $\endgroup$ – Ben Crowell Dec 1 '15 at 16:18
  • $\begingroup$ Somewhat related: matheducators.stackexchange.com/q/10540 $\endgroup$ – Tommi Brander Feb 14 '16 at 17:13
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Perhaps not a perfect answer, but http://scimath.net/articles/32/323.pdf is a study of whether the number of maths modules students take affects how good they are at proofs. The result is roughly that students start off bad at proofs and end up slightly less bad at proofs, but those who do more maths modules (being on a maths degree rather than a teaching degree) do improve more.

This looks at how students go about reading a proof; the result is they tend to focus on the mathematical expressions and ignore the logical structure, so they cannot easily identify whether a proof is correct.

I can't think of other references right now. But the main message I've picked up is that students don't by default know what to do when you show them a proof, so unless you actually teach them how to benefit, they will not gain much from reading/watching a proof.

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