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I have a hypothesis that I have not yet implemented, and am seeking guidance before I do. I have always taught algebra (and above) in such a way that the students understand the derivations. I check their understanding through discussion. The problem is that I have found that these students are unable to recreate the derivations, nor do anything on their own. My hypothesis is that if, in addition, I require them to copy my on-board derivations exactly in their notebooks, that, although this might distract from understanding somewhat, it will give them a good sense of the mechanics, and so might better support their being able to create, or at least recreate, their own. I resist this for obvious reasons, but I’m at whit’s end with students that clearly understand, but cannot actually do derivations.

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    $\begingroup$ "I resist this for obvious reasons" - like what? This was an accepted practice when I was a student, and I think it worked well. $\endgroup$
    – Rusty Core
    Sep 6 at 18:33
  • $\begingroup$ @rusty: I assume that I would have to go slower, which may be good, but will take away from the discussion that I believe leads to depth of understanding. But I may be wrong about the importance of that discussion v copying the mechanics. $\endgroup$ Sep 6 at 20:14
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    $\begingroup$ Unless they use the methods in the derivation in a few contexts then they will surely forget the argument in a year or two. So ask them to work out the derivation in several actual examples to reinforce the method. For example, don’t ask them to regurgitate the derivation of the quadratic formula in the general case, but in a few special cases. $\endgroup$
    – KCd
    Sep 6 at 22:17
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Mere discussion is useless. You need to actually force students to produce proofs, if you want them to ever be able to prove anything. In particular, from my experience I can be quite sure that most students cannot produce perfectly correct mathematical proofs of any newly encountered theorems (say at the undergraduate level) unless they have spent at least roughly 1000 hours practicing proving. Of course you need to in the first place provide them some kind of deductive system, for which the only practical style seems to be Fitch-style (e.g. this one or the one in LPL).

So what you need is literally to have them do many proof exercises, many times. To give a very simple analogy, you can discuss all the details you know about building a steam engine with a student, but if that student never tries to build a steam engine, you can be very sure that the student will utterly fail on an actual attempt, no matter how much they 'understood you through discussion'.

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In general, trying to learn formal language or algorithms leads to many pupils not understanding what is going on and thus making bizarre mistakes, because they have a rough sense of what the result should look like, but little sense as to why.

This might be counteracted by the understanding created through discussion, but that is hard to say without further details.

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