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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 ...


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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 ...


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