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Proofs, or any mathematical derivation, appearing in any real setting, such as a book or textbook or talk, or even when we're teaching it in class, includes a great deal of surrounding explanation. But when we ask students to regurgitate proofs, we ask for what is merely the skeletal core of the proof in this rib-like two column format. At the very least, it seems to me, proofs ought to have three columns, including a new multi-row lefthand column that describes (at least) the approach being taken in that section of the proof. By "multi-row" I mean that the proposed new lefthand "row" can encompass multiple rows of the basic two-column proof. The resulting format would look like this:

    =================|===============|==============
      What we're     |   <c=<c       |  Reflective Property
       up to in      |  tABC~=tDEF   |  SAS
       this section  |    etc...     |  etc...
    =================|===============|==============
      What we're     |    etc...     |  etc...
       up to in      |    etc...     |  etc...
       this section  |    etc...     |  etc...

The context would, I think, largely reflect the reasoning and planning that went into (goes into) the proof, and would commonly, I think, end up representing lemmas that participate in the larger proof. One could say that these lemmas ought to be rolled off into prior proofs of their own, and I would agree. But we do not provide a way, in geometry, of naming and organizing proofs usefully so that prior short proofs (technically lemmas) can be looked up and referred to easily.

Because we do not have a clear naming scheme for proofs, we cannot call upon them as one would functions in a programming language. Indeed, one might wonder why student of geometry aren't being taught geometry like one would teach a programming language: Here's a bunch of functions (lemmas) you can use, and here's how to use them. We do do this for some things, like the triangle congruence lemmas (SAS etc), and for some logical rationales (CPCTC, etc), but the dozen random theorems (lemmas) regarding parallelograms, mid segments, and so on aren't ready-to-hard functions with clear naming, so we end up re-deriving/proving them in the middle of other proofs, which makes the proofs into these long-winded, un-memorable and ultimately unwieldy things.

The three-column format I'm proposing at least offers a way to internally organize proofs into logical segments so that even without addressing the problem of the previous paragraph, at least there is a a way of making the substructure explicit.

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    $\begingroup$ I'm just happy if I can get the students to write complete sentences with actual words. $\endgroup$ Jan 12 at 5:01
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    $\begingroup$ The rigid 2-column proof format along with abbreviations for axioms or already proven theorems is characteristic for American high schools. In other parts of the world they use free-flow text form. Your three-column proposal is of the same "8 steps to solve a linear equation" sort of schemes that American edu-consultants like so much because they can copyright them and sell them as a "method". $\endgroup$
    – Rusty Core
    Jan 12 at 17:28
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    $\begingroup$ As a Russian, I don't understand the question. What columns, what table, what the hell are you even talking about? $\endgroup$
    – CrabMan
    Jan 12 at 19:13
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    $\begingroup$ Well, then do not ask for proofs to be in the rib-like two column format - problem solved. $\endgroup$
    – Rusty Core
    Jan 13 at 8:35
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    $\begingroup$ The "basic two-column format" is peculiar to elementary education in the US. It is certainly never used by practicing mathematicians, at least not since duality was assimilated and the need to prove every result in projective geometry twice, once for points once for lines, was abandoned. It should be purged from US education too. It does not aid in teaching careful thinking or writing convincing arguments. $\endgroup$
    – Dan Fox
    Jan 13 at 17:22

2 Answers 2

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OP: What seems to be missing as taught is an explicit indication of the plan.

Even though this is far afield from your concentration on geometry, it illustrates your point. Michael Sipser's text, Introduction to the Theory of Computation, includes many "proof idea" sections prior to launching into each formal proof. I've taught from this text and this really works. Such "proof idea" thumbnails can be presented in any format (Sipser just uses prose paragraphs), a 3rd column if you prefer.

        Cover

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I think maybe the correct answer to this question is something simple like

Sure, do something else if you like.

I also recommend reading this question because the good answer there may give you some insights into how imposing a structure on proofs can help and how it can hurt.

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