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Should proofs include a third "context" column?

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$\begingroup$

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.

secondary-education proofs
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edited 8 hours ago
 
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J W
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asked 18 hours ago
 
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$\endgroup$
22

  * 8
    $\begingroup$ I'm just happy if I can get the students to write
    complete sentences with actual words. $\endgroup$
    - James S. Cook
    18 hours ago
  * 5
    $\begingroup$ What is your question? $\endgroup$
    - J W
    16 hours ago
  * 3
    $\begingroup$ Please put your question in the title, so that this
    reads as a question (and not as a rant). You might consider
    actually asking a question. Are you interested in other folks'
    ideas about how to help students write proofs? If you just want
    to know whether this could be useful, I think it's a reasonable
    thing to use. $\endgroup$
    - Sue VanHattum 
    16 hours ago
  * 5
    $\begingroup$ I think this question needs a country-tag. $\
    endgroup$
    - Jochen Glueck
    13 hours ago
  * 4
    $\begingroup$ What level are you talking about high school or
    college? $\endgroup$
    - Amy B
    10 hours ago

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