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When presenting a proof, there are usually a lot of parts which look like "obvious", "routine" manipulation to me, and between zero and two genuinely insightful steps. I want to point out the difference between these two steps and the others.

I worry that calling things like this "obvious" or "routine" makes students who don't see them feel like idiots. Is there a way to discuss this distinction without making the students who are still working on these steps feel dumb?

The same issue arises when helping students work out their own proofs, rather than presenting one.

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    $\begingroup$ I usually tell students that a standard technique is just a dirty/insightful trick that you've seen used more than twice. $\endgroup$ – Aeryk Mar 28 '15 at 20:20
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    $\begingroup$ I love the notion that there are "between zero and two genuinely insightful steps" to a proof. It seems so true (and new!) to me that this insight should be preserved regardless of how you respond to the other issues raised. $\endgroup$ – Joseph O'Rourke Mar 29 '15 at 0:15
  • $\begingroup$ Is this not overly paranoid? I can't remember a single time when a proof said that something was obvious but which I myself didn't find obvious. $\endgroup$ – Jack M Mar 29 '15 at 23:56
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    $\begingroup$ I like to ask students to reflect on their struggles years ago when learning how to tie their shoe laces. $\endgroup$ – user52817 Mar 30 '15 at 16:16
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    $\begingroup$ Just from the question title, "this will eventually be easy" doesn't seem insulting to me at all. I don't teach math, but in situations where I've thought 'this will eventually be easy for you' and I needed to pad the person's ego (or their confidence in his/her abilities), I've usually told them a story about how hard it, or something like it, was for me, and how it became easy. I also usually tell them about how cool the experience of it 'clicking' was for me, which I sometimes convey by telling them what else I immediately apprehended as a result of it clicking. $\endgroup$ – Hal Mar 30 '15 at 16:47
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One thing that you might want to do early on in your course is think about the classroom norms that you wish to establish. From your post, it seems like an example of a norm in your class is that it is okay to ask a question of the form:

What do you mean by obvious (routine, etc)?

If you do not try to make it clear to students from the outset that such questions are totally on task, then it is much less likely for them to ask it. (After all, "What is obvious?" sounds silly in most contexts.) Similarly, you may wish to establish the related norm that students are expected to know what is meant when such words (obvious, routine, etc) are used. To this end, perhaps you may want to call on students occasionally (or often, depending on how you teach...) to have them chime in to fill in those details.

Note that classroom norms are typically established early on in the term; if the current norm in your classes is that you will lecture unless a student asks a question, then cold-calling on someone halfway through the semester to ask "what did I mean by obvious candidate?" could be a bit jarring.

For a concrete example of the type of questions you might ask in a math class, here is a handout I used at the start of a Topology course that used Munkres' text (mentioned earlier in MESE 2189). Although this handout pertains to reading mathematics, I think it is relevant for listening to mathematics, as well.

As one more concrete example, consider the following problem for a first Real Analysis course:

Prove: Any infinite, closed set $A \subset [0,1]$ contains a limit point.

What do students need to know (or figure out) as they come to understand a proof, or generate a proof, or prove a more general statement (e.g., replace $[0,1]$ with any bounded set; or consider the analogous proposition in $\mathbb{R}^n$ rather than just $\mathbb{R}$)?

If you are curious about frameworks from the mathematics education literature that could help with thinking about such questions, then Dubinsky's APOS theory is one place to start; essentially, what is asked above amounts to providing a "genetic decomposition" (terminology originating from Piaget's work as a "genetic epistemologist"). Alternatively, you may wish to look at Simon et al on hypothetical learning trajectories.

I think there is about one genuine insight needed for the problem above, which is bisecting repeatedly to pick a side, each time, with infinitely many points. (As mentioned in MESE 2226, I think about this as specifying the limit point by finding its binary representation.) Perhaps this is really two insights that I have combined together. In any event, there are many details to be filled in:

How were the givens used? Must the set by infinite? (Yes!) Must the set be closed? (In general, yes: Consider $A$ as the set of unit fractions, for example.) Was bounding $A$ necessary? (In general, yes: Consider $A = \mathbb{N}$, for example.) Are these parenthetical counterexamples clear? Is it obvious why the point produced through repeated bisection is a limit point? (Perhaps not the first time through; can the student show as much for any given $\epsilon > 0$?)

I think that unpacking questions/proofs by writing down (for yourself) what you consider to be the key insights and the routine details is a good way to prepare for a topic. This prepares you to emphasize the insights, and perhaps to pose follow-up problems in which they are used or adapted, and to ensure students can fill in the details, and perhaps to call on them occasionally to do as much (both for their sake, as learners, and for your sake, as you check in with your class).

Summary:

  1. Establish classroom norms around questions and expectations with respect to that which could be called (by some) obvious, routine, etc.

  2. Try to partition material into key insights and routine details beforehand; emphasize the former, and check in to make sure the class is on board with the latter.

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    $\begingroup$ I liked a lot of the answers here, but this seemed like the one that is most likely to affect my future teaching. Thanks! $\endgroup$ – DES-SupportsMonicaAndTransfolk Mar 31 '15 at 12:24
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Perhaps not pointing out that the obvious steps are obvious but that the insights are insights.

I believe students don't feel bad for not seeing the "magic steps" by themselves, so pointing out that those are hard is not a problem. The opposite is what you mention: they would feel bad for not seeing the obvious. Hence, only treat the difficult as difficult.

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For presenting proofs, I would only use terminology such as "obviously", "clearly", "routine", etc if it would be for a course just below the class had the student had 'perfect' recollection. For instance, in a introductory commutative algebra class, if the proof should be routine for the previous regular algebra class (i.e., something involving a course below algebra), then I would use such terminology. It should be used only as an indicator for how difficult the proof is relative to their expected skills. Another example, in Multivariate Calculus, I would say "obvious", "routine", or whatnot to Intro. Calculus concepts, but not necessarily to Calculus 2 concepts.

If it is not at this level, I would just present it without comment or just state the result without explanation. Many students at the level of proofs would question it if it wasn't obvious.

In general, I use such terminology to indicate what I expect my students to have as prior knowledge that they should recall pretty quickly. I do this even in the calculus sequence, quickly going through prior knowledge that should be clear quicker.

Using the measure of "material from two classes prior in a course series" as "obvious" has worked for me so far in indicating expectations to students while not perturbing them. Sometimes it takes a semester or two for things to "click."

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    $\begingroup$ I think there is a big difference between "routine" and "obvious". In the mentality of a student, if I missed something "routine", then that's a gap in my knowledge (which will be fixed after I work through a few examples) and nothing to be embarrassed about. If I missed something "obvious", then there must be something wrong with me. Experts in a field exhibit arrogance when they label manipulations as being "obvious" when they are merely "routine" in their field. $\endgroup$ – Atsby Mar 30 '15 at 4:49
  • $\begingroup$ The criterion in your first paragraph is that the material is from the previous course; the criterion in your last paragraph is that the material is from two courses prior. Moreover, I think that your final sentence is spot on ("Sometimes it takes a semester or two for things to 'click'") which is why I think this terminology can be so harmful -- even when arguments were presented in the previous couple of courses (though perhaps it is more harmful for earlier math learners than those who have reached a course in commutative algebra...). $\endgroup$ – Benjamin Dickman Mar 30 '15 at 7:53
  • $\begingroup$ @Atsby Good point, but I never said that they were similar in any way, just that when I would use such terminology. $\endgroup$ – Chris C Mar 30 '15 at 13:52
  • $\begingroup$ @BenjaminDickman Eh, I think I was a little unclear with my wording there. In a perfect world, all students would know the previous course material making it "obvious" or "routine" or whichever is applicable to us in the current. But I would only use such terminology in the next course, or two past where they first learned it. $\endgroup$ – Chris C Mar 30 '15 at 13:57
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It may help your students if you make explicit why it is obvious/routine/etc. Have they seen it in the previous example? Does it just depend on unpacking definitions? It is a calculation/result/etc that was covered in a prerequisite class?

I don't think there is anything wrong with pointing out which parts are routine, but saying why it's routine will help them more, since this gives them some place to start when they get stuck in a proof, eg instead of asking themselves "is this routine?" they can ask themselves "is this a matter of unpacking definitions?" or "have I solved a similar problem before?" I think being able to ask yourselves these questions (and more!) is a big part of learning to write proofs.

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I find the easiest way is to just straight out say that it's not something you expect them to know offhand or immediately understand, but that as their studies advance it will become trivial. That way they understand that it's new, but nothing to be scared of. Something along the lines of:

"This may not make sense to you yet, but trust me it will soon be second nature!" For now, just try to learn it/remember it/keep it in mind [as applicable]."

or

"Don't worry if you can't do this immediately or it take a few tries, it isn't always something you "just get"... you'll get faster and better at it with practice/experience."

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  • $\begingroup$ "Second nature" is definitely the appropriate expression, here. $\endgroup$ – David Richerby Apr 1 '15 at 8:10
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I see the question cross this and academic sites a lot about whether "trivial" is insulting. Trivial, taken in the context of its literal meaning and explained as such, is exactly the word for these steps you're describing.

Wiktionary says the following:

trivial (comparative more trivial, superlative most trivial)

1. Ignorable; of little significance or value.
2. Commonplace, ordinary.
...
5. (mathematics) Of, relating to, or being the simplest possible case.

When referring to steps of a proof, any of those three definitions apply to many of the steps.

Commonplace steps which are found in any proof of its kind, I suggest you explain thoroughly and mention that they are in fact "routine" and the student will see such steps frequently.

There are also those steps which are required to provide foundation to a proof and assume familiarity with prior proofs. They are present for the sole purpose of indicating that assumptions or entire branches of reasoning present in the proof are already well-established in earlier proofs. I suggest those steps be pointed out as familiar once the reader is familiar with those prior proofs, and explaining what those proofs demonstrate, even if you don't go into how those proofs demonstrate it.

Alternately, they could be proving the most basic case of a proof, which to a student may seem like a red herring, but are foundational to understanding later steps. I'd suggest that you not characterize anything in this latter category as routine or obvious. Rather I'd suggest explaining them fully in the context of the proof the first few times you come across them in your teaching materials, as well as explaining the role that those steps play in proofs of that form.

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You can say, "This is the straightforward part; vs. this is the less obvious part.... The straightforward steps will become easier with practice. The more experience you have with proofs, the more comfortable you'll get."

You are right, that it is not helpful to say, "This part is obvious/routine/trivial."

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