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:**

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

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.

11I usually tell students that a standard technique is just a dirty/insightful trick that you've seen used more than twice. – Aeryk – 2015-03-28T20:20:02.560

4I 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. – Joseph O'Rourke – 2015-03-29T00:15:12.650Is 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. – Jack M – 2015-03-29T23:56:26.310

5I like to ask students to reflect on their struggles years ago when learning how to tie their shoe laces. – user52817 – 2015-03-30T16:16:18.453

2Just 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. – Hal – 2015-03-30T16:47:09.330