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I'm sorry if this isn't an appropriate place to ask this.

I taught a couple years in a secondary school, where teachers are supposed to explain every little step in the simplest but accurate way.

In high school things don't change much. Perhaps the pace is a little faster but still the goal is to be as clear as possible. Teachers are supposed to use pictures, animations and justify every step when solving problems.

In the university things are a lot different. We get exercises every week and solutions are extremely compact. I don't mean to show every little step when computing a determinant of a $3\times 3$ matrix but often one reads "one implication is trivial, one can (easily) see that, it is clear that..." and so on, which makes me personally feel incompetent because most of the times I don't easily see what I'm supposed to. At least in secondary school, it is mandatory to avoid using words such as "trivial, easy" and so on because of that reason. Still, some teachers do it and some believe this to be one of the causes of the spread of anxiety for mathematics, which certainly hinders the learning process even more.

This applies to this platform as well. The style completely changes. Why is that so? Are compact solutions more elegant? Are mathematicians just lazy?

I'm very curious about it.

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    $\begingroup$ The second paragraph is cut off. Do you want to edit it? $\endgroup$ Dec 28 '20 at 20:35
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    $\begingroup$ It depends on context. For a first year course in math (like an introduction to proof based math type of course) I think it is extremely bad practice to use words like "trivial" or to leave too many things to the reader, because indeed it is a drastic change compared to how most people are previously taught things. But I think as one progresses, it becomes more necessary to skip some of the smaller details because they really are trivial/simple enough that with a little bit of thought can be solved (often it's a matter of unwinding definitions so there's nothing much to write) $\endgroup$
    – peek-a-boo
    Dec 28 '20 at 20:38
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    $\begingroup$ “The proof is left as an exercise for the reader”. 🤣 $\endgroup$
    – BobaFret
    Dec 28 '20 at 20:40
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    $\begingroup$ I think it depends. Partially, you're just describing different types of communication: when it's from teacher knowing it all (lol) down to a student, you need another voice than between two researchers knowing the subject fairly well and won't bore each other with insignificant details. Nonetheless, quite a few of those "trivial" and "obvious" can mean that the writer is just too lazy to write it up. $\endgroup$
    – user436658
    Dec 28 '20 at 20:43
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    $\begingroup$ For example, in a first year analysis course, you may learn about continuity, uniform convergence etc, and you may learn the proof that if $f_n$ is a sequence of continuous functions which converge uniformly to $f$, then $f$ is also continuous. In first year, you may see this proof in great detail, but if something similar comes up in a more advanced real analysis course (as a small part of some bigger theorem), people would just write something like "this is a simple $\frac{\epsilon}{3}$ argument". So, the level of elaboration depends very much on the maturity of the intended audience. $\endgroup$
    – peek-a-boo
    Dec 28 '20 at 20:46
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In school, mathematics lessons largely focus on teaching you how to follow certain algorithms, so it's important to spell out every step well enough each student will get the hang of it. But from the undergraduate level onwards, you're being trained to write proofs for mathematicians. There are two key challenges here: inculcating the right vocabulary, and ensuring a proof's author and reader see the forest for the trees. As I've discussed here, the latter goal often requires concision, some of which makes the reader write half the proof in their head.

Some concision techniques serve both goals simultaneously. For example, "it's trivial" not only keeps the write-up short (partly by leveraging the reader as aforesaid), but also trains the reader to do the same when they write a proof. For example, if I have to use a fact of the form $\sum_{k=1}^na_k=b_n$ in a proof, I'm not going to write out a proof by induction before I move on. What would be the point? If anything, the reader might get lost. "Wait, why do we care again? Wait, what were we working on? Wait, what was it we proved before they're using now?"

Teachers at every level train students to write what they're expected to. In an exam, you show your work so if you make a mistake you can get partial credit. (That's what I was already told, anyway; how you're supposed to work out the answer without some work you might as well write down is a separate issue.) In a paper you're preparing for publication, you keep it short because they don't want to allow you many pages, nor do readers want you to waste their time. If I saw someone waste four lines on a trivial induction proof in a 7-page paper that's trying to teach the mathematics community something new, I'd be gobsmacked. That's four lines that could have gone towards one of the less trivial things that made the paper worthwhile.

Much of this comes down to chunking. Once you've learned how to do something that has several parts, it can become one part in learning to do something else etc. This is as true of writing proofs as it is of executing algorithms. Before you know it, 1,000 pages of secondary school textbooks will feel like the contents of a post card in your mind. So don't sweat what you're struggling with now; you'll be surprised how quickly it all becomes second nature.

Edit at @BCLC's suggestion: the concision I described can be preferable to relegating details to appendices because, whereas certain steps can break the flow even if brief, appendices are reserved for examples that nevertheless cannot be omitted due to being non-obvious.

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    $\begingroup$ 'I'm not going to write out a proof by induction before I move on.' --> How would you respond to someone who says something like 'What about putting in appendix?' ? I mean you're responding to why the proof is postponed from discussion but not really to why omitted $\endgroup$
    – BCLC
    Dec 29 '20 at 6:50
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    $\begingroup$ Disagree with, "But from the undergraduate level onwards, you're being trained to write proofs for mathematicians...". There are many (most) people who take Calculus I, II, III, differential equations, and linear algebra, who are not engaged in writing proofs. (U.S. here) $\endgroup$ Dec 29 '20 at 7:14
  • $\begingroup$ @BCLC Appendices are more for "too long not to break the flow, but too non-obvious to omit" than "obvious how to prove (given the audience), so would break the flow even though brief". $\endgroup$
    – J.G.
    Dec 29 '20 at 7:44
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    $\begingroup$ @DanielR.Collins The exact cutoff will depend on many factors. UK here. But undergrads may experience writing styles based on optimistic assumptions about their future in the subject. $\endgroup$
    – J.G.
    Dec 29 '20 at 7:46
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This is more of a comment than an answer and might be seen as some sort of addendum to J.G.‘s answer.

It is completely correct that in school one is trained to follow algorithms, while in university the focus switches to writing correct algorithms (=proofs). Moreover it is very tempting (if not necessary in the case of research papers for example) to gloss over standard details to get to more interesting stuff.

Yet I feel like using words like trivial, obvious or easy is very bad practice, especially in the context of mathematical education. It unnecessarily makes students feel bad, if they don’t get the detail, sometimes even after hours of staring at it. It always is a triviality at the end, but most of the time it needs that one standard trick, which just isn’t memorized yet.

I don’t understand, why it isn’t more common to give concise sketches of the idea required to prove a result:

  • by definition says plugging in the definitions will make the result obvious
  • by iteratively using lemma X on Y is a pretty concrete recipe
  • by strong induction on $n$ gives information about which type of induction hypothesis is needed

et cetera, et cetera. I don’t think this requires that more time and space and most importantly it gives some sort of reality check: If you cannot summarize the triviality in like one sentence, it most likely is not so trivial at all...

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    $\begingroup$ (+1) I typically tell my student that when a mathematician says that something is "clear", "obvious", or "trivial", it typically means one of the following: (1) the author is too lazy to explain the details; (2) the author hasn't actually checked the details, any maybe doesn't know how (I once fell prey to this, and got rightly reamed by my master's advisor for it); or (3) the author thinks so little of the reader that they don't think it is worth their time to explain the reasoning. $\endgroup$
    – Xander Henderson
    Dec 29 '20 at 5:31
  • $\begingroup$ Amen to that. I think there should be some kind of proof sketch/outline or even some kind of 'proof preview' (something less than a sketch/outline, but at least gives an idea what the proof is going to be like) Like I think I can work out the details myself IF i know what details i'm supposed to work out. $\endgroup$
    – BCLC
    Dec 29 '20 at 6:52
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    $\begingroup$ Unfortunately, I think a few times 'trivial' is justified eg if a statement's assumption is like 'for all charts' and then the conclusion is 'there exists a chart' $\endgroup$
    – BCLC
    Dec 29 '20 at 6:53
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    $\begingroup$ @BCLC regarding the charts I would agree that this is a case where using trivial is alright, yet I feel like by definition is more polite. $\endgroup$
    – PrudiiArca
    Dec 29 '20 at 10:54
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    $\begingroup$ @BCLC But even in a case when the result is trivial, what is gained by adding that to the text? If the result is clear, how is the exposition improved by adding in one of these phrases? Why say "It is clear that there exists a chart" when you could say "By definition, there exists a chart" or even more simply "There exists a chart"? The latter two phrases convey the same information, but without being condescending. :) $\endgroup$
    – Xander Henderson
    Dec 29 '20 at 11:55
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Levels of clarity/terseness and the accompanying motivations behind them will vary a lot from context to context.

  • For terse teacher-written solutions to homework, it may just be due to time constraints, with the expectation that if something is unclear you would go ask for help/clarification.
  • Sometimes books are written very compactly partly for elegance, but also partly because they are aimed at an audience who are able to fill in the details. (For example, imagine reading a physics textbook that wrote out the details of long division every time they divided two numbers.) For this reason, it can be difficult for newcomers to learn effectively purely from reading a book since they are still learning how to fill in the gaps
  • Teachers also have the opportunity to be more clear in person/lecture (faster mode of communication) than in writing (more time/effort needed to write things down formally), and you have the opportunity to directly ask questions when things are not clear. Take advantage of having an expert available by asking them questions even if you think it is a "stupid question."
  • Even at high levels of math, everyone appreciates having concrete examples to make abstract concepts more tangible, especially when communicating about research in a niche area to a broader audience. Terseness is not "more acceptable" as math gets more advanced, but the level expected of the audience may be higher.
  • The use of "trivial" in pedagogy is not helpful in my opinion. But more generally, it can be a helpful indication that the justification for some claim follows from some simple definition rather than some complicated argument.
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