It would be impossible to spell everything out, though. Then you would have to start every proof by inventing the concept of natural numbers and so on.
While I agree that you can't reasonably implement a complex subject from first principles every time you want to talk about something, I think the core of the complaint was more in frequently calling something trivial when addressing a room full of students with varying ability.
Trivial is, I suspect, best expressed in terms of inferential distance that the student has to cover. Education naturally works on the borders of what people know: too far and showing people something is incomprehensible to them, too close and they're learning nothing they couldn't have found themselves simply by looking at your powerpoint stack.
I suppose, to develop that line of reasoning, it might go something like this:
Given that you're operating on the edge of people's concept space to be teaching them something worthwhile, if you're saying that something's trivial (i.e. that they should find it trivial) a lot, you're either:
A) Wasting people's time, (it's too close to their known concepts)
or
B) Confusing people, (it's too far from their known concepts)
If you're in the goldilocks zone for learning, it shouldn't be trivial. Might it rely on things that are trivial to them? Sure. But I don't see any value in mentioning that they're such, and if you're dealing with varying ability it's worth keeping in mind that some of the things you think are trivial aren't going to be to everyone.
Basically, my question to you, would be: What value is added by calling something trivial to justify the harm to those who don't find it such?
You could after all just not mention that the thing is trivial, perhaps more students would have the courage to ask you about it if they don't understand it that way.
I think you interpret too much into this question. If you do a proof, at some point you have to say something. Either "it's trivial" or "it's obvious" or "it is known" or whatever. Otherwise, how would you stop expanding the steps of the proof?
I also don't think a university course has to hold every students hand. For example if you offer Advanced Calculus or whatever, it is fair to expect students to know that 2+2=4. If some step is too far from a students known concepts, they can either invest extra time trying to catch up (Google is your friend), or they can switch courses.
You could also debate which approach is better for learning maths. In my time, there were comparatively thin books about calculus and comparatively thick books about calculus. The latter spelled everything out. I hated them - too many words really made it hard to focus. I personally much preferred the dense books that left more thinking to the reader.
Maybe for some students the thick books are better, but I don't think a teacher has to please everybody. Students should have the opportunity to switch to another teacher or subject if they can't cope with the current teacher.
> I think you interpret too much into this question. If you do a proof, at some point you have to say something. Either "it's trivial" or "it's obvious" or "it is known" or whatever. Otherwise, how would you stop expanding the steps of the proof?
Just stop when you've got as deep as you care to go. I don't see the necessity to say anything there. If it is trivial, then it's not going to be challenged because everyone will know that they'd lose, and if it's not trivial and it is challenged, then you've identified that at least one of you's going to learn something in the exchange.
> I also don't think a university course has to hold every students hand. For example if you offer Advanced Calculus or whatever, it is fair to expect students to know that 2+2=4. If some step is too far from a students known concepts, they can either invest extra time trying to catch up (Google is your friend), or they can switch courses.
There's failure on both sides if students are being entered for a course that's significantly beyond their ability in that sense.
On the one hand, the student needs to try if they expect to get anything out of it, and persistent focused effort should have equipped them for most of what they can reasonably expect to run up against. On the other hand, the college shouldn't be taking people on who are manifestly unsuited to the subject they're being admitted to.
Perhaps in first year, that's understandable. Testing what people know for admittance, especially given the pitiful standard of secondary education and testing, is a non-trivial task. However, by the time you're getting into second year, if the university has passed them, and they're under-equipped for the second year... well, why the ever loving spaghetti monster did you pass them from first year?
I think we'd probably both agree that both parties in education need to make reasonable effort. If the student isn't willing to try, then there's nothing that can be done. If the college isn't putting its best foot forwards, then the student may as well just be paying for the right to sit the exam for all the value the college is providing.
"Just stop when you've got as deep as you care to go."
Really, I don't think saying "it's trivial" is such a big ordeal for students. In maths at least, they'll quickly learn to get over it. I think in the beginning I was surprised a couple of times. Then I thought about it, and then after a while I realized it is trivial. It is also a pointer for the students so they can see "I should know this".
As for passing students from first term to second term - I am not sure if I care. Why not let students enter any course they wish?
If we're talking about college math courses, then presumably there are prerequisites for any moderately advanced course. So "trivial" would be in the context of students with the appropriate mathematical background to be taking this course.
Its a meta problem. Both math and physics have their own little language which is only vaguely inspired by standard English, and its a popular meme to complain they're just similar enough to cause emotional anguish.
For example a pathological problem doesn't mean you'll get a nasty MRSA infection from working on that problem. At least not directly...
Of course it's subjective. Why is that relevant? Math teachers are teaching a course with certain prerequisites, so in that context "trivial" means something like "it should be easy/obvious for people with the appropriate mathematical background for this class."
Many of the students in the class got through those prerequisites with, say, 70% on the final exam. That means they have a very tenuous grasp on much of the content, and so teaching with at least some overlap with the prerequisite is useful.
The prerequisites may have been taken 6-12 months earlier. Without constant use, a lot of the concepts that were learned in the prerequisites will be lost over that time period. Many classes will indeed start with the first lesson or two as a revision of the prerequisites for this reason. It possibly annoys those who have a firm grasp, but not everybody learns mathematics (or language, or anything else) the same way.
I've never taken a math course that didn't contain some overlap. But you simply can't realistically teach everything from every prerequisite in every class. There may always be some students who snuck through a prerequisite without learning or retaining the info. That's what office hours are for.
http://en.wikipedia.org/wiki/Triviality_(mathematics)
(Obvious/easy to prove)