Has anyone tried as an additional technique the "fill-in" method?
This is based on the tried and tested method of teaching called "reverse chaining". To illustrate it, if you are teaching a child to put on a vest, you do not throw it the vest and say put it on. Instead, you put it almost on, and ask the child to do the last bit, and so succeed. You gradually put the vest less and less on, the child always succeeds, and finally can put it on without help. This is called error-less learning and is a tried and tested method, particularly in animal training (almost the only method! ask any psychologist, as I learned it from one).
So we have tried writing out a proof that, say, the limit of the product is the product of the limits, (not possible for a student to do from scratch), then blanking out various bits, which the students have to fill in, using the clues from the other bits not blanked out. This is quite realistic, where a professional writes out a proof and then looks for the mistakes and gaps! The important point is that you are giving students the structure of the proof, so that is also teaching something.
This kind of exercise is also nice and easy to mark!
Finally re failure: the secret of success is the successful management of failure! That can be taught by moving slowly from small failures to extended ones. This is a standard teaching method.
Additional points: My psychologist friend and colleague assured me that the accepted principle is that people (and animals) learn from success. This is also partly a question of communication.
Another way of getting this success is to add so many props to a situation that success is assured, and then gradually to remove the props. There are of course severe problems in doing all this in large classes. This will require lots of ingenuity from all you talented young people! You can find some more discussion of issues in the article discussing the notion of context versus content.
My own bafflement in teenage education was not of course in mathematics, but was in art: I had no idea of the basics of drawing and sketching. What was I supposed to be doing? So I am a believer in the interest and importance of the notion of methodology in whatever one is doing, or trying to do, and here is link to a discussion of the methodology of mathematics.
Dec 10, 2014 I'd make another point, which is one needs observation, which should be compared to a piano tutor listening to the tutees performance. I have tried teaching groups of say 5 or 6, where I would write nothing on the board, but I would ask a student to go to the board, and do one of the set exercises. "I don't know how to do it!" "Well, why not write the question on the board as a start." Then we would proceed, giving hints as to strategy, which observation had just shown was not there, but with the student doing all the writing.
In an analysis course, when we have at one stage to prove $A \subseteq B$, I would ask the class: "What is the first line of the proof?" Then: "What is the last line of the proof?" and after help and a few repetitions they would get the idea. I'm afraid grammar has gone out of the school syllabus, as "old fashioned"!
Seeing maths worked out in real time, with failures, and how a professional deals with failure, is essential for learning, and at the reasearch level. I remember thinking after an all day session with Michael Barratt in 1959: "Well, if Michael Barratt can try one damn fool thing after another, then so can I!", and I have followed this method ever since. (Mind you his tries were not all that "damn fool", but I am sure you get the idea.) The secret of success is the successful management of failure, and this is perhaps best learned from observation of how a professional deals with failure.
When I am working on my research, or on MO/math.SE questions, I often find myself thinking in a way which reminds me of the feel of solving Olympiad problems. If I then solve the problem, I try to find a very special case of my problem which is still challenging, and can be stated and solved using only material on the Olympiad curriculum. I then e-mail Kiran Kedlaya and say "Hey Kiran, do you think this would be a good Olympiad problem?" If he thinks so, he proposes it to the USAMO committe.
I wrote Problem 2 on the 2010 USAMO in this way; it is Theorem 3.2 of this paper specialized to the case that $W_0$ is the group $S_n$. The fact that the "total number of moves" referred to in the theorem is at most $\binom{n}{3}$ is computed in Section 5.2.
I think I send about one problem a year; but most of them get rejected.
UPDATE 2014 Problem B4 of the 2014 Putnam was mine. Let $F(x,y,z) = \sum F_{ijk} x^i y^j z^k$ be a homogenous polynomial of degree $n$ with positive real coefficients. We say that $F$ is hyperbolic with respect to the positive orthant if, for all $(u_1,v_1,w_1)$ and $(u_2,v_2,w_2) \in \mathbb{R}_{> 0}^3$, the polynomial $f(t) = F(tu_1+u_2,tv_1+v_2,tw_1+w_2)$ has $n$ negative real roots.
In this paper, I show that there are constants $V_1$ and $V_2$ (dependent on $n$) so that,
(1) if $F$ is hyperbolic with respect to the positive orthant, then $F_{i(j+1)(k+1)} F_{(i+1)j(k+1)} > V_1 F_{i(j+1)(k+1)} F_{(i+2)jk}$ and the same for all permutations of the indices
(2) if $F_{i(j+1)(k+1)} F_{(i+1)j(k+1)} > V_2 F_{i(j+1)(k+1)} F_{(i+2)jk}$ and the same for all permutations of the indices, then $F$ is hyperbolic with respect to the positive orthant.
The proof is nonconstructive; I also (Theorem 20) give an explicit value of $V_1$. I was thinking about whether I could give a concrete value for $V_2$. The problem was too hard, so I thought instead about homogenous polynomials in two variables, which is the same as inhomogenous polynomials in one variable. At this point, I was basically looking for a converse to Newton's inequality: I wanted a constant $C$ so that, if $a_k^2 > C a_{k-1} a_{k+1}$, then all the roots of $\sum_{k=0}^n a_k z^k$ are real. The result in one variable wasn't worth publishing, but I figured I could make a nice problem by choosing a particular polynomial and asking people to prove the roots were real.
UPDATE 2020 Problem 6 of the 2020 USOMO was mine. It is the key computation from this MO answer, specialized to the case that the matrix $X$ has rank one, so $X_{ij} = x_i y_j$. The rank one case turned out not to be easier than the general case, which is why it doesn't show up in that MO thread, but it is one of the things I thought about when working on that answer, and I noticed at the time that it looked like a strengthening of the rearrangement inequality.
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I think I know how you feel. I have had the frustrating feelings with unsolved problems. I am an university math student and have encountered a lot of problems that I couldn't solve. Sometimes I've spend days thinking about them. I have also had the problem that sometimes I don't want to ask for help, because I feel like this is my problem to solve and getting help would be cheating.
I can't say for sure if great mathematicians feel this way sometimes, but I would imagine they do.
Even though I myself don't ask for help often, I think that if you have given a specific problem a lot of time without results, you should ask for help. (for exemple, on this site) Whether or not you ask for help, sitting alone and looking at a specific problem can get counter productive after a while. It would simply be better for you math education if you concentrated either on new stuff or different problems. Hope this helped a bit :)