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Math is a deductive science, that one starts from basic premises to prove more complicated results. From doing questions and reading solutions (at least at the elementary level eg: Olympiad questions), I find that there is always a dichotomy between a 'solution as it is presented' and 'a solution about how it is motivated', and sometimes when I am reading a solution I find myself saying to myself 'yes, this is true' without understanding how it is motivated until the end.

In short, I am wondering how to get a more intuitive/and motivated understanding of a solution. So far, I have found that playing with examples and then trying to generalize the result is helpful.

Any comments?

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What do you mean by "motivated"? – Warren P May 27 '12 at 16:09
Some writers are good at explaining the idea behind the proof, or how one can take an intuitive idea and turn it into a proof (evading the pitfalls along the way). Other writers are not, or prefer to present the polished jewels rather than the raw material. Often, you have to find your own way of "how to think about the result" and understand the solution, and that comes from practice and reflection. A good exercise is to take a proof of a theorem and see exactly how and where each hypothesis comes into it, and try to come up with examples where the argument fails if the hypothesis fails. – Arturo Magidin May 27 '12 at 16:19
Well, it might happen that even the author of the proof does not recognize the idea behind it. It had happened for me (more than once) that I reinvented some well-known methods, but only a few months/years later, when introduced to whole theory, I was able to grasp the bigger picture and understand my own proofs. Of course, there probably are even better perspectives that puts all this into new contexts, however, let us not dissolve into philosophy, I do like using a trash can ;-) – dtldarek May 27 '12 at 17:00
up vote 3 down vote accepted

In addition to the great comments above (I especially agree with Arturo's):

Let us suppose for a moment that we are considering math at the level of olympiad-style questions. Then the single most important method of developing intuition and motivation for such problems that can think of is to attempt, and I mean really attempt, to do the problem first.

I bring this up largely because of your phrasing with respect to 'readng solutions.' One should not, perhaps, just read one solution after another. There is a large amount to be gained from struggling with a problem. This is said every now and then, but it is really true.

So let's say that you are working on developing olympiad inequality intuition. Then what do you do? Find a set of olympiad inequality problems, and start to try them. Maybe you'll have no idea what to do. That's alright - push along. Try things. Play, struggle, move to other problems. Maybe you'll make progress on one that might give you ideas on others. If you're lucky, or perhaps brilliant, then you will motivate some of the solutions on your own. But it's likely that some will elude you.

And in that case, the time you spent struggling on the problem will have given you a certain roadmap of the problem, so that when (if) you start to parse a solution, you might be able to see exactly what obstacles different aspects of the solution overcame, or at least start to develop such an understanding.

Yet still, this does not give a full intuition. It is when you repeat this, trying more problems, struggling, etc., and when you are able to push a bit further, modify your previous methods of attacking the problems, and see how ideas of previous problems present themselves in present problems that the intuition starts to settle.

The fact is that it's hard to see what bits are important, or how to apply them. So you must struggle in order to see what different things are useful, struggle some more to see how they are useful, and struggle yet more to make it worthwhile for you to learn it.

It reminds me of the language teacher who lamented the prevalence of easy, online dictionaries. The idea is that when a student has to look through a paper dictionary for words all the time (which takes a bit, and a bit of effort), they remember the words more than when they are able to instantly look up anything with just a few effortless keystrokes.

Now let's leave behind elementary or olympiad-style questions, and consider the question again. It is an unfortunate aspect of the math publishing world that people often do not explain how they came up with the ideas behind their papers, or how they thought to cross whatever gap they crossed. (their motivation, to use your word). There are many causes - one that's perhaps undermentioned is that often the birth of an idea is not the clear, clean, seemingly divinely inspired flash of insight, but instead something that comes out of a very messy, circuitious, roundabout jumble of work. And the act of cleaning up the idea into something presentable manages to obfuscate the origin of the idea.

This aspect of math is very hard to overcome, I think. In a certain sense, struggling with similar problems and trying to generalize are still a good idea.

Two examples come to mind. One might show that $\displaystyle \sum_{p \; \text{prime}} \frac{1}{p}$ diverges by many different elementary techniques. Dirichlet's result on the infinitude of primes in arithmetic progressions can be recast in a similar statement about the divergence of $\displaystyle \sum_{\underset{(a,b) = 1}{an + b \; \text{prime}}} \frac{1}{an + b}$. Of course, the proofs are radically different. But the heart of the idea is sort of there.

A bit easier to fathom is Sophie Germain's identity. For a long time, I never saw how someone might come across it. But then one day I realized that it comes sort of naturally if you complete a square $a^2 + b^2 = (a+b)^2 - 2ab$ and wonder what it would take to use $x^2 - y^2 = (x-y)(x+y)$. You might quickly ratonalize that having $a^4 + 4b^4 = (a^2 + 2b^2)^2 - 4a^2b^2$ lets you proceed, and leads immediately to Sophe Germain's identity. Somehow, this thought process gives me peace.

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Training for Olympiad is more reducing the problem to pre-existing methods rather than blindly approach the problem. Quoting from Terrence Tao's Solving Mathematical Problem: "The journey of a thousand miles begins with one step." Terrence Tao: As long as one steps in a firm, clear direction, with long strides and sharp vision, one would need far, far less than the millions of steps needed to journey a thousand miles. And of course, one should endeavor to solve the problem before reading the solution. – Mat.S May 27 '12 at 17:38

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