[As you may have already guessed, I'm going to be tying this in with the ongoing Pólya discussion]
This is another reason why it is important for math teachers to work so hard on building your student self-confidence. Whether we are talking about calculus or golf or playing the guitar, failure usually comes when we hit one of the following walls:
Lack of ability;
Lack of time;
Lack of patience and self-discipline;
Lack of resources.
It is often the fear of these walls that prevents us from investing the time and effort into mastering a skill we would very much like to have. That is a perfectly rational attitude. As mentioned before we always judge the time and effort needed to do something against the expected returns. Unfortunately, people often have a very unrealistic concept of these returns, particularly when it comes to the placement and order of these four walls.
Students generally expect "lack of ability" to be the first wall that they hit and yet this almost never happens. After years of teaching and working with students, I honestly can't think of an example where this was the case. They run out of time; they run out of patience; they run out of resources. These are things you see all the time, but I don't know that I have ever seen a student who simply put not handle the material. I am not saying that this does not happen but I am saying that it is extremely rare.
This does not mean that the "ability wall" isn't out there somewhere. It's important to realize that even with a tremendous amount of effort and support, some goals will still be beyond you. For the extraordinarily (or perhaps more accurately, obsessively) driven, this can be a problem. I'm sure instructors at Julliard encounter this all the time. However, for those teaching math on the primary, secondary, even undergraduate level, this is probably not something you will ever have to worry about.
Kids will not be receptive to instruction, they will not respond to incentives, they will not focus on material, and they will not put forth serious effort unless you can convince them that they are not about to hit the ability wall. This may not be the most important part of mathematics instruction, but it is the first part.
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