As a future math teacher there are quite a few issues I am thinking about these days. One of them is “crackpots”. Academic outcasts if you prefer. The point I’m going to develop is that a person may evolve into a crackpot not because of lack of intelligence or self-criticism, but more because of lack of knowledge — or more precisely of guidance to acquire knowledge.
Let’s start from the beginning. We’ve all read papers which contained serious errors. This can happen even to the best minds (think Poincaré and the discovery of chaos). So as long as there is later recognition that the idea was wrong, and if it happens only a few times in a career, there’s no problem.
Less often, but still often enough, one encounters what can be called ‘a bold claim’. Again nothing wrong with this if it is acknowledged as such and thrown out only in an attempt to stimulate the reader’s imagination or to justify an otherwise perfectly rigorous definition.
Then there’s the issue of expository style. That’s usually when one starts viewing a paper as crackpotish. But there are many aspects here, and one needs to be careful. John Baez has a semi-serious crackpot index which does a good job at pointing out the kinds of behaviour that usually occur. Most of them are well-taken. My point is that some perfectly sane people could yet have a fairly high such index due to a succession of misunderstandings.
When I was in high school, I naively thought that papers published in La Recherche (roughly similar to American Scientist) were very technical. I then heard about Nature as being a “professional publication”, but could not find it at the local newsagent and felt puzzled. Now suppose I had left high school and instead of going on at the university I had opted for some two-year professional course and found a job. At that point my view of science as it is actually done was completely flawed: papers written in discursive style with very few equations, bigger fonts to emphasize the main arguments, and so on. I’m not sure I might not have turned into a crackpot myself in that case, for example plotting some graph related to prime numbers like those seen in La Recherche and starting to believe that I had done work on-par with these researchers!
Of course these days there’s the internet. But what may still be lacking is a less flawed presentation of actual research results in the media, and in the classroom. That’s something I’d like to test next year with a class.
. First a few sentences with inline code: suppose I’d like to talk about a manifold 
and its cotangent bundle 
. But in fact its a very simple case, namely 
. And now consider the Schrödinger equation for a wavefunction 
defined on 
. At this point all the 
$. Ok that last bit must be centered manually it seems, but so far so good!
episodic thoughts 