Mathematical reasoning as such (and how exactly humans perform it) is extremely fascinating, as is the article. I offer a tentative explanation of why people who are slow to pick mathematics up at first later go on to dominate it: the search algorithm (if you’ll tolerate a loose metaphor) their cognitive software is running is breadth-first. When they first begin to learn mathematics their neurons are assaulted with a slew of possible interpretations—assigning a clear semantics to the notation through a haze of conflicting ideas is difficult. In fact, it can be intellectually paralyzing. Repeatedly investigating faulty interpretations due to assigning a slightly wrong semantics will leave you intellectually exhausted and seemingly no closer to a solution.
Mathematics may be easier on first introduction if you completely ignore the semantics of your notation, and reason strictly within it. People who are capable of doing this would seem to be quickly mastering the subject, while what they’re really doing is rigid symbol-shifting rather than getting beneath the notation. If you ask such a person to reason outside the notation, they’ll founder.
An attendant explanation is that slow-learning mathematicians synthesize their symbol-manipulation procedures from the ground up. Simply following instructions produces intellectual discomfort, so they have to understand small things totally before they can proceed to justify using those small things in more complex ways. They’re driven by their own instinctive desire for rigour to learn the hard (but ultimately more thorough and edifying) way.
I’m not a mathematician, but this was definitely what it felt like when I first attempted to learn computer programming, and what it felt like when I started taking mathematics seriously in school.
“I’ve asked them all, and I have nothing to show for it.” … “Let me disclose first that I have no idea how to fix this problem.”
I’m unsure what “nothing to show for it” means? I want to recommend that you have intensive (in the sense of contentful rather than combative) arguments, try to identify weakpoints in their reasoning (or excuses, as the case may be), and then write up your analysis for others to read. However you may already have tried to argue the issue, and came out the other end with nothing worth analysing. On the other hand, you may have had unintense arguments, in which case it seems like a substantive discussion is the first port of call. On second thought, a substantive discussion is called for in both cases, since if you come out with nothing worth analysing then the discussion wasn’t substantive after all!
PS. This is my first comment on this site, I’m not familiar with the etiquette surrounding introductions and (http://wiki.lesswrong.com/wiki/FAQ#Site_Etiquette_and_Social_Norms) doesn’t contain any information on that point. Let me know if there’s any introductory rituals I would be remiss to ignore.