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> One of the barriers I've had trouble piercing, and that wouldn't really go away in a more traditional academic context is how to learn the 'language' of math used in academic computer science papers.

Interesting that you say that. I taught English for 5 years in Japan (without an education degree, so I suppose it's kind of related to the conversation ;-) ). I did a considerable amount of study on language acquisition. Although I find the research "sketchy", the work of Stephen Krashen is very compelling. I tried to apply the principles in my classroom and in my own acquisition of Japanese. I found it to be quite enlightening. (Fair warning: Krashen's work has a following that pretty much thinks "This is the way it is -- for sure!" and are trying to prove it. On the other side are people who think he's definitely wrong. As in many fields of psychology, it is hard to build good studies, so people mostly shout at each other ;-) ).

Anyway, if you are interested, take a look at: https://en.wikipedia.org/wiki/Input_hypothesis

In a nutshell: There are 2 types of learning: learning and acquisition. Learning is about remembering stuff. Acquisition is about being able to use stuff without really thinking about it (i.e. it comes unbidden to your mind when you see a reference).

A good example of the difference is reading. Take a look at an English word:

Apple

When you read it, what came to your mind? What thought process did you follow to get there. Now if I told you that in Japanese sakana=fish, inu=dog, ringo=apple and then write:

Inu Sakana Ringo

How does your thought process change? Now go back and look at the English word I wrote earlier, but this time try not to remember what it means. Can you look at the word and not read it? How difficult is it? Now look at the Japanese words and try not to remember what they mean. Can you notice a difference?

I hope you can. That difference is the difference between acquisition and learning. With English, the meaning comes unbidden to your mind. With Japanese, you have to actively try to associate the word in your brain with the reading -- even if you can remember that meaning already.

Being able to look at something and instantly recognise it, or to have it spring to your mind automatically when you want to use it requires acquisition. So when you speak English, you don't sit and plan the grammar of the sentences that you want to say. You have a thought and the English just shows up in your brain. In fact, it shows up so conveniently that you can actually use it as part of your thought process (i.e. thinking in English). When you get fluent in another language you can do the same in that language.

Finally, I get to the punchline. I have been told by some people that mathematics works the same way for people who are fluent in it. It's just another language and they can think in math because they have acquired it.

How do you acquire a language? That's the $64,000 question. According to Krashen, acquisition is a function of input that you can understand. Output seems to be unnecessary (although, personally I find it helpful even if it is unnecessary).

It's a difficult topic and I've already typed too much, but if you would like to try something, this is what I did for learning Japanese. Do not memorise vocabulary or grammar. Do not try to reason the whys and wherefors of the system. Instead memorise exemplars of the language. Just an example sentence for things you come across that you don't know. Make a flashcard (spaced repetition software is useful: http://ankisrs.net/). Read appropriate level material and keep making/memorising cards for anything you don't know.

The memorisation builds a fast lookup table that allows you to work out your own comprehensible input. Then by reading, reading, reading, reading, you get lots of comprehensible input. Finally you can test whether or not you have acquired the input by trying to produce output.

Hopefully this will help you with your math.