I've had - and still have, to a degree - similar troubles with understanding math notation. Some 9 years ago a friend, solidly mathematician, gave a good advice

I've had - and still have, to a degree - similar troubles with understanding math notation. Some 9 years ago a friend, solidly mathematician, gave a good advice to me - at least try to pronounce the math expression aloud. That forces you to pay attention to each symbol - instead of trying to immediately find word-like "phrases", failing that and missing the expression as a whole. APLs are - as Arthur Whitney likes to employ - supposed to be read symbol-by-symbol, since each one represent a whole operation. That's why APL program are so dense - you may have plenty of work done in a short line (but many APL writers prefer to keep lines short, to help with understanding). A screenful of APL can be moderately sized program, without need for scrolling... When you write J, you might be tempted to add more and more operations to the left (APL and J work from right to left), until it's hard to "explain" what the whole expression does - because it does so much. With an alternative - short, understandable expressions - you have another problem, that of naming :) - trying to give a traditional, long-worded name to an intermediate result of expression doesn't fit well with the rest of the style... So a good sense of balance is very valuable. Another saying which may help to understand APL's mindset is that APLers spend 5 minutes to write a program, then spend next 1 hour to write the same program better and cleaner. That's what I understand as refactoring; the idea is to make the code more readable, the expression more obvious, more obviously correct, more reuseable... Properly done, the expression is a pleasure to look at and well worth the efforts to decipher it symbol-by-symbol. And another great option is documentation - lots of comments, just as in good math texts, where the idea is explained in plain English and then succinctly put to code. This way I better see the math background behind the problem. I certainly have an easier time to change something here and there - the changes required are so small. I also see similarities between different pieces of code, if they are put as similar expressions close to each other, on the same screen. Yes, the notation is terse... but it at least has good sides.