In a Stack Exchange answer (https://math.stackexchange.com/question ... 410#181410), the mathematician Qiaochu Yuan explains how he remains motivated to study:
I start with an explicit and reasonable mathematical goal in mind. By that I don't mean "do a certain number of problems from this book," I mean "learn the material necessary to prove this interesting result" or "learn the material necessary to understand how to interpret this interesting computation." The keyword is "interesting": if I can't drive myself to work using my curiosity, I admit that I usually can't do it.
On another page (http://math.stanford.edu/~vakil/potentialstudents.html), Ravi Vakil gives this observation:
Here's a phenomenon I was surprised to find: you'll go to talks, and hear various words, whose definitions you're not so sure about. At some point you'll be able to make a sentence using those words; you won't know what the words mean, but you'll know the sentence is correct. You'll also be able to ask a question using those words. You still won't know what the words mean, but you'll know the question is interesting, and you'll want to know the answer. Then later on, you'll learn what the words mean more precisely, and your sense of how they fit together will make that learning much easier. The reason for this phenomenon is that mathematics is so rich and infinite that it is impossible to learn it systematically, and if you wait to master one topic before moving on to the next, you'll never get anywhere. Instead, you'll have tendrils of knowledge extending far from your comfort zone. Then you can later backfill from these tendrils, and extend your comfort zone; this is much easier to do than learning "forwards".
The common idea here is that there are (at least) two approaches to learning math: 1) working through textbooks linearly, mastering every concept along the way, and 2) starting with a particular concept you want to understand or problem you want to solve, and letting that guide what you learn. And, to me, this distinction also applies to language learning.
I think most learners have some (roughly) concrete goals. For a mathematician, they may be something like "I want to understand this concept" or "I want to solve this problem." For a language learner, they may be "I want to understand these particular books/movies/TV shows" or "I want to be able to converse fluently with a native." And I think these concrete goals usually imply a more abstract goal. If you want to understand a bunch of books which are all in the same language, you probably also have the abstract goal of learning that language.
To me, "learning backwards" means attacking your concrete goals directly. If you want to understand a movie, you would start by watching that movie and trying to understand it. This would then guide what you learn. Maybe you'll go look up unknown words in a dictionary, or look for concepts in grammar books, or make cards in Anki, etc. The point is that in the "learning backwards" approach, you are doing everything for the sake of the concrete goals.
"Learning forwards" means focusing on your abstract goal. This change in focus might broaden the sort of activities you may choose to do. Maybe you'll work through a course, or study a grammar book, or learn vocabulary from a list or dictionary. Even the process of trying to understand a movie, like mentioned above, could count as "learning forwards." But the change in focus is significant: you are doing it for the sake of learning the language, rather than simply for the sake of understanding the movie.
In my opinion, the "learning backwards" approach is more effective in terms of motivation. I am not personally motivated by large abstract goals like "I want to learn mathematics" or "I want to learn this language." There are specific concrete things I want to be able to do, so it makes sense to center my studies around them. However, I think this approach should also include activities that look a lot like "learning forwards" activities. It is obvious that if I want to understand a lot of media in some language, then I should try to develop the more general skill of understanding that language. This means studying textbooks, learning vocabulary, and all the rest. This has worked quite well for me because the formal studies "feed into" and help me during my more "concrete" studies, and vice versa. I would consider AJATT a very "learning backwards" method, and even it acknowledges that having basic knowledge of grammar will assist you when you are immersing in native material.
In mathematics, concepts build upon each other, but this isn't 100% hierarchical. It's more like a large, distributed graph. I think this is what Vakil means when he says that mathematics is rich. And the same goes for languages. There are many different things to learn which don't necessarily depend on each other. They don't need to be studied in a linear, systematic fashion, mastered one after the other. You can have less-than-perfect understanding in some areas while building better understanding in other areas. So while I think studying grammar books, etc. is important, these are really just more methods of filling in the "tendrils" - it doesn't have to be systematic or linear.
What do you think about all this? Do you tend to organize your studies around "learning these languages" or more around "learning to do these certain things in these languages"? I could certainly see some people simply being interested in the languages themselves, rather than wanting to do anything with them in particular. I also think the goal of "I just want to be in the process of learning these languages, because it's fun" is also valid, and probably a lot easier to achieve!
