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As an alternative I would suggest a top-down approach. Start with the theorems/results you truly wish to understand and work backwards. There was a great quote

As an alternative I would suggest a top-down approach. Start with the theorems/results you truly wish to understand and work backwards. There was a great quote from an interview of Peter Scholze (one of last year's Fields' Medallists), which has really changed how I view learning: At 16, Scholze learned that a decade earlier Andrew Wiles had proved the famous 17th-century problem known as Fermat’s Last Theorem, which says that the equation xn + yn = zn has no nonzero whole-number solutions if n is greater than two. Scholze was eager to study the proof, but quickly discovered that despite the problem’s simplicity, its solution uses some of the most cutting-edge mathematics around. “I understood nothing, but it was really fascinating,” he said. So Scholze worked backward, figuring out what he needed to learn to make sense of the proof. “To this day, that’s to a large extent how I learn,” he said. “I never really learned the basic things like linear algebra, actually — I only assimilated it through learning some other stuff.” [1]https://www.quantamagazine.org/peter-scholze-and-the-future-of-arithmetic-geometry-20160628 https://www.quantamagazine.org/peter-scholze-and-the-future-...

sweeneyrod 8y ago
An approach might work well for a Fields medallist but less so for normal people.
- empath75 8y ago
  I dunno, it's worked for me. I have a problem I want to solve, then learn the tools I need to solve it. Not just math, but just as a software developer.
- kowdermeister 8y ago
  I love generative art, so I quite often run into things like Lorenz attractors. I looked up how they are made and that lead me to differential equations which lead me to better understand calculus. So my love of pretty graphics lead me to learning math with a purpose. I haven't won any awards yet.
tzs 8y ago
I've long wished for a series of interactive ebooks or websites based on that type of approach. Each would be devoted to one great theorem or problem. Each would start with a presentation of the theorem and proof, presented how it would be presented today if it were a newly discovered research result being presented by professionals to professionals in the field. At each step of the presentation, there would be two expansion options. One is to ask for filling in the details. A detail expansion keeps the presentation at about the same level of required knowledge, but takes smaller steps. You use a detail expansion when you understand where a step starts and end, but you just don't quite see how it made the connection. The other expansion option is to ask for background or prerequisites. A background expansion is for when you don't have the background to even understand the start and end points of a step. It opens up material to teach you the background necessary to understand what is going on. A key aspect is that this would all be recursive. You could do a background expansion on a background expansion, and so on, all the way back to common high school math. The background expansions would just teach enough of their subject to support the step above. So, for example, if you used one of these interactive books to learn an analytic proof of the prime number theorem, and you started knowing nothing beyond high school algebra, you would end up learning all the calculus and complex analysis needed to prove PNT, but only such calculus and complex analysis as are needed. What I wonder is if you could pick a set of theorems and problems for such books such that (1) someone could go through them all in about the same time as a conventional math degree takes, and (2) combined, the background expansions would have covered as much as a conventional degree. If so, that could be an interesting way to keep motivation high because everything you are learning has a direct, visible, connection to advancing the proof of the interesting theorem at the top.
ivanhoe 8y ago
The issue with this approach is that if you learn just "the fun parts" you might be left with some huge gaps in your knowledge, all that in-between stuff - especially if you're not a genius like Scholze. Standard approach is perhaps less motivating and you learn a lot of stuff that frankly you'll never need and you'll probably forget most of it, but it ensures that you've at least heard about all the major ideas. One day when you run into a problem you'll know where to look for more details. This is IMO a common problem with self-thought programmers as well, they often end up inventing a wheel simply because they just never heard that solution to their problems already exist in some 70s CS textbook.
- tbabb 8y ago
  I think fun is an extremely important part of keeping up the pace of learning. I would rank that much higher over "methodically combing through everything", which could cause you to quit from boredom long before you reach your goal. I'd hypothesize that the problem you describe is due to insufficient curiosity about the way things are already done. Especially with Google, Wikipedia, and well-populated internet forums, it'd be hard not to find "foundational solutions" after a little research. I had a music teacher who was obsessive over minutiae, and fussing about the exact position of each finger on a piano key was much less fun than loosely noodling around with chords and making things that sounded closer and closer to real music. I made little progress with the lessons and quit, but since I started noodling on my own I've been getting much farther. I propose that "intelligent" is a synonym for "gets enough pleasure from learning to do a lot of it", so I would overwhelmingly optimize for that.
  - hyperpallium 8y ago
    I have a lot of sympathy for this approach, and of course you learn more by having fun and continuing, then being meticulous and stopping. However if at each fork, you choose the fun road, you might get a long way until you run out of options (which may be fine). But the terrible difficulty is necessary knowledge that is not explicitly stated. These gaps ("mathematical maturity") are difficult to even identify, let alone fill. Story time: --- In high school, I missed a week or two, and later got stuck on a problem in calculus. Discussed it with the teacher, and he eventually seemed to see my difficulty was, but wouldn't tell me, instead saying "you can work it out". I couldn't. Years later, I started asking online, and people responded helpfully, but didn't actually help. I read wikipedia and math websites about it. Watched videos about it, that were interesting and gave a new perspective, but didn't help my particular issue. I looked it up in famous textbooks (Spivak, Hardy), still nothing. Finally, I did all the problems in the derivatives section of Khan Academy, but when I came to this problem, it still didn't address my specific issue. Thinking I had gaps in my knowledge, some known to me, and suspecting others unknown, I went back to even earlier material. Some time later, I reviewed the problem again, and realized that the issue was a completely trivial part of limits - an earlier section. Which was what I missed in those 2 weeks of high school, all those years ago. A nice maths teacher can save you years.
selimthegrim 8y ago
This works right up until someone makes you have to pass a qualifying exam. Ask Terry Tao - he failed his first time.

sweeneyrod 8y ago

An approach might work well for a Fields medallist but less so for normal people.

empath75 8y ago
I dunno, it's worked for me. I have a problem I want to solve, then learn the tools I need to solve it. Not just math, but just as a software developer.
kowdermeister 8y ago
I love generative art, so I quite often run into things like Lorenz attractors. I looked up how they are made and that lead me to differential equations which lead me to better understand calculus. So my love of pretty graphics lead me to learning math with a purpose. I haven't won any awards yet.

tzs 8y ago

I've long wished for a series of interactive ebooks or websites based on that type of approach. Each would be devoted to one great theorem or problem. Each would start with a presentation of the theorem and proof, presented how it would be presented today if it were a newly discovered research result being presented by professionals to professionals in the field. At each step of the presentation, there would be two expansion options. One is to ask for filling in the details. A detail expansion keeps the presentation at about the same level of required knowledge, but takes smaller steps. You use a detail expansion when you understand where a step starts and end, but you just don't quite see how it made the connection. The other expansion option is to ask for background or prerequisites. A background expansion is for when you don't have the background to even understand the start and end points of a step. It opens up material to teach you the background necessary to understand what is going on. A key aspect is that this would all be recursive. You could do a background expansion on a background expansion, and so on, all the way back to common high school math. The background expansions would just teach enough of their subject to support the step above. So, for example, if you used one of these interactive books to learn an analytic proof of the prime number theorem, and you started knowing nothing beyond high school algebra, you would end up learning all the calculus and complex analysis needed to prove PNT, but only such calculus and complex analysis as are needed. What I wonder is if you could pick a set of theorems and problems for such books such that (1) someone could go through them all in about the same time as a conventional math degree takes, and (2) combined, the background expansions would have covered as much as a conventional degree. If so, that could be an interesting way to keep motivation high because everything you are learning has a direct, visible, connection to advancing the proof of the interesting theorem at the top.

ivanhoe 8y ago

The issue with this approach is that if you learn just "the fun parts" you might be left with some huge gaps in your knowledge, all that in-between stuff - especially if you're not a genius like Scholze. Standard approach is perhaps less motivating and you learn a lot of stuff that frankly you'll never need and you'll probably forget most of it, but it ensures that you've at least heard about all the major ideas. One day when you run into a problem you'll know where to look for more details. This is IMO a common problem with self-thought programmers as well, they often end up inventing a wheel simply because they just never heard that solution to their problems already exist in some 70s CS textbook.

tbabb 8y ago
I think fun is an extremely important part of keeping up the pace of learning. I would rank that much higher over "methodically combing through everything", which could cause you to quit from boredom long before you reach your goal. I'd hypothesize that the problem you describe is due to insufficient curiosity about the way things are already done. Especially with Google, Wikipedia, and well-populated internet forums, it'd be hard not to find "foundational solutions" after a little research. I had a music teacher who was obsessive over minutiae, and fussing about the exact position of each finger on a piano key was much less fun than loosely noodling around with chords and making things that sounded closer and closer to real music. I made little progress with the lessons and quit, but since I started noodling on my own I've been getting much farther. I propose that "intelligent" is a synonym for "gets enough pleasure from learning to do a lot of it", so I would overwhelmingly optimize for that.
- hyperpallium 8y ago
  I have a lot of sympathy for this approach, and of course you learn more by having fun and continuing, then being meticulous and stopping. However if at each fork, you choose the fun road, you might get a long way until you run out of options (which may be fine). But the terrible difficulty is necessary knowledge that is not explicitly stated. These gaps ("mathematical maturity") are difficult to even identify, let alone fill. Story time: --- In high school, I missed a week or two, and later got stuck on a problem in calculus. Discussed it with the teacher, and he eventually seemed to see my difficulty was, but wouldn't tell me, instead saying "you can work it out". I couldn't. Years later, I started asking online, and people responded helpfully, but didn't actually help. I read wikipedia and math websites about it. Watched videos about it, that were interesting and gave a new perspective, but didn't help my particular issue. I looked it up in famous textbooks (Spivak, Hardy), still nothing. Finally, I did all the problems in the derivatives section of Khan Academy, but when I came to this problem, it still didn't address my specific issue. Thinking I had gaps in my knowledge, some known to me, and suspecting others unknown, I went back to even earlier material. Some time later, I reviewed the problem again, and realized that the issue was a completely trivial part of limits - an earlier section. Which was what I missed in those 2 weeks of high school, all those years ago. A nice maths teacher can save you years.

selimthegrim 8y ago

This works right up until someone makes you have to pass a qualifying exam. Ask Terry Tao - he failed his first time.