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Some credit needs to be given for just jumping in. Just analytically breaking down complex problems into pieces that are understandable. It's often faster to w

Some credit needs to be given for just jumping in. Just analytically breaking down complex problems into pieces that are understandable. It's often faster to work top-down and turn unknown unknowns -> known unknowns -> known knowns.

deleted 2y ago
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JustinSkycak 2y ago
It can definitely be helpful to take a top-down approach in planning out your overarching learning goals. However, the learning itself has to occur bottom-up. Especially in math. Math is a skill hierarchy, and if you cannot execute a lower-level skill consistently and accurately, you will not be able to build more advanced skills on top of it.
- MichaelNolan 2y ago
  I think these conversations of top down vs bottom up, are frequently a case where people are talking past each other due to using different definitions, and having different goals. And I think this is especially common when one of the people is a software engineer, and the other is not. For software, the tooling and the abstractions are so powerful that you can make incredible software without knowing what's really happening under the surface. Imagine an "abstraction ladder", with each level building on the previous level. Frequently in software development, you only need to understand a single level n, and maybe a little of n-1, to make great software. So the advice of "just jump in" is often great advice because you're jumping into a "shallow pool". (P.S, I love the work you're doing with MathAcademy, though I wish there was discounted price for "casual" learners that only have an hour or two a month)
  - JustinSkycak 2y ago
    Agree, it depends highly on goals. Using off-the-shelf ML/AI models (to make great software) requires far less background knowledge than implementing new models being introduced in papers, which in turn requires far less background knowledge than producing new models that improve upon the state-of-the-art. Thanks for the kind words about Math Academy! It's true that we focus on students who are trying to acquire math skills to the highest degree possible -- we teach math as if we were training a professional athlete or musician. We maximize learning efficiency in the sense that we minimize the amount of work required to learn math to the fullest extent. I realize that there are many learners who only want to devote an hour or two per month, but, at least right now, such learners would be better served elsewhere. It's a totally different optimization problem -- maximize surface-level coverage subject to some fixed, miniscule amount of work -- and as a result it would require different different curriculum and possibly different training techniques (or at least, differently calibrated techniques). But it's definitely an idea to think about in the future. :)
- ocean_moist 2y ago
  > Especially in math. Math is a skill hierarchy, and if you cannot execute a lower-level skill consistently and accurately, you will not be able to build more advanced skills on top of it. To be frank, I am only a math hobbyist and studying CS. I haven't taken any proof based courses (only calc 3, linear algebra). However, it is my experience that once you get the initial proof based stuff down and are familiar with common proof writing techniques, you can pretty much learn anything. I often start on an nLab page for a mathematical structure and just click on links in the definition till I see something I know. I then try and make sense of it and go back up the chain. I try to solve some problems and see some invariants. If I want to go more in depth I watch lectures on youtube and then read a book about it and do some more problems. I'm not really sure if this is a top down or bottom up process, but it seems more top down to me. Of course, if I encounter a whole field I don't know then I need to start from scratch and go bottom up. This is also for essentially a late-undergrad math major understanding of topics at most.
- danaris 2y ago
  The top-down part is determining what skills you'll need, starting from the highest level and breaking them down as you go. Then the bottom-up part happens as you actually start learning them, methodically, building back up toward those high-level skills.

deleted 2y ago

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JustinSkycak 2y ago

It can definitely be helpful to take a top-down approach in planning out your overarching learning goals. However, the learning itself has to occur bottom-up. Especially in math. Math is a skill hierarchy, and if you cannot execute a lower-level skill consistently and accurately, you will not be able to build more advanced skills on top of it.

MichaelNolan 2y ago
I think these conversations of top down vs bottom up, are frequently a case where people are talking past each other due to using different definitions, and having different goals. And I think this is especially common when one of the people is a software engineer, and the other is not. For software, the tooling and the abstractions are so powerful that you can make incredible software without knowing what's really happening under the surface. Imagine an "abstraction ladder", with each level building on the previous level. Frequently in software development, you only need to understand a single level n, and maybe a little of n-1, to make great software. So the advice of "just jump in" is often great advice because you're jumping into a "shallow pool". (P.S, I love the work you're doing with MathAcademy, though I wish there was discounted price for "casual" learners that only have an hour or two a month)
- JustinSkycak 2y ago
  Agree, it depends highly on goals. Using off-the-shelf ML/AI models (to make great software) requires far less background knowledge than implementing new models being introduced in papers, which in turn requires far less background knowledge than producing new models that improve upon the state-of-the-art. Thanks for the kind words about Math Academy! It's true that we focus on students who are trying to acquire math skills to the highest degree possible -- we teach math as if we were training a professional athlete or musician. We maximize learning efficiency in the sense that we minimize the amount of work required to learn math to the fullest extent. I realize that there are many learners who only want to devote an hour or two per month, but, at least right now, such learners would be better served elsewhere. It's a totally different optimization problem -- maximize surface-level coverage subject to some fixed, miniscule amount of work -- and as a result it would require different different curriculum and possibly different training techniques (or at least, differently calibrated techniques). But it's definitely an idea to think about in the future. :)
ocean_moist 2y ago
> Especially in math. Math is a skill hierarchy, and if you cannot execute a lower-level skill consistently and accurately, you will not be able to build more advanced skills on top of it. To be frank, I am only a math hobbyist and studying CS. I haven't taken any proof based courses (only calc 3, linear algebra). However, it is my experience that once you get the initial proof based stuff down and are familiar with common proof writing techniques, you can pretty much learn anything. I often start on an nLab page for a mathematical structure and just click on links in the definition till I see something I know. I then try and make sense of it and go back up the chain. I try to solve some problems and see some invariants. If I want to go more in depth I watch lectures on youtube and then read a book about it and do some more problems. I'm not really sure if this is a top down or bottom up process, but it seems more top down to me. Of course, if I encounter a whole field I don't know then I need to start from scratch and go bottom up. This is also for essentially a late-undergrad math major understanding of topics at most.
danaris 2y ago
The top-down part is determining what skills you'll need, starting from the highest level and breaking them down as you go. Then the bottom-up part happens as you actually start learning them, methodically, building back up toward those high-level skills.