Comment by elliekelly | Hacker News Reader

Comment by elliekelly | Hacker News Reader

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I've done this and would echo the good advice you've already gotten about Khan Academy but I think the comments in this thread are skipping over the most important part: pace. Work slowly. Don't move on until you really "get" the concept you're working with. Some topics might "click" right away and others might take weeks of bashing your head against the wall until your brain makes sense of it. That's okay. It can definitely be tedious at times (so be kind to yourself!) but a lot of learning happens when you're struggling to make sense of something so it's worthwhile to lean into the trouble you're having and figure out what it is that you're struggling to understand.

The downside of formal education is that the teacher has a timeline that needs to be kept and the class will move on without you. It might not seem like a big deal in the moment but in math the foundation is so important because the concepts build on one another. Sometimes in really subtle ways. If you have the motivation (and it sounds like you do!) then the huge upside to self-paced learning is that you can build a really strong foundation and the more complicated advanced topics will be so much easier to master. Khan Academy is really good at breaking subjects down into tiny bites and then slowly building on what you've learned but if you don't quite understand a lesson or if you haven't properly paid attention don't be afraid to do it again. If you guessed the right answer but don't understand why your answer was right, don't be afraid to go back and do it again.

So just because one person here could do all of the Khan Academy 9th grade algebra in a week that doesn't mean you should set that timeline for yourself. Everyone is starting from a different place. So maybe it'll take you a day, or a week, or a month, or even months. But however long it takes is how long it takes.

I'll just mention: From a scientific perspective, this is absolutely the wrong advice.

If you couldn't understand something yesterday, that's very good evidence you won't make sense of it today. You should work hard, go deep, and try hard problems, but if you fail to understand something after putting in the effort, just move onto something else.

Learn a few unrelated things, and come back. Much of the time, if you come back in weeks or months, it will just click into place.

You also want to visit topics /many/ times. Truly understanding many topics in math should take months or years. You can do it a lot more efficiently if you visit a topic for e.g. 5 hours each every 6 months over 2 years than a single 80-hour cram. You'll put in less time and understand more deeply.

A good starting point for references is Bruner (and specifically under Spiral Curriculum), but there's a whole literature on this. There has been a slow transition from seeing math as a set of bricks which stack on each other to an interconnected network of knowledge. The spacing effect literature is good too.

Math professor here.

I see students hit a wall in linear algebra, very good at "trained seal" but profoundly resentful that the time has come to go meta: Part of one's learning has to be investment in reflection and experiment on how one learns.

They generally believe that intelligence is innate, not deliberate gardening. There were thousands of people with Michael Jordan's body (alas, I'm not one) but he spent decades creating the athlete he became. Einstein found math difficult. It's like running: If you find running boring, you're not running hard enough. Math is a constant struggle, like mountaineering, and you shape, invent your mathematical mind. Those of us who love math (or mountaineering) are drawn to the struggle.

One syndrome I've seen often enough to classify: A student is uncomfortable with their calculus background. They want to pick the hardest calculus book they've heard of (often Apostol) and retreat to a desert island till they've mastered every word. I cringe in horror, and try to explain what's wrong with this approach.

It is widely observed that grad students learn four times faster than undergraduates. My first few years of grad school, it felt like I doubled what I knew every year. It was profoundly depressing when this subsided; I could have "been someone" with just a couple more doublings. What was going on?

Grad students learn what they need any given day, for goals they've set for themselves. Undergrads are taking on faith what authorities say is good for them. It's far easier to understand something when you see the point as you're learning it.

I know few mathematicians who read math comfortably, as if reading a novel. We mostly get angry and go think on our own, then return to realize that's just what the article said. Communication conventions in math are horrendous; one writes crappy machine code, then asks the reader to reverse engineer one's thoughts. Anyone in the tech sector knows how easily reading someone else's code can be "just kill me now" territory. As horrendous languages go, mathematical notation is a profound achievement. Nevertheless, beginners feel inadequate when faced with the inadequacies of mathematical notation to convey intuition.

If one wants to learn math, one needs to learn how to play, whatever that means. Find anything whatsoever in the text that leads to an example one can expand beyond the text. Does the text remind you of a pattern you've seen elsewhere? Perhaps the author just isn't saying. Play on your own, trying to decide if the ideas are in fact related.

>> As horrendous languages go, mathematical notation is a profound achievement.

Are there any resources or strategies you would recommend to help with this?

> From a scientific perspective, this is absolutely the wrong advice.

Those are unnecessarily harsh words. You and elliekelly are not contradicting each other, but talking about different things.

If you didn't understand chapters 1 and 2 from a textbook, reading chapter 3 from the same book is usually a BAD idea. Sadly, this is what formal education often tells you to do, because it follows a predetermined schedule.

If you didn't understand chapters 1 and 2 from a textbook, opening a different textbook and reading chapters 1 and 2 is usually a GOOD idea. Two authors describing the same idea from slightly different perspectives is better than reading the same text twice; at least it makes it easier to pay attention during the second reading, but the second book might also answer a question you had when reading the first book.

I personally have always had a hard time inhaling chapters of a textbook sequentially. It’s often easier to skim it first, attempt the problem set, and then go back when you get stuck. Forces you into a more active mode of looking for particular information instead of just trying to absorb everything.

The same works for the humanities too, once my attention starts waning I’ll start skimming and rewind when necessary. Oftentimes authors are setting up a ton of boring but necessary context upfront, but it’s not obvious to a reader why such context is necessary until you get to the point first.

A smarter person would probably be able to anticipate what they’re trying to learn the first time around and avoid my flipping back and forth, but that’s hard.

I believe this works because you're creating a demand in yourself for the information. I suspect it's something like natural selection: you will remember and understand what you "need to" in order to cope with the environment you're in and for studying you should create such an environment for yourself. The more problems and advanced texts that you supplement your readinv with, the more demand there is for the more basic information you're trying to retain.

This is also known as order control in inquiry-based learning.

This also, while common sense, is unfortunately often incorrect -- at least the part about avoiding chapter 3 (a different book is a good idea).

If you didn't understand chapters 1 and 2 in a textbook, reading -- or at least skimming -- chapters 3 and 4 is usually helpful. It doesn't make sense to read them for mastery, but you'll see context where material from chapters 1 and 2 is applied, or concepts that build on them. You have to be comfortable with confusion, since most of it will go over your head. You will pick out some parts.

Once you've done that, go back to chapters 1 and 2.

Think of calculus: limits -> derivatives -> integrals.

However, you can understand a Reimann sum without derivatives or limits. It helps understand and motivate both.

This is called a whole math approach. You make successive passes with increasing depth. I've explained calculus concepts to kids under 10 with no problems, as have many others. Over time, you want to develop:

- Mechanics of integration, differentiation, and other computation

- Applications (e.g. debt versus deficit)

- Intuition

- Formalism

- And so on...

All of those support each other.

You see this in how you read research papers too. Novices read them linearly, and experts absorb them nonlinearly.

As a footnote, Vygotsky (in the original, e.g. 1978 translation) is probably the first person to discover that doing things beyond your level of ability accelerates learning. Not aiming for mastery means you're free to fail, free to try harder things, and learn faster (if more painfully).

85% of American textbooks about Vygotsky, though, always say exactly what he was trying to debunk. He pushed very hard for learning being harder and more abstract than most thought possible.

To me, this has been the largest advantage of learning things at my own pace instead of in a classroom environment. Even as a student I recognized early on that topics that'd stump me would often click months later. By then it was too late for my grade, but I'd go back and realize that other things I'd learned in the interim made it easy.

So now when I find something blocking me, I just go learn something else instead. And what do you know, some combination of more wisdom, broader knowledge, or perhaps just a different mental state often works.

I personally found that I would have to keep exposing my brain to the same mathematical concept many times over days or weeks by doing problems when I could and visualizing the math in my brain when I was doing other things. Over time it felt like my brain was gradually getting "imprinted" with the shape of the math ideas, but you had to be gentle and do it over a period of days to weeks, otherwise it's like the math shape would just squish your brain instead of letting your brain adapt to it :)

Care to share the science? This sounds more like a learner's internal/external locus of control. If you believe you are capable of figuring something out you're probably right and likewise if you believe you can't.

I'd recommend rereading my post. The problem with education research is that there's a hundred years of it. I listed the two in my original post. A longer list:

- Bruner's work on spiral curriculum, and follow-up work. Spiral curricula are mis-implemented in most schools which claim to use them; it's not review. It's successive passes going deeper.

- Cognitive work on progressions such as surface->deep->transfer learning (Hattie) or chunking

- Physics education research on complex multiconcept problems

- Spacing effect. You can go all the way back to 1885 with Ebinghaus, but the esoteric author of supermemo seems to have done the best work here. If you prefer the academic establishment, there's a ton.

- There's a bunch of science that about 2/3s of what we learn, we're not aware we're learning or teaching. Concepts support each other, deeply. Kaplan, back when Bror was there, did a bunch of nice work there. It's easier to learn long division if you know a bit of algebra, and vice-versa. Chemistry helps learn physics, and physics helps learn chemistry. This was supported with pretty independent methodologies in independent domains (data analysis from tests, cognitive task analysis, etc.).

... and so on. This has very little to do with affect. It's a very well-established, rarely-applied set of results.

A complete lit review here would be book-length, at least.

> Don't move on until you really "get" the concept you're working with. Some topics might "click" right away and others might take weeks of bashing your head against the wall until your brain makes sense of it.

Careful. This can sometimes be counterproductive. It is often the case that one concept taught when learning a subject will be applied as part of teaching the next concept.

The best way to learn that first concept in many cases is not to focus on it until you really "get" it but rather to just get enough of it for how it is used for teaching the next concept. Then you move on to the next concept and the application of that first concept there gets you experience using the first concept which solidifies your grasp of the first concept and makes you truly "get" it.

The application of earlier concepts throughout the subsequent material can be in effect a kind of natural spaced repetition for learning the earlier concepts.

I think that deciding when to move on is one of the hardest things in self-learning, especially if the subject is new enough to you that you can't use your experience in a related subject to help guide you. You don't know enough to know when you've learned enough to go on.

Khan has problem sets right? That was always my issue, that and math "teachers" who'd just offer to reread the textbook or point to the index and smirk.

But don't feel bad if you struggle, it's really hard to learn math concepts without an incentive. I'd just pick up a book on python and learn algerbra and a few other things along the way.

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