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Comment by aphextron | Hacker News Reader

>My current proficiency level is pre-high school mathematics as I didn't pay much attention in high school, learning effectively nothing.

Advice from someone who was in the same position: Take a class. Multiple classes. Go sign up for a Mathematics AS at your nearest community college right now. You will never know enough of what you don't know to learn this stuff on your own. A lot of it is just doing the painful repetition work of practicing problems over and over again, which is hard to force yourself into without a "coach" pushing you. Having a cohort of students to work through problems with is also priceless. And the drive of having accountable grading will keep you at it regularly.

It can be a bit awkward at first feeling stupid not knowing what a logarithm is in a room full of 18 year olds. But it's the only way to really get there. I went from high school dropout who didn't know how to add fractions to passing calculus in 18 months.

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Depends what your goal is. I cant imagine anything worse than High School style teaching of mathematics. Its awful. That repetitive calculation, entirely unnecessary for almost any real world situation. Almost every time you are doing repetitive calculation what you should have been taught was a second or third conceptualisation of whatever maths you are looking at. So linear algebra I was only ever taught numerically in computer science. WHICH IS CRIMINAL. It has such an obvious and more easily understood geometric interpretation. The same is true of calculus, learn it in physics. Seriously. Do high school physics /and/ maths if you wish to understand calculus. Otherwise you've got someone who probably did a pure maths degree teaching you something that was created by physicists for a reason, that reason has a physical and real world interpretation not to mention a geometric one. Another good example is in chemistry you might learn statistical mechanics, in computer science or electronic engineering you might learn information theory. If those two classes were taught back to back for both groups of people they would actually deeply understand the mathematics behind it. Personally I've always found learning maths in a pure way incredibly challenging. As an adult I think its a lot easier to learn maths through something that has physical or meaningful interpretation. You can then draw on your actual life experience to understand the maths.

Thank you for saying this. I've been learning math from Khan Academy for the last 200 days. I've reached calculus and I feel more and more the need for examples of how to apply the math that I'm learning to truly understand what it's useful for.

I was actually thinking of learning how to build video games (or at least parts of it) to properly understand linear algebra. Someone else also suggested physics for calculus.

> I've been learning math from Khan Academy for the last 200 days. I've reached calculus and I feel more and more the need for examples of how to apply the math that I'm learning to truly understand what it's useful for.

Just so that you know this was my impression at a university as well. In fact, calculus courses are translated into "basic mathematics." I think this underlines how much these courses are about learning the tooling for practical applications taught in later classes.

Hmm. Which kinds of math? Might be useful to begin a compendium of examples, a problem-to-solution (and vice-versa) phrasebook

Fwiw, I found electromag (typically undergrad physics 2) the best application of calculus.

Classical physics of motion (physics 1) used some superficial calculus, but none of the use cases that would let you spread your mathematical wings cropped up in the material we studied.

I disagree about the repetitive calculation being "entirely unnecessary".

Background: I (probably like many here) excelled at high school and undergraduate mathematics. I have a graduate degree in a heavily mathematics-focused branch of engineering (digital signal processing) and much of my work involves applying that math at a conceptual level.

I'm currently tutoring an adult I'm close to who is approximately at GP's starting level and has a strong anxiety reaction to math. I'm finding that the repetitive calculation aspect is important to being able to developing an intuition for the concepts.

The process that's working well for us so far involves alternating between practical word problems (to establish a motivation for learning the material), theoretical/background explanations (to hit the understanding at a high level) and repeated simple problems (to practice the mechanics).

I'm finding that, in terms of process, the repeated problem-solving is critical for two reasons. 1) It helps to build a sort of mental "muscle memory", and 2) it helps with developing intuition, since your mind eventually gets bored with the mechanics and starts to notice patterns.

Remember that mathematics at that level is all about building blocks. Every concept/problem-solving practice you learn is part of a tower of strategies; if earlier mechanisms aren't almost mindless, it's MUCH harder to build on top of that knowledge.

One of the earliest memories I have of really enjoying math was a unit on modular arithmatic from my 6th grade math class. The teacher introduced the concept with a hands of a clock metaphor, gave us a whole bunch of drill problems, and told us to watch for patterns. We had an easel in the class room where he collected the insights students had gained. Discovering the relationship with remainders by ourselves was incredibly cool.

I had a similar experience when I learned about factoring (not sure what age). I got a little obsessed with finding prime numbers by hand, and eventually figured out that I only had to search for factors up to the square root. I distinctly remember that as the first time I realized how powerful a simple mathematical insight can be to speeding up a process.

Now, of course, I use mathematical insight to speed up processes occupationally; I guess that experience stuck with me.

> 1) It helps to build a sort of mental "muscle memory", and 2) it helps with developing intuition, since your mind eventually gets bored with the mechanics and starts to notice patterns.

muscle memory is super important in math.

the last thing you want is to be struggling with foundation when you're trying to progress through concepts. everything needs to be available to you at the snap of your fingers.

trying to progress math knowledge without the basics is doable, but it's a ton of mental overhead and frustration that you can cut down if you just get the practice out of the way.

plus it's super cool if you're one of those guys who have the ability to quickly apply 10-20 different models on a real world problem. that's what we think of when we think "good at math".

Having another interest to tie it to (especially if it's something you're more comfortable with) is huge. I have a CS degree, and for me, what bridged the gap into being comfy with "pure" mathematics (and appreciating it outside of any specific application), was my theory of computation class.

My high school calculus and physics teachers synchronized their classes, so shortly after we learned something in calculus, we would use it in physics. Since then I’ve learned this is not as common as it should be!

Perhaps it works especially well in that case because calculus was literally invented to do the kind of physics you learn in high school, so it’s a very natural combination. Linear algebra (e.g.) is so general-purpose that the overlap with another course might not be so continuous.

My high school only taught physics without calculus. In 12th grade when I took calculus separately I was sort of annoyed to realize how much easier physics would have been using calculus.

This was my experience too.

I remember doing integrals in high school, but didn't take calculus-based physics until my second year of college.

Jesus christ, it explained a lot of things.

It's always a good idea to anchor new knowledge to pre-existing knowledge. (Perhaps this could be called coral reef learning?)

Adult brains, with less plasticity, are able to synthesise new concepts that build on what they already know.

I second this. Having studied to the postgrad level, I've come to appreciate just how broad the subject is. Looking back, I feel it would have been near impossible for me (personally) to get here without the help of lecturers and tutors - often prerequisite knowledge creeps in unexpectedly, and you come to rely on them to guide you through this.

I agree also that regular grading and a lesson schedule help to both push you to learn, and for you to assess your progress in an unbiassed fashion.

Personally, though, I am biassed - and am thrilled when people say they want to learn more maths. I love maths, and find it very rewarding. The best of luck with your journey!

As someone that has loosely tried to learn Mathematics over the past decade, I second this.

Self-learning mathematics is really difficult, especially when you don't really know where you stand any more in terms of pre-requisite education.

It's easy to say you're going to dedicate x of hours of time to study Mathematics, but it's easy to drop off, ditch it in favour of the next thing to learn, and essentially forget what you tried to learn. Taking a class will at least physically allocate that time to maths.

Do you also think that one can only learn Computer Science effectively by taking classes?

Yes, for many people they will not become as strong without classes even in CS.

By starting code so young, I had years of experience before college, serious enough to work part time developing on commercial products.

I was cocky, thought CS 101 was a joke, and it mostly was. But it didn’t take long before vast areas of theory and practice I wasn’t even aware of became part of the classes.

At the end of an undergrad degree you don’t master all these new concepts, but you do become aware of them, you practice them enough to know where to dig later in your career.

I just see it as very unlikely that most jobs will make it easy to establish this same broad foundation, and it’s a powerful thing.

Surely for a few people it may not make a difference, but even for geniuses who turn out to have wildly successful careers, who’s to say it wouldn’t have enabled more insights or productivity?

Can I ask where you are now in life? Im a high school drop out as well and have found it very tough to get back into things, crippling anxiety only driven further by lack of participation :/

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