Ask HN: What's the best way to learn mathematics outside of formal education?
19 comments
I know someone who wanted to build up a solid mathematical foundation after years of considering themself to be bad at it, and ended up loving Spivak's book on calculus. (Another well-written book on calculus, in a very different style, is the classic book "Calculus Made Easy" - this book is old enough to be available on Project Gutenberg.)
You can learn linear algebra without learning calculus, but to really care about linear algebra you have to know why it's useful. The main lesson from (multivariable) calculus is that you should approximate functions by linear functions whenever possible - so the basis of almost everything else in math is a deep understanding of linear functions (and the occasional quadratic when you need that little bit of extra oomph). Once you manage to convince yourself to care about linear functions, the subject magically becomes easier to learn. I don't have any especially great textbook recommendations - perhaps Strang's book?
You can learn linear algebra without learning calculus, but to really care about linear algebra you have to know why it's useful. The main lesson from (multivariable) calculus is that you should approximate functions by linear functions whenever possible - so the basis of almost everything else in math is a deep understanding of linear functions (and the occasional quadratic when you need that little bit of extra oomph). Once you manage to convince yourself to care about linear functions, the subject magically becomes easier to learn. I don't have any especially great textbook recommendations - perhaps Strang's book?
Do the problem sets.
YouTube has a tonne of great presentations on different subjects like these, otherwise you could try the Khan Academy.
You might even find some decent text books with exercises online but if you prefer to watch etc, YT is great.
However, if you want to be able to actually use and apply it to a decent standard, you are probably going to need something more formal by way of teaching.
You might even find some decent text books with exercises online but if you prefer to watch etc, YT is great.
However, if you want to be able to actually use and apply it to a decent standard, you are probably going to need something more formal by way of teaching.
Time and effort.
True story: I college I had an upper level math course. On day-one the prof comes in and says (paraphrasing) "You've got the book. Read it. If there are questions, I'll help. Otherwise, you're on your own."
We left that class pretty angry. He's not going to teach us? Etc.
Long to short, it was one of the best things that ever happened to me. Up til that point, I was always taught such things (i.e., math and technical material). From that class forward I had the attitude, confidence and mindset to teach myself. Let's just say the number of technical lectures I attended the rest of my college career was cut in half.
Dive in. Find the medium your prefer (e.g., video, books, etc.) and keep going :) Ideally, find someone else who shares your curiosity so you have share the highs and lows.
True story: I college I had an upper level math course. On day-one the prof comes in and says (paraphrasing) "You've got the book. Read it. If there are questions, I'll help. Otherwise, you're on your own."
We left that class pretty angry. He's not going to teach us? Etc.
Long to short, it was one of the best things that ever happened to me. Up til that point, I was always taught such things (i.e., math and technical material). From that class forward I had the attitude, confidence and mindset to teach myself. Let's just say the number of technical lectures I attended the rest of my college career was cut in half.
Dive in. Find the medium your prefer (e.g., video, books, etc.) and keep going :) Ideally, find someone else who shares your curiosity so you have share the highs and lows.
Definitely steeling myself for a long hard slog to get to a decent standard, but fortunately I can be very motivated when it comes to things this interesting!
Take university classes - it's the same material, you just won't get any academic credits. MIT Opencourseware has Single Variable Calculus (18.01), Multivariable Calculus (18.02) and Linear Algebra (18.06). That's plenty of material to get started with. Watch the lectures, follow along with the textbook, do the exercises.
If you're concerned that you don't have some of the prerequisites like high school algebra or trigonometry, Khan Academy have a pre-calculus course - start there.
I switched to data science from an unrelated field and had to do a lot self-study like this to catch up on my math.
If you're concerned that you don't have some of the prerequisites like high school algebra or trigonometry, Khan Academy have a pre-calculus course - start there.
I switched to data science from an unrelated field and had to do a lot self-study like this to catch up on my math.
Cheers for the advice!
All of the comments here are valuable advice. Perhaps the only things to add are that learning math is cumulative, and learning math is like learning a new language. In other words, the more difficult concepts are usually built on other fundamental concepts. Be sure to build your foundation carefully, and if you have difficulty with any concept, look for your holes in understanding and don’t be deterred or discouraged from going back and relearning the fundamentals. Find your process and stick with it. Good luck.
Where OP is at, I wouldnt say it's necessarily cumulative. To learn linear algebra you dont really need trig or calculus
My method has been to audit a university class, do lots of practice problems, and find a support group of people to regularly work with
If youve taken say, trig, but are rusty, dont bog down on trying to relearn upfront. If it comes up when learning something else, then relearn whats necessary. The problem with this is if you're in a scheduled class where you might get behind. The benefit is that when you go on to upper level math, its a necessary skill to relearn some things quickly
If youve taken say, trig, but are rusty, dont bog down on trying to relearn upfront. If it comes up when learning something else, then relearn whats necessary. The problem with this is if you're in a scheduled class where you might get behind. The benefit is that when you go on to upper level math, its a necessary skill to relearn some things quickly
Wouldn't it be Magic to interact with Siri, verbally and visually, on the iPad to get to the gist of what you need in 15-30 minutes and continue on your quest?
It's a hard subject to teach yourself. I was in a similar situation to you, and after starting with various books I ultimately ended up enrolling with the Open University for a BSc in Mathematics.
Before that though, the most useful thing I did was to get the OU course books for their introductory courses (MST124 and MST125). They're written for self-study, and they're extremely comprehensive. These books filled in all the gaps and ironed out all the kinks in my fundamental mathematical knowledge. Things that used to baffle me make perfect sense now, like why the rules for non-integer exponents work the way they do.
Another book I'd recommend for someone in this situation is Measurement by Paul Lockhart. It won't exactly teach you maths, but it'll teach you what maths is, which is an important thing that most people are missing.
Before that though, the most useful thing I did was to get the OU course books for their introductory courses (MST124 and MST125). They're written for self-study, and they're extremely comprehensive. These books filled in all the gaps and ironed out all the kinks in my fundamental mathematical knowledge. Things that used to baffle me make perfect sense now, like why the rules for non-integer exponents work the way they do.
Another book I'd recommend for someone in this situation is Measurement by Paul Lockhart. It won't exactly teach you maths, but it'll teach you what maths is, which is an important thing that most people are missing.
Thanks for the advice! This sounds very much like the sort of thing I'm looking for.
Books from Dover are really cheap! You can also find lots of old, cheap, high quality math textbooks on Amazon and ThriftBooks.
The problem with self-learning is lack of feedback. You read something, you think you understand, but you might not, and you don't even know it. I once read Capablanca's "Chess Fundamentals" book, and I felt like I made a quantum leap in my chess understanding. One day later a friend of mine beat me 5 times in a row, so I learned the super-valuable lesson: self-assessment is not always very reliable.
Given that, how can you get feedback when you learn math on your own? I suppose there are various ways, but you can't go very wrong with Khan Academy. I just went to their website and practiced limits a bit [1]. I've known limits for 30 years, in and out, but I still found these 3-4 minutes practicing their exercises to be quite fun. I even got some stars and badges :)
I'm not affiliated with them in any way. The site just looks good to me. Give it a try.
[1] https://www.khanacademy.org/math/calculus-1/cs1-limits-and-c...
Given that, how can you get feedback when you learn math on your own? I suppose there are various ways, but you can't go very wrong with Khan Academy. I just went to their website and practiced limits a bit [1]. I've known limits for 30 years, in and out, but I still found these 3-4 minutes practicing their exercises to be quite fun. I even got some stars and badges :)
I'm not affiliated with them in any way. The site just looks good to me. Give it a try.
[1] https://www.khanacademy.org/math/calculus-1/cs1-limits-and-c...
The nice thing about math is that once you reach a certain level, self-assessment becomes extremely reliable. From that point on you can learn math on your own without worry!
The key is to always have some concrete procedure available to check the results of what you studied. If you are solving a system of equations, the concrete procedure is to check if your answer actually solves those equations. If you are taking a derivative, the concrete procedure is to numerically compute the expression (f(x+h) - f(x))/h for a really small value of h, and check if it is approximately the same as your computed value of f'(x). If you are integrating a function, the concrete procedure is to take the derivative of your answer and see if you got the original function back, or to numerically compute a Riemann sum.
If you are writing a proof... that is harder to verify on your own, but there are still ways to do it. You could think of a particular example and follow the steps of the proof in that particular situation and see if they work as advertised. You could try to rephrase the proof as a method for doing computations, and check that the method of doing computations gives correct results. You could attempt to transcribe the proof into a formal language, or even all the way to an interactive theorem prover. (Of course, someone who can successfully use an interactive theorem prover probably doesn't need advice on verifying proofs - they aren't very usable pieces of software yet.)
When learning a new area of math, your first question should be: how do I concretely check whether basic claims people are making are true or not? For this you need to know a few examples that illustrate different behaviors that can occur, and you need to understand these examples like the back of your hand. You also need at least one reliable method of concretely computing with whatever objects are being studied. The moment you don't feel like the subject is grounded in something concrete anymore, stop, try to find some examples, look for another book, maybe just put the topic to the side and study something else.
The key is to always have some concrete procedure available to check the results of what you studied. If you are solving a system of equations, the concrete procedure is to check if your answer actually solves those equations. If you are taking a derivative, the concrete procedure is to numerically compute the expression (f(x+h) - f(x))/h for a really small value of h, and check if it is approximately the same as your computed value of f'(x). If you are integrating a function, the concrete procedure is to take the derivative of your answer and see if you got the original function back, or to numerically compute a Riemann sum.
If you are writing a proof... that is harder to verify on your own, but there are still ways to do it. You could think of a particular example and follow the steps of the proof in that particular situation and see if they work as advertised. You could try to rephrase the proof as a method for doing computations, and check that the method of doing computations gives correct results. You could attempt to transcribe the proof into a formal language, or even all the way to an interactive theorem prover. (Of course, someone who can successfully use an interactive theorem prover probably doesn't need advice on verifying proofs - they aren't very usable pieces of software yet.)
When learning a new area of math, your first question should be: how do I concretely check whether basic claims people are making are true or not? For this you need to know a few examples that illustrate different behaviors that can occur, and you need to understand these examples like the back of your hand. You also need at least one reliable method of concretely computing with whatever objects are being studied. The moment you don't feel like the subject is grounded in something concrete anymore, stop, try to find some examples, look for another book, maybe just put the topic to the side and study something else.
You are absolutely right, but this is quite an advanced level of mathematical maturity, I would say Ph.D. level.
Someone who is not pursuing a math education can still get to this level, especially if they want to get a deep understanding of ML. You simply try to implement some of the ML algorithms using the math that you learned. If your math is wrong, the ML that you implement does not work. Debugging that is difficult, but once you get good at that type of debugging, you acquire the skills to self-check your math.
Until then, just successfully getting the answers to the Khan Academy exercises should do the trick.
Someone who is not pursuing a math education can still get to this level, especially if they want to get a deep understanding of ML. You simply try to implement some of the ML algorithms using the math that you learned. If your math is wrong, the ML that you implement does not work. Debugging that is difficult, but once you get good at that type of debugging, you acquire the skills to self-check your math.
Until then, just successfully getting the answers to the Khan Academy exercises should do the trick.
I've recently been getting deeper into interests where this lack of a mathematical background is becoming a bit of a problem. I've been interested in machine learning concepts for a while so I looked into linear algebra, I've been interested in signal processing in music so I looked into things like Fourier transforms, the list goes on and I keep running into issues with my poor fundamentals. I just don't have a solid enough background with subjects like trigonometry and calculus to understand a lot of what's going on in many cases, I can kind of mentally "bodge" things together sometimes but it's a very inelegant way of going about it.
What I have found quite useful is the kind of tutorials where concepts are given a visual, geometric interpretation which is quite strange as I'm not usually a particularly visual person. I'm still running into issues with my fundamentals though, so I guess my question is: assuming nothing more than mostly-forgotten A-level maths how can somebody who works full time build up a solid mathematical foundation so that they can progress to more advanced topics?
Thanks in advance.