Ask HN: How Do I Get Math?

[todayiamme]

I just graduated out of high school and I want to learn loads of stuff in math, but there's a problem. I don't get it. Where I live it is expected of me to mindlessly crunch questions and output solutions by rote#. Practice is an important part of learning anything, and I know that I have to put my hours in. However, I need to understand why I am doing what I am doing.

Proofs of key concepts are like magic to me. I can follow the logic, but I cannot derive them on my own without seeing them before hand. I know that I lack understanding somewhere down the line, but I don't know where. Moreover, I don't know how to cure it.

I don't want to just rote up stuff. I want to appreciate the beauty of what I am learning, but I simply don't know how.

Any suggestions?

Thank you in advance.

# Most teachers tell me to practice in order to memorize "problem solving techniques". I don't want to do that. I want to see the logic on my own, follow it through and think about what I am solving. I want to see stuff for what it is, and engage it on that basis.

[Vargas]

Pick something "easy" (high school trigonometry did the trick for me) and try to understand it completely. Don't look at hte formulas or the proofs, just look at the problem space itself.

You have a circle with radius=1 centered in (0,0). Draw a line from the center of the circle to any point in the circle. Now you build a triangle using this line as your hypotenuse. Inmediately, you will see that the heigth and the width of the triangle will be in (-1, 1). That is why -1 <= sen(x) <= 1 and -1 <= cos(x) <= 1. You can see that your triangle is always a rigth triangle so you can apply Pitagoras. That is why sen^2(x) + cos^2(x) = 1. Keep thinking about it for a few weeks and you will realize that all the formulas you memorized in high school are just common sense, you can deduce all of them just by drawing a circle with a triangle in your mind.

Try to do the same with other stuff (set theory is a good second thing to look at). You will discover that math notation is just notation and that many proofs (at high school level) are just common sense written in a very formal language.

Think about what the maths are about, not about formulas or notation.

[todayiamme]

Thanks a lot!

You nailed it. This is exactly what I want to do!

Can you suggest some good resources and a sort of rough path for me out of your experience? I know that it may not be possible for you to do so and it is quite okay if you can't do the latter.

Thank you for replying.

[thefool]

Take a proof based math class in college. They will likely start at a low enough level to make you happy.

Once you get to the point where the subjects start dividing, you'll see that the real skill that math teaches you is to look at a problem, and create a set of definitions which frame all the properties of the problem and make them make sense in that framework.

You get this skill by both playing with problems and trying to define frameworks (theorems, lemma's, axioms, ect) and by reading about the techniques that others have developed. It isn't about rote memorization, its about playing with abstract concepts till you have a way of understanding them. One thing that playing will teach you is that it will show you that the path you take to understanding something is not the same as the simplest path for explaining something once you understand it.

In contrast, in things like physics you are generally given the frameworks (by the mathematicians) and the cleverness comes from understanding how particular situations can be described by the existing (pretty abstract) frameworks (like newtons laws).

If you are bothered by how some of the ideas seem to appear from nowhere, notice that there is a lot of math out there that you don't know, and even basic statements about open and closed sets build on a lot of more basic set theory and group theory.

[todayiamme]

Thanks a lot for commenting. Your comment was pretty insightful to me and validated a few of the assumptions I had made.

Although, it is highly unlikely that I will end up at a college which offers such a class (indian education pretty much sucks) I'll search OCW and iTunes U until I find something like that.

I want to play with abstract concepts like you've said, but I have internal problems which I need to overcome before reaching that goal. I can't sit down at stuff at any lengths of time like that until I am really engrossed stuff from my past comes up and I tend to get emotionally strained whenever this happens. I remember stuff that shouldn't be there in my mind in the first place and it is a constant battle to get stuff done.

So, is there any structure you use to get stuff done?

[silentbicycle]

You might find Jan Gullberg's _Mathematics: From the Birth of Numbers_ helpful - it's a big book, but it's fairly light reading. Instead of a dense book of problems, proofs, exercises, etc., it gives an overview of the different kinds of math, their historical context, what kinds of problems they're good at solving, etc. More the 'what' and 'why' than 'how'.

It might help you get a better sense of what you're more interested in learning. (You might really like probability and statistics but not trig, for example.) You can probably find a copy for under $10, and a decent public library is likely to have it as well.

I've had a lot of fun with Project Euler (http://projecteuler.net/), a math puzzle website for programmers, though it's not clear from your post whether you're also into programming or not.

Also, another hard part of learning math is the notation. There's no getting around that. The Gullberg book will introduce it, but it won't drill it into you the way working through exercise problems will.

[jcdreads]

You may find that undergraduate physics is a good match for your learning style. There is plenty of ridiculous math, but all of it is based in simple (!) models of the way the world actually works. Differential equations and vector calculus are relatively easy for visual thinkers, and the need to get the math to tell you precise answers drives you pretty naturally to some reasonable approximation of mathematical rigor. Where heavy mathematics demands insane amounts of rigor to make progress, physics is merely descriptive, so you can enjoy learning about the symmetry and beauty of things (like how all of chemistry falls out of quantum mechanics) without (all of) the mindlessly mechanistic formality (like proving that legendre polynomials are really a complete basis set).

Also know that math (and physics for that matter) is almost universally poorly taught, so don't get discouraged by that.

[Arun2009]

I am NOT a Mathematician, but FWIW, here are my antidotes to the crisis in meaning that you are facing:

1. You most likely wouldn't have had a problem solver's education. So the first thing to do would be do start understanding how problem solvers approach Mathematics. I suggest two books:

- The Art and Craft of Problem Solving by Paul Zeitz

- How to Solve It by George Polya

There are also others, e.g., http://www.amazon.com/Math-Olympiad-Resources/lm/1SWDJ5NA047...

Keep in mind that most proofs you see are highly polished for the sake of clear presentation. The route Mathematicians take to reach the proofs are almost always messy. And, unfortunately, they like to clear their tracks.

A great way to learn how to solve problems is to find a good problem solver and asking them to think out loud while they are solving problems. Also, try to find alternative proofs of any proof you learn.

2. Understand the history behind what you're studying. How did the ideas you're trying to learn (Abstract Algebra, Topology, Vector Analysis, etc.) come about? This will add an immense amount of meaning to your subjects.

I recommend the following books:

- "Men of Mathematics" by ET Bell. Also "The Development of Mathematics" by the same author.

- "A Concise History of Mathematics" by Dirk Struik,

- "A History of Mathematics" by Boyer

- And of course, the internet.

3. Read a few "big picture" books side-by-side. A few suggestions:

- "Foundations and Fundamental Concepts of Mathematics" by Howard Eves

- "Concepts of Modern Mathematics" by Ian Stewart

4. IMHO - and this is just my preference - study application of Mathematics to Physics and Engineering. Physics especially was an inspiration for a lot of Mathematics, and in addition, this will also let you solve concrete problems using the tools you learn.

5. During my undergraduate and graduate periods, I found several Schaum's and Dover books helpful. They're usually short and pretty cheap.

6. Initially, do not let rigor get in the way of understanding the content. Fourier, Newton, Euler, etc. weren't all that rigorous by modern standards.

[kenjackson]

Great suggestions!

Two more must read books: 1. "The Mathematical Experience" by Davis and Hersh (http://en.wikipedia.org/wiki/The_Mathematical_Experience#cit...) 2. "I Want To Be a Mathematician" (http://www.amazon.com/I-Want-Be-Mathematician-Automathograph...).

Probably the key thing for me in helping my math ability was to actively try to prove theorems. Before reading a proof, I always try to solve the theorem myself first. And then after reading the actual proof... try to prove it again. You'll be surprised how many times you can't prove a theorem for which you just read the proof!!!

But this will help you get better at doing proofs, and understanding math. And it will also help you appreciate good proofs, because you would have already tried to solve it. You'll say, "Ahh... I didn't even think to try that, but that was exactly the step I would have needed".

Lastly, as someone else mentioned -- the proofs you read in texts are polished proofs. Often those theorems proved have been attempted by famous mathematicians who failed to prove it in their lifetimes. Take your time, be rigorous, and thoughtful. If you do that, you come out ahead regardless.

[todayiamme]

Thanks a lot!

Can I ask you a more detailed question? Can you suggest resources to tackle machine learning and statistical inference of patterns in data? I am currently going through (http://www.amazon.com/Data-Mining-Techniques-Implementations...) and there is so much depth in it that I want to understand. This is why I wanted to strengthen my fundamentals first before moving on to the fancy stuff.

If you're in this field or have any idea of it can you suggest me a gentle introduction to the depths of it?

[Ben_Dean]

I was in the same position as you recently (and still am to some degree) though in worse shape, since I decided I needed math at the end of college. The points people made previously about working at it and repetition are critical. I absolutely understand that you don't want to do things by rote, but what's important to realize is that like ANY other skill, there is one best way to improve, and it's called practice. You'll just need to do exercise after exercise, and read and re-read proofs, even when you don't feel that you "get" it. First you do it, then you'll get it.

The other thing that works the same for every other skill is to teach it. Get a friend who feels the same, and teach each other math. This works pretty well when you're not even talking to a real person, even. Just having to formulate your understanding solidly enough to convey it to someone else reveals where you get it and where you don't.

The personal trick that will help though I find is the hardest to do consistently is to be OK with not getting it. You'll spend a lot of time not understanding, so get comfortable with that fact and keep going. For myself, I tend to back off the subject when I don't get it right away, but I learn much faster when I stick with the stuff that makes me uncomfortable.

Prescriptively, try looking at How To Prove It for an explanation of what's going on in proofs, and how to engage with them.

[Open-Juicer]

I've got lots of good math books to recommend. However, giving beginners too much information would make them more confused.

So, only two recommendations for you, one book, one website:

1) If I'm only allowed to recommend one math book to beginners, It'll definitely be:

What is Mathematics - by Richard Courant and Herbert Robbins

Take a look at the review by Albert Einstein. Yes, Albert Einstein!

http://www.amazon.com/Mathematics-Elementary-Approach-Ideas-...

2)BetterExplained

This site explains math intuitively unlike the traditional formal approach.

http://betterexplained.com/archives/

[Tichy]

Practice and have some patience. I still remember the first year of my maths degree. The exercises seemed rather hard. A couple of months later, they seemed easy (the ones from the early months, not the new ones from the later months).

I don't think rote learning is the solution, but there are common problem solving techniques.

Also, my experience was that you don't understand maths so much as that you get used to it. I think I know how you feel - I also expected to logically process a section of the maths book and then having understood. In practice it didn't work that way. I remember at some point I suddenly had a grasp of "Gaussian Manifolds" , which seemed completely intractable at first glance (don't ask me now - it might not have been 'Gaussian Manifolds' but something else, I forgot the name and the concept).

I think in reality as a mathematician you just have to get used to feeling stupid.

it also frustrated me how little we really understand. For example sometimes we prove something by some really clever way, some border case contradiction or whatever. But to me that was not understanding - understanding would be "seeing it", not proving it.

[todayiamme]

Wow. Thanks a lot for that insight.

You know I still think sometimes that I am a retard and I can't possibly understand stuff. I know it's rather self-defeating, but when I used to see a few teachers handle stuff with amazing dexterity I used to just gape at the whiteboard. Aware that something beautiful was going on, without being able to put my finger on it.

Whenever I used to talk about this to people they would say that I don't work hard enough. Or, that I am a fool to expect anything more than just get the marks which are expected of me. Or, that this is not the time or place to do such stuff. When I told my mother right now about this amazing place called HN and this thread she told me to shut up and stop being so impractical (don't judge her on this she's quite amazing).

Long story short, in the end I assumed that the problem was with me and that I was deficient in some way or the other. Now, I'm learning that is not so and hence this thread. So, your comment was an eye opener.

I am so sorry that I put proofs in there. To me that was the biggest symptom of not seeing stuff, you know. So, I thought that people may understand from that. My bad.

[ajdecon]

The kind of understanding and appreciation you're talking about, where you "see" the math, can be hard to get to. For me I never found it until I started doing math in applied contexts such as physics or computer science.

My "eureka" moment was in a physics course where we were looking at the derivation of the differential equation for a damped harmonic oscillator. One moment the equation was something I had seen many times and knew how to solve problems with; the next, something "clicked" and I understood it, where the terms came from, why it all made sense, in a way that's hard to describe. Visualizing the oscillator and knowing the equation were the same thing in a weird way.

After that, and a couple more similar moments, I found it much much easier to understand and appreciate not only applied math but also purer concepts.

[pragmatic]

I suggest Project Euler http://projecteuler.net/. There are a series of (interesting?) problems to solve that will require you to learn a little math and a little programming.

For me, the hardest part of Math is the Byzantine formula. My brain just doesn't look at them and go "oh, that's how it works." Instead, I need to translate that into some nice executable code (Python, etc).

I stumbled through Calculus not really getting what it was about. Physics on the other hand, was real world, here is the formula; here is how to apply it. That made it click for me.

TL;DR: Physics and Project Euler.

[todayiamme]

>>>For me, the hardest part of Math is the Byzantine formula. My brain just doesn't look at them and go "oh, that's how it works." Instead, I need to translate that into some nice executable code (Python, etc). I stumbled through Calculus not really getting what it was about. Physics on the other hand, was real world, here is the formula; here is how to apply it. That made it click for me.<<<

I know exactly what you mean. I can see stuff in physics and I intuitively know what is going to happen when I take a metallic hollow piston with air inside of it and place it partially in a magnetic field. It's observation of Solenoids and other stuff that allows me to do that, but I really couldn't do the same with math, and now I am trying to do that.

[cvore]

When I came out of high-school some years ago I had the same problem. I really wanted to learn the math needed to do 3d-programming so I could to do fancy demo-scene stuff (see www.scene.org), but the high-school level math just didn't cut it and I didn't manage to learn it by my self. I started at a university to study CS, and took some calculus courses there. It was a fairly advanced calculus course which focused on proofs, and I worked really hard on it through cooperation with others and the help of great teachers. Although this experience lead me to end up doing a PhD in math, it was this first more advanced mathematical experience which made me able to really _get_ it. It took a lot of hard work, and after helping some younger people with high-school level stuff, I now see that the math-books at this level often suck. The "proofs" are more heuristic justifications that unfortunately often simply hide the real point of what is going on. My advice would be to take a proof-oriented math-course at university level. Even if the curriculum is not the math you would like to learn (for example a calculus course might be ideal even though you would like to do discrete math stuff for CS), I do not think that mathematical maturity needed to go on by your self is possible to get in any other way (except if you are a genius).

[sammyo]

Lemma: More coffee.

Derived from the theorem that a mathematician is a machine that transforms coffee into theorems.

Ok other thoughts. Go listen to some hard stuff. I loved a talk by Ron Graham about problems that would never be solved by computation. I remember feeling like I followed but all that remains is awe and images of infinite arrows.

Knowing I'll never be a mathematician but intrigued by topology, I thought about trying to write “Idiots guide to Algebraic Topology” as a way to push beyond mobius strips and klein bottles. I've made it through an intro and some insights seem sort of trivial, more along the lines of how to write down precisely an action like a twist. We know intuitively what a twist is, what a bounce is, getting it exact is hard work. But it relates to a lot of stuff we grasp as denizens of a 3D world.

Draw lots of pictures, get really good a graphing. Graham seemed to dash off quick graphs that gave quick insight, I expect he has incredible amounts of practice.

Reading some history of math has helped me.

I also think talking/arguing/chatting/ranting with peers that are very interested in topics in the general range will go along way.

[will_critchlow]

Pick something elementary to prove (by which I mean requires little prior knowledge) - like the fact that the square root of 2 cannot be expressed as a fraction and look up the proof. Spend time understanding not only the proof itself, but also, crucially, why it is important to prove things in this kind of formal way. Here lies the beauty of mathematics at first. Later you will find exciting results, at first, there are only beautiful proofs.

You are then probably going to have to go back and do some rote-work. I think you probably need to crunch some calculations to get comfortable enough to ever get really good.

The good news is that with the excitement in hand, you should be able to carry on through the rote stuff and then start on some undergrad courses (plenty are published for free online) that have no prerequisites. Logic and number theory are good for lack of existing knowledge - both come from sets of axioms.

[Obligatory, 2. ???, 3. Profit]

[dododo]

maths can be like programming: it can seem pointless without a purpose. too much is made of proofs and too little of intuition.

so pick a project, something you'd like to know. then apply math as, and when, needed.

what are you interested in? (other than learning math) if we know, then maybe someone can suggest a project related to something you'd care about :)

[todayiamme]

Awesome. You nailed how I wanted to approach it. I am sorry if this comment sounds like a bunch of gibberish, but I am half asleep down here.

I actually want to learn maths because I want to make this beautiful thing I have in my head. It's basically a user friendly data mining app that learns as you go along and it gives you the stuff you need. Not what I think you need.

Basically I am trying to make a system that can collect data, and analyze it for someone who doesn't know what parameters to set in the first place. After that I want to apply what I learn from those interaction to trim the excess fat or give anything more if required. I know that this is a really lofty goal, but if I make it then I would be the happiest person on earth. As this is a part of an even bigger thing I want to make, which is brewing in my head. :)

I haven't been able to get much mileage as I have concentration issues and there are environmental pressures which expect me to conform to the rat race of indian society. So, I figured that this was the best way to kill two birds with the same stone; I master my fundamentals as well as create the foundation for my real life later on. I won't give up on my idea though and I try to force stuff through my head at each sane moment of time I get.

Any suggestions?

Thank you for commenting.

[thesnark]

I agree with this. For me, math started to make sense when I saw it applied to problems, especially simple physics problems that explain things you can directly observe. The Feynman lectures were an especially rich source of these types of problems.

My favorites: at what angle will a marble rolling down the side of a bowling ball leave the surface?

when you observe water flowing out of your tap, the stream tappers and becomes more thin as it falls, can you derive the formula for that?

[jacquesm]

http://news.ycombinator.com/item?id=755043

[todayiamme]

I got a lot of mileage out of that thread thank you, but my other problem still remains; how do I see it and appreciate it?

There's something beautiful going on over here and I want to be able to peel it apart like I do with my ideas. It's like I can learn the formula of force and how it is applied day to day, but I don't necessarily see it around me unless I can visualize it and internalize what it means.

I know I am quite dumb and somehow there are like layers in my understanding. I can't vocalize it but there is a huge difference to me between learning about stuff the way I am supposed to and learning it the way I want to.

Thanks a lot for replying! :)

[hackinthebochs]

Speaking as someone who always "got" math, the understanding really needs to come from you. You can't expect to get it from a book or from the teacher. I had two tricks that has served me well: always ask yourself "why" until you reach that critical "a-ha" point, and secondly, visualize visualize visualize.

What I always did for new concepts was to ask myself "why". And I'd just think about it and visualize it until a reasonable explanation came to me. It doesn't necessarily have to be completely "right", but it just needs to make sense to you.

As an example from high school, everyone seemed to struggle learning basic algebra. The basic idea to learn from algebra is that you can do the same thing to both sides of an equation and the "truth" of it remains in-tact. This fact covers about 90% of everything you'll learn in Algebra I and II. The rest is basically just learning tricks for different situations.

The problem is you can't really be made to "get" this critical idea, the understanding has to come from you. The takeaway from all this is to focus on finding that a-ha moment for every new concept you learn.

As far as proofs go, again you can't be taught to do them well. You have to find your way on your own. The first step is to just convince yourself what you're trying to prove is true. Either by working out many different special cases and finding some pattern, or by visualizing whatever process you're trying to prove. Then you "formalize" your intuitive understanding by writing a proof. Convincing yourself that something is true is the crucial part.

[pmb]

You want to go take a proof-based college math class. Or sit in on one. Or find videos of one. It is hard to guarantee that a course will be like that, unless its name is either "Analysis" or "Abstract Algebra". Don't sell Indian universities short - it's so hard to get a good job in mathematics that you have to go very far down the hierarchy to find a place with bad mathematicians.

And you do need to memorize techniques - the only issue is that the techniques you need to memorize are at a much higher level than you are currently being taught. Techniques include "induction", "contradiction", "diagonalization", "construction", etc. Mathematics is a toolset, and the techniques and facts you learn are the tools required. They are beautiful in themselves, and can bootstrap to better tools, but if you aren't a mathematician then you should be memorizing solution techniques, just at a higher level than you currently are.

[sp332]

You will have to learn some problem solving techniques. In fact, the more the better. You will find that some of them will be intuitive for you, and many of them won't. Don't think of these as "rote". It's training the basic mental skills you'll need to call on without thinking about it. It's like learning to walk: the exploration isn't in finding new ways of walking, but in finding new places to do the same old "left, right, left, right".

I wouldn't worry too much about not being able to derive a proof yourself. At least you can follow a proof - that's still a useful skill. Deriving a proof is much, much harder.

[golwengaud]

I've found Terry Tao's blog (in particular the stuff linked at http://terrytao.wordpress.com/career-advice/ ) fascinating and helpful.