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Ask HN: How Do I Get Math?
64 points by todayiamme on July 27, 2010 | hide | past | favorite | 76 comments
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.




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.


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.


As far as online resources, http://khanacademy.org/ has in my opinion, a knack for letting you into the underlying principles. They don't yet cover extremely advanced topics, but have helped me to grok some concepts that I has previously only had mechanical knowlege of.

http://news.ycombinator.com/item?id=1059076


Out of my experience, you only 'get' math when your boss gives you a paper on a subject you have absolutely no clue and says: "I want this implemented in our system by next week" :-P

On a more serious note: Other people have recommended very good books that will teach you problem solving so I would like to give you a feeling of what "real" math is. In high school and in most University degrees you only do applied math => how to use math to solve this or that problem. The more math you know, the more tools you have to solve your problem. For a given problem: could I solve it with a partial differential equation? with a Bayesian approach? etc. You can see how the more math you know, the wider your search for a solution can be.

However, most people never get to do "pure" math => to create your own new math. When you read a book or attend to a class you only get to see the finished product. When you see a finished building, they have removed all the scaffolding, the cranes, and all the other tools that were used to build it. If you spend some time creating your own new math you will get a very deep understanding of what math is and it will widen your learning abilities.

Of course I don't expect a high school student to create something truly new, this is just a learning exercise. I just want to provide a 'feeling', I won't be formal or rigorous.

Definition 1: An object of class TIAM is composed of an integer and a set. The set can be empty or contain letters from the English alphabet with no duplicates.

Definition 2: Operation "+" This operation will take two TIAM objects and produce a new TIAM object. The integer part of the new object will be the sum of the cardinalities of the sets. The set part of the new object will be the union of the two sets with all the letters shifted as many places as specified by the integer part of the second TIAM object.

Example 1: [1, {a,b,c}] + [2, {c, d}] = [cardinal({a,b,c}) + cardinal({c,d}), {a,b,c}U{c,d}>>2] = [3+2,{a,b,c,d}>>2] = [5, c,d,e,f]

Example 2: [0,{z}] + [1, {}] = [cardinal({z}) + cardinal({}), {z}U{}>>1] = [1+0,{z}>>1] = [1,{a}]

* TIAM objects are closed under +. (See http://en.wikipedia.org/wiki/Closure_%28mathematics%29). Do you understand why? Can you prove it?

* Is the operation + conmutative or non conmutative? (See http://en.wikipedia.org/wiki/Commutativity) Why? Can you prove that it is / it isn't?

* Can you find a neutral element? (See http://en.wikipedia.org/wiki/Identity_element) I mean, can you find a TIAM object (I will call it TIAM-ZERO) so that TIAM-A + TIAM-ZERO = TIAM-A for any given value of TIAM-A?

Etc.

This is just an example, try to build your own stuff and reason about it. Ask yourself questions about it and answer them. Once you get used to this kind of thinking, you will learn faster and deeper.


I am sorry if I make a fool out of myself, and I am really sorry if they are some mistakes over here (it's 4 AM and I need to sleep), but I'll try to do this.

#TIAM objects are closed under + as it is a union of two sets and we are not including any element outside that set. So basically I get a bigger set that includes all the elements of the previous set sans repetition. Now, since I am including all the unique elements of the previous set this means that no new symbol is over there.

However, the last operation (the shift one) isn't one with closure.

#It is commutative as it is the addition of the two sets with an algebraic operation. So it doesn't matter what order in which I add the stuff together. The results always remain the same.

#Yeah,

[0, {}]


4) Can you find an identity element TIAM-ZERO' such that TIAM-ZERO' + TIAM-A = TIAM-A for any TIAM-A

5) Is question 4 different than question 3 and why or why not?


4) The identity element is the same as that in question 3.

5) It's the same question as you've asked me to come up with something that when it's put through a machine that computes the '+' operation with another thing it comes out the same on the other side. In this operator it's limited to this {0, []}. I don't know how I know this, and I don't know if I can prove this but I think that this is the answer.


Are you sure?

When I think on this, let's say TIAM-A is

[1, {a}], and you add [0, {}]

that's [card({a})+card({}), ({a}U{})>>1], or [1,{b}]

so that can't be the neutral element.

I think that there must not be an identity element, because the >> operator forces a change under any circumstance which is dependent upon TIAM-A's contents.

Similarly (I haven't thought about this enough yet), I suspect that the operation is non-commutative, because unions can change the number of elements (deletion of duplicates), and the shift is involved (which changes elements, as opposed to just combining them). So let's say you shift and then union, you could get a different result than if you union and shift.

I'm gonna see if we can get a concrete example here...

at somewhat random:

TIAM-A (now referred to as A): [0, {a,b,f}]

TIAM-B (now referred to as B): [1, {b,c}]

TIAM-C (now referred to as C): [2, {d,e}]

A+B+C=T1, A+C+B=T2, if T1 != T2, not commutative (pardon me as I riff on notation for ease of writing)

A+B = [0, {a,b,f}] + [1, {b,c}]

A+B = [3+2, (abf U bc)>>3+2]

A+B = [5, (abcf)>>5]

A+B = [5, fghk]

A+B+C = [5,fghk] + [2, de]

A+B+C = [4+2, (fghk U de)>>4+2]

A+B+C = [6, (defghk)>>6]

A+B+C = [6, jklmnq] = T1

then:

A+C = [0, {a,b,f}] + [2, {d,e}]

A+C = [3+2, (abf U de)>>3+2]

A+C = [5, abdef>>5]

A+C = [5, fgijk]

A+C+B = [5, fgijk] + [1, {b,c}]

A+C+B = [5+2, (fgijk U bc)>>5+2]

A+C+B = [7, (bcfgijk)>>7]

A+C+B = [7, ijmnpqr] = T2

T1 = [6, jklmnq], does not equal T2 = [7, ijmnpqr]

hence, non-commutative - but I could have misunderstood something entirely here


Hi, yes, I made it non-conmutative to make it a little bit more difficult, I think you don't get to see a non conmutative algebra in high school, it was a way to take her out of her comfort area.

However you made a sligth mistake with the shifting part:

"The set part of the new object will be the union of the two sets with all the letters shifted as many places as specified by the integer part of the second TIAM object."

I think you are shifting a little bit too much.


Oh. I just noticed it right now.

I should have done it on paper and worked out it out more extensively. Is there anything else conceptually wrong in my reply?

Also, is there any way at all I can contact you? I would love to learn more from you. If possible can you please drop me a line at yesthisisananonymousid at gmail?


aha!

I interpreted "the second TIAM object" as "the new TIAM object", as opposed to the second of the original objects. Thank you for the clarification :)


I learned something just the same. Thanks a lot!


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.


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?


Get this book: http://matrixeditions.com/UnifiedApproach4th.html and work through it if you want a rigorous treatment of MV calc and intro analysis.

Or this book: http://www.amazon.com/gp/product/0914098918/ref=pd_lpo_k2_dp... for regular calculus. Read them, work through the problems, get an answer key, email your solutions to professors asking them to look at them.

You will get a good introduction to analysis (and exposure to many other parts of math) from those two books.


Don't sell Indian universities short. It is very hard to get a job in mathematics, which means that even at "bad" universities the mathematics faculty can be quite good.


That's the problem you see.

No one takes a job in mathematics, down here the trend is to do engineering get a MBA and work in Goldman Sachs. I wish this wasn't so, but it's quite true. There are precious few souls who try to buck the norm, but they are far and few in between.

Most mathematicians end up in coaching institutions which exist for one thing; to "crack" the IIT-JEE for the aforementioned rat race. I really wish I could sell them short, but outside the IISc and TIFR along with a few other research organizations there is hardly any innovation taking place. Most universities are intellectually dead and students study only to get "placements" in some company so that they can prepare for their MBA later on.

I know that this sounds like an excessive generalization, but I live near one of those prestigious IITs and I've worked with a PhD student as well as solicited the advice of a few professors. You know what shocked me? One of them, who is one of the original brilliant researchers and an ex-dean, told me point blank to get the hell out of this country and that my future isn't over here.

I really want to change things, but since I am in this system I simply can't do it. However, at least I can speak the truth. Please try to understand that I am not judging anyone in this context. It isn't anybody's fault. It's just a reflection of how the majority of this society thinks.


You misunderstand me. I am not saying that Indian Universities properly prepare you for a lifetime of success. I am saying that it is certain you can get the mathematical knowledge you want from them. As to whether that is the career you want or whether you want to be a mathematician - I have no idea. But if you want to learn and understand proof-based math, Indian universities are as good as everyone else's, and the good ones are quite good.


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.


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.


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.


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.


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?


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.


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/


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.


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.


Nothing against proofs, of course proofs are the basis of maths. What I meant is that typically I couldn't just read a mathematical text, follow every step logically, and then have an understanding of that mathematical subject. Rather, the understanding would sink in after a while of trying to follow the proofs and solving exercises.

Could you give examples of the kind of problems that stump you, and the things you try to do? I am sure a lot of people, including me, would be glad to help.

To me it just sounds as if you are trying to take too many steps at once, if you can not follow some of the proofs.


For example a few days ago this teacher wrote this problem in class;

(x^2 - 2x + 2^|a|)/(x^2 - a^2) > 0

I had no clue how to do it. When he wrote the solution on the board only then could I follow it. I didn't understand why I couldn't see the solution on my own. It looks so simple later on and I just can't understand why I couldn't get it on my own in the first place. Can you help me out with that?

Thank you.


Here's hoping I won't embarrass myself too much...

Since presumably your teacher already wrote the solution onto the board, I'll just talk about how I would arrive at a solution.

So first of all, it is a very common problem. We must have solved countless polynomial problems at school. So usually, seeing this kind of problem should already ring several bells.

So what I associate with polynomials (from school day) is

- zero points

- factorization

- quadratic equation

- "curve discussion" (don't know the english name for it, determining minima and maxima by calculating differentials)

Did you have the same associations? If not, then that is simply a matter of experience that would come with practice. Without that experience, naturally arriving at a solution would be hard.

It should be immediately obvious that we can factorize this polynomial without problems: simply because we know that we can factorize every second degree polynomial, thanks to the quadratic equation. This knowledge is basically "rote learning", except that you'll probably pick it up without decidedly doing rote learning. The polynomial presented here is already factorized into two second degree polynomials, so no problem there - do you agree so far? (Bonus points for seeing immediately that (x^2-a^2) = (x+a)(x-a) - this is a really common and basic formula that you should probably just "know". However, in principle you can also derive it from the quadratic equation, knowing this formula is just a shortcut).

So, are these zero points of use after all? Yes!!! We can simply check some values of the function between the zero points and determine if they are greater or less than zero. Since the function can only "switch sides" at the zero points, we then know it has to have the same sign (negative or positive) everywhere between those zero points (and same for outside of all the zero points).

Do you agree so far? If not, where is that you were stuck?

With that idea, let's just determine the zero points. From the (x+a)(x-a) part we already know that a and -a are zero points.

Using the quadratic formula on x^2 - 2x + a^|a| I get -1-sqrt(1-2^|a|) and -1+sqrt(1-2^|a|) - just ask if that is too fast, also be aware that I am very prone to make errors in such calculations.

From there on, I admit it is not much fun. We have to somehow order the zero points, so that we now when the sign could change. So we have to make case distinctions.

In fact, I haven't yet thought further than this, and now I am looking for ways to make it easier (I hate case distinctions). Laziness is the main motivator for maths, therefore I am not that much in favor of the rote learning approach...

So one step back: we have a products of two polynomials, poly1*poly2 with poly2 = x^2-a^2. That product is > 0 if poly1 > 0 and poly2 > 0 or poly1 < 0 and poly2 < 0 (if this is not obvious, please ask).

It is clear that poly2 < 0 for |x| < |a| and > 0 for |x| > |a|.

So let's look at poly1 = (x+1-sqrt(1-2^|a|))(x+1+sqrt(1-2^|a|)) (if my calculation was correct).

Since 1-sqrt(1-2^|a|) < 1+sqrt(1-2^|a|) we get

poly1 < 0 for -(1+sqrt(1-2^|a|)) < x < -(1-sqrt(1-2^|a|)) and > 0 for x < -(1+sqrt(1-2^|a|)) or x > -(1-sqrt(1-2^|a|))

So alltogether, the (intermediate) solution is

(-(1+sqrt(1-2^|a|)) < x < -(1-sqrt(1-2^|a|)) and |x| < |a|) or (x < -(1+sqrt(1-2^|a|)) or x > -(1-sqrt(1-2^|a|))) and |x| > |a|

I have a feeling this should be simplified further, but I'll post it like this because it becomes a little bit intractable in the small comment box of HN. Maybe I'll find time to finish later.

Anyway, I probably made mistakes, and overlooked some much easier approach. The main point is that most of the steps are very common approaches, and I think/hope I demonstrated dividing the problem into smaller sub-problems.

One advice I would give for maths is to not be intimidated - you need a little bit of faith... If you fear the equations, you'll be too paralyzed to find a solution.

Also, I used to cheat a little bit on the problems in school: usually chances are high that the problem at hand has something to do with a recent thing the teacher has taught us. So it pays to look for ways to use the knowledge from the latest lessons.

Also, this problem seems comparatively ugly.

Sorry that this got rather complicated. Let me know how I can help better. I'll try to create a shorter summary of the steps involved. (Meanwhile, please post the shorter solution if there is one and you find it).


Awesome reply.

Thank you.

I'll reply when in the evening. Right now I need to crash and it's 4 AM and I'm mentally impaired.


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.


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.


>>>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.


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).


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.


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]


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 :)


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.


often the simplest things in machine learning/data mining work best---so maybe come up with a very simple way of achieving what you want, try it out. work out what's not so good, and improve. repeat.

an important thing is to work out what maths you can just rationalise and ignore the details of, and what maths you need to know in detail. for example, you probably at this stage don't need to know how to prove the central limit theorem, but it might help you to know the intuition. it might be a good idea to know, in detail, where linear regression comes from though.

warning: in my experience, data mining books tend to provide poor explanations of the mathematical justifications of what's going on. if you can, get to a library and get some more theoretical books on machine learning or statistics. in stats, larry wasserman's all of statistics is great, in machine learning, hastie et al's the elements of statistical learning, mackay's information theory book, bishop's machine learning book, etc, etc...

with any text book, don't read them cover to cover. just get what you need and move on. if it's not obvious what you need from the book, try another one and go back. books can be expensive, but time is more precious. it's more important to do math than read math.


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?


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


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! :)


Appreciation comes with understanding, there is not much I can do about that for you. I'm not too good at math (I know a lot of tricks to use a computer to work around my lack of math skills though ;) ), so plenty of the beauty of math is hidden from me too, but the root of it for me is that math is an absolute, something that stands all by itself and that does not need a physical universe to manifest itself in order to be 'true', even though some of the mathematical constants do seem to be rooted in our physical environment.

(pi for instance).

I don't like the 'I am quite dumb' assessment you give yourself, everybody is born 'dumb', it's those that spend the time to educate themselves that rise above that and you seem to be willing to do so.

Everybody, no matter at what level of knowledge they currently reside has the power and the potential to rise above that level, maybe not much, but you can always improve.

I'd say work through some of the stuff mentioned in that thread and see if it helps you to gain understanding, you're almost certain to pick up something and that in return should help with your other problem, the appreciation.

Best of luck with all this, think of it as enjoying the journey, not necessarily to only enjoy it all upon reaching a destination.


Thank you for replying! It's always awesome to talk to you.

>>>I don't like the 'I am quite dumb' assessment you give yourself, everybody is born 'dumb', it's those that spend the time to educate themselves that rise above that and you seem to be willing to do so.<<<

You're right I am really sorry. It's just that old habits and learned helplessness dies hard.


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.


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.


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.


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


I'm going to play devil's advocate and say that you should listen to your teachers.

I've seen plenty of students "diagnose" themselves, only to neither mindlessly crunch questions nor appreciate the beauty of mathematics.


My teachers tell me to just solve questions and get good marks so that I can move ahead in the rat race of life.

They are quite awesome, but unfortunately my paradigm doesn't exactly superimpose upon theirs. So, I don't get the mileage I should be getting from their advice. My goals and aspirations are simply too different. The ones that do understand, smile and tell me to just tow the line.

I don't want a job in some big company. Neither do I want truckloads of money. I just want to create beautiful things.


Start with this - http://www.geniebusters.org/Riemann_intro.html and then http://www.schillerinstitute.org/fid_97-01/982_Gauss_Ceres.h...

It follows Vladimir Arnold's method of re-coupling physics with mathematics as the original way of learning it... since observation of planetary physics was how a lot of mathematics was derived.

It is a very, very interesting read.


Watch these videos and understand them: http://www.khanacademy.org/

You can start at as basic level as you like.

Expose yourself to lost of proofs of simple theorems. Try to reconstruct proof after single reading (with understanding why steps follow one another).

After a while you will see how information flows within a proof.

Basically expose yourself to a lot of math proofs and be sure that you understand why every element of each single proof has to be where it is.


Yes, there is beauty in math and in proofs. I was able to see this beauty after reading the wonderful book by George Polya called "How to solve it?" I understood math intuitively, after learning many proofs like you, but the key point that I learnt from the book was "review your work". When you review, you reinforce how a particular problem was solved and learn from the method that was used. This forms as the basis for your next problem and so on.


Since you are also into programming, I would recommend studying some discrete math. Discrete Mathematics and Its Applications by Kenneth H. Rosen is rather friendly and instructive on how to do proofs.


Get "How to Prove It" by Daniel Velleman.


I'm going to try to give you something totally off the wall. Don't forget that mathematicians were people. Here's three biographies of some of my favorite mathematicians:

http://www-history.mcs.st-and.ac.uk/Biographies/Poincare.htm...

http://www-history.mcs.st-and.ac.uk/Biographies/Peirce_Charl...

http://www-history.mcs.st-and.ac.uk/Biographies/Smale.html

Also, please check out some of the giants on that site and wikipedia: Gauss, Laplace, Euler. Why were they interested in what they were? What techniques did they develop? A very crucial question that very few ever ask: Would they have done much differently had they had access to a computer? (Of course).

Focusing on proofs to learn math is very perverse. A proof in a textbook is a proof that has been refined an almost ludicrous number of times. Definitions and axioms have even been optimized for the sake of elegance of said proofs. Don't get me wrong, a connoisseur appreciates what has been done. However, removed from the problem that motivated this way of thinking, mathematical techniques can often confuse more than enlighten.

What would you like to learn exactly?


In this thread I realize that I have been a bit vague. I want to develop the skills of visualization and in depth understanding of math for a reason. I want to learn how to make systems that exploit machine learning and use statistical techniques to infer patterns from data.

In the future some day I want to work in A.I. and create something beautiful with the knowledge I have gained. This is why I want to lay down the foundations to understand the beautiful advances in it I see around me.

Thanks a lot for commenting!


Cool, although I'd suggest not getting to excited about AI itself. It's cool your already focusing on machine learning.

A great way to get going in that direction is to check out the state of the art as expressed by AT&T labs Netflix Prize solution. To get very far you'll want to have some background in linear algebra. However, theory isn't the way to go (although you'll probably need it eventually). Do a little digging and you'll find code for svds, the simplex method and all sorts of super important algorithms that are used all over the place.

You'll find if you ask questions like: How does x work in y algorithm? people will have a better idea of where to start. When you actually are coding and able to see numbers for a particular problem, you'll be able to wrap your mind around what's going on.


Thank you.


after review, i'm not sure that this post answers your question, and because i'm very tired it may simply be self-serving and totally useless to you. however, i am taking a gamble that it could answer your question and am posting it anyway.

it's normal to not be able to derive proofs on your own without seeing them beforehand. you don't want to "rote up stuff"? then don't do math, bitch.

seriously.

the only way that you're going to develop the ability to prove things on your own is by reading over and memorizing a lot of other proofs. math is a skill to be built and you will notice the repeating patterns in proof techniques, and then you will be able to prove new solutions yourself.

i totally killed all my math classes in college because i wrote down the proofs on index cards and drilled myself until i memorized them. i spent tons of time in the library doing ROTE MEMORIZATION of all the various proofs. does that make me a "boring tool" who can't "think creatively" and only relies on "lame rote repetition"? no, not at all, i was consistently a creative problem solver. WHY?? because i built the base of explicit knowledge upon which to draw, and THEN i was able to apply it in novel ways. DOING math is INTUITIVE. that it means it's SKILL-BASED and NOT RATIONAL. FOLLOWING a proof, on the other hand, requires LOGIC.

if your problem is understanding proofs, then, just make sure you have a thorough understanding of the building blocks and concepts.

do you know how i made it through abstract algebra? before school started, i sat down at the bookstore and read through the first 5 chapters of the book. then, when we discussed the chapters in class, it was my second exposure and it was much easier to memorize. consequently, i was way ahead of everyone and did minimal work. use your prefrontal cortex and think long-term.

and if you're having trouble visualizing things, try to reduce them into their smallest possible visual elements. for example, if you're trying to understand a proof of something in R^n, then start by understanding how to do it in R^2. Then R^3.

always try to look for concrete examples first. it's often much easier to go from concrete -> abstract than abstract -> concrete. it's how the human brain works. our consciousness is just a series of metaphors for real-world operations.


Aside from the part about "bitch", I agree with pretty much everything zackattack said. Key points:

1. On your second exposure, things will make MUCH more sense. It's just how I've found the mind works. The first time you read something, you're reaction is: what??? Come back to it in a few days and you feel more comfortable with it.

2. When I first started reading proofs, I felt so clueless as to _why_ they were doing the steps that they were. Basically, I felt like I was trying to scale a sheer cliff that was smooth as glass, with no handholds. I had no idea where to _start_ if I wanted to prove something. However, after reading a lot of proofs, you start to understand some tricks, concepts, and approaches on how to tackle them.

3. Going from concrete --> abstract (i.e. generalizing) is the way to go. As zackattack said, always try to work out cases in lower dimensions, or even "trivial" cases. After doing one or two of those, it's a lot easier to guess at the general form and verify that, than to derive the general form from the get-go. Also, doing the lower-dimension case will help you build _intuition_. Mathematics is all about formalizing intuition. The symbols and equations will often obscure that, but always try to understand the concept behind what the equation means. That will make most proofs much more amenable to understanding.


> it's normal to not be able to derive proofs on your own without seeing them beforehand

I STRONGLY disagree.

Let's address this with an example: "Two planes are either parallel or they intersect in a line" Someone who just finished high school should be able to "prove" it informally without even opening a text book. Of course high school education doesn't give you the tools to write a formal proof, but if you think about it for a while, then write an informal proof, when you finally open your book and read the formal proof it will be pretty obvious to you. Once you learn proper maths in University, you should be able to sketch a formal proof very quickly.

Obvioulsy you cannot reinvent all the math from the last 5000 years all by yourself, so you need massive amounts of effort and study to learn (a part of) it. In my view, learning the basics of your problem space, then thinking informally about it, then (informally) writing your own theorems and proofs is the best way of learning. You won't come up with everything, but when you go to your book and read that all-important theorem you will think: "Aaaahhh! of course! how didn't I think of it!?"


my point was that he should not trip about it, but instead be learning more techniques. in the future when he has sufficient skill, then OBVIOUSLY he should attempt to prove them on his own.


I am a she by the way. :/


There's a cultural issue in affect over here. From where I come from all I am expected to do is rote stuff up, not understand it. My teachers explicitly say that I'm wasting my time, but I frankly disagree. The why is not important for this discussion.

So a lot of math is fairly non-intuitive for me. My problem wasn't understanding the proof, but visualizing it and deriving it on my own without any aid.

The last bit of your comment is spot on though. I do have tons problems in visualizing stuff, but there's more to it than that. It's a learned skill that depends upon the knowledge of the fundamentals I have and how I have to apply them. I lack that at some level and I am trying to understand how to correct it.

Thanks for replying!


I am telling you. You will develop intuition, and the ability to derive proofs on your own, only after you spend lots and lots of time reviewing other proofs. Memorizing them and their techniques will accelerate your learning process. If you are ever in doubt about a technique used in a proof, study it further until you are confident you understand.


[deleted]


@zackattack I understand. It is of no use to argue over this stupid straw man, but I want you to know something. I firmly stand by Voltaire's school of thought and I will defend your right to say this, but I will not defend the ethical implications of it. It's a subtle distinction, but quite an important one at that.

Please try to understand that this wasn't about political correctness, but genuine understanding about the human being on the other side. I don't mind, but someone else can. One can argue that they're weak. That may be the case, but they're still human beings and they deserve respect.

P.S.- I thought that this was HN not 4chan where I have to use such stuff like; watcha doing ya mother-fucking dawg? (Ugh) :)


That sounds like a horrible experience. Depending on the country you live him, you could have sued that teacher.

Unfortunately a lot of maths teachers are really bad. It's great that you kept your interest in maths despite of such experiences!


I can't and I couldn't even tell my parents. The thing is that I am intersexed and hence I am socially unacceptable. My parents don't want anything to do with me, and, well, it's a long sob story that isn't worth a damn.

In a way it was a good thing. It taught me a lot about human nature and the implications of our behavior, look don't judge the man. He did it as a joke and didn't mean any harm. So, it is better for me to forgive and forget and just move on. It's human nature to make mistakes and I am pretty sure that someday he will realize this and correct it. So, who am I to judge?


Good if you can see it that way. I hope that you won't let your environment bring you down! There are > 6 billion people on this planet, somewhere there are people who are more tolerant and would not consider you socially unacceptable. I hope you can give yourself the freedom to go out into the world and find the place where you fit in, rather than fighting the useless fight in the "small world" at home. While in high school, sometimes that small town (or wherever you live) seems like it matters so much, but believe me, it can feel very good to just leave it all behind and go away.


I didn't realize. I apologize. Nice to meet you. Feel free to email me if you need me to point you towards textbooks or other such resources in the future.


> does that make me a "boring tool" who can't "think creatively" and only relies on "lame rote repetition"?

I strongly disagree. It does make you such. Memorization is overrated.


And I strongly suspect that you actually don't know what you're talking about. (How far did you get in math?)


Indeed. I never memorized anything, then I got to uni. It's just a waste of time rederiving 5000 years worth of mathematics for the fun of it, and pretty much impossible to do in exam conditions. You need to know the theorems backwards in order to be able to intiutively apply them.

It's exactly the same as how basic calculus is taught...integration at low levels is a HELL of a lot easier when you know that say, 1/n integrates to ln n, and you can then just use basic algebra to get your integral into that form. Without having that core intuitive knowledge obtained by practicing and learning, you can't make those jumps.


What you got about the role of rote learning from your teachers is bad and appears not to be from teaching in the US.

The 'understanding' you explained you wanted want in math IS the right direction. The proper role for 'rote' learning is meager.

To get anything new, or even new to you, in math, first essentially have to guess it. For this you need some intuition, that is, some ways to guess. You should be free to take wild flights of intuition where only one in a million guesses is correct. Proofs, then, add discipline to your intuition. Slowly your intuition becomes more accurate.

E.g., given a guess, do some 'thought checks': So, check some extreme cases. Guess what some of the consequences might be and see of those, first-cut, seem to be true or false. Then will have guesses (1) very likely true, (2) very likely false, (3) don't much know yet. Once you narrow down to what you need, you will be well on your way to success.

Actually for anything new, the harder work is guessing it's true and being fairly sure, based mostly on intuition and your checks, it is. Then usually a proof is the less difficult work. That is, by the time you start a proof, you might have some quite good intuitive evidence that what you are trying to prove is true and, also, some good hints on how to prove it.

For more, read the now famous comment by A. Wiles on how he wanders in dark rooms in an old mansion, bumps into furniture, slowly figures out where everything is, finally finds the light switch, then clearly sees everything in the room, and then starts on another dark room. Practiced intuition provides the guesses, and proofs provide the correctness.

What I am saying here about guessing, Wiles, etc. is good for research, e.g., your Ph.D., and some of the more difficult text book exercises. Mostly in the exercises can do well enough with less than a million wild guesses!

Yes, there is some 'sense' to at least the 'analysis' (i.e., calculus and beyond) part of math. E.g., 'half' of calculus is finding areas under curves. Okay, what curves have properties sufficient actually to HAVE area well defined in this sense? Okay, the now classic answer is that the curve has to be 'continuous'. So, when you learn about continuous curves, you are learning about the classic hypothesis that makes area under the curve well defined.

Okay, why? Well, that's because are trying to find the area under the curve over a 'closed' interval such as all the numbers x so that 0 <= x <= 1, that is, the interval [0,1]. Okay.

What's so great about such a close interval? Uh, it's 'compact' so that any infinite subset has a limit point -- that is, if take infinitely many points, they MUST bunch up at least somewhere. We know that [0,1] is compact because it is closed and bounded. Also, a continuous curve on a compact set is 'uniformly' continuous, that is, is highly restricted in how fast it can 'wiggle'. So, it has to be somewhat 'well behaved'. Also, that curve has to be bounded -- can't run off to infinity, either positive or negative. Also the curve has to actually have a largest value and a smallest value.

Then, consider all such continuous curves on [0,1]: Let the 'distance' between any two be the absolute value of their maximum difference, which we now know has to exist. All those curves form an infinite dimensional vector space, and that distance is a 'norm'. Next, if a sequence of such curves appear to converge, then they really do, that is, that vector space is 'complete' (much as you learned for the real numbers) in that vector space and, thus, is a Banach space. Now you have a nice list of surprising Banach space properties -- Hahn-Banach, open mapping, closed graph, uniform boundedness.

Now you have some of the high points, intuitively, of Baby Rudin and some what makes 'sense' there.

I very much disagree with most of the advice in this thread: E.g., I read Bell's 'Men of Mathematics' and CANNOT recommend that book: (1) It won't give you the understanding you said, appropriately, you wanted. (2) It's a BIG distraction, misleading, even dangerous, from anything like math now. Polya is okay but a bit off the track. Maybe read Polya about half way through your college level studies.

Instead, for what you want, the usual courses are FINE and are:

HIGH SCHOOL:

first year algebra

plane geometry (with all the emphasis on proofs)

second year algebra

trigonometry (with all the emphasis on proving the identities)

solid geometry (with all the emphasis on proofs), optional but good if you can get it.

COLLEGE:

freshman and sophomore calculus from any of the famous books, e.g., Protter and Morrey, Thomas. Here seek to get some of the understanding you want via pictures and applications to physical science and engineering.

abstract algebra -- emphasis on proofs and a really good start on the understanding you want.

linear algebra -- emphasis on proofs.

Baby Rudin (W. Rudin, 'Principles of Mathematical Analysis') -- calculus with the proofs.

ordinary differential equations -- with Baby Rudin and linear algebra, can do well with nearly all the proofs. I like Coddington's book. Then have some fun with deterministic optimal control, A/C circuit theory, and more.

advanced calculus -- various applied topics in 'analysis', often without the proofs, e.g., Hildebrand's book long popular at MIT.

BIG Note: One of the most important applications of calculus in several variables is to Maxwell's equations. For that, you need Stokes' theorem in 1, 2, and 3 dimensions. For that, I STRONGLY recommend that you get an appropriate, OLD treatment. The best I know of is

Tom M. Apostol, 'Mathematical Analysis: A Modern Approach to Advanced Calculus', Addison-Wesley, Reading, Massachusetts, 1957.

He has a more recent book where he left out the good stuff; f'get his more recent book.

What you want from that old book is only about 30 pages. I got the whole book used -- no, I won't sell it! So, find the book in a library and make copies of the pages. Then can cover those pages in one very pleasant evening. Then will have quite well the calculus of several variables you need, and that nearly all of the physical science community STILL uses, for Maxwell's equations and more. For the math stuff with exterior algebra, etc., that's mostly for later and mostly for modern approaches to relativity theory.

Uh, Stokes' theorem done that way is just some simple pictures and a simple application of the fundamental theorem of calculus you learned in freshman calculus and proved in very fine detail in Baby Rudin.

By then you should have all you want and more.

A high school plane geometry course with all the emphasis on proofs is one of the best starts for the understanding you want.

In the books where the emphasis is on proofs, nearly all the exercises are proofs. The easy exercises you should be able to do easily enough, and as you work on the more difficult exercises you will get good at proofs. By the time you can write a good proof of, in any separable metric space, each closed set is the union of a perfect set and a set that is at most countable, you will be well on your way.

Generally, as you start to write proofs, you should have at least a semester where a good mathematician reads and corrects your proofs. The usual place for this help is in the course on abstract algebra.




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