I almost quit self-studying mathematics, but should I continue?

Question

Before I move on to the main idea of this post, I need to tell you some background information about myself. Hopefully, it proves useful for you in giving me advice. I'm a 16 year old high school student who just recently got interested in mathematics a couple of months ago. I was never interested in mathematics in elementary school and middle school. However, all of a suden in the middle of my high school journey I got interested in it.

Now back to the point.

Currently my mind is in a state of conflict! I do not know how I should self-study mathematics!

My original approach to self-studying math was to merely solve problems that caught my interest. For instance, I would search online (like this http://maaminutemath.blogspot.com/ ) or in a book for a problem that caught my attention. Then I would do my best to solve it. I would play with it like a toy. I would even try to create my own problems similar to the problems I solved. However, I realized this approach leads me to a lack of foundation. In other words, I have gaps in my knowledge!

Because of this worry, I set myself a goal to focus on filling my gaps and building foundation. I bought an Art of Problem Solving book called Introduction to Algebra because of this. After awhile on working on the problems in the book, I got bored. The problems weren't challenging or interesting. It was the usual find x and applied to some word problems. In fact ,I already did algebra before in middle school, yet I felt like I should have continued to work on the problems no matter how dull to fill my gaps. UGH! It's so fustrating.

Afterward, my mind diverted to something else. It was the Putnam problems like these:http://www.math.niu.edu/~rusin/problems-math/MP1975.html .I wanted to know how to solve them! They were very mysterious to me (and still are) because I do not know how you solve them, yet I want to solve them. So, I bought a book on how to form proofs since the Putnam problems involve prooving stuff. I wanted to know how to solve these problems, but I lacked foundation(and still do). I did not know where to start(and still do). I did work with the book I bought but eventually I reached that same experience as the Introduction to Algebra book.

For some time, I think I burnt myself out, and I quited solving problems. Currently my head is in chaos with all this overwleming information in front of me(the internet and masses of books at library), yet not knowing where to start or where to end or am I researching too much on this problem that's way above my head like the Putnam.

How do you guys self-study mathematics, especially you very experienced mathematician out there? Do you force yourself to solve dull or uninteresting or easy or unchallenging problems in order to have a firm understanding of things?

Please mathematicians with great experience I need some guidance, tips, or advice. I did ask my math teacher for advice but she said not to worry, which is not the kind of advice I was looking for.

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Addendum

I'm sorry it took me time to respond to this thread. I was sorting thoughts in my head. Now I decided to post my thoughts as it might be helpful for others.This might not be the finest addendum for this thread as it seems to me to be fragmented, but I did my best shot in writing it. Another thing, I may sound egotistic or cocky in this. But remember these are my thoughts, and my thoughts could change in the future.

First I'd like to thank you guys for responding to this question. Your guys replies are informative and useful. Nevertheless, I think there is no best answer in this thread, although I already chose one based on how agreeable it is to what I'm saying. I think it's fair to say that advice given in mathematics(or in other fields) is complicated. Why? It's because everyone is different. Some piece of advice might work for one person but at the same time might not work for the other. In the grande scheme of things, the world is too complicated.

Currently I'm ignoring all advice because I think the best advice comes from oneself. In addition, I have been keeping myself an intellectual journal. Whenever a negative emotion appears in my mind, I describe it in my journal. Then I do my best to replace that emotion with a positive emotion. In other words, I redefine how I see things.

For instance, if I approach a math problem that causes me anxiety, I would find a way to change that emotion to a positive one. Suppose I find that other emotion that I want to use to replace the negative emotion , and it's the emotion of assurance. Specifically,change my negative emotion so I can think that something good will happen. I would say to myself, "although the problem may be tough or frightening, if I do my best to solve it even though I might fail, I will gain better insight into the problem than if I didn't attempt to solve it." I know this is cheesy. But this method has worked for me for the past two days. It's practical for me.

In conclusion, this how I'm approaching mathematics in a general sense:

First of all when I do mathematics, I should just do it and approach it. Secondly, I should treat mathematics like a game. It shouldn't be extremely serious. Curiosity should be my driving force in solving problems . Plus, I should be more playful to gain that "likening/love" as a motivation with the problems even if they're boring, dull or easy. However, if the problem takes too much energy and time, I'll think of this nursery rhyme:

"For every problem under the sun, there's a solution or there's none. If there be one, think till you find it. If there be none, then never mind it." That last bit of the quote I do not take 100% serious. If I put the problem off in the side, I won't put it off forever. Instead when the time comes when I acquire the necessary knowledge(either be it a month or year), I will get back to it.

Lastly, I should ignore all given advice on how I should go about doing math(I know that sounds cocky or arrogant). But I am I and they are they. What works for me may not work for them and vice verse. Another thing, is that I found that I was too focused on getting advice in mathematics instead of actually doing mathematics. Here's a quote that can perhaps add to what I mean:

"It's one of the best theories that when people give you advice , they're really just talking to themselves in the past."- Austin Kleon

You probably got the whole point of my strategy; I'm redefining the way I approach things and getting rid of the negative emotions in my head. This strategy is what I have been doing for the past two days. As of now, this strategy is still working for me.

Again, I want to emphasize these are my thoughts, and my thoughts are pretty messy. Do not take it all too serious what I wrote as an overall character of me. Feel free to critique what I wrote as well. I just wanted to share it as it might help other people.

Answer 1

Do the things you like. If something you're doing isn't fun, stop, and do something else.

Mathematics is hard enough that if you are doing it it should be because you enjoy it. There is no point in doing it to become a mathematician if you don't enjoy it; you'd be in the position of someone who is trying to become a rabbi, but isn't Jewish, or maybe someone who is wants to become a professional violinist but doesn't like playing the violin. It's not just silly, it's insane.

If you find some topic uninteresting, skip it. Perhaps someday it will become interesting to you, perhaps because you need it to solve some problem you want to solve, and you can learn it then. Or perhaps it never grabs you; so what? You aren't going to run out of things to learn. (Or if you do run out of subjects that seem interesting, perhaps you should take that as a sign that you don't really want to be studying mathematics.)

If you find you're burned out and you don't want to do any more problems, stop and do other things for a while. I would not be surprised if you found that that after a few weeks off you discovered you were no longer burned out, and you wanted to start again. At sixteen, a few weeks or months seems like a long time. But when you look back on it, it seems a lot less important. When I was a teenager I loved programming the computer and I did it all the time. But I burned out on computer programming when I was sixteen and stopped doing it for a year and a half. Then one day it was interesting again and I went back to programming. My 18-month vacation did not prevent me from having a productive career as a programmer, or from becoming a well-known expert on computer programming. Do I look back on my life and wish I had that extra 18 months, do I wonder if I wouldn't have been more successful with 39 years of practice instead of only 37½? Of course I don't. Instead I look back and it seems that taking a year-and-a-half vacation from programming was most likely the right thing to do at the time.

But suppose you did burn out and didn't want to solve math problems ever again? What then? All I can think is "So what?" The world would have to do without your mathematical contributions. That's okay; you can find something else to contribute instead.

Answer 2

It sounds like the mathematics you've been exposed to so far has been very problem/solution driven. If that's not floating your boat at the moment, why not try some math that is theorem/proof driven? Here are a few recommendations.

1. A Book of Abstract Algebra
2. Classic Set Theory
3. How to Think Like A Mathematician
4. How to Prove It

I've read a fair chunk of books 1 and 2, and they're great! They might be a bit advanced for a sixteen year old with no background in proof, so perhaps a better starting point would be books 3 and 4. I haven't read them, but the amazon reviews speak for themselves.

Hope that helps!

Remark

I've also heard that introductory number theory books are great for people in your situation; that is, bright high school students who are looking for a bit more of a "kick" and a bit more insight. Perhaps someone else will be able to recommend something in this domain.

Answer 3

I self studied at your age. But I progressed through high school at a regular rate. Looking back, the self-study was very helpful and useful. Maybe I didn't necessarily learn things in the right order, but the insights I developed with my self-study were invaluable.

This is what I would do. I would focus less on where you want to go, and more on how to enjoy getting there. In time, assuming you major in Math at college, you will be told lots of things in the right order, and the foundations will become strong. But in the meantime, find books that you enjoy reading from the library.

Also, one thing I did (and still do at the tender age of 50) is to sometimes speed-read a book. Rather than doing each and every exercise, I try to read it and get the flavor. Maybe there are things I don't fully understand, but later - sometimes years later - I look at it again, and then it makes so much more sense.

I did have an extra advantage. My mother had taken Math at college. And she did provide me with some of the books I read in my earlier years. But later on, I started to self-select. And it sounds like you are in the same place I was at your age.

I really think you should continue to self-study. I think you should focus on enjoying it. And also, try those impossible problems (like the Putnam problems). Don't feel bad if you cannot get most of them to work out. Rather do it because it is fun.

Answer 4

Math is a truly enormous subject, and learning even one small area can take a lifetime. I'm sure this will be said in other answers, but 16 is very young. You are in high school; I am in graduate school. Yet, there is still so much that I do not know. The key is persistence. Working at learning something new or sharpening your skills every single day. Some days will be more productive than others, but it takes a lot of hard work. Especially in the beginning, where many of the problems don't seem to be motivated. You want to jump right into doing Putnam problems. We all do. However, you can't expect to hit a homerun your first time at bat. It takes hours and hours (and hours and ...) of practice. If you truly have the desire, I guarantee that, if you stick with it and are persistent, you will amass more knowledge than you can imagine right now.

As an answer to your question about how I study math: for me it depends on the subject. For algebra, I have to do a lot of problems because I don't particularly care for it (no offense to those algebraists :)). A lot of the problems are 'boring', but I have to do them if I want to get better. I set myself of goal of completing a section of exercises in a book that I have every night. Two weeks and 70 pages of problems later, guess what? I knew the material cold. Now, I will have to review periodically, but the hard work is essentially done. Analysis results, on the other hand, tend to stick with me better because that is the subject I like. Notice how you never try to memorize the lyrics to your favorite song? It works the same with any other piece of knowledge. Find something that you truly like to do, and the 'work' will seem like play.

Answer 5

Too long for a comment:

The problems weren't challenging or interesting. It was the usual find x and applied to some word problems.

$$\frac4\pi=\sum_{n=0}^\infty\left[\frac{(2n-3)!!}{(2n)!!}\right]^2\qquad\iff\qquad C_{2n}^n=\sum_{k=0}^n\left(C_n^k\right)^2$$

Personally, I adore calculus (analysis), and abhor combinatorics... and yet, the first formula above, related to the topic I cherish so much, is nothing more than a direct implication of the second one, belonging to the area of mathematics I so utterly and sincerely despise. Ironic, isn't it? Life is full of such little ironies, and has a very sick, dark, and twisted sense of humor. More to the point, the x you mentioned might change its nature, and the formerly dull and boring formula might take on new depths and completely unexpected meanings.

Take for instance the notation above. It refers to combinations of n taken by k, and its usual meaning refers to how many distinct subgroups of k objects can be taken from a larger set of n elements. Obviously, given this understanding, n and k cannot be anything other than plain natural numbers, correct? Expressions such as $C_1^\frac12$ seem to border on the absurd, defying all sense and logic, wouldn't you agree? And even if one were to play such a silly game, of trying to pointlessly extend the natural meaning, intent, purpose, or definition of combinations to non-natural arguments, how on earth could all this be related to π, the famous constant that defines the circle?

I mean, when you say combinatorics, you think of probabilities, which, for the most part, are characterized by the Gaussian normal distribution, $\int_{-\infty}^\infty e^{-x^2}dx$ which obviously has nothing to do with pies and circles, am I right? I mean — you know — apart from the fact that its value happens to be $\sqrt\pi$, and the exponent inside the integral bears some very faint resemblance to the algebraic equation of the circle, $x^2+y^2=r^2$. But that's probably just one of those weird coincidences... Right?

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It was the Putnam problems like these. I wanted to know how to solve them! They were very mysterious to me (and still are) because I do not know how you solve them, yet I want to solve them.

Try and prove the first formula above without any knowledge of the second one. See how far you get. Stay on this course, and I can guarantee you that all of mathematics will continue to be very mysterious for you, until the end of your life.

Now, can you tell me the volume determined by the intersection of two cylinders? Or the value of the area trapped between the graphic of the function $f(x)=e^{-\sqrt[n]x}$ and the horizontal axis? Perhaps something a bit easier, like finding the solution to $x^3+y^3=z^3$, with $x,y,z\in\mathbb{N}^*$? Or maybe compute the value of $a^b$, where $a=\pi+i\cdot\ln(-1)$ and $b=\ln2+\sum_1^\infty\frac{(-1)^n}n$? How about $\tan\left(\sum_0^\infty\frac{(-1)^n}{2n+1}\right)$?

If anything I wrote here spikes your interest, know that the only way to be able to finally taste the rewarding sweetness of mathematics is by eating its veggies first. As far as books are concerned, I'd like to recommend to you Charles W. Trigg : Mathematical Quickies: 270 Stimulating Problems With Solutions for its beauty, elegance, ingenuity, and simplicity.

Answer 6

You always have to go through the"boring" stuff in order to get to the really interesting mathematics. That being said, the boring stuff is of extreme importance to truly understand. Even in basic algebra there is some amazing things. Read about irrational numbers, read about the intermediate value theorem, read about the history of algebra. Remember that this basic math was not basic at some point. Learn about where the Pythagorean theorem comes from. Believe me, in the "trivial" math there is plenty of amazing stuff.

Answer 7

Buy some of the "classic" math texts and work through them, making sure you're completely justified in all the moves you're making. Write up formal proofs, show them to friends or post them here for comments/criticism. It may seem silly at low levels, but when you progress to higher mathematics, you'll be glad you have a solid foundation. One majorly important aspect of being a good mathematician is being able to effectively express what you mean to say. As well as present proofs and provide expositions in a swift manner. In essence, learn the language of mathematics.

As for specific books, I would suggest Spivak's Calculus and Axler's Linear Algebra Done Right. Many people suggest Velleman's How to Prove It as an introduction to writing proofs, but I've never used that book myself, so I cannot personally vouch for it. Work through them and do all of the exercises, especially from Spivak. It will take time, and you will be busy with problems, but you will advance your mathematical maturity tremendously in doing so. Moreover, you will be prepared to begin learning more sophisticated mathematics, as well as solve and create many interesting problems. Not to mention, you'll also be well ahead of the game for college-level mathematics.

I got a late start on mathematics, so I try to encourage any younger people who have an interest to really pursue it. Mathematical maturity takes time, and the more effort you put into the more results you'll see. That is, if you study well and use your time to work effectively. Study hard, but most of all, enjoy it. You'll never get far if you don't enjoy what you're doing.

Answer 8

I suggest that you read the answers to this question Advice for Self-Study that was asked by someone who is about the same age as you. His situation is not identical, but the answers are applicable.

You did the right thing, by asking your teacher first. But what did you ask her? If you expressed those feelings, of being overwhelmed with the vastness of mathematics, she was probably correct in saying not to worry. Did you ask her for advice about what you can do to pursue your interests in mathematics, beyond the classroom? Try that.

If that is not helpful, ask your guidance counselor at school, if you are in the U.S.A. and attend a public school. He or she may be able to arrange something for you, that is beyond anything you'd imagine. That's what happened to me. My guidance counselor called up someone at the university that he knew, who had a PhD in mathematics from the University of Chicago. He asked me if I liked chess. I didn't. He asked me, then what about contract bridge or other card games? I learned all sorts of things about statistics and probability that summer.

• Thank you, Professor Fred Richman of New Mexico State University.

Answer 9

If you're in high school and really enjoy your solving harder problems, I can absolutely identify with you. My solution was to (rather coincidentally) get involved with math olympiad problems. These are very challenging, don't require unreasonable amounts of background knowledge, and you get to meet (other) very smart people.

If you're from the US You can check out http://www.maa.org/programs/students/student-resources/high-school Nearly every other country has something similar.

Answer 10

After a lifetime of doing mostly other things (I'm 51) I've been engaged in a somewhat systematic self-study of math (mostly geometry, but also including topology and number theory and history of math) for the past several years. I'm doing this entirely on my own (i.e. I'm not in any kind of course). The key to me is to find high-quality texts that really resonate with me (list I'm currently working with is at the end of this post). As an addendum, since originally posting this, I wound up taking 6 undergraduate math courses at the U. Colorado / Boulder, which was very fun and helpful in my self-study; even though I am now back to self-study again, I feel much stronger as a result of going through a set of coursework, and much more able to tackle what is emerging as my main math goal, which is to read classic works in their completeness (e.g. disuisitiones arithmeticae, the elements, conics, original writings of galois, abel, etc.). Nonetheless I still feel the same way regarding my comments below about coursework, despite its usefulness!

There is a liberating aspect to self-study: You are less tied to a particular text (during my time as a math student I found that in a course there is a tendency to be completely fixated on the text that the course is built around). You can also slow down (or speed up) as needed without being forcibly pegged to the pace of the course (I often felt that courses try to cover so much material that they are more surveys than in-depth treatments).

I spent some time searching and examined a considerable number of books, looking for ones that really clicked. When I find a well-written book I almost feel like I am communing with the author(s), like I can anticipate the next sentence before I read it. I can also interleave different books together so that I simultaneously get different authors' treatments of the same subject, or get one author's background material for a prerequisite while also reading the text for which I need the background (going back and forth as necessary).

As a self-studier, I prefer books that have hints or answers, and as someone interested in geometry, I like lots of diagrams. While almost without exception the books I'm studying are what would be considered introductory material I'm surprised at how deep these books can go if one turns over every stone in them.

I've amassed a lot of books (they're mostly quite inexpensive if one shops astutely, so I've gotten quite a lot more than I will ever get through - I have no problem with that). These are mostly unordered, except the ones at the top tend to be the ones I'm more actively focused on (i.e. mostly more elementary), and some of the ones at the bottom I haven't even looked at yet. Many I know are way beyond me and always will be, but I still like to look at them. Some are more curiosities than anything else. If nothing else I've rescued a lot of classic old texts that someone else will maybe appreciate someday. I can't vouch for the quality of all of these, but most are classic or historical and hence probably are pretty good since they've stood the test of time. I only started buying them within the last five years or so, it's amazing what's out there if you look for it! A cliff note to my list though could be, if Coxeter wrote it it can't be a bad place to start...

$$\text{Calculus and Analytic Geometry (Thomas).}$$ $$\text{Geometry Revisited (Coxeter and Greitzer).}$$ $$\text{Projective Geometry (Coxeter).}$$ $$\text{Euclid's Elements (Green Lion edition).}$$ $$\text{Introduction to the Theory of Numbers (Hardy and Wright).}$$ $$\text{Introduction to Algorithms (Cormen, Leiserson, Rivest).}$$ $$\text{Geometric Transformations I, II, III (Yaglom).}$$ $$\text{Elementary Geometry from an Advanced Standpoint (Moise).}$$ $$\text{Linear Algebra and its Applications (Strang).}$$ $$\text{Statistics (Freedman, Pisani, Purves).}$$ $$\text{A First Course in Probability (Ross).}$$ $$\text{Continued Fractions (Olds).}$$ $$\text{Introduction to Geometry (Coxeter).}$$ $$\text{Sequel to the Elements of Euclid (Casey).}$$ $$\text{Geometry and the Imagination (Hilbert) - Also Anschauliche Geometrie German edition.}$$ $$\text{Elementary Mathematics from an Advanced Standpoint (Klein).}$$ $$\text{Diophantische Approximationen (Minkowski - German edition).}$$ $$\text{Die Geometrie der Zahlen (Minkowski - German edition).}$$ $$\text{Episodes from Early Math (Aaboe).}$$ $$\text{Non-Euclidean Geometry (Coxeter).}$$ $$\text{Conics (Appolonius).}$$ $$\text{Modern Geometries (Henle).}$$ $$\text{The Enjoyment of Mathematics (Rademacher and Toeplitz).}$$ $$\text{Irrational Numbers (Niven).}$$ $$\text{Inequalities (Hardy)}.$$ $$\text{Patterns of Plausible Inference I, II (Polya).}$$ $$\text{Projective Geometry I, II (Veblen and Young).}$$ $$\text{Directional Calculus (Hyde).}$$ $$\text{First Six Books of Elements of Eudlid (Byrne).}$$ $$\text{Arithmetische Untersuchungen (Gauss - German edition).}$$ $$\text{Disquisitiones Arithmeticae (Gauss - English translation).}$$ $$\text{A Course of Geometry for Colleges and Universities (Pedoe)}.$$ $$\text{Regular Complex Polytopes (Coxeter).}$$ $$\text{A First Course in Abstract Algebra (Fraleigh).}$$ $$\text{Algebra (VanDerWaerden)}$$ $$\text{Engineering Drawing (French).}$$ $$\text{What is Mathematics (Courant and Robbins).}$$

And too many others to list ... but any one or two of these is likely to lead you to your own list. The good news is that after awhile, you will have enough that you will be able to stop before you have recreated the national library...

Answer 11

First of all you start with the 96 page book: Halmos's Naive set thoery. This requires almost nothing as prerequisite, even counting, but an inquisitive mind. I think, one year should be at your disposal. After that you will need no more guidance.

Answer 12

My personal opinion is that you should try relatively more abstract stuff now. Topics in Algbera by I.N.Herstein is what comes to my mind.