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Im a junior in high school, since I was young I've had a profound interest in Math, and I'd always looked forward to high school mathematics. Little did I know, Special education had taken away what thought I'd be entitled to- A quality education. I had once been surrounded by brilliant minds that were far more capable than I was, and the rivalry between us had motivated us to get ahead of the class. I came from a private school in which the average student was 2-3 years ahead of any public school curriculum. Long story short, I'm in a public school with Adolescents that abhor learning, on a daily basis, they have the audacity to disparage their education. Though there are individuals who share a similar passion to my own, they are few in number. my high school years are drawing to an end, and I feel that its in my own hands to educate myself in the beauty that is math. In every class I understand the subject before the teacher can finish stating it, I learn much faster and retain much more info than my peers. They take notes - I don't, because I feel that learning comes from paying attention to whats being taught/conveyed to us during class.

My question is how would I teach myself? Only recently have I found out that there are holes in my knowledge due to my time in special education. I want to teach myself algebra 2 from the ground up, but where would I start? Every book I can get a hold of offers problems rather than concise explanations. I want to find a book, and price isnt an issue, that can teach me EVERYTHING about algebra 2. Afterwords, I'd like to know If I'd be able to do the same for pre-calculus and Calc itself. I feel that with dedicated study (2-3 hours a day) I may be able to finish calc by July/August... Maybe even sooner. But the books I have don't explain WHY you do things, just how. There are multiple occasions in which I question something I dont understand, but there is no explanation. The books I have rely on being active in class. I want to teach myself these simple mathematics, so if possible, please point me in the direction of a book that can convey algebra 2 from the ground up (not that I don't HAVE knowledge of the subject, It's just filled with holes of which I can't locate. I dont know what Im supposed to know, but I dont know WHAT IT IS im supposed to know. Its like finding a needle in the wrong haystack)

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Something that I would keep in mind is that there are different philosophies about how one does mathematics depending on whether or not you are looking for applications in science/engineering/economics, OR if you want to study mathematics from a proof-based approach (which is how mathematicians typically approach the subject). It sounds like you want to understand why certain things are true as opposed to how to carry out computations. In order to do this (at least for calculus) you're going to want to study a subject called real analysis, which is essentially a proof-based approach to understanding calculus except that you look at topics from a much broader perspective. To learn this from self-study may honestly be an up-hill climb, but certainly not impossible depending on your interest and dedication. An important part of it is learning how to write mathematically. For a good gentle introduction to proof-writing and analysis I would recommend this this book. There are other good books on calculus which focus on a computational approach (I definitely would recommend both approaches to the subject), a great free one is by the famous MIT mathematician Gilbert Strang.

In terms of learning algebra II, I would just jump ahead in the book you're in. It's really not a subject that is understood from a proof-based mathematical approach (there is a subject called abstract algebra, but I really doubt this is what you want to learn right now; it's quite different than anything one typically encounters in high school, though I do know a good introductory book for it).

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  • $\begingroup$ +1 for the application vs proof based approach. But it could be the case that analysis is too hard for him since he might not have background about concepts like set, logic and functions and an analysis book might not be sufficient for proving such a background. Therefore, I think a self learner can start from the foundation here: amazon.com/Foundations-Mathematics-Ian-Stewart/dp/019870643X $\endgroup$ – Li Chun Min Apr 29 '17 at 3:24
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First, here's a book recommendation. Try Precalculus by Cynthia Young. It is very thorough for a high school level textbook. Its proofs, especially in the sections on analytic trigonometry, are lucid and instructive. Of course there are many exercises, but those are an important part of learning as well. Reading this text carefully ought to give you an excellent foundation for further mathematics.

Personally, I would not venture into textbooks dedicated to calculus just yet. Books with real analysis and abstract anything will likely be beyond your level, if your education has been anything similar to that of the students I tutor.

It seems that you are rather inquisitive. If reading is not enough, you may want to consider hiring a private tutor and devising a curriculum of chapters to read and be lectured on. Alternatively, you can look to see if any local colleges are offering courses at your level. Email a professor who is teaching an appropriate course and ask whether it is possible to unofficially audit his or her class. Many lecturers are cool with an eager student sitting in the class, and as a student, you'd be receiving free lectures!

And as a final note, be wary about your attitude for note-taking and exercises. I get it. I've been there. I've met many precocious students who are or have been there. At some point, though, the material will become demanding, and hopefully by that point you will develop organized note-taking and studying habits. So while you're fine for now, seek to maximize your learning in all possible ways. Seek to cultivate your study habits sooner than later.


I did see that you want to start with Algebra 2. Most of those topics ought to be revisited in the Precalculus textbook I've written about. If my recommendation seems too advanced, then please comment on the post. Truth be told, though, there's no section of mathematics that's formally defined as algebra 2 or precalculus. They're just course names. I think this book will give you a good mix of theory and practice.

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  • $\begingroup$ First I'd like to thank you for such a detailed answer, and I sincerely appreciate your advice. I understand how audacious my attitude sounds towards my curriculum, but that's only because Im tired of having this "easy a" system... I've yet to be academically challenged in years. I'll take a look at this book you've recommended, and If I can hire a tutor of some sort. I'd also like to ask, if this website allows you to reply to this comment, How difficult would Proof analysis be..? Keep in mind that Im still a junior in high school with a basic geometry/algebra education. $\endgroup$ – Ethan Singer Oct 18 '15 at 2:16
  • $\begingroup$ An introductory class about proof and exploration should be accessible. It might see different at first, but I do not think you'll be needing many advanced algebraic techniques. I don't have any book recommendations, though.. Did your geometry class involve proof? If not, you could look for a rigorous text on Euclidean geometry. $\endgroup$ – zahbaz Oct 18 '15 at 2:22
  • $\begingroup$ Also, it's not audacious at all. I completely get it. I'd be frustrated too. Could you do me a favor, though, and unmark my answer for the time being? If you unmark it, others might give you responses in the next few days as well. If you thought my post was helpful, an upvote is a good way to show it. $\endgroup$ – zahbaz Oct 18 '15 at 2:24
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The other answers are very good. I would like to offer more resources which may give you more latitude. In particular:

The OpenStax project is an excellent resource.

https://openstax.org/subjects/math

The PDF's for the books are Free, DRM free, and they come with links to videos which provide you with worked examples and explanations.

You should work out example problems, post them here, or other places, and get feedback as much as possible. Doing math in a bubble is the worst thing you can do. Make sure you work on communication, and logical deduction most of all.

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