I am a high school student with two years left until I graduate. I managed to move through my high school math curriculum quicker than expected and am on track to take AP Calculus AB this coming school year. I talked to my guidance counselor and the school's calculus teacher and they both agreed it wouldn't be necessary for me to take Calc BC due to the similarities to AB. So I may selfstudy just to take the AP exam. It's only recently that I realized that after this school year, I won't have any math classes left to take and I'm not comfortable with having a math "gapyear" for the rest of high school. Are there good options for math classes going forward? I'm not sure where else to ask since I don't have access to a guidance counselor over the summer.

1Perhaps helpful: https://math.stackexchange.com/questions/1714966/whatbooksshouldigettoselfstudybeyondcalculusforsomeoneabouttostart/1714986#1714986 – Ethan Bolker Jul 11 '18 at 16:40

5Some high schools have programs allowing you to take college courses in math if your school doesn't have any higherlevel math courses left for you. You might ask about this possibility. – Jair Taylor Jul 11 '18 at 17:16

This also definitely seems worth pursuing as an option, I just don't know how feasible it will be moneywise. – CaptainAmerica16 Jul 11 '18 at 17:29

Rather than racing onward through the standard curriculum, there is a lot to be said for developing problem solving ability and learning to prove things. Check out artofproblemsolving.com (but don't feel obligated to do contest math). Look into attending a math summer program. Consider working through a book on number theory or reading Spivak's book Calculus. In a more applied direction, you could take a mooc on machine learning and try to understand the vector calculus and linear algebra you need for that. But most of all, learn whatever you think seems exciting. – littleO Jul 11 '18 at 20:47

1Check your state laws. Growing up 30 years ago in Wisconsin, I "ran out" of math courses to take in high school. The high school was compelled (as I was) to have a math course for the year. They paid tuition at a local college and I received college credit for the math course I took there. Your state of residence may be important to answering this question. – Keeta  reinstate Monica Jul 11 '18 at 20:53
6 Answers
CaptainAmerica16, it's great that you've asked for advice on this.
The first thing I would say is that if at all possible, you should ask if your school can set you up with a mentor, such as a math professor at the college or university level. That may or may not be possible, but if it is, you'll benefit from advice tailored to your situation from someone who knows you.
The second thing is to be aware that the normal progression in math in the North American system is not well suited to the highest achievers. That is, a typical curriculum goes through calculus, multivariable calculus and linear algebra, perhaps even more, with no serious focus on proofs and mostly routine problems. Typically, people are in their junior year of university before they get a taste of what math is really like, when they take abstract algebra or analysis.
This may be a good way to do things for those who need to become competent in some of the routine calculations of calculus for use in other fields (although that is debatable), but it is certainly not right for young people who show promise in math.
For talented highschoolers, the thing to do is to focus first on developing an understanding of proofs and the habit of working on harder problems at the precalculus level, and then to study singlevariable calculus in a fully rigorous way and go from there. The fact that those kinds of students usually just get plain acceleration rather than a highly enriched curriculum results not from this arrangement being better for the students educationally, but from it being simpler on an institutional level.
For rigorous calculus, I would recommend either Spivak's Calculus or Apostol's Calculus, Vol. 1. Spivak's book has a solution manual. It may be the case that you are already ready for this level of math  you can see by starting to read one of these books. If not, adequate preparation would be furnished by working through some of the following (short) books aimed at high schoolers.
The Method of Coordinates, Functions and Graphs and Algebra, by Gelfand.
Numbers: Rational and Irrational and Mathematics of Choice, by Niven.
An Introduction to Inequalities, by Bellman and Beckenbach.
Mathematical Circles, by Fomin, Genkin and Itenberg.
I haven't listed any books on geometry here. The books by Gelfand cover central parts of the precalculus curriculum in a rigorous way. The book on inequalities gives you practice in an area that is often neglected in high school but is crucial in rigorous calculus. The book by Fomin has interesting problems in it. The books by Niven are on number theory and combinatorics. You may find them to be of great interest even if you are already at the level to start rigorous calculus.
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If you are interested in math or a science that uses math you could do self study; however, you could also enroll part time in a community college ( if that's a option) and then take multivariable calculus, linear algebra, and differential equations. But I really recommend checking out some books on multivariable calculus, linear algebra, proofs, and first year university physics to see whats out there. Once you've done that you can really get some books and start self studying. I recommend at the beginning to read Div, Grad, Curl, and All that: An Informal Text on Vector Calculus (its an informal text so no big proofs and introduces some physics to see if you like that) and How to Prove It: A Structured Approach (To see if you like and can get through writing proofs). I don't know anything about ap calculus (as i dropped out and went to college in a very similar situation to yours) but if you think it might not have covered enough you can always use mit ocw to review topics and see some new ones. (They have nice video lectures on single variable and multivariable calculus)
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I will definitely check out mit ocw. I've heard about it, but I didn't realize there were things on there I would be able to comprehend, ha. Thank you for showing me a new source :) – CaptainAmerica16 Jul 11 '18 at 17:25
Good for you for wanting to stick with math! I was in a very similar position to yours at the end of my sophomore year, and I was lucky enough to have a college right across the street from my high school, but lacking that, I'd be more than happy to point you towards some helpful resources.
The natural continuation after AP Calculus AB is Calculus II, which is (roughly speaking) the last third or half of AP Calculus BC. It's sometimes taught as an entirely separate course at universities, and if you want to go that direction, I'd strongly recommend James Stewart's Single Variable Calculus: Early Transcendentals. After that, you can move on to Calculus III, or multivariable calculus, for which I'd recommend Jon Rogawski's Calculus: Early Transcendentals. Both of those textbooks are available for purchase online, and if you look hard enough, I'm sure you could find pdf's of them, too.
Alternatively, if you're tired of calculus, you can explore a different field of mathematics, like linear algebra. It's got a very different flavor than calculus, but its applications are too numerous to list here (see the answers to this question for what I mean by that). Jeffrey Holt's Linear Algebra with Applications is a really great way to approach the subject. If you do try linear algebra and find it too weird or too confusing, just try a different subject for a while and come back to it. It's well worth learning, but you've got plenty of time. By the way, all these textbooks I've been mentioning are textbooks I used and liked, so trust me when I say they're wellwritten.
And of course, there are other options, too. You could try number theory, like Ethan Bolker suggested in the answer he linked to, or differential equations, or analysis (though I would strongly recommend a bit more calculus before you tackle analysis), or something else entirely. As long as you're interested in learning more math, there'll be more math for you to learn. And I'd be more than happy to offer more suggestions if you'd like!
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Thank you for this! I looked at the link and I think Linear Algebra definitely seems worth checking out. Especially since I have a strong interest in Computer Science as well. I'm always looking for more suggestions, so I would appreciate anything else you can point me to! – CaptainAmerica16 Jul 11 '18 at 17:23

2Some other possibilities that come to mind  "discrete mathematics" (e.g. combinatorics, graph theory), abstract algebra (which encompasses the mentioned number theory), complex analysis (as opposed to the real analysis mentioned in the answer, a beginning course in complex analysis could be done without too much of the advanced $\epsilon$$\delta$ type arguments that are the bread and butter of real analysis). – Daniel Schepler Jul 11 '18 at 17:24

1Analysis sounds interesting, I didn't know it could be accessible at my level. – CaptainAmerica16 Jul 11 '18 at 17:27

1@Robert Howard Just an update...I was able to get in touch with my school and they're currently looking into me taking Multivariate Calculus at a local university! I've also started studying linear algebra for fun since I was able to get a grasp on multiple calculus concepts since I asked this question. Thank you for all of your suggestions! – CaptainAmerica16 Aug 04 '18 at 04:02

1@CaptainAmerica16 That's great! I really hope that works out! I've read some of the other questions you've posted here and I can really tell you have a passion for math, so I'm sure you're going to love multivariate calc. What do you think of linear algebra so far? – Robert Howard Aug 04 '18 at 04:08

1@Robert Howard My favorite part is the geometric representations of vectors. I literally made a 3D model of R^3 space as soon as I saw it. I'm really into geometry so the visualizing part is fun. – CaptainAmerica16 Aug 04 '18 at 04:21

1@CaptainAmerica16 I know exactly what you mean; I loved that part of linear algebra too. I've always especially enjoyed anything in math that I can visualize clearly, so all the ideas from linear algebra that deal with vectors and how you can apply any transformation you can imagine to them with the right matrices still fascinate me. – Robert Howard Aug 04 '18 at 04:49

1That's so exciting. The more I progress through math, the more I see it's not just the plugandchug formulas that I'm used to. It's kind of arty I guess. – CaptainAmerica16 Aug 04 '18 at 04:58

1I would certainly agree that math is like an art. It's intricate, beautiful, occasionally infuriating (which makes the beauty all the more satisfying), and rewarding all at the same time. – Robert Howard Aug 04 '18 at 05:01
I like Robert's answer, and I see you've accepted it but I'll add a couple remarks. When I was in high school, I had a similar issue (I knew I was going to be a math major also), and I wanted to continue taking math classes that would help me place further when I went to university. My community college only went up to differential equations, and I was unable to study there for a semester due to musical commitments. There are online mathematics courses you can take (they aren't great at all, but at least you get college credit), but I highly recommend you continue self studying.
You don't mention what you want to study in university, or if you even have any idea (this is totally fine, most people don't), but if you find yourself interested in mathematics, there are lots of summer programs and resources you can use to get ahead. If you find yourself enjoying deeper mathematics after calculus and multivariable calculus, linear algebra, etc., try real analysis, abstract algebra, number theory, and basic topology, and if this peaks your interest, great! Keep going. Otherwise find something else you really love, you have a great (scholarly) start already.
Good luck!
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Someone else mentioned taking classes at a community college and I think that may be worth pursuing. I'm really interested in computer science and up until recently, I was certain that would be my major now I'm also considering majoring in math or maybe even double majoring (stress!). – CaptainAmerica16 Jul 11 '18 at 19:26

Right now, I would worry less about exactly what your college career path will be and instead expose yourself to mathematics and computer science that you _really_ like. Getting ahead will be helpful and important, but nothing will serve you better than finding something you're truly passionate about. Another answer suggested finding a mentor. If you're fortunate enough to find one, this will also be extremely beneficial. – Nico Jul 11 '18 at 19:32
This happened to me in high school too. I was advised to take a "math gap year", which I did. To this day I think it was a mistake, so .. avoid it if you can! A book I enjoyed recently and recommend is John Derbyshire's "Unknown Quantity: A Real and Imaginary history of Algebra." It really lays out a wide range of math: I wish I had had it way back then.
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I know several people who had a similar experience at my high school Calculus AB junior year, what do we do for the last year? We had a variety of different things we all did. I took statistics my senior year still math, but a break from the high theory of calculus. This seemed to be the most common thing to do. A couple of other students managed to take their math class at a local university instead. And lastly several others took a break senior year and didn't take math at all. I don't think any of these options are terrible.
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