<p>I am a high school junior and I have taken Geometry Honors, Algebra 2 Honors, Pre-calculus Honors, AP stats and AP Calculus AB. My high school has no more math classes to offer me. I thoroughly enjoy math and would like to pursue it in college so i obviously want to take an advanced class senior year. What are my options and what would look the best for colleges especially MIT?</p>
<p>I could either take classes at a community college, online classes, or self study. (These are the only options i am aware of) so my question to you is what option would look the best on an application? Is there any other options that i was unaware of? and what class should i study/take (calculus BC, calculus 3 etc.)</p>
<p>Thanks in advance for your responses!!! (i have also posted this in other forums)</p>
<p>Any of those three options would reflect well on you, and there isn’t one that’s automatically preferable over the others. </p>
<p>Which option would be best for you depends on the options available to you and on your personal preferences – with self-study or an online class, would you be able to pace yourself and keep up with the course, even when you’re busy with other things? A community college course (or a well-structured online course) might provide you more structure, which might or might not be helpful for you. Or, if your local community college involves a commute, taking an online course or self-studying might be more efficient. </p>
<p>If you’re interested in attending MIT, it will be very useful to you to take Calc BC-level calculus. Multivariable calculus or differential equations would also be very useful.</p>
<p>Obviously this will be depend on how convenient taking classes at the community college is and whether anyone at your school would be willing to supervise your self-study. I think Calc BC should be your first priority. After that it depends on how sure you are about doing math versus other science subjects. If you are sure about going into math I would recommend some kind of discrete math. Ideally, you can do something proof-oriented although that would probably require someone helping you learn how to write proofs. Another option is math competitions. I generally think math competitions are not ideal but they have the significant advantage of being conducive to independent study. If you might go into other science subjects calc III or linear algebra is probably the logical choice after Calc BC.</p>
<ol>
<li>Real Analysis - learning calculus in the rigorous way </li>
<li>Multivariable calculus/linear algebra (community college)</li>
<li>Competition math (AMC, AIME, USAMO, Putnam <- only requires #1 to score half-decently, ARML, etc)</li>
</ol>
<p>I don’t think you have enough time to prepare for competition math so I’d go with real analysis if I were you. Check out “Principles of Mathematical Analysis” by Rudin and subsequently “Complex Analysis” by Alfhors. Since it sounds like you plan to pursue math in college, these two books are the classics in the respective subjects. I’d estimate that each would take about half a year to learn (or one summer).</p>
<p>Possible plan:
Summer: Summer program + Real Analysis (Rudin develops linear algebra and multivariable calculus so you’ll learn a bit here)
Fall: Complex Analysis (I learned most of my multivariable calculus from complex analysis so… yeah)
Spring: Topology by Munkres (Topology is used in almost every field of math)
Summer: Differential Manifolds by Lee or Spivak/Modern Algebra by Dummit and Foote (manifolds are an extention of the previous three ‘courses’)
Dummit has linear algebra and a bit of algebraic topology so yeah, you’d be covering the basic undergrad curriculum here + a bit of graduate-level stuff: MV calculus, Linear algebra, real analysis, topology, modern algebra.
Fall '14: College! Take the honors courses/graduate courses at whatever school you go to!</p>
<p>Of course, the above plan assumes that you want to study math in college. If you start studying in the spring and go through all of the above very quickly, then start going into homology and Lie Groups/differential equations (though diff eqs is more applied math) and whatever else you’re interested in. There are way too many math classes to take and not enough time to take them all (as I have quickly found out…)</p>
<p>Although, I love real analysis and Rudin’s book it does require a decent amount of mathematical maturity and familiarity with proofs. The OP stated that AP Calc AB was the most advanced math class he or she has taken. Unless the OP is particularly brilliant or already proficient with proofs reading Rudin’s Principles of Mathematical Analysis is likely to be a waste of time.</p>
<p>Also I don’t think Rudin is a good book for self-study as Rudin is very terse and doesn’t provide a lot of motivation. In a class, lectures can provide that stuff but in self-study it’s more important for the proofs to be more detailed and motivated.</p>
<p>Eh, if OP goes to any top-50 college and takes an honors calculus class, he/she will encounter either Rudin/Apostol/Carothers/Spivak/some much more advanced book written by the professor. I see no reason why he/she would be more prepared for a Rudin-level book after taking Calc BC than after taking Calc AB =p</p>
<p>Many students in this honors sequence have only taken Calc AB/BC and nothing further. Many other students have done exactly what I have told OP to do. Regardless, exposure to real analysis and all of the ‘basic’ subjects is important if the OP truly intends to study math.</p>
<p>Rudin is not really comparable to Spivak/Apostol. It is considerably more abstract [more topology, includes Lebesgue integration, etc.]. I suspect that most of the students in Stanford’s math 51-3H series have considerable exposure to proofs. Also the students in Stanford’s math 51-3H are likely to be rather exceptional being among the best math students at a top 5 university. It is possible that the OP is quite gifted and would succeed but also very possible that OP is not quite as gifted/prepared and would struggle. There is also quite a bit of difference in taking a class and self-study. In a class you have classmates and professors to help while in self-study you or more less on your own. Although any serious math major will eventually want to learn those topics they don’t have to do it in high school.</p>