Should a High School Junior Know Proofs Before College?

[zcd]

Hi physicsforums, this is my first time posting here, but I've lurked here and there before. While reading over a lot of the threads, I noticed how some people advised to not rush the learning process and to take things as they come, but at the same time I see a lot of other high school students probing deeply into very high level math and physics like PDE's or quantum theory. My question is: if I want to major in physics during college and eventually move onto graduate school to pursue a PhD in physics, where on the spectrum should I be at right now?

I wouldn't describe my mathematics background as shallow, because I'm currently excelling in my school's AP Calculus BC course. But at the same time I have to admit that I find mathematics here less than satisfying, because there has been almost no emphasis placed on proofing since geometry back in middle school. Should I be very knowledgeable of how to proof by the time I enter college? If so, where can I find resources to build this skill? I have checked my local library, but none of the mathematics books there help in this specific area.

Thank you for your time.

 

[eri]

At your age, I was taking pre-calc and honors physics. Now I'm finishing a PhD in physics. You're fine. Believe me, those kids you're talking about are mostly talk - they think they're doing a lot more than they really are, or understanding a lot more than they're really getting.

 

[Chewy0087]

I'm sort of in the same boat as you, but atm I'm just reading into things that really interest me (like relativity, some optics) and if you have the right book a lot of the stuff is made clear to you, and learning the other stuff is no problem because it's exciting :P (well at least that's how I find it), although my further pure math course will start in a month or so (not sure of the US equivalent), and just looking at the spec. is scary

 

[djeitnstine]

You should be fine. Depending on your program, you won't even need proofing skills until second year or third year university.

 

[Astronuc]

I did Calculus BC as a senior in high school. We covered some proofs, but not as much as I would have liked. Nevertheless, I think one is at the right point with Calc BC.

If one can find a some introductory books on elementary analysis and linear algebra, that might be a good complement to Calc BC. I've found quite a disparity in high school programs across the country, and I've noticed some students have an advantage in their early years, based on their experience in high school.

One might check the texts in this thread - https://www.physicsforums.com/showthread.php?t=310681

 

[mbisCool]

So long as you can start taking calculus and physics shortly after you enter college you should be fine. My high school only offered about 5 AP classes in total. No physics or calculus, so that's where I started as a freshman.

 

[Pengwuino]

I agree with a previous poster, there's almost no chance that they're seriously doing quantum mechanics or partial differential equations with any real rigor or actual understanding.

One thing I think a lot of high schoolers don't understand is that your education is not a race. There's no cut-off for how long you have to get your PHD (although don't take 20 years!). You can start college barely out of algebra 2 and 1 physics course and eventually get a phd in physics. At my university, for example, people change majors on average, 3 times during their college career so being "behind" is somewhat nonsensical. Study and advance at your own pace. The only real dangers is if you go way too fast and you just burn out and become uninterested, or you are one of those 6-8 year undergraduates that just can't finish for whatever reason. However, that "whatever reason" is never because you had to start at a lower math course because at the most, you're 2 or 3 math courses behind the average college freshman in the sciences and that can be made up easily.

 

[jtbell]

Keep in mind that the introductory two-semester calculus-based physics course that college freshmen physics majors take, generally has calculus 1 and 2 as co-requisites, not pre-requisites. There are actually people who arrive in college in the USA without having studied calculus in high school, who successfully complete a bachelor's degree in physics. Maybe not at Ivy League schools, though...

 

[Howers]

You should find this very useful:

https://www.physicsforums.com/showthread.php?t=307797

 

[zcd]

Thanks everyone for the replies. I should have a fairly busy summer now

If I may ask a few more questions:

Is a goal of being a physicist practical? With today's recession and everything I'm worried that my opportunities will be very limited. After reading http://docs.google.com/View?docid=df5w5j9q_5gj6wmt aboutt the graduate school level, it also seems as if the qualifying exams are very selective. If this is the case, should I consider having a second major in an engineering (probably electrical) as backup?

My main target colleges are going to be UC Berkeley and UC Santa Barbara (more or less guaranteed by ELC program), but I will also try for Caltech and MIT. From the UCB website, I've learned that they only offer BA for physics; will having a BA instead of a BS negatively impact my graduate school application? Also, how good is the engineering program at UCSB in case I want to double major?

Thanks again

 

[jtbell]

From the UCB website, I've learned that they only offer BA for physics; will having a BA instead of a BS negatively impact my graduate school application?

No. Certainly not from Berkeley!

Note that Harvard also offers only the BA.

 

[Pengwuino]

With today's recession and everything I'm worried that my opportunities will be very limited. After reading http://docs.google.com/View?docid=df5w5j9q_5gj6wmt aboutt the graduate school level, it also seems as if the qualifying exams are very selective. If this is the case, should I consider having a second major in an engineering (probably electrical) as backup?

You'll be getting your bachelors at the least, in 4 years, economic conditions change. Of course, no field is immune to economic downturns.

 

[qanx]

I didn't read all the responses in depth, but I can tell you about what I did when I was your age. I was atypical, but one of the things I did was start to take all my classes via a local community college (they had a few other honors HSers around, so they were used to "my situation"). If you have the option (either financially or otherwise), I would recommend getting a few college credits out of the way (i.e. instead of "AP Calc" take "College Calculus I" or even take something that your school may not offer, like an intro programming course depending on what your needs and such are :) ).

I will say that some of the best experiences I had when I was your age and younger were the nerd camps I attended. The one I attended toward the end of my HS career was called the Summer Science Program. It's worth looking into as well:
http://www.summerscience.org/home/index.php

Even without these things you are on track, these are just some things I enjoyed and had fun with that I thought would be beneficial :)

 

[qntty]

I wouldn't describe my mathematics background as shallow, because I'm currently excelling in my school's AP Calculus BC course. But at the same time I have to admit that I find mathematics here less than satisfying, because there has been almost no emphasis placed on proofing since geometry back in middle school. Should I be very knowledgeable of how to proof by the time I enter college? If so, where can I find resources to build this skill? I have checked my local library, but none of the mathematics books there help in this specific area.

It's not expected that you'll have experience with proofs as a freshman but I'd imagine that It'd give you a big advantage as you enter higher level courses. Not only that but you'll learn the material better by tackling proofs. As for a general intro to proofs, "How to prove it" is good and doesn't assume anything beyond high school math (not including calculus). To build the skill requires more than reading a book about proofs though, you have to actually do proofs, and lots of them. One of the "Baby analysis" Calculus textbooks is a good place to do this but you'll probably want to start with something more elementary (eg number theory).