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Ideal Undergraduate Sequence Main question:

I looked above and what I'm interpreting out of it is that one should be able to do proofs after studying some intro to proofs class, calculus multivariable+vector calculus, and differential equations linear algebra.

Is it possible that one won't be able to do proofs after finishing doing calc 3/vector calc and differential equations?

Background: (context)

I really want to be a pure mathematician and not some high school teacher in the future. I suck at other subjects. math's all I got left, i.e. I can't write, I can't code, manual labor or mcdonald's job sounds possible future. I'm taking calculus 3 and differential equations right now and I'm wondering whether there's no hope in me becoming a mathematician if I fail to be able to do proofs like the ones in algebra or analysis like weistrass M-test and something banach after finishing calculus 3 and differential equations.

I cannot prove theorems at this point independently and can't study some class like real analysis or number theory right now.

Optional questions: (ignore if you're busy) Is what the link above says that if you can't do proofs after calc 3 and differential equations, you'll never be a mathematician?

Does finishing taking calculus 3/vector calculus and differential equations transform your mind to being able to do proofs? I mean why did we suggest (in the hyperlink above) that people should take multivariable calculus and differential equations before a serious proof class? Are calc 3/ differential equations special?

Are math teachers in high school the people who took calculus 3 and differential equations but just didn't have the intelligence or weren't transformed from such an experience to be able study proof writing?

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    $\begingroup$ About the last paragraph: You must be kidding. You might be surprised that there are very skilled people who actually want to teach to high school kids. And for your information, in some places proofs are taught also at the high-school level: I certainly studied the proof of the Weierstrass M-test at high school. $\endgroup$
    – Massimo Ortolano
    Oct 28, 2017 at 9:05
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    $\begingroup$ I sense a certain disdain towards high school teachers. You don't need "advanced" subjects to teach proof writings, nor to severely challenge yourself. Mathematical olympiads then to use very little vector calculus or differential equations and involve several proofs, proof writing, concept comprehension and mathematical thinking and flexibility. $\endgroup$
    – Mark Fantini
    Oct 28, 2017 at 12:56
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    $\begingroup$ Calc 3 and ODE haven't been proof-focused classes in any of the institutions I've been at, so I wouldn't really expect students to be all that good at proofs afterwards. Some proofs might have been presented and a there might have been a light dusting of computational-style proofs in assignments, but that's it. Also, while I understand not wanting to be a h.s. teacher, they aren't dumb and there are more options out there than those two. $\endgroup$
    – Adam
    Oct 28, 2017 at 13:38
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    $\begingroup$ FYI, I've taught 4 years of high school, 3 years of which were after my Ph.D. and a few publications, and my most enjoyable teaching experiences were doing this. However, for me at least, the workload was nearly off the chart (80-90 hour weeks being common). $\endgroup$
    – Dave L Renfro
    Oct 28, 2017 at 18:41
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    $\begingroup$ Math is ultimately about writing (i.e., writing clear explanations). So "math's all I got left, i.e. I can't write" doesn't synch up logically. One or the other needs to change. $\endgroup$
    – Daniel R. Collins
    Oct 28, 2017 at 21:48

2 Answers 2

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It seems to me that you don't really understand how mathematics education works (possibly education in generally). You don't magically acquire the ability to do anything by taking a course. You acquire the skills by practising it yourself, which hopefully you will do in the process of taking (at least some) courses.

In particular, you learn to write proofs by getting on and trying to write them. You won't be able to do analysis proofs without studying some analysis, because you won't have a clue what you are trying to prove or how to go about doing it. But an Intro to Proofs course should give you a good idea of how to go about proofs in general.

It is certainly not true that everyone who takes Calc 3 will be able to write proofs. That happens all the time. Calc 3 is not particularly intended to teach rigorous proof. That will happen in the algebra and analysis courses. But I would expect someone who is going to go on to be a pure mathematician to be showing signs of it even in non-proof-based courses.

Note that being a pure maths academic is extremely competitive (academia in general is, and it's harder to move to industry from pure maths than from many other areas of science). As well as generally requiring long hours, it restricts how much say you can have over where you live (often to essentially none). People might reasonably choose to teach at school level instead because they want more flexibility, or because they actually enjoy doing so.

I think you should be wary of saying you definitely want to be a pure mathematician, given you have done very little pure maths yet. Also, some of the skills you seem to have given up on are necessary for that as a career path. As someone has mentioned, writing is very important as a researcher. Without writing papers, you will not have a job. The style is different to what you are probably referring to, and learning more maths will help, but it is writing none-the-less. Coding is not essential, but understanding algorithms and logic are pretty important. And for the vast majority of academics, teaching makes up a large part of the job.

On the apparent underlying question: it sounds like you are feeling rather down at the moment. It might be a good idea to visit your careers service, and find out about other jobs that are open to you. There is no reason to think that the options are only academic, teacher or McDonalds. There will be lots of jobs you haven't heard of yet. It might take some work to acquire the right skills/work experience, but that isn't a reason to give up. It may also be that you can do more than you think, and it is feeling down that is limiting your achievement.

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I think you are reading a lot into the answers to that question which is not there.

Whether vector calculus and diff EQ prepare you for proof based courses is entirely dependent on the content of the particular course you have taken. These can be quite rigorous courses, or they can be "cookie cutter" pattern matching types of courses depending on the instructor. There is no special power of Calc 3 and Diff EQ to transform people into proof-ready mathematics students. In fact, I would say that it is much more typical that Calc 3 and Diff EQ are the last "service courses" to other departments. These courses are incredibly valuable from the perspective of real world applications, because the vast majority of mathematical problems in the real world involve multiple variables, and often involve relationships between quantities and rates of change in those quantities. They absolutely should be included in the training of any mathematician for that reason. They are placed first in these lists because they have less prerequisites, and because they often ``weed out'' students who are not willing to put in hard work mastering difficult content.

In general, learning how to do proofs requires a lot of hard work. It doesn't happen all at once. You can expect to get better and better throughout your entire career!

Not knowing you beyond this question, there is no reason to think that you couldn't become a decent mathematician. There is also little harm in trying and failing: a math degree is pretty valuable for building your logical reasoning capability, and this is useful for lots of careers outside of being a professional mathematician.

As for your final question: There are lots of incredibly gifted people who pursue math education to the exclusion of math research not because they ``couldn't hack'' pure math, but because they are more motivated to teach than to do research. You will find many such people on this website.

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