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I am a student, in my last year of school(17 years old)

When I was about 13 years old I fell into the calculus trap by starting off learning trigonometry on my own, when I was supposed to factor equations or solve basic probability questions. Being good at math(fast arithmetic skills, could grasp new concepts easily etc.) I came across an interschool math competition at that time which was purely based on geometry. I studied by myself for that comp, mainly the properties of circles, triangles and as I earlier mentioned, was introduced to trigonometry. Even though I didn't score well in that comp, I was mesmerized studying those new topics which led me to discover more and more, and finally started learning calculus just one year after that time.

I soon realized that I have fallen into a large hole with no end, though going deeper and deeper was becoming more interesting, it started affecting my performance in other subjects. I also realized that instead of starting with calculus, I could have solved much harder problems related to my school curriculum and hence could score better in Olympiads.

Unfortunately I am still in that hole, going deeper and deeper, studying more abstract, higher level concepts. But I will be passing out of my school next year and will get admission in a nice college aiming to study higher maths and everything will become normal.

My main question is that should I encourage my juniors to do the same thing which I did? Or should I guide them to study maths in a more systematic way.

The link that I had provided is from art of problem solving and hence I found it a bit harsh towards my situation, so I decided to ask here for guidance. I also read this question but that is not exactly my case, I was thorough in my study, also took help from my teachers if I did not understand anything.

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    $\begingroup$ With all due respect to "problem solving", you can't beat calculus. $\endgroup$ – Peter Saveliev Jun 11 at 20:33
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    $\begingroup$ @PeterSaveliev If you have an answer to the question, even a small one, can you post it as an answer? Mini-answers in comments can't be voted on, can't get their own discussion threads, and suppress other answers from being posted. $\endgroup$ – Chris Cunningham Jun 11 at 21:43
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    $\begingroup$ In 7th grade, I discovered how wonderful math is and started reading all sorts of math, neglecting the math I was supposed to be doing in school. So I was in danger of failing 8th grade math. Fortunately, my parents wouldn't have tolerated that, so I shaped up quickly. You're older than I was and therefore less likely to be rescued by your parents. Take this comment as an attempted rescue. $\endgroup$ – Andreas Blass Jun 12 at 0:00
  • $\begingroup$ @Andreas thank you for your comment sir, my performance in the ninth grade also fell drastically. My parents tried to rescue me but I did not take it seriously, I am scoring above 90% score in maths after that but the impact it has done on other subjects is indeed large. My ability of rote learning has fallen, hence my scores in school also fell. $\endgroup$ – Pj30 Jun 12 at 6:48
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    $\begingroup$ I think that believing that a fall in school scores can be attributed only, or perhaps even mostly, to a reduced ability for rote learning is probably missing your reduced focus on the topics that you should be studying. It is great always to be reaching beyond what you know now, but it's also too easy to lose yourself in the higher reaches before you have a solid foundation, and this can lead to poor understanding that shows up in lower grades as you've indicated even while feeling like you're learning well. $\endgroup$ – LSpice Jun 16 at 16:38
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Echoing @AndreasBlass' remark, and having experienced somewhat similar episodes, it is already precarious enough to make such choices _for_oneself_. So, to directly answer your question: I think "no, do not encourage others to (too violently) disconnect from the math curriculum at school". I don't think it's about problem-solving versus calculus, at all. And, no, I'm not a fan of typical school math curricula.

If nothing else, given the way many kids think about things (as I did, for sure), there is the risk of providing a convenient rationalization for disconnection and non-conformity that is not replaced by anything constructive. And then it's hard to get back in sync even if one wants to.

E.g., when I learned about the expressibility of trig functions in terms of exponentials, via (basic things about) complex numbers, the game/challenge of proving trig identities lost its charm. I could not make myself care very much. But the instructor for that class did not care about "better alternatives", so, after some earlier class-room debacles, I decided to play along to a sufficient degree to avoid trouble. And "physics without calculus" seems ridiculous, too, but, ... and "economics without calculus"...

So I'd recommend maintaining at least two threads: one following your own curiosity and "genuine" mathematics and science (rather than school curricula), but another to maintain "presentability", in effect showing that you can understand the ambient social constructs and cooperate with other people at least minimally.

No, I am no fan of conformity for its own sake! :) But it is certainly very convenient for individuals to be able to "code switch", to survive. Or "to survive long enough to get to a position to have less obligation to conform"? A tricky balance, for sure.

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The article at artofproblemsolving seems silly to me. The author's idiosyncratic opinion seems to be that students who are ready to take calculus should refrain from taking calculus and instead do math contests. People are all different, and there is not just one appropriate path for a mathematically precocious student. Some people might want to take calculus and do math contests -- it's not like they're mutually exclusive.

One good argument for learning calculus early is that at many universities, the quality of instruction in first-semester calculus is atrocious. It might be better to learn it on your own or in an AP high school class than to be subjected to that.

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[VERY LONG ANSWER, needs patience to read through]

I feel this is a problem many students who are good at maths face. They understand the simple tricks and patterns which are present in the school syllabus and so it is simple for them and after some practise and memorisation they are done. Then they seek out more maths and find out about topics like trigonometry and calculus.

One thing which I feel is a big factor causing the drive and a part of the problem which many people ignore is the 'need to be a genius'. Growing up we all hear stories about Einstein and his legendary E=mc^2 formula which he dreamt up because his brain was just so big. In mathematics we have role models like Euler, Gauss and Ramanujan who seemingly picked out amazing results from thin air. These stories make up our perception about scientists who are introverted geniuses which seem to know everything except how to talk to people and conform to normality.

So students such as myself begin to rebel against the system and find out more advanced maths on their own. However, this is much more difficult than just following the school curriculum. Even if the topic is within your intellectual capacity, having no one to explain it to you is very discouraging and you give up easily. Since you are talking about studying high level mathematics in school you definitely would have experienced finding out about some topic or the other in which you have no idea how to even process the proofs and theorems related to it after a certain basic point. This is very frustrating because you have an image of being good at maths in your mind and you cannot meet it if you don't immediately understand your textbook or whatever you are studying.

This then becomes a trap: you don't study topics which you don't immediately understand or have good intuition for, and so you keep going down and down into deep rabbit holes where you go into one sub-topic after another without pausing at any level to expand your knowledge to related topics and building a firm base before you go on into more specialization.

This is why there is a need to tell students to be firmly confident in their own stupidity. They don't get things immediately and make like 10 mistakes while solving a question, but they can explore and work harder until they are proficient in that topic. Not only their own stupidity, they should know that everyone else is also stupid. Even Euler, Gauss and Ramanujan were stupid, in the sense that each one of them must have struggled with some topic or the other, and they must have felt frustrated and incompetent many times because of that.

The best way I have found to overcome this inferiority complex is to let students make something original of their own. If you know about the process of making an entirely new discovery (new to them, maybe not to the world) without always relying on thought patterns and tricks which are just programmed into you after solving many school and Olympiad level problems (which do not test your mathematical ability accurately), you learn to appreciate many things. You realise how random and arbitrary making progress in a problem is. You can be stuck for days on a single lemma but come up with a one liner that completely resolves it while having a bath. It is also extremely non-linear, which means you can take long detours without coming close to the correct method. However, you also realise that progress in a small or big form will always come if you try hard for long enough (and take enough breaks to reset).

When you realise that every one of those geniuses went through this same random, frustrating but highly satisfying process every time they solved a hard problem, the illusion of being smart only if you are lightning fast in solving and understanding is quickly broken. The only thing you need to make an independent, worthwhile discovery is lots of studying and lots of thinking. That is how maths in the real world is done. You may never get even close to a level like that of Euler, but it is unreasonable to hold such an expectation.

So I think you should encourage your juniors to explore maths: not just topics which are at a higher level, but to truly explore and wonder about things and try to find out more about them. Learning a new topic should be an interest rather than a habit or a drive. It is psychologically very unhealthy and dangerous to have such an unfounded fear of failure and lack of confidence in your abilities. Expanding your horizons above school, competitions and Olympiads helps with that.

As an example, when I proved the formula for the sum of a geometric series in 9th class months before it was taught, I felt much more proud and confident in my abilities than when I solved much harder Olympiad type problems which tested not my independent thinking but my ability to remember and apply obscure formulas and patterns. It also probably helped me much more in developing mathematical thinking than those problems.

As for school, unfortunately, it is a necessary part of life and you have to devote some time to study school maths and other topics according to what marks you want. It is not all bad: if you look close enough, there are many things to be explored in school mathematics.

Finally, when talking about studying other subjects, you should have practicality in mind and know about the consequences of your actions when you avoid studying for Olympiads or school. If you are okay with sacrificing that to fulfill your interest in maths, only then should you do that, otherwise you should search for a compromise.

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  • $\begingroup$ "You realize how random and arbitrary making progress in a problem is." +1 for that alone ^^ $\endgroup$ – FiMePr Jul 17 at 9:39
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I will also chime in and say that the argument on the linked Art of Problem Solving site is unpersuasive, and somewhat misses a broader point.

Ultimately, the real point of the mathematical discipline is to identify patterns in systems and prove their correctness (hopefully in an insightful, persuasive, explanatory style). The "trap" that I would identify is the standard calculus-track being about numerical calculations for grades K-14, and then mathematics majors needing to switch tracks to proof-based courses, which is the essence of the profession of mathematics. It's really a huge irritation that so many students are "tricked" into thinking that being good at following rules for calculations means that majoring in math is a good choice. (E.g., in reality, computers can compute any such K-14 exercise instantaneously, so humans doing calculations is not really in itself of any practical use.) I may argue that "problem solving" in terms of applications with numerical answers at the end is not any better in this regard.

So my top suggestion would be to supplement the time between high school geometry and the 2nd year of college with some kind of practice at reading and writing proofs (number theory being a common starting sandbox). I might recommend Richard Hammack's free, open-textbook Book of Proof as a good starting point, even though it's written for a college audience -- if others have better suggestions I'd like to hear them.

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