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Joined: Dec 2021
Posts: 34
Junior Member

OP
Junior Member
Joined: Dec 2021
Posts: 34 
Having gone through a number of posts here, I felt like it would be nice to compile some of my observations in case it might help someone down the road.
School age:  School math education is usually quite insufficient to prepare kids for solving any sort of difficult math problems. Usually, it is predicated on memorizing formulas and being able to directly apply something which is already taught. Simply advancing kids to more advanced courses in the sequence will not teach these skills.
 SAT math is very easy, relatively speaking, and for a gifted math student, the difficulty is trying to 100% the test under time pressure, and not losing those few marks due to avoidable errors. However, it does not use a lot of real mathematical thinking ability. Standards for math in standardized testing across the western world are typically not that high. Even someone with a moderate amount of mathematical ability can do well on those tests  there is a low "ceiling".
 Real skill development in math is about depth of understanding, and problem solving ability. It's not that hard to teach a semester's worth of calculus in a couple weeks to a fast learner; the problem is that they are unlikely to be able to apply it to any sort of nontrivial problems.
 Relatedly, it is much more beneficial for a student of school age to do difficult problems involving easy mathematical concepts, rather than easy problems involving difficult concepts. School calculus is typically quite watered down. This is, in my opinion, partly because schools can not assume that kids have a very solid background when it comes to algebra and trigonometry before they start calculus. Classic mathematics books from the early 20th century are often aimed at working at these fundamentals with fairly difficult questions. It's typically more productive to "pay your dues" with difficult geometry, trigonometry and algebra problems before moving on to the calculus sequence. Calculus is not that hard; the real question is at what level you would learn it at once you get to it, and that will be predicated by the depth of your understanding of everything that precedes it.
 Math Olympiads are a very good way to meet likeminded peers when it comes to math. You're almost guaranteed to meet people who are equally talented at math, as well as likely those who are considerably better than yourself at it. This is true even if you are very highly gifted. (Basically, it's true unless you are a Grigori Perelman or Terence Tao, and even then, you will meet those kinds of people at the IMO.) You will likely not get a similar peer group for mathematics at university, even if you major in mathematics, unless you are at a very top tier program.
 More advanced courses aren't *necessarily* harder in a sense. Differential equations and multivariable calculus may well be easier than math olympiad problems. Higher level courses are still designed for average or slightly above average students with several more years of experience, and experience doesn't necessarily substitute for skill or talent. Advanced courses aren't typically designed to bring out high level problem solving ability, but to teach difficult concepts  there is a difference. Likewise, first year undergrad courses at a top university may be harder than graduate courses at an average university, or in a less challenging subject.
 I think for most highly gifted kids, middle school is about the time they can start trying out more advanced math (algebra, trigonometry etc.) with a decent degree of success.
 Books oriented towards Olympiad problem solving are also quite good at teaching many of the fundamental skills involved in math problem solving (which are not taught in school).
 That said, we need to keep in mind that there is a distinction between recreational mathematics (such as math olympiads), and actual mathematics which you study at the undergraduate level and beyond. Doing well at one doesn't necessarily mean you'll do well in the other. Recreational mathematics involves pattern matching and understanding how to apply certain tricks very well under time constraints. There are other factors such as the depth of understanding of a topic, facility for understanding higherlevel abstractions, and so on, which affect your ability to advance in mathematics past the high school level.
College age:  Read Terence Tao's blog. There's tons of good advice there.
 From what I've seen, unless you're at a top 20 college (and perhaps even then), you're not getting a good math education from your classes which would allow you to get into a math or closely related grad school. The level of the students and the instruction simply isn't up there. Try to get in touch directly with the professors. It is often the case that the professors themselves don't understand highly talented math students and what they are capable of. But you can likely find some good professors who can give you good advice. Because of the selectivity of academia, it's likely that the professors themselves are at a much higher level than the students.
 If you're not at a top college, think of doing internships at good labs, at good universities. They can give you a leg up and expose you to the culture at the top places. There is a difference in philosophy, where top colleges actively aim to facilitate those kids and allow them to push themselves by taking harder courses and doing research, while lesser colleges are often cash cows and can get you stuck doing courses which you find easy without much possibility for advancement.
 If you are at a top college, other students and professors can be really good guides to help you achieve your goals. Not all of them, necessarily, but they are likely to have seen different levels of gifted students, and know better how to work with them. The top few students at the top universities in math are really, really smart almost without exception.
 There comes a point where taking courses will not help you much. People reach this point at different stages. It's more likely that you'll reach this point quicker if you're doing anything mathematically oriented other than pure math. I have seen students get there as soon as their sophomore or junior year, more commonly in anything other than pure math (statistics, computer science, etc.) Once you reach that point, even graduate level courses in your domain might not be difficult for you, and you may be able to learn simply by reading textbooks and research papers pretty much equally well if not better. At that point, it's important to seek out possible options for independent study and research because those will be the things which really push you forward.
 First year undergrad students who are very gifted can often find graduate courses at average universities to be quite manageable/easy. What's ideal in this kind of situation is to find an undergraduate course which offers more challenge. It depends on the ability level to which the class is geared more than where it is in the sequence. In my experience, a graduate class where the instruction is delivered in a way tailored to someone 1SD above average will likely not challenge the undergraduate student who is 3SD above average.




Joined: Feb 2020
Posts: 201 Likes: 5
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Joined: Feb 2020
Posts: 201 Likes: 5 
It’s good of you to take the time to compile your observations for the benefit of younger numberphiles. What you have described rings true from personal experiences and observations of professional mathematicians.
I would agree that the content and pace of delivery of most of primary and secondary school maths education is not well suited to young gifted mathematicians but that is because the curriculum is designed to fit as many as possible and therefore targeted at some notion of an average student. I have widely expressed praise for the digital resources which allow young numberphiles to quickly yet rigorously cover all the foundational materials.
Calculus, in particular, seems to be frequently debated with respect to whether it ought or not to be included in the high school syllabus, with many secondary school educators advocating that it be replaced by maths topics with greater apparent day to day relevance, but fortunately it remains in the secondary curriculum as it is an integral part of the foundation for applied mathematics and physics, for those who later decide to take up further studies in these fields. My son was fortunate enough to have been in a radically accelerated stream for maths so he finished high school extension maths in year 11  the calculus and vector maths he learnt stood him in very good stead for his in depth physics investigations this year, which I hope will give him a strong start for his planned R & D engineering studies and possible career in the renewable energy sector.
Everything in our physical reality obeys universal laws and rules which requires an understanding of their mathematical bases to understand and work with  whether or not the majority appreciate this truth, it’s great the maths foundations are laid at each stage of education, to enable those who are interested in understanding physical reality, to go on to further studies.




Joined: Dec 2021
Posts: 34
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Joined: Dec 2021
Posts: 34 
Calculus, in particular, seems to be frequently debated with respect to whether it ought or not to be included in the high school syllabus Just to clarify, I think that calculus should be in the high school syllabus. But if you want to properly learn math, it's better to introduce calculus after covering more basic math at a deep level. This may be impractical for a school syllabus but will be really important if you want to gain a solid foundation of math which will allow you to be able to properly get to more advanced math later on (for math, statistics, physics, etc.). Geometry, trigonometry, permutations and combinations, some basic number theory, etc. are things I would suggest working on first. Calculus is not such a big deal as such for a gifted math student in high school, but I see people trying to complete the "sequence" quicker, with multivariable calculus and linear algebra in high school, and wonder what the point of it is. Edit: I understand that the college credits transfer and it's a good idea from a college admissions perspective. However, what they teach at high school will be so much less rigorous than it should be at a good university anyway, so you will probably need to relearn a lot of it. The better your skill at problem solving and conceptualization, the better you'll fare once you get to the actual hard stuff.
Last edited by giftedamateur; 09/08/22 12:12 PM.




