Ask HN: A math study program?
I took math courses through high school up to calculus in college and a course on discrete math, which i did well in. I just got John Stillwell's "Mathematics and It's History" and I'm dazzled by the way math is presented and the beauty inherent in it, unlike the way it was taught to me in school. However, I'm starting to struggle in some of the early geometry exercises like with regular polyhedra and conic sections, and later with exercises in projective geometry. Is there a course or series of courses I can take that can build my math skill level to solve such problems with ease? Stillwell mentions using this course to teach senior-level math undergrads.
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[ 2.7 ms ] story [ 125 ms ] thread1) Art of Problem Solving - all of the books to review mathematics at middle/high school level 2) Roughly follow the plan mentioned in https://www.susanrigetti.com/math to redo an undergraduate degree 3) Also complete the books here - https://foundations-of-applied-mathematics.github.io/ 4) Move on to more advanced graduate level mathematics
I have a PhD in engineering that was fairly mathematical, but I was missing lots of fundamentals and working through those books has changed my life, and the way my brain works, in a massive way.
They aren't cheap, but they are worth every cent. I am very grateful for having discovered them.
We are using the printed books, but there are online resources available.
The AoPS books jassume that if you are smart enough to learn the tricks and hard problems, you are smart enough to learn deep math later.
Which is a strange premise, because the PhD geniuses who created AoPS, base their credibility on the fact that they aced all these contests when they were kids... But they did it without any of these materials!
They created a "teaching to the test" curriculum, which, while mathematically sound, created a rat race that makes the contests harder every year, requiring more memorization and silly speed training, as more people memorize all the strategize that the original contests expected student to try to creatively discover on their own.
There is an old Schaum's series book on Projective Geometry that was very useful to get the basic ideas quickly. It may not be an elegant presentation of the subject, but it was very quick, and to the point.
All said and done, synthetic geometry (not analytic geometry) is not very easy, and it might help to create models (wire-frame or 3d cardboard or plasticine/play-dough) to visualize things. They may not help to solve the problem rigorously, however.
By coincidence, I am slowly working through another of Stillwell's books, on Reverse Mathematics. You are right, his way of presenting things suddenly makes things you know snap into place.
https://brilliant.org/
The first step was a non-profit program started as part of the Pasadena public school system's offerings that other schools are free to adopt. It's taken students from basic arithmetic to calculus by 9th grade and through a fully undergraduate curriculum by the time they finish high school. His son was one of those students and must be part of why he's been willing to invest so heavily for so many years.
There's now a commercial online version open to the public. The founder hasn't done any marketing of it yet, but I found out about it through a mutual friend / podcast co-host. Math Academy is very comprehensive and the most streamlined way I know of to learn, or in my case, relearn an undergraduate applied math curriculum. It's not as polished, but the content and the actual academic results make offerings like Brilliant.org look like a joke.
I requested a life-time deal last year for access myself and intend to make the most of it, likely in binges between busy periods at work.
https://www.washingtonpost.com/local/education/ap-calculus-e...
https://www.mathacademy.us/beta-test-information
Which doesn't mean such programs are useless, just that they're typically far less effective if you try to apply them to a representative slice of the overall student population.
but
> a couple of the top kids in the original cohor
Are their parents from a tight-knit immigrant community, highly educated in their previous country, with economic prospects in the US dimmed by language barrier and professional certifications that got dropped at the border?
My son, who's majoring in CS, is starting with 300 (junior-level) courses in both math and CS - although the departments were happy to allow him to start even deeper in the curriculum if he wanted. Another student, who collaborated with one our our math PhD instructors on some original research that's about to be published, is starting with at least one graduate math course this year (the last I heard, anyway).
Just imagine how much more quickly a student could progress through a course if they were to work one-on-one with an expert tutor 5 days per week. Quite a bit more quickly in most cases. Well, our system effectively serves as an expert AI tutor and based on our calculations is on the order of 4 times more efficient than a traditional math class.
But I get the skepticism. I'd probably be skeptical myself.
But we've found the concern over the multiple-choice format to be overblown. People like to believe they can outguess a multiple-choice question by being clever, but that's not reality on our system or elsewhere such as the AP exams, the AMC exams, or the GRE Mathematics Subject Exam.
Later this fall we're going to be introducing a UI for constructing proofs that's looking really cool and should take things up a notch for the more abstract subjects like Abstract Algebra and Real Analysis. Teaching university-level proof techniques is extremely challenging and time-consuming process (most never really get it) even for undergraduate math majors at university, but I think our new tech will make it much less painful and with a much higher success rate.
Wow, that's excellent. I can imagine that would have been a lot of work.
> People like to believe they can outguess a multiple-choice question by being clever
Personally I think it adds a bit more complexity and toughness if I can't see the answer in advance. But that is purely my individual style and opinion and I have not seen any research either proving or disproving my hypothesis.
Start with a book you want to read. If you get stuck, then buy another book (hopefully aimed at a lower level) on that topic and repeat the process with the new book.
Don't be afraid to read "easy" books. You should probably aim to start reading books where you look at the contents page and think you know 80-90% of the material already. I've wasted a lot of time trying to read books that were above my level. The path of least resistance is longer, but in my experience it pays off.
"Do the exercises" is good advice, but don't be too obsessive about it. Be more obsessive about regularly working on the topic, even if that means skipping exercises or jumping between books (on the same topic). You can often find the answer to an exercise in one book in a different book's presentation of the same topic, or on a website or in a paper. As long as you can integrate these discoveries into your conceptual framework of the subject, that's not cheating, it's success.
Writing things out in a lot of detail and working out examples in a lot of detail in a notebook can really help. This is like designing your own exercises and can be better than doing exercises in a book sometimes.
Come on. This is totally not what OP was asking for. Pithy adages might seem wise and helpful, but you're dismissing the fact that OP very much asked for the work and the hard road. They want to backtrack and fill gaps in their skill, and just want recommendations on the path to take to get there.
> Is there a course or series of courses I can take that can build my math skill level to solve such problems with ease?
It seems to me that this question assumes that there is a "royal road" of courses that will turn him into the mathematician he wants to be, but I don't believe that's the case. The only way to get where he wants to be is through a long and difficult process. And in my comment I tried to give what advice I could, however inadequate.
(Here's where I really go off the rails, but trust that I mean this in the same good humored sense with which I would point out that say, vinegar catches flies better than honey.)
We should also cast doubt on the original quote. One, we only have it according to Proclus, roughly eight centuries after Euclid. Two, Ptolemy I was brilliant in his own right, even among the Diadochi. Three, we don't teach anyone straight from the Elements anymore and have learned a great deal about ways people learn. So while I agree there's no substitute for rigorous practice if you're looking to understand a mathematical concept on an intuitive level, we've certainly found some "highways" since 300 BCE.
Despite studying engineering up to postgraduate level, I tended to avoid any concept that required a deep understanding of the Mathematics behind it. Now, I love Math and feel like I could express most problems using it, as well as make sense of papers and text books that were closed off to me previously.
And to answer your other question, here's the stats module: https://www.khanacademy.org/math/statistics-probability
I am personnaly on a mathematics path, starting with geometry.
Email in bio if you want to exchange.
https://www.uc.pt/en/congressos/thedu/ThEdu22
https://itpconference.github.io/ITP22/
Here's a good list of books: https://github.com/ystael/chicago-ug-math-bib
When it comes to exercises, brilliant.org is lacking in volume. Khan Academy is a great supplement for geometry, single and multivariable calculus although brilliant.org goes a bit further than Khan Academy with Linear Algebra, Group Theory and more.
Khan Academy also has a linear algebra course, but I found it to be kinda crap with no exercises. For linear algebra it's better go with brilliant and 3blue1brown's linear algebra videos, then a good continuation would be Linear algebra done right by Sheldon Axler and also fast.ai's free online course Computational Linear Algebra for Coders.
Axler is very mathematical, focusing on vector fields in the abstract as opposed to matrices, Savov is what you need to get through engineering school, Strang straddles both
also ulaff.net ... more in the realm of, all about matrices, as opposed to pure math
It is part of AOPS and it gives step-by-step solutions to problems.
You need to create an account, but it is free.
The best places that I am aware of for self-studying university-level math are university websites (like MIT OCW) or just going through undergrad textbooks on your own. Someone made an imho decent guide for the latter [0], curious if HN users have other recommendations.
[0] https://www.quantstart.com/articles/How-to-Learn-Advanced-Ma...
If you want to broader review of undergraduate math and physics, then check out the longer book: https://minireference.com/static/excerpts/noBSmathphys_v5_pr...
Both books have hundreds of exercises, which, as other have pointed out, are the most important part of any learning resource. Several readers have said they appreciate how complete the curriculum presented in these books are (based on years of experience helping people review math needed for university-level courses).
Forgetting the book, but I have a book on solving problems which I studied from for the Putnam. The premise was to take someone seemingly around your level and build them up to problem solving machines. Probably math competition textbooks would be a good source for you.
The book starts from axiomatic arithmetic and works all the way up to what you would need as a mathematics major with a focus on pure mathematics. He also manages to touch some beautiful areas like geometry, abstract algebra, symmetry, linear algebra, set theory and more.
There are only a handful of exercises at the end of each section and they are very good at locking in the concepts. I finished a mathematics degree and realized afterwards that there just wasn't enough of a focus on the concepts and beauty. Even with 4 years of math experience, this book still managed to open my eyes
You can message him on Reddit if you'd like and see if he'd give you an invite: CheapViolin