HACKER Q&A
📣 agomez314

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.


  👤 AlchemistCamp Accepted Answer ✓
IMO, the clear choice for this is Math Academy. It was started by a very, very early Uber contractor who built the early versions of their dispatcher, their profiler and also the grid system that helped them scale and pretty much saved them during new year's in 2013. Since becoming wealthy from Uber's IPO, he's been pouring money into hiring experts and building the curriculum.

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


👤 t_mann
My impression of sites like brilliant.org or Khan Academy (or the Math Academy mentioned in another post here, which sounds like an absolutely brilliant resource for curious high school students) is that they're geared towards "pre-university" math, which is decidedly different from the mathematics in a university curriculum. Really just different, not even necessarily easier, because it's focused a lot on "mechanical" computations, which can get really gnarly, versus the more "intuitive" proof-based courses at university (tbf, you still have to master the gnarly mechanical topics at university, but you'll also have markedly different courses from the start, and their prominence just grows the further you go).

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...


👤 Aromasin
Khan Academy. I completed all their Math modules over the course of a Summer and, frankly, learnt more from their content than I had in 10+ years of mainstream Math education. I feel like if I had done it sooner I would have been an A* student, which haunts me a little.

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.


👤 Penyngton
There's no royal road to geometry. You just have to keep trying things and keep suffering. Sadly, I'm not familiar with that book or particularly with the topics you've mentioned, so I can't recommend specific books, but this is the basic recipe that I've used to teach myself some amount of mathematics:

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.


👤 dr-neptune
A book I always recommend to folks looking to learn mathematics with lots of beauty and from first principles is Serge Lang's Basic Mathematics.

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


👤 equilibrium
You might want to have a look at The Art of Problem Solving - https://artofproblemsolving.com/

👤 gavinray
The best resource I've found is Brilliant.org's courses:

https://brilliant.org/


👤 sn41
I think H. S. M. Coxeter's books are good sources for training in geometry, leading to ideas in groups. I haven't looked at this book by Stillwell, so I am not aware of the difficulty of problems presented in this book.

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.


👤 macrolime
If you want to get better at geometry a very good start is brilliant.org. They have five great geometry course that's good for learning concepts visually and has some basic exercises, but not enough for drilling.

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.


👤 dekervin
We have a discord HN-learn community, where we share our learning journeys and interesting ressources. https://discord.gg/TFMvt9vh

I am personnaly on a mathematics path, starting with geometry.

Email in bio if you want to exchange.


👤 dorchadas
I'm part of a Discord server that does exactly this. It's run by a math PhD, and he takes you through proof writing and up through the basic fields (algebra, analysis, etc) checking and vetting your proofs. It's great, and I've gone through multiple courses worth of stuff in abstract algebra, linear algebra and analysis. Currently working on some geometric algebra stuff.

You can message him on Reddit if you'd like and see if he'd give you an invite: CheapViolin


👤 ianai
Write in vote for devoting time to actually work problems routinely. Ie every day, every other day, at least 3 days per week. It’s working problems and across topics that builds intuition and ability.

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.


👤 ivan_ah
I have a bit of early geometry (polyhedra and formulas from high school) in Chapter 6 of my book, see preview here https://minireference.com/static/excerpts/noBSmath_v5_previe... It's pretty basic though... you might need another book for more details, especially if you want to do geometry proofs.

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).


👤 mejutoco
I like the exercises at https://artofproblemsolving.com/alcumus

It is part of AOPS and it gives step-by-step solutions to problems.

You need to create an account, but it is free.


👤 haskellandchill
I swear as soon as we unlock the educational power of theorem proving software it's going to be amazing. We're close. I've been following conferences like ThEdu and ITP religiously.

https://www.uc.pt/en/congressos/thedu/ThEdu22

https://itpconference.github.io/ITP22/


👤 sambapa
It may be an overkill, but here you go: https://github.com/TalalAlrawajfeh/mathematics-roadmap

👤 soheil
3b1b YT channel has some of the best math explanatory videos some very advanced.


👤 Hardwired8976
I really enjoyed calcworkshop.com with the long videos with a lot of examples and concise explanations.

👤 lupire
Have you tried reviewing a regular highschool geometry / algebra 2 /precalculus textbook?

👤 jovial_cavalier
People are going to tell you to go to X mooc or Y youtube videos. Read books, do the exercises, and if you can, have a project where you can apply what you learned.

Here's a good list of books: https://github.com/ystael/chicago-ug-math-bib


👤 winkeltripel
you may want to go through this other post on the homepage right now: https://news.ycombinator.com/item?id=32916994

👤 sailorganymede
OSSU