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Ask HN: A math study program?
134 points by agomez314 on Sept 21, 2022 | hide | past | favorite | 60 comments
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.



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


Good morning, HN! It's Jason (the founder). My wife (co-founder and co-conspirator) just woke me up to alert me to a sudden spike in demo requests, so I'm a little blurry-eyed at the moment (was up late last night trying my best to get the marketing site functional), but I'd be happy to answer any questions you might have.


My biggest worry with these kind of programs is that the questions are invariably multiple choice ones. How would we practice proofs or questions where the answers are not "seen"?


We started with the multiple-choice format as it's a good 90-10 solution in terms of technology, but we now have free=response questions that are automatically evaluated and can handle fairly complex mathematical expressions - man, was that a lot of work!

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.


> free=response questions that are automatically evaluated and can handle fairly complex mathematical expressions - man, was that a lot of work!

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.


Hey Jason, would also like the idea to have a lifetime subscription model for working folks who can use the platform based on time availability.


It's funny you bring that up as it seems to be a common request from adults interested in leveling up their math. I'm sure we could come up with something reasonable.


You are going to spoil Jason's HN announcement post Mark :)


jeez, what kind of 11th and 12th grade curriculum teaches topology, real analysis etc...? Is this a syllabus for math whiz kids?


Well, it turns out that when you have mathematically-talented students completing Calculus in 8th-grade, and move on the Linear Algebra and Multivariable Calculus in 9th, you eventually reach Real Analysis and Topology by the end of high-school - although, we haven't gone quite as deep in those particular subjects as the others mentioned. But we do comprehensively cover Differential Equations, Discrete Mathematics, Probability & Statistics and Abstract Algebra.


Honestly that looks like superficial rushing for rushing's sake. There is so much advanced and deep elementary math kids could learn.


I understand why it might seem that way, but it's really not. We've just put a premium on learning efficiency, and most math classes are highly inefficient - whether K-12 or university.

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.


The "High School" Curriculum is basically an undergraduate mathematics degree.


Yeah, pretty close, although it's not like most universities will allow students to skip straight to graduate school. Our first cohort of students just graduated this past year, so we're just now getting a sense of how this is likely to play out.

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).


I can't tell if they believe they have cracked the code that makes learning these concepts accessible or if they are suffering from a severe case of expert syndrome, totally unable to put themselves in the shoes of a novice.


Through trial and error and the application of some rarely employed, but highly-effective educational strategies like active learning, distributed practice, mixed review, mastery-based learning, interleaving, layering, etc., you end up with a pace of learning that's a little shocking. I realize it's hard to believe -extraordinary claims require extraordinary evidence and all that, but that's what got us started having our 8th-graders take the AP Calculus BC exam as as an outside, objective measure.


Opt-in educational programs of all sorts are usually heavily selection-biased toward students that are well-motivated and generally exceptionally easy to teach—good support from home, probably above-average SES, above-average IQ, et c.

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.


It's true that the students in the school program are high-aptitude, but something like 70% of the students in the middle school where this all started are on the federal free and reduced lunch program (low SES) along with a couple of the top kids in the original cohort, who I taught personally and they were amazing.


> 70 % of the students in the middle school where this all started

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 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...


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.


Seconded. I started learning calculus for the first time in my late 20s (about 7-8 years ago) and Khan Academy was great for that. Once you get into Linear Algebra, Differential Equations, etc., find Gil Strang's lessons through MIT's YouTube channel. He is truly a master teacher and I learned a ton from him without ever meeting him or setting foot on MIT's campus.


What made you want to learn calculus later in life?


I was a self taught programmer/dev and started getting interested in signal processing, systems of pulse coupled oscillators, and machine learning. So I figured I should "learn the language" so to speak.


did khan academy give you enough knowledge to have the background required to understand ML ? My company heavily focuses on ML/Statistics and i want to understand what they are doing so i'm trying to self study statistics as a first start, any courses you recommend?


Not the poster you're replying to, but yes definitely. I worked through the Algebra 1, 2, Differential Calculus, Differential equations, Integral Calculus, Linear algebra, Multivariable calculus, Statistics and probability, and Trigonometry course.

And to answer your other question, here's the stats module: https://www.khanacademy.org/math/statistics-probability


Could you maybe share the order you did them? Or maybe the order you would do them knowing how it went?


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.


> There's no royal road to geometry.

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.


I'm sorry that the phrase riled you. I didn't intend it to be pithy, but simply to demonstrate that it's well known that what OP is asking for doesn't exist.

> 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.


"I'm starting to struggle [how do I get better?]" is a lot different than Ptolemy telling Euclid the Elements is too hard and asking for a shortcut. That much is obvious because OP pretty clearly enjoys the work and wants to do more of it. There are going to be right and wrong ways to go about this.

(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.


Not the answer OP was hoping for, but definitely the right one.


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


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


Came here to recommend this. I completed my bachelors in mathematics back in 2003. However, recently I am thinking of going back to school to study graduate level mathematics or higher, and possibly do a PhD in a related field. My tentative plan over the next couple of years is as follows

1) 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 purchased the pre-algebra course a couple of weeks ago and I'm planning on working through the books as well. Would you be interested in some sort of study group?


Ok. I am based out of India though, so not yet sure if we can make the time zone work. You can reach me in Whatsapp or Signal - my 10 digit mobile number is 10^10 - 119214203 (excluding country code).


I've tried to reach you via whatsapp. Shoot me an email - email is in my profile


I also came here to say this.

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.


AoPS leans a little too hard into silly contest math / trivia tricks, not into deep understanding of the underlying mathematics. Also, they have low production values, lacking the nice graphics used in professional school textbooks (before Eureka and the greedy cult of throwaway monochrome textbooks ruined everything) to help students visualize 2D/3D functions and geometry.

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.


That's all well and good. Could you provide your insight of what you consider to be a good resource?


+1 What I find really useful about these books is they allow self-teaching. (in my case a 13 year old kid) The books follow a methodology of presenting problems and having you try to solve them before walking you through the explanations. The chapters build up to "challenge problems" which are carefully crafted to require you to apply what you've learned from the current and earlier chapters. Most of the challenge problems are lifted from math competition problems (with citations), but their use is very well curated.

We are using the printed books, but there are online resources available.


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

https://brilliant.org/


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.


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.


for linear algebra, maybe this one too, along with Axler, and Strang https://minireference.com/

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


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.


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


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.


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).


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.


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/


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


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


https://news.ycombinator.com/item?id=30591177


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


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


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


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


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