Relearning math as an adult

About a year ago my wife and I made the joint decision to get her into college for a bachelor's degree in either English or computer science. Here in Finland, university admissions are based around how well you did on various end of school exams, and she fortunately did quite well on all of them a decade ago - except for math. She bombed math due to health and family problems as a teenager decimating her ability to pay sustained focused attention.

Luckily, soon after the decision was made, I finally got my first full time job in Finland. I could easily support the two of us and give her an uninterrupted life to focus on grinding up for the math exam. We took the last ~10 years' worth of math exams online, turned their problems into Anki cards, and set her nose to the grindstone. I had had tremendous success with this in college with abstract algebra and real analysis, so I figured the same methodical approach + a very stable living situation was bound to work for algebra through basic calculus.

Five months later, she retakes the high school exam and gets the highest score possible! Her rapid success at this convinced her to give CS a serious try, and indeed her improved math exam was the differentiator - without it she would not have been accepted to the CS program. She got top marks in her first semester at CS as well, I couldn't be more proud. My wife is truly an incredible person.

set her nose to the grindstone

Worth calling out the key observation: it takes practice, and lots of it (for most people, anyway) to get good at math. Just reading about it does zilch.

OP described a specific type of practice to get good test grades using 'Anki cards' . Standardized tests test for grinding and dedication not expertise or 'getting good'.

There is no (multiple choice) standardized testing in the Finnish school system and I cannot imagine passing the math tests based on rote learning. I have to assume she took the test somewhere else.

EDIT: The exams at the end of high school can be considered standardized tests, but they are taken at your own school, graded by your own teacher and only verified by the national test organisation. They are not multiple-choice tests.

I cannot imagine passing the math tests based on rote learning.

why not. Its not exactly rote learning like a parrot. You just learn tricks and patterns in problems. Tests usually have a limited amount of patterns.

These are not tests generated from patterns. You may be asked e.g. to solve an applied problem you've never heard of or to sketch a proof to a given (simple) lemma you've never proved before.

OTOH, you are not supposed to solve every problem in the exam, so perhaps you can get the best grade even if you skip all the problems where there's no pattern to apply. In that case, that's a loop hole which the exam creators should plug.

solve an applied problem you've never heard of or to sketch a proof to a given (simple) lemma you've never proved before

Can you link me to finnish high school question that is asking for 'sketch a proof to a given (simple) lemma you've never proved before '

I agree that this cannot be rote learnt .

I hear these problem types are not as common as they used to be, but here's some recent examples in this direction.

Fall 2023: "Prove that 2^12345678910 - 1 is divisible by 1023."

Spring 2022: "Using induction, show that the sum of the numbers on the line n of Pascal's triangle equals 2^n."

Here's the full exam from fall 2023: https://yle.fi/plus/abitreenit/2023/syksy/matematiikka_pitka...

I cannot seem to access that link outside finland.

That is indeed a tough proof to solve sight unseen. Any reason you say that students are seeing that question for the first time in the test. Seems like a famous questions, even chatgpt got the proof correctly .

Thinking about how to learn is fascinating. A few short things that helped me:

Essentially we need to get fundamentals first.

Then apply the knowledge and get challenged (feedback loop) -> build a project, play in front of others, speak the language.

Improve - not by force, but by understanding (remembering something and understanding something are two different things).

Synthesise - learn about a topic in a different manner or try to find similar concepts in completely different context /

Use mentors to amplify the knowledge and get feedback quicker.

Immerse yourself in practicality (work in a field, live in a country etc.)

This is a great summary. Thank you!

It depends entirely on the cards and what's done when a card is missed. If the cards are straight definitions then probably not the best way to "get good at math". But if the cards (you can do this with plugins) are things like generated problems using specific mathematical techniques (solve a system of linear equations, for instance) and after failing a card you can go back and study the material, then it's effective. You can also throw textbook problems into cards if you don't want to use a generator, but that's a lot more effort than most people are willing to put in. You don't even need to put in the solutions if you're willing to keep the book around. It does mean cards take longer than a few seconds, but it still has the effect of taking advantage of spaced repetition to develop mastery of a technique.

I had great success using Anki cards during my undergrad to fully internalize important definitions (i.e. trig identities, etc.) for utilization within practice problems.

Sometimes, you just know a thing or don't, and for me, spaced repetition cards helped a ton with the static things you just have to memorize (or derive). Knowing definitions is important—you're absolutely right that it's just one piece of the puzzle though.

It does a little more than zilch. Exercise is important. Reading is also important.

If you want to learn fast, you need a balance of both that works for you. Some people will spend months grinding hard problems on their own in a single chapter to make sure they really understand. Some people will read too fast and have to go back because they don't have solid foundations, and only thought they understood.

Reading is about as important as doing exercises. You're not going to reinvent all of math on your own. You get insights from both.

This is just not true. Reading is necessary but hardly about as important. If you are lucky and get a good text, the exercises guide you to "invent" the important parts of the theory. Otherwise its just definitions and theorems, and its up to you to make your own examples to gain an intuition, which to be fair is closer to life outside of a classroom.

Could you elaborate on this please: "the exercises guide you to "invent" the important parts of the theory."

BTW: I think time not doing exercises is just as important; it's when your mind tries to piece together the data. Coincidentally(?) time resting, after physically exercising, is when your muscles strengthen.

Sure, texts like "linear algebra done right" or "Understanding Analysis" do a unusually good job of integrating large multi part examples where the reader works through them and proves the theorems themselves before they are explained in the book. The nominal case for most math texts is definition, lemmas, theorems, and sometimes they provide examples, and other times, readers are expected to make their own examples. For example, basic topology really only needs 10-20 pages to completely define, but to really understand why the axioms are chosen and the implications, you must work through examples, it is the only way. In fact there is a good book on topology specifically due to this " Counterexamples in Topology"

Thanks! I think I now recognize this from Spivak (Calculus), where much of the teaching is literally in the exercises. You are guided along, deriving/proving many things along the way, some incidental, some cumulative. (There's also important exposition in the exercises.)

A downside is you lose the thread if you skip exercises (e.g. do alternate ones) - the exercises are an integrated whole. But it's a lot to do all of them.

I hadn't gotten the impression that these helped show why exactly the axioms were choosen - though could well be there and I just didn't see it.

It really depends on the text, it doesn't have to be just a list of theorems and definitions.

Some intuition can be conveyed through text, or geometry and visualization. A good explanation can make things click.

I think you may be right about your own experience, that you get most of your intuition from exercises, but I'm confident this varies. Some people get much more than zilch from a good text, in addition to doing exercises.

The real world is harder than a classroom, but we don't have to make classroom learning as hard as research. It's okay to start with help and increase the difficulty gradually. You don't have to do everything on your own!

The "if you are lucky and get a good text" is carrying a lot of weight here. My school identified certain young students as having high mathematical aptitude. While other students were learning the basics of addition and multiplication, the advanced students spent hours practicing multiplying nine digit numbers by hand.

The school was then always disappointed by by their performance in mathematics competitions. After all, the other teams were "wasting" their times unimportant reading about algebra, geometry, and combinatorics while our team was "practicing" math with hours of manual long division nightly.

The practice is certainly vital, but it's useless without a good text to guide you. Unless you're lucky and grab a good one on the first go, you'll need to read a few texts to find the good one.

I'd say that reading is as important to learning mathematics as breathing. You'll be a lousy mathematician if you spend all your time focused on your breathing, but you'll be worse one if you skip breathing entirely.

It depends a bit on what you call "reading" (i.e. whether you are reading passively or actively). If you stop and think through each proof yourself before reading the one in the book, that more-or-less turns the main content into a series of exercises.

please tell me that how can I get question for practice in math of university level? most of textbook even don't contain answers of practice in the book...

If you have a textbook you're interested in studying from, you can usually find courses from searching on Google that use that text. If you're lucky, they will make the homework and answers available online.

If you search long enough and look for older courses (before all of these online learning platforms like Canvas became popular), you can usually find lots of worked examples. There is no central location to find what you're looking for, but you should be able to find supporting resources.

You just find books with the solutions.

Some like Hubbard and Hubbard's Vector Calculus come with a solutions manual.

Some like Knuth's Concrete Mathematics contain full solutions.

There's also whole genre of books like Schaum's Problem books and their Outlines which contain thousands of solved problems. And they're quite cheap.

For any given math subject X, you can probably search for an "X problem book".

Hop on the mathematics discord and ask in their book channel. It's the big server with a picture of a torus as the icon. There are some very knowledgeable people there

That's mostly because proofs are too long to write out easily in the back of a textbook, imo. Your best bet here is to learn the proofs of the theorems proved in the book first - they're usually much harder to come up with at first glance than the problems one bases on them. Or, if you're using a commonly used textbook, you can usually find people's own proofs online. Sourcing an easy to grok and easy to check proof is a very good skill to build in any case, it minimizes wheel reinvention at the neural level.

Read the book and whenever you come to a theorem, stop and try to prove it on your own. If you can't, read some of the proof, then try to continue on your own.

yes. learning to doing maths is a bit like learning to play an instrument in that you have to spend lots of times practicing scales or other simple stuff (doing the exercises) before you can really be creative on it.

or like learning to write - you got to spend lots of time practicing and memorizing your letters before you can start getting to words and sentences and novels.

It's also a bit different from learning to play music in that almost everyone appreciates music even without being able to create it themselves. There's a much more direct connection between the craft and its outcomes than with math.

Exactly. I'm not a programmer because I don't program. I can't play guitar because I don't play guitar. I can't speak another language because I don't practice one. It sounds obvious but doing the thing you want to do is critical to doing it.

It seems not to be entirely obvious to the students in my classroom. Today second day of the semester, job one is to explain that they need to have a notebook. They need to write in the notebook. And they need to do the homework in the notebook. They seem not to know that, many of them.

Very true.

I would love to know more about the study process you followed and resources because my wife is in a very similar situation, but here in Norway where they require her Math R1+R2. Great story, and congratulations!

Feel free to email me, let's see if we can help her out :) check my bio

It is truly amazing, what some people can do, when they are enabled. Now imagine, if somehow we could get this for more people, as a society, not just on a partner/SO level. How much more human potential we could unlock.

A society is just a collection of individual human relationships. We need to teach not just knowledge but character.

Great story and I'm sure the new math chops help her a lot during her studies.

In case anyone else is interested, it is possible to study computer science without any entrance exams due to the Digital Education for All initivative (https://www.helsinki.fi/fi/projektit/digital-education-all in Finnish, sorry). You get the full study right after completing 60 credits (out of a total 180 credits) worth of courses in the first year.

Cool comment, it's great seeing people being supportive of their SO =)

I may be biased as I am a trained Mathematician, but I always feel when someone says "Math is Hard", that is because they had bad teachers.

Math is easy if you build up from fundamentals, not like physics education where you say "but lets delete everything before because it had an oversimplifying assumption", rather if you build your knowledge entirely sequentially from things you know or assume, you build up a toolbag that applies literally everywhere.

So math isnt hard. Learning random bits of math out of context is hard. Climb the ladder once, you have it for life.

Hopefully for this person that sticks.

I may be biased as I am a trained Mathematician, but I always feel when someone says "Math is Hard", that is because they had bad teachers.

You're biased.

I've had excellent teachers, math was - and still is - hard. Especially when you get into the more complex stuff. Not everybody is as gifted at math as you are.

Completely disagree. Problem with (at least) math is that you want a teacher is not super good at math, but still knows what they are teaching. When you get taught by a brilliant match wiz teacher they skip over the stuff that is obvious to them, but what is probably crucial for mere mortals.

I have had teachers who just blew over the simple stuff because they didnt care about it and focused on the interesting hard stuff, which felt a lot of people behind and also with actually good teachers who focused on the "easy stuff" to build a strong foundation before moving to the harder stuff.

I wrote "I had excellent teachers", not that they were super good at math.

you also wrote other stuff.........

Which I still stand by. Math is easy for people that are good at math, computer programming is easy for people that are good at programming etc. For the rest of the world those things are not so easy, even if they do have good teachers. To assume that everybody can be equally good at math or computer programming is denying reality. I am a pretty good teacher and have found that some kids take to this stuff like fish to water and for others it is a serious effort with everything else being more or less constant. It would be great if we could identify that one single factor of 'the teacher' as the root cause of all of the trouble but unfortunately that's an oversimplification. Sure, there are bad teachers, and some of those are really good at math themselves. But that's just a fraction of the problem.

Math is easy if you build up from fundamentals

To a certain point, I guess. Most people hit a wall of abstraction at some point, either because the abstraction is too hard or because the abstraction stops being relevant so the person loses drive to learn. For me, the wall is model theory and the second course of abstract algebra. They are both too hard and too abstract for me to push through.

abstraction stops being relevant

I found this to be the points where abstractions being learned today are only precursors for abstractions that will be learned tomorrow. Another way to put it is at the stage where you're learning to make tools that are themselves only used to make other tools, not used to get results outside of the domain of tool making.

These stages have no apparent relevance outside of math, and if your style of memory formation depends on making many inferential links to laterally associated concepts, moreso than making a few direct links between vertically associated concepts, it can be rough going. A lot of what feels like following memorized pirate treasure map directions in the dark, with no sense of what obstacles you're working around or even the general direction where the treasure lies to give you a sense of bearing and progress.

I remember memorizing multiplication tables in school.

I learned that 3 x 9 = 27. You just had to memorize that, right? Well then I realized that if 3 x 10 = 30, then 3 x 9 must be one fewer '3' added together by the multiplication, which means take out one '3' from the set of 3s you are adding together by multiplication when doing 3 x 10, which comes to 30 - 3 = 27.

That means I didn't really need to memorize 3 x 9, I needed the above simple rule in addition to the fact that n x 10 is always what you get when you take the digit 'n' and add a 0 after it.

So learning multiplication tables was hard, until I learned the rule of looking for an easier-to-remember result and then adding or subtracting something to it. Of course I also had to understand that multiplication is really just repeated addition.

My teacher never taught me this trick, just told us to recite the multiplication tables in out heads again and again. But after doing that for some time I figured out the above trick myself.

Learning math beyond multiplication is hard if you cannot multiply numbers in your head, because lots of math presentations assume that of course you know that 3 x 9 = 27. Or something similar. It is not just about understanding the concepts, it's about being able to perform calculations, in your head. Else you cannot understand the explanations of new concepts. Even though we have pocket-calculators, we still need to be able to do calculations in our heads to understand new topics. in math.

So, learning what is 3 x 9 is not hard AFTER you have learned n * 10, and this trick. I assume something like that happens in the minds of mathematicians. They know a lot of math already which makes it easier to understand new results when they already know a lot. To learn what is n * 10, you had to learn 1 x 10, 2 x 10, 3 x 10 etc. and then understand the pattern in there.

Learning something is easy if you already know lots of related stuff. So it's not about learning more and more difficult things, it is about just learning more and more, related things. It is about having more and more (learned) data in your head.

I assume that is also why LLMs work so well: They have lots of data.

In summary: Learning math is not "difficult", it is tedious.

So, learning what is 3 x 9 is not hard AFTER you have learned n * 10, and this trick.

The tricky thing here is that you have a limited amount of working memory, energy, and focus.

To do well at math you need:

- practice at being focused and confronting things that are hard

- an understanding of the problem space you are facing and how your tools work

- enough stuff memorized so that you don't have to context switch too much

You can have some missing pieces in the third area and do okay. But for a lot of students, needing to context switch to do simple arithmetic throws them off. I encounter students who can do any step of a problem, and can even describe the steps of what to do, but when I observe them thunk down to arithmetic and struggle, they aren't able to find their place again and make mistakes.

Most students are better served by getting their multiplication tables firmly committed to memory; perhaps a mnemonic or a simple algorithm of multiplying by 9 helps them get there. But you still don't want to be leaning on that when you're trying to factor a quadratic or cancel things in fractions or whatever.

(Seeing patterns, and learning why the pattern works is perhaps more valuable than multiplication tables... but that doesn't mean you don't need the multiplication tables.)

Good point about working memory. And you are right if it is in memory you can read stuff that assumes you know it and just glide through without stopping.

For me the tricks like above were like a backup solution, using it a few times it became obvious that 9 x 3 == 27. Indelible. It is. For some cases it was like "It can only be 27 OR 26" and then I would use the trick figure out which.

But whether you use a simple trick and a trivial calculation or don't have to do that at all the point is the same it should not take much thinking which would cause you to lose your focus and train-of-thought, as you say.

This assumes that climbing the ladder comes easy. To me at least, it doesn't. It requires tedious labor, and lots of repetition for every single step. That's the main difference I keep noticing between me and people who say they like math or find it easy. They just look at each step of the ladder once, and immediately "get" it, sometimes even skipping steps. In contrast, I need to repeatedly step up and down the ladder multiple times, until I can take the next step.

My theory is that people who like math have a reward system that responds well to gaining an understanding on empirical concepts. I have that, and it does drive me to keep studying math. Not that I find it easy though, I don't think I'm able to skip steps, and I often have to repeat things I've already done before they sink in. The difference is that I find this process enjoyable, so I don't mind spending the time.

If I can compare to another activity, I've always wanted to be an artist as well, and have spend quite a bit of time trying to build up the skills. The problem is that, if I'm honest with myself, is I just don't enjoy the process of creative expression, it doesn't trigger any reward system that means anything for me. I wish it did but there's just nothing there. It was a hard pill to swallow, but I realized I like the idea of being an artist, but I don't enjoy the process. Hence my ultimately crummy artwork!

Sorry, I realized I'm talking about myself more than you, but I hope it's some help. The point I hope it makes is that everyone has a different personality, and from that different reward systems. It sounds to me like yours doesn't align with math, and that's fine. I wouldn't try to force yourself to study something which you don't love, at least if it's optional self study. Find subjects that you love learning, and the results will come naturally.

The difference is that I find this process enjoyable, so I don't mind spending the time.

This is definitely the difference for at least some of the people out there, however...

Imagine however that you do enjoy it at the start so you move on from topic Y to topic Y+1, then to to Y+2. However you find that you no longer understand Y and you need Y when you are doing trying to learn Y+3 so you study Y+3 and Y, now your progress in Y+3 has been slowed down.

Really your goal was to get o Y+7 though that is where you can start breaking new ground and contributing but as you try Y+4 and Y+5 the gains stop and maybe even reverse. You are now on a learning treadmill(perhaps sometimes falling off and having to restart too) redoing Y-1,2,3,4,5 not moving forward. Often it is possible to find a trick/skill/simplification/etc to continue moving forward to get to Y+6,7.

How long would you find the process fun on that treadmill though? I think it is common to not find covering the same ground over and over fun or never being able to make it to the point where you are part of peer group where you can contribute. An understandable result is when those people invest elsewhere, where they see better returns.

I think this is an excellent observation. Interest definitely matters, but so does talent.

I probably had crummy teachers in some places, but my experience was that math up through linear algebra made sense and wasn’t all that bad, but that calculus was a huge bag of “if it looks kinda like this try this thing, and if the result looks kinda right it probably worked, if not try this other thing” such that I could never form a framework for it in my head. Also didn’t help when teachers in some things would say “oh this is much easier and more straightforward with calculus”, even without a prerequisite for it, and proceed to only explain concepts with calculus half or more of the class had never learned. One of these days I need to find a way to get it the right way.

In hindsight, that’s because high school calculus doesn’t teach you how things work and just teaches you a bag of tricks so you can grind through problems. There’s a certain number of tricks you should know, I.e., you should be able to take some simple integrals and derivatives, but for higher math, you run into complicated things where the tricks don’t work or don’t exist. Some of the tricks are actually really useful, but you have to fully internalize where they come from, e.g., integration by parts just comes from rearranging the chain rule, and, if you know that, you can apply it to more exotic derivatives.

I did well in HS calculus but struggled in college math because the bag of tricks approach doesn’t work there. It took a lot of effort for me to undo the bad habits I learned from K-12 math and learn the good stuff, but it paid off.

Also, it’s well known that eventually professional mathematicians hate certain kinds of math. There’s the classic divide between analysists (those that do calculus-type stuff) and algebrists (those that do things like group theory, and linear algebra goes here). You don’t have to like it all, and something you don’t appreciate the first time you see it, you may enjoy later

IMHO when people say 'math is hard' they mean 'it takes more work than other subjects to be good at' - you're either a prodigy or you grind problems until you get the intuition. The easier subjects you can usually talk everyone and yourself into thinking you know them, or perhaps the ratio of memoization to practice is skewed more towards memoization. Maths is practice, practice, practice and then some more practice - blood, sweat and tears.

I’m not even sure it takes more work than, say, getting good with language. Hell, it might take less!

I think the main difference is that practicing language is far more rewarding for most people, than practicing math. They also have way more opportunities to practice it naturally, without even intending to do so.

... it is because they had bad teachers.

Math is easy if...

The one constant I observed in most parts of my mathamatics journey (math major in college, software engineering & computer science at university) was the lack of understanding by the person doing the math teaching that not everyone will be able to follow along if steps in the ladder are missing.

Words and sentences like 'it is obvious', 'clearly', 'as can be seen' should be avoided when teaching someone a subject as abstract as mathematics as inevitably you are not fully realising the size of the gap in knowledge between you and your students and how such statements can leave them feeling frustrated.

The recurring phrase in physics was "deriving this is left as an exercise to the reader."

"Bad teacher" often strikes me as a face saving excuse. Not that having a bad teacher will not make it harder to learn, but there seem to be a lot of bad maths teachers out there, if I go by how many times I heard that.

I mean it's okay to be bad at something. I sucked in history class, and I'm not blaming the teachers. I simply had zero interest in it as a teenager, unlike maths and physics.

Sure, but it's like saying "not dying of dehydration in a desert is easy. Just drink water!". Where do you find the water?

The clarify my terrible analogy, where do you find a curriculum that tells you exactly what to learn in what order? When you don't know math, you can't even tell if you ladder is missing steps.

Now tell us where to find the ladder with all the steps in order.

What do you mean by "trained mathematician"? I ask this because I always think that mathematicians are simply people do research in mathematics, if not they aren't mathematician. So no need to add "trained", what is an "untrained mathematician" btw?

I saw some people claimed on their twitter/blog that they are "trained mathematician" but I cannot find any single published contribution of them in mathematics.

And everyone I know who do research in math seems to agree all that "math is hard".

I'll bite, I think this is mostly bias. Strong evidence against this is that the average IQ of a mathematics undergraduate is, like, 125 or so, compared to ~115 for the average college graduate - that is just way too sizeable a difference to be explained by chance.

Math really does seem just plain hard for a great many people. It seems to me from having done some math on the inside like it also would get harder with each point downward in IQ than at a faster rate than most other valuable things in life.

I may be biased as I am a trained Mathematician, but I always feel when someone says "Math is Hard", that is because they had bad teachers.

I dunno, man. I have a PhD in complex differential geometry and think math is pretty hard.

One advantage math (and computer science) has is it’s entirely man made. You can start at the beginning and follow every step of the way.

The "math" used in ML/AI papers is usually just a sort of 'whiteboard math' which is a domain-specific mishmash of linear algebra, calculus, set theory and statistics.

If you could find a book just going through the relevant bits you wouldnt really have to "learn math again", it can be translated into english straightforwardly -- very very few ML papers relevant to industry have extended proofs, etc. that require eg., even being able to differentiate anything yourself.

90% of it is: here's the domain (ie., type) of our variables, here's the formula of our functions, we're taking a weighted average with some inner products involved.

It might sound like a lot of math, but it's really all doable in semester-1 of an undergrad course, were it focused enough.

I think you don’t realise how much math you know. I’ve heard of set theory, but never studied it.

sure, but you dont need to understand actual set theory -- it's just notation for mostly obvious programming stuff,

R means float, Z means int

R^2 is actually notation from linear algebra, but here it means a point is 2 floats with a measure of distance between points etc.

A lot of this could just be given in a "crib sheet" for tech people, and you'd get 80% of it straightaway.

It's years of work to understand this notation as used by the professional domains it was invented by, but it's a couple weeks for most "good, technical, software engineers" -- since they arent really using notation in much more than superficial ways.

Of course there are hard mathematical papers, etc. but they're rare in ML/AI mainstream papers; and not something most would read.

If you need to be able to read, eg., some sort of adv. statistical time series research in economic modelling, you'd already have the background to do that. If all you want is to be able to read 90% of the popular papers, the notation in them is just syntax sugar for things you could state easily in english or python

sure, but you dont need to understand actual set theory

How would you know? You sound like you understand set theory. Ironically, a body needs a rather extensive grounding in the theory and practice of set theory to look at something and think "I don't need to use the tools of set theory here" with deserved confidence.

A lot of maths is about having a huge repertoire of tricks that you know won't help solve the problem at hand. Saves weeks of fruitless attempts. But I'll bet you didn't notice the things you know not to do. That does matter for interpreting these papers.

> sure, but you dont need to understand actual set theory

How would you know? You sound like you understand set theory.

Easy: most mathematicians don't really care about set theory (beyond the fact that it exists and is the standard foundation).

Understanding set theory goes deep. For example: why does the Banach-Tarski paradox exist in ZFC (C: Axiom of Choice (https://en.wikipedia.org/wiki/Axiom_of_choice)), but not in ZFD (D: Axiom of Determinacy (https://en.wikipedia.org/wiki/Axiom_of_determinacy))? Or another one: why does the statement "every field has an algebraic closure" not hold in ZF (but in ZFC)?

Never heard of this stuff despite having studied mathematics? Just like I said: most mathematicians don't really care about set theory.

Not every mathematician needs to understand axiomatic set theory, but probably every working mathematician needs to understand at least basic set operations, simple identites like De Morgan's laws (I don't care if you know the name or not), as well as what cartesian products, relations and functions are.

every working mathematician needs to understand at least basic set operations, simple identites like De Morgan's laws (I don't care if you know the name or not), as well as what cartesian products, relations and functions are.

This is stuff that you learn in the 5th or 6th grade in school, and is about as far removed from what mathematicians call "set theory" as basic arithmetic operations are from college math courses for soon-to-be mathematicians.

that is a little bit of hyperbole. No school in 5th or 6th grade is giving formal definitions of relations and functions. There is a reason nearly all undergrad math books in analysis, topology, algebra, all devote an entire first chapter to it.

No school in 5th or 6th grade is giving formal definitions of relations and functions.

I remember that my math teacher pretty surely did.

There is a reason nearly all undergrad math books in analysis, topology, algebra, all devote an entire first chapter to it.

Indeed there exist multiple good reasons:

- recapitulation

- setting up the notation

- clarifying how the textbook defines the relevant mathematical objects, because the definitions of some concepts might differ depending on the textbook

- making clear what existing standard the textbook expects from the learner

- ...

how do you motivate formal functions and relations to 5th graders, and more importantly why? Relations are important because of the natural partitions of a set they create and the development of group theory. Functions are useful in calculus, but not really the algebraic properties, those are glossed over, i.e kids learning calculus are usually not learning the formalization of functions. That isn't important until analysis or abstract algebra, hence why its included in the textbooks

how do you motivate formal functions and relations to 5th graders, and more importantly why?

The word "you" in English has two meanings:

1. how would I motive this to 5th graders ("you" as "tu/vous" in French or "du/Sie" in German)?

2. how is this topic motivated in school to 5th graders ("you" as "on" in French or "man" in German)?

For 2: Well, it isn't. The pupils have to accept that in future, they will hopefully get why it is useful. Until then, better learn the material so that you won't fail on the tests.

For 1: If the prophet does not come to the mountain, the mountain must come to the prophet. If you need group theory to motivate functions (as you implicate in your answer), then teach group theory to 5th graders, so that the pupils get the motivation that they desire. If you additionally need to motivate group theory: well, I do know some quite interesting applications of group theory. :-)

Just to make it clear: I do have quite some experience in teaching mathematics, but to highly gifted students.

Functions are useful in calculus, but not really the algebraic properties, those are glossed over, i.e kids learning calculus are usually not learning the formalization of functions. That isn't important until analysis or abstract algebra, hence why its included in the textbooks

In Germany, there is no distinction made between calculus and analysis. At the university, this subject is taught from beginning on in the abstract way. In school, what you call "calculus" is often taught in a more "hand-waving" way by bad math teachers. Good teachers rather attempt to teach calculus/analysis in the abstract way in school.

fair enough, I was only speaking for the US, which has a slower pace and lack of motivation throughout k-12 + calc series. I still have no idea why they don't teach basic algebra with matrices, so it doesn't seem like a giant bad of tricks, or why linear algebra is so separated from multi-dim calculus, it really makes it more difficult with busy work and you never really comprehend anything.

It would be dangerous to teach anyone in ML what a function was, since they might realise, the premise of the entire activity is false.

(ie., that in the vast majority of cases there are no empirical functions to model, no f: Pixel -> Animal)

So better continue the current practice where few could distinguish a relation and a function; and fewer still are aware that they arent approximating empirical functions.

You may have learned this in 5th or 6th grade, but I certainly didn't.

The point is that any working mathematician is comfortable manipulating sets in algebraic expressions and that's not something you expect from your average high schooler.

I could have added that mathematicians need to know at least about different cardinalities, but I guess strictly speaking you could be working in discrete maths and not care about any of this.

He's saying you don't have to understand set theory to use and understand set notation.

In the same way that you don't have to understand number theory and abstract algebra to use and understand numbers and their basic operations (such as +, -, ·, /).

It would help if instead of saying "set theory" we emphasize the algebra of sets. Not by that name, but this level of teaching sets was common in middle school math late last century, not sure about this century, in the US. And this would be enough for most people trying to read technical documents using set notation.

https://en.wikipedia.org/wiki/Algebra_of_sets

What they’re saying is that you may be falling for the curse of knowledge. From Wikipedia:

The curse of knowledge is a cognitive bias that occurs when an individual, who is communicating with others, assumes that others have information that is only available to themselves, assuming they all share a background and understanding. This bias is also called by some authors the curse of expertise.

You know which parts of the AI math notation is shallow and which not. Someone else might not have that same knowledge, so they might not know exactly what to put on the crib sheet and what they need to go deeper on.

By the way, if you wanted to make such a crib sheet and publish it online, I think a lot of people would be very grateful!

Normalizing flows and neural ODEs are becoming a bigger thing, and they do involve some more heavy maths.

Yes, and if you look at neural operators, then you're in for something that even math majors need to put a bit of effort into.

Oh, I read that paper about a year ago. I feel like it could have been explained better, but the concepts aren't that hard if you've taken numerical methods.

While I think, that you may be right, that most papers are not about proofs, and that higher math is mostly about proofs, I still think it is math and important math at that. Perhaps it is a link in the chain between mathematics and engineering skills.

The proof centered math is indeed the difficult one. I struggled with a lot at university and was completely unprepared for that finishing school. So I think I have a good idea about the difference in difficulty.

We probably think fluently of these mathematical structures in tangible, real world terms. Still it requires some practice translating to the mathematical abstractions, and still further practice thinking in terms of the abstractions altogether. I don’t know how relevant it is that the textbook content can be covered in a single semester.

This is definitely true. It's largely just notation for communication.

I just paste sections of dense math from AI/ML papers into chatgpt and it explains it. Almost none of it is complicated. It's just really awful notation.

If you could find a book just going through the relevant bits you wouldnt really have to "learn math again"

Hard to find a book or a course doing applied math just for AI and ML. I was interested in learning the math needed to understand quantitative analysis and I was lucky to find some resources doing just that.

I've been doing similar for about a year. My target is to learn the math needed to make 3d games, so basically algebra, geometry, calculus and linear algebra.

I started with brilliant.org, and while I liked the level of polish in the interactive lessons, I found the lesson structure to be out of sequence, often referring to things that haven't been covered yet. They didn't seem to have put as much thought into pedagogy as Math Academy as described in TFA.

So I gave up on that and instead have been shipping several kilograms of dead tree across the pacific in the form of The Art Of Problem Solving series of textbooks. They are great, the lesson structure and building up of complex ideas from first principles is outstanding. They will humble you though, as the exercises are tough. They're also quite expensive but IMHO worth it.

Math Academy does look interesting, If I was not halfway through my series I would probably take a look. But I do enjoy having reference books on hand. Many times I've jumped back to brush up on a topic that has slipped from memory.

I solve my exercises with the most low tech solution possible, but I like the freedom it gives me to try new approaches, and nothing beats the latency between idea to ink on paper.

edit: also wanted to add that I've enrolled Chat GPT4 as my tutor. Contrary to many other's experiences that I've read, I find it to generally be very good at reasoning in this level of mathematics. It's helped me many times when I've gotten stuck. And on the occasions where it bullshitted its way to an incorrect answer, I always challenge it if I don't understand, and we ultimately find out if it hallucinated something (rare, can usually be fixed by restating the problem), or I gave it the wrong input to start with (unfortunately more common than I'd like)

I totally agree with you on the value in using Chat GTP when stuck.

What's the scope of The Art of Problem Solving? How far does the series go?

AOPS audience is gifted high school kids, so it doesn't get up to the college level. The core texts are:

- Prealgebra

- Intro to Algebra

- Intro to Counting & Probability

- Intro to Geometry

- Intro to Number Theory

- Intermediate Algebra

- Intermediate Counting & Probability

- Precalculus

- Calculus

Are you doing the online classes or only the books? I wanted to register for the online classes but they seem to be heavily oriented towards interactive learning.

Just self study with the (physical) books. I did also try the ebook combo for the Prealgebra book, but I found typing latex in the answers to the exercises was cumbersome.

I think the online classes with interactive lessons is a separate thing, but I don't have any experience with that.

The ones that have “instructors” and class times have chat-based sessions that you can skip if you prefer. Part of the homework is based on an adaptive problem system (Alcumus, which you can actually use for free) and part is weekly problem sets mostly based on the textbook. Writing (proof) problems are graded by a human so it is a useful way to get feedback on your proof-writing skills (if you know you are worse at it than a college math major).

I think it's a weird way to learn math, and I learned it this way in school. Most of these courses just teach information memorization and recall. sin(x)^2 + cos(x)^2 = 1, etc.

I would start with something like Elementary Analysis: The Theory of Calculus, and work from there. You'll eventually arrive at the same place -- Calculus but from a much stronger mathematical foundation.

I have not found that to be the case, the books I have read have gone into deep foundational detail to build up knowledge. Perhaps you're referring to Vol 1 & 2 of "The Art Of Problem Solving"? I haven't read them but from what I know they are a distillation of core concepts for students looking to do competitive maths.

It's confusing because that title is also the name of the publisher / website of the series of the books I'm reading.

You learned using the AOPS books? Don't be fooled by the titles, these books exclusively use a proof-based approach to construct a pretty wide foundation around these topics.

Ah, okay. I actually took calculus in 8th grade. I studied another two years past that, dropped out, and then later did a complete 180 and graduated with a literature degree.

I'm now over 40 and interested in relearning the math I learned long ago and pushing a bit further than I had before.

There’s also an intermediate number theory class that’s basically at the level of a college elementary NT course (one that does not assume abstract algebra), an Olympiad geometry class, and a group theory class. The first two do not have a text, the third has a text but you can’t get it without enrolling.

I bought the whole set for my kid. He's also doing Brilliant.

It starts at somewhere that the kids are at the end of primary school (at least in the UK) and ends somewhere in high school. My kid could already do all the pre-algebra stuff, so that book went fast. The way I see it, the kids waste a lot of time in the middle years when they already know the arithmetic and pre-algebra, but might as well be doing a bunch of more interesting things.

Are you using any of the stuff you're learning for whatever practical 3d game-making things you're working on? Just curious how it's working out, you've picked a pretty broad foundation as a starting point.

I took a brief detour late last year to study "Linear Algebra: Theory, Intuition, Code", and to my surprise it stuck pretty well. The author said the pre-reqs were just "basic high school math", but I'm glad I had recently done lots of algebra and geometry, as the difference between that and some vague memories of stuff I did 30 years ago in school is pretty wide.

I haven't started any 3d game projects yet. For that, my plan is to do the webgpufundamentals.org course first. Scanning the TOC, I think I would be able to attempt it from what I learned from the linear algebra book.

That said, I'm doing AOPS Intermediate Algebra at the moment, and the Precalc text covers more advanced trig and matrix stuff, so I'm thinking it would be good to finish at least to there before starting to apply the knowledge.

Yeah, it sounds like you're not far from the point where you can start jumping ahead and working backwards to fill in the bits that you're missing - that's what many people naturally and instinctively try and it can work but can also be frustrating if one misjudges one's degree of proficiency. You don't often see 'I'm just going to give myself a full secondary school maths refresher' which is more demanding on time and self-discipline but at least we know it's pretty reliable given those things.

I'm in that camp and can suggest a few recommendations in order of:

https://d3dcoder.net/ -- The DX12 book is the latest edition. The books have several chapters at the beginning covering 3d transformations.

https://foundationsofgameenginedev.com/ -- The first installation, Mathematics. This will cover a lot more ground and derive things from first principles while not being overly formal.

https://www.mathfor3dgameprogramming.com/ -- A lot more formal than most game/graphics math books, and goes into more depth, particularly on the linear algebra.

i was motivated by the exact reasons you are but after a few years of maths i started to like that more than the 3d games and programming :(

It looks how do do proofs is at university level in the US. I wonder why other countries start to teach kids how to do proofs from grade 7. They start with rigorous proofs in Euclidean geometry, the move to solid geometry, then to proofs in elementary functions, sets, number theories, and etc, then to polynomials and simple discrete maths and analytic geometry. By the time a kid graduates high school. using things like proof by induction or by contradiction is like a second nature. And no, I'm not talking about elite kids, but curriculum requirements for all the STEM students.

We did proofs in 7th grade as part of geometry, in a semi-rural United States middle school.

Can't say I remember 7th grade ... but we were doing proofs in Geometry in High School. I also lived in a semi-rural US neighborhood.

10th grade is the “on level” time to take geometry.

So US school kids don't encounter any geometry in school till they are ~16?

Shapes, areas of simple polygons and circles, Pythagorean theorem, volume of solids, graphing, translation, all that stuff and more that one might call “geometry”, is scattered between roughly ages 6 and 12.

I think the high school level “geometry” class—which may be the only one named such, but primary school math is full of geometry—is an atrophied organ left over from when it was still common to teach directly from Euclid, which is why it tends be about introducing proofs more than covering new abilities and techniques in geometry (though it may cover some of that, too)

Thanks for the explanation! That is a lot less surprising.

No. The formal geometry class is usually age 14/15 (first or second year of high school). They should have seen a less formal treatment of geometry before then, though they may not know it.

Good to know. I guess education in the US has local standards? I have this impression of the US not teaching proof because in a number of introductory math classes in my college, the profs dedicated chapters to teach basics of induction, how to write proofs, and etc.

I guess education in the US has local standards?

Very much so! The US didn’t have a Department of Education until the 90s.

90s? You're off by a bit. The current Department of Education started in 1980, but it came out of the Department of Health, Education, and Welfare, founded in 1953. That, in turn, came out of the FSA which included an Office of Education (1939). Prior to that, the Office of Education had been part of the Department of the Interior. Before that it was its own Department, starting in 1867.

Though "90s" is delightfully vague. You're either off by nearly 1800 years, you're very old and meant 1890s and were only off by a few years, or you really meant the 1990s and were off by nearly 130 years.

Thanks. I have no idea where I read that, but have been carrying around that “fact” for over twenty years!

I would have still appreciated that refresher in a college class on content I learned 7-8 years prior.

Every state sets its own curriculum, with varying degrees of freedom for the districts and schools within the states. The federal level has standards, which students are generally tested against in various standardized tests, that are tied to federal funding.

Multiple curricula can satisfy the same standards, at least on paper if not in practice. So states are, more or less, free to teach things how they want. However, they're also strongly driven by the textbook industry, which turns on the two biggest textbook purchasers: Texas and California. So a lot of the textbooks (and associated curriculum material) available for purchase in the rest of the states are driven by whatever those two states are pushing.

The “standard” place for geometry is 10th grade (after algebra 1 in 9th grade). A few geometric ideas have been moved from geometry to algebra 1 so that slope can be explained using similar triangles but this is just about memorizing explanations instead of doing proofs.

Today, if you aren’t in an honors geometry section, you likely learn a handwavy version of two-column proofs and do some pretty linear proofs that way. No symbolic logic, no mathematical writing.

which other countries? that's certainly not the case in Germany. you might see some proofs in geometry, and maybe some informal derivations of things later on, but "induction" is a university level thing (it's possible that some kids that focus more on maths in school see this kind of stuff earlier, but it's definitely not a universal experience).

I’m a thirty-something with an arts degree who decided to learn math. Basically I was tired of reading popular science and being fed metaphors to understand concepts. I wanted to “see” it for myself. I spent time online at Khan Academy and friends for a year or so on and off. It was fine but meandering?

So I enrolled in community college! It’s great. I have a clearer path, immediate feedback, teachers, and an obligation to do work that keeps me on it.

Ultimately my plan is to get enough transfer credits for university and spend this decade slowly working toward a bachelor of science in physics.

working toward a bachelor of science in physics.

Thats cool, I kind of want to do that. But also Im stuck with wondering, I put all this work into that, what do I do at the end?

Then you will have gained insight into how the world works on a fundamental level. Isn't that something in its own right?

I can only speak for myself, and did go on to get a PhD, but even on a bachelor level, studying physics changed how I see the world and how I think.

It is, which is why I was interested in the first place. But physics the way it is taught is pretty rigorous and study is intense for someone with a passing curiosity. And idk if there is a way around the math for some of the stuff.

You could get a PhD in physics, and if you time it right, you will die right after graduation, the moment when all of your classmates realize they're staring down 40 years of software engineering. The tombstone could read, "I figured it out."

Some things are enjoyable in themselves, without needing a “so that”.

100% Agree. Im a big proponent of developing self-thought skills and making the best of e-learning, but I think math (alongside languages and other) are one of those areas where the real time feedback makes traditional education still worthwhile.

It's hard to replace it with just books and online courses. Not impossible but harder.

Funny. As a college student I found face-to-face interaction was much more important to studying social sciences and humanities than to studying math or theoretical physics or computing which could pretty effectively be studied in a book.

I actually find the immediate feedback from khan academy exercises better than school actually. The feedback loop is much more tighter instead of waiting an entire week later for results and not really getting another go at it. I also tend to zone out or miss something in the lecture, and that missed thing is what builds the entire foundation for everything else and then the entire lecture is pointless. With video based lectures, I can pause, go back, play 2x speed, look at multiple well produced explanations, etc. Ultimately I don't find face to face all that valuable a various amount of subjects.

Replacing lectures with video lectures is fine, even an improvement. But not such much for exercises/problem-solving/practical classes.

Khan used to have something called "world of math", which would take you through the entirety of Khan academy "backwards". Eg., you would do problems, and only take lessons when you couldn't complete a problem.

While the first 1 or 2 weeks felt more like a case study on how to teach number literacy to children ("how many elephants are there in the picture?") it soon became real calculations and later on all the branches of math. I did it for a few month leading up to my masters, and it was great.

Videos are good to get some barebone knowledge, but are too superficial or unstructured to replace classroom settings, with graded exams and homework.

That's awesome that you're working towards this goal!

Side note: community colleges are an incredible deal, and since starting a woodworking class at one, I could easily see myself taking one or two classes at a time basically for decades.

It was fine but meandering?

I think that's a good point you sound like me I need to be pushed. I sued to read a lot of magazines and books on all kinds of subjects. Then came the Internet it has everything available but for me structure is missing.

If OP still wants to learn the mathematical foundation of transformers, I built a free alternative learning tool: https://afaik.io/nebula?mode=nebula&category=blueprint&id=ed...

It's also based on an underlying knowledge graph, connecting concepts across various subjects like maths, machine learning, physics, etc. You can check the graph for transformer here: https://afaik.io/nebula?category=brickset&id=VLlOnZLl&mode=d... (only available on desktop...

Basically, it frees you from learning maths from scratch and just learning the prerequisites required to grasp the concept, and there are free resources attached.

Don't get me wrong, I can totally relate to the desire to relearn maths. One of the reasons that I'm building this tool is for me to relearn physics and know how to get there with my maths and cs background. I just feel in this specific scenario there might be more effective ways to learn in depth and have fun at the same time.

I find the concept of "underlying knowledge graph" interesting. What does it mean?

I assume it means such a graph connects the topics together as "pre-requisites". To understand A you need to already understand B and C, and to understand B you need to understand D and ... etc.

But the thing about such a graph is that really it must be a tree, not just a directed graph. Why? Because there cannot be cycles in it. If to understand A you need to understand B, and to understand B you would need to understand A, you could never understand either of them. Right?

it must be a tree, not just a directed graph

It may be a tree. But it must be a DAG (directed acyclic graph).

Heh knowledge graphs sometimes don’t feel acyclic, at least not to me anyway. Sometimes I’m stuck bouncing back and forth :)

If it's prerequisite relationship, you need to make sure that when A points to B, and B points to C, C doesn't point to A. Otherwise you're creating a loop.

it may be because the the "graph" is not accurate

100% agreed

Right, it must be acyclic. Which means it can be presented as a tree with some duplicate nodes. The important thing is the student must understand in which order they can try to understand the topics.

If to understand D you need to know both B and C, each of which requires familiarity with A, the graph is not a tree

Right, but if it is not acyclic, in which order should I try to understand them all?

If it's not acyclic then you haven't broken down the knowledge graph enough. But that's probably a waste of time, trying to come up with a perfectly ordered plan of study for all of mathematics. When you find an apparent cycle, it means the two domains are strongly interrelated and you'll be studying part of one, then the other, then the first again, repeat until you're done (whatever that means to you). No need to try and break every subject down into one-week or one-day chunks and finding a perfect ordering, just figure out the roughly course-lengthed chunks of study and start working through them, concurrently if needed as described.

Right when you read something you don't need to understand it all to understand something which may be needed to understand something else elsewhere.

But still I think it would motivate me to keep on learning if somebody could show me an accurate acyclic pre-requisites graph and tell me: "These are the thing you need to understand before you should go to the next topic. If someone could come up with the time to come up with an accurate acyclic "knowledge-graph" it would help millions of students of mathematics.

If you try hard and long enough you will understand what you're trying to understand, you will. The question is what would make that more fun and less tedious. It is about precision and not needing to learn something you don't need to learn, to understand something that you need to learn. Spend your time on learning stuff you need to learn to understand what you want to learn.

"underlying knowledge graph" is a directed acyclic graph (DAG), based on prerequisite relations among topics. So you are right that there cannot be cycles but it's not a tree either because a tree (technically) only allows one parent.

So... basically an ad for Math Academy?

How about some free resources like Khan Academy?

That's basically what it is. There is nothing to learn from this post other than "smash that beta sign-up button".

Has anyone tried that course? Is it any good?

I have been doing the Math for ML course and would recommend.

I have UK A level math but not Further Math, so up to basic calculus. But I forgot most of it and so Math Academy has me going through a lot of the Math Foundation units along the way.

I was initially put off by the monthly price, as it is quite steep. The clincher is that about a year before starting Math Academy I had gone through the Open University’s MST124/125 textbooks (covering the same stuff as Foundations). Except even after a year I’d already forgotten most of it.

Math Academy learning feels much more robust, since it includes spaced reviews and regular tests. I record things in Anki but it’s useful to have regular practice questions too. I also use ChatGPT to spell out things and find it works well at this level.

Some things I’d like Math Academy to have:

- ability to skip lessons (I don’t want to spend ages going over symbolic integration again)

- a reference page to track unlocked material, maybe with Anki integration

- fewer multiple choice questions and more in depth problems

- proof-based math. I’m told this is coming but the degree-level courses have missed their estimated due dates.

I will definitely finish Math for ML and then do linear algebra and multivariate calculus. You’d still need a good textbook to do them rigorously, but I think Math Academy sets you up well.

Hi, I'm Alex, curriculum director at Math Academy.

Thanks for your comments. In response to the things you'd like us to have:

"ability to skip lessons" - we plan on introducing "mini-diagnostics" sometime soon, hopefully within the next few months. This will allow students to "place out" of certain content they know. The primary diagnostic assessment will have done most of the grunt work here, but mini-diagnostics can be used for fine-tuning the knowledge frontier.

"a reference page to track unlocked material" - This is an interesting idea that we can discuss.

"fewer multiple choice questions" - We're actively introducing "Free Response" across the entire curriculum. Complete coverage across all courses will likely take several months, maybe over one year. Many of our lower-grade students should be seeing lots of free-response questions already.

"more in-depth problems" - we have multipart problems in most courses. We plan to add many more. Introducing "challenge problems" into the curriculum is also something we have planned for the near future.

"proof-based math" - We plan on launching our "Methods of Proof" course within the next 6-8 weeks. This course is designed to introduce students to all fundamental concepts related to proof building: sets, logic, functions, relations, cardinality, proof by induction, direct proofs, counterexample, contrapositive, contradiction, and trivial and vacuous proofs, to name a few. Most of the content is already ready. We have a few technical challenges to overcome before it can be launched due to our new "proof" question format, but we have a clear idea of how these challenges are to be resolved, so 6-8 weeks is certainly realistic.

"a reference page to track unlocked material"

If this is implemented to resemble an "upgrade tree" found in games, I bet it could work as an extra motivator for the kid audiences.

...and for some adults. As a gamer particularly when I was young, this is catnip for me :D

Yes, it's very good.

Math Academy is much more dense and on-point that Khan's. You don't have to sit through 15 minutes of video when 2 minutes worth of text explanation does it.

It uses spaced repetition for topics that you aren't good at, and for recently learned subjects.

The topic dependency tree and automatic progressing to "unlocked" topics is obvious in retrospect, but here it's done very cleanly and unobtrusively.

The initial evaluation test is worth its weight in gold. It eliminates the need to grind through things that you already know, but still covers any gaps.

I had kids on Khan for few weeks and it was a hassle. The pace was too slow, too much time sunk into trivialities and they were bored most of the time. With Math Academy they sit down, they do their 20-30 min of focused hands-on effort and they are done for the day.

Math Academy is much more dense and on-point that Khan's. You don't have to sit through 15 minutes of video when 2 minutes worth of text explanation does it.

This is precisely what bothers me about KA. I guess they're trying to ease into the topic, but I find that kind of repetition annoying and distracting.

It's really good for its goals. I've used it for a few months and was really happy with the results. The spaced repetition aspect worked perfectly. The courses are still being worked on - already 99% there with quality, but you can report any issues and they get fixed. Just keep in mind that the target is largely students, (at least at the moment) so the aim is mastery of the subject - if you're interested in learning the concept but not actually doing a lot of practice of using it, then it may not be the right service. And there are magic internet points / leaderboards if gamification is something that works for you.

The exercise sizes are also very small almost all the time. That means instead of a whole topic at the time and figuring out where you left the last time, you can do as much as you want at a time and not be restricted by artificial "chapters".

Replying to myself since I can't edit: thanks for the feedback. I am compelled to look into the course after reading all the replies here.

It's very good. I've tried it after hesitating a bit because of the price tag compared to Khan Academy — no regrets.

K.A. is great and I still use with my kid, but M.A. is more condensed and to the point for my needs. I was properly guided through the first program choices according to my profile, and the diagnostic exam you start with was perfect to highlight what I actually need to work on given my limited time.

Explanations and courses are super condensed, with the right amount of example and pedagogy that clicks for me.

I did a year and a half ago before getting too busy with work. I found it to be a bit spartan, but still the most efficient tool for math study I've used.

I was a math major long ago, so it was more a case of relearning than initial learning for me but the built-in SRS helped a lot and so did the granularity of the lessons. It's head and shoulders above Brilliant, IMO.

If it didn't exist or I couldn't afford it, I'd probably go the OpenCourseWare approach. https://ocw.mit.edu/

This is basically an unsolicited ad post. It communicates nothing of substance and the entire thing amounts to "click the sign-up button". It doesn't even cover the course contents. How this makes it to the HN front page is mind-boggling. Goatse would at least be more entertaining.

Thank you for bringing that visual memory back.

Why do I struggle to remember my anniversary date but have that image burned into my brain.

You need an AI art of your anniversary date in the form of a goatse

Adding this to my arsenal of memorization tricks.

They don't teach that in schools.

Goatse set up shop on a plot of mental real estate nothing else would go near. its in a bad part of town, but acts as a kind of landmark being the only object anything like its kind for miles in all directions.

edit: removed the unintentional reference to the url; I did not expect it would still be active!

On a very apt topic for HN readers, though, who likely enjoy math to some degree and have heard of LLMs. What if the post inspires someone to look into revisiting actively learning math as an adult? Is that so bad?

Yes, I agree, it reads like an ad. But I did in fact find it inspiring.

For all I know, maybe it is a targeted ad. I am a software engineer at a large company. I was selected and flown to HQ for training on integrating LLMs into applications. I am currently building systems that support our data scientists.

I’ve tried picking up more math skills a few times. But I’ve never taken trig or calc.

I’d like to understand ML and LLMs better, but I feel like I’m not even sure where to start with trying to learn math. For I have a family and a job as well.

So the adult track of that Math Academy site does seem like something I would try.

but I needed a good reason that would justify the time investment

It should be understood though that there are cases when math(-like) language is abused, leading to overcomplication and obscurantism [1]. In mathematics, there is always the temptation of formalizing for formalization's sake. Indeed, 99% of pure math is non-constructive ("there exists a group such that", "the algorithm converges in O(N) steps"), as opposed to the practical CS and applied math ("here are the runtimes on real world data") which are likely the primary concerns of the HN crowd.

None of this can diminish the sheer impractical appeal of pure math and pure CS, not unlike that of poetry, but I would rather not oversell either of the two.

[1] A good illustration is a rant by Cosma Shalizi at http://bactra.org/notebooks/nn-attention-and-transformers.ht..., recently posted on HN.

leading to overcomplication and obscurantism

Mathematicians attempt to express ideas in the most readable and clear way possible.

It's actually code that must be obfuscated by the constraints of the language and computer. Mathematicians have no constraints preventing them from presenting something in the way that makes the most sense.

The part that can be called "Obscurantism" is when they use a high-level abstraction you are unfamiliar with. This is mostly driven by the audience.

Indeed, 99% of pure math is non-constructive

Citation needed?

the algorithm converges in O(N) steps

That doesn't sound non-constructive. Even the group example is usually done by constructing such a group.

Also the best part about math is you can use it to approach the problems you want with constraints you want. Knuth uses math to solve real CS problems.

No, "there exists a group" is not constructive, in general. You just need to prove that non-existence leads to a contradiction. The whole deal about "constructive mathematics" is to have that if you can prove something exists, you can also construct it.

I think the success of the constructive mathematics program is really debatable, but in any case I don't think it leads to more 'natural' mathematics.

(The terms used by GP are very confused and I agree with most of your reply)

You just need to prove that non-existence leads to a contradiction.

Indeed however this is the exception, not the rule. The general way to do an existence proof is to construct it.

Mathematicians attempt to express ideas in the most readable and clear way possible.

I buy this from the mathematicians and scientists that I know and have interacted with.

I also think mathematicians spend more time trying to discover/play with new math than optimizing the communication of what already exists and is communicable.

My speculation is that this naturally leads cruft that needs to be worked through by people entering the field. The cruft can not get too big or people don't enter the field so people are motivated to keep the cruft below a certain level but not the minimum.

The cruft makes it harder to enter the field and once you have over come that hurdle you move on to do things in the field not reduce the cruft.

Other things that make it hard to reduce cruft

1. not everyone is going to agree what is cruft

2. Person X spend time on reducing cruft in sub field Y may find out that Y is no longer hot topic so while there is less cruft there are not many people taking advantage of the reduced cruft in Y.

3. Mathematicians and scientists are reward more for new and interesting things than better pedagogical practice/techniques.

4. Optimizing for communication/pedagogy is mostly a different skill than science/mathematics so you have to split your focus or not dive as deeply into one or both.

5. I am sure there are others.

This seems reasonable to me. It is s system where most everyone is well meaning and want to improve things and where things do improve over time, but where it is still easy to find areas that would benefit from substantial from improvement.

Man, this article is an ad for Math Academy. I'm 100% sure that at the level of what Math Academy teaches, you don't need to worry about "non-constructive" or "pure math".

Indeed, 99% of pure math is non-constructive ("there exists a group such that", "the algorithm converges in O(N) steps")

Almost all mathematicians work with classical logic, but that doesn't mean that they always use all of its power. On the contrary, most of what you would see in an undergrad math program goes through constructively with at most a few minor modifications.

To this day, I fail to understand why some people cannot enjoy constructive mathematics or practical engineering without shitting on traditional maths or CS.

The "runtime on real data" thing is a trope by now, an algorithm that is exponential is in general not going to miraculously be very fast on "real-world" data, and even if it is, chances are, it won't be anymore once you change your data (with some few exceptions like the Simplex algorithm).

Can anyone help me with some questions about this program? (I assume the founders will see this thread once they notice the HN hug of death)

1. How exactly is AI being used here? Is there an AI chat-bot that I can ask for help? Do you generate problem-sets with AI? Check answers with AI? Is it GPT-4?

2. Do you utilize Spaced-Repetition in any way? Have you found that to be useful?

Thank you

Hi there, my name is Justin Skycak, I'm the Director of Analytics & Algorithms at Math Academy, I developed all of our quantitative software, and I'd be happy to answer your questions.

1. The AI is more like an expert system that emulates the decisions of an expert tutor with regard to what tasks a student should work on at any given point in time (what should the student learn next, what do they need to review). There's a knowledge graph that encodes structural relationships between thousands of math topics (such as prerequisite relationships, but also other types). And then there's an algorithmic reasoning system that looks at a student's answers, overlays them on the knowledge graph, figures out what the student knows (and how well they know it), and decides what learning tasks are going to move the needle most given their personal knowledge profile. The decision-making is inspired by cognitive learning strategies such as mastery learning, spaced repetition, interleaving, minimizing associative interference.

2. Yes, spaced repetition is a core part of the system. Each student has a personalized spaced repetition schedule that adapts to their performance on each topic, and when choosing what topics a student should review or learn next, we're always trying to implicitly "knock out" as many due reviews as possible to maximize learning efficiency. (For instance, if a student is due for a review on one-step ax=b equations, we can implicitly "knock out" that review by having them learn two-step ax+b=c equations instead.)

From a quantitative standpoint, the spaced repetition model was one of the more challenging (but equally fun) parts to build. You normally think of spaced repetition in the context of independent flashcards, but in a hierarchical body of knowledge like mathematics, it gets really complicated because repetitions on advanced topics should "trickle down" to update the repetition schedules of simpler topics that are implicitly practiced (while being discounted appropriately since these repetitions are often too early to count for full credit towards the next repetition).

Our spaced repetition model not only accounts for implicit "trickle-down" repetitions but also minimizes the number of reviews by choosing reviews whose implicit repetitions "knock out" other due reviews (like dominos), and calibrates the speed of the spaced repetition process to each individual student on each individual topic (student ability and topic difficulty are competing factors).

Why am I not surprised that a marketeer shows up in an 'organic' posting about a company.

FYI this is the guy that wrote the comment you replied to: https://www.justinmath.com

He's true math nerd, and definitely not a marketer. In fact, I don't think a single marketer works at the company. It's a bootstrapped labor of love and I've been following their journey for almost a decade.

I get the skepticism, but some things are legit.

I'm sure there are math nerds that work in marketing.

Your whole blog post comes across as an advert, that may not have been your intention but that's what it looks like to me, legit or not.

Don't be nasty.

Especially ridiculing a tech person giving detailed and interesting answers as a "marketeer".

"Director of Analytics"

Very cool! Thanks for the reply.

It would be awesome if you could also add GPT-4 as a kind of helpful tutor. Not sure if you're already experimenting with that.

Our spaced repetition model not only accounts for implicit "trickle-down" repetitions but also minimizes the number of reviews by choosing reviews whose implicit repetitions "knock out" other due reviews (like dominos), and calibrates the speed of the spaced repetition process to each individual student on each individual topic (student ability and topic difficulty are competing factors).

That's super interesting and definitely one of the issues I faced while building anki cards for math classes I took in undergrad. Thanks again!

$49/student/month? That's excessive.

Compared to what? They have a track-record of students not even old enough to be in high school yet passing AP Calculus BC exams.

Here in Taiwan, where I live, it's not that uncommon for people to pay 5x that price per month on supplementary math courses for their kids.

https://twitter.com/_MathAcademy_/status/1708542077695574292

Compared to Kahn Academy, which also has a proven track record, is free, and has been around for a long while.

If your time is worth even $1/hour, the "free" Kahn Academy option will be far more expensive than this program.

I respect what Sal Kahn built, especially in the early days, but it's just not anywhere near as time-efficient.

The tool might be good but there’s also strong selection bias due to its price.

This is cheaper than lessons with a teacher, so it's hardly excessive. (Even if higher than other learning services)

If it works, it is a bargain

It's not cheap. Given that it sounds like this is something you want to deliberately spend time on rather than just an app to use when you're bored it might be worth it but probably not so much if you can't use the knowledge professionally somehow.

> The ‘Foundation Series‘ is what I’m starting with. It’s for adults to help streamline learning (it skips the stuff that kids need, but adults don’t) and work back up through college-level math relatively quickly (emphasis on relatively ).

I'm curious to know what 'stuff' he's referring to. And what about it makes it such that kids need it but adults don't. And if that's true, are we SURE kids need it?

I had horrible math teachers growing up and always thought "I just don't have the 'math gene'." I eventually disabused myself of that thought and set out on my own (re)learning journey. Could it have been less arduous had I skipped the stuff I didn't need to know because I was an adult?

Hi there, my name is Justin Skycak, I'm the Director of Analytics & Algorithms at Math Academy. I can speak a bit as to the stuff that's skipped in the Foundation Series.

After developing a curriculum that covers all the standards for 4th grade through AP Calculus BC, as well as plenty of advanced university courses (many of which are still under construction, but the structure is mapped out pretty comprehensively), we found that roughly a third of 4th grade through AP Calculus BC topics were not actually prerequisites for university math. So, we created a streamlined Mathematical Foundations course sequence that cuts out those topics. Those topics are necessary to check the box on grade-level / common core standards, but they're not really necessary for adult learners who want to pursue advanced university courses as soon as possible but lack the necessary foundational knowledge.

I'll also send your question to my colleague Alex Smith, our Director of Content, who designed the Mathematical Foundations courses himself and can elaborate more on the specifics.

Thanks for responding.

What are some examples of topics that you cut out from the high school math curricula? I have seen modern Algebra II courses remove conic sections in order to make more room for probability and statistics.

Hi, I'm Alex, curriculum director at Math Academy.

As Justin mentioned, there are several criteria that we must meet in our high-school pathway that aren't needed for studying higher-level (e.g., undergraduate) math, or they can be postponed. We decided to remove some of these in the Foundations series.

The idea behind the foundations series is to provide adult learners with the most efficient path possible to get onto the higher-level material.

Examples of topics that were removed from the high-school series to create the foundations series include some of the following:

* Various Geometry topics: All of the _essential_ geometry is covered. However, we removed topics on inscribed angles, Thales' Theorem, Triangle congruence, and similarity criteria (apart from the AA, which is the only one that seems to come up in practice), midpoint and triangle proportionality theorems, a fair amount of solid geometry, except what's fairly standard for calculus (volumes and surface areas of spheres, volumes of cones), lots of stuff on different types of quadrilaterals.

* Conic sections: The essentials are covered in both pathways. But in the high-school path, we go into a little more detail about foci, directrices, eccentricity, and utilizing their geometric definitions (e.g., focus-directrix properties).

* Trig identities and Equations: Covered in both pathways, but the high-school versions go into more detail and consider more cases.

* Some word problem/modeling topics.

* Other arbitrary Prealgebra topics: Divisibility rules, going into more detail about ratios in contextual settings, scientific notation, and some basic data representation topics that one would normally meet in Prealgebra.

* Slope fields. This will be covered in our upcoming differential equations course.

* Some analytical applications of differentiation that are quite specific to the BC Calculus exam: Identifying and removing point, jump, and infinite discontinuities and analyzing graphs of first and second derivatives.

* There are also fewer topics on related rates and optimization, though these topics are still covered.

* Some contextual applications of integration, like volumes of revolution and volumes of known cross-sections.

* Convergence tests for infinite series. When we get to that, these will be covered in real analysis, but other than infinite geometric series (which _is_ covered in Foundations), these tests don't show up too often anywhere else.

* Some ODE models, such as exponential and logistic growth and decay. We cover ODE basics in the foundations course, but particular models will be covered in the differential equations course.

* Taylor series. Again, this can be covered in the differential equations course for anyone wishing to take that course when it's ready.

Happy to answer any further questions you may have.

Thanks for the detailed reply.

This largely makes sense to me. Stuff like jump discontinuities I've only seen as an exercise for calculus classes.

Sad to see Taylor series go but that is kind of a dangling topic in an intro class and could be picked up later when there is a need for it.

Removing Taylor series was a tough call. It's one of my favorite calculus topics topics. Something had to give. However, those topics will still serve as prerequisite material for courses that explicitly need them.

> I'm curious to know what 'stuff' he's referring to. And what about it makes it such that kids need it but adults don't. And if that's true, are we SURE kids need it?

OP here. My understanding is that it's the stuff kids are tested on in school to pass (like standardized tests), but not necessarily needed for an adult to meet their learning goals.

Like, if your kid was using it they'd take take the grade level courses, but if you wanted to work up to Math for Machine Learning like I am you'd take the Foundation courses, which are streamlined.

One cool thing I like that I shared a screenshot of in the post is the knowledge graph that shows all the topics and how they are connected to make all of the lessons feel more purposeful. And if you get stuck somewhere there's an easy way to brush up on past lessons (dependencies).

I tried math academy for this exact purpose, but I had to give up because it didn't let me test out of lessons that I was already good at (Similar to how duolingo would allow you to do). So I ended up wasting a lot of time sitting through all the exercises which I didn't need.

You could've asked them to add this option. They are very responsive and reasonable.

I did. They replied a generic corporate answer that we do the best for our customers and they said I’m free to cancel my subscription

Hi, I'm Alex, Curriculum Director at Math Academy.

It is on our radar to allow students to "place out" of certain topics and modules if they feel ready. We'll call them "mini-diagnostics" or something similar when they're ready. I believe it will be worked on within the next few months.

Unfortunately, these things do take time to implement, but we are listening. FWIW, we're a tiny, bootstrapped company with literally two programmers (Justin, our ML/backend dev, and Jason, our founder and solo UX/UI) working on the entire codebase.

The content team is a little larger. We have around 12, mostly PhD mathematicians, working on the content, which is why the context is probably a little further ahead in some respects.

Nice Ad post, there are A LOT of free resources to learn math on the web, why go through $49/month ? People are citing Khan Academy which I warmly recommend, there are a lot of good Math teachers teaching math on YouTube,

and there is also OpenStax.org [1] which releases free public books on different subjects including... you guessed it... math !

Go check it out ! But maybe for some people, spending money is a necessity because it motivates them to finish the courses. It happens.

[1] - https://openstax.org/subjects/math

Do you know how much people pay for tutoring for their kids IRL? You might find a college student willing to charge $25 an hour or whatever, but if you take them to one of those corporate tutoring places you’re looking at $400/mo at minimum.

$50 is a bargain when you have nervous parents whose kids are struggling and Math is hurting their GPA.

$0 is a bargain too :)

There are plenty of resources that will go through proofs and tell you about math. Much fewer resources that will give you exercises and some sort of framework to help you move through them.

Khan does this really well for up to high school. Not really sure about beyond that.

I think a good service could be worth $50/month, but I agree it’s a tough ask.

I feel like the author, and did like the author for like year now. And I know lots of people feel this way, and we're telling each other to like "learn math, man, coding is something LLMs be doin in 10 years".

Indeed, these ML/AI papers are indeed full of cryptic writing, but in essence they are not so difficult compared to like... understanding what a FSM is and how to produce a minimal automata it from regex (hey ML guys, would like to see you do this on paper!). The greek-letters-infused-notation is what scares most people, and you know what - this notation has its origins in pre-computing age. Perhaps ppl need a new notation, or like magazines should require math geeks to also provide pseudo-code for dummies.

I also note that lot of things in ML seem to be about algorithms really, and very much about how things are engineered when implemented. Then I love graphs, discreet math, et. and were surprised to relearn that stuff like Markov chains is something very natural to me. Then the linear algebra needed for ML is not so much also - matmul, diagonal matrix, eigenvalues, inverse matrix, Hessian... wait! that's a lot already. But not so difficult, it is basically a lot of definitions. And some of these were not in the curriculums back in the day. Like... my mother does not have any recollection of learning about median and mode in statistics, even though they (with my father) attended a technical (by nature) university in the 70s.

You know, I'm starting to realize that even our professors in the university back in the day did not fully grasp what all these things were about, because I remember them reading from educational sources, and also being very punctual about the material, which is not something that s.o. who groks certain domain is going to do. And only few of them gave some actual examples why all the math nonsense could come handy. Well perhaps they are to blame that we have to le-learn, or perhaps it is the natural thing to happen in this new brave world.

Do "current gen" llms use Markov Chains? I have been wanting to look into trying to understand this stuff but I honestly don't know where to begin.

I guess I should clarify that I prefer being able to purchase textbooks or use free online sources.

There are many attempts at using attention-less approaches to the architecture, and Markov Chains being one of them. Nothing to compete with transformers yet though, AFAIK. Some people experiment with graph-based rules systems, like very smartly compressed grammars (otherwise the task is NP-Hard i think).

Tangential, but: the notes that this post refers to are probably not the best way to learn how transformers work. If you want mathematical precision, those notes are based on this paper from DeepMind:

https://arxiv.org/abs/2207.09238

The paper provides mathematically precise definitions of all the parts of a transformers, though it's showing its age (ha!) in that it doesn't include some formalizations that are common in, for example, Llama.

Thanks for this. What I included was just one of the ones I came across while researching. But given my lack of proficiency I couldn't tell up from down. I will bookmark this and dig in after I get a better handle on the fundamentals.

I work in IT as a network architect and when I tell people I graduated high school with one F on my final list being math they frown a bit. In my high- schooldays (90'ies era) they made a distinction between "math-A" which was more statistics and "math-B" which was more algebra and calculus. I really sucked at math-A which was considered by some as not even real math. I will definitely check this one out and hope to improve this even now in my 40'ies..

Sounds like The Netherlands, no? At the very least, the same system is used in The Netherlands to this day.

I graduated high school with a barely-passing grade in Math A, and years later was able to finish exams in Math B with a passing grade. All it took was dedicated practice, and I suck at math! You can do it as well.

Always wanted to try this, but I've got a hangover from my highschool days. I was bad at math, or it was taught very badly, or both. I'm worried I'd need to start from zero, like Algebra 1 and Geometry

I'm in the exact same boat. I'd literally have to relearn basic algebra and geometry.

Before you look into paid options for your tuition needs I'd look into the Khan Academy first. It's free, very high quality and it's more than just math if you want it. Sal Khan is one of the greats and the Khan Academy, together with WikiPedia is one of the best things to come out of the internet.

Khan Academy is good and free, but it is inferior to Math Academy in many aspects. It's more tedious, takes more time and, most importantly, it's not as streamlined, as guided and as fuss-free as the MA. The latter got the process and the UX absolutely nailed down. All the praises you read and hear are 100% deserved.

For anyone coming back to math I can't recommend enough the book "Journey through Genius: The Great Theorems of Mathematics". It's interesting how much of algebra came originally from geometry and the path to developing these fields makes it so much more intriguing / understandable vs just learning math to learn math.

Similarly, I've been enjoying "The Joy of X" as additional context around the fields and how they connect in an entertaining wyay.

Can OP now understand the decoder architecture? I would be interested in a follow-up post.

Not likely yet, I'm still working through the Mathematical Foundations courses before I earn my way up to the Mathematics for Machine Learning course.

I would love to do a follow-up post at some point.

Since we're doing some ad posting anyway, I am available for online undergraduate and graduate mathematics tutoring one-on-one. So we can solve problems together, I can give you roadmaps for specific math topics, or we can analyze mathematical aspects of software development. You are not getting any certifications, though! (´• ω •`)

Given that khan academy is free, I think 50/month is a steep ask for trying something out. Especially when it is labeled as beta.

Ad in disguise.

Parsing math notation has been a big weakness for me, especially with AI stuff. If I can see it in code, or at least pseudocode, stuff clicks much more intuitively. Wish there was a more efficient way to bridge that skill gap. Maybe condensing variables to single letters that have to be kept in mind just annoys me too much.

When I went back to school for CS, I used this book https://a.co/d/7hlRdnK to relearn math. I couldn’t recommend it more.

It starts with algebra and works through calculus. There is a pretest before every section, so you know what you need to focus on and what you can skip.

Self-learning is the sort of thing where you really want to be pointed to the right books. Some books are really good if you have people you can ask questions too, but those might not work at all without that. Meanwhile, others truly are a "lock yourself in a room for a couple of weeks and process it" thing.

One side thing I've been thinking of to try and tackle this is an autodictact's version of letterboxd: have people talk about books and resources they're using, offer help to one another, and maybe help people discover interesting things to poke at. At the very least it would help me track my own in-progress material

This is inspiring! I think your realistic, keep-plugging-away attitude will lead to continued success.

The cost is $49/month/student.

There is so much interest in self-learning math or physics. We're in something of an autodidacticism boom. The amount of work to even have any hope of being proficient at this is substantial. For some reason it's always self-learning math or physics, which are among the two most difficult subjects. You got your work cut out. Even if you understand the basic concepts, understanding papers is another level above that.

50 a month for what Khan academy does for free?

Ads do be getting smarter, albeit more annoying

I'm a 38yo male and was (in my high school at least) some math prodigy. But hormones got in the way, I wasn't a great student at university and decided to study economics. I have a pretty good job at a well-known software company but it still itches to do something with my talent. My dream right now is to become a Math professor. I'm really considering starting a Math degree at my local university and see where it takes me. If I'm just mediocre or even bad, or don't enjoy it, I'll stop pursuing it, but if I'm great than I really hope to get a kick out of it and just keep going.

I've kept my Math skills a bit up to date, so I won't have to overcome a 20 year pause. I've got some cash to keep me going for a couple of years, but would need a plan to generate money in meanwhile, that's what's holding me back somewhat at the moment.

Khan Academy worked for me and has a bunch of other fun classes too.

Learning grade school math won’t really help you understand these things. Yes, you will learn to think numerically and practice applying opaque algorithms, but you’d be better off starting with basic set theory and logic and learning “real” math. Book of Proof is one of my personal favorites. Then you can move onto some Real Analysis while brushing up on Calculus, then maybe consider formal probability starting to learn the foundations

Some years ago I got back into math basics, so I could do mental math as a hobby. I'm still doing it today, nearly daily. I've also added crosswords. I've found that this helps to keep me sharp (I'm in my 40s), and should help me even more as the years go by.

I'm curious how many days before author can read a mathematical description of an algorithm (let's say expectation-maximization algorithm) into code?

This post couldn't be any more timely. I dropped out of school years ago (16 years ago to be exact) to take care of my sick mother when she was dying.

I never went back. I just started working.

I am happy to report I am back in school and will be FINALLY finishing my Computer Science degree but I have a very long 4 years ahead of me. Math is going to be hard.

What is encouraging is the thrill of when I get the answer right and most importantly knowing HOW I got there. It's (almost) better than sex.

People here mention KhanAcademy and AOPS series for self learning. I've used both when relearning Math as an adult. But there is one more resource which is absolutely terrific: Henry Sinclair Hall's books. Not only they are good (way better than the aforementioned ones), but being published in the 19th century, they are in public domain now and can be downloaded from the Internet Archive free of charge: https://archive.org/search?query=creator%3A%22Hall%2C+H.+S.+... And, also, How to Prove it by Velleman is a must read.

Use chatgpt. The math isn't actually hard, it's just the person who wrote it wants to look smart so he makes it hard by writing it in a mishmash of formal notation.

There's a lot of domain specific stuff too that a straight up math major won't understand.

I will recommend a couple great channels for Math:

1. https://www.youtube.com/@SawFinMath : She is a great Professor that walks through problems/solutions one step at a time, so that you can follow along. Has a bunch of Under Graduate course playlists like discrete math, calculus, (abstract or linear) algebra, statistics. highly recommended for the great pacing and solving a lot of problems live.

2. https://mathispower4u.com/ : Another professor who takes an open math textbook (free), and makes a course out of it. Its a little bit more difficult because of denser "pure math" material, but in return, you basically cover everything that a college usually would. He has courses like discrete math, calculus, graph theory, trigonometry, statistics, algebra, geometry etc. Also solves a lot of problems live.

I haven't really tried it, but apparently there is https://www.myopenmath.com/index.php which is sort of like exercism in that you follow along a textbook and solve problems. If someone has tried this, maybe they can share their experience.

I was literally doing random exercises out of Apostol’s Calculus the other night. But I’d love to do good exercises from something more recent and advanced… gotta find my copy of Chris Bishop’s pattern recognition.

I've been looking for exactly that, for exactly the same reasons, for a few years now. Never got the motivation to start digging back into my old math university books.

Thanks for the post. It does read a bit like an ad, but it feels authentic, and meets exactly my current need, so i'll just jump right into it !

Thanks again !

Since the original motivation from the OP was to understand maths in ML papers, has anyone tried the free Maths for ML book, https://mml-book.github.io/, and if so what are your thoughts?

I've got an engineering degree but I've been thinking about trying to relearn this stuff. Sometimes it felt like I was just surviving rather than getting any of the skills to stick.

Well, it's essentially an ad/shout-out for the mate's product, but if the value is there, the value is there. I can't afford the subscription price and don't care much for subscriptions these days anyway, but I'd love to found my own education company in the future, so wishing nothing but the best, and happy to see someone hopefully being able to make some money in the spacez even when free alternatives like Khan Academy are already established.

Edit: The original post has no value in explaining how the author is learning maths other than to say that they're using the Math Academy platform, and taking notes. Useless to anybody not interested in a $49/month subscription to a semi-open beta. I would almost characterise the title of the post as a bait and switch after further consideration.