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 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.

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

- Prealgebra

- Intro to Algebra

- Intro to Counting & Probability

- Intro to Geometry

- Intro to Number Theory

- Intermediate Algebra

- Intermediate Counting & Probability

- Precalculus

- Calculus

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

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.

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.

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

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

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 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?

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.

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

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 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.

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.

How about some free resources like Khan Academy?

Has anyone tried that course? Is it any good?

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/

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.

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".

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.

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.

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.

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.

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.

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

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.

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.

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)

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.

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.

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

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.

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.

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?

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.

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

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.

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.

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.

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

The cost is $49/month/student.

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.

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.

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.

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.

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!

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.

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.)

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.

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.

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.

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.

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 .

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...

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.

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".

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.

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.

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.

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.

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.

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)

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

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).

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.

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?

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.

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.

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.

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.

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

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 !

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.

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.

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

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.

> 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.

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.

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.

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

- ...

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.

(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.

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 +, -, ·, /).

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

> 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!

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.

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.

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.

Ads do be getting smarter, albeit more annoying

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

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).

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.

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.

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

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!

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

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

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

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

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

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.

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

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

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.

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.

> 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.

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.

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 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.

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.)

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.

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

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.

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 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".

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 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.

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.

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

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.

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.

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.