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.
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.
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 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.
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
> 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.
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.
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.
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.
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.
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".
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.
"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.
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.
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)
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).
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 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.
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.
AoPS are among my favorite math books, but they're definitely not proof-based or particularly rigorous in terms of formalism.
They do focus on complex problem solving, which is equally important. The key value-add of AoPS are interesting, often beautiful examples and problems.
However, they don't do proofs or formalism much. They don't do applications or show what math is useful for. And they completely, totally, and universally screw up units (you'll have problems trying to equate a length with an area and similar; that's true of their classes as well, and RSM is similar).
I don't think there's a one-stop-shop for math, though, which does everything right. AoPS is at the peak of their particular game (which is right in the name: problem-solving).
That's best complemented by:
- Something which does data, applications, visualizations, and storytelling well.
- Something which does early exposure / surface learning well
- Something which is more formal and rigorous in terms of proofs and derivations
- Something which touches on a broad set of interesting topics (graph theory, oddball parts of geometry, etc.)
- In 2024, I would add something which does computational mathematics well
Nothing I know of does all those well in a one-stop-shop.
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.
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.
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?
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).
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.
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!
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.
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.
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.
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.
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.
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)
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).
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.
> 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.
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.
> 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.
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)
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).
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.
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/
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".
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.
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.
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.
>> 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.
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.
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’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.
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.
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.
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.
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."
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.
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.
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.
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.
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.
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:
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.
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.
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.
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?
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.
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.
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.
"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.
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.
This is inspiring to me, as I'm in a similar boat and while I'm pretty okay at my job in practical terms, I often feel as if I lack a certain mathematical foundation. May I ask how old you are, whether you are enrolled in a full-time course (w/ much younger other students, I suppose?) and how that has been for you?
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
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.
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.
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[ 3.6 ms ] story [ 276 ms ] threadThe cost is $49/month/student.
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.
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.
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
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.
> 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.
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.
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 dunno, man. I have a PhD in complex differential geometry and think math is pretty hard.
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".
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 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.
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.
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)
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 think the online classes with interactive lessons is a separate thing, but I don't have any experience with that.
I'm now over 40 and interested in relearning the math I learned long ago and pushing a bit further than I had before.
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.
They do focus on complex problem solving, which is equally important. The key value-add of AoPS are interesting, often beautiful examples and problems.
However, they don't do proofs or formalism much. They don't do applications or show what math is useful for. And they completely, totally, and universally screw up units (you'll have problems trying to equate a length with an area and similar; that's true of their classes as well, and RSM is similar).
I don't think there's a one-stop-shop for math, though, which does everything right. AoPS is at the peak of their particular game (which is right in the name: problem-solving).
That's best complemented by:
- Something which does data, applications, visualizations, and storytelling well.
- Something which does early exposure / surface learning well
- Something which is more formal and rigorous in terms of proofs and derivations
- Something which touches on a broad set of interesting topics (graph theory, oddball parts of geometry, etc.)
- In 2024, I would add something which does computational mathematics well
Nothing I know of does all those well in a one-stop-shop.
It's confusing because that title is also the name of the publisher / website of the series of the books I'm reading.
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 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.
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.
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!
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.
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)
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.
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.
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.
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/
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".
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
https://www.amazon.com/dp/0521017076
Maths: A Student's Survival Guide: A Self-Help Workbook for Science and Engineering Students 2nd Edition by Jenny Olive
ISBN-10: 0521017076 ISBN-13: 978-0521017077
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.
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).
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.
However, I start a math course that meets two days a week in person soon.
I look relatively young with a hat on, that will keep me from feeling insecure about my appearance. I'm bald as Dwayne Johnson.
Remains to be seen but I won't let anything stand in my way of finishing. Just going to put my head down and do the work. Not socialize.
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