I had a similar experience where math went from being easy and fun to an intimidating and painful slog. My problem was just how focused most courses are on learning techniques for solving problems. I found all those endless substitutions that you learn in calculus to be infinitely dull and so it was difficult to do a good job. Ditto for the solution techniques for differential equations. Don’t get me started on matrix inverting. I think I had to do a 5x5 matrix once for a homework assignment. What a colossal waste of time.
Proof-based math classes came like a revelation to me. When I took Real Analysis, for the first time in over a decade, math was fun. You weren’t just memorizing and reapplying recipes. You were seriously thinking about unique problems and devising solutions. And all the while, you were learning where all these techniques actually came from and how everything connected together.
I don’t understand why we can’t have more proof heavy math in high school. Who cares whether you remember the arctan substitution or whatever in an integral; I’d always just use a solver for that anyway. I’d rather be learning about what an integral is in the first place.
I'm a private tutor who works with adults on proof-based math. I've often had a similar thought to the one you're expressing here --- I also found proofs pretty revelatory when I first exposed to them and wondered where this magical tool had been all my life --- but I wonder how well this experience would scale to the mass of students in high school math classes.
After teaching proof-writing to my students for several years now, I've seen a lot of variation in how quickly students take to the skill. Some of them have the same experience that it sounds like you and I had, where it "clicks" right away, some of them struggle for a while to figure out what the whole enterprise is even about, and everything in between. Basically everyone gets better at it over time, but for some that can mean spending a decent amount of time feeling kind of lost and frustrated.
And this is a very self-selected group of students: they're all grown-ups who decided to spend their money and spare time learning this stuff in addition to their jobs! For the kind of high school student who just doesn't really think of themselves as a "math person", who isn't already intrinsically motivated by the joy of discovering what makes integrals tick, I think it would be an even harder sell. High school math teachers have a hard job: they have to try to reach students at a pretty wide range of interest and ability levels, and sadly that often leads to a sort of lowest-common-denominator curriculum that doesn't involve a lot of risk-taking.
A more rigourous approach was tried after WW2, when Americans feared the Soviets were edging ahead mathematically/scientifically. It was called "New Math" [0]. For an example of the type of textbook high school students were taught from, check out Dolciani's Modern Introductory Analysis (the 1960s and 1970s editions only; the later editions were dumbed down, especially when Dolciani died) [1], which starts with set theory, logic, field axioms, and proof writing techniques.
> I don’t understand why we can’t have more proof heavy math in high school.
proof based math requires critical thinking and its a lot harder to scale the teaching of critical thinking. We dont' pay enough for teachers of quality to be able to do this at the public school level. Its also much harder to test for in standardized tests.
> Its also much harder to test for in standardized tests.
You could test it using interactive proof verifiers. This would also make it a lot easier to teach, since proof verifiers can handle even very complex mathematical proof via the repeated application of a mere handful of rules. (The rules are also surprisingly similar to the familiar "plug and chug" workflow of school-level math, only with different underlying objects - lemmas and theorems as opposed to variables and expressions.)
Great post! It's always interesting to see the experiences of fellow peers going through Math Academy.
It took myself 2 1/2 months to complete Mathematics for Machine Learning on Math Academy last year (2024) working through reading material, taking notes, and completing all the exercises took all day everyday I loved it, this was after I completed Khan Academy (starting from the beginning of mathematics negative numbers, to the end differential equations) because I kept putting it off for years when I got to busy.
The main thing for me was learning not to get too frustrated when getting an answer wrong. If I made a mistake, I focused on understanding what went wrong, looking up youtube videos on the topic if it was confusing, and then trying again.
At the end of a lesson I wish I had someone to bounce questions off of but thats when I used chatGPT.
Yeah! Unfortunately at the time I had gotten laid of from work so I had extra time just not extra $ to keep paying monthly subscription, also some courses still said coming soon at least the ones I wanted to take when I was taking M4ML.
Good Luck with M4ML its a great course! Covers a lot, I was impressed, wish there was some videos or more visuals but it doesn't hurt to use youtube. I took maybe 7 pages of notes on my github and each over 4000-8000 lines (I used the notes to do the step by step exercises it was easier for me to type notes and do the exercises on computer than pen and paper this is what I used to do).
I take the notes because I will probably forget, but I think its key to always be learning and keep practicing even when your done the course.
Once I get hired again I will def take Discrete Mathematics. In the mean time I've just read books on ML and LLMs, free online courses, youtube videos etc.
I really want to do Math Academy and even briefly tried it a year ago. It's absolutely great but it's also very expensive. I know that math skills are invaluable, it's far cheaper than schooling, and that long term the investment is likely to pay for itself but when you're skint $49/month is still a pretty hefty sum, especially if you live outside of America. For context in the UK, a basic gym membership (£17/month) and a SIM only phone plan with unlimited data (£22/month on a two year contract) only costs £1 more in total than Math Academy (£38/month). I can't help but feel that the people who would benefit from it the most are also the people least likely to be able to afford it.
Go on eBay and buy the following Open University book sets. They go for around £30-50 a pop: MU123 (basics), MST124 (more complex). 6 months worth of study in each book set. If you like it do MST125 (even more complex) and M140 (stats) after. That's the first year of a mathematics degree literally from the ground up through GCSE and A-level stuff. If you really like it, get a student loan and do the associated accredited degree.
£30 for 6 months is pretty damn cheap and you get to keep it forever!
This is a proper accredited course developed over 50 years or so with its own textbooks and material from a respectable university, not a gamified subscription portal experiment put together by god knows who that can disappear in a puff of smoke at no notice.
I'm studying the Q31 (BSc Maths) on Open University.
I can second this recommendation. The maths books are _excellent_.
It's hard to explain how, but let me try: most of the maths textbooks I possess (plenty of them) are written with the assumption that you attend lectures at a classroom and use them for extra material/exercises/reference.
The OU books are written with the assumption that you learn from them as the primary material, so they go a lot further with regard to explaining things as well as producing them from first principles.
The way I come to look on such offers (monthly unlimited subscriptions) is not the net price itself, and not future supposed returns to it (who knows what they be, and they for sure will depend on many other things), but how many hours a week I am willing to spend on that service.
If you can and willing dedicate on average 2 hours a day (a big commitment but I think I was able to hold it for several month with them) the cost of mastering, say, Linear Algebra will be ~4 less then if you subscribe and will be spending ~30 minutes a day.
I guess it depends on where you are at in the world, but in our neck of the woods $50/month is an absolute bargain compared to using a tutor. Not to mention you get to work at your own pace and to practice spaced repetition consistently.
It is. But I don't think there is an alternative way to make it sustainable. There are just not that many people who are serious about self-education, and you won't like it to cater to the less dedicated customers.
> I can't help but feel that the people who would benefit from it the most are also the people least likely to be able to afford it.
Even if it were $10/mo, the people who would benefit from it the most (around the world) still can't afford it.
my read as a US person is that math academy is optimized towards students who would otherwise be well served by an in-person supplemental math program. at the earlier grades for math academy (grades 4-5 etc) the main competition i've encountered are in person programs like AoPS, Russian Math, or Kumon. The prices for those range between $450-$100/mo and for a student or student and parent combo that may be looking to supplement their math classes or for somebody who needs to home school for a period of time, mathacademy at $50/mo is a steal.
I wonder if they could charge lower rates for people who live in poorer parts of the world.
$49/month is almost nothing to me now, but it would be prohibitively expensive for a 15 y.o. me in freshly independent Czechia.
I suspect it would also be prohibitively expensive for most 15 y.o.s in the developing world today, and these are the guys and gals who stand to gain the most.
I'll call out 3b1b and khan academy for me. Especially over covid. Made math fun again.
My middleschool principal thought it'd be a good idea to skip me over pre-algebra into alg 1.
Turns out that doesn't work great, and I still have confidence issues because I have a hard time remembering the properties of addition & multiplication by name. I know the rules.
A noun that only refers to one thing isn't a real word, so if you want to cure yourself of "the associative property" being meaningless, you could study other algebras where the rules are different.
My middleschool principal thought it'd be a good idea to skip me over pre-algebra into alg 1.
Next time you read a novel, try this:
1. Read each sentence at half your normal reading pace
2. Skip every other chapter.
Sounds ridiculous, right?
That's my reaction when people propose grade skipping as the only solution for a child whose natural pace is 2x the 'standard' pace at which math is taught in school.
Yes, it's ridiculous. They should really only grade-skip in math after giving the student take-home exercises during the current year that will serve as a replacement for the skipped grade. It's irresponsible to do otherwise, imo.
Over the past few years, while homeschooling my daughters, I've come to see the way math is usually taught as horribly pathological. In the US, where we live now, it's often seen as a competitive activity -- almost like a sport. In the UK, where I grew up, that wasn't the case but still it was taught as this huge body of knowledge and skills with almost no motivation.
My daughters are so advanced in math and I really don't believe it's even mostly due to innate ability. It's because, just to take an easy random example, when we studied geometry our very first lesson was me pointing out that the word "geometry" just means "earth measuring", and it was useful for farmers to be able to do that. Or, when we proved the irrationally of sqrt(2), of course I entertained them with the tale of Hippasus being thrown into the sea by the Pythagoreans. For basically everything we've learned there are so many fun stories. It makes me sad that most students of math never get to hear them.
As a b and c grade student, who messed about, stumbled through a not very good info technology degree at university I definitely agree with this. The stories and lore are what makes me now so interested in programming and software engineering. I've pretty much taught myself everything programming related and that's what I work as too. I desperately want to learn math up to and including calculus as I feel like it's a hidden shame that I'm a programmer with not much math ability. I'm actually considering signing up for math academy.
I am currently studying for our country's version of the SAT and, having tried Math Academy — having been convinced there is nothing anywhere as polished and developed on the market — I still had to cancel my subscription after the first month. The price just wasn't worth it; over a single year, it translates to a cost greater than one-on-one tutoring.
Small companies have to understand the value of local pricing — nobody is willing to pay above h percent of their salary for a service X, and there's only so much that rule can be bent. I understand that, at the end of the day, the company still has all their expenses in USA prices, but for digital services with no manufacturing or logistic costs, it can be better to make a modest profit than none at all.
Wow that’s a pretty glowing review of the service. Sucks about the pricing though.
I haven’t really looked at math academy, but I was in school (including college) I probably learned 40% of math from khan academy, 40% from textbooks, and maybe 10% from lectures.
Math Academy uses spaced repetition for skills with tiny, to-the-point interactive lessons (typically following "theory, some exercises, theory, some more exercises" formula) based on an initial diagnostic test, where the skills are structured as a graph of dependencies.
I didn’t, at the time, appreciate how challenging a problem it was until I started researching Bayesian Knowledge Tracing. While their definition of a skill can be a bit narrow, thus putting more time into reviewing things I'd rather move on from, it does work from what I've observed.
I recall they had a course on Abstract Algebra and other more advanced subjects, so if you're really interested, the great thing about subscriptions is that you can afford to try it.
Always great to hear from people on the far side of the valley of despair. I don't think it is pointed out enough that people who fall off of "mount stupidity" can sometimes get really really stuck. In my experience when they do that at work it is quite traumatic.
Another good book for the author and others is "5 Elements of Effective Thinking" by Burger & Starbird. It thinks about thinking which can sometimes side step the depression of suddenly not thinking you know anything about anything that accompanies that big drop off mount stupid.
It was discussed in the article, but to be more explicit, sometimes a person who is sure of their understanding of things learns a new thing and that new thing opens their eyes to a huge amount of complexity they were missing. They go from feeling like they knew everything there was to know about a thing to feeling like they know little to nothing about the thing. (this is the "Falling off Mount Stupidity")
Depending on how senior they are at work, that can be quite traumatic. A lot of people in tech sort of base their self image on how smart they perceive themselves to be with respect to their peers. When that perception inverts their own world model makes them feel worthless.
In the two cases where people I was managing this occurred (that I knew of) their productivity dropped like a rock and they became seriously depressed. One I managed to get back on track, the other left tech and I have lost track of where they ended up.
Mount Stupidity is the peak of overconfidence greatly outpacing your competency level. So, falling off is essentially being humbled by expreiences that make you realize you do not know as much as you think you do, and your confidence takes a major dive as a result.
Math Academy is awesome, I'm fully hooked, but, repeating something I wrote elsewhere: it is a bleak existential confrontation with your ineptitude with fractions.
I'm signing up like, oh, I have a lot of gaps I can fill in with calculus, and it's like, no, you got a lot of gaps you need to fill in with simplifying cube root expressions. The best is every once in awhile it double checks to make sure I still know what multiplication is, with like Dick and Jane bought 10 apples problems. I have given it no reason to believe otherwise! But I trust the algorithm.
Readit News
Proof-based math classes came like a revelation to me. When I took Real Analysis, for the first time in over a decade, math was fun. You weren’t just memorizing and reapplying recipes. You were seriously thinking about unique problems and devising solutions. And all the while, you were learning where all these techniques actually came from and how everything connected together.
I don’t understand why we can’t have more proof heavy math in high school. Who cares whether you remember the arctan substitution or whatever in an integral; I’d always just use a solver for that anyway. I’d rather be learning about what an integral is in the first place.
After teaching proof-writing to my students for several years now, I've seen a lot of variation in how quickly students take to the skill. Some of them have the same experience that it sounds like you and I had, where it "clicks" right away, some of them struggle for a while to figure out what the whole enterprise is even about, and everything in between. Basically everyone gets better at it over time, but for some that can mean spending a decent amount of time feeling kind of lost and frustrated.
And this is a very self-selected group of students: they're all grown-ups who decided to spend their money and spare time learning this stuff in addition to their jobs! For the kind of high school student who just doesn't really think of themselves as a "math person", who isn't already intrinsically motivated by the joy of discovering what makes integrals tick, I think it would be an even harder sell. High school math teachers have a hard job: they have to try to reach students at a pretty wide range of interest and ability levels, and sadly that often leads to a sort of lowest-common-denominator curriculum that doesn't involve a lot of risk-taking.
[0] - https://en.wikipedia.org/wiki/New_Math
[1] - https://archive.org/details/modernintroducto00dolc
proof based math requires critical thinking and its a lot harder to scale the teaching of critical thinking. We dont' pay enough for teachers of quality to be able to do this at the public school level. Its also much harder to test for in standardized tests.
You could test it using interactive proof verifiers. This would also make it a lot easier to teach, since proof verifiers can handle even very complex mathematical proof via the repeated application of a mere handful of rules. (The rules are also surprisingly similar to the familiar "plug and chug" workflow of school-level math, only with different underlying objects - lemmas and theorems as opposed to variables and expressions.)
It took myself 2 1/2 months to complete Mathematics for Machine Learning on Math Academy last year (2024) working through reading material, taking notes, and completing all the exercises took all day everyday I loved it, this was after I completed Khan Academy (starting from the beginning of mathematics negative numbers, to the end differential equations) because I kept putting it off for years when I got to busy.
The main thing for me was learning not to get too frustrated when getting an answer wrong. If I made a mistake, I focused on understanding what went wrong, looking up youtube videos on the topic if it was confusing, and then trying again.
At the end of a lesson I wish I had someone to bounce questions off of but thats when I used chatGPT.
Congrats!
Good Luck with M4ML its a great course! Covers a lot, I was impressed, wish there was some videos or more visuals but it doesn't hurt to use youtube. I took maybe 7 pages of notes on my github and each over 4000-8000 lines (I used the notes to do the step by step exercises it was easier for me to type notes and do the exercises on computer than pen and paper this is what I used to do).
I take the notes because I will probably forget, but I think its key to always be learning and keep practicing even when your done the course.
Once I get hired again I will def take Discrete Mathematics. In the mean time I've just read books on ML and LLMs, free online courses, youtube videos etc.
£30 for 6 months is pretty damn cheap and you get to keep it forever!
ebay example of the latest edition for sale: https://www.ebay.co.uk/itm/197011707080
On archive.org too if you are happy with PDFs: https://archive.org/search?query=creator%3A%22The+MU123+Cour...
First MU123 book A: https://archive.org/details/BookAMU1232ndedOU2014/MU123-Book...
This is a proper accredited course developed over 50 years or so with its own textbooks and material from a respectable university, not a gamified subscription portal experiment put together by god knows who that can disappear in a puff of smoke at no notice.
I can second this recommendation. The maths books are _excellent_.
It's hard to explain how, but let me try: most of the maths textbooks I possess (plenty of them) are written with the assumption that you attend lectures at a classroom and use them for extra material/exercises/reference.
The OU books are written with the assumption that you learn from them as the primary material, so they go a lot further with regard to explaining things as well as producing them from first principles.
- 5th grade math
- prealgebra
The book does look high quality. But I'm surprised it covers these fundamentals, given it's for a university course.
If you can and willing dedicate on average 2 hours a day (a big commitment but I think I was able to hold it for several month with them) the cost of mastering, say, Linear Algebra will be ~4 less then if you subscribe and will be spending ~30 minutes a day.
I guess it depends on where you are at in the world, but in our neck of the woods $50/month is an absolute bargain compared to using a tutor. Not to mention you get to work at your own pace and to practice spaced repetition consistently.
> I can't help but feel that the people who would benefit from it the most are also the people least likely to be able to afford it.
Even if it were $10/mo, the people who would benefit from it the most (around the world) still can't afford it.
$49/month is almost nothing to me now, but it would be prohibitively expensive for a 15 y.o. me in freshly independent Czechia.
I suspect it would also be prohibitively expensive for most 15 y.o.s in the developing world today, and these are the guys and gals who stand to gain the most.
I wish there was PPP for the subscription, i tried for a few months but stopped the subscription recently.
Dead Comment
My middleschool principal thought it'd be a good idea to skip me over pre-algebra into alg 1.
Turns out that doesn't work great, and I still have confidence issues because I have a hard time remembering the properties of addition & multiplication by name. I know the rules.
1. Read each sentence at half your normal reading pace
2. Skip every other chapter.
Sounds ridiculous, right?
That's my reaction when people propose grade skipping as the only solution for a child whose natural pace is 2x the 'standard' pace at which math is taught in school.
skipping chapters of a novel doesn't work very well, but it works great for the encyclopedia, and pretty well for a lot of textbooks
it's also not that hard to use khan academy or wikipedia to fill in the gaps, if you did miss something
Over the past few years, while homeschooling my daughters, I've come to see the way math is usually taught as horribly pathological. In the US, where we live now, it's often seen as a competitive activity -- almost like a sport. In the UK, where I grew up, that wasn't the case but still it was taught as this huge body of knowledge and skills with almost no motivation.
My daughters are so advanced in math and I really don't believe it's even mostly due to innate ability. It's because, just to take an easy random example, when we studied geometry our very first lesson was me pointing out that the word "geometry" just means "earth measuring", and it was useful for farmers to be able to do that. Or, when we proved the irrationally of sqrt(2), of course I entertained them with the tale of Hippasus being thrown into the sea by the Pythagoreans. For basically everything we've learned there are so many fun stories. It makes me sad that most students of math never get to hear them.
Small companies have to understand the value of local pricing — nobody is willing to pay above h percent of their salary for a service X, and there's only so much that rule can be bent. I understand that, at the end of the day, the company still has all their expenses in USA prices, but for digital services with no manufacturing or logistic costs, it can be better to make a modest profit than none at all.
It would be impossible for me to have one-on-one tutoring for a year at only €465 ($499 but I'm in EU). And that's regardless of the tutoring quality
I haven’t really looked at math academy, but I was in school (including college) I probably learned 40% of math from khan academy, 40% from textbooks, and maybe 10% from lectures.
How does math academy compare to Khan academy?
I didn’t, at the time, appreciate how challenging a problem it was until I started researching Bayesian Knowledge Tracing. While their definition of a skill can be a bit narrow, thus putting more time into reviewing things I'd rather move on from, it does work from what I've observed.
I recall they had a course on Abstract Algebra and other more advanced subjects, so if you're really interested, the great thing about subscriptions is that you can afford to try it.
Another good book for the author and others is "5 Elements of Effective Thinking" by Burger & Starbird. It thinks about thinking which can sometimes side step the depression of suddenly not thinking you know anything about anything that accompanies that big drop off mount stupid.
Depending on how senior they are at work, that can be quite traumatic. A lot of people in tech sort of base their self image on how smart they perceive themselves to be with respect to their peers. When that perception inverts their own world model makes them feel worthless.
In the two cases where people I was managing this occurred (that I knew of) their productivity dropped like a rock and they became seriously depressed. One I managed to get back on track, the other left tech and I have lost track of where they ended up.