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Ask HN: How to learn math from zero for adults?
559 points by stArrow on May 29, 2022 | hide | past | favorite | 172 comments
I am a 26 year old learner who is really into Machine Learning. But my lack of understanding in math has held me back. Skipping and hating math classes in high school have been my biggest regret.

Now, I am slowly trying to learn, but I don't know where to start. I need some guidance.




I was a C student in high school. I never took a math class beyond basic algebra. I started from Khan Academy's first exercise when I was 33 years old. It seems silly looking back that I was solving problems on the number line as an adult.

At that time, around 10 years ago, Khan Academy had excellent coverage through trigonometry and single variable calculus. Once I reached that point I went to my local community college and took all of their math classes. I transferred to University of Illinois at Urbana-Champaign and continued onward to get a BS CS, BS EE, and MS EE. I finished at 41 years old and landed a dream job that I would have never thought possible when I started.

I guess my advice is to start from the beginning and see where it takes you.


I was an A+ student up to Sophomore year in HS. I downgraded to A or B+ student after I knew how much fun actual learning is.

Before that, all I was doing was memorizing and practising heuristics to get A+.

I actually started learning Science since 11th grade. I grew and grew. And my scores tanked.

Got only 6.xx/10 in college as well.

But no regrets. Learning Math properly gave me a super solid foundation to learn Deep Learning very quickly. Now I do my semi-dream job as a Deep Learning Research Engineer building actual products with Deep Learning.

I have a truly realistic chance of getting my dream job within the end of this calendar year.

No regrets. Screw scores and learning systems designed for the mediocres. The education system has no care for the poor learners with no parental care. Neither it is good for future scientists and geniuses.


This, 100%. I couldn’t agree more. I’m currently a Software Engineering undergrad in Europe and, at least by my personal experience, the whole education system just focuses on preparing people to pass an exam, not to properly learn a subject.

IMHO, the problem starts when the students get used to memorizing and repeating sample exercises which involve 0 creativity and then graduate and get an actual job.


What I said shouldn’t really be taken as an advice.

If you think that getting good scores will open doors for you, and you are in need of those, then, by all means, work hard to get good scores.

What many people like me do not realize is that learning something properly is among the material luxuries. You can only afford to do this if you do not have socio-economic disadvantages (like most people need jobs straight out of college- mostly for the money, but social norm is also a factor).

I am a person of simpler needs and I do not care about socio-economic norms. You might not be like me.

So, sticking to social design might achieve you benefits.


This nails it, and is something that took me some time to identify in myself. I was first in my family to finish college (and first to consider grad school, let alone go), my grandparents were refugees, and for most of my life my family was lower middle class. I went to school on scholarships and lots of loans, and that weight (plus the financial stresses we went through when I was growing up) led me to focus a lot on grades and 'playing the game' to the detriment of fully learning the material, exploring the material when my curiosity pushed me, and taking risks.

As I've understood that part of myself, I've had to regularly and actively push myself out of my comfort zones (studied neuroscience in grad school, then became a software engineer and have worked at startups for the majority of my career, founded a couple of tiny ones that went nowhere with some friends too) and still need to as a mid-30s adult today. It's much easier now with a steady income, but seeing my friends who had more stable homes take risks that I'd never be able to stomach or recover from really imprinted how important financial stability is to a child growing up.

Hopefully despite all this, I can raise my own kid in a stable environment that allows them to learn and grow without worrying about their life being completely ruined if they don't make perfect grades or don't follow obvious well-trod paths in life.


Fwiw it is not necessary to throw your college scores to actually learn as well: they just feel orthogonal sometimes. You can do both!


It depends where you study and what courses you take / who teaches you. Sometimes the professor will drive a ferocious pace of instruction and assessment and you will simply not be able to meaningfully keep up unless you sacrifice the depth of your learning in exchange for getting everything done. I have been in that position and watched my grades tank. Then, a year later, I watched my peers struggle with follow-up material that seemed to make a lot more sense to me.


I was always pushing to take 16-18 units (5-6 classes), so never any extra time to "savor" the details, or deep dive into areas of interest.

If I did it again, I would take the minimum of 12 units (3-4 classes)... assuming I could afford to stay in school for 6 years. At the time, I was racing to graduate and get a job.


12 units was the sweet spot for me. I was able to digest and master the material because I had the chance to put in the necessary work to truly understand the material and still have a life outside of school. I decided to take more classes (21 units or 7 or 8 classes) and although I was still able to get very good grades (graduated with about 3.8/4.0), I wasn't able to master the material and I had no life outside of school.

> I was racing to graduate and get a job

This is also why I increased the number of units that I was taking.


> You can do both!

Not if you have other interests!

I am a very cultural person (reading books, writing, playing music, etc.). I volunteer after natural disasters, too. And I also value interpersonal relationships with people I care about.

It felt bad in HS, but in college, I was mature enough to simply not care. (I also gained enough skills in the meantime to be certain that I won't ever starve.)


I understood it as that learning math is easy and fun in high school but becomes hard and boring work at university level.


> I actually started learning Science since 11th grade.

what event caused this? was it specific material or a specific teach or approach?


A teacher, or two.

One taught me about what rigor was. In so many ways.

Another taught me that there is another side of Math. The non-heuristic parts. The stuff where is there is reason, logic, etc.

Dad took a broadband connection, and watching YouTube became realistic. I got 3b1b, Numberphile, MIT OCW.

Edit: My discovery of Halliday, Resnick, Walker physics textbook and Feynman lectures also was helpful. It was mostly- the Internet.


> Edit: My discovery of Halliday, Resnick, Walker physics textbook and Feynman lectures also was helpful. It was mostly- the Internet.

thank you for the tips! much appreciated


> Now I do my semi-dream job as a Deep Learning Research Engineer building actual products with Deep Learning.

> I have a truly realistic chance of getting my dream job within the end of this calendar year.

So what’s your dream job? :-)


It’s a cliche, but yeah, Google.

Or any co/startup doing "deep tech" in the area of Deep Learning.

I am doing _applications of DL_ now, want to do actual DL.


Side-related note: You'll get there (actual DL), because your "want" is close enough to determination (as opposed to mere motivation, i.e. conscious decision versus mere fleeting feelings).

On academia and grades: yes, it's a "dark time" (as in "Dark Ages" for what we mean by it, regardless of the veracity of that historical perception). I often summarize all those collective F-ups by remarking that most people, either individually or collectively as institutions, have lost their purpose, they "have no mission in life" so to speak. Thus, we lose sight of the endgame, of the actual purpose of education for instance.

That's not about to change if the now widespread intuition that most institutions have been rotting too much for too long is true, so the solution might be some historical disruption of the domain — I think Khan was an early ripple in those foundations.


> Or any co/startup doing "deep tech" in the area of Deep Learning.

Send me a message, my email is in my profile.


I would love to chat sometime if you’re available. I’ve been teaching middle school math for the past 6 years. I’m now trying to transition into tech.


What do you want to do in tech? You want to do software development (like building apps, websites, etc) or something else? I pivoted into software after starting somewhere different and happy to give advice.


@Perez418, this person can give you better advice than I.

I have been in a tech since the last year of college.

This person has more relatable career trajectory as yours.


Send me something, like a Discord username, or an email.

Best would be Libera.chat, though.

Update your profile, and I will check back after 24 hours.


Absolutely need to check out 3Blue1Brown on youtube, especially for visual learners.


I think that the fact that there are special types of learners have been debunked thoroughly.

Veritasium recently made a video, too [0].

3b1b is a fantastic supplementary resource. But you can't start with it, or use it as your primary resource. That would be a mistake.

[0]: https://youtube.com/watch?v=rhgwIhB58PA


That video is not convincing, at least for me, wrong style learning is more tiring. I can read a book about a subject all day, but I can only listen to about an hour of lecture a day.

I'll absorb the same amount - during that first hour, but after that nothing further is getting in my head.

Also the computerized thing about a diagram vs words isn't a learning style difference - both of those are visual, just one is more effective (obviously a diagram is better).

His on-the-street test is also bad, someone with a different style will, for example, say the image name out loud, if they need it auditory.


What you say and the video says are not conflicting.

These are just personal preferences. There are no inherent style at the biological level. That's what "learning style" s are about.

I personally have found a combination of all (with more focus in applications and teaching the topic to others) works best for me.

> His on-the-street test is also bad

Agree. It was amusing in a bad way.


Mate, you've no idea how much I could related to your story. I've a bachelor degree in Electrical Engineering. Looking back at those subjects, I feel like I didn't have the maturity needed to appreciate the depth and meaning of it. I was too busy just passing exams and working to get a degree and eventually land a job. I did that and I have no regrets. But those concepts taught to us are the foundations of almost everything. The more I think about it the more respect I have for scientists who came up with those before 1950s! They had nothing and they solved their own questions. Bloody legends.


Yes - imagine inventing Fast Fourier Transforms without a computer.

Or the original continuous Fourier Transform, with just a pen and paper.

Or Heaviside's simplification of the Maxwell equations. Or calculating planetary orbits with perturbations.

Incredible leaps of imagination.


True indeed. I was patting myself on the back for graduating engineering school without any help from WikiPedia or YouTube. But these legends wrote all those universe describing equations without the help of a computer.


>It seems silly looking back that I was solving problems on the number line as an adult.

I study biostatistics and dabble in population genetics research. I still resort to number lines and venn diagrams frequently. There is a reason we teach number lines - they are very useful!


Where exactly do you use number line? Thanks.


I don’t recall ever learning number lines in school. Was in eighties. Are number lines a new teaching tool or am I just not remembering this?


You might be forgetting them, it's literally just a line with numbers marked on it. I learned them in primary school in the 90s, further classes never used them but they've always been useful tools for visualizing relationships between numbers.


Usually just when I am having trouble with an idosyncratic number scale. So usually when I need to make sense of dates and the time between them. It's often got nothing to do with work, but I do use them in my personal life.


Above is not advice but I would echo it, with emphasis that higher education makes much more sense when you go there as an adult with other stuff to do outside of it so uou just focus on learning things.


The counterpoint is that that maturity is much more valuable to industry, which is why they don't incentivize us to return to school.

Some balance needs to be found.


Wow. I was going to come here and suggest them as well but your story does an even better job. Congrats to you and all the hard work you must have put in over the 10 years!


Khan academy math by grade level. For each grade level you can do a 30 minute exam.

It identifies weak points and updates your progress on areas where you are successful.

Then you can focus on areas where you are weak instead of doing the entire grade again.

Then move to the next grade.


Germane: https://nautil.us/how-i-rewired-my-brain-to-become-fluent-in...


Wonderful story--thanks for sharing! I love stories like this.

Maybe off-topic: But did you run into any difficulties landing your first job after finishing your MS in EE? I'm tempted to pursue something similar, but I worry about ageism and other factors when going into the job marketplace at that age in a brand new field. Would love to hear more about your experience.


Almost same thing. Khan Academy is amazing.


Thanks for sharing! Your story is an inspiration to me.


What kind of job do you go into with a math degree?


Math degrees open an amazing numbers of doors.

Data science, analysts, strategy, legal research (or research in general), many CS jobs, etc.


I'm in my 40s and I started a machine learning degree in Nov 2020, so it's been about 1.5 years for me till now. I am a slow learner and my recommendations may reflect the same.

1) Maths (precalculus and calculus) - I started with Khan Academy 10th grade onwards. I finished till grade 12 in a week. By this time I had gone through content elsewhere, so 1 week may (not) be enough. Regardless, Khan Academy app comes highly recommended.

I did realise that there were a few gaps (more basic in nature), I covered those with Eddie Woo on YouTube (e.g. why is zero factorial 1). For others I just looked up relevant searches online.

2) Maths (calculus) - MITs videos on YouTube. That is the pace of content I really love. Lots of overlap with Khan academy calculus, but do go through both. Also 3blue1brown "essence of calculus" playlist.

3) Maths (Basic Linear Algebra) - although if someone were to say Gilbert Strang's MIT videos, they would be bang on perfect, I had to start slower. Bingewatch (with popcorn and beer) 3blue1browns "Essence of linear algebra" on YouTube. Then move ahead to the channel "Math the beautiful" which has a slower pace. You would also wish to visit their website www.lem.ma where they have exercises. Then of course come back and start Prof Strang's lectures (you're delving into heavier stuff midway through his course).

4) Statistics - hands down professor Leonard on YouTube, he is the statistics equivalent of Eddie Woo. Slow, smart, funny and he has biceps too ;-) After this Prof Tsitsikilis (MIT) on YouTube.

It goes without saying that you'll need to practice problems (ironical, coming from me). You can download question sets of your country from online.

5) When you've done the above, your search for "linear algebra" and "calculus" on HN will yield a lot of lovely results. Hidden gems will be there in comments too. Check those books, interactive books, websites, etc out. Your pace will be good by this time, but you will occassionally come across something which you have not come across before.

If there's anything else, feel free to ask.


3blue1brown … that is it, one if the gem of internet. So wish I have that when I am young.


Wow, thanks so much for sharing your learning journey here. Gonna bookmark this comment.


I found Khan Academy's lectures on Linear Algebra closer to the pace I needed.


had me at biceps


I've done this and would echo the good advice you've already gotten about Khan Academy but I think the comments in this thread are skipping over the most important part: pace. Work slowly. Don't move on until you really "get" the concept you're working with. Some topics might "click" right away and others might take weeks of bashing your head against the wall until your brain makes sense of it. That's okay. It can definitely be tedious at times (so be kind to yourself!) but a lot of learning happens when you're struggling to make sense of something so it's worthwhile to lean into the trouble you're having and figure out what it is that you're struggling to understand.

The downside of formal education is that the teacher has a timeline that needs to be kept and the class will move on without you. It might not seem like a big deal in the moment but in math the foundation is so important because the concepts build on one another. Sometimes in really subtle ways. If you have the motivation (and it sounds like you do!) then the huge upside to self-paced learning is that you can build a really strong foundation and the more complicated advanced topics will be so much easier to master. Khan Academy is really good at breaking subjects down into tiny bites and then slowly building on what you've learned but if you don't quite understand a lesson or if you haven't properly paid attention don't be afraid to do it again. If you guessed the right answer but don't understand why your answer was right, don't be afraid to go back and do it again.

So just because one person here could do all of the Khan Academy 9th grade algebra in a week that doesn't mean you should set that timeline for yourself. Everyone is starting from a different place. So maybe it'll take you a day, or a week, or a month, or even months. But however long it takes is how long it takes.


I'll just mention: From a scientific perspective, this is absolutely the wrong advice.

If you couldn't understand something yesterday, that's very good evidence you won't make sense of it today. You should work hard, go deep, and try hard problems, but if you fail to understand something after putting in the effort, just move onto something else.

Learn a few unrelated things, and come back. Much of the time, if you come back in weeks or months, it will just click into place.

You also want to visit topics /many/ times. Truly understanding many topics in math should take months or years. You can do it a lot more efficiently if you visit a topic for e.g. 5 hours each every 6 months over 2 years than a single 80-hour cram. You'll put in less time and understand more deeply.

A good starting point for references is Bruner (and specifically under Spiral Curriculum), but there's a whole literature on this. There has been a slow transition from seeing math as a set of bricks which stack on each other to an interconnected network of knowledge. The spacing effect literature is good too.


Math professor here.

I see students hit a wall in linear algebra, very good at "trained seal" but profoundly resentful that the time has come to go meta: Part of one's learning has to be investment in reflection and experiment on how one learns.

They generally believe that intelligence is innate, not deliberate gardening. There were thousands of people with Michael Jordan's body (alas, I'm not one) but he spent decades creating the athlete he became. Einstein found math difficult. It's like running: If you find running boring, you're not running hard enough. Math is a constant struggle, like mountaineering, and you shape, invent your mathematical mind. Those of us who love math (or mountaineering) are drawn to the struggle.

One syndrome I've seen often enough to classify: A student is uncomfortable with their calculus background. They want to pick the hardest calculus book they've heard of (often Apostol) and retreat to a desert island till they've mastered every word. I cringe in horror, and try to explain what's wrong with this approach.

It is widely observed that grad students learn four times faster than undergraduates. My first few years of grad school, it felt like I doubled what I knew every year. It was profoundly depressing when this subsided; I could have "been someone" with just a couple more doublings. What was going on?

Grad students learn what they need any given day, for goals they've set for themselves. Undergrads are taking on faith what authorities say is good for them. It's far easier to understand something when you see the point as you're learning it.

I know few mathematicians who read math comfortably, as if reading a novel. We mostly get angry and go think on our own, then return to realize that's just what the article said. Communication conventions in math are horrendous; one writes crappy machine code, then asks the reader to reverse engineer one's thoughts. Anyone in the tech sector knows how easily reading someone else's code can be "just kill me now" territory. As horrendous languages go, mathematical notation is a profound achievement. Nevertheless, beginners feel inadequate when faced with the inadequacies of mathematical notation to convey intuition.

If one wants to learn math, one needs to learn how to play, whatever that means. Find anything whatsoever in the text that leads to an example one can expand beyond the text. Does the text remind you of a pattern you've seen elsewhere? Perhaps the author just isn't saying. Play on your own, trying to decide if the ideas are in fact related.


>> As horrendous languages go, mathematical notation is a profound achievement.

Are there any resources or strategies you would recommend to help with this?


> From a scientific perspective, this is absolutely the wrong advice.

Those are unnecessarily harsh words. You and elliekelly are not contradicting each other, but talking about different things.

If you didn't understand chapters 1 and 2 from a textbook, reading chapter 3 from the same book is usually a BAD idea. Sadly, this is what formal education often tells you to do, because it follows a predetermined schedule.

If you didn't understand chapters 1 and 2 from a textbook, opening a different textbook and reading chapters 1 and 2 is usually a GOOD idea. Two authors describing the same idea from slightly different perspectives is better than reading the same text twice; at least it makes it easier to pay attention during the second reading, but the second book might also answer a question you had when reading the first book.


I personally have always had a hard time inhaling chapters of a textbook sequentially. It’s often easier to skim it first, attempt the problem set, and then go back when you get stuck. Forces you into a more active mode of looking for particular information instead of just trying to absorb everything.

The same works for the humanities too, once my attention starts waning I’ll start skimming and rewind when necessary. Oftentimes authors are setting up a ton of boring but necessary context upfront, but it’s not obvious to a reader why such context is necessary until you get to the point first.

A smarter person would probably be able to anticipate what they’re trying to learn the first time around and avoid my flipping back and forth, but that’s hard.


I believe this works because you're creating a demand in yourself for the information. I suspect it's something like natural selection: you will remember and understand what you "need to" in order to cope with the environment you're in and for studying you should create such an environment for yourself. The more problems and advanced texts that you supplement your readinv with, the more demand there is for the more basic information you're trying to retain.


This is also known as order control in inquiry-based learning.


This also, while common sense, is unfortunately often incorrect -- at least the part about avoiding chapter 3 (a different book is a good idea).

If you didn't understand chapters 1 and 2 in a textbook, reading -- or at least skimming -- chapters 3 and 4 is usually helpful. It doesn't make sense to read them for mastery, but you'll see context where material from chapters 1 and 2 is applied, or concepts that build on them. You have to be comfortable with confusion, since most of it will go over your head. You will pick out some parts.

Once you've done that, go back to chapters 1 and 2.

Think of calculus: limits -> derivatives -> integrals.

However, you can understand a Reimann sum without derivatives or limits. It helps understand and motivate both.

This is called a whole math approach. You make successive passes with increasing depth. I've explained calculus concepts to kids under 10 with no problems, as have many others. Over time, you want to develop:

- Mechanics of integration, differentiation, and other computation

- Applications (e.g. debt versus deficit)

- Intuition

- Formalism

- And so on...

All of those support each other.

You see this in how you read research papers too. Novices read them linearly, and experts absorb them nonlinearly.

As a footnote, Vygotsky (in the original, e.g. 1978 translation) is probably the first person to discover that doing things beyond your level of ability accelerates learning. Not aiming for mastery means you're free to fail, free to try harder things, and learn faster (if more painfully).

85% of American textbooks about Vygotsky, though, always say exactly what he was trying to debunk. He pushed very hard for learning being harder and more abstract than most thought possible.


To me, this has been the largest advantage of learning things at my own pace instead of in a classroom environment. Even as a student I recognized early on that topics that'd stump me would often click months later. By then it was too late for my grade, but I'd go back and realize that other things I'd learned in the interim made it easy.

So now when I find something blocking me, I just go learn something else instead. And what do you know, some combination of more wisdom, broader knowledge, or perhaps just a different mental state often works.


I personally found that I would have to keep exposing my brain to the same mathematical concept many times over days or weeks by doing problems when I could and visualizing the math in my brain when I was doing other things. Over time it felt like my brain was gradually getting "imprinted" with the shape of the math ideas, but you had to be gentle and do it over a period of days to weeks, otherwise it's like the math shape would just squish your brain instead of letting your brain adapt to it :)


Care to share the science? This sounds more like a learner's internal/external locus of control. If you believe you are capable of figuring something out you're probably right and likewise if you believe you can't.


I'd recommend rereading my post. The problem with education research is that there's a hundred years of it. I listed the two in my original post. A longer list:

- Bruner's work on spiral curriculum, and follow-up work. Spiral curricula are mis-implemented in most schools which claim to use them; it's not review. It's successive passes going deeper.

- Cognitive work on progressions such as surface->deep->transfer learning (Hattie) or chunking

- Physics education research on complex multiconcept problems

- Spacing effect. You can go all the way back to 1885 with Ebinghaus, but the esoteric author of supermemo seems to have done the best work here. If you prefer the academic establishment, there's a ton.

- There's a bunch of science that about 2/3s of what we learn, we're not aware we're learning or teaching. Concepts support each other, deeply. Kaplan, back when Bror was there, did a bunch of nice work there. It's easier to learn long division if you know a bit of algebra, and vice-versa. Chemistry helps learn physics, and physics helps learn chemistry. This was supported with pretty independent methodologies in independent domains (data analysis from tests, cognitive task analysis, etc.).

... and so on. This has very little to do with affect. It's a very well-established, rarely-applied set of results.

A complete lit review here would be book-length, at least.


> Don't move on until you really "get" the concept you're working with. Some topics might "click" right away and others might take weeks of bashing your head against the wall until your brain makes sense of it.

Careful. This can sometimes be counterproductive. It is often the case that one concept taught when learning a subject will be applied as part of teaching the next concept.

The best way to learn that first concept in many cases is not to focus on it until you really "get" it but rather to just get enough of it for how it is used for teaching the next concept. Then you move on to the next concept and the application of that first concept there gets you experience using the first concept which solidifies your grasp of the first concept and makes you truly "get" it.

The application of earlier concepts throughout the subsequent material can be in effect a kind of natural spaced repetition for learning the earlier concepts.

I think that deciding when to move on is one of the hardest things in self-learning, especially if the subject is new enough to you that you can't use your experience in a related subject to help guide you. You don't know enough to know when you've learned enough to go on.


Khan has problem sets right? That was always my issue, that and math "teachers" who'd just offer to reread the textbook or point to the index and smirk.

But don't feel bad if you struggle, it's really hard to learn math concepts without an incentive. I'd just pick up a book on python and learn algerbra and a few other things along the way.


You don't say how much you already know.

Your first stop would be Khan Academy and knowing your gaps.

Fill your gaps.

Learn HS level Calculus, Linear Algebra, and Statistics.

You will need more Calculus and Linear Algebra later. But not now.

Then try studying "Machine Learning for Absolute Beginners" book. It not very mathy.

Then just keep going through ML courses. Learn what you need on the way.

The "way" of math needed in Machine Learning is not the same "way" that brings you scores in school/college exam.

You need absolutely crystal clear concepts in Linear Algebra, Multivariable Calculus, and in some areas of ML, Statistics.

Corporate "Data Science" and Machine Learning research/projects are wildly different beasts. Learn what you will pursue, and decide your path based on that.

And most importantly, you have to be patient. Machine Learning and Math for it takes time- not days or weeks, but months and years.


Although both are rooted in statistics, Data Science or Data Mining and ML are very different fields even if they might share some concepts and methods.

I did different courses for each in University. Data Science is concerned with extracting patterns from existing data.

Like you have some papers for exam and you want to know if students cheated. Or you have the results from a poll about people hobbies, income etc and you want to correlate that with voting for party X. Or you want to correlate the race of canines with their abilities.

ML is also mostly about patterns but in a different way. You want something to tell how likely a comment is spam, if an article is positive towards a politician, if a picture is of a cat or dog.

So, to get to get a fundamental understanding you will need to learn statistics. Which in turn will require some calculus and algebra, but nothing too difficult.

Although I have the basic math knowledge and I have the basic knowledge of ML and Data Mining, I quit trying to do things in those fields because they are really vast, especially ML. Knowing the math and the basics of ML is required but far, far from enough to get good results. The people who work in the field are focused on it. I like ML but I like software architecture and development more, so I did my choice.

That being said, I still got some benefits from basic ML knowledge when I used ML libraries such as ML.NET in my day job. Knowing what a SVM or random forest is and how to tune parameters to improve my model was helpful. It was just a simple usage case like suggesting to customers what they might want to acquire based on their past purchases.


Honestly, I almost never need Statistics.

More than basic notations and explanatory parts of research papers, I never need even iota of Statistics.

When you need to learn something specific, like, say, KL divergence, if your math is solid, then you can pick it up in 10 minutes.


What helped me get a grip on learning mathematics was learning to prove theorems.

Working through Daniel Velleman's book "How to Prove It" (the only pre requisite is that you can understand boolean logic, which programmers have no problem with), and then a Set Theory book (I used Enderton) set me up to tackle (proof based) Linear Algebra, Analysis etc.

Just my personal experience. Hope this helps.


I think theorems already presuppose some basic understanding of algebra, but I might be wrong.

It really depends on what OP actually knows and how deep he wants to learn and in what direction


I wonder about this myself. I've always had a much easier time learning things that makes sense from first principles rather than something that I need to just take for granted, the latter being the case with the first 12 years of my own math education. The latter is much more difficult to form a mental model around.

Would it be possible to teach mathematics by theorem proving ab initio? I guess conventional algebra would be hard to digest for a first grader, but I maybe something like the Peano axioms can be thought of as rules for a game that the students can play, where subsequent arithmetic lessons will be about finding shortcuts to the tedious application of the rules in order to solve problems.


You may find this resource useful. It was written by Susan Rigetti (née Fowler) [1]:

So You Want to Study Mathematics…

https://www.susanrigetti.com/math

She also wrote:

So You Want to Learn Physics…

https://www.susanrigetti.com/physics

[1] https://en.wikipedia.org/wiki/Susan_Fowler


> You may find this resource useful. It was written by Susan Rigetti (née Fowler):

> So You Want to Study Mathematics…

> https://www.susanrigetti.com/math

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


Pretty sure a lot of people in the comments have already suggested helpful online resources however I'd recommend you start right where you left from; your mid/high school textbooks. I'm kind of old-school and I think a pencil and paper approach w/o online distractions is how you really get closer to understanding math.

Of course you can use various tools (e.g. Wolfram Mathematica) for any fancy visualizations and maybe some tedious calculations. Just don't rely too much on them while learning the fundamentals (don't skip these boring "find the following indefinite integral" problems).


This recent Ask HN covered similar ground:

* https://news.ycombinator.com/item?id=31488608

Older discussions I shared on that one:

* Susan Rigetti’s “So You Want to Study Mathematics…”: https://news.ycombinator.com/item?id=30591177

* Terry Tao’s “Masterclass on mathematical thinking”: https://news.ycombinator.com/item?id=30107687

* Alan U. Kennington’s “How to learn mathematics: The asterisk method”: https://news.ycombinator.com/item?id=28953781


* Subscribe to Beast Academy, and do the interesting problems (puzzle and challenge ones). Cost is $15/month, and you should be able to do this fast, so it shouldn't be too high

* Sign up for Alcumus. This is free. Set it to easy. Try problems in each section, and use this to identify gaps. If you find a gap, do a deep dive to address it. If you can't solve a problem independently, solve it using online resources (but put in the wrong answer / give up -- you don't want the ITS thinking you could do it).

* Watch videos on 3B1B, and sign up for Brilliant. Take a few courses.

* This will sound silly, but do sinerider, nandgame, and the fun Khan Academy courses (Advanced JavaScript and Pixar-in-a-Box), and things like this. Math is broad, and those give helpful exposure to a broader range of topics.

https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw

https://artofproblemsolving.com/alcumus/


Ask HN: Serious mathematics books that can replace a good teacher?

https://news.ycombinator.com/item?id=31488608 169 comments, 5 days ago


Learning math is a personal journey, it can be slow and difficult at times. No matter how smart one is, there will always be mathematics that one finds completely baffling. Generally, curiosity, motivation, patience, perseverance, maturity, and perspective are necessary ingredients to get very far.

An extraordinarily accomplished mathematician replied, when I asked him what books to study, “You don’t solve problems by reading, you solve problems by thinking.” Thinking (hard) is the key, the practice of mathematics requires and unlocks a level of clarity far beyond ordinary everyday thinking.

Try mulling over simple facts which even children know, like the Pythagorean Theorem or the Quadratic Formula, and see if you could explain them so clearly and convincingly that any reasonable person would immediately and plainly see that they must be true.

Learning is spiral. You will almost never understand everything there is to know about a subject, but you will learn more as you return to that subject as you’ve grown and matured, and have acquired more knowledge, skills, and perspective.


Run through the math on Khan Academy to fill in your gaps for at least Algebra I, II and Precalculus. Then you need to Calculus I-III, Linear Algebra and basic Statistics which are also on Khan Academy.

Also I have created some youtube channels aggregating quite a bit of the quality university courses organized into playlists of playlists.

https://youtube.com/channel/UCjgQ2pJDjZlhdI4Ym7NQdUw

Note: You have to click on the titles of the topics on the home page that slide left/right (or up/down on a phone) to see the whole list of courses because YouTube truncates the lists on the home page.


Besides Khan Academy (which is excellent) and other online resources, I would buy text books, and then do everything in the textbook. I mean everything. Read everything in the chapter, work through every example problem, do every practice exercise. Do not skip ahead, or assume you understand without actually doing. And find someone who can answer your questions. I don't like asking for help, but that's held me back. If you have to, pay someone.

I prefer hardbound textbooks, because computers are a big source of distraction. But there are also lots of generally very good free and open-source textbooks you can find online.


https://openstax.org/subjects/math

For free online textbooks these ones seem to have good reputation on the Internet. I did the 'pre-algebra' and 'algebra and trigonometry' books, found them good. Do all the excersices (very important), sometimes I got stuck (every books misses edge cases) so be prepared to look at other resources.


I’m in a very similar position. I didn’t go to uni and somehow found myself in a programming career. In a few weeks I’ll be sitting A-level (UK advanced high school) Mathematics exams for the same reasons as you. I’m 32 years old.

I’ve found having formal exams to study for very motivating, and humbling. I thought I knew things I’d read in books, but it turns out I just recognised them when I came across the same topic again. Being able to recall concepts and use them without help under time pressure is a different level of mastery.

Can you sit high school exams as an adult in the US? There might be private exam centres in a city. That way you’ll _know_ you’ve learned the math(s) you need, on top of all the great learning resources linked here.

—

Edit: Cambridge International do A-level exams, with centres accepting private candidates globally: https://www.cambridgeinternational.org/programmes-and-qualif...


There are no standardized high-school exams in the USA. The closest things would be SAT/ACT which are taken by college-bound high school students, but they are separate from high school and have zero bearing on graduating from high school.

I don't know of any reason an older person could not take the SAT but also most older students interested in going to university would be admitted under different critera than a 17 year old, and high school grades and SAT might not be relevant at all for someone in his 30s or older.


I don't understand, why are you taking high school exams? Is it entry requirements for university?


I want to verify for myself that I _know_ the material. A series of challenging exams will do that more thoroughly than completing exercises from a textbook or online course (where, let’s be honest, we’re more forgiving of ourselves than we should be).

Check out Barbara Oakley’s Coursera course called Learning How To Learn. It turns out exams are actually an excellent tool for learning, more so than learning without assessment.

A side effect has been that having a deadline and a measurable outcome has been incredibly motivating. I’d have probably been distracted by side projects if I hadn’t committed to the exams.

And yes, it’s accepted for uni entrance here. STEM courses at top unis will expect A-level grades in maths/science subjects, even for mature students with industry experience.


Check out ossu/math. Good list of resources and community on discord https://github.com/ossu/math or ossu/datascience has suggested math prerequisites for data science https://github.com/ossu/data-science


I would advise you to download[1] free maths class 10,11,12 books which is taught to indian students. They are well written, covers calculus, lots of exercises to practice as well to test your knowledge.

[1] https://ncert.nic.in/textbook.php?kemh1=6-16


The beautiful thing about learning math is it is incredibly well curated up to about 3rd year university. Suggest re-taking the high school courses over again via correspondence. Since you are interested and motivated it will not take you long at all. Then move on to first year calculus and linear algebra - either by correspondence or audit the actual class. I don't think there are any shortcuts up to this point. After this it is reasonable to self-curate the material based on your particular interests.


Some standard steps would be:

basic arithmetic

ratios, proportions, percentages, roots and exponents, compound interest, basic lengths, areas, and volumes

Then to get to computing topics with a minimum of work:

some basic algebra, think of it as arithmetic but with symbols instead of actual numerical values

can pass up plane geometry, or just learn the Pythagorean theorem and do see a good, simple proof (will see it again with some nice generalizations in linear algebra)

should touch on trigonometry, such as can do in just a few hours, that is, without a whole one semester course

for calculus, here is the world's shortest but still basically correct course in calculus:

There are two parts to calculus, differentiation and integration. In a car, look at the odometer and a from it construct what the speedometer reads. That's differentiation. Then look at the speedometer for, say, a minute, and from those readings construct the change in that minute in the odometer. That is integration. So, each of these two undoes what the other one does, and that is the fundamental theorem of calculus.

For more, maybe do some linear algebra -- the above will give you sufficient prerequisites. For linear algebra, look on the Internet for a text that is highly recommended (by a professor at a famous university, e.g., MIT, Princeton, Harvard, Stanford, Berkeley) and claimed to be a relatively elementary view.

That should be enough for a lot in the early parts of computing, computer science, and machine learning.

With a good teacher, might get through the whole thing in a month, a really good teacher, in a week.

Uh, I know VERY well what I'm talking about: I hold a Ph.D. in math with a lot in computing from a world famous research university. I taught computer science at Georgetown and Ohio State and math at Indiana University and Ohio State. I've published in pure and applied math, mathematical statistics, computer science, and artificial intelligence. I've done serious applications of math and computing to US national security. Twice I saved FedEx from going out of business, once with some computing with a little math and once with just some math and a little computing.


Get a copy of the Open University (UK distance learning uni, been around since the 70s) books on EBay.

A full set of books for MU123 (The most basic maths course, basically school [age 16] level books) can be had for under £50 on Ebay. Work through them, then move on to the MST124 books which are college [age 18] level), and are also widely available on EBay. Those begin to cover cover calculus, vectors, etc...

I'm 1/3 of the way through doing a maths degree with them, having scraped a C at school almost 20 years ago.


I'm studying math at Cambridge University right now, and a lot of resources for our courses are available for free online. I do think that learning on your own is just not the same as learning for a degree, but if you have the motivation then you may find it useful to follow along with parts of our course.

Dexter Chua was a Cambridge math student who started in 2014, TeXed all his notes and shared them on his website [0]. More people have followed in his footsteps, and you can find most of them by googling `Cambridge math notes site:srcf.net`. Many professors also put notes up on their websites, which can also be found by googling `Cambridge math notes`. I find that for many courses the notes are just as good as lectures, even notes written by students, and sometimes they're even better. They're certainly a lot faster.

You'll probably want to start with our first year courses, which are designed to take bright highschoolers and teach them how to think like mathematicians. If you're interested in the math of machine learning, you'll probably want to look at our courses on linear algebra, probability, optimization, and statistics.

[0] https://dec41.user.srcf.net/notes/


Once we're outside a formal academic setting, the way to start learning is to start learning.

The first high hurdle is accepting that starting out everything (to a first approximation) is over our heads.

There's no perfect first resource because hard subjects are hard and take time.

But because we are out of school, we have decades to learn.

There's no final in sixteen weeks and only a pop-quiz tomorrow if we are in the middle of applying what we learned.

So just start learning math and figure out what works for you as you go along.

Good luck.


It is perfectly natural to hate math classes if you weren't the smartest kid in the class. It's basically a guessing game where you constantly get zapped for not knowing something you had no way of knowing.

It's not the teacher's fault, it's not the student's fault, they both could address the problem on their own, but really it's that education receives very low priority. Children don't matter because they don't have money, and teachers don't matter because they teach children.

Plus, there's no scholarships or opportunities for remedial students who don't have that much talent but want to learn and put in the effort. All the big math opportunities are elitist, great mathematicians only go out of their way for exceptional students, which is fine, but never for an ordinary student who just wants to be smart and works for it. Studies. Well one time, I actually did that, I tutored two really disadvantaged kids and one of them pulled his grades up from like a C to a A-, and they awarded him a prize for most improved, due to his effort. His name is León. When I got to where our classes were, he was always acing the marshmallow test, just eating the bad parts of his food and leaving the best parts for last. Not like anybody was going to give him an extra marshmallow, it was just natural for him.

I designed lessons, for instance a first one about multiplication with just paper, no pencils. Multiply with a blank sheet of paper.

But in general, that charity only goes to the richest.

If you want you can write me, I can give like a 2 paragraph rundown for you in particular, just because you want to learn. Email in profile.


You have most of the mathematics needed to understand ML in one free book, Mathematics for Machine Learning [1].

It covers just about anything you need.

It would be better to study algebra, calculus, some trig and then statistics, it will give you a better knowledge, but this is a more condensed and a faster way to put you on track.

If you have time, you can go through it in one month.

[1] https://mml-book.github.io/


Thank you for this, it seems like a great resource, and completely free too.


Everything that can be learned can be studied through some mix of these techniques:

1. Rote learning/memorization. Copying, tracing, flash cards and so forth. This is how you learned to read and write, and while in school math it tends to be applied to calculation(memorizing results from adding and multiplying and so on) it can also be applied to build up recall of mathematical concepts like postulates and theorems.

2. Logic and problem solving strategies. Math "homework" is usually about finding a result through a mix of deductive, inductive and abductive strategies. When the result is calculation-focused it becomes very mechanical and "follow the steps you've memorized", and so can usually be delegated to a computer program now, but higher level math is more about integrating the concepts together to prove something is correct, which means having a really clear understanding of the definitions you're working with.

3. Dividing and conquering. Sometimes it's hard to see a concept in totality but you can understand a particular limited context and then generalize on it. This is typically where math research starts: there's a flash of insight into a concept and then progressive attempts to generalize it and reuse it to solve more problems or define its relationship to other concepts, like how there are multiple ways to define coordinate systems in geometry.

When reading a math text, it can be hard to get started because skimming the text doesn't really grant any access to the concepts: you have to follow through on internalizing them first, which means a mix of the rote learning and posing problems for oneself to solve, and looking for analogies in things you already know to find the differences and so gain more detailed understanding. By the time you've done that, you probably have read the same words hundreds of times and "slept on the problem" for weeks.

This quality of not really understanding math until you've grappled with the problems means that research mathematicians tend to only have a really detailed understanding of their own specialty, but have a more limited background in others, enough to communicate a little bit but not necessarily participate in the discussion substantially. To get "there", look at what's offered in college courses: you can reuse their textbooks and problem sets. Following an online course is also a valid method. You don't have to attend classes or lectures to study math, although sometimes you may want to ask questions to clarify - but the internet exists for that and lots of people are willing to help, at least up until you actually get to a research level problem.


I am going to make a suggestion. Try speaking with the head of a math department at a local community college or at the nearest college. Explain what your goals are and ask for guidence in putting together a plan. I think you will be suprised. You may well find a mentor who can help you. Everyone learns at their own speed and many aren't ready for some subjects until later in life. For example, I was a C student in high school and college in chemistry despite the attempts of my teachers to get me interested. In my 40's all that learning that I didn't care about started to surface and I found that I now remembered and understood more than I actually thought I learned. The same thing happended with Calculus. I barely passed it but when I needed it for my work it became fascinating. I believe you are about to experience the same sort of revelation. I am in my 60's and I can't wait to learn new things. A completely different person than who I was when I was younger. Become curious, it will lead you to a life of wonderful fullfillment.


I've been in a similar boat. Khan Academy has been helpful, as have some other resources:

* Gelfand (et al.)'s HS math book series: System of Coordinates, Functions And Graphs, Algebra, Geometry, Trigonometry

* The problem collections Challenging Problems in Geometry and Challenging Problems in Algebra, available from Dover

* The math and logic courses on Brilliant.org

* Smullyan's Introduction to Logic

* Balakrishnan's Introduction to Discrete Mathematics

* For a bracingly irreverent "skip the bullshit" perspective, George Simmons' Precalculus Mathematics In A Nutshell

* The A-Levels A* prep sequence from Imperial College London, available on edX (this is actually roughly equivalent to a first year undergrad curriculum at US unis)

General advice:

* Practice, practice, practice - solve lots of problems & push yourself to repeatedly prove and reprove things. 'Getting it' once is not enough, math is like basketball. Something all the resources above have in common: tons of exercises, with solutions

* Keep mixing up difficulty so that you get some easy wins for confidence and motivation but also challenge yourself and keep yourself humble. Occasionally dip way down to remind yourself how far you've come, and on the other hand sometimes dip into something like Concrete Mathematics to remind yourself how far you still need to climb

* Go over the same material many times from different authors/teachers/sources

* Take your time. As long as you keep challenging yourself, and keep putting in those 10-20 hours a week, every week, you'll get to a good place and then keep going


I recommend this book:

https://www.glassner.com/portfolio/deep-learning-a-visual-ap...

It goes over the basics in the first half to make sure you have a foundation to understand the deep learning stuff.

And don't feel bad if there are gaps in your math. Go to khan academy or youtube and fill in the gaps. The most important thing is to find exercises and do them yourself, watching videos and lectures help understand the big picture but you'll never internalize them without doing the work.

I think in any kind of knowledge, math, music, art, etc... the biggest limitations come from people not knowing the basics deeply. Don't just work on stuff until you get it, or it clicks, keep going until it's so boring it's automatic. It will feel slower but you'll ultimately learn faster as you won't eventually hit walls.


Ciao! Six years ago, I decided to try to get a Bachelor's degree in CS while working as a software developer for eBay (I had a tremendous impostor syndrome).

In Italy, most Universities, even the public ones, require passing one or more tests before attending the courses; since a CS degree required a math test, I decided to prepare myself one year in advance. But unfortunately, I barely remembered how to do even basic algebraic operations, so I practically started from a secondary school program, fighting hard to reach a "last year high school" math level.

I decided to hire a teacher for private lessons because I found online courses very dispervise, and I felt that having a clear, structured path to the goal was a better approach for myself instead of struggling with different platforms and books. I studied 4 days per week circa, and the hard work paid back since I was admitted to the course!


Some good links have been posted. I'll post another, but it's just the first link I saw for what it is I think you should learn first: the Distributive, Associative, and Commutative Laws. They're taught in middle school, and easy to get. But you need to get them _deeply_. Spend hours internalizing them. Play around in a spreadsheet, making 2 or 3 formulas at a time that all do the same calculation but look different because you distributed, associated, and commuted the cell addresses differently in the formulas. Learn them so well you might forget the names but you can't forget the concepts, ever. Make them part of your being. https://www.mathsisfun.com/associative-commutative-distribut...


KhanAcademy and 3blue1brown (https://www.youtube.com/c/3blue1brown/playlists). Remember, it will take time. It won't be fast. Especially if you really want to understand ML algorithms.


I prefer traditional textbooks, so I've always been turned off by suggestions to use Kahn Academy. My preferred math resource is OpenStax, an open-source textbook publisher. The books only go up through calc 3, so if you're beyond that level, you will need to find a different resource.


I generally agree, as videos tend to take a long time to convey information. Have you actually worked through the OpenStax books?


I've used them for reference but haven't fully worked through them chapter by chapter.


Paul's Math Notes https://tutorial.math.lamar.edu/


This is what got me through my university math courses. Unfortunately the notes for his Linear Algebra course were taken down at some point.


I would say err on the side of too easy to begin with. Start with textbooks for high school (that you skipped). And master that before moving on.


I'm restarting my own math education from the point of Calculus 1 so I can learn more about AI/ML, and want the college credits so that if I decide to do more graduate school, I will have solid prerequisites. I searched pretty exhaustively and settled on NetMath (https://netmath.illinois.edu/) which is a UIUC program. Most classes are self-paced and use Mathematica extensively, but you can go amazingly far in the program. I hope to get through their Calculus-based probability by about a year from now.


I neglected math a lot in HS too and tried the Khan Academy route, but figured out over time that you really can find your gaps and fill them on demand by focusing on the work you want to do.

I wouldn’t discourage you from trying a more comprehensive approach to building a great foundation in math. Math is awesome and if you enjoy it, go for it. If you want to stay focused on ML, you might do alright by figuring out what you need as you go. Just walk back from each problem until you find your bearings, then dig in.

Math is huge and you could find it takes forever to arrive at the skills you need to do the specific thing you want to do.


The “Mathematics for the Practical Man” series, although old, might be a good resource to consider. These are the titles I know are out there:

- “Mathematics for the Practical Man” - “Arithmetic for the Practical Man” - “Algebra for the Practical Man” - “Geometry for the Practical Man” - “Calculus for the Practical Man” -> this last one was the one Richard Feynman used to teach himself calculus (https://physicstoday.scitation.org/do/10.1063/PT.5.9099/full...)

good luck!


Start with reading https://en.wikipedia.org/wiki/Playing_with_Infinity it might help with hating math.


You’re not alone but most of the advice in this thread is very overwhelming and wholly unnecessary.

Start with ISLR

It’s a very well done book to quickly get anyone up to speed.

https://www.statlearning.com/

I then recommend this linear algebra book.

https://web.stanford.edu/~boyd/vmls/

The beauty and effectiveness of these two books is that they are applied. Applied learning is the most motivating way to learn IMO.

After you go through these two books You will have a very strong background!


I’m in a very similar boat. I didn’t find math interesting in any capacity until I read Frege’s foundations of mathematics when I was about 21 or so. Unfortunately, at that point my skills had degraded so much that I could barely get through basic geometric and algebraic problems. Since then, I’ve managed to greatly improve my understanding through self-directed learning to the point that I’m able to use mathematical concepts to make sense of other domains.

My approach is a bit less structured than what others have recommended. I typically start by gathering tons of materials on the domain, say, digital signal processing. I then read things carefully, accepting that I won’t understand everything right away. Whenever I encounter something I don’t understand that seems important, I’ll sit down with a pen and paper or open vim and quite literally go point by point and try to write an explanation of the concept and make sense of it for myself. This method has served me pretty well for learning about new things.

I think one thing you cannot get around, which others have also mentioned, is that the process is necessarily slow. trying to come up with detailed explanations for yourself takes time, as does rereading material, practicing, and learning to apply what you’ve learned. I think the biggest mistake autodidacts can make is rushing through things. It can be hard to resist the temptation to go fast sometimes, but it’s really important to dwell on things and go slowly


I have a similar problem but I am a bit older than 26 years old by 2. I took math in community college upto Trig and Technical Math for Electronics which included Determinants and Matrices. Did not help at all with finding an electronics job as only theory was taught. Fast forward to present and lo and behold, Linear Algebra, Determinants and Matrices are where it's at. Now I'm also looking for a solution to the forgot my math problem issue, but now I'm working full time and well my learning environment is noisy.

Were I to be given another shot at learning math knowing what I know now, I would have first found a way to learn learn Vedic Mathematics so that I don't have to waste so much brain cells on intermediate steps when solving math problems and second, I would have concentrated on learning Technical Mathematics with Calculus and Linear Algebra since those concepts, particularly Linear Algebra, are directly applicable to Machine Learning and possibly Quantum Computing, from what I've read but don't quote me on that last point.

Would it not be great if somebody were to design a game that would teach you all the concepts of ML Math, and ML without the victim, er learner, knowing that such knowledge was the goal of the game at least towards the beginning?

Hard probem I know as it requires a totally different way of thinking about math and science in general. I mean, who ever heard of math being fun? ಠ_ಠ


This is a good map of Mathematics to get an overview of where you can go... https://www.youtube.com/watch?v=OmJ-4B-mS-Y

Then Giles McMullen-Klein has an awesome recommended list for data science (your mileage may vary). https://www.youtube.com/watch?v=V2aIDbpESyU


High-school dropout with a phd in a social science here. My grades weren’t good enough to study math during undergrad and I was dealing with a chronic illness that meant I didn’t have my shit together enough to transfer into a useful degree either. I’m a machine learning scientist at a company you’ve heard of and probably spent money on (not FAANG). Math is a big part of my day job.

Here’s what I did to learn math. I wouldn’t recommend this path. You will have a much easier time getting a job with a quantitative degree. You’re only 26, so you can go back to school without really losing much time. If you must do it this way, do the exercises, do the exercises, do the exercises, build a portfolio, and do the exercises.

Calculus/Analysis

Stewart single var calculus, then the multivar book. These are easy starters

Real analysis: Series, Functions of Several Variables, and Applications. Miklós Laczkovich, Vera T. Sós

Spivak Calc and Calc on Manifolds books

Bonus: Advanced Calculus A Geometric View, Callahan. This is what I turn to when I want to punish myself or remind myself how certain analysis proofs go.

Linear Algebra

Strang, Intro to LA is great. Start with this one

Strang, LA and learning from data. Will be tough without the first book

Stats

Hogg, Introduction to Mathematical Statistics

Gelman et al, BDA3 for Bayes

ML

Bishop PRML and Elements of statistical learning. Do the exercises. Build the algos in Python.


I went a similar route. What did it for me was conceptual analysis. You know how to program already. Maybe you did object oriented analysis? Once I understood, that creating a sample space can be understood as creating a class of the samples you expect, and then putting the attributes on an axis of a space, it clicked. Depending on the space, you then can visualize the outcome probabilities as weight of a point or area in the space. A random variable is then a mapping of that space to R etc. Or that differentiation is basically a way to get the slope of a function’s tangents at any point (it is differentiation of a function!) and its converse, integration, helps you find the area under a function graph. The important point is here that many interesting points can be mapped to doing this (for example calculating the probability from A to B, if you decided to represent it as area). I had worked with UML, OPM (object process methodology) and BFO (basic fundamental ontology) before. Asking “what is it (attributes, parts) and what can I do with it helps me a lot. The most important trait however was coping with frustration, sometimes it took me months to understand a concept.


Browsing https://math.stackexchange.com/questions can be a nice way to learn math notation and to see which math topics you find interesting.

(also, occasionally a question or answer will be so good you'll instantly grok the math concept even if you haven't learned it formally before; it's rare but magical when it happens).


I would suggest to start with reading The Foundations of Mathematics by Stewart and Tall: https://global.oup.com/ukhe/product/the-foundations-of-mathe...

This will give you an understanding of which topics make sense to you and which you can barely follow. From that, you will need to catch up but where from depends only on your level. I guess, you might have some Khan Academying to do to catch up with the school program.

After you are done with The Foundations of Mathematics, I think you can take Andrew Ng's courses and go from there. I also use https://www.routledge.com/A-Concise-Handbook-of-Mathematics-... as a general reference when I need to look up a topic a have a rather vague recollection of.


There are cheap soft-cover math workbooks for just about every grade level, which are good to keep a kid busy over the summer and not forget everything they learned in that grade. We have one here ... "Spectrum Geometry, Grade 5," "Pre-Algebra, Grade 8."

If you want something more hi-tech there is Khan Academy of course.


I think we do a terrible job in the US education system of explaining why and how concepts are important. When I entered the University of Michigan for Computer Science, I had to take many math classes in my early years that seemed at the time very unrelated to programming (derivatives, integrals, multi-variate calculus, differential equations, linear algebra). I did what was required to pass these classes but didn’t understand their importance. Then, my senior year of college I took computer graphics and machine learning courses. I was so disappointed that I didn’t realize the real-world applications of math and how it would be helpful to my career. I feel like these math concepts could have been more integrated into the degree program. At the very least, a better explanation of the applications for these concepts would have been helpful.


I'd recommend going through the Khan Academy curriculum and filling your gaps - only study the topics you are struggling with. Once you do that, I'd suggest studying this https://teachyourselfcs.com/#math


Since you mentioned ML, I assume you'll need linear algebra (LA). I hated it for a long time since the notation abstracts away too many things. You can look at an equation involving matrices, but you cannot imagine how it would be coded. Unlike calculus where numerical approximations and code implementations are relatively obvious.

But I eventually needed to learn it as I have to code something not in Python/Matlab/etc as part of an app and would like to postpone using LA libraries unless absolutely needed for performance. What helped me get the grips on LA is Jeff Chasnov's lectures on youtube and Mike Cohen's book. I would also recommend 3blue1brown for appreciation, if he happens to cover the topic you wish to dive into.

I'm many years older than you, by the way.


YMMV and I am not paid to do ML, but diving deep into 3D game programming forced me to learn quite a bit about linear algebra which is useful for ML.

I found it very useful having visual uses that solved actual problems instead of just theoretical uses (or the old “trust me this’ll be important later”). For example, trying to figure out how light would bounce of an object put me down the vector, normal and dot product path (http://learnwebgl.brown37.net/09_lights/lights_diffuse.html)

After building a custom play around game engine, I can actually have discussions the ML people at work and I have a rudimentary understanding of what they are talking about.


This won't help in a practical sense, but Norm Wildberger has a very different take on mathematics and maybe this could be a way of coming to peace with what you hated: https://cosmolearning.org/courses/math-foundations-with-norm...

For a more practical direction, you should start with Guesstimation if you are not already versed with it. It is an indispensable tool for getting people to think you know some math as well as helping you detect really bad ideas in addition to helping you get playful with mathematics, which is the only way to really master it as a tool.


In high school my favorite resource was the Calculus (9E?) textbook by Larson. Super thorough in the explanations, there were tons of practice problems and step by step solutions on their website (no subscription needed iirc) for odd # problems.

My biggest struggle was skipping some of the basics and having to go back and re-learn after really struggling. But struggling helped in a way because then I really understood why X or Y solution didn’t work. If there’s anything i’d suggest, it’s that once you get some confidence in solving problems, try to approach new ones without looking through the entire lesson first, and that will help the actual way to solve the problem really stick. (And it’ll help you ask more focused questions!)


I always recommend courses on Udemy, Coursera, etc. Anywhere you have to pay for the knowledge. The money seems to be an important filter for quality. Not always, and there are certainly exceptions, but in my experience, it's highly predictive of useful knowledge.


This is subject to Goodhart’s Law. I’m not sure how to avoid that except by deleting this comment so nobody knows what measure you use.


I don't think it is, because I'm talking about a marketplace where goods and services can be exchanged for a price as being a filter for quality; otherwise known as capitalism.


If you assume price=quality and therefore always buy the most expensive one, you’re not participating in price discovery and they can just raise it, producing Veblen goods.


Oh I think I wasn't clear earlier. I'm not actually arguing that the higher priced the good, the higher the quality, since luxury items are one obvious exception. I agree with you there. I was making a comparison between educational lessons that are priced for free (like say YouTube tutorials) versus anything priced above $0.

I think just having a price tag at all incentivizes a marketplace where the best lessons compete. Now how one determines that quality though...lots of different heuristics and I have a few thoughts on how it could be done better but reviews, especially negative ones do a fairly decent job of assessing for qualities.

My apologies if it was not clear what I meant earlier.


For me "For dummies" series of math books works well: https://www.dummies.com/category/books/math-33720/ Start with "Basic Math & Pre-Algebra For Dummies" if you do not understand math at all. After get Algebra I and Algebra II. After Geometry, after Trigonometry after Precalculus or Directly Calculus. I am now at Algebra II and going well with self-study.


If I could relearn mathematics starting from elementary school, I'd get my hands on the AOPS books (Art of Problem Solving) and go through them on my own. They inspired my daughter sufficiently to pursue a math major in college.


I did the same as you, literally starting with the number line and basic factorisation around 10 years ago, using this book: https://www.bloomsbury.com/us/engineering-mathematics-978135...

It's designed for self-study, and it together with "advanced engineering mathematics" by the same author covers all the math from the first two years of a 3 year UK engineering degree (they don't have any gen ed requirements so my sense is the material covered is equivalent to the math from a US engineering degree.

At some point, if you want to learn even more advanced math, you need to learn to do proofs, but you don't need a lot of mathematical knowledge to learn how to prove things, probably someone with a good high school math knowledge coupdnprsrn the basics, using a book like chartrand "mathematical proofs" which for the first five or six chapters only requires algebra knowledge.

I now have an MS in math and statistics -self-learning is great, but once you get to the "proofy" math, it's really helpful to get feedback on your proofs from someone who knows what they are doing. I took some summer math classes at berkeley, and then did my MS at cal state east bay, and found that the teaching was much better at cal state, which I would guess is because they mostly don't get paid to do research (small sample I know, but talking to many friends doing math classes at berkeley, my experience seemed representative). So that kind of school (cal state) might not be the best if you wanted to do a top tier PhD (though every year we have a couple of people getting admitted somewhere for a math PhD program), but for actually just learning and intellectual satisfaction, it was great!

I would second Khan academy, to quickly get an overview of a topic, it's great. But you don't really know something mathematical until you've done enough problems to be fluent in that area, which I think causes problems for some people who "understand" a topic but move on before they have "mastered" it IMO.

Feel free to contact me if you want to chat or have questions


What do you mean by starting from 0? Are you literally at the 2+2=idunno level? Understand some basic algebra and calculus? Or where? I think in any case there is no substitute for working through textbooks and writing out a lot of exercises. Grasping concepts is only part of it. Like a CNN in machine learning, you actually have to "train" and not just "learn".

If there are particular subjects you want to get up to speed with (basic probability might be a good place to start), this thread can probably suggest resources, and it might help to work with a tutor.


I repeat my comment from https://news.ycombinator.com/item?id=11274264

„A single book is enough to learn mathematics: Riley, Hobson, Bence: Mathematical Methods for Physics and Engineering: A Comprehensive Guide It has a whopping 1300 pages, but it has everything you need.

And if that is not enough for you get Cahill: Physical Mathematics This will give you advanced topics like differential forms, path integrals, renormalization group, chaos and string theory.“


There are plenty of places on the web that will teach you math. Follow the usual high school college path. Alg1,2,Trig, Calc1... Grab the related text books so you can practice what you learn. Also set deadlines. It helps you move forward.

The primary thing you need to remember is that you have be persistent. Only you can be successful at learning. You have to be persistent and not give up. When things get hard just look at it as a challenge that you need to get through.

You've taken the first step and that's wanting to learn. Now it just a matter of getting it done.


What machine learning? For what use case? Hot take, you probably don’t need to know the math. It’s mildly illuminating to derive that the MLE of a Gaussian distribution is the mean, but it’s also irrelevant to solving problems. Statistical reasoning is important. Being able to think about things in terms of vector spaces is good. But do I really use calculus or linear algebra much as a data scientist? No. Basically never.

It’s way more impactful to say… visually and manually make a decision Tree on a 2D scatter plot of data points to grok the algorithm.


I’m in this boat as well. I dropped out of first year university some years ago and have subsequently forgot all the higher level math that I spent years learning. I can’t even understand the notation which used to be so familiar.

From my experience you have to start from scratch since some concepts are foundational and will block progress later if you don’t know them.

I have started with Khan Academy just answering questions on my phone. It’s a grind but I do it later in the day when my cognition is reduced. I imagine later on I will have to get a tutor.


If you are far behind start with Khan Academy. Then there are MOOCs on Coursera and eDX as well. Less interactive but 100% Real deal (and it doesnt mean harder) are course materials often with videos from MIT Open CourseWare. If you are into ML do Gilbert Strang Linear Algebra. It will develop you so much that it can’t be understated. Buy his book and do exams and assignments. If you want to take pleasure in math there are brilliant channels on yt, for example Numberphile. GL man.


Ps. Strang comes with recordings which are on yt as well.


Maybe I'm conceited but I think curiosity should guide you. Obviously something about ML inspires you. Envision what you want to achieve and fill the gaps as you go. Yes, there are more or less optimal ways to learn, but maybe it's more important to keep the fire alive than to push through quickly. The harsh truth is that maybe you still harbor an aversion for math. Most ML practitioners have dark spots, that's fine. But you need a genuine interest to navigate.


This. Flame that curiosity into an intense burning desire. It's worth at least a x10. Another order of magnitude comes from talking one-one with someone who is living and breathing the work. I just don't get these "give me a list of books to read about mathematics" questions that pop up regularly on HN. Yes it's good to read & do your homework, but that shit is slooow. Even the professionals don't "read" mathematics, they talk to other mathematicians, and they work it out themselves. Understanding is not memorizing. I don't know what it is but it's not that.


You're still young and given you have an already existing interest/reason why you want to learn. You'll learn so much faster and you will enjoy it.

I wrote this guide to help from 0 to advanced math. I hope it helps. https://juandavidcampolargo.substack.com/p/juan-davids-newsl...


Not sure if learning from scratch as an adult would be much different than learning it from scratch at any other age. Learning math always took a lot of intentional work and focus for me and I would presume it’d be the same for a student of any age. There’s no silver bullet to learning a topic but I saw Khan Academy mentioned which is a good way to receive condensed “lectures”


The short answer: Go to https://www.khanacademy.org/. Click "Courses" on the top left. Start working your way through the pre-calculus course. Review Algebra if you're struggling with its prereqs. Continue down the course list (Jump around as required) until satisfied.


There are a lot of great introductory mathematics courses available at coursera.org[0] for free, without the certification.

See this thread[1] for a list of great math book resources.

[0]: https://coursera.org

[1]: https://news.ycombinator.com/item?id=30485544


there are fundamental math that appear in all other math. i think calculus is one such math, and perhaps probability (even more true given you’re pursuing machine learning). for both subjects i almost self-teach using calculus made easy (https://www.calculusmadeeasy.org/) and probability for the enthusiastic beginner (https://scholar.harvard.edu/david-morin/probability). both are fairly self-contained although the sometimes take shortcuts that can be dizzying at times. but since you’ll proceed at your own pace, you’re about to make a diversion to investigate, further, why they were able to take those shortcuts. check them out and see if they help. otherwise i’d like to know what worked for you :)


I'm a web developer turned digital marketer and I know exactly what you mean here. Even I used to suck at maths to such an extent that it haunted me for a very long time. Honestly, sometimes it still does.

Anyways, I started learning maths through Khan Academy and edx. Also, Math24 is really good as well.


Khan Academy is a pretty solid starting point. There's also a lot of great stuff on Youtube. I'm a fan of a Professor Leonard. He covers from pre-algebra through differential equations.

https://www.youtube.com/c/ProfessorLeonard


I'm 42 and just getting started in ML and "the math" myself. I never got beyond algebra in High School but I have made a successful career in programming. I'm taking it day by day, I dedicate an hour each morning to plugging away. It'll happen eventually...


I feel people ask "how to learn math?" every other week. Probably a good thing more people want to learn math.

Anyways, I have reasons to believe Susan Rigetti's recommendations [1] are good.

[1] https://www.susanrigetti.com/math


Try https://betterexplained.com


Asess where you are. Define goal(s) as per comments below. And start the heck climbing a hill. Good luck.


Why not practice "Just in Time Mathematics"? Start studying Machine Learning and only study the minimum mathematics necessary to understand particular problems. In other words learn by doing. Do we memorize the entire C library before starting to code in C?


I truly learned math useful for software developers while studying “Software Foundations”: https://softwarefoundations.cis.upenn.edu/


I'd recommend starting from Terence Tao's Analysis I book. It really starts you from "zero" (literally), and constructing natural numbers, all the way to real numbers and integrals, and beyond.


khan academy is the best for ANY age. Start from basic algebra and progress through trig, geometry, calculus 1,2,3 and linear algebra, and you'll be fine. That's what I did. Should take a few months depending on work ethic


Depending on your level, you can get through high school level using https://www.houseofmath.com

Topic for topic, read and do exercises.


I think in principle it boils down to two things: finding the right material, and proper time management because it is hard to do anything if you have a job and other responsibilities


There are a lot of these threads (though I for one welcome more because there are always new resources).

It would be cool to see more follow-up posts about how it's going or where they ended up.


I recently saw this one https://youtu.be/pTnEG_WGd2Q , but I'm not sure it is a good one


i too want to learn math, but being more advanced i m looking at specific fields like number theory or differential geometry. I just need some math to advance in physics… i came to the conclusion while self learning that a tool like a “google translate” for math or a hypertexted math would be of great use to find the right curriculum… pitched it to a foundation / but will def try to push it at one point of my life… Math education could go a lot faster than it does now


Study slowly

Don’t skip through anything

Do the exercises


The MIT online lectures are amazing. I can't recommend them more highly. They got me through an engineering degree.


why do you need math for machine learning? Code first, learn the math as you go. Otherwise, you will learn a bunch of math you don't need when your actual goal is machine learning.

Do you know pandas and scikit learn? If not, start there.


Read the "mathematics for machine learning" book.


Actually, it's better to start with one not zero.


Coincidentally I'm doing the same thing right now. The plan I'm pursuing is to design a multi-month course in a particular mathematical subject I feel I need to learn, as if I was going to teach it to someone else. I would need a course syllabus as a framework for the material (in my case, this is something called complex analysis, because I need to understand how it all works, and unfortunately neglected it in the past).

Step One: Find some good source material. Many people have suggested Khan Academy, in my particular case I've discovered this excellent Herb Gross lecture series, 'Calculus of Complex Variables' provided via MITOpenCourseWare. Then, review all the source material and make a list of the topics:

Part I: Complex Variables:

The Complex Numbers; Functions of a Complex Variable; Conformal Mappings; Sequences and Series; Integrating Complex Functions;

Part II: Differential Equations:

The Concept of a General Solution; Linear Differential Equations; Solving the Linear Equations L(y) = 0 with Constant Coefficients; Undetermined Coefficients; Variations of Parameters; Power Series Solutions; Laplace Transforms;

Part III: Linear Algebra:

Vectors Spaces; Spanning Vectors; Constructing Bases; Linear Transformations; Determinants; Eigenvectors; Dot Products; Orthogonal Functions;

Step Two: Plot a timeline. Here I'll give myself one week per lecture. I'll set aside 1-2 hours per day, at least four days a week, to work on the material. This might be inadequate. Staying motivated without having some external pressure will be a bit of a problem, but noting that from the beginning helps.

Step Three: Devise problem sets to test my understanding of the material. Here is where having a good teacher on hand would be invaluable, but that's not an option, so I'll have to find example problems. One option is to find a complex analysis textbook somewhere, and also find its solution manual, and use the problems provided there. Having the solutions around to check results can be very useful. A websearch like "complex analysis problems and solutions site:.edu" turns up a lot of results, just look for the simplest introductory ones (i.e. not the advanced proofs!).

So, I think this is the way to go for self-learning in math. If you have a more introductory level subject, say Linear Algebra (w/o complex), or Differential Calculus, just try to do the same thing.

P.S. I find Jan Gullberg's "Mathematics, From The Birth of Numbers" to be a great overview of the whole mathematical world, kind of like a guidebook: https://www.goodreads.com/book/show/383087.Mathematics


Get a private teacher.


I'm 25 and I studied a computer science BSc. After I graduated I worked as a data scientist for 2 years and now I'm a machine learning engineer at a small start up. I have also been feeling held back from the more senior roles by my lack of mathematics and toyed with the idea of MSc or PhD. I applied to Imperial and UCL Machine Learning MSc on 3 seperate occasions and was rejected from all because of the lack of Maths. In the end it would have cost £16k (course fees for 1 yr) + £26k (loss of earnings for the year after tax) and so I haven't applied since because, although studying those masters programmes would give me a gread mathematical foundation, I can also 100% self-study and save £££.

The habit of studying is the most important habit you can learn at this early stage. If you haven't read it, I strongly advise you read Atomic Habits. It's really helped me. Here are some great mathematics for machine learning courses, some I have taken and some I am currently studying.

Study tips: - Always show up even if you only study for 5 minutes, it's more important to show up than to be perfect - Make studying a daily habit even if you only do it for a short amount of time - Stack studying onto the end of another daily habit, e.g. study after you've eaten breakfast before you start work (this is what I do) - Make studying satisfying (I bought a small calendar and cross of each day I succeeded with a sharpie and it's super satisyfing crossing them off) - Remove distractions by putting your phone on charge in another room - Make notes on new concepts in the form of questions, these can be used later on flash cards to help refresh the material and avoid the effects of the forgetting curve - Write code to see the math in action. Running code is immediately gratifying and makes the value of the maths real and tangible.

- https://www.coursera.org/learn/machine-learning (free, approachable) - https://www.udemy.com/course/machine-learning-data-science-f... (like £10 on sale, very approachable) - https://www.udemy.com/course/mathematics-statistics-of-machi... (like £10 on sale) - https://www.coursera.org/specializations/deep-learning - https://mml-book.github.io/book/mml-book.pdf [textbook, not very approachable until you have some background] (free)


I've never been to high school or college. The last school grade I completed with 7th -- or really just 6th. I can go into that in more detail if you like but there's a lot of personal drama that is irrelevant.

I will say that math is one of those things that you don't need to go to school to learn if you are persistent and passionate enough to learn on your own. There are tons of ways to learn.

First is to determine where you are. For that I highly recommend taking some of Khan Academy's free math courses. They start at the very basics. You will get bored of those.

But after that... I would say to take some things in your life and determine how to apply math. Create your own questions and try to answer them.

When I was 26 I was at the tail end of a janitorial job. I had applied (and learned) math to determine how much cleaning stock would likely be in use each week. Sometimes the math was "wrong" in so much as cleaning stock would be used more frequently or less frequently depending on what events there were that week. But it was a good exercise.

I'd also been writing some software on the side. Software and math are like two peas in a pod. A lot of people will tell you that you don't need to learn math to write software and they might be right for certain kinds of software. But even if you don't need math for software, it will certainly help. I now write software to fly drones and geometry is one of the core requirements.

Beyond the basics are Khan Academy there are plenty (!) of youtube videos from mathematicians who describe some of the simple [0] and more complex [1] concepts. Those have also been handy to me. All you need to do is search one math term that involves what you think you want to learn. A lot of the videos will include links to other videos that are prerequisites. Almost all of them will mention math words that will be new to you and if you search for those math terms you are likely you find other useful information.

Ten years ago I would have recommended that you search Google. Now? Don't do that. Google is full of trash and shit that want to take your money. Use Wikipedia, youtube, and reddit.

I've found 3blue1brown's lessons [0] [1] to be quite insightful for myself and have successfully used knowledge learned from his videos.

[0]: https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x...

[1]: https://www.youtube.com/playlist?list=PLZHQObOWTQDNPOjrT6KVl...




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