Ask HN: How to learn math from zero for adults?

[stArrow]

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.

[zerohp]

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.

[rg111]

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.

[jp0d]

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.

[ImaCake]

>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!

[xoac]

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.

[gigatexal]

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!

[99_00]

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.

[jseliger]

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

[BigHatLogan]

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.

[saminjerseycity]

Almost same thing. Khan Academy is amazing.

[badrabbit]

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

[justsomeuser]

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

[pseudostem]

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.

[ngcc_hk]

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

[kamilafsar]

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

[jonnycomputer]

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

[867-5309]

had me at biceps

[elliekelly]

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.

[blagie]

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.

[tzs]

> 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.

[dontbenebby]

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.

[rg111]

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.

[DeathArrow]

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.

[plinkplonk]

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.

[shp0ngle]

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

[Arjuna]

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

[webmaven]

> 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

[antegamisou]

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).

[macintux]

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

[blagie]

* 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/

[password4321]

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

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

[dougabug]

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.

[arisbe__]

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.

[jonnycomputer]

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.

[misterpurple45]

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...

[SoftTalker]

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.

[triyambakam]

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

[joebo]

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

[shmde]

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

[osigurdson]

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.

[graycat]

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.

[em10fan]

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.

[broses]

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/