I have yet to find a guide that does not start with the assumption that you graduated highschool.
That is a very reasonable assumption to make. We are in a community of technology and engineering, it would be a bit ridiculous to assume the people you are surrounded by did not have a fundamental base of mathematics.
But the times I have tried to go through these teach-yourself materials, it went from zero to draw-the-rest-of-the-fucking-owl real quick. 
I have been programming for 14 years, but stopped doing schoolwork around age 12, and never did any math beyond pre-algebra.
Does anyone know of materials for adults that cover pre-algebra -> algebra -> geometry -> trigonometry -> linear algebra -> statistics -> calculus? At a reasonably quick pace that someone with a family + overtime startup hours could still benefit from?
(Also, curse the Greeks for not using more idiomatic variables. ∑ would never pass code review, what an entirely unreadable identifier)
As you finish a subject, see if there's a corresponding book in the Art of Problem Solving store ; you can revisit the subject at a deeper level that will strengthen your foundation. The AoPS books will also expose you to areas useful in programming like discrete mathematics.
Before any of the above, take Coursera's Learning How to Learn course. You'll learn lots of effective strategies to get the most out of your efforts. For example, you can use Anki  to remember definitions and concepts you've managed to understand and to schedule review of problems you've already solved.
Myself and my daughters use Anki every day. It has been the defining factor in turning my mediocre job into a career. And Ankidroid on Android is open source.
I like Anki but I have never been able to figure out how to capture complex and often long business related information in tiny cards.
Is it though?
In this particular case, for this particular person, may be. But personally, I always found myself bored and ADHDly switching to something else in 5 minutes if I wouldn't drop myself right in the deep end. And from my interactions with different software engineers over the years, I doubt I'm the only one.
More like the most foolproof way.
Disclaimer: I was a full-time community college professor for a decade. I had no idea what a resource they were. It's small money compared to either the alternative of a university or not succeeding. If you use them you will succeed. It's what they do, and they've been doing it for a very long time.
After I went back to school, I tutored in the math study center to pay the bills, which really helped cement not just the learning but also the notion that I could survive academia. I'd gone in with a plan to study engineering, but after I transferred to university, I kept dawdling on the math prerequisites and not taking the engineering courses that needed them. So it kinda gradually dawned on me that math was what I loved, and away I went.
I never strayed too far from computers. I'm a graph theorist, specializing in computation; had I known better I'd have gone into computer science because that's where I see the most progress being made.
Interestingly this is similar to what my two advisors (one from the math department and one from CS) suggested to me. It would be easier to do the math I like in a CS department than it would be to do the CS I like in a math department. Do you feel like math departments are more conservative when it comes to working outside the discipline?
Maybe an even better way to start would be with trying to take a math class during the summer semester. 8 weeks of intense study to kick things off. I bet after doing some programming algebra would be a breeze.
My experience with community college math developmental classes was: "Go to this AV room, watch these videos, do these workbooks. If you have questions, I have office hours on these days.". Which was terrible. I did have someone I could come to with things I didn't understand, but it was obviously not something they liked doing and you had little choice about the material used.
This was decades ago, but I'm betting that now it's: "Watch these Youtube videos, do these workbooks. If you have questions, ask them in our online forums." Which is likely worse than just doing your own thing.
Anyway, my recommendation is to check out "the vibe" of the mathlab or whoever are the folks doing the tutoring. If they engaged and love answering the same questions all day long, then definitely sign up. If the tutors are like watch this video and do this quiz, if you have a problem sign up on the sheet kinda attitude, then find another place.
One thing I tell my high-school students: mathematics always looks harder than it actually is. One of the essential skills in succeeding in math is looking at a page of arcane "stuff" and having your reaction be, "Whoa! Can't wait to learn what this means," rather than, "Whoa! This looks so complicated!"
Mathematical is its own language that has developed across continents and millennia. It has its quirks and foibles, but overall, community consensus has guided its notation. Mathematicians want things to be simple and "make sense", especially the notation they use. It's never as terrible as it looks.
Sigma specifically is a Greek letter, but the notation is not Greek. Like a large amount of modern mathematical notation, the convention came from Leonhard Euler in the 18th century. It was a disambiguation choice because the letter S was overloaded.
Single-symbol identifiers are enormously popular in mathematics because mathematics is not computing. Because math is (even now) essentially a handwritten subject, its design plays to the strengths of handwriting. Line size, height, and character layout are essentially freeform. Character accents and modifiers are easy. discrete_sum would never fly in a handwritten world, just like ∑ wouldn't pass code review.
In attempt #1 I was jumping ahead to read the interesting stuff (calculus), and while I could make some progress, it was needlessly difficult because I didn't start from the basics.
It attempt #2 I started from the very beginning (course 1 out of 10 mandatory high school courses), and focused on doing exercises. However progress was slow, because I would just continue forward when I felt like it.
Finally attempt #3 was successful. I committed to doing exercises in order consistently every day after waking up. This felt great, as every week I was making noticeable progress, and having all the prerequisite knowledge for each next step made progress much easier than I had imagined it could be.
With the slow start but gaining pace towards the later courses, I finished this self-study project in 2 years (could have been close to half that, had I gotten into the groove from the beginning), and found it quite enjoyable. It didn't feel like a chore at all, more like the highlight of each day.
So in my first year of studying maths, I had 8 hours of maths lectures (and 4 hours of a minor which was of negligible effort). The exercises that came with the maths lectures made this a full-time program (and I estimate that while I didn't usually study all weekend, I typically only had 1-2 weekends per year in which I didn't look at anything at all). So one thing that can easily go wrong is underestimating that for every minute spent reading / listening to a class, one would want to spend 4 minutes working the problems.
The other comment I would have is that, yes, university level mathematics is (at least it was for me) incredibly hard. The reward is also astonishing: All these hard exercises I struggled with one year, are easy to do on a napkin during breakfast in the next year.
Doing it every day on the other hand, I would quickly need the thing I just learned, both reinforcing the learning and making it easier to apply it. Then as progress was much faster, it was more motivating as I could see myself making gradual progress each day, such that completing each course seemed like a doable undertaking.
I completely ignored math during high school due to a number of reasons (bad influences, even worse teachers...). I then went to college and managed to pass through calculus classes, mostly thanks to pure mechanical memorization and professors turning a blind eye to my lack of understanding.
Since my graduation (~5 years ago) I've been trying to fill this gap, but like you perfectly described, all materials expect you to have a solid basis. I think the problem is that math is huge and people spend a good chunk of their lives learning it (4-17 for the fundamentals alone!), so we fail to see how much it involves and how hard is for somebody that didn't have a proper education to learn it.
I have been making solid (although slow) progress with https://www.khanacademy.org/. I tried to learn from the top a bunch of times, but always hit a wall and dropped it. I only started moving forward when I decided to go through the basics, algebra and trigonometry 101. It has been a hard and slow journey, but each step comes faster and becomes more rewarding.
Gelfand also has some nice texts on Algebra, Trig, and Geometry that are reasonably cheap, especially if used.
I'm older and went back to school later in life to study math, and these are the books I learned that material (for the first time -- I flunked math thru high school) from.
Here are the exact titles and ISBN-10s:
Ron Larson, Calculus: An Applied Approach, ISBN: 0618218696
Ron Larson, Trigonometry, ISBN: 1133954332
Israel Gelfand, Trigonometry, ISBN: 0817639144
Israel Gelfand, Geometry, ISBN:1071602977
Israel Gelfand, Algebra, ISBN: 0817636773
And as others have mentioned, Khan Academy is pretty good, although I tend to prefer patrickJMT's explanations a bit more: http://patrickjmt.com/
* Guesstimation:Solving the World's Problems on the Back of a Cocktail Napkin. Math is a tool. Start using it with some simple arithmetic and scientific notation. Once it becomes something you can use and play in that context, everything else becomes a lot easier. This is water cooler talk and is something actually usable immediately.
* Speed Mathematics Simplified. From the 1960s. Wonderful book about doing arithmetic from left to right. Also has some good stuff about decimals/fractions/percents as well as checksums. Being quick with arithmetic and getting that number sense makes everything else easier.
* Burn Math Class. Gives an appropriate viewpoint for a lot of math. Gets a little whacky as it goes on, but the core ideas should help you take ownership of math.
* ... gap not sure what to put in ... Maybe Precalculus in a Nutshell... But play around with GeoGebra. Exploring geometry, trigonometry, and precalculus visually is key to getting an intuition about. Get to know the behaviors of the functions, but don't get lost in trig identities or solving random algebraic equations. Things like Newton's method (or the Secant Method) are more important for learning about than lots of arbitrary algebraic simplifications (they can be important too)
* Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach. At some point, if you mastered K-12 math and want to get a good mix of theory and application and efficiency, this book by John and Barbara Hubbard is really quite nice. It puts linear algebra in one of its primary contexts of being the main foundation for solving nonlinear systems.
One thing I’ve realized from experience: most books with lots of pictures and thick stacks are faking it (i.e. most college and school math/physics textbooks are not even worth the paper they’re printed on). The whole conceptual basis of these subjects is to distill everything down to a few simple ideas, which can then be applied in different contexts. The concise books typically tend to be much better at conveying the essence without bullshit. You just need to read a couple of hundred pages without getting stressed, rather than getting lost in an 800 page book and losing the big picture.
Good luck in your journey. I know how frustrating it is to not be able to find the math resources you need at an awkward level. If you happen across even better resources please share.
Also a tip that really has been helping me: when you "read" a math equation, don't simply recite the variable names and numbers. Try to say out loud what they represent. I've found that if I can't then I don't really understand the concept I'm working with.
Of course you get the books when you sign up for the course, but it's way cheaper to get the books and study on your own, and you'll get 90% of the information that way.
I don't know if you'll find the pace quick or not; personally I would say that learning maths is very hard, and the materials you use are unlikely to prove to be the botteneck (spoiler: it's you).
Can you share the names of the textbooks?
The courses the comments refer to are the level 1 courses listed here: http://www.open.ac.uk/courses/maths/all-modules
The text books are the texts for these courses. I don't know if you can get the books from OU without registering for the courses.
I took a Masters in Maths with them and it was brutal. Serious exams are a youngster's game. I got a glimpse into why so many Cambridge wranglers were also serious atheletes.
Second-hand OU books are usually bought from https://www.universitybooksearch.co.uk/
> Question 1: Sara wants to buy a desk before she starts her Open University course. She has chosen a suitable place for it but needs to measure the space before going to buy it.
> Which of these units is the most suitable for measuring the width of the desk? (a) millimetres, (b) metres, (c) centimetres, (d) kilometres.
Seems pretty arbitrary to me. Okay (d) is arguably out. (b) is maybe out? (c) vs (d) seems a toss up to me. Who cares if it's 1234mm or 123.4cm? How is this even a good question? Not a good sign
Note: the answer they want is (a). I don't think I have a tape measure I could get accurate millimeters on especially given measuring a space where I need to bend the tape at 90 degrees like next to a wall.
Imagine being someone who cannot produce an answer to this question. Who cannot discuss it.
The vast majority of people for whom this entry-level course is aimed at do not think like you. The purpose of the question is not to elicit a correct answer, although I note the effect intended actually worked perfectly on you and the question has been a success in your case. Maybe you are exactly the kind of person this question is aimed at!
The reason is to learn why there is an answer, at which point you can give an opinion as to whether cm/mm are better. You've missed the primary step, and almost certainly are - as others point out - over qualified for this question (which is a precursor to discussions of decimal precision).
Often when helping the kids with homework I find myself answering "well this is the answer they want ... but the real answer is ...".
tl;dr the question is there to make you think, not because there's only one answer.
Not taught courses, took its courses himself, in mathematics.
It went up in my estimations then for sure, because he was already a great teacher, intelligent and knowledgeable in the subject (certainly more than enough for up to 'Further Additional' at A level) and studying for his own interest (or I suppose a career change or further study for all I know).
From my own personal experience, I would recommend getting to the high-school graduate level and then taking some classes at your local university or community college for topics beyond trigonometry. You'll likely be able to handle everything up to that point on your own without much support, but many people struggle at that point and benefit from having people to ask for help when they need it.
I would wish you the best of luck, but I have no doubt you'll get to where you want to be without it.
I bought the Math and Physics copy because it has an ebook option and the first chapter is the Math guide. I’m going through a few pages a day and it’s crisp and straightforward. There is a sample of the first chapter on the site, I suggest you check it out to see if this is what you are looking for.
Found via an HN thread from last year on this topic.
Here is a direct link to the preview: https://minireference.com/static/excerpts/noBSguide_v5_previ...
I won't claim that Chapter 1 is complete and detailed review of all of high school math, but I did my best to cover all the essential topics needed for mechanics and calculus so that the book will be self-contained.
The preview linked is a little outdated so it shows the old figures, but the latest version has all the figures done in TikZ (a vector drawing package using LaTeX syntax, see https://www.overleaf.com/learn/latex/TikZ_package for examples).
There is no "tooling" per se for the book production (just run pdflatex to get the PDF, then upload to lulu.com and amazon.com, and they take care of the printing). I do have some scripts to enforce naming and notation conventions though, and there is some advanced git-rebase kung fu going on that allows me to reuse the high school prerequisites from Chapter 1 of the MATH & PHYS book for the LINEAR ALGEBRA book as well (basically when I fix typo in the master branch, I have to rebase the LA branch).
One thing that has been tremendously useful and I highly recommend for any authoring task, is using text-to-speech for proofreading: https://docs.google.com/document/d/1mApa60zJA8rgEm6T6GF0yIem...
I'm going to release one on Maxwell's equations next week, and I started working on a Calculus and General Relativity guides as well, so hopefully it helps!
Linear algebra is quite a beautiful, approachable subject; and a certain amount of it is necessary to make the leap from single variable to multi-variable calculus. Without a good grip on calculus, you can’t really what’s going on under the covers with linear algebra. What you need to do is precalculus (Simmons) -> single variable calculus -> very introductory / elementary linear algebra -> multi variable calculus (Apostol) -> less introductory linear algebra but still fairly basic (Gilbert Strang Intro to Linear Algebra) -> mathematical analysis (Apostol) -> linear algebra done right (Axler). You have to apply a spiral method where you return to subjects as you gain the tools you need to understand them better. You’ll never be done understanding geometry, algebra, or analysis.
Also, math is a problem solving art, and you can’t solve problems by reading, you solve them by thinking. Seek out problems that challenge and consolidate your understanding. You should be able to prove everything in Simmons and it should seem totally natural and intuitive. Then you’re ready to struggle with calculus, which is a subject humanity struggled with for centuries before getting a rigorous handle on. You probably want to get a handle on the mechanics and intuition, first, and for that I’ve heard that “Calculus Made Easy” by Silvanus Thompson is good.
Don’t try to eat too much all at once, you’ll make yourself sick. Don’t try to cheat yourself of the patient struggle to understand, confusion is completely natural when striving to really know something.
George F. Simmons,
Introduction to Topology and Modern Analysis.
I studied it one summer in an NSF program at Vanderbilt. Then I concluded that he is one heck of a good math writer.
• 3Blue1Brown has great introductory series on linear algebra and calculus.
• Khan Academy covers pretty much all of US high school mathematics, and you can go through it at whatever pace you want.
• I can send you a few Australian high school textbooks if you want.
"Who is Fourier: a mathematical adventure"
It is a simply brilliant book that takes you from basic trigonometry, logarithms and so on through calculus and finally fourier series.
> Each grade folder has a number of chapters, each chapter with a number of exercises, and answers to these in a single file.
> Exemplar Problems (for in-depth learning) with Answers
> Try solving the exercises and problems using pen and paper.
I found Khan to have too many endless lists of equasions, while Brilliant is much better at building intuition.
Please do email me silas [at] brilliant.org, so I can figure out which failure mode you hit and evaluate if it has improved at all since you quit.
In addition, as others have noted, our courses have no community aspect to them at all.
the books by sanjoy mahajan are also a treat and teach real-world applications of mathematical and scientific thinking.
Speaking as someone who was kicked out of precalc and then never touched math again despite working as a dev, your point resonates with me :)
i guess its idiomatic if you know "Greek".. ∑ is sigma, greek letter S, so Summation. And Π is pi, for Product..
Consider discarding that requirement.
> that someone with a family + overtime startup hours could still benefit from?
I suggest drilling fundamentals with easy exercises in moments of low quality time. (I often do a few Khan Academy skills during bouts of insomnia. For others it might be the commute, or just before bed, or...) Periodic repetition over the long term is more powerful than cramming.
Save your best quality time for your family and your job. Accept that you will progress in math at a slow pace. Before too long you will nevertheless end up ahead of many successful (!) software engineers who do not have strong math foundations.
Restructuring code, for example, often needs a good grasp of negation in logic.
Do you need formal training to do it? Not really. It’s advantageous to have a good math grounding though. My colleagues that have a good math education can often reason and communicate using graph theory, especially when it comes to architecture. Set theory is also super useful - I’ll often see people writing crappy algorithms because they don’t know about using sets (again, understanding computability and complexity would have helped here).
Maths is all around us, it is really just the study of patterns after all. That applies more so in software, even if it’s not immediately apparent.
Basic arithmetic comes up all the time: pro-rating a monthly plan, figuring out how much to scale up or down a system in response to changes in data, figuring out when your system will run out of memory/disk.
Statistics and probability are also pretty common. I'm often calculating standard deviations and finding expected values of non uniform random variables. For example, how fast does a queue have to be to handle 1 second tasks 90% of the time, but 30 second tasks 10% of the time?
Derivatives comes with many graphics tasks, such as 3D graphics or animations.
[EDIT] and then there's "what is math?". The memorization from early grades that everyone shits on, with some simple algebra, is what I actually, ever, use in my life, plus some very basic geometry when doing stuff around the house. If it's for work it's some practical application thing. "Real" math like proofs? Never, ever.
Good remark. I wouldn't call arithmetic math, nor would I call using a Boolean expression math. I am currently working on a compiler bug that has to do with liveness analysis. That is an algorithm, which kind of is math, but the actual bug is just 'oh, for some reason the registers that the function arguments are passed in are not marked as live', and I wouldn't say that I had to use any math.
In my job, I'm usually either fixing bugs, parsing formats, making different API's work together by converting stuff or writing wrappers around API's. I wouldn't call any of this 'math', and if you'd ask me I have never used math at work (which is a shame really, because I really love math).
Can you share example code? I can't see how that works.
I'm studying CS in University now and will have to do a math-related subject and I'm quite nervous about it, because my mathematics skill is extremely low, and has been my entire life.
I've also been bookmarking guides like this but haven't gotten to looking into them (pure haziness) other than reading the introduction, which usually says "this guide assumes highschool level mathematics".
 (amazon: https://www.amazon.com/Calculus-Easy-Way-Douglas-Downing/dp/...)
S for Sum, T for Total, N for Number, I for Index, etc. might though.
I only meant that criticising anyone for using them in the first place is unfair/hypocritical, because we'd quite readily do the same with the Latin alphabet.
There's also Prof Leonard YouTube channel, get the book(s) he uses, covers exactly what you want:
tl;dr - don't be afraid to go back to math concepts from elementary school to help you along the way to learning more math.
You have a gaggle of responses to go through, but I want to put this out there anyways.
Algebra is talked about as a 'breaking point' for many Americans; however the solutions rarely look at what transpired (or didn't) all those years before a student reached algebra.
Math standards in the United States are set so that ideally:
Kindergarten: learn to count
1st and 2nd grade: learn to think additively (+ and -)
3rd and 4th grade: learn to think multiplicatively (x and division); learn fractions
5th and 6th grade: learn to think in ratios and proportions; learn to think algebraically
Throughout all of those grades you are also supposed to be learning the properties of operations.
By the time you reach an algebra course in 8th grade or 9th grade, it requires you to call upon all of that previous knowledge.
- learning the properties of operations by rote and thus not understanding how to use them to manipulate algebraic equations
- not making the leap from additive to multiplicative reasoning, which hurts a students ability to understand fractions, which hurts a students ability to understand ratios and proportions, which hurts a students ability to reason with algebraic equations
- I forgot exponents. Most students only know those by rote or a bit about them before suddenly seeing huge exponents and negative exponents attached to variables in algebra.
Algebra itself may not be a problem. It is however a strong indicator of knowledge of the above. It's also where the house of cards falls down for students like you and me.
Source: was student who math fell apart for in school. I learned all about this when I left the business world to teach 4th grade, eventually created and piloted an "Arithmetic to Algebra" course for students to put all of this into practice, students learned, we rejoiced
If you do Book of Proof first you will find Spivak much easier, since Spivak is very light on using set theoretic definitions of things. Even the way he defines a function pretty much avoids using set terminology. Book of Proof on the other hand slowly builds up everything through set theory. It was like learning assembly language, then going to a high level language (Spivak) and I could reason about what’s going on “under the hood”. Book of Proof is such a beautiful book, I wish I had something like it in high school, mathematics would have just made sense if I had that one book.
I read a quote somewhere, think it was Von Neumann that said, you never really understand mathematics, you just get used to it. Keep that in mind.
Ah Spivak...yes, I absolutely agree it's one of the best books to build up that mathematical maturity everyone talks about.
For me Spivak took about 6 months and I managed to do almost all of the starred exercises - Gifted? No. Brutally determined: yes. And I was quite fortunate to be in a place in life where I could put serious hours in to it at the time.
After that, I learned to relax a bit more as I realised I had pushed myself way too hard and was close to burning out. I still love looking back at that damn book though. There's just something that's so special about it...the way the exercises build upon each other and connect together. It's really unique.
Although rarely, some authors do provide solutions, like Knuth's books, Stephen Abbott's Understanding Anaylsis, etc.
For immediate feedback, maybe you can checkout  to learn some formal proof by doing interactive proving.
BTW, you can always ask questions on https://math.stackexchange.com
It's a little like saying "learn programming well enough and you'll know if some piece of code works as expected without running it."
That is 100% true.
Obviously the code needs to be self-contained (not calling into other unknown code) but so do mathematical proofs.
The hardest part I think is understanding and measuring your progress. In school you've got exams and classmates to compare against, profs to talk to. Alone it's much harder. "Do I understand this well enough?" "Did I do the problems right?" (Especially with proof problems, how do you know you're right?). "I can work through some problems one by one, but it feels like something fundamental I'm missing. Am I, or is this chapter really just about some tools?"
Then it's way too easy to say well I'm never actually going to use any of this so why am I doing it ... and take a few months off and come back forgetting what you'd learned.
1. Don't do exercises unless you want to. Completionism is a trap.
2. Take notes. Rewrite things in your own words. Imagine you're writing a guide for your past self.
3. Ask questions. Anytime you write something down, pause and ask yourself. Why is this true? How can we be sure? What does it imply? How could this idea be useful?
4. Cross-reference. Don't read linearly. Instead, have multiple textbooks, and "dig deep" into concepts. If you learn about something new (say, linear combinations) -- look them up in two textbooks. Watch a video about them. Read the Wikipedia page. _Then_ write down in your notes what a linear combination is.
Anyway, everyone's different of course, but these practices have been helping me get re-invigorated with self-learning math. Hope they help someone else out there. I welcome any feedback!
especially the part about skipping the exercises. if you're not trying to write a dissertation or pass a qual (and you're just interested in learning and being exposed) then you don't need to do them. a lot of exercises are a hazing ritual or imagined by the author to be a dose of bitter medicine (i'm looking at you electrodynamics by jd jackson) since they mistakenly believe all readers are formal students.
the most important exercise is to mull over and consider what you're reading/learning. naturally dovetails in to asking question: what happens if i remove a hypothesis from a theorem, what happens if i add one, is there an analogy to another object/group/measure/etc, etc.
also read multiple books (http://libgen.is/ is your very very good friend and generous friend). a lot of math authors (no matter how esteemed they are) are terrible writers or make mistakes (look up errata for previous editions of your favorite book).
the only thing i'd add is to learn to use LaTeX to take notes - it is much easier and faster and neater.
I think this is deeply mistaken. In a well-chosen book, such as the ones in the submitted article, doing the exercises is not to test your memorisation, it's to develop your understanding.
Math is not a spectator sport. Reading about math is fine, but it will not take root and develop unless you engage with it, and the exercises are the way to do that.
Ignore the exercises if you want, but you almost certainly will end up knowing about the math, but not able to do it.
This is a great point and example of the problem with a one-size-fits-all strategy. For some books, exercises are an essential part of comprehension. For others, not so much.
> Math is not a spectator sport. Reading about math is fine, but it will not take root and develop unless you engage with it, and the exercises are the way to do that.
My experience is that by taking excellent notes and asking why, you engage with the material to a similar degree, if not a greater degree, than by doing exercises. (Once again, depending on the book, as you mentioned.)
> Ignore the exercises if you want, but you almost certainly will end up knowing about the math, but not able to do it.
I would argue that's the point. Usually self-taught math is about self-growth. Getting new ideas, being exposed to new concepts, recognizing patterns. Being able to actually "do it" on-the-spot is beside the point (and is the quickest level of skill to evaporate once you stop focusing on that material, anyway.)
Isn't that literally exactly what I said?
> if you're not trying to write a dissertation or pass a qual (and you're just interested in learning and being exposed) then you don't need to do them
There's ground in the middle, and this thread is about that. This thread is not about learning for tests and qualifications, nor is it about "being exposed", it's learning how to do the math.
And for that you need to do the exercises. You don't need to do all of them, you don't need to be completionist about it, but if you don't do the exercises, if you don't actually do the math then you won't actually be able to do the math.
Specifically, you said (quoting again):
> if you're ... just interested in learning ...
There's a difference between learning about and learning to do. If you meant just "learning about" then you are at odds with the entire thread. True, in that case you don't need to do the exercises, but I don't think that's what people are talking about here. I think people are talking about being able to do the math.
And if you meant "learning to do" then in my opinion you are wrong, and one needs to do a large slab of the exercises.
Otherwise it's fairy floss, and not steak.
My apologies if all this seems overkill, but there's a real danger of talking past each other and being in violent agreement, and I wanted to state explicitly and clearly what I mean, and why I thought you said something different.
but i'm not a mathematician. i don't need to be able to do math anymore than i need to be able to do history (while reading serious history books).
>And if you meant "learning to do" then in my opinion you are wrong, and one needs to do a large slab of the exercises.
no i didn't. that's precisely why i used the word "exposed".
we don't agree but i'm not being violent. but my responses are short and yours are long.
i do not see the exercises as essential for anyone other than practicing mathematicians. i have read a great many serious math books (i just recently finished Tu's Manifolds book and am now reading Oksendal's SDEs). i read them without doing absolutely any exercises but following the rest of the guidelines in the post i responded to. the experience is gratifying because i learn about new objects and new ways of thinking about objects i've already learned about. that's absolutely the only thing that matters to me.
but let me ask you something
>That, to me, implies learning to do, not learning about.
here's a fantastic explanation of the topological proof of Abel-Ruffini
would you say that I don't understand that proof if i haven't done any exercises related to it? and therefore would you say I didn't learn any math by having watched that video?
Do we agree that if you don't do the exercises then you probably won't actually be able to do the math?
You are discussing learning about the math, and not eventually being able to do it, because you say that you don't care about becoming a mathematician, therefore you don't need to do the math. Fair enough.
But my reading is that that's not what this thread is about. This thread, and the original submission, is about learning how to do the math.
> i do not see the exercises as essential for anyone other than practicing mathematicians.
I think you're wrong. Knowing how to actually do the math has proven useful to many people for whom it is a tool in their craft/job/employment. Learning Linear Algebra properly, being able to actually do it rather than just talk about it, can be enormously useful in Machine Learning.
>> That, to me, implies learning to do, not learning about.
> here's a fantastic explanation of the topological proof of Abel-Ruffini ... would you say that I don't understand that proof if i haven't done any exercises related to it? and therefore would you say I didn't learn any math by having watched that video?
Understanding a single proof implies very little about one's ability to actually do the math. I've met many people who are math enthusiasts and who have watched hundreds of math videos. They say they understand all of what they've seen, and yet they are unable to do the simplest proofs, or the most elementary calculations.
My experience of people's abilities is that if they haven't done the exercises, they usually can't actually do the math.
But you complain about the length of my replies, so I'll stop. I think I've made my position clear, and I think I understand what you're saying, even if I don't agree with it.
You keep repeating this but you're evading the question about abel-ruffini and the question about whether reading a history book is "learning about history" as opposed to learning history.
You're making a weird distinction. People learn in different ways. Some by doing exercises and some by just playing with the objects. I wonder how you think actual research mathematicians learn new math from papers that don't include exercises lol.
You edited your response.
>I've met many people who are math enthusiasts and who have watched hundreds of math videos
There's a difference between watching numberphile or whatever and essentially watching a lecture on a proof. Very few people are watching/consuming rigorous expositions. I think that's the difference not the lack of exercise.
> You're making a weird distinction.
As someone who has done a PhD, done research in math, done research in computing, worked in research and development in industry, taught math, and headed a team doing research in technology, this is a distinction that I can clearly see. My inability to explain it to you is regrettable.
> People learn in different ways.
Yes they do.
> Some by doing exercises and some by just playing with the objects.
Doing the exercises is playing with the objects to try to answer specific questions. Good exercises are carefully constructed to help the reader learn how those objects work in an efficient manner.
> I wonder how you think actual research mathematicians learn new math from papers that don't include exercises lol.
In my experience research mathematicians learn now math from papers by, in essence, constructing their own exercises based on what they're reading. In general it takes significant experience and training to be able to do that.
Clearly you don't think one needs to do the exercises subsequently to be able to do the math. Good for you.
Me too so now what? I don't think your credentials give you any real authority but just make you look like you're gatekeeping.
>Doing the exercises is playing with the objects to try to answer specific questions.
Great so then we're in agreement: playing with the object is doing the exercise.
The funny thing is that at one time I actually did all of the exercises in volume 1 of apóstol's calculus. You know what effect on me it had? I was so bored I didn't read volume 2. And today I'd still need to look up the trig substitutions to do a vexing integral.
It wasn't intended to, it was to provide a context for my opinion.
So let me state my opinion as clearly as I can, and then I'll leave it.
* Math is a "contact sport" ... you have to engage with it;
* Reading books is not, of itself, engaging with the math;
* Watching math videos is not, of itself, engaging with the math;
* Well designed exercises are a valuable resource;
* If you can easily do an exercise, skip ahead;
* If you can't do an exercise, persist (for a time);
* Ignoring the exercises is ignoring a resource;
* For the vast majority of people, doing the exercises is an efficient way to engage with the material;
* To say "ignore the exercises" is, for the vast majority of people, an invitation to not bother engaging with the subject;
* Doing all the exercises is probably a waste. Doing none of them is an invitation to end up with a superficial overview of the subject, and no real understanding.
Just reading doesn't get much at all. Not even a superficial overview. I tried it. It's essentially a meaningless combination of words after a certain point.
Reading extremely thoroughly is actually marginally useful. Stopping to think, do all these assumptions matter, why, what if one of them changes, etc, pencil in hand, making notes, testing things out. I've managed to "understand" the topics when doing this, and so far it's been the highest ROI method. But it does still leave one feeling like something is missing. Just because you can sight read music doesn't mean you're an expert on the piano.
Doing exercises is a huge jump on investment, and the return on that investment is a bit questionable from my experience. A couple reasons: first you don't know if you did them right. If you did them wrong then that's negative ROI. Second you don't know what a "reasonable" workload is. It varies by author. Is it three problems per chapter, is it all of them, are some orders of magnitude more difficult than others? Without some guidance it's hard to know if your difficulties are due to not understanding basic material, or due to that problem being a challenge geared toward Putnam medalists. So they may cause you to question your understanding and thus mentally roadblock you unnecessarily. And finally with proofs (and this may be a me thing), it's pretty easy to say "I guess this is okay(?)" and move on, even if you're not sure. Since nobody is ever going to review it, and it's just a homework problem, it's very very hard to will oneself to make sure every assumption is correct and you're not missing anything, even if you feel like there's a good chance you are. Or perhaps I just don't have the constitution to do so.
So while I think doing exercises is necessary for a deeper understanding, I don't know whether the ROI is worth it outside of a classroom perspective. You need feedback for exercises to be beneficial. At least, I feel like I do.
Finally, is even taking a class that useful if the end state is that two years from then you'll have forgotten most of it and so what was the point. Can you claim knowledge of a subject that you've never actually used beyond some homework problems and exam questions, or is this still a superficial understanding? Having an ends where that knowledge gets used seems critical.
I feel like I have some knowledge but I don't feel like I'm there yet. But I don't know if I know where there is. Maybe that's the biggest challenge. Does completing a Ph.D. even get you to there? No idea. But, I guess it's up to the individual to decide what they want out of it. Nobody can determine that for you.
Yes, it is really important to learn math with study-mates. Just like in code, we do reviews, in math too, we need someone else who can review our proofs. It is even easier to make an error in a proof and believe that something is proven when it isn't. A study-mate helps to prevent us from fooling ourselves.
Mailing list: https://groups.google.com/d/forum/integerclub
We pick up old concepts from popular textbooks and literature as well as new stuff from new literature in both mathematics and computer science. We plan to have online meetings periodically to share what we learn, work through popular literature, and have a few talks on interesting topics.
It is a tiny community right now that hangs out at Freenode IRC but the Slack channel is there too if you are more comfortable with that.
I realize how lucky I was that I found a Discord server ran by a math PhD graduate who is willing to help us guide our learning. From this, I've started learning Algebra and Analysis (just starting with the latter). It's nice to have someone to discuss problems with when you get stuck and to guide you. Likewise, he can suggest exactly which problems I should do for a give chapter, that way I don't spend my time doing ones that just repeat the same simple things over and over and can focus on nice, conceptual ones. So, if you can, please try to find someone to help guide you, or be that guide for someone else! Having it has made me seriously consider going back for a mathematics masters (and maybe PhD), switching from my physics background.
Even the professionals try to find someone to learn a new concept from. There's something about mathematical writing that is too fragile/brittle for wetware.
One other strategy: I've noticed the really smart (arrogant?) people just don't bother reading new mathematics, they re-invent it themselves. It's actually worth trying if you can stomach it.
I always felt like maths was too abstract to keep me engaged, but when the output of your work is immediately observable visually it becomes a lot more engaging. There's just something so much more satisfying being able to "see" the results.
Plus as a self-taught programmer, I find it much easier to learn front-to-back by deciding on a desired outcome and working towards it, rather than progressively building up abstract fundamental skills that can later be combined to achieve a desired outcome (which is essentially the traditional academia path for learning STEM fields)
Keeping it as low-level as possible, I'm using CycleJS for dataflow management and Regl.js for drawing via a CycleJS-Regl.js driver.
All state is explicitly managed observables/streams in CycleJS, which maps out to Regl.js draw commands, which are basically raw frag/vert shaders with some bindings mapping my state from CycleJS to appropriate uniforms/attributes.
I probably would be able to produce some usable output much faster if I used an engine like Unity or a framework like Three.js, but I feel like I would have missed out on gaining so much knowledge by only working with high-level abstractions and never having to even touch GLSL code.
I really enjoy how the subject is divorced from a lot of the modern attention demands and encourages more of a 'zen' thinking style.
As others have highlighted, it can be difficult. I work full-time as a software engineer and at the end of the day there's usually not much left in the tank in terms of "creative work". The morning is usually more productive for me - generally I'll spend 10-15 minutes on the commute in reading over the proof of some lemma or working through some computational exercise.
Things that have helped me:
- Focusing on a particular problem area rather than just "mathematics". The classical problems of Gauss and Euler tend to be more my speed than the modern mathematical problems of Hilbert or beyond. What started my journey was looking into the insolubility of the general quintic polynomial equation, something you learn in high school as a random factoid but has a lot of depth.
- Studying from small textbooks that I can fit in a backpack, so I can "make progress" during my commute. Dummit + Foote might be a great algebra reference but it's just too bulky to transport.
- Limiting the scope of how I think about the activity - my goal isn't to master these concepts on the level of a mathematics graduate student, it's more on the order of Sudoku. If I don't get something, that's okay. People spend their whole lifetimes learning this material and I'm just trying to fit this into whatever creative time I have left after the full-time job is done.
I try to read papers now and again with a math orientation, and I quickly get lost when trying to translate the concepts into cryptic formulas, and often when they make the "obvious" transition from step 3 to step 4 I just have no idea how they got there.
I feel this is by far my biggest barrier to understanding most mathematics, and I have thus far found no way to overcome it.
Any statement in math is meant to be directly translatable to human language. You should be able to read it out loud in English and know exactly what you mean when you say it.
Unfortunately, sometimes math uses awful notation. For example, df/dx. This is a case where df doesn't mean anything (or at least it's not normally well-defined), and dx doesn't mean anything either (same comment). But the notation as a whole means something. If we write g = df/dx, then we can understand that g is a function whose input is x and output is the slope of f at x.
Time is valuable, it's the most valuable thing a human being has, I understand it's the hobby of OP to learn all this math, but unless you are going to use it why wasting all the time?
I will say I don't feel like single-variable real number calculus tells the whole story. I had taken that and linear algebra in undergrad but never any further, and now that I've taken single and multiple variable calculus, with real and complex numbers, plus integration of linear algebra ideas, the mathematical model feels a lot more like a cohesive whole to me, highlighting fundamental ideas that only barely peek through in a typical Calculus I class. I would encourage anyone talking to calculus to at least do the typical Calc II class, if not Calc III/multivariate. There is a beauty and structure to building up from calc I through III that I was missing before.
I have my own "best of" list that is very different to this list, although there are a couple of crossovers.
If you are fortunate enough to have access to a university library (or libraries) I would _highly_ recommend inquiring about access to their general collection. I was also fortunate enough to study mathematics to a university-level three-year degree at a research university. So I had an excellent head start.
A HUGE part of my journey of collecting my "perfect library" of mathematics self-tuition and reference books (and course books) was to do my own research on collecting the perfect titles. I started when I was in the early days of my mathematics degree and I used resources like Amazon, Usenet, libraries (already mentioned), and ... that was about it.
Another important question to ask yourself is the following:
"Why am I doing this?"
Life is short and by the time you hit middle age, if you have a family or bills to looks after, are you REALLY going to want to lock yourself away in your study room to learn Lebesgue integration instead of focusing on the rest of your life?
Consider that people fail to emphasise is that mathematics is a social activity much more than many people realize.
Exercise: Find the topics of mathematics that are important to your goals and are missing from the list and find your favorite books or two that cover/s these topics.
Exercise: Consider whether your interest in (self-directed) mathematics is so sincere such that you have a serious application in mind, that you might be better off enroling in a course? Even if it's a night course that last a couple of years, you will meet a LOT of people who can help in ways that are immensely more productive than trying to do this all by yourself.
I recently purchased volume 1 of my favorite calculus and analysis book. It's an incredible masterpiece. The coverage of topics is much broader and more interesting than Aposotol or Spivak. The latter books are both very good but they also have myopic, one-track pedagogical approaches and limited themes in their coverage.
Exercise: Find your own favorite introductory calculus book that is suitable for the motivated student.
Which book would that be if I might ask? I'm wagering... Courant? ;)
Courant & John, to be exact.
I am of the opinion that notation is a very powerful tool for thought, but the terseness of mathematical notation often hides the intuition which is more effectively captured through good visualizations. I would really like to take self-driven "swing" at signal processing, this time approaching it through the lens of solving problem on time-series data, since as a programmer I believe that would be quite useful and relevant.
I put it in context because it's not quite a direct comparison since I have been in greatly different situations and ages between studying math and CS, but putting that aside, I have to say I have greatly enjoyed the computer science means of teaching more than math, doubly when it comes to self-learning. Concepts in math are generally taught entwined with the means of proving those ideas. That's important if you're a grad student looking to be a math researcher, but (IMO) it is not so great if you're a newer student or learning on your own and trying to grasp the concept and big picture. A proof of a theorem can be (and too often is) a lot of detail that really doesn't help you grasp the concept the theorem provides or is used towards, often because it involves other ideas and techniques from higher levels or just different types of math, both of which are out of the scope of the student learning the topic. Worse yet, it is standard for a proof to be written almost backwards from how it would be thought out. Anyone from a math educational background has the experience in homework of solving a problem, then rewriting it almost in total reverse to be in the proper form to submit. This means not only is the proof of the theorem not useful towards conceptual understanding, reading the proof doesn't show you chronologically how you would discover it yourself. That is a lot of overhead cost to break through to get to real understanding, real learning. As you mention, notation as well is another thing you need to break through.
I have found computer science and related classes to be taught more constructively. Concept is given first, and then your job as student learner is to construct it. Coming from the ML field, I love comparing math and CS proofs of topics here. Explanations from CS people of back propagation, for example, are always visual, and books/courses will have you construct a class and methods to do the calculations. Someone with a bit of programming knowledge can follow along in their language of choice. Math explanations get into a ton of notation from Calc 3+, and it's going to take a lot of playing around and frustration to get a working system out of the explanation. Even the derivation section on Wikipedia is not something most people will understand and be able to turn into useful output.
The more I see other ways concepts are taught, the more I wish math had been taught a different way. There is a lot to break through in order to get to real understanding, just by the way it's formed and taught.
My strategy to get back to study math these days is getting to learn Wolfram Mathematica and Sage. Once I can move around those two, I feel like I will be able to create a tighter feedback loop on whatever Math subject I'm happen to be studying at the time.
I might give it a go anyway, but do you know of a more advanced version of this book as well?
It is written by a true pioneer. And also, you will impress your friends by your hipster foray into category theory.
However, this book is far from being hipster. Also, I would not be surprised if a high school student would be able to follow this book over the course of a year or two.
If you titled the book: Sarcastic introduction to how simple set theory is then I would actually be fooled that it were the correct title.
There is maybe nothing wrong with being thorough with the elementary topics if you're studying for fun. But if you're studying for applications, I think you should cover the basics only adequately, and then quickly move on to more advanced topics. Basic Calculus is only the foundation, stuff that is actually useful in applications comes later. Basic Linear Algebra can be useful in its own right, but the advanced stuff is even more useful.
I suggest building an adequate foundation, not a comprehensively thorough foundation, and then moving on to the more powerful stuff. Which varies depending on what you actually want to use math for.
You can take a look at it, at the Internet Archive,
So grateful. The world is wide open to the self learner in this day and age.
We are very lucky.
Even something purely in the modern era, learning about Fourier and Weiner, harmonic analysis, etc.
I know that Freeman Dyson attributes his proficiency in math to his love of math, which he claims was kindled as a teenager by reading Bell's "Men of Mathematics"
Do not do it alone. I mean, it is okay to self-learn mathematics as much as possible but don't let that be the only way to learn. Find a self-study group where you can discuss what you are learning with others.
I think the social-effect can be profound in learning. I realized this when I used to learn calculus on my own. My progress was slow. But when I found a few other people who were also studying calculus, my knowledge and retention grew remarkably. I think the constant discussion and feedback-loop helps.
With round the clock internet connectivity, it is easier to find a self-study group now than ever.
I'm glad to see there are online options for groups like Stack Exchange or tighter group's like the one integerclub mentions, but I still seem to run into the same problem. For example, I'm not sure how to get a group of people that are interested in reading book X when I want to start it. If anyone has advice on that, please share.
Having said that, I think it probably would be sufficient to find _just one_ other person who is at the same level of mathematical maturity and has the same degree of commitment to change the entire learning experience for the better. You don't need a big group.
After that I worked through my classes with a tutors help. Everything up to and including linear algebra and numerical analysis with the help of a extremely kind PhD student named Adnan. He had the patience of a saint and ended up becoming a very good friend.
The most valuable part of having someone like this available for an hour or two every week is that it increases your knowledge or understanding/minute rate dramatically. It's like having Google or Khan academy on steroids. Someone that did everything already and knows exactly what page of a text book to look lat to help you understand, but they don't even need the textbook, because they know how to explain the concept you're having trouble with.
To this day I work as one of many data lscientists on a team where we all have fairly diverse backgrounds. In fact I'm the only person that is only CS and does not have a graduate degree. I have a teammate that did her undergrad and Masters in mathematics and if I'm having a hard time with something math heavy after some googling, the first thing I do is ask her for a quick explainer. She does the same with me for CS or programming issues as well and I help her with informal code reviews.
I know this will seem like basic teamwork to a lot of folks, but far too often I see people in our industry exert huge amounts of effort to understand a difficult concept that likely someone they're sitting a few feet away from has a very good understanding of and would be happy to help them with, so they're not banging their head against the wall for hours. I had to have a similar conversation with my intern a couple years ago. She was spending hours doing pen on paper math to understand Kalman filters. Things went much more quickly after I talked to her about my process of working with my colleagues and asking for help when I didn't understand something.
TL;DR Ask for help sooner rather than later. We all stand on the shoulders of giants.
See the user profile of my account for more details.
Number System, Algebra, Geometry, Trigonometry, Calculus, ...