As a self taught programmer I feel like I have a big gap on my math knowledge, where can I start learning math concepts that will help me in my career?
I agree with this. I think that the way we teach math is a disservice to the kids. When you watch a talented mathematician's lecture you always see how relatable and intrinsically motivating it is.
On the flip side, I think that when people write about math in scientific reports for example, they don't explain the math they use either.
Teaching by key-value pair (which is what far too many people default to) is a horrible way to teach
Concepts may need to be presented propositionally, but there [almost] always needs to be a "why" (even if the "why" isn't yet understood) paired with the "what" - the Pythagorean Theorem isn't just about memorizing 3-4-5 right triangles, it's about figuring out the shortest distance between two points, knowing how much fencing you need around your property, and on and on
Identify the concept you don't get and read! Specifically read a _variety_ of explanations on every concept you struggle with. Eventually you will find or form a point of view that will make it clear. There's no easy way, no one source.
Chances are you actually suck at algebra. I went through this issue when I started electrical engineering as part of an overly elaborate midlife crisis. I got the book in the bookstore that was supposed to bring you up to high school level and just did all of the exercises. Then I did a whole lot of exercises for the introductory university math courses.
Basically it is something you have to grind at. Once you do enough problems all algebra will seem easy and you are done.
I am not sure that you actually need math for programming. Code is its own algebra.
I second the grind. There's not really a shortcut... like lifting weights or practicing scales/rudiments.
I returned to grad school at 27 for math and had trouble passing qualifying exams. I ended up using spaced-repetition to memorize large sections of books and old questions. I had previously shunned memorization... but coupled with focused practice it is a powerful tool. Memorizing forces you to distill the material and patterns to their essence and allows you to recognize the patterns in new situations.
There's not really a shortcut... like lifting weights
And just like lifting weights, you will lose it when you stop doing it.
I remember doing the CFA program which is math and formula heavy. I had hundreds of formulas memorized. I couldn’t tell you the formula to calculate the standard deviation of a portfolio of three assets now if my life depended on it.
This is my main qualm with testing as a means of validating understanding. If you truly understand a topic, you will probably pass a test on it. However passing a test most definitely does not guarantee understanding.
I remember very, very little from the CFA exams (passed L3 about 10 yrs ago), because it tested surface level knowledge of an incredibly broad array of subjects. Far too broad to allow a deep-dive in any one thing. At the time, I could fit a bunch of formulas in my head and regurgitate them on command, but it was like filling up a leaky bucket. I had to load that sucker up and run into the exam center before it poured out.
IMO testing well is it's own skill, and has little to do with understanding a topic. Understanding is when you've forgotten all the stuff you've memorized, but you can work your way back to a solution from first principles. I think the only way to get to that point is genuine interest and sustained study.
And just like lifting weights, you will lose it when you stop doing it
While this is 100% true, I hate putting it that way because it makes weight training (and mathematics) sound like drudgery. The way I like to think about it is that if you want to stay strong, you have to push against resistance on a constant basis.
It's the same thing with math. You have to exercise that part of your brain with algebraic resistance.
I agree there are no shortcuts, but just like the scales will come much more easily to someone naturally gifted at music, so will algebraic operations to someone gifted at math.
We can all improve our abilities, but those not gifted has a much steeper hill to climb.
I am a big fan of the working out analogy. In all cases you need to have a decent plan then execute-- basically this means showing up and putting in the work. If you do it right it should be a bit painful at times.
And just like working out some people will progress more rapidly than you and have an easier time at it. So what? As long as your goal isn't to 'place' at the elite level, this is irrelevant. You really just need to show up and embrace the struggle. (For elite level tasks including grad school, its a bit more involved.)
3B1B has stated that the series is mostly useful for summarizing or augmenting existing knowledge after taking the class; it's not for learning Linear Algebra fresh. If you sum all the minutes in the 3B1B videos it's basically the length of one video from a university course.
I'm not sure about recommending video courses because... there are so few people who have taken up the burden of producing a full Linear Algebra video course. It's sad. But I would at least say that whether you're following Gilbert Strang or someone else, you should be following with a book as well.
There's another guy, Jim Hefferon of St Michael's College, who has written a full book for his own class and produced a full video course on Linear Algebra (all free).
I totally agree with you and that is why I have also added to the post the link to the open course MIT site.
I also believe that is quite naive thinking that you can learn math in some depth just watching videos (at least with my capacity). For me it is a way to keep some pace and order when learning and, as you say, to find some bibliography and material that is being used by the professor.
It's good for concepts but you can watch all his videos, think you know what linear algebra is about, and still not know how to calculate anything. You have to do exercises if you want to actually learn math.
I made that mistake when I started university. I managed quite well in high school without doing lots of exercises. Didn’t work quite as well later and I had to sit and really work through the exercises.
I think the sort of visualizations that are around on Youtube are great for those who don't "get" maths easily and need another angle to help them understand. Although 3Blue1Brown is not for beginners there are other beginner resources.
For example, this goes right back to basic algebra but is a very well explained with the afore mentioned visualizations.
Not to pick on your response, but every time I see a thread about not understanding math, I think *my people!!!*. I come to these threads hoping to find answers, and all I see are “read this calculus textbook” or “3blue1brown”! I watched this whole linear algebra series about 7 times and I just don’t get it. There’s something about the way people explain math that just isn’t intuitive to me, like it’s too disconnected from reality or something. I wonder if I’m not the only one…
I really really think math is in fact harder than other things people do, and math people are using weird definitions of words to hide it.
Like, when they say something is easy, they might mean "It's easy if you know all this foundational stuff that you could learn in six months".
Everything else technical has shortcuts. Nobody knows how WiFi works, they use a module. If they do, they're probably a specialist.
But math has almost no shortcuts almost by definition. If it did we wouldn't call it math we'd call it using a math library.
It's the same shock I have practicing music, or why I can't drive, or why it takes me 2 whole minutes to line up a jig to cut a 2x4.
What programmers do is on the screen more than in the head. There's no "practice this kick 10000 times". When you see drawPixel(x,y, hexcolor)... that's... all there is to know about that function besides how long it takes.
Things that DO require skills that have to be learned, rather than just facts you have to be familiar with, seem shockingly difficult compared to anything in a JS app.
100% agree this is something you grind at. Calculus, even multivariate, and even the stuff beyond that, still grind. I used to be really good at it, definitely not the best but among the higher scores in my college math classes. But 10+ years later, when I go and look at stuff that would have barely fazed me then, I go cross eyed now. I remember the concepts, vaguely, but following the math and linking together the various theorems and transformations feels like wading through syrup. It’s a problem for me as I’m trying to dig into data science stuff, and it feels like getting back into marathon shape after 10 years of not running.
There’s absolutely a component of learning the concepts, but a huge part of being good at the math is just building up the mental muscle memory for symbolic manipulation. Sure there’s innately talented people out there, but the consistent application of incremental effort will get you very far regardless of talent.
Agree with this and would also add: in addition to being able to do algebra fast and have it be second-nature, at higher levels it helps if you are able to do it 100% accurately. 5 nines isn't enough.
I didn't appreciate this until I was a graduate physics student and was frequently frustrated by minor algebra errors in long, complex problems.
I've known a lot of mathematicians, some of whom were really good, and five nines (one-in-one hundred thousand) is an implausibly high standard for humans to reach. I'm certainly nowhere near there.
What's worked for me is to develop an intuition about how things 'should' look, so that if I do make a mistake I'll get a 'hey, that's not right' feeling a couple steps later.
Just to expand on this, when I worked as a grader in college I noted on engineering test keys the professor's answers were always roughly three short lines of handwritten equation. But the student's answer section were often totally darken with pencil they wrote so much. The heuristic was definitely the longer the answer the more wrong it was.
This is something I've taken to heart professionally too. If the equations are getting out of hand it's either wrong or I need some simplifying assumptions (like that cow needs to be spherical). Otherwise one just can't keep track of it and reason about it.
If I need a more precise answer, that's what computer numerics is for.
What's important isn't being able to do it right quickly the first time. It's having a strong habit of checking your work as you go, both at a low level and a high level. (Kinda like unit testing and integration testing...)
Fast arithmetic and algebra are fairly particular skills. As one progresses into new world of math, the calculations and operations may be completely different from what's been practiced, making the fast math less useful. But a good habit for careful progress is completely general!
Perfect accuracy is an insane goal. Much better is to develop the skill of finding and correcting your mistakes. Part of this is doing frequent "sanity checks" - for instance, if you are computing an area, and you get the answer -3, probably you made an algebra mistake somewhere earlier, and you should find and fix it before going on (some ways of checking your work also help you pinpoint where the mistake occurred, others are less helpful for that). Another way to check your work is to try plugging in numerical values for all of the variables and going through the arithmetic to check all of the steps (if you need to check your arithmetic, try something like "casting out 9s").
Learning to find and correct your mistakes will at first make it take ten times as long to solve any problem as just scribbling your way to an answer. But if you practice it, you will discover two things:
- with practice, the overhead of finding and correcting mistakes drops significantly,
- the feedback you get from noticing your mistakes helps you avoid making those mistakes in the future. Eventually mistakes become extremely rare, and you can go through a very long computation without making a mistake, even if you don't bother checking for them any more!
I‘ve enjoyed revisiting topics I learned in school through the Kahn Academy[1]. You can pick any topic you want to start with and follow it through as far as you want.
I went through precalc in school, then promptly forgot most of it once I started working. All of the bits I had learned felt rushed, so a lot of them fell out of my brain even while in school. I only had a basic understanding of most of what I "learned".
About a year ago I decided to go back to basics, and start over from zero on Khan Academy. I've been v.e.r.y slow, maybe a lesson a month on average, but have been grinding through the lowest level math to really understand and remember it. Down to addition, subtraction and multiplication even - I've relearned how to multiply small numbers in my head, and gotten much better at quick mental addition and subtraction.
I have scheduled and dedicated more time for it this year, so I hope to catch up to my original math basis. This time, hopefully, remembering all of it instead of skimming. That should set me up well for linear algebra, and then calculus next year and beyond.
One thing that may help regardless of how you choose to learn: pick a problem you want to tackle that needs Fancy Maths. It doesn't need to be anything useful, or anything difficult like "prove this unproven theorem". It will motivate you as long as it keeps your brain engaged. For me it's factoring and primality proving, where I can't even read the equations / algorithms because I don't have the basic tools. As I get further along I keep looking back to those problems and seeing how the bits I'm learning apply around the edges or to simpler factoring algorithms. It keeps me motivated to learn more to be able to dive deeper into the juicy parts.
The problem for me is identifying where exactly in my mathematical fundamentals things fall apart (like with prime numbers, some basic arithmetic, geometry) that makes higher-level math a struggle conceptually. It's always felt like there were holes in my fundamentals. I suspect the only way to figure this out is to go back and do everything again either through Khan Academy or books. Does anyone else have this issue?
When it comes to math, I try to avoid self-diagnosing with dyscalculia or with poor working memory. Instead, I think I've gone so long with math being an anxiety-inducing subject that any time I need to solve math under the tiniest amounts of pressure, I fall apart.
Schaum's Outlines are like that "Learn X in Y Minutes" site for school subjects. They give the clearest, most bare bones explanation so you can quickly identify gaps. I had the college algebra book for the required math class in technical school. Between that and YouTube, I was able to pass the class.
Try Khan Academy. The exercises (especially with how they focus on mastery), coupled with the videos will help you root out your weak points and even master them. Do not be ashamed in starting with the kindergarten math topics and moving upwards from there. If you are able to power through it, good on you. If you get stuck on something, even better -- now is your chance to master it.
KA helped me go from hating math in high school to double majoring in Math & CS in college, and graduating with honors. I donate to KA now.
"Mathematical fundamentals" is a sliding scale. You have to define your end goal. What do you want to understand, that you are currently unable to understand due to your incapacity with mathematics?
If you want to understand computational complexity theory, for instance, you need a different set of "fundamentals" than you do if you want to understand high-school physics. It'll be a different thing if you want to read econ papers, and a different thing if you want to study machine learning.
Within the intersection of all these fundamentals is probably basic arithmetic and algebra. Those tools are basic requirements for everything else. Beyond that, you need to define your goal before you decide what math to learn.
For books on strong fundamentals, you can try the Art of Problem Solving series [0]. They suggest a curriculum to start with prealgebra, move to algebra, then counting & probability, then geometry, then precalculus, and finally to calculus (though a regular calculus book like Thomas Calculus/Stewart Calculus/even Spivak/Apostol would work fine).
The main advantage of these books are its focus on building intuition by visualizing shapes or immediately rephrasing notation (e.g. 4/2 is better understood as 4*(1/2), which better explains why you should avoid cases where you divide by zero; I also found their exponent rules easier to understand, because it encourages visualization instead of just memorizing the rules).
The downside is that they're time-consuming due to a large number of exercises (I'm currently still trying to slowly work through them when I can, but if you need higher-level math in the short-term, it's probably better to start there). They're also not a free resource.
For free lecture videos, I've found Professor Leonard's lectures to be excellent, and equivalent to lectures at a university classroom [1].
Okay. Been there. Done that. Learned a
lot of it. Got a Ph.D. in it. Taught it.
Applied it. Published peer-reviewed
original research in it. Had a good
career applying it. Am using it as an
advantage in the core of my startup.
Broadly, for a career in computing, at
times math can be an advantage, one that
might be significant, e.g., get you
founder's stock in a startup that becomes
successful.
Math and computing can be a career
one-two punch: With some math you might
find an application, maybe a valuable one,
and then with some computing you get to do
the associated programming. Maybe then
you can show up at work one morning, maybe
after doing an all-nighter, and show the
final, useful, maybe quite valuable
results -- done deal, no waiting,
meetings, project approvals, etc.
This is a great time for both math and
computing, no doubt unique in all of
history. We are awash in what is in
historical terms just astounding
computing, and part of that is that a lot
of math is just a few clicks away at
Wikipedia, YouTube, in PDF files from word
processing with TeX, etc.
The first thing in math is arithmetic. Of
course, current computing eats arithmetic
problems much faster than Godzilla eats
fish.
You should know basic arithmetic for whole
numbers and fractions.
Then you should know the basics of ratios,
proportions, percentages, square roots and
exponents, logarithms, compound interest,
areas, and volumes. E.g., on my instance
of Windows 10 Home Edition (that I have as
a result of a sad situation, long story),
the key in the upper right corner of the
keyboard runs (opens, launches -- maybe
computing will think of more silly
synonyms) a version of an old
scientific-engineering pocket calculator
that has a lot of such arithmetic and
math.
Uh, that software is harder to learn to
use than the math it does! If you can
find out how to use such software in less
than a few hours of clicking guesses, you
can also learn the associated math!
Then on to algebra: That subject is
just doing arithmetic with symbols instead
of specific values, and that should be
really easy for anyone who can write math
expressions in a computer language.
Then on to plane geometry: The most
important idea there is triangles,
especially ones with one angle 90 degrees
-- right triangles. Then, sure, the
biggie result is the Pythagorean theorem
-- it gets applied throughout our economy
and has surprisingly far reaching
generalizations. For a proof, take 4 of
the right triangles and arrange them so
that they form a square where each side of
the square is the longest side of one of
the triangles and all the triangles are
inside the square. Then will also see a
square in the middle. Then write out the
area of the squares and, presto, bingo,
get the theorem. There are also 149 or so
other proofs.
For a while, I taught trigonometry (about
triangles) at Indiana University. The
best student in the class was a pretty
girl, and later I dated and married her --
see, math can be useful!
Then there is second year algebra where
learn some more, e.g., about, say,
(x + y)^n
for numbers x and y and a positive integer
n. From that can learn a lot about how
many HEADS might get if flip a fair coin
1000 times and can understand the math
shown in the baseball movie Moneyball.
Also that way can start to understand the
bell curve of Gauss and the powerful
law of large numbers.
Might study solid geometry, that is,
planes, lines perpendicular to planes,
spheres, circles on spheres, etc.
Next up, calculus: As you already know,
in a car the speedometer is the rate of
change of the odometer. The rate of
change of the speedometer is
acceleration. From Newton's law of
motion F = ma, that is, force is mass
times acceleration, in a car you feel the
force as you are pressed back in your seat
when your Tesla does 0 to 60 MPH in less
than 4 seconds! Going around in a circle
is also acceleration, and that's why when
you make a fast ...
If you do much with computer graphics you
will encounter matrix theory. That takes
you into linear algebra; next to
calculus it is likely the most useful
math. Evidence: There are a lot of
downloads of LINPACK.
Can start a course in linear algebra by
considering solving several equations in
several unknowns. The standard technique
is Gauss elimination, and can program
that in about one page of code. Linear
algebra is a good start on curve fitting
in statistics and the math of quantum
mechanics.
If you want to understand more about
cryptography and error correcting codes,
you should study abstract algebra. Here
I would suggest that you actually take a
course (a) to help you get through that
quite different world of thought and (b)
especially to learn how to write proofs.
And for (b), take a course where the prof
is really good and also carefully reads
and comments on your proofs. Abstract
algebra is the easy place to learn to
write proofs.
Somehow long, maybe still, knowledge of
both math and computing can be welcome and
lucrative in parts of US national
security. That was the case early in my
career when my annual salary was 6+ times
the cost of a new high end Camaro.
Soon FedEx had what their founder, COB,
CEO called their "most important problem"
-- fleet scheduling. The BoD was
concerned, and crucial funding was at
risk. I typed furiously, wrote some
software, the output "solved" the problem,
enabled the funding, and saved FedEx.
There, sure, needed to calculate great
circle distances so used the law of
cosines for spherical triangles -- solid
geometry can be good stuff! Also had to
handle wind vectors -- linear algebra can
be powerful stuff. Then I went off to do
much more, integer linear programming set
covering where can discover much of the
motivation for currently the most
important problem in computer science, P
versus NP.
Later the BoD wanted some revenue
projections. I did a little with some
calculus and got a nice answer. Long
story short, that work saved FedEx a
second time.
For another long story -- I needed to be
better at office politics -- I just missed
out on some FedEx stock that should be
worth ~$500 million now.
The US Navy was collecting ocean wave data
at sea, and I was in a software house
bidding on writing some software to
analyze the data. One customer engineer
wanted (a) to know the power spectrum of
the ocean waves (that is, what
frequencies have the power) and, then,
(b) to generate synthetic, random
ocean waves with that power spectrum. I
quickly read a book by Blackman and Tukey,
typed in some software, showed the
engineer the results on how to find the
power spectra (with an important point
about handling low frequencies) and how to
generate the synthetic waves, and our
company got "sole source" on the software
work.
Later at IBM's Watson research lab, we
were doing AI for monitoring of server
farms and networks. I thought of another
way, for some of the monitoring much more
powerful than the AI, based on some
original math, and published the results.
Net, some math, especially through
calculus and linear algebra, can at times
be an important career advantage. For
more, get good with probability theory, if
you can, the version based on the subject
measure theory. Then learn some about
stochastic processes. E.g., once the US
Navy wanted an evaluation of the
survivability of the US SSBN (missile
firing submarines) fleet under a special
scenario of global nuclear war limited to
sea -- in two weeks. From some old work
by B. Koopman, I saw a continuous time,
discrete state space Markov process
subordinated to a Poisson process, wrote
some software, and was done on time. My
work got reviewed by a famous
mathematician, and he questioned how my
...
Try ALEKS. It's basically an online grind session that identifies which underlying concepts you've got wrong or haven't got, and builds you toward them organically. I recently used it to get myself sufficiently intimate with trigonometry that I don't have to stop and look things up all the time "just in case." It's not free but it works well: https://www.aleks.com
There can be quite a few techniques, theorems, and facts at play while working through any given problem.
For those who are uninitiated all of these tools can be difficult to remember (long-term) and correctly apply (sometimes creatively) to arrive at a solution.
Dyscalculia and poor working memory can be issues if your aim is speed and conciseness which is critical during an exam but not as necessary if your goal is to simply understand at a deep level.
I have the same problem as you, I just don't know the fundamentals at math. So I decided to start from the beginning with basic arithmetic. You probably should ignore book recomendations like Spivak for now. The books that I use right now are "Arithmetic for the practical man" by "J. E. Thompson" and "The Number Devil: A Mathematical Adventure" by "Hans Magnus Enzensberger" which is a childbook.
IF!! You have kids learning math is right there with a full program. I forgot all my math so I keep up with my kids and help at least 15 minutes per day. I have been tripped up with 5th grade word problems - it is fun. Follow along with your kids math - it will move quicker than you think.
I remember my last math class: Number theory. I got through the final, shook the professors hand and told him, "this is the end of intellectual ability". I have never seen a professor laugh that hard. Once you get past abstract math classes thin out fast and it just gets weird.
Anyone who sucks at algebra and has a low grind tolerance should try https://dragonbox.com/products/algebra-12 It’s very well made. And, it’s designed for 12 year olds. So, adults can usually manage to keep up.
I hadn't thought about it that way. I was good at algebra, but can't get through calculus to save my life. I've mostly found that coding is more about logic than math. When math has come up in my work, it's generally a formula or calculation that's already defined and I just have to code it into a system, which has never been a problem for me.
Same happened to me. I completed a BS in CS and learned just enough algebra, trig & calculus to pass the classes and then forgot almost everything. Many years after graduation, I came across an interesting post in comp.lang.java - somebody was asking how to create a java applet that could draw a 3D block arrow that could orient itself in any direction. I'm pretty sure somebody was asking for help with his homework, but I thought it was an interesting problem so I started looking at it and the first issue I ran into was drawing a line perpendicular to another. I vaguely remembered that there was a formula for computing the slope of a line perpendicular to another and when I looked it up - in context to a problem I was trying to solve - it just kind of "clicked" and I found it interesting. From there I fell down the rabbit hole of re-learning everything up to differential equations that I was supposed to be learning when I was an undergraduate.
My second year of university is where i fell off math.
Several people have told me that once they reached post graduate
level at university that math made sense to them.
They saw the perspective of unified math.
How it all fits together.
If anyone is looking to brush up on their Algebra fundamentals, I built an app for that and it's free to use: https://apps.apple.com/us/app/pensend/id1571322730. I had a similar problem where I didn't understand where my fundamental knowledge gaps were so I ended up starting from the beginning (pre-algebra). I hope this is helpful for people!
Silvanus P Thompson | Calculus Made Easy (html book) - https://calculusmadeeasy.org/ (This shouldn't be your only exposure to Calculus. It is more for building intuition.)
The Discrete Mathematics course above is probably the most important for your work. In fact I would look for more Discrete Mathematics courses if I were you as it is far more important than anything else here.
Do you struggle with mathematical reasoning in itself, or with different aspects of mathematics (aspects that are built on, and work based on, said reasoning)?
I second this recommendation. It's a lovely way to explore some of the beauty of mathematics.
I have a BA in math and I think the most-useful mathematics for everyone is probability (math theory and basic calculations). This is because you'll be better able to deal with (quantify) uncertainty. The concept to aim for is (learn enough so you can understand and use) expected value calculations - which are, I think, the foundation of rational decision making under uncertainty.
Time. Be honest about how much time you’re willing to invest into understanding math. You can understand math if you’re willing to give it the time, but without that time you’re likely setup for frustration.
Where to start?
Math is huge. I suspect discrete math may be the most useful to a programmer who’s looking to just be a more theoretical programmer. Proof by induction stands out as a core, helpful concept too.
Otherwise I suspect it’s about problem domain. Geometry has its uses as does algebra. It all kinda branches from there.
Journey through Genius: The Great Theorems of Mathematics - William Dunham
How to Bake Pi - Eugenia Chang
These are all sort of 'pop-math' books -- that is, they're more intended to spark a joy & love for math than teach rigorous mathematics. Great Theorems and A History of Pi include a lot of history (edit to add: in addition to covering the math involved!) -- did you know some mathematicians in history would duel over their theorems? That theorems were a carefully guarded secret instead of something you shared?
Introduction to Graph Theory is specifically intended as an introduction to mathematics for 'the mathematically traumatized'.
In my opinion, after reading these, if you've sparked a joy for the puzzles and fun of mathematics, then I would then suggest branching out into more formal presentations of them relevant to your interests... it's much easier to slog through a book on abstract mathematics when you receive from enjoyment from the puzzles presented.
Agree! I’ve seen many a mathematics professor struggle with math. It’s not innate to any of us. But certain of us have spent more time with it and seen more of it. The time and familiarity eventually goes a long way - to a phd, even.
Agreed, although it's good to recognise what your strengths and weaknesses are. I would argue it's often better to focus on being great at the things you're good at than trying to be decent at stuff you really struggle with. If OP struggles with math they may be better off focusing on something else, at least if they're optimising for career success. If you're optimising for happiness and enjoy math problems then ability doesn't really matter.
I suppose depending on how extreme we're going with unfocusing on math.
But one doesn't really get to ignore math in life because of money. You'll still need to figure some things out. I can say I'm a poor painter so I'm just going to ignore that and things will be fine. The cost of ignoring math can be quite high.
I am beginning to think “I suck at math” less and more of “I’ll never be good at it”, from the comments on this thread.
I absolutely despise grinding and drilling uncontexualized and contrived problems and have never been able to maintain something stuff like that for very long. Which of course becomes problematic at I find myself interested in problems that require a decent grasp on some math.
The grinding is, by analogy, equivalent to the "reader" books in many languages. "See Spot. See Spot run. Run Spot, run!" level and then up. It's very hard (for most people) to jump from "I now know the English alphabet" to "I can read any article in a literary magazine." Many people forget that they went through that process because they did it so young, but also because language (written and spoken) is far more pervasive than mathematics. This pervasiveness means that you have less "grinding" to do, because you get to learn just by being in the environment the language is used in (if it's your native language, or if you can afford or are otherwise forced to move into an environment where it's the local native language).
Unless you're like me as a kid, you don't look around and start counting, adding, and multiplying based on the objects or scenes that you see, and "thinking in numbers". Thus the need for grinding, you need some math fluency (which takes practice to develop and maintain, like any other domain) before you can move on to more advanced topics (at least easily). If you lack fluency, even at a low level, then you have to do a refresher every time you try and get to something moderately advanced. Which is frustrating, at least, if not demotivating to the point of causing most people to stop. The more fluent you are, the easier it is to pick up an arbitrary advanced topic (at least to read, if not to apply or extend).
I think when people say they suck at math what they mean is; I find math homework extremely boring and unrewarding. Which perpetuates sucking at math and only compounds. Then things get worse as modern society works better for the individual when one does not totally suck a math.
Kinda like, I suck at piano because I find practicing the piano boring and unrewarding, but without the ramifications of sucking at math.
I really like the "placement exam" in this, that tells you where your weaknesses are and what to focus on. I feel like this is the kind of tool I want when someone says "I suck at X".
I also recently released shorter book focussing just on high school math chapter, for people who are not interested in calculus but still want a refresher on the basics. You can check it out here with fancy new cover design and website: https://nobsmath.com/ (it's basically Chapter 1 of the big book) Aussi disponible en Français https://nobsmath.com/fr/
Are you sure you suck at math? I used to think I sucked at math (now I think I'm merly bad at it). Turns out I suck at numbers. I can't get a feel for them. Manipulating equations and such though I found quite easy back when all that was fresh in school.
As for career advise, depends on what you want to do: There is not much use in learning calculus if you end up in astatistics-heavy field like data science. I'd say figure out what kind of computing you want to work with and see if there is a specific part of math that are useful there, if there even is one.
I've done work for finance and never had to go much deeper than simple multiplication.
I thought I sucked at math for a long time, reinforced because my math teachers in school believed it too.
Turns out I suck at spatial thinking. I can’t manipulate things like plots and curves in my head and they don’t help me understand mathematical principles. Algebra and logic though come easy to me. So learning math became an exercise in translation.
That is to say, please don’t think you suck at math and definitely look at different ways to learn the same concepts if something is not clicking.
Agreed, I've used trig once since I left college for an image manipulation problem. Discrete mathematics, Algebra, statistics, and depending on the roles ... linear algebra are far more likely to be useful.
I'm currently going through "Math for programmers" (Manning Publ.) and I enjoy it immensely! A lot of math books can feel too abstract and dry but this book was perfect for me coming mostly from a programming background.
Start learning learning linear algebra and you will soon find yourself in graphics programming and game engines. That's where I am at the moment and I feel I regained my passion for programming after years in webdev..
I completely agree about a lot of math books seeming too abstract. I hated math until I took an interest in physics and realized I needed math to understand the ideas being presented. Suddenly it clicked that the math was actually a way to better represent the ideas, not just a trick for getting answers. Seeing how math applies to create and represent knowledge gave me a huge appreciation for it.
Personal advice: unless you have a very specific reason to do so, completely pass on advanced calculus, real/complex analysis etc., and continue with discrete maths instead - you will find it immediately useful. Only later, when you find you need it, choose other topics, carefully selected. We only have so much time in life, and the range of fields to study is enormous, so choose wisely. Don't blindly follow the advice of people who will tell you to study everything, you will soon realize it's impossible and it will only leave you sad.
Why has there been such a proliferation of these Reddit-style obvious questions upvoted? The front page is full of "Ask HN: what can I do about <insert obvious problem>?"
But to be fair, HN is not aiming to be a wiki. It's a vivid forum that discusses topics that are relevant to its participants. Looking at the amount of comments and upvotes on this thread, it's clear that a lot of people are willing to help or engage in some way. Also, people often not only discuss the initial topic (or question) but something else that arose from it. As I said, it's a vivid place. And that's for good.
Any edition of "Mathematical Ideas"[0] might be a good place to start. While it may seem too basic for some, I feel that it covers a lot of concepts and material that are very useful when programming - problem solving, set theory, logic, number theory, basic algebra, etcetera - and does so in a way that is gradually cumulative and not so daunting.
Maybe this book will help you to realize you don't really suck at math, you just had some terrible teachers or whatever. It's also a great introduction to many different mathematical subfields so you can see which ones are most interesting/useful to you for future study.
So far I've only read the first few chapters of the book, and the exercises often feel too difficult to me. But I think he does a great job of easing into mathematical notation, pausing to reflect on what a seasoned mathematician might be thinking when they come across that notation. He also makes a lot of analogies to programming, and has example programs that are easy to follow. It's helpful to have that angle to understand things from.
A related question: which part of math can be used to model object-oriented programming? AFAIK set theory deals purely with associations of objects, not how you make new ones. HS algebra is concerned with simplifying and inverting numeric equations. Meanwhile abstract algebra (and number theory) is concerned with finding talking about broad categories of numbers, relating those categories to each other, and making new, presumably interesting (read: surprising) statements about categories of numbers. Math is a study in "going meta" since each component of each activity gets a name, which are further grouped and named into other categories, limited only by what the thinker can stomach/finds useful.
But I am hard-pressed to think of any mathematical construct that reflects, even a little, the reality of OOP. This is evidence that OOP is an engineering concern, not a math concern. That is, OOP is one method to help humans deal with the complexity of a large amount of shared, mutable state (SMS), by partitioning it into smaller units of SMS. But math itself doesn't care about the scale of anything, and will happily encode any state into a single, very large integer, if you let it.
Some parts of programming are better grounded in math, like functional programming and relational algebra. Some distributed programming problems have some nice, ad hoc mathy treatment (e.g. Paxos), but don't really have a clear correspondence to anything.
Interesting the field that is closest to programming in real life, IMHO, is statistical thermodynamics. This is usually taught as part of the physics curriculum, and is pretty math intensive, and the field's remarkable job is to generally model microscopic behavior and then predict macroscopic behavior of huge aggregates. Programs always deal with huge numbers of tiny things, each having unique degrees of freedom, (alternatively, which have unique constraints), so there is some connection there. ST is also the field most closely related to certain "quant" jobs in the finance field, AFAIK, since the same tools let you model individuals in an economy and from that predict markets.
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[ 465 ms ] story [ 5644 ms ] threadOn the flip side, I think that when people write about math in scientific reports for example, they don't explain the math they use either.
Concepts may need to be presented propositionally, but there [almost] always needs to be a "why" (even if the "why" isn't yet understood) paired with the "what" - the Pythagorean Theorem isn't just about memorizing 3-4-5 right triangles, it's about figuring out the shortest distance between two points, knowing how much fencing you need around your property, and on and on
Maths for high school: https://studyflix.de/mathematik-schueler
Maths for university: https://studyflix.de/mathematik
Basically it is something you have to grind at. Once you do enough problems all algebra will seem easy and you are done.
I am not sure that you actually need math for programming. Code is its own algebra.
Did it ease your midlife crisis? Would you do it again?
I returned to grad school at 27 for math and had trouble passing qualifying exams. I ended up using spaced-repetition to memorize large sections of books and old questions. I had previously shunned memorization... but coupled with focused practice it is a powerful tool. Memorizing forces you to distill the material and patterns to their essence and allows you to recognize the patterns in new situations.
And just like lifting weights, you will lose it when you stop doing it.
I remember doing the CFA program which is math and formula heavy. I had hundreds of formulas memorized. I couldn’t tell you the formula to calculate the standard deviation of a portfolio of three assets now if my life depended on it.
I remember very, very little from the CFA exams (passed L3 about 10 yrs ago), because it tested surface level knowledge of an incredibly broad array of subjects. Far too broad to allow a deep-dive in any one thing. At the time, I could fit a bunch of formulas in my head and regurgitate them on command, but it was like filling up a leaky bucket. I had to load that sucker up and run into the exam center before it poured out.
IMO testing well is it's own skill, and has little to do with understanding a topic. Understanding is when you've forgotten all the stuff you've memorized, but you can work your way back to a solution from first principles. I think the only way to get to that point is genuine interest and sustained study.
While this is 100% true, I hate putting it that way because it makes weight training (and mathematics) sound like drudgery. The way I like to think about it is that if you want to stay strong, you have to push against resistance on a constant basis.
It's the same thing with math. You have to exercise that part of your brain with algebraic resistance.
We can all improve our abilities, but those not gifted has a much steeper hill to climb.
And just like working out some people will progress more rapidly than you and have an easier time at it. So what? As long as your goal isn't to 'place' at the elite level, this is irrelevant. You really just need to show up and embrace the struggle. (For elite level tasks including grad school, its a bit more involved.)
https://youtube.com/playlist?list=PL0-GT3co4r2y2YErbmuJw2L5t...
For more advanced linear algebra I would recommend the class from Prof. Gilbert at MIT.
https://youtube.com/playlist?list=PLE7DDD91010BC51F8
https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra...
What is your recommendation?
There's another guy, Jim Hefferon of St Michael's College, who has written a full book for his own class and produced a full video course on Linear Algebra (all free).
https://hefferon.net/linearalgebra/
https://www.youtube.com/watch?v=JnTa9XtvmfI
https://www.youtube.com/watch?v=DJ6YwBN7Ya8
I also believe that is quite naive thinking that you can learn math in some depth just watching videos (at least with my capacity). For me it is a way to keep some pace and order when learning and, as you say, to find some bibliography and material that is being used by the professor.
For example, this goes right back to basic algebra but is a very well explained with the afore mentioned visualizations.
Veritasium - How Imaginary Numbers Were Invented https://www.youtube.com/watch?v=cUzklzVXJwo
Like, when they say something is easy, they might mean "It's easy if you know all this foundational stuff that you could learn in six months".
Everything else technical has shortcuts. Nobody knows how WiFi works, they use a module. If they do, they're probably a specialist.
But math has almost no shortcuts almost by definition. If it did we wouldn't call it math we'd call it using a math library.
It's the same shock I have practicing music, or why I can't drive, or why it takes me 2 whole minutes to line up a jig to cut a 2x4.
What programmers do is on the screen more than in the head. There's no "practice this kick 10000 times". When you see drawPixel(x,y, hexcolor)... that's... all there is to know about that function besides how long it takes.
Things that DO require skills that have to be learned, rather than just facts you have to be familiar with, seem shockingly difficult compared to anything in a JS app.
There’s absolutely a component of learning the concepts, but a huge part of being good at the math is just building up the mental muscle memory for symbolic manipulation. Sure there’s innately talented people out there, but the consistent application of incremental effort will get you very far regardless of talent.
I didn't appreciate this until I was a graduate physics student and was frequently frustrated by minor algebra errors in long, complex problems.
What's worked for me is to develop an intuition about how things 'should' look, so that if I do make a mistake I'll get a 'hey, that's not right' feeling a couple steps later.
This is something I've taken to heart professionally too. If the equations are getting out of hand it's either wrong or I need some simplifying assumptions (like that cow needs to be spherical). Otherwise one just can't keep track of it and reason about it.
If I need a more precise answer, that's what computer numerics is for.
Fast arithmetic and algebra are fairly particular skills. As one progresses into new world of math, the calculations and operations may be completely different from what's been practiced, making the fast math less useful. But a good habit for careful progress is completely general!
Learning to find and correct your mistakes will at first make it take ten times as long to solve any problem as just scribbling your way to an answer. But if you practice it, you will discover two things:
- with practice, the overhead of finding and correcting mistakes drops significantly,
- the feedback you get from noticing your mistakes helps you avoid making those mistakes in the future. Eventually mistakes become extremely rare, and you can go through a very long computation without making a mistake, even if you don't bother checking for them any more!
[1] https://www.khanacademy.org
I went through precalc in school, then promptly forgot most of it once I started working. All of the bits I had learned felt rushed, so a lot of them fell out of my brain even while in school. I only had a basic understanding of most of what I "learned".
About a year ago I decided to go back to basics, and start over from zero on Khan Academy. I've been v.e.r.y slow, maybe a lesson a month on average, but have been grinding through the lowest level math to really understand and remember it. Down to addition, subtraction and multiplication even - I've relearned how to multiply small numbers in my head, and gotten much better at quick mental addition and subtraction.
I have scheduled and dedicated more time for it this year, so I hope to catch up to my original math basis. This time, hopefully, remembering all of it instead of skimming. That should set me up well for linear algebra, and then calculus next year and beyond.
One thing that may help regardless of how you choose to learn: pick a problem you want to tackle that needs Fancy Maths. It doesn't need to be anything useful, or anything difficult like "prove this unproven theorem". It will motivate you as long as it keeps your brain engaged. For me it's factoring and primality proving, where I can't even read the equations / algorithms because I don't have the basic tools. As I get further along I keep looking back to those problems and seeing how the bits I'm learning apply around the edges or to simpler factoring algorithms. It keeps me motivated to learn more to be able to dive deeper into the juicy parts.
When it comes to math, I try to avoid self-diagnosing with dyscalculia or with poor working memory. Instead, I think I've gone so long with math being an anxiety-inducing subject that any time I need to solve math under the tiniest amounts of pressure, I fall apart.
Mind naming what book you used?
Schaum's Outlines are like that "Learn X in Y Minutes" site for school subjects. They give the clearest, most bare bones explanation so you can quickly identify gaps. I had the college algebra book for the required math class in technical school. Between that and YouTube, I was able to pass the class.
KA helped me go from hating math in high school to double majoring in Math & CS in college, and graduating with honors. I donate to KA now.
If you want to understand computational complexity theory, for instance, you need a different set of "fundamentals" than you do if you want to understand high-school physics. It'll be a different thing if you want to read econ papers, and a different thing if you want to study machine learning.
Within the intersection of all these fundamentals is probably basic arithmetic and algebra. Those tools are basic requirements for everything else. Beyond that, you need to define your goal before you decide what math to learn.
The main advantage of these books are its focus on building intuition by visualizing shapes or immediately rephrasing notation (e.g. 4/2 is better understood as 4*(1/2), which better explains why you should avoid cases where you divide by zero; I also found their exponent rules easier to understand, because it encourages visualization instead of just memorizing the rules).
The downside is that they're time-consuming due to a large number of exercises (I'm currently still trying to slowly work through them when I can, but if you need higher-level math in the short-term, it's probably better to start there). They're also not a free resource.
For free lecture videos, I've found Professor Leonard's lectures to be excellent, and equivalent to lectures at a university classroom [1].
[0] https://artofproblemsolving.com/store/recommendations
[1] https://www.youtube.com/c/ProfessorLeonard/playlists
How to get good with math?
Okay. Been there. Done that. Learned a lot of it. Got a Ph.D. in it. Taught it. Applied it. Published peer-reviewed original research in it. Had a good career applying it. Am using it as an advantage in the core of my startup.
Broadly, for a career in computing, at times math can be an advantage, one that might be significant, e.g., get you founder's stock in a startup that becomes successful.
Math and computing can be a career one-two punch: With some math you might find an application, maybe a valuable one, and then with some computing you get to do the associated programming. Maybe then you can show up at work one morning, maybe after doing an all-nighter, and show the final, useful, maybe quite valuable results -- done deal, no waiting, meetings, project approvals, etc.
This is a great time for both math and computing, no doubt unique in all of history. We are awash in what is in historical terms just astounding computing, and part of that is that a lot of math is just a few clicks away at Wikipedia, YouTube, in PDF files from word processing with TeX, etc.
The first thing in math is arithmetic. Of course, current computing eats arithmetic problems much faster than Godzilla eats fish.
You should know basic arithmetic for whole numbers and fractions.
Then you should know the basics of ratios, proportions, percentages, square roots and exponents, logarithms, compound interest, areas, and volumes. E.g., on my instance of Windows 10 Home Edition (that I have as a result of a sad situation, long story), the key in the upper right corner of the keyboard runs (opens, launches -- maybe computing will think of more silly synonyms) a version of an old scientific-engineering pocket calculator that has a lot of such arithmetic and math.
Uh, that software is harder to learn to use than the math it does! If you can find out how to use such software in less than a few hours of clicking guesses, you can also learn the associated math!
Then on to algebra: That subject is just doing arithmetic with symbols instead of specific values, and that should be really easy for anyone who can write math expressions in a computer language.
Then on to plane geometry: The most important idea there is triangles, especially ones with one angle 90 degrees -- right triangles. Then, sure, the biggie result is the Pythagorean theorem -- it gets applied throughout our economy and has surprisingly far reaching generalizations. For a proof, take 4 of the right triangles and arrange them so that they form a square where each side of the square is the longest side of one of the triangles and all the triangles are inside the square. Then will also see a square in the middle. Then write out the area of the squares and, presto, bingo, get the theorem. There are also 149 or so other proofs.
For a while, I taught trigonometry (about triangles) at Indiana University. The best student in the class was a pretty girl, and later I dated and married her -- see, math can be useful!
Then there is second year algebra where learn some more, e.g., about, say,
(x + y)^n
for numbers x and y and a positive integer n. From that can learn a lot about how many HEADS might get if flip a fair coin 1000 times and can understand the math shown in the baseball movie Moneyball. Also that way can start to understand the bell curve of Gauss and the powerful law of large numbers.
Might study solid geometry, that is, planes, lines perpendicular to planes, spheres, circles on spheres, etc.
Next up, calculus: As you already know, in a car the speedometer is the rate of change of the odometer. The rate of change of the speedometer is acceleration. From Newton's law of motion F = ma, that is, force is mass times acceleration, in a car you feel the force as you are pressed back in your seat when your Tesla does 0 to 60 MPH in less than 4 seconds! Going around in a circle is also acceleration, and that's why when you make a fast ...
If you do much with computer graphics you will encounter matrix theory. That takes you into linear algebra; next to calculus it is likely the most useful math. Evidence: There are a lot of downloads of LINPACK.
Can start a course in linear algebra by considering solving several equations in several unknowns. The standard technique is Gauss elimination, and can program that in about one page of code. Linear algebra is a good start on curve fitting in statistics and the math of quantum mechanics.
If you want to understand more about cryptography and error correcting codes, you should study abstract algebra. Here I would suggest that you actually take a course (a) to help you get through that quite different world of thought and (b) especially to learn how to write proofs. And for (b), take a course where the prof is really good and also carefully reads and comments on your proofs. Abstract algebra is the easy place to learn to write proofs.
Can get more guidance on how to learn math at
https://news.ycombinator.com/item?id=28215105
Somehow long, maybe still, knowledge of both math and computing can be welcome and lucrative in parts of US national security. That was the case early in my career when my annual salary was 6+ times the cost of a new high end Camaro.
Soon FedEx had what their founder, COB, CEO called their "most important problem" -- fleet scheduling. The BoD was concerned, and crucial funding was at risk. I typed furiously, wrote some software, the output "solved" the problem, enabled the funding, and saved FedEx. There, sure, needed to calculate great circle distances so used the law of cosines for spherical triangles -- solid geometry can be good stuff! Also had to handle wind vectors -- linear algebra can be powerful stuff. Then I went off to do much more, integer linear programming set covering where can discover much of the motivation for currently the most important problem in computer science, P versus NP.
Later the BoD wanted some revenue projections. I did a little with some calculus and got a nice answer. Long story short, that work saved FedEx a second time.
For another long story -- I needed to be better at office politics -- I just missed out on some FedEx stock that should be worth ~$500 million now.
The US Navy was collecting ocean wave data at sea, and I was in a software house bidding on writing some software to analyze the data. One customer engineer wanted (a) to know the power spectrum of the ocean waves (that is, what frequencies have the power) and, then, (b) to generate synthetic, random ocean waves with that power spectrum. I quickly read a book by Blackman and Tukey, typed in some software, showed the engineer the results on how to find the power spectra (with an important point about handling low frequencies) and how to generate the synthetic waves, and our company got "sole source" on the software work.
Later at IBM's Watson research lab, we were doing AI for monitoring of server farms and networks. I thought of another way, for some of the monitoring much more powerful than the AI, based on some original math, and published the results.
Net, some math, especially through calculus and linear algebra, can at times be an important career advantage. For more, get good with probability theory, if you can, the version based on the subject measure theory. Then learn some about stochastic processes. E.g., once the US Navy wanted an evaluation of the survivability of the US SSBN (missile firing submarines) fleet under a special scenario of global nuclear war limited to sea -- in two weeks. From some old work by B. Koopman, I saw a continuous time, discrete state space Markov process subordinated to a Poisson process, wrote some software, and was done on time. My work got reviewed by a famous mathematician, and he questioned how my ...
For those who are uninitiated all of these tools can be difficult to remember (long-term) and correctly apply (sometimes creatively) to arrive at a solution.
Dyscalculia and poor working memory can be issues if your aim is speed and conciseness which is critical during an exam but not as necessary if your goal is to simply understand at a deep level.
I remember my last math class: Number theory. I got through the final, shook the professors hand and told him, "this is the end of intellectual ability". I have never seen a professor laugh that hard. Once you get past abstract math classes thin out fast and it just gets weird.
Same happened to me. I completed a BS in CS and learned just enough algebra, trig & calculus to pass the classes and then forgot almost everything. Many years after graduation, I came across an interesting post in comp.lang.java - somebody was asking how to create a java applet that could draw a 3D block arrow that could orient itself in any direction. I'm pretty sure somebody was asking for help with his homework, but I thought it was an interesting problem so I started looking at it and the first issue I ran into was drawing a line perpendicular to another. I vaguely remembered that there was a formula for computing the slope of a line perpendicular to another and when I looked it up - in context to a problem I was trying to solve - it just kind of "clicked" and I found it interesting. From there I fell down the rabbit hole of re-learning everything up to differential equations that I was supposed to be learning when I was an undergraduate.
My second year of university is where i fell off math.
Several people have told me that once they reached post graduate level at university that math made sense to them. They saw the perspective of unified math. How it all fits together.
Sadly, all I did know I have forgotten.
- you don't really "learn" math, you just get used to it
- math is really mental gymnastics, and "learning math" actually is doing a lot of math exercises
(sorry for the shameless plug)
Bill Shillito | Introduction to Higher Mathematics (YouTube lecture course) - https://www.youtube.com/playlist?list=PLZzHxk_TPOStgPtqRZ6Kz...
Richard Hammack | Book of Proof (pdf book) - https://www.people.vcu.edu/~rhammack/BookOfProof/
Taylor Dupuy | Fundamentals of Mathematics (YouTube lecture course) - https://www.youtube.com/playlist?list=PLJmfLfPx1OedcIUn5nSCZ...
Silvanus P Thompson | Calculus Made Easy (html book) - https://calculusmadeeasy.org/ (This shouldn't be your only exposure to Calculus. It is more for building intuition.)
Dana Mosely | Understanding Basic Statistics (YouTube lecture course, no calculus) - https://www.youtube.com/playlist?list=PL9Wxhr5qVFN0WY2CXB4tR...
Gilbert Strang | Highlights of Calculus (YouTube lecture course) - https://www.youtube.com/playlist?list=PLBE9407EA64E2C318
Josh Starmer | StatQuest (Short various statistics videos) - https://www.youtube.com/c/joshstarmer/playlists
Bob Franzosa | Introduction to Topology (single public lecture) - https://www.youtube.com/watch?v=zsN_guq__Ac
Socratica | Abstract Algebra (short videos) - https://www.youtube.com/playlist?list=PLi01XoE8jYoi3SgnnGorR...
MIT Calculus Revisited (Single Variable Calculus): https://www.youtube.com/playlist?list=PL3B08AE665AB9002A
MIT Calculus Revisited (Multivariable Calculus): https://www.youtube.com/playlist?list=PL1C22D4DED943EF7B
MIT Calculus Revisited (Complex Variables, Differential Equations, Linear Algebra): https://www.youtube.com/playlist?list=PLD971E94905A70448
Matthew Macauley | Visual Group Theory, Differential Equations, Discrete Mathematical Structures, Advanced Linear Algebra, and Advanced Engineering Mathematics (YouTube lecture courses) - https://www.youtube.com/channel/UCH1cV4RtgI_N97M8jepiUzw/pla...
The Discrete Mathematics course above is probably the most important for your work. In fact I would look for more Discrete Mathematics courses if I were you as it is far more important than anything else here.
Open University (BBC) | Geometric Topology (YouTube lecture course) - https://www.youtube.com/playlist?list=PLKB3Q5Oyy_RNBrS3V2WbO...
Joel David Hamkins | Philosophy of Mathematics (YouTube lecture course) -
I have a BA in math and I think the most-useful mathematics for everyone is probability (math theory and basic calculations). This is because you'll be better able to deal with (quantify) uncertainty. The concept to aim for is (learn enough so you can understand and use) expected value calculations - which are, I think, the foundation of rational decision making under uncertainty.
I wouldn’t worry about it. The amount of math that a working programmer needs is minuscule.
Where to start?
Math is huge. I suspect discrete math may be the most useful to a programmer who’s looking to just be a more theoretical programmer. Proof by induction stands out as a core, helpful concept too.
Otherwise I suspect it’s about problem domain. Geometry has its uses as does algebra. It all kinda branches from there.
A History of Pi - Petr Beckmann
Journey through Genius: The Great Theorems of Mathematics - William Dunham
How to Bake Pi - Eugenia Chang
These are all sort of 'pop-math' books -- that is, they're more intended to spark a joy & love for math than teach rigorous mathematics. Great Theorems and A History of Pi include a lot of history (edit to add: in addition to covering the math involved!) -- did you know some mathematicians in history would duel over their theorems? That theorems were a carefully guarded secret instead of something you shared?
Introduction to Graph Theory is specifically intended as an introduction to mathematics for 'the mathematically traumatized'.
In my opinion, after reading these, if you've sparked a joy for the puzzles and fun of mathematics, then I would then suggest branching out into more formal presentations of them relevant to your interests... it's much easier to slog through a book on abstract mathematics when you receive from enjoyment from the puzzles presented.
But one doesn't really get to ignore math in life because of money. You'll still need to figure some things out. I can say I'm a poor painter so I'm just going to ignore that and things will be fine. The cost of ignoring math can be quite high.
Wholeheartedly agree, and this applies much more broadly than math.
Once you've built the mental fence, breaking it down is a giant PITA.
I absolutely despise grinding and drilling uncontexualized and contrived problems and have never been able to maintain something stuff like that for very long. Which of course becomes problematic at I find myself interested in problems that require a decent grasp on some math.
Unless you're like me as a kid, you don't look around and start counting, adding, and multiplying based on the objects or scenes that you see, and "thinking in numbers". Thus the need for grinding, you need some math fluency (which takes practice to develop and maintain, like any other domain) before you can move on to more advanced topics (at least easily). If you lack fluency, even at a low level, then you have to do a refresher every time you try and get to something moderately advanced. Which is frustrating, at least, if not demotivating to the point of causing most people to stop. The more fluent you are, the easier it is to pick up an arbitrary advanced topic (at least to read, if not to apply or extend).
I think when people say they suck at math what they mean is; I find math homework extremely boring and unrewarding. Which perpetuates sucking at math and only compounds. Then things get worse as modern society works better for the individual when one does not totally suck a math.
Kinda like, I suck at piano because I find practicing the piano boring and unrewarding, but without the ramifications of sucking at math.
Also: what kind of programming are you doing? E.g. working with 3d games relies on different math skills than dealing with AB testing.
I've been keeping the book up to date and "maintaining" by fixing typos and improving certain explanations. If anyone else is interested, you can see a extended preview here: https://minireference.com/static/excerpts/noBSmathphys_v5_pr... and the concept map from the book is here: https://minireference.com/static/conceptmaps/math_and_physic...
I also recently released shorter book focussing just on high school math chapter, for people who are not interested in calculus but still want a refresher on the basics. You can check it out here with fancy new cover design and website: https://nobsmath.com/ (it's basically Chapter 1 of the big book) Aussi disponible en Français https://nobsmath.com/fr/
As for career advise, depends on what you want to do: There is not much use in learning calculus if you end up in astatistics-heavy field like data science. I'd say figure out what kind of computing you want to work with and see if there is a specific part of math that are useful there, if there even is one.
I've done work for finance and never had to go much deeper than simple multiplication.
Turns out I suck at spatial thinking. I can’t manipulate things like plots and curves in my head and they don’t help me understand mathematical principles. Algebra and logic though come easy to me. So learning math became an exercise in translation.
That is to say, please don’t think you suck at math and definitely look at different ways to learn the same concepts if something is not clicking.
Why has there been such a proliferation of these Reddit-style obvious questions upvoted? The front page is full of "Ask HN: what can I do about <insert obvious problem>?"
But to be fair, HN is not aiming to be a wiki. It's a vivid forum that discusses topics that are relevant to its participants. Looking at the amount of comments and upvotes on this thread, it's clear that a lot of people are willing to help or engage in some way. Also, people often not only discuss the initial topic (or question) but something else that arose from it. As I said, it's a vivid place. And that's for good.
Maybe this book will help you to realize you don't really suck at math, you just had some terrible teachers or whatever. It's also a great introduction to many different mathematical subfields so you can see which ones are most interesting/useful to you for future study.
[0] https://www.amazon.com/Mathematical-Ideas-14th-Charles-Mille...
So far I've only read the first few chapters of the book, and the exercises often feel too difficult to me. But I think he does a great job of easing into mathematical notation, pausing to reflect on what a seasoned mathematician might be thinking when they come across that notation. He also makes a lot of analogies to programming, and has example programs that are easy to follow. It's helpful to have that angle to understand things from.
But I am hard-pressed to think of any mathematical construct that reflects, even a little, the reality of OOP. This is evidence that OOP is an engineering concern, not a math concern. That is, OOP is one method to help humans deal with the complexity of a large amount of shared, mutable state (SMS), by partitioning it into smaller units of SMS. But math itself doesn't care about the scale of anything, and will happily encode any state into a single, very large integer, if you let it.
Some parts of programming are better grounded in math, like functional programming and relational algebra. Some distributed programming problems have some nice, ad hoc mathy treatment (e.g. Paxos), but don't really have a clear correspondence to anything.
Interesting the field that is closest to programming in real life, IMHO, is statistical thermodynamics. This is usually taught as part of the physics curriculum, and is pretty math intensive, and the field's remarkable job is to generally model microscopic behavior and then predict macroscopic behavior of huge aggregates. Programs always deal with huge numbers of tiny things, each having unique degrees of freedom, (alternatively, which have unique constraints), so there is some connection there. ST is also the field most closely related to certain "quant" jobs in the finance field, AFAIK, since the same tools let you model individuals in an economy and from that predict markets.