This is half a rant and half a cry for help. I see topics like this on HN often; "Anyone can learn math!" but I really don't think I can. Not because I'm a defeatist and have given up but because I've tried basically all my life to understand math and I've never managed to grasp anything but the most basic concepts.
I've tried different teachers, my friends have tried tutoring me, I've tried Khan Academy. No matter what I do, the information just won't stick. The connections in my brain aren't made. What I don't understand is I learn other subjects relatively well. It's just math I can't grasp which really sucks because I love science and cryptography; two fields I imagine I could appreciate more with a solid mathematical background.
It's worth noting I have some of the symptoms of dyscalculia, so perhaps my brain isn't really built to do math and is why I struggle so much?
It's frustrating when I see "anyone can learn math!" because I've gotten shit from people in the past like "you can't be a good programmer if you're bad at math". I feel like we need to be more accepting that people have strengths and weaknesses in different areas.
I'm tired of feeling ashamed to be bad at math, especially as I'm not sure it's even my fault anymore.
I'm sorry math has brought you frustration and shame. I agree that it seems much easier for some people than others, though there's so much room for improvement in how it's taught.
About the cry-for-help part: I recently started reading Visual Group Theoryhttp://web.bentley.edu/empl/c/ncarter/vgt/ which looks to be gentle and illuminating. The style so far is quite different from a typical math textbook. Another book I've read more of was Turtle Geometry by Abelson and diSessa (the same Abelson who cowrote SICP). It can get difficult, but it's also very unusual: it's about exploring mathematical ideas for yourself by programming, and includes both hints and solutions to its suggested problems.
Being ashamed of not being able to do X is not a good reason or motivation for learning it!
Another mistake is the idea which the article rightly denies, namely that you have to be smart enough to learn X.
The author makes a mistake too: she thinks she can predict the rest of her life. ("I realized I actually wanted to study physics for the rest of my life.") This last idea is related to the question adults frequently ask children, "What do you want to do when you grow up?"
The limiting factor in all these cases is how genuinely interesting you find the field and how much you want to learn it now. Which is a property of the ideas and the problems as much as of your personality.
It's not your fault...her IQ is probably one in 100 million ...it's called winning the genetic powerball. Like why is Ed Witten so smart at math and physics? His brain is wired that way to understand those concepts very easily relative to the amount of effort required. If we break down IQ into verbal and math components, some people have very high verbal IQ and only average math; some both.
Learning QED and general relativity is often unobtainable even for people who are 'good at math'.
> If we break down IQ into verbal and math components, some people have very high verbal IQ and only average math; some both.
Yes but note that verbal and math scores are highly correlated. Likewise, basketball and football are different skills, and some people are much better at one than the other, but if we look over a population, being very good at one is a strong predictor that you are much better than average at the other.
> It's worth noting I have some of the symptoms of dyscalculia, so perhaps my brain isn't really built to do math and is why I struggle so much?
Does discalculia have to do with numbers and calculations exclusively? It's not very hard to cook up math problems where the numbers are well-hidden so that you don't touch them directly. A lot of induction arguments are like that. Have you ever seen a math proof? The reason I am asking this is because your writing seems lucid enough and math proofs are nothing but tight explanations why a statement is true.
Thanks for the link, I skimmed through the Direct Proof PDF - it seems very complicated. I have done some Discrete Math which I found fairly understandable once I went over it a few times as I could link it back to computer science. It was kinda like learning a very terse programming language.
One of the things that I used to do with knew math ideas was to write a program that illustrates them, then tweak values in the program to get a better intuitive feel for how the math works.
I usually come to understand the concept through experimentation that way, and from that point the actual notation doesn't matter as much (although it's still like reading a foreign language, to a certain degree).
I have a form of dyscalculia that affects my ability to read, write out, and process basic numerical operations. While I know that x^2 * x^2 should equal x^4, when I see a math problem like this, I often screw it up for no damn good reason. It's also not a matter of care; I can review and review my work and I simply do not see issues.
What seems to have helped me was to drill on an abacus, specifically an IOS app called Know Abacus. I also had a physical abacus to play with and that was really fun. I'm brushing up my skills with Khan at the moment, and I've noticed that after the abacus drills, my gut instincts were better about spotting issues or knowing what to do.
I think the abacus helped by making math operations a physical action (move 2 beads then move 2 beads to add, then see & count 4 beads) compared to a mental projection action (think of 2 things then think of 2 more things and remember I have 4 things because 2 + 2 = 4).
Edit: for reference, I am American who went through US public schools in a variety of states. The abacus training was completely absent from my primary education unlike perhaps some of our fellow HNers from Asia.
Interesting. I'd be happy to give you one or two (free) online tutoring sessions. I've always been able to get my students past their roadblocks so far, and I'm interested to see what a more difficult case might look like.
Thank you! I might take you up on that offer! How would the sessions be structured? It's been a while since I tried learning any math so I'll probably be extra rusty.
Do you have some contact information? I couldn't see anything in your profile. Mine isn't great but does have my Keybase which should lead you to my Twitter and website.
That's funny, I thought I had added email to my HN profile. My name is Tashi Daniels and I have a Gmail account under my name.
First session would be talking generally about how much you want to learn and figuring out what your definition of "success" would be, possibly with a timeline, and quizzing you with a ton of simple problems to see how much you know now and what confuses you. Ideally we'd find out in the first session if there's one concept or procedure you have the most trouble with and we can start hacking away at it.
In modern society we forget that math is a language, just like spoken language, because it is taught with an emphasis on the physical. Women are better at learning languages than men so why should they be worse at math? The answer is because men's struggle to grasp abstract concepts like language and emotion and their firm nature boxed math into the visual/auditory dimension instead of an abstract/spiritual dimension.
You must rethink how you think about math fundamentally. This video and paper are like poetry - short but demanding. Watch this video and read the paper. Really think about it in your own way, don't try to force your mind into an unnatural construction.
I felt the same way until I got to undergrad and relearned math from the basis of naive set theory and spent time working through the basics (proper definitions, proofs, Dedekind cuts etc.).
I've tutored math for the SAT. My anecdotal impression is that many people missed a few bits of math in the early grades, then spent the entire rest of their math careers hopelessly behind.
So, for example, algebra might be difficult because of a gap from grade two about not knowing part of a times table.
If my hypothesis is correct, that would call for approaching math from a total beginner mindset, and taking nothing for granted. For example, there is much to contemplate in a triangle, or in mental tricks for adding small numbers.
Practice is important. That's why I like Khan: for every concept, they give you exercises you can do to the point of mastery. I believe it also does review exercises so you keep working on the knowledge.
A sense of play and fun is important, and helps with practice. My dad is excellent at mental arithmetic: when he sees a license plate, he'll add the numbers together for fun, or multiply them, etc.
Another app I like is Dragonbox. I'm not sure how far it will take you, but the goal is building intuition. Again, beginner mindset is important. It's intended for kids, but for all intents and purposes you are on the same level as a kid – and that's fine!
Another important element is that people who are good at math have usually just figured out small tricks that make certain calculations easier, and help them visualize concepts.
I'm not entirely sure how to teach them, but while reading Isaac Asimov's autobiography I came across this reference to a short book he wrote on exactly that topic. Asimov is a wonderful writer and this may be an excellent primer:
Finally, I recommend studying for the SAT math section. The questions test math in somewhat novel ways. They're fun and reward creative thinking. And as a bonus, there are books written to help you figure out this kind of math, quickly. I really, really like Pwn the SAT. My students got a lot better just by using it: it teaches you how the author thinks, and really breaks things down simply.
(Note: The SAT changed in 2016. You could also use the earlier SAT test booklet (the blue book) and the earlier Pwn the SAT for this exercise)
I don't think any single concept in math is particularly hard. What's hard is that even high school math requires mastery of a few hundred concepts. So you can learn one, but it goes away.
The tools I outlined above are the best I know to solidify math knowledge and make it habitual. I believe any reasonably intelligent adult could use these to learn math.
It is, however, a large subject, and I expect it would require months of focussed work practicing these for every day. But I believe it's doable. I'm basing this on my experience with students who were "bad at math". They made great strides.
I hope this may be of some use! Feel free to reach out if you want to talk about it further. My email is on my profile.
I want a better, deeper understanding for things that are built on math, like cryptography and physics. Math should help me understand topics I enjoy more.
Lately I feel like I'm hitting a brick wall with cryptography. For example, I know what algorithms are secure but I can't tell you why as I don't understand the math behind it. I took the Cryptography 101 course on Stanford which used discrete probability. I didn't get very far into the course.
It's possible some of this just comes from me trying to fix the feeling ashamed problem but it's just making me miserable. What I ought to do is stop beating myself up.
"Learning math" to understand "cryptography" seems to me as a narrow goal. I have an impression that observing it that way you can get disinterested as soon as "the math" is doing its "mathy" things of developing itself just for the sake of it. Because that's math in essence. You want to be on the level of "applied" math but you have to be ready to "dirty" your hands with math "just so" too. Once you are proficient enough to feel comfortable of approaching math as something that doesn't have one purpose and not even some universal consistency (e.g. not in all cases is the notation the same) it will be easier for you: then you use math as a tool, but you aren't afraid of that tool. But a lot of math exists for itself.
My suggestion, it worked for me: buy a lot of paper, like, thousands of sheets. Take some books with the problems (and also with the solutions) then work through, maybe more than once. Don't say you haven't tried until you actually used all these sheets to write your own derivations of the solutions. If you once see the solution, you have to try the second time without looking. I could not learn by not really doing it, a lot, and I don't think anybody can. You can't just "read" it as you read history books, you have to "work" it.
A lot of people simply can't imagine themselves sitting and filling the papers with formulas (also in the comments here), but don't have problems spending months doing video games, for example. It's this "belief" that's limiting.
On another side, you need to have some kind of context too, but you have to find the interactions of the context and the actual work you do yourself. I like the "historical" context, because the big steps in math were traditionally not accidental. Archimedes invented the kind of "infinite" methods to calculate volume of some solids, and the methods remained forgotten, people remembering just results, Newton also invented a new math just to solve the problems he considered etc. For you, you have to adjust your steps to the knowledge you already have: if you know little, you can't avoid spending time learning the basic stuff. You have to work through, step by step.
Maybe also watch the documentary about Fermat's Last Theorem as an example of how it looks like doing math.
Yeah, this was my problem as well (if I'm understanding your point correctly).
I could only do very basic math because I got bored, or my retention rate was very low. I'd go through the excercises but it wouldn't stick in my brain.
Eventually I figured out I just couldn't handle theoretical, like on the paper, stuff. So I started making up scenarios where what I were trying to learn would actually be applicable and create small programs that illustrated it, and suddenly I started to actually remember the stuff.
I've considered this, I was planning on reimplementing `bc' or something on those lines. Perhaps some kind of crypto suite or physics sandbox would work for me? What sort of programs did you write, I'd love to hear more!
What I'm looking for is theoretical knowledge, and that's something else I struggle with. Can I still get that with more hands on experience?
For instance, something I find fascinating are One Time Pads. They're perfectly secure, given len(k) >= len(m); so the key must be the same length as the message.
As far as I understand, it's because each possible key may produce any possible message which I think is expressed:
Pr(x) = 1/|U|
Where U is the universe of all possible messages: {0, 1}^n where n = len(k)
I probably wrote some mistakes there. I was able to pick this concept up, I think as it just made sense to me; maybe I was able to visualize it or I could just think it through and realize the conclusion made sense.
When I was in my second year of high school, I was put up a class for math and science. I immediately found myself unable to cope, and my grades dropped from As to Es, in a few weeks. Two years later, the math teacher for that year gave up on me in the first couple of weeks, and refused to teach me or acknowledge I was even there, after role call.
As punishment for my laziness, my legal guardians beat the hell out of me. Maybe I would learn to apply myself, they said, between blows.
10 years later, I was diagnosed with a non-verbal learning disorder/disability. I ///couldn't/// learn math the way it was being taught, nor science, economics, english, history, or any other topic I'd tried my hand at. In reality, I had burned out in the first few months of being elevated from one class to another and never had the chance to recover (just ignore the physically and psychologically abusive home life I had).
In spite of this, I struggled my way through a CS degree, which is effectively worthless to me because of the disability. The whole time, I had friends, colleagues, family, tell me that I'm lazy and just need to suck it up and do the work. Not one of them had the slightest idea of what I was going through and what was necessary to complete the courses of study - approximately 3x the work required for the same grades. The disabilities support office provided me with only written notes, absolutely inappropriate assistance given my difficulties with written materials.
My degree was a waste of time, money, and effort, because nobody will ever look twice at me. I'm overqualified and no support services are able or willing to help me, many employers won't hire me because in their minds, they'll train me up (and pay minimum wage) then ungrateful-old-me will jump at the first decent job to come my way. I've previously noted interviewers would shout and swear at me for wasting their time, threatening to bill me at their consultant rates. One employer who knew of my disability later told me that I couldn't have told him about it, because if I had he would never have hired me. My present employer hired me for a non-IT role, and then later on told me it was expected that I would provide IT support for all the office staff for no extra money - I was already working 30 extra hours a week without pay.
You may be able to bring your abilities up to a useful level, I don't know, but the question you should consider is whether it's worth the time and effort. It may well be that any perceived benefit is far outweighed by the costs and struggle. I would strongly recommend professional assessment, as being diagnosed with my disability has probably saved my life, and very much my sanity. I do get tired of people telling me that it's just an excuse and that I'm really just lazy - non-verbal learning disorders are far more than simply "I don't understand what I don't read." One genius once told me that everybody had to learn body language, I just had to force myself to do it, work harder and I'll get it. Apparently he didn't.
Don't be ashamed about your lack of ability, be ashamed for the people who think you can because they could, that you just need to apply yourself, because they never will be ashamed of their prejudice.
I've tried different teachers, my friends have tried tutoring me, I've tried Khan Academy. No matter what I do, the information just won't stick. The connections in my brain aren't made. What I don't understand is I learn other subjects relatively well. It's just math I can't grasp which really sucks because I love science and cryptography; two fields I imagine I could appreciate more with a solid mathematical background.
It's worth noting I have some of the symptoms of dyscalculia, so perhaps my brain isn't really built to do math and is why I struggle so much?
It's frustrating when I see "anyone can learn math!" because I've gotten shit from people in the past like "you can't be a good programmer if you're bad at math". I feel like we need to be more accepting that people have strengths and weaknesses in different areas.
I'm tired of feeling ashamed to be bad at math, especially as I'm not sure it's even my fault anymore.