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.
Where I live, physics and math are a part of obligatory general education, so you'll definitely have to study them until you're 15. After that, if you stay in the general education system (which is true for 95% of people who plan to go to college), you'll have to learn them for 4 more years. Sure, some people will develop a likeness/talent for it and others will hate it, but everyone needs to pass. Only when you go to college at 19 will people start telling you that perhaps you need to be a physics person to study physics and that it's not for everyone.
Art, on the other hand, is considered something magical and depending entirely on your talents right from the start. Art subjects barely touch on any practice, you have to learn a bit about the history and you pass. But I feel it's absolutely the same. Everyone can (and IMHO should be forced to) learn to read sheet music, play a simple tune on the recorder, draw still life with correct shading and proportions, write a short poem etc.
Personal involvment would give people a much deeper understanding of the topic. I mean, imagine if physics only consisted of everyone learning about Newton, Galileo and Einstein with just the smart kids doing calculations.
Interestingly, there's a paper "Lockhart's Lament"[1], on math education written about 15 years ago that proposes an absurd world from a musician's nightmare where formalized music education is mandatory but playing music is generally discouraged.
The point, poorly summarized, is that it resembles math education where a soltary focus on calculation distracts from learning how to do the type of math that only professional mathematicians seem allowed to do now.
It's worth reading if only to see how it often seems like the grass is greener for the educational environment of other fields.
Can we take a step back and make the physics/math optional instead of forcing art too on people :) I never understood why things are mandatory in education. Given a choice I would skip history (apart from reading on my own time what ever interests me),civics and anything that end up quizzing my memory instead of skill (aka memorized processes).
I never understood why things are mandatory in education.
Perhaps it's to combat entrenched prejudice. Back then we would make jokes about "Physics for Women", i.e. the non-calculus kind for non-majors. Everyone who would joke like that would know that physics is for anyone, just look around the lecture hall where 25 % of the audience are female, but talking like this he would pretend to be a Neanderthaler from an era thankfully past.
Forward to the present, where in Britain they make physics more "accessible" by watering the subject down. That doesn't help groups that are traditionally excluded, they need to be included, forceful if necessary.
I think it's a false premise that the subject is hard in the first place. Why is it hard ? Because you have to get the "right answer" on your test/homework. Ok, but why do you have to get the "right answer" ? Why do I get zero points for a problem when I put 0.33 and the answer was 0.145 ? That is utterly ridiculous. Why are we using logical AND on all a problems steps ? Oh, you made one mistake, no points for you!
And before someone brings up "partial credit", That's myth that meant "pity points", not "9/10 points, minus one because you f-ed up your algebra or minus sign". You should be tested on the process of solving a problem, not whether you got 0.33 as your final answer. In the real world, even if you do exactly what the book says(!) you won't get 0.33. Your measuring apparatus wont be calibrated, there will be noise, other sampling errors, defects in materials, etc. If you start the test out with this bullshit premise that IF( ANSWER == 0.33 ) THEN grade='a'; then you're not testing them on physics, you're testing them on how detail oriented they are and how well they can concentrate without making a mistake (like the Japanese show "Unbeatable Banzuki"). If that's your class, then call the class "Following Detailed Processes I" or "Doing N steps without ever making a mistake II", dont call it "physics". You should be able to get an A without ever getting the "right answer", so long as you demonstrate that you know the concepts and what the equations represent and how to apply them. Not contorting yourself to do the work of a machine, like a sadistic hazing ritual.
Edit:
And I almost forgot about the fun little thing they do in college where the course covers topics {A,B,C}, yet they test you {A, B^-1, and C-d}, you know, just a little bit different* material or problem formats than what they lectured about. Just different enough that they dont resemble any of the quiz or homework problems. If you want to test me on Ax^2+bx+c, then test me on Ax^2+bx+c, not on problems I've never seen before.
When I went to uni (physics) an exam had four/five questions that would take about 40 minutes each, and the "final answer" was worth about 5/10% of the question. So it really was about the process.
Sorry but education serves society and not you personally. It serves society that you know certain things. Civics for example is extremely important in a free democratic society. I think of all my schooling it was singularly the most important. This is in part because a good civics class doesn't have you memorize the 1st Amendment (which BTW allow you to post your opinion on HN) but understand why free speech is necessary, the limitations on free speech, and what can happen when free expression is limited. I think few things are more relevant today.
“Right. I don’t believe in the idea that there are a few peculiar people capable of understanding math, and the rest of the world is normal. Math is a human discovery, and it’s no more complicated than humans can understand. I had a calculus book once that said, ‘What one fool can do, another can.’ What we’ve been able to work out about nature may look abstract and threatening to someone who hasn’t studied it, but it was fools who did it, and in the next generation, all the fools will understand it. There’s a tendency to pomposity in all this, to make it deep and profound.”
– Richard Feynman, Omni 1979
A big problem with schools in the US is that grades carry so much weight. Taking a class which you don't have an immediate aptitude for is frowned upon.
Kids should be allowed to take more risks to learn different things.
I would be curious to see a study of people with average IQs, to see if they can learn very advanced math and physics concepts, with large monetary rewards for successful completion. An offer of $100k to learn General Relativity may entice someone with only an IQ of only 90-100 to be so motivated as to learn it and understand it. The large monetary reward is an important component, because people won't be motivated to something unless it's worth their time. If it's successful, the learning techniques could be applied to the general population.
Many studies show monetary rewards make it less likely people will learn things. So you've basically set things up for failure. As Susan points out in the article learning physics for her is about mental and spiritual enrichment. By introducing money into the mix you are changing the framing and putting things into an economic frame. The rules are different in economic frames.
This article makes a major point that a lot of comments seem to be missing.
She's criticizing how people get pigeonholed -- not just early in their careers but early in childhood -- as being "math people" or "physics people" or "humanities people". And that those categories and labels limit people who in fact are capable of learning and enjoying STEM subjects.
It's like how there's an unspoken rule in some corners of the tech industry that someone who first picked up programming later than high school can never be a "real programmer". Which is bullshit and does both individuals and the industry a major disservice.
Does anyone have any ideas how to persuade someone that they are capable of learning things? My mother, ex-girlfriend, and now wife have all been firmly convinced that they were too math-dumb to learn physics or programming. Its fine if people don't have an interest in something of course. What I'm wondering is that if you know someone to be intelligent and you are able to offer support in learning a subject, is it possible to persuade them that they are mentally capable of learning it?
I worry that after I adopt, my daughter is going to fall into the same trap.
"is it possible to persuade them that they are mentally capable of learning it?"
Depends on why they think they are incapable of doing it. Whatever the reason, it is possible that they do not see value in learning it and putting in effort. It might be that they failed in the past - or never tried because they always assumed they can not do it.
However, if your worry is the kid, then talk about that worry with them directly. Don't go around trying to manipulate them into learning something they do not care about. Just say that you worry about the baby picking up the attitude and thus having less options in life (and growing up less capable as a result). Healthy kids are super good in picking up attitudes like that, so the worry is well grounded.
Your wife might be more open to not make herself less capable then she is in front of kid for the sake of kid then to spend a lot of effort on something she does not care about.
Don't underestimate the drive for people to lie or greatly exaggerate their achievements over the internet.
I personally find the post not entirely believable. E.g.,
- First she says "I had learned nothing beyond sixth grade math: no algebra, ...". Then she says "I had been lucky enough to be introduced to ... some algebra ..."
- "You see, I had no formal education". Oh wait, that still leaves out the possibility of thousands of hours of khan-academy, and youtube and what not, for math, physics, programming, during high-school, summers, evenings, freshman college, sophomore college, etc, etc. (Yeah no.).
- She was ignorant in math and physics in June 2012, took a QFT class around Dec 2012, and had an eye opening experience. It's good poetry but not clear what she meant by that. (could be as simple as sitting in a QFT class without credits, and being able to follow some of the discussion)
- She was taking QFT in Dec 2012, and yet she graduates with BA in physics as last as May of 2014 (from her profile page). So very likely QFT was a giant namedrop, and if you ignore that, spending 2 years on undergrad math and physics coursework, when you've already taken logic and set theory, leading up to a bachelor's degree is pretty routine for most students.
- She worked on the ATLAS project. But. Doing data analysis and "helping design electronics". That sounds like a computer nerd who liked to dabble into physics just for the sake of picking a major for the degree (not that hard).
Each of us has the unfortunate case of being a population sample size of n=1. I think each of us, on some deep level , have met and recognized our own learning rates- and not only the linear rate, but derivative- and I agree with you that this rate of change is deeply linked to fluid intelligence
Wherever she started, I'm impressed that her first reaction to QFT was anything other than "I thought this would blow my mind, but, dear God, it's impenetrable!"
Because, in several years at a highly-regarded physics program, I've met very few people who understood QFT after their first (or second...) brushes with it.
She may not have been "a physics person" for a long time, but she certainly became one!
It takes time to understand a new topic, and sometimes more than one try. However, that is the expected difficulty level for someone with an undergraduate degree in a related field. Going from middle school algebra to QFT in 1.5 years is technically possible, I suppose, but it sounds like a big load of bullshit to me.
it's odd how her blog posts from 2013 have amazon aff links in them ..but I think her story is real, its just that her iq is astronomically high. John Moffat, the inventor of the theory of Modified Gravity, went from learning calculus to general relativity in a year, and this was in the 50's, so there was no internet to help him.
I didn't notice about the affiliate links, but her story is undeniably real: reading some of the posts on her fledling physicist blog (previous one) , one is struck with the sense of an immediate and lucid understanding of deep physics problems that probably take other early stage academics many years to grok. And , as GP stated, QFT is notoriously thorny, and an immediate comprehension of that is the hallmark of something very special.
I don't know what her IQ really is, I don't think astronomical, but it is certainly very anomalous. I only have a truly deep perspective on my own abilities, and just based on this I think I can say somewhat confidently she is at least a generation above mine in general ability-, for myself, having jumped from standard engineering undergrad, to software engineering and algorithm design, to applied physics, and now finally to nanoelectronics , each of which I can say I somewhat mastered within the space of a year-1.5yrs, and statisically I konw I am 3.5-4 standard deviations above mean (at age 10 and 14 I tested 164 and 158 respectively), but all of that did not give me the power to so quickly grasp QFT- remained incomprehensible to me in my first year grad school classes, and probably always will, unless I invested massive time into mastering it.
But I want to comment on another thing I find highly strange, which is that someone of such rare intelligence would leave academia (ostensibly she was at one time considering staying in Physics as a grad student) back to software architecture, rather than gravitating to the edge of mathematical physics- or continuing to follow that road to reality, to riff on Penrose. Indeed, I found myself following the inverse path. And ultimately, Moffat & his correspondents in our generation,- possibly another lady, like Pasterski looks like one possible member of this cohort- will be the ones to pave the next part of it.
Don't know why you're getting downvoted. This is the correct attitude. All human invention and creativity by definition is a human activity. If one human can do it so can another to various degrees of success. Just because you're not the next Einstein or Feynman doesn't mean you should not even bother trying.
I feel the same with Computer Science. It is so freaking complicated! I work in IT, I can program, I am totally immersed in UNIX culture. Yet I don't manage to sit down and study CS as I would like to, as I fear I won't get over the rather theoretical subjects. But you know what? Somehow this short article really resonated. I will sit down and hunt my degree, starting today.
(Disclosure: I am already enrolled in a online-university program here in Germany, which is almost free, but so far I only took two exams on rather practical subjects I had no problems with).
Not speaking to her case, but it happens. My family moved frequently, which meant hopping schools in different states. As a result, I missed many 'required' math classes, simply due to coincidences of scheduling.
(This was in the 80's, well before the current attempts at cross-state standardization.)
In my case, it did not cause me any particular trouble, as I was able to teach myself what I missed. My parents had predicted that this could be an issue, so they made sure I had the resources I needed. But if they had not been proactive, I could have easily graduated HS without ever having taken certain core classes.
Homeschooling is fairly common in the US. Only the end of year state assessments are required for primary and middle school. Instead of highschool I took the GED, which is typical.
While I was in college I didn't pursue Physics , math or or computer science due to bad entrance advisement and knowledge of English language. Two years ago, I decided that I want to code and pursue computer science, I loved every minute of it, from there I gained interest in physics and math. Now I am back in college pursuin computer science and taking physic and math classes. I really enjoy my classes, great stuff
I wholeheartedly endorse the idea that people should learn more physics and math, but I'm not sure the author's personal story supports her apparent message.
She apparently went from 6th grade math to graduate quantum mechanics in a year and a half. This is highly atypical and is surely not an example of a "not smart" person overcoming and managing to learn some physics.
I find that progression of mathematical maturity highly unlikely. It is possible and I won't discount it, but compressing 4+ years of study into 1.5 is not something that more than a few people can do.
I tend to agree. Having gone through a full undergraduate and graduate physics program, and having met more than a few Nobel Laureate level physicists, I don't think I've met a single one that could go this fast. This is 5 to 6 years of just concentrated college-level physics, to say nothing of all the mathematical prerequisites.
As a recently minted theoretical physics PhD, I feel rather the opposite way on this. The vast majority of Nobel laureates still went through the usual university progression, which isn't at all designed to cater to the smartest students, but rather to ensure that a substantial part (~15-20% or so in my program, but this varies greatly) of the student population can actually graduate and do at least some research work afterwards. Notwithstanding the actual thesis research, I feel like my program could have easily been compressed by a factor of at least 2x (dispensing with much of the repetition that goes into making sure most people can do the exercises and internalize the concepts) without compromising the final state of my education.
Furthermore, if your goal isn't to do research work but rather to be able to follow along an average research article using the basic framework of QFT, you can also cut out much of the lab work. Susan is clearly considerably above average in intelligence and a quick study, and I'm impressed by her perseverence, but I don't think this is at all beyond what any good theoretical physicist could have done. It's just that they went a different path, going through the university system, a path with its own advantages, but brevity is not among them.
In spite of Susan, seems you're hell for leather determined to set low levels on people's expectations. Albeit in the physical domain (actually it's more restrictive for reasons of actual capacity unlike mental activity)https://en.wikipedia.org/wiki/Four-minute_mile has something to say on the subject.
There is something to be said for realism.
The four minute mile didn't just topple for psychological reasons. It happened due to the development of better strategy (pacers), and continued to drop from there due to better training, better track materials, better shoes. The idea that it was purely a mental barrier is popular mythology.
It def. seems unlikely to be able to learn advanced math and physics so quickly unless you have an IQ of 160+ or something, in which case why in the 21st century America with modern schooling and education, why would someone with such a high IQ enter college not knowing 6th grade math. I'm not a teacher, but aren't there clues of profound giftedness that manifest early in life?
The question is, is her almost overnight success reproducible by other smart people. Let's say you know calculus and linear algebra, QM is doable, I suppose. But still...mind blown. Even Ed Witten with a physicist for a dad still took 4-5 years to learn the stuff.
I was under the impression that, due to the changes in the brain's plasticity, if you didn't pick up maths before a certain age you probably wouldn't be able to pick it up with ease after that. If this is correct, and she was older than that age, then there's a good possibility that there's not just IQ involved here.
The author's story is undoubtedly inspirational, to the point of being on the verge of unbelievable; and yet, the titular goal seems positively unattainable for not only average folk , but even above average folks , and possibly reserved just for those with an iq of 125/130+ (at least 2stdevs+) to even have a go at such an auto-didactic trial by fire at the frontiers of physics.
What may be MORE interesting to see, not just for HN folks but in general, would be a parameterized set of possible/reachable (as well as 'stretch') goals, for folks interested in learning more about math/physics/cs at various bands of natural/innate ability. While these bands may not be set in stone and raw effort may help level up someone up one tier above their 'home' range, it is for instance frankly impossible for at least 99% of folks to ever become competent at QFT - and likely, 95-99% cannot even grok it in part.
I dont think this is fair. Susan is a genius, she has multiple degrees from top universities.
The average person hears things like this and only gets demoralized. Its just like the lean in campaign. What worked for the 1% isnt going to work for the average person.
Hrm. Learning anything requires sustained and consistent focus on the subject. I don't think I can.
I have tried learning physics, linear algebra, calculus, abstract algebra, discrete maths, proofs, drawing, game development, compiler construction, operating systems, to write, sound synthesis, music in general and many more. In the past year. I'm not thick so I have varying degrees of success but I never make much progress. The interest lasts from a few days to a few weeks.
Computer science and programming are ones I just so happen to keep coming back to at closer intervals. I just wish I wrote more code, built more projects. There's a lot to learn and that is what lies in larger, more important projects. I know if I could get past this, I could make a positive, if only somewhat, contribution to our field.
I have ADHD, I take my medication everyday. It helps but it isn't a silver bullet. I have learned more in the past year with medication that I have the years before. It is a problem that does not seem to be going away.
Sorry Susan, I really admire your path. You've done well and it's inspirational. I just don't think I can learn psychics like you.
I think consistent focus might be a bit overrated, and bouncing between topics can be a nice way to learn for some people. For example, yesterday I looked at the type signature of runST in Haskell and it was completely obvious how it works and how to implement my own. I had stumbled on that weird type signature a few times before, filed it away as incomprehensible and moved on to other things. But I guess my mind was processing it in the background somehow.
That happens to me a lot with other topics as well. I get intrigued by something, play with it enough to memorize a few details without really understanding them, then forget about the whole thing for awhile, and then come back to find a deeper understanding without apparent effort. Does anyone else feel that way?
Yes, all the time. I read a book called refactor your wetware, and despite the patronizing title, it's about how our brain processes things in 'real time' but also in a sort of longer term, background data crunching sort of way and we should take advantage of this.
Have you ever you've ever had a problem you just couldn't solve, that was really hairy and beat you up all night that was suddenly trivial and obvious when you woke up the next morning? Same process, apparently. Basically your background subconscious or a lower level part of your brain goes and picks through all your knowledge that isn't readily available to your consciousness.
Absolutely. I notice it most with math. I try to reconstruct my intuition by sheer force of will, then give up in frustration because my mind will not move. When I wake up, the intuition is just there, as if something was updated or placed. It's not always overnight. Sometimes it's days or weeks, and some things I'm still working on after many years.
Actually yes. This is exactly what happens to me. Even after months, I come back and have an easier time. I think this touched on in Barbara Oakley's stuff though she seemed to criticise the extreme lengths between my learning.
My other problem is I spent tonnes of time learning and then never do anything with it. A lot of the stuff I learn often becomes useful mental models - calculus and the idea of continuity and change did this. It felt like a new way to look at the world.
The problem is you can have infinitely many ways of looking at the world but it means nothing if you don't do anything with it!
When you learn something, you can try using your newfound skills to answer questions on subreddits, mathoverflow, etc. It feels great to satisfy other people's interest and get feedback on it.
I feel like I'm similar. In the past couple of years, I've tackled most of the stuff on your list (I didn't touch compiler construction, sound synthesis and writing). My problem is that I pursue these as an alternative career to being a regular dev, and am pretty quickly being dissuaded by the vision that, in the end, my work life will not improve that much.
For example, artists and gamedevs go through tons of anxiety and (usually) ultimately fail and need to find an alternative career. Or, even if they succeed in some manner (i.e. an artist getting a AAA concept artist position), they may become so overworked that they reach burn out pretty fast. Another alternative career I have in mind: being a developer/researcher in an area that's a blend between programming and maths (signal processing, CAD, computer vision etc.) looks interesting on the surface (i.e. I love to study the concepts, implement them in matlab etc.) but I'm afraid that I would end up in some company as an overworked and unhappy faceless cog. That's partly because I feel there's only so much truly interesting work available in this field, and it will most likely go to extremely talented people who have PhDs from MIT in this field - leaving "normies" like me to just do the mundane implementation, bug fixing etc.
I feel like I'm chained by the proverbial golden handcuffs. I can make six figures working on boring and tedious code remotely from my home in a very cheap country (I'm easily in the top 1% of income here). Doing the researcher/developer route would probably require moving to some other country, and thus trading some possible increase in job satisfaction for loneliness and alienation.
I've tried that a couple times (and my saving are growing, I'm at 30-50% of early retirement goal already). I just find it super-hard to hold on a boring job for longer than 6-12 months, and I convince myself that I should quit and do something else (my CV is pretty spotty because of that).
If I had a guy that comes and whacks me in the head every time I think about quitting a job, I'd probably already be retired.
How much time have you devoted to learning these? I felt like I was in the same boat until I was able to fully devote myself to learning things at my pace. The worst part about the structured environment is that you go through things only once.
1. Read into something for a while
2. Realise I'm deficient in X
3. Repeat 1 with X until I get into a state where I feel "blah I can't focus on anything long enough to get ahead" by which point I either:
A. Indulge in a stimulating video game (complex ones from paradox are best but I enjoy Civ too).
B. Explore my spiritual side and try to get a grip on whovi really am and why I am this way until...
I span out of my rut and get back on the mouse wheel.
#1 and #3 could also mean that you need more "junior games" right away that kind of sort of simulate the entire experience instead of learning from elements. I suggest picking up a copy of "Making Learning Whole" by David Perkins. I personally don't quite like the writing style but the content is pretty interesting and important. Super Cliffnotes version: try to frame everything you learn in some sort of mini-version that is kind of sort of like the whole...yes it's hard for some (especially intellectual) domains but there's usually ways to do it.
Absolutely, I think that the failure to exploit spaced repetition is perhaps the most obvious and easily correctable flaw of most formal education.
I strongly suspect that even just mixing questions from previous parts of the course into the homework would be enough to make a substantial difference; grinding through a sequence of problems that all make use of essentially the same techniques before moving on to another set is a completely brain-dead way of trying to learn.
It seems most people who are curious experience the same thing.
I'm interested in all those things even though I wouldn't want to pursue a career in most of them. But I realize I spend X hours a day doing nothing, and I could spend those hours learning all kind of stuff, including for a career, even if I didn't to do it.
Instead of meaningless TV shows and movies I'll forget, and infinite amounts of Internet distractions, I could be an expert in Python or welding, for that matter. I could be making 5 times my income and change my life dramatically within 2-years, guaranteed. No more hoping, wishing, wanting, dreaming.
But I won't. I can't. I'm not.
Do I have ADHD? Do I need an amphetamine to focus me? If it doesn't work for you that well, what's the cure?
I don't believe in ADHD but I don't know anything for sure anymore. I am a pragmatist and open-minded. I'd gladly do anything to just focus on one thing.
ADHD is a real thing. Maybe not the same as procrastination and lack of self-managment.
I've known ADHD people with extreme self-control (amazing time planning), persistence and will to learn but when they tried concentrating it wouldn't happen. Either something in their brain can't click to grok the subject or they get constantly distracted by their thoughts. The time they put into the subject is huge but they get so little from it.
I've tutored several and am amazed at how well they try to avoid the problem by being better organized but their brain sometimes just can't focus on the important stuff which limits their ability to learn stuff that requires serious attention as quickly as others.
It is because of ADHD that I pursue these. It's a constant stream of stimulation. If my gut says no, it's impossible to focus on it no matter how hard I try.
But I do spend time on them. People think I'm a weirdo for even reading a math textbook outside of uni. I don't care: I am who I am.
No absolutely not. To learn something, you learn by doing. Writing code is doing, reading code is like learning math by reading how others have solved math problems rather than work on math problems yourself.
> I spent every minute of my days trying to learn everything I had never been able to learn from 6th-12th grade physics and math. I had the most difficult time possible taking intro physics and the beginning calculus courses. I kept going. I knew that if I was ever going to learn this stuff, I had to learn it now.
If you have the drive to put in this "most difficult" effort, utilizing "every minute" of your days closing the gap then you can learn like Susan.
I think the people who aren't "math people" or aren't "physics people" just aren't willing to put in much effort to learn it. They probably aren't all-consumed by it like Susan is.
It's not that these subjects are inaccessible to some people, it's that some people don't want to learn these subjects.
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.