Now, I am slowly trying to learn, but I don't know where to start. I need some guidance.
At that time, around 10 years ago, Khan Academy had excellent coverage through trigonometry and single variable calculus. Once I reached that point I went to my local community college and took all of their math classes. I transferred to University of Illinois at Urbana-Champaign and continued onward to get a BS CS, BS EE, and MS EE. I finished at 41 years old and landed a dream job that I would have never thought possible when I started.
I guess my advice is to start from the beginning and see where it takes you.
Some balance needs to be found.
I study biostatistics and dabble in population genetics research. I still resort to number lines and venn diagrams frequently. There is a reason we teach number lines - they are very useful!
Or the original continuous Fourier Transform, with just a pen and paper.
Or Heaviside's simplification of the Maxwell equations. Or calculating planetary orbits with perturbations.
Incredible leaps of imagination.
Before that, all I was doing was memorizing and practising heuristics to get A+.
I actually started learning Science since 11th grade. I grew and grew. And my scores tanked.
Got only 6.xx/10 in college as well.
But no regrets. Learning Math properly gave me a super solid foundation to learn Deep Learning very quickly. Now I do my semi-dream job as a Deep Learning Research Engineer building actual products with Deep Learning.
I have a truly realistic chance of getting my dream job within the end of this calendar year.
No regrets. Screw scores and learning systems designed for the mediocres. The education system has no care for the poor learners with no parental care. Neither it is good for future scientists and geniuses.
If I did it again, I would take the minimum of 12 units (3-4 classes)... assuming I could afford to stay in school for 6 years. At the time, I was racing to graduate and get a job.
> I was racing to graduate and get a job
This is also why I increased the number of units that I was taking.
Not if you have other interests!
I am a very cultural person (reading books, writing, playing music, etc.). I volunteer after natural disasters, too. And I also value interpersonal relationships with people I care about.
It felt bad in HS, but in college, I was mature enough to simply not care. (I also gained enough skills in the meantime to be certain that I won't ever starve.)
what event caused this? was it specific material or a specific teach or approach?
One taught me about what rigor was. In so many ways.
Another taught me that there is another side of Math. The non-heuristic parts. The stuff where is there is reason, logic, etc.
Dad took a broadband connection, and watching YouTube became realistic. I got 3b1b, Numberphile, MIT OCW.
Edit: My discovery of Halliday, Resnick, Walker physics textbook and Feynman lectures also was helpful. It was mostly- the Internet.
thank you for the tips! much appreciated
> I have a truly realistic chance of getting my dream job within the end of this calendar year.
So what’s your dream job? :-)
Or any co/startup doing "deep tech" in the area of Deep Learning.
I am doing _applications of DL_ now, want to do actual DL.
On academia and grades: yes, it's a "dark time" (as in "Dark Ages" for what we mean by it, regardless of the veracity of that historical perception). I often summarize all those collective F-ups by remarking that most people, either individually or collectively as institutions, have lost their purpose, they "have no mission in life" so to speak. Thus, we lose sight of the endgame, of the actual purpose of education for instance.
That's not about to change if the now widespread intuition that most institutions have been rotting too much for too long is true, so the solution might be some historical disruption of the domain — I think Khan was an early ripple in those foundations.
Send me a message, my email is in my profile.
Veritasium recently made a video, too [0].
3b1b is a fantastic supplementary resource. But you can't start with it, or use it as your primary resource. That would be a mistake.
I'll absorb the same amount - during that first hour, but after that nothing further is getting in my head.
Also the computerized thing about a diagram vs words isn't a learning style difference - both of those are visual, just one is more effective (obviously a diagram is better).
His on-the-street test is also bad, someone with a different style will, for example, say the image name out loud, if they need it auditory.
These are just personal preferences. There are no inherent style at the biological level. That's what "learning style" s are about.
I personally have found a combination of all (with more focus in applications and teaching the topic to others) works best for me.
> His on-the-street test is also bad
Agree. It was amusing in a bad way.
IMHO, the problem starts when the students get used to memorizing and repeating sample exercises which involve 0 creativity and then graduate and get an actual job.
If you think that getting good scores will open doors for you, and you are in need of those, then, by all means, work hard to get good scores.
What many people like me do not realize is that learning something properly is among the material luxuries. You can only afford to do this if you do not have socio-economic disadvantages (like most people need jobs straight out of college- mostly for the money, but social norm is also a factor).
I am a person of simpler needs and I do not care about socio-economic norms. You might not be like me.
So, sticking to social design might achieve you benefits.
As I've understood that part of myself, I've had to regularly and actively push myself out of my comfort zones (studied neuroscience in grad school, then became a software engineer and have worked at startups for the majority of my career, founded a couple of tiny ones that went nowhere with some friends too) and still need to as a mid-30s adult today. It's much easier now with a steady income, but seeing my friends who had more stable homes take risks that I'd never be able to stomach or recover from really imprinted how important financial stability is to a child growing up.
Hopefully despite all this, I can raise my own kid in a stable environment that allows them to learn and grow without worrying about their life being completely ruined if they don't make perfect grades or don't follow obvious well-trod paths in life.
Best would be Libera.chat, though.
Update your profile, and I will check back after 24 hours.
I have been in a tech since the last year of college.
This person has more relatable career trajectory as yours.
Data science, analysts, strategy, legal research (or research in general), many CS jobs, etc.
It identifies weak points and updates your progress on areas where you are successful.
Then you can focus on areas where you are weak instead of doing the entire grade again.
Then move to the next grade.
Maybe off-topic: But did you run into any difficulties landing your first job after finishing your MS in EE? I'm tempted to pursue something similar, but I worry about ageism and other factors when going into the job marketplace at that age in a brand new field. Would love to hear more about your experience.
1) Maths (precalculus and calculus) - I started with Khan Academy 10th grade onwards. I finished till grade 12 in a week. By this time I had gone through content elsewhere, so 1 week may (not) be enough. Regardless, Khan Academy app comes highly recommended.
I did realise that there were a few gaps (more basic in nature), I covered those with Eddie Woo on YouTube (e.g. why is zero factorial 1). For others I just looked up relevant searches online.
2) Maths (calculus) - MITs videos on YouTube. That is the pace of content I really love. Lots of overlap with Khan academy calculus, but do go through both. Also 3blue1brown "essence of calculus" playlist.
3) Maths (Basic Linear Algebra) - although if someone were to say Gilbert Strang's MIT videos, they would be bang on perfect, I had to start slower. Bingewatch (with popcorn and beer) 3blue1browns "Essence of linear algebra" on YouTube. Then move ahead to the channel "Math the beautiful" which has a slower pace. You would also wish to visit their website www.lem.ma where they have exercises. Then of course come back and start Prof Strang's lectures (you're delving into heavier stuff midway through his course).
4) Statistics - hands down professor Leonard on YouTube, he is the statistics equivalent of Eddie Woo. Slow, smart, funny and he has biceps too ;-) After this Prof Tsitsikilis (MIT) on YouTube.
It goes without saying that you'll need to practice problems (ironical, coming from me). You can download question sets of your country from online.
5) When you've done the above, your search for "linear algebra" and "calculus" on HN will yield a lot of lovely results. Hidden gems will be there in comments too. Check those books, interactive books, websites, etc out. Your pace will be good by this time, but you will occassionally come across something which you have not come across before.
If there's anything else, feel free to ask.
The downside of formal education is that the teacher has a timeline that needs to be kept and the class will move on without you. It might not seem like a big deal in the moment but in math the foundation is so important because the concepts build on one another. Sometimes in really subtle ways. If you have the motivation (and it sounds like you do!) then the huge upside to self-paced learning is that you can build a really strong foundation and the more complicated advanced topics will be so much easier to master. Khan Academy is really good at breaking subjects down into tiny bites and then slowly building on what you've learned but if you don't quite understand a lesson or if you haven't properly paid attention don't be afraid to do it again. If you guessed the right answer but don't understand why your answer was right, don't be afraid to go back and do it again.
So just because one person here could do all of the Khan Academy 9th grade algebra in a week that doesn't mean you should set that timeline for yourself. Everyone is starting from a different place. So maybe it'll take you a day, or a week, or a month, or even months. But however long it takes is how long it takes.
If you couldn't understand something yesterday, that's very good evidence you won't make sense of it today. You should work hard, go deep, and try hard problems, but if you fail to understand something after putting in the effort, just move onto something else.
Learn a few unrelated things, and come back. Much of the time, if you come back in weeks or months, it will just click into place.
You also want to visit topics /many/ times. Truly understanding many topics in math should take months or years. You can do it a lot more efficiently if you visit a topic for e.g. 5 hours each every 6 months over 2 years than a single 80-hour cram. You'll put in less time and understand more deeply.
A good starting point for references is Bruner (and specifically under Spiral Curriculum), but there's a whole literature on this. There has been a slow transition from seeing math as a set of bricks which stack on each other to an interconnected network of knowledge. The spacing effect literature is good too.
So now when I find something blocking me, I just go learn something else instead. And what do you know, some combination of more wisdom, broader knowledge, or perhaps just a different mental state often works.
- Bruner's work on spiral curriculum, and follow-up work. Spiral curricula are mis-implemented in most schools which claim to use them; it's not review. It's successive passes going deeper.
- Cognitive work on progressions such as surface->deep->transfer learning (Hattie) or chunking
- Physics education research on complex multiconcept problems
- Spacing effect. You can go all the way back to 1885 with Ebinghaus, but the esoteric author of supermemo seems to have done the best work here. If you prefer the academic establishment, there's a ton.
- There's a bunch of science that about 2/3s of what we learn, we're not aware we're learning or teaching. Concepts support each other, deeply. Kaplan, back when Bror was there, did a bunch of nice work there. It's easier to learn long division if you know a bit of algebra, and vice-versa. Chemistry helps learn physics, and physics helps learn chemistry. This was supported with pretty independent methodologies in independent domains (data analysis from tests, cognitive task analysis, etc.).
... and so on. This has very little to do with affect. It's a very well-established, rarely-applied set of results.
A complete lit review here would be book-length, at least.
I see students hit a wall in linear algebra, very good at "trained seal" but profoundly resentful that the time has come to go meta: Part of one's learning has to be investment in reflection and experiment on how one learns.
They generally believe that intelligence is innate, not deliberate gardening. There were thousands of people with Michael Jordan's body (alas, I'm not one) but he spent decades creating the athlete he became. Einstein found math difficult. It's like running: If you find running boring, you're not running hard enough. Math is a constant struggle, like mountaineering, and you shape, invent your mathematical mind. Those of us who love math (or mountaineering) are drawn to the struggle.
One syndrome I've seen often enough to classify: A student is uncomfortable with their calculus background. They want to pick the hardest calculus book they've heard of (often Apostol) and retreat to a desert island till they've mastered every word. I cringe in horror, and try to explain what's wrong with this approach.
It is widely observed that grad students learn four times faster than undergraduates. My first few years of grad school, it felt like I doubled what I knew every year. It was profoundly depressing when this subsided; I could have "been someone" with just a couple more doublings. What was going on?
Grad students learn what they need any given day, for goals they've set for themselves. Undergrads are taking on faith what authorities say is good for them. It's far easier to understand something when you see the point as you're learning it.
I know few mathematicians who read math comfortably, as if reading a novel. We mostly get angry and go think on our own, then return to realize that's just what the article said. Communication conventions in math are horrendous; one writes crappy machine code, then asks the reader to reverse engineer one's thoughts. Anyone in the tech sector knows how easily reading someone else's code can be "just kill me now" territory. As horrendous languages go, mathematical notation is a profound achievement. Nevertheless, beginners feel inadequate when faced with the inadequacies of mathematical notation to convey intuition.
If one wants to learn math, one needs to learn how to play, whatever that means. Find anything whatsoever in the text that leads to an example one can expand beyond the text. Does the text remind you of a pattern you've seen elsewhere? Perhaps the author just isn't saying. Play on your own, trying to decide if the ideas are in fact related.
Are there any resources or strategies you would recommend to help with this?
Those are unnecessarily harsh words. You and elliekelly are not contradicting each other, but talking about different things.
If you didn't understand chapters 1 and 2 from a textbook, reading chapter 3 from the same book is usually a BAD idea. Sadly, this is what formal education often tells you to do, because it follows a predetermined schedule.
If you didn't understand chapters 1 and 2 from a textbook, opening a different textbook and reading chapters 1 and 2 is usually a GOOD idea. Two authors describing the same idea from slightly different perspectives is better than reading the same text twice; at least it makes it easier to pay attention during the second reading, but the second book might also answer a question you had when reading the first book.
The same works for the humanities too, once my attention starts waning I’ll start skimming and rewind when necessary. Oftentimes authors are setting up a ton of boring but necessary context upfront, but it’s not obvious to a reader why such context is necessary until you get to the point first.
A smarter person would probably be able to anticipate what they’re trying to learn the first time around and avoid my flipping back and forth, but that’s hard.
If you didn't understand chapters 1 and 2 in a textbook, reading -- or at least skimming -- chapters 3 and 4 is usually helpful. It doesn't make sense to read them for mastery, but you'll see context where material from chapters 1 and 2 is applied, or concepts that build on them. You have to be comfortable with confusion, since most of it will go over your head. You will pick out some parts.
Once you've done that, go back to chapters 1 and 2.
Think of calculus: limits -> derivatives -> integrals.
However, you can understand a Reimann sum without derivatives or limits. It helps understand and motivate both.
This is called a whole math approach. You make successive passes with increasing depth. I've explained calculus concepts to kids under 10 with no problems, as have many others. Over time, you want to develop:
- Mechanics of integration, differentiation, and other computation
- Applications (e.g. debt versus deficit)
- Intuition
- Formalism
- And so on...
All of those support each other.
You see this in how you read research papers too. Novices read them linearly, and experts absorb them nonlinearly.
As a footnote, Vygotsky (in the original, e.g. 1978 translation) is probably the first person to discover that doing things beyond your level of ability accelerates learning. Not aiming for mastery means you're free to fail, free to try harder things, and learn faster (if more painfully).
85% of American textbooks about Vygotsky, though, always say exactly what he was trying to debunk. He pushed very hard for learning being harder and more abstract than most thought possible.
Careful. This can sometimes be counterproductive. It is often the case that one concept taught when learning a subject will be applied as part of teaching the next concept.
The best way to learn that first concept in many cases is not to focus on it until you really "get" it but rather to just get enough of it for how it is used for teaching the next concept. Then you move on to the next concept and the application of that first concept there gets you experience using the first concept which solidifies your grasp of the first concept and makes you truly "get" it.
The application of earlier concepts throughout the subsequent material can be in effect a kind of natural spaced repetition for learning the earlier concepts.
I think that deciding when to move on is one of the hardest things in self-learning, especially if the subject is new enough to you that you can't use your experience in a related subject to help guide you. You don't know enough to know when you've learned enough to go on.
But don't feel bad if you struggle, it's really hard to learn math concepts without an incentive. I'd just pick up a book on python and learn algerbra and a few other things along the way.
Your first stop would be Khan Academy and knowing your gaps.
Fill your gaps.
Learn HS level Calculus, Linear Algebra, and Statistics.
You will need more Calculus and Linear Algebra later. But not now.
Then try studying "Machine Learning for Absolute Beginners" book. It not very mathy.
Then just keep going through ML courses. Learn what you need on the way.
The "way" of math needed in Machine Learning is not the same "way" that brings you scores in school/college exam.
You need absolutely crystal clear concepts in Linear Algebra, Multivariable Calculus, and in some areas of ML, Statistics.
Corporate "Data Science" and Machine Learning research/projects are wildly different beasts. Learn what you will pursue, and decide your path based on that.
And most importantly, you have to be patient. Machine Learning and Math for it takes time- not days or weeks, but months and years.
I did different courses for each in University. Data Science is concerned with extracting patterns from existing data.
Like you have some papers for exam and you want to know if students cheated. Or you have the results from a poll about people hobbies, income etc and you want to correlate that with voting for party X. Or you want to correlate the race of canines with their abilities.
ML is also mostly about patterns but in a different way. You want something to tell how likely a comment is spam, if an article is positive towards a politician, if a picture is of a cat or dog.
So, to get to get a fundamental understanding you will need to learn statistics. Which in turn will require some calculus and algebra, but nothing too difficult.
Although I have the basic math knowledge and I have the basic knowledge of ML and Data Mining, I quit trying to do things in those fields because they are really vast, especially ML. Knowing the math and the basics of ML is required but far, far from enough to get good results. The people who work in the field are focused on it. I like ML but I like software architecture and development more, so I did my choice.
That being said, I still got some benefits from basic ML knowledge when I used ML libraries such as ML.NET in my day job. Knowing what a SVM or random forest is and how to tune parameters to improve my model was helpful. It was just a simple usage case like suggesting to customers what they might want to acquire based on their past purchases.
More than basic notations and explanatory parts of research papers, I never need even iota of Statistics.
When you need to learn something specific, like, say, KL divergence, if your math is solid, then you can pick it up in 10 minutes.
Working through Daniel Velleman's book "How to Prove It" (the only pre requisite is that you can understand boolean logic, which programmers have no problem with), and then a Set Theory book (I used Enderton) set me up to tackle (proof based) Linear Algebra, Analysis etc.
Just my personal experience. Hope this helps.
It really depends on what OP actually knows and how deep he wants to learn and in what direction
Would it be possible to teach mathematics by theorem proving ab initio? I guess conventional algebra would be hard to digest for a first grader, but I maybe something like the Peano axioms can be thought of as rules for a game that the students can play, where subsequent arithmetic lessons will be about finding shortcuts to the tedious application of the rules in order to solve problems.
So You Want to Study Mathematics…
https://www.susanrigetti.com/math
She also wrote:
So You Want to Learn Physics…
> So You Want to Study Mathematics…
> https://www.susanrigetti.com/math
HN discussion here: https://news.ycombinator.com/item?id=30591177
Of course you can use various tools (e.g. Wolfram Mathematica) for any fancy visualizations and maybe some tedious calculations. Just don't rely too much on them while learning the fundamentals (don't skip these boring "find the following indefinite integral" problems).
* https://news.ycombinator.com/item?id=31488608
Older discussions I shared on that one:
* Susan Rigetti’s “So You Want to Study Mathematics…”: https://news.ycombinator.com/item?id=30591177
* Terry Tao’s “Masterclass on mathematical thinking”: https://news.ycombinator.com/item?id=30107687
* Alan U. Kennington’s “How to learn mathematics: The asterisk method”: https://news.ycombinator.com/item?id=28953781
* Sign up for Alcumus. This is free. Set it to easy. Try problems in each section, and use this to identify gaps. If you find a gap, do a deep dive to address it. If you can't solve a problem independently, solve it using online resources (but put in the wrong answer / give up -- you don't want the ITS thinking you could do it).
* Watch videos on 3B1B, and sign up for Brilliant. Take a few courses.
* This will sound silly, but do sinerider, nandgame, and the fun Khan Academy courses (Advanced JavaScript and Pixar-in-a-Box), and things like this. Math is broad, and those give helpful exposure to a broader range of topics.
https://news.ycombinator.com/item?id=31488608 169 comments, 5 days ago
An extraordinarily accomplished mathematician replied, when I asked him what books to study, “You don’t solve problems by reading, you solve problems by thinking.” Thinking (hard) is the key, the practice of mathematics requires and unlocks a level of clarity far beyond ordinary everyday thinking.
Try mulling over simple facts which even children know, like the Pythagorean Theorem or the Quadratic Formula, and see if you could explain them so clearly and convincingly that any reasonable person would immediately and plainly see that they must be true.
Learning is spiral. You will almost never understand everything there is to know about a subject, but you will learn more as you return to that subject as you’ve grown and matured, and have acquired more knowledge, skills, and perspective.
Also I have created some youtube channels aggregating quite a bit of the quality university courses organized into playlists of playlists.
https://youtube.com/channel/UCjgQ2pJDjZlhdI4Ym7NQdUw
Note: You have to click on the titles of the topics on the home page that slide left/right (or up/down on a phone) to see the whole list of courses because YouTube truncates the lists on the home page.
I prefer hardbound textbooks, because computers are a big source of distraction. But there are also lots of generally very good free and open-source textbooks you can find online.
For free online textbooks these ones seem to have good reputation on the Internet. I did the 'pre-algebra' and 'algebra and trigonometry' books, found them good. Do all the excersices (very important), sometimes I got stuck (every books misses edge cases) so be prepared to look at other resources.
I’ve found having formal exams to study for very motivating, and humbling. I thought I knew things I’d read in books, but it turns out I just recognised them when I came across the same topic again. Being able to recall concepts and use them without help under time pressure is a different level of mastery.
Can you sit high school exams as an adult in the US? There might be private exam centres in a city. That way you’ll _know_ you’ve learned the math(s) you need, on top of all the great learning resources linked here.
—
Edit: Cambridge International do A-level exams, with centres accepting private candidates globally: https://www.cambridgeinternational.org/programmes-and-qualif...
I don't know of any reason an older person could not take the SAT but also most older students interested in going to university would be admitted under different critera than a 17 year old, and high school grades and SAT might not be relevant at all for someone in his 30s or older.
Check out Barbara Oakley’s Coursera course called Learning How To Learn. It turns out exams are actually an excellent tool for learning, more so than learning without assessment.
A side effect has been that having a deadline and a measurable outcome has been incredibly motivating. I’d have probably been distracted by side projects if I hadn’t committed to the exams.
And yes, it’s accepted for uni entrance here. STEM courses at top unis will expect A-level grades in maths/science subjects, even for mature students with industry experience.
basic arithmetic
ratios, proportions, percentages, roots and exponents, compound interest, basic lengths, areas, and volumes
Then to get to computing topics with a minimum of work:
some basic algebra, think of it as arithmetic but with symbols instead of actual numerical values
can pass up plane geometry, or just learn the Pythagorean theorem and do see a good, simple proof (will see it again with some nice generalizations in linear algebra)
should touch on trigonometry, such as can do in just a few hours, that is, without a whole one semester course
for calculus, here is the world's shortest but still basically correct course in calculus:
There are two parts to calculus, differentiation and integration. In a car, look at the odometer and a from it construct what the speedometer reads. That's differentiation. Then look at the speedometer for, say, a minute, and from those readings construct the change in that minute in the odometer. That is integration. So, each of these two undoes what the other one does, and that is the fundamental theorem of calculus.
For more, maybe do some linear algebra -- the above will give you sufficient prerequisites. For linear algebra, look on the Internet for a text that is highly recommended (by a professor at a famous university, e.g., MIT, Princeton, Harvard, Stanford, Berkeley) and claimed to be a relatively elementary view.
That should be enough for a lot in the early parts of computing, computer science, and machine learning.
With a good teacher, might get through the whole thing in a month, a really good teacher, in a week.
Uh, I know VERY well what I'm talking about: I hold a Ph.D. in math with a lot in computing from a world famous research university. I taught computer science at Georgetown and Ohio State and math at Indiana University and Ohio State. I've published in pure and applied math, mathematical statistics, computer science, and artificial intelligence. I've done serious applications of math and computing to US national security. Twice I saved FedEx from going out of business, once with some computing with a little math and once with just some math and a little computing.
A full set of books for MU123 (The most basic maths course, basically school [age 16] level books) can be had for under £50 on Ebay. Work through them, then move on to the MST124 books which are college [age 18] level), and are also widely available on EBay. Those begin to cover cover calculus, vectors, etc...
I'm 1/3 of the way through doing a maths degree with them, having scraped a C at school almost 20 years ago.
Dexter Chua was a Cambridge math student who started in 2014, TeXed all his notes and shared them on his website [0]. More people have followed in his footsteps, and you can find most of them by googling `Cambridge math notes site:srcf.net`. Many professors also put notes up on their websites, which can also be found by googling `Cambridge math notes`. I find that for many courses the notes are just as good as lectures, even notes written by students, and sometimes they're even better. They're certainly a lot faster.
You'll probably want to start with our first year courses, which are designed to take bright highschoolers and teach them how to think like mathematicians. If you're interested in the math of machine learning, you'll probably want to look at our courses on linear algebra, probability, optimization, and statistics.
The first high hurdle is accepting that starting out everything (to a first approximation) is over our heads.
There's no perfect first resource because hard subjects are hard and take time.
But because we are out of school, we have decades to learn.
There's no final in sixteen weeks and only a pop-quiz tomorrow if we are in the middle of applying what we learned.
So just start learning math and figure out what works for you as you go along.
Good luck.
It's not the teacher's fault, it's not the student's fault, they both could address the problem on their own, but really it's that education receives very low priority. Children don't matter because they don't have money, and teachers don't matter because they teach children.
Plus, there's no scholarships or opportunities for remedial students who don't have that much talent but want to learn and put in the effort. All the big math opportunities are elitist, great mathematicians only go out of their way for exceptional students, which is fine, but never for an ordinary student who just wants to be smart and works for it. Studies. Well one time, I actually did that, I tutored two really disadvantaged kids and one of them pulled his grades up from like a C to a A-, and they awarded him a prize for most improved, due to his effort. His name is León. When I got to where our classes were, he was always acing the marshmallow test, just eating the bad parts of his food and leaving the best parts for last. Not like anybody was going to give him an extra marshmallow, it was just natural for him.
I designed lessons, for instance a first one about multiplication with just paper, no pencils. Multiply with a blank sheet of paper.
But in general, that charity only goes to the richest.
If you want you can write me, I can give like a 2 paragraph rundown for you in particular, just because you want to learn. Email in profile.
It covers just about anything you need.
It would be better to study algebra, calculus, some trig and then statistics, it will give you a better knowledge, but this is a more condensed and a faster way to put you on track.
If you have time, you can go through it in one month.
1. Rote learning/memorization. Copying, tracing, flash cards and so forth. This is how you learned to read and write, and while in school math it tends to be applied to calculation(memorizing results from adding and multiplying and so on) it can also be applied to build up recall of mathematical concepts like postulates and theorems.
2. Logic and problem solving strategies. Math "homework" is usually about finding a result through a mix of deductive, inductive and abductive strategies. When the result is calculation-focused it becomes very mechanical and "follow the steps you've memorized", and so can usually be delegated to a computer program now, but higher level math is more about integrating the concepts together to prove something is correct, which means having a really clear understanding of the definitions you're working with.
3. Dividing and conquering. Sometimes it's hard to see a concept in totality but you can understand a particular limited context and then generalize on it. This is typically where math research starts: there's a flash of insight into a concept and then progressive attempts to generalize it and reuse it to solve more problems or define its relationship to other concepts, like how there are multiple ways to define coordinate systems in geometry.
When reading a math text, it can be hard to get started because skimming the text doesn't really grant any access to the concepts: you have to follow through on internalizing them first, which means a mix of the rote learning and posing problems for oneself to solve, and looking for analogies in things you already know to find the differences and so gain more detailed understanding. By the time you've done that, you probably have read the same words hundreds of times and "slept on the problem" for weeks.
This quality of not really understanding math until you've grappled with the problems means that research mathematicians tend to only have a really detailed understanding of their own specialty, but have a more limited background in others, enough to communicate a little bit but not necessarily participate in the discussion substantially. To get "there", look at what's offered in college courses: you can reuse their textbooks and problem sets. Following an online course is also a valid method. You don't have to attend classes or lectures to study math, although sometimes you may want to ask questions to clarify - but the internet exists for that and lots of people are willing to help, at least up until you actually get to a research level problem.
* Gelfand (et al.)'s HS math book series: System of Coordinates, Functions And Graphs, Algebra, Geometry, Trigonometry
* The problem collections Challenging Problems in Geometry and Challenging Problems in Algebra, available from Dover
* The math and logic courses on Brilliant.org
* Smullyan's Introduction to Logic
* Balakrishnan's Introduction to Discrete Mathematics
* For a bracingly irreverent "skip the bullshit" perspective, George Simmons' Precalculus Mathematics In A Nutshell
* The A-Levels A* prep sequence from Imperial College London, available on edX (this is actually roughly equivalent to a first year undergrad curriculum at US unis)
General advice:
* Practice, practice, practice - solve lots of problems & push yourself to repeatedly prove and reprove things. 'Getting it' once is not enough, math is like basketball. Something all the resources above have in common: tons of exercises, with solutions
* Keep mixing up difficulty so that you get some easy wins for confidence and motivation but also challenge yourself and keep yourself humble. Occasionally dip way down to remind yourself how far you've come, and on the other hand sometimes dip into something like Concrete Mathematics to remind yourself how far you still need to climb
* Go over the same material many times from different authors/teachers/sources
* Take your time. As long as you keep challenging yourself, and keep putting in those 10-20 hours a week, every week, you'll get to a good place and then keep going
https://www.glassner.com/portfolio/deep-learning-a-visual-ap...
It goes over the basics in the first half to make sure you have a foundation to understand the deep learning stuff.
And don't feel bad if there are gaps in your math. Go to khan academy or youtube and fill in the gaps. The most important thing is to find exercises and do them yourself, watching videos and lectures help understand the big picture but you'll never internalize them without doing the work.
I think in any kind of knowledge, math, music, art, etc... the biggest limitations come from people not knowing the basics deeply. Don't just work on stuff until you get it, or it clicks, keep going until it's so boring it's automatic. It will feel slower but you'll ultimately learn faster as you won't eventually hit walls.
In Italy, most Universities, even the public ones, require passing one or more tests before attending the courses; since a CS degree required a math test, I decided to prepare myself one year in advance. But unfortunately, I barely remembered how to do even basic algebraic operations, so I practically started from a secondary school program, fighting hard to reach a "last year high school" math level.
I decided to hire a teacher for private lessons because I found online courses very dispervise, and I felt that having a clear, structured path to the goal was a better approach for myself instead of struggling with different platforms and books. I studied 4 days per week circa, and the hard work paid back since I was admitted to the course!
I wouldn’t discourage you from trying a more comprehensive approach to building a great foundation in math. Math is awesome and if you enjoy it, go for it. If you want to stay focused on ML, you might do alright by figuring out what you need as you go. Just walk back from each problem until you find your bearings, then dig in.
Math is huge and you could find it takes forever to arrive at the skills you need to do the specific thing you want to do.
- “Mathematics for the Practical Man” - “Arithmetic for the Practical Man” - “Algebra for the Practical Man” - “Geometry for the Practical Man” - “Calculus for the Practical Man” -> this last one was the one Richard Feynman used to teach himself calculus (https://physicstoday.scitation.org/do/10.1063/PT.5.9099/full...)
good luck!
Start with ISLR
It’s a very well done book to quickly get anyone up to speed.
I then recommend this linear algebra book.
https://web.stanford.edu/~boyd/vmls/
The beauty and effectiveness of these two books is that they are applied. Applied learning is the most motivating way to learn IMO.
After you go through these two books You will have a very strong background!
My approach is a bit less structured than what others have recommended. I typically start by gathering tons of materials on the domain, say, digital signal processing. I then read things carefully, accepting that I won’t understand everything right away. Whenever I encounter something I don’t understand that seems important, I’ll sit down with a pen and paper or open vim and quite literally go point by point and try to write an explanation of the concept and make sense of it for myself. This method has served me pretty well for learning about new things.
I think one thing you cannot get around, which others have also mentioned, is that the process is necessarily slow. trying to come up with detailed explanations for yourself takes time, as does rereading material, practicing, and learning to apply what you’ve learned. I think the biggest mistake autodidacts can make is rushing through things. It can be hard to resist the temptation to go fast sometimes, but it’s really important to dwell on things and go slowly
Were I to be given another shot at learning math knowing what I know now, I would have first found a way to learn learn Vedic Mathematics so that I don't have to waste so much brain cells on intermediate steps when solving math problems and second, I would have concentrated on learning Technical Mathematics with Calculus and Linear Algebra since those concepts, particularly Linear Algebra, are directly applicable to Machine Learning and possibly Quantum Computing, from what I've read but don't quote me on that last point.
Would it not be great if somebody were to design a game that would teach you all the concepts of ML Math, and ML without the victim, er learner, knowing that such knowledge was the goal of the game at least towards the beginning?
Hard probem I know as it requires a totally different way of thinking about math and science in general. I mean, who ever heard of math being fun? ಠ_ಠ
Then Giles McMullen-Klein has an awesome recommended list for data science (your mileage may vary). https://www.youtube.com/watch?v=V2aIDbpESyU
Here’s what I did to learn math. I wouldn’t recommend this path. You will have a much easier time getting a job with a quantitative degree. You’re only 26, so you can go back to school without really losing much time. If you must do it this way, do the exercises, do the exercises, do the exercises, build a portfolio, and do the exercises.
Calculus/Analysis
Stewart single var calculus, then the multivar book. These are easy starters
Real analysis: Series, Functions of Several Variables, and Applications. Miklós Laczkovich, Vera T. Sós
Spivak Calc and Calc on Manifolds books
Bonus: Advanced Calculus A Geometric View, Callahan. This is what I turn to when I want to punish myself or remind myself how certain analysis proofs go.
Linear Algebra
Strang, Intro to LA is great. Start with this one
Strang, LA and learning from data. Will be tough without the first book
Stats
Hogg, Introduction to Mathematical Statistics
Gelman et al, BDA3 for Bayes
ML
Bishop PRML and Elements of statistical learning. Do the exercises. Build the algos in Python.
(also, occasionally a question or answer will be so good you'll instantly grok the math concept even if you haven't learned it formally before; it's rare but magical when it happens).
This will give you an understanding of which topics make sense to you and which you can barely follow. From that, you will need to catch up but where from depends only on your level. I guess, you might have some Khan Academying to do to catch up with the school program.
After you are done with The Foundations of Mathematics, I think you can take Andrew Ng's courses and go from there. I also use https://www.routledge.com/A-Concise-Handbook-of-Mathematics-... as a general reference when I need to look up a topic a have a rather vague recollection of.
If you want something more hi-tech there is Khan Academy of course.
But I eventually needed to learn it as I have to code something not in Python/Matlab/etc as part of an app and would like to postpone using LA libraries unless absolutely needed for performance. What helped me get the grips on LA is Jeff Chasnov's lectures on youtube and Mike Cohen's book. I would also recommend 3blue1brown for appreciation, if he happens to cover the topic you wish to dive into.
I'm many years older than you, by the way.
I found it very useful having visual uses that solved actual problems instead of just theoretical uses (or the old “trust me this’ll be important later”). For example, trying to figure out how light would bounce of an object put me down the vector, normal and dot product path (http://learnwebgl.brown37.net/09_lights/lights_diffuse.html)
After building a custom play around game engine, I can actually have discussions the ML people at work and I have a rudimentary understanding of what they are talking about.
For a more practical direction, you should start with Guesstimation if you are not already versed with it. It is an indispensable tool for getting people to think you know some math as well as helping you detect really bad ideas in addition to helping you get playful with mathematics, which is the only way to really master it as a tool.
My biggest struggle was skipping some of the basics and having to go back and re-learn after really struggling. But struggling helped in a way because then I really understood why X or Y solution didn’t work. If there’s anything i’d suggest, it’s that once you get some confidence in solving problems, try to approach new ones without looking through the entire lesson first, and that will help the actual way to solve the problem really stick. (And it’ll help you ask more focused questions!)
I think just having a price tag at all incentivizes a marketplace where the best lessons compete. Now how one determines that quality though...lots of different heuristics and I have a few thoughts on how it could be done better but reviews, especially negative ones do a fairly decent job of assessing for qualities.
My apologies if it was not clear what I meant earlier.
It's designed for self-study, and it together with "advanced engineering mathematics" by the same author covers all the math from the first two years of a 3 year UK engineering degree (they don't have any gen ed requirements so my sense is the material covered is equivalent to the math from a US engineering degree.
At some point, if you want to learn even more advanced math, you need to learn to do proofs, but you don't need a lot of mathematical knowledge to learn how to prove things, probably someone with a good high school math knowledge coupdnprsrn the basics, using a book like chartrand "mathematical proofs" which for the first five or six chapters only requires algebra knowledge.
I now have an MS in math and statistics -self-learning is great, but once you get to the "proofy" math, it's really helpful to get feedback on your proofs from someone who knows what they are doing. I took some summer math classes at berkeley, and then did my MS at cal state east bay, and found that the teaching was much better at cal state, which I would guess is because they mostly don't get paid to do research (small sample I know, but talking to many friends doing math classes at berkeley, my experience seemed representative). So that kind of school (cal state) might not be the best if you wanted to do a top tier PhD (though every year we have a couple of people getting admitted somewhere for a math PhD program), but for actually just learning and intellectual satisfaction, it was great!
I would second Khan academy, to quickly get an overview of a topic, it's great. But you don't really know something mathematical until you've done enough problems to be fluent in that area, which I think causes problems for some people who "understand" a topic but move on before they have "mastered" it IMO.
Feel free to contact me if you want to chat or have questions
If there are particular subjects you want to get up to speed with (basic probability might be a good place to start), this thread can probably suggest resources, and it might help to work with a tutor.
„A single book is enough to learn mathematics: Riley, Hobson, Bence: Mathematical Methods for Physics and Engineering: A Comprehensive Guide It has a whopping 1300 pages, but it has everything you need.
And if that is not enough for you get Cahill: Physical Mathematics This will give you advanced topics like differential forms, path integrals, renormalization group, chaos and string theory.“
The primary thing you need to remember is that you have be persistent. Only you can be successful at learning. You have to be persistent and not give up. When things get hard just look at it as a challenge that you need to get through.
You've taken the first step and that's wanting to learn. Now it just a matter of getting it done.
It’s way more impactful to say… visually and manually make a decision Tree on a 2D scatter plot of data points to grok the algorithm.
From my experience you have to start from scratch since some concepts are foundational and will block progress later if you don’t know them.
I have started with Khan Academy just answering questions on my phone. It’s a grind but I do it later in the day when my cognition is reduced. I imagine later on I will have to get a tutor.
I wrote this guide to help from 0 to advanced math. I hope it helps. https://juandavidcampolargo.substack.com/p/juan-davids-newsl...
See this thread[1] for a list of great math book resources.
[0]: https://coursera.org
Anyways, I started learning maths through Khan Academy and edx. Also, Math24 is really good as well.
Anyways, I have reasons to believe Susan Rigetti's recommendations [1] are good.
Topic for topic, read and do exercises.
It would be cool to see more follow-up posts about how it's going or where they ended up.
Don’t skip through anything
Do the exercises
Do you know pandas and scikit learn? If not, start there.
Step One: Find some good source material. Many people have suggested Khan Academy, in my particular case I've discovered this excellent Herb Gross lecture series, 'Calculus of Complex Variables' provided via MITOpenCourseWare. Then, review all the source material and make a list of the topics:
Part I: Complex Variables:
The Complex Numbers; Functions of a Complex Variable; Conformal Mappings; Sequences and Series; Integrating Complex Functions;
Part II: Differential Equations:
The Concept of a General Solution; Linear Differential Equations; Solving the Linear Equations L(y) = 0 with Constant Coefficients; Undetermined Coefficients; Variations of Parameters; Power Series Solutions; Laplace Transforms;
Part III: Linear Algebra:
Vectors Spaces; Spanning Vectors; Constructing Bases; Linear Transformations; Determinants; Eigenvectors; Dot Products; Orthogonal Functions;
Step Two: Plot a timeline. Here I'll give myself one week per lecture. I'll set aside 1-2 hours per day, at least four days a week, to work on the material. This might be inadequate. Staying motivated without having some external pressure will be a bit of a problem, but noting that from the beginning helps.
Step Three: Devise problem sets to test my understanding of the material. Here is where having a good teacher on hand would be invaluable, but that's not an option, so I'll have to find example problems. One option is to find a complex analysis textbook somewhere, and also find its solution manual, and use the problems provided there. Having the solutions around to check results can be very useful. A websearch like "complex analysis problems and solutions site:.edu" turns up a lot of results, just look for the simplest introductory ones (i.e. not the advanced proofs!).
So, I think this is the way to go for self-learning in math. If you have a more introductory level subject, say Linear Algebra (w/o complex), or Differential Calculus, just try to do the same thing.
P.S. I find Jan Gullberg's "Mathematics, From The Birth of Numbers" to be a great overview of the whole mathematical world, kind of like a guidebook: https://www.goodreads.com/book/show/383087.Mathematics
The habit of studying is the most important habit you can learn at this early stage. If you haven't read it, I strongly advise you read Atomic Habits. It's really helped me. Here are some great mathematics for machine learning courses, some I have taken and some I am currently studying.
Study tips: - Always show up even if you only study for 5 minutes, it's more important to show up than to be perfect - Make studying a daily habit even if you only do it for a short amount of time - Stack studying onto the end of another daily habit, e.g. study after you've eaten breakfast before you start work (this is what I do) - Make studying satisfying (I bought a small calendar and cross of each day I succeeded with a sharpie and it's super satisyfing crossing them off) - Remove distractions by putting your phone on charge in another room - Make notes on new concepts in the form of questions, these can be used later on flash cards to help refresh the material and avoid the effects of the forgetting curve - Write code to see the math in action. Running code is immediately gratifying and makes the value of the maths real and tangible.
- https://www.coursera.org/learn/machine-learning (free, approachable) - https://www.udemy.com/course/machine-learning-data-science-f... (like £10 on sale, very approachable) - https://www.udemy.com/course/mathematics-statistics-of-machi... (like £10 on sale) - https://www.coursera.org/specializations/deep-learning - https://mml-book.github.io/book/mml-book.pdf [textbook, not very approachable until you have some background] (free)
I will say that math is one of those things that you don't need to go to school to learn if you are persistent and passionate enough to learn on your own. There are tons of ways to learn.
First is to determine where you are. For that I highly recommend taking some of Khan Academy's free math courses. They start at the very basics. You will get bored of those.
But after that... I would say to take some things in your life and determine how to apply math. Create your own questions and try to answer them.
When I was 26 I was at the tail end of a janitorial job. I had applied (and learned) math to determine how much cleaning stock would likely be in use each week. Sometimes the math was "wrong" in so much as cleaning stock would be used more frequently or less frequently depending on what events there were that week. But it was a good exercise.
I'd also been writing some software on the side. Software and math are like two peas in a pod. A lot of people will tell you that you don't need to learn math to write software and they might be right for certain kinds of software. But even if you don't need math for software, it will certainly help. I now write software to fly drones and geometry is one of the core requirements.
Beyond the basics are Khan Academy there are plenty (!) of youtube videos from mathematicians who describe some of the simple [0] and more complex [1] concepts. Those have also been handy to me. All you need to do is search one math term that involves what you think you want to learn. A lot of the videos will include links to other videos that are prerequisites. Almost all of them will mention math words that will be new to you and if you search for those math terms you are likely you find other useful information.
Ten years ago I would have recommended that you search Google. Now? Don't do that. Google is full of trash and shit that want to take your money. Use Wikipedia, youtube, and reddit.
I've found 3blue1brown's lessons [0] [1] to be quite insightful for myself and have successfully used knowledge learned from his videos.
[0]: https://www.youtube.com/playlist?list=PLZHQObOWTQDPD3MizzM2x...
[1]: https://www.youtube.com/playlist?list=PLZHQObOWTQDNPOjrT6KVl...