Mastering Linear Algebra in 10 Days: Astounding Experiments in Ultra-Learning

Cal Newport has the funniest definition of "mastering" and strangest definition of "world’s most efficient studiers" (another blog post of his from a couple years ago)<p><a href="http://news.ycombinator.com/item?id=2658927" rel="nofollow">http://news.ycombinator.com/item?id=2658927</a><p>I have ever seen. The shtick is getting old. Gee-whiz posts about a dilettante ramping up to a beginner's knowledge of a subject with little time and effort have nothing to do with the really challenging learning tasks in this world.<p>I'll be impressed when I see a headline like "Middle East diplomatic issues resolved by undergraduate who completed one course in international relations" or something like that. Show me someone who has solved a genuinely hard problem before proclaiming a new breakthrough in learning. For a refreshing change of pace from the usual blog post on quick-and-dirty learning, see Peter Norvig's "Teach Yourself Programming in Ten Years"<p><a href="http://norvig.com/21-days.html" rel="nofollow">http://norvig.com/21-days.html</a><p>or Terence Tao's "Does one have to be a genius to do maths?"<p><a href="http://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-genius-to-do-maths/" rel="nofollow">http://terrytao.wordpress.com/career-advice/does-one-have-to...</a><p>for descriptions of the process of real learning of genuinely challenging subjects.

There was effectively nothing about linear algebra on that page. After some link followings, it appears that <a href="http://www.scotthyoung.com/mit/1806-exam.pdf" rel="nofollow">http://www.scotthyoung.com/mit/1806-exam.pdf</a> is the exam which the student was happy with (having done worse on the first version). The final score appears to be 66 out of 100.<p>Based on that test, I think the title is link-bait as it isn't "mastering linear algebra" but "passing an introductory algebra course."

This is written from a student perspective, where "mastering" means passing the exam. I'll stick with Norvig's 10 000 hours.

I suggest reading very carefully.<p>I absolutely believe what he writes, because he's quite precise about his experiment and how he did it and this really works for a couple of reasons:<p>* This guy isn't 20 anymore. He has actually explored and learned and trained "productivity and focus" which he blogs and writes books about - so he doesn't start like a 18 year old directly from school, unexperienced maybe in this level of focus and discipline.<p>* He was pragmatic in his goals - very much so. He didn't write "becoming the world's foremost expert in linear algebra" but "passing an exam". And so he did. He also didn't write "passing everything with a top grade" but "just pass, if better - wonderful".<p>* He actually did his math on "hours to put in" - a semester doesn't take full 6 months, you usally don't attend lectures/lab every day 3 hours a day but 1-2 times a week, 2 (university) hours plus preparation. If you carefully add this up, you actually get a surprisingly low count of actual course/lesson hours.<p>* Taking in a course in a focused manner is actually quite efficient and helps you (at least it does for me) follow the material without interruptions. You also can repeat as often as you like (he mentions a fast forward and replay button in his TEDx talk) - which btw. makes part of the success of e.g. Khan university material.<p>* He also put some effort and training into the right way of learning and _that_ pays off massively in terms of speed.<p>Also, one of the points he is actually making is part of what most of you critizise: Going through the list of MIT requirements is something different compared to "becoming an expert in X" - don't mix that up.

Learning for exams and learning for yourself are obviously different kinds of activities, even if the level of depth and rigour are similar.<p>For maths-heavy subjects, I'm not really inclined to believe that traditional exams are the best way to assess a student's knowledge and understanding of the material (especially with regard to rote memorisation). Exams in such subjects haven't changed fundamentally in many many decades, even though we now have lots and lots of new things we could do with them.<p>For instance: do more with computers - like getting the students to solve real-world, many-tentacled, hairy problems by numerical methods, rather than giving them some carefully pruned equation that just happens to have nice analytical solutions. Or introduce more computer-assisted mathematical modelling (e.g. use classical mechanics, to start with). Or on the pure front, teach students to write or at least understand some interesting automated theorem prover.<p>Stuff like that.<p>I suspect that traditional exams have survived simply because they serve their purpose: a percentage of exam-takers fail the exam (which allows the exam-setters to claim that their standards of assessment are rigorous), and a fair percentage will pass the exam, some with flying colours. Whether or not the actual learning goal was achieved has not been determined, since the exam is deemed to be the only instrument that can measure that.

My best bursts of rapid learning are almost always project/puzzle driven. I didn't, for instance, set out to master FFT directly, but it seemed like something that could improve my abysmal performance on a Project Euler that I was working on, so I looked into it. I question (open-mindedly, not snarkily) the efficiency of ploughing through a course or series of courses. On the one hand there's the possibility of cross-pollination that having all sorts of cool bits of knowledge and techniques fraternizing in one's head for as long as possible promotes. On the other hand, there's the sense that the most efficient learning sequence is the one that matches the actual sequence of problems as they present themselves. Just-in-time learning of helicoptering, but if and only if you find yourself in a rooftop gunfight, as it were. Of course, then the issue becomes predicting forthcoming problems with enough lead time to learn the solution.

I also am inclined to believe much of this. I actually did the same thing while at MIT for chemical engineering. Took all the required freshman through senior level classes at the same time, each semester, and finished all the requirements for a chemical engineering degree in a year. I loved MIT for this reason - they had no rules/regulations on the number of classes you could take in any semester, and they didn't enforce prerequisites/corequisites. Very different than other institutions I've trained at. I was still able to participate in extracurricular activities and develop relationships with lots of people.<p>It's true that I didn't attend a lot of classes (since they all overlapped anyways), and had 2-3exams virtually every week. The only issue I see is that there is only so much you can do online. I also did the same thing with Chemistry and Biology, which had lots of laboratory classes, and I don't see how one could gain the practical experience of putting knowledge to work in those fields without a wet lab class. EECS however is amenable to this (for the most part - likely hard for an optics laboratory), and most of my EECS labs were really done in Athena clusters instead of a distinct laboratory.

People are kind of picking apart his use of the word "mastering," but I'd say that the crux of the article is spot on when it comes to learning techniques.<p>As an aside, I've never heard it called the "Feynman Techniques." However, one of my favorite things in the world is the so called "Feynman's Algorithm": (1) Write down the problem. (2) Think very hard. (3) Write down the answer. I just found to hilarious, but I digress.<p>There are two points of his with which I agree 100%.<p>Firstly, the process of writing a short summary paragraph of what you just read after reading a chapter or big section of a technical book. There is actually a fantastic book -- maybe one of my favorites of all time -- called, somewhat strangely, How to Read a Book. It's all about <i>very</i> active reading over passive, almost to the point of having a "conversation" with the text you're reading.<p>Ever since reading that book, I've gotten into the habit of writing a summary of each thing that I read. It really forces you to confront whether or not you "got" the point of what the book is saying. I usually find that there are quite a few bits that I either missed, or didn't quite understand, at which point I go through and search for the pieces I'm missing.<p>Secondly, looking at all of the low level pieces to understand the whole. This is something Salman Khan, of the Khan Academy talks about in (I believe it was) his TED presentation. Quite often, I find that there is some early concept that I glossed over which is slowing my understanding of the current material significantly. For me, doing this makes me being 'honest' with myself over the state of my current understanding -- which was kind of hard at first when I took this new approach to learning. So much of my 'ego' seems to be unfortunately wrapped up in 'what I know,' and thus I convince myself incorrectly that I do understand something, even when I don't, just because it's something that I "should" already know. Admitting to myself that I didn't understand, for instance, some basic math concept that I should have learned in high school was somewhat difficult -- as odd as that may sound. I suppose I have a fragile ego! But sometimes, getting a good grasp on my modern course work, meant stopping what I was doing, and going back a couple of levels and starting at the beginning.<p>The question of "What do I need to know in order to understand this" is, I find, an extraordinarily powerful one.

Although this is an accomplishment and there is some practical advice of value in the post, the "rules" he posts include correcting his own papers and tests and a minimum 50% passing grade.<p><a href="http://www.scotthyoung.com/blog/mit-challenge/" rel="nofollow">http://www.scotthyoung.com/blog/mit-challenge/</a><p>Would be more compelling if he was not selling books. Nothing wrong with making a profit but I'm just saying...

Ignoring the semantic controversy on 'mastery' and 'expertise', here's my story.<p>I dropped out of a CS program after first year. I was the classic case of a student who had always been told he was brilliant, so I never worked very hard. In high school, I coasted along simply on a fantastic memory, often 'studying' for the final exams that determine graduation the night before. I never learned how to learn.<p>Going to college was like being thrown into a bath of cold water. I had never been particularly conscientious, so being in an environment where I was now responsible for my learning was new to me. I skipped lectures, forgot homework that was due, turned in coursework late; the usual suspects. On raw talent though, I qualified for 2nd year, only failing Pre-Calculus. (I skipped the classes and tried to learn math from 1st principles. Ugh...)<p>I got a summer job at a small telecom startup. By time 2nd year rolled around, my student loan was denied, so I dropped out. I'd always hated school, so I didn't care. I never applied for leave of absence, nothing. I just didn't show up in September. That was 2006.<p>I was 20 then. I'm 26 now. I've had a lot of time (6 years!) to reflect on why I did so poorly despite being talented (not being conceited; my lecturers in 1st year said as much). There are quite a few reasons; but the major one is that <i>I didn't know how to learn.</i> So if something didn't immediately click, I'd give up in frustration, and decry the teacher as an idiot who couldn't teach (oftentimes true; but irrelevant). I didn't know there was another way.<p>Being around HN and places like LessWrong which exposes you to so many thought-leaders brought about some interesting side-effects, which culminated earlier this year. Upon reading an article on LW entitled "Humans are not automatically strategic", which was a reply to a Sebastian Marshall article "A failure to evaluate return on time fallacy", I had an epiphany that being <i>systematic</i> about things was the route to accomplishing great things. "Rationalists should win", the LW meme goes, and it's correct. I came to realize that <i>for every goal, there exists an efficient path to achieve it.</i> My task was to find that path, and execute ruthlessly upon it.<p>Since then I've made leaps and bounds in my personal development. I still slack off sometimes, but I won't fall into my old perfectionist way of thinking that I'm a failure. It's better to be 80% there than 0%.<p>I made the decision a few weeks ago to get my CS degree, albeit at a different, larger university. Since then, I've been devouring articles like this one. I recently bought two of Cal's books and wanna sometimes slap myself when I realize that if I had had this knowledge and the discipline to implement it 6 years ago, my life would be so much better. But c'est la vie. These articles on meta-learning are priceless.<p>So if you're in school now, or are going soon, pay attention to articles like these, Here are a few gems I've dug up recently: <a href="http://news.ycombinator.com/item?id=3427762" rel="nofollow">http://news.ycombinator.com/item?id=3427762</a><p><a href="http://news.ycombinator.com/item?id=818157" rel="nofollow">http://news.ycombinator.com/item?id=818157</a><p><a href="http://www.quora.com/The-College-and-University-Experience/How-do-some-people-get-near-4-0-GPAs-in-college" rel="nofollow">http://www.quora.com/The-College-and-University-Experience/H...</a><p><a href="http://www.quora.com/Harvard-College/What-are-the-best-Harvard-College-study-hacks" rel="nofollow">http://www.quora.com/Harvard-College/What-are-the-best-Harva...</a><p><a href="http://www.quora.com/How-do-top-students-study" rel="nofollow">http://www.quora.com/How-do-top-students-study</a><p>Thanks to knowledge like this from Cal Newport and others, I'm going back to college full-time as someone with an above-average cognitive toolset, and a myriad of experiences that will suit me. I'm <i>much</i> more sociable, have a great eye for design having moonlighted as a freelancer some years back, and will now know how to engage my lecturers on an adult level rather than the kid I was 6 years ago. I'm going for a 4.3 GPA. I'm tempted to say wish me luck, but with tools like these, I'll <i>make</i> my own luck.<p>This rationalist will win.<p>PS If y'all have more articles like this, let me know. If you wanna chat privately, email's in profile.<p>EDIT: formatting; clarity

This person, Scott Young, did not "master" linear algebra. If anything, he mastered the <i>curriculum</i>. There's a difference, and in a year he won't remember a word of it.<p>He is, however, a master of self-marketing:<p><i>"To find out more about this, join Scott's newsletter and you'll get a free copy of his rapid learning ebook (and a set of detailed case studies of how other learners have used these techniques)."</i>

I'm inclined to actually believing a lot of this.<p>When I got into university I found every course very easy, didn't attend any lectures, got all my workshops to run on the same day to reduce my face time and maxed out my free time to do whatever I wanted (work/friends/extra/etc). I'm a STEM major at a top 30 world ranked engineering school with good grades.<p>I've often asked if I could max out my classes and finish a degree within a year and a half - but I've never been allowed to skip more than a few subjects (tests/bugging the heads of departments).<p>University shouldn't be time capped or subject load restricted - people should be allowed to do as many as they wish - or you'll find more and more moving towards MOOCs instead.

I'll leave the discussion of how embellished this post/blog/exercise of MIT Comp Sci in 1 year is to other comments. But! The explanation of Fourier transforms from Scott's notes (<a href="http://www.scotthyoung.com/mit/fourier.pdf" rel="nofollow">http://www.scotthyoung.com/mit/fourier.pdf</a>) is one of the must succinct ones I've read. I've always understood what the transform does, but the nitty gritty on how the equation works was awesome

&#62; However, eventually you’ll reach a stopping point where you can’t explain. That’s the precise gap in your understanding that you need to fill.<p>This <i>is</i> a useful technique, giving motivation and focus. Though imperfect: it can't detect incorrect understandings that seem consistent. But to be fair, that's a tricky case.

One of the achievements I am most proud of was doing a full year's university course in computer science in 6 weeks and passing. It was pure cramming though and very hard work. I got into a routine of full-on study from 9am-1am with short breaks every hour or so. 16 hours a day for 6 weeks.<p>Not something I would ever want to repeat and was first year level courses. Basically I was doing a correspondence 3 year degree while working full time. I got heavily involved in my work and decided that I wouldn't continue studying. Then with about 4 weeks to go to the 2 week final exams period I thought, what the heck let's give it a shot...<p>Amazing what focus and hard work can achieve!

&#62; he completed all 33 courses (...) in less than one year.<p>&#62; That works out to around 1 course every 1.5 weeks<p>WTF? What kind of university imposes that you take only one course at any given time? It's not just linkbait, it starts from a wrong assumption. When you take many related courses simultaneously, you see the pieces meshing together and that helps learning. That's different from taking them in a serial manner.

It's funny because the students at the competitive schools work their ass off for the entire semester/ quarter to learn this material. If you use the weakest possible definition for "learn," then you can claim you have learned anything you want. But that doesn't mean your skill will be comparable to someone who spent 3-5 months practicing non-stop.

...this guy really knows how tomuch puts the "bait" in "link bait" ...nothing about linear algebra in the article but the perfect title to hook the bank of HN fish ...congrats to the OP for pulling this one off :)

I would imagine that the vast majority of students at Universities around the world who take Linear Algebra "master" it in 10 days. That is, the ten days before the exam, having spent most of the term drinking, socialising, falling asleep in lectures (or just staying in bed and skipping the lecture bit). I certainly know <i>I</i> "mastered" elementary linear algebra in about 10 days.

Not seeing this as link bait at all - more method for than what was being learned. Great read that gives a lot of interesting insights and methods - definitely not for everyone. A lot of the same information and ideas have been discussed on Study Hacks though it is great to see the provided examples.

He may have accomplished something impressive, but I had trouble appreciating it because the article seemed so pretentious, and I found that distracting.

Dazzling title, even no Linear Algebra at all, but I like the systematic introduction about "Feynman Technique"

I thought Linear Algebra was the easiest math-class?

Master it in 10 days? Forget most in 10 more.