
Mastering Linear Algebra in 10 Days: Astounding Experiments in Ultra-Learning - phenylene
http://calnewport.com/blog/2012/10/26/mastering-linear-algebra-in-10-days-astounding-experiments-in-ultra-learning/
======
tokenadult
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)

<http://news.ycombinator.com/item?id=2658927>

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.

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"

<http://norvig.com/21-days.html>

or Terence Tao's "Does one have to be a genius to do maths?"

[http://terrytao.wordpress.com/career-advice/does-one-have-
to...](http://terrytao.wordpress.com/career-advice/does-one-have-to-be-a-
genius-to-do-maths/)

for descriptions of the process of real learning of genuinely challenging
subjects.

~~~
tom_b
I have several concerns about the original article, but if the point of this
guest post on Newport's blog is "look, I was able to grind through the MIT
requirements in a year with reasonable success" it is not particularly damning
that all the person achieved is a beginner's knowledge in several courses.
Hopefully readers realize that this is not necessarily such a wonderful
achievement while simultaneously respecting the effort made.

My bigger concern is that the overt focus on "time-compressed" learning is
that it will devolve to pure hucksterism with no attempt to move beyond
anecdote to real research and thought as to how teaching (either by self-
directed methods or to students in a traditional way) can be improved. People
seem so interested in short-cuts. I personally want to an "efficient studier,"
but what I mostly mean is that I do not want to waste time using bad learning
methods or materials.

We do know that improvements in the learning process are out there - better
books, spaced repetition for recall, better guidance and opportunity for early
learners than they get in schools in subject-specific areas, etc.

These types of posts along with my reading of immersive learning experiences
(like your own math programs, dual language programs, "programming school"
opportunities) has me thinking about the innate value that immersion offers to
the learner.

Would linear algebra learners be better off to spend 4 weeks focused on only
that subject with a professor/mentor, a textbook, and deliberate practice than
a single semester of normal lecture instruction?

------
dalke
There was effectively nothing about linear algebra on that page. After some
link followings, it appears that
<http://www.scotthyoung.com/mit/1806-exam.pdf> 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.

Based on that test, I think the title is link-bait as it isn't "mastering
linear algebra" but "passing an introductory algebra course."

~~~
stephencanon
I am quite confused looking at that final; it seems incredibly basic for even
an introductory course at MIT (that _might_ have been a midterm when I was the
graduate instructor for linear algebra at Berkeley).

~~~
jaredlwong
18.06 is the very basic, more applied version of linear algebra at MIT. About
half of the students that take it aren't in the math department. The more
theoretical option is 18.700. And then there's 18.701, Algebra 1, taught by
Artin using his book. The latter two are almost completely theoretical/proof
based.

~~~
stephencanon
To be clear: I mean that it seems too basic even for a final exam of an
introductory applied linear algebra course for non-majors.

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

~~~
47uF
Agreed. I had an interesting experience recently when I decided to "relearn"
math. I had taken calculus courses in high school and university, and aced all
the exams. But when I eventually came back to calculus out of personal
interest, I realized that I didn't know what a derivative was! I didn't know
what a limit was! How did I pass those classes? And if someone who got the
highest marks didn't learn anything, what about the people who were actually
struggling? How many people in that class actually learned anything?

And I could say the same for my science classes, foreign language classes,
etc. I don't think you really learn anything unless you really want to
understand the material and you work hard to do so. And if that were the case,
hour for hour you'll get what you put into it. You wouldn't be studying to an
exam. You wouldn't be satisfied with 60%+ on that exam. You wouldn't restrict
yourself to a specific curriculum. You wouldn't put time limits on your
learning. Instead you would learn the thing that you decided you wanted or
needed to learn, and however long it took for you to really understand it,
that's how much time you would spend.

~~~
rtkwe
The real test is how well you were able to pick up the concepts again. Not
using it for, how ever many, years it's natural to forget. However if you can
quickly reacquire the knowledge then you did learn it atrophied so to speak.

~~~
47uF
But I really think that I never learned those concepts in the first place. All
I learned were rules to solve the problems given to me. For example I actually
did remember the "chain rule". But those rules had no meaning to me, just
moving numbers around. Which means I would never be able apply it to anything.
For example, calculus clearly has applications all across basic physics. But I
never really made that connection. All this makes me think that exams are
much, MUCH less effective at measuring comprehension than we think.

------
Su-Shee
I suggest reading very carefully.

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:

* 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.

* 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".

* 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.

* 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.

* He also put some effort and training into the right way of learning and _that_ pays off massively in terms of speed.

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.

------
hdivider
Learning for exams and learning for yourself are obviously different kinds of
activities, even if the level of depth and rigour are similar.

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.

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.

Stuff like that.

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.

~~~
goldfeld
Yeah, I find the best learning happens when you put that knowledge immediately
to work in a real world problem, and preferably one that's meaningful to you,
as opposed to most term projects. And that's where learning for yourself makes
so much more sense; if you're learning for yourself then you probably already
have a real world application for it which got you into the goal of learning
it in the first place.

------
gall
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.

------
RVijay007
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.

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.

------
goostavos
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.

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.

There are two points of his with which I agree 100%.

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 _very_ active reading
over passive, almost to the point of having a "conversation" with the text
you're reading.

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.

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.

The question of "What do I need to know in order to understand this" is, I
find, an extraordinarily powerful one.

------
EzGraphs
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.

<http://www.scotthyoung.com/blog/mit-challenge/>

Would be more compelling if he was not selling books. Nothing wrong with
making a profit but I'm just saying...

------
oz
Ignoring the semantic controversy on 'mastery' and 'expertise', here's my
story.

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.

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...)

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.

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 didn't know how to learn._ 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.

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
_systematic_ 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 _for every goal, there exists an efficient path to achieve it._
My task was to find that path, and execute ruthlessly upon it.

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%.

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.

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:
<http://news.ycombinator.com/item?id=3427762>

<http://news.ycombinator.com/item?id=818157>

[http://www.quora.com/The-College-and-University-
Experience/H...](http://www.quora.com/The-College-and-University-
Experience/How-do-some-people-get-near-4-0-GPAs-in-college)

[http://www.quora.com/Harvard-College/What-are-the-best-
Harva...](http://www.quora.com/Harvard-College/What-are-the-best-Harvard-
College-study-hacks)

<http://www.quora.com/How-do-top-students-study>

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 _much_ 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 _make_ my own luck.

This rationalist will win.

PS If y'all have more articles like this, let me know. If you wanna chat
privately, email's in profile.

EDIT: formatting; clarity

~~~
ArbitraryLimits
> I didn't know how to learn.

I'm not trolling here, but don't you think it's more accurate to say you
didn't know how to work?

~~~
forrestthewoods
I disagree. College is the first time many students have to learn how to teach
themselves.

I did quite well in a small town high school with minimal effort. Not perfect,
but I graduated 11 out of 399. I can probably count the number of times I did
_any_ work or studying at home on my fingers. During the day I listened in
class, focused on school, and gave forth effort. It's pretty easy when classes
move at a snail's pace lest any student get left behind.

When I went to college my first physics class kicked me in the face so damn
hard and I didn't even see it coming. Lectures moved at blistering pace and
entire chapters were covered in just one or two class periods. I remember
going into the first big test, thinking I did well, and getting a grade so low
the teacher pulled me aside and asked if I should drop it. Holy crap that was
embarrassing.

After that I realized I needed to learn how to learn. I had to learn how to
take the book, break everything down, ask questions when necessary, and master
the material without guidance. It was a slow process but my scores got better
with every single test to the point I tied for the highest score on the final.

You could argue that I had to learn how to work but I don't think that's
accurate. I knew how to put in work rate. I was an Eagle scout and played on
the high school soccer team. Both required tons of work. Learning how to learn
certainly requires work and effort, but I think it's a skill the same as any
other.

~~~
tubbzor
I can't speak more to the 'learn how to learn' statement.

I have a very similar story (Eagle scout/football and wrestling team) and had
good marks in high school. I arrived at college and was destroyed by Calc 2
and having to learn at that pace when in high school I never studied at all.

After a rough freshman year I knew I had to change how I work and learn. My
dad handed me 'Moon-walking with Einstein' and I found calnewport.com (along
with other study blogs) and developed a study plan that I still use (I'm
currently a Junior).

------
peripetylabs
This person, Scott Young, did not "master" linear algebra. If anything, he
mastered the _curriculum_. There's a difference, and in a year he won't
remember a word of it.

He is, however, a master of self-marketing:

 _"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)."_

------
confluence
I'm inclined to actually believing a lot of this.

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.

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).

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.

~~~
gus_massa
(I'm a professor in a University, mostly for first year students.)

The problem with allowing the students to take all the courses they want is
that some of them can't correctly measure how much work they can handle. So
you will get horror stories like: "A student get enrolled in 3 times the
expected courses, has too many homework, too many midterms the same week, and
finally don't approve any of them."

Now we have a lost of students that come with questions like "I didn't approve
Calculus I last semester. Can I enroll in Algebra I, Calculus I and Calculus
II, so I don't lose a semester?" And it is hard to tell them, but most of the
times the best thing is to take only "Algebra I" and approve that than taking
the three classes and approving none of them.

I also understand that there are special cases, but should be evaluated case
by case.

~~~
confluence
Agreed. I'd like to note I literally have zero problems with either system
really.

I'm rather enjoying university right now and the lifestyle I have and am
perfectly fine taking the scenic route. I learn plenty on my own and basically
consider it a semi retirement stage :-)

~~~
drharris
I'm glad you feel that way. Those of us with jobs and families snicker at
college students who claim to be "so busy". Life is good at this time; do your
studies and use those massive amounts of free time to socialize and create new
things.

~~~
acuozzo
> do your studies and use those massive amounts of free time to socialize and
> create new things.

I never had massive amounts of free time at college. Maintaining a 3.9 in
Computer Science without having the same amount of raw talent (and previous
experience) as your peers is really hard, so I had to work my ass off. My
schedule was close to 16 hours per day, 7 days per week (including classes).
It was "Hell Week" for the ~4 years I was there. I worried my family and
friends. It was unhealthy.

Trying to keep up with, frankly, __smarter__ (and, most likely, more
intelligent) peers was a real challenge. I tried "Studying Smarter, not
Harder", but lacked the skills from K-12 education to do so. Attending classes
on "How to Study" and the like really didn't help and, no matter what I tried,
I always fell back into investing more time as a solution.

I lacked the bio-hardware (read: brains) to go into Computer Science. I did it
anyway. Don't forget about people like me!

> Life is good at this time

No, it wasn't. Not for me.

> Those of us with jobs and families snicker at college students who claim to
> be "so busy".

I don't. I have more free time now that I'm married and working (a stable and
well-paying job, thankfully).

------
navpatel
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 (<http://www.scotthyoung.com/mit/fourier.pdf>)
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

~~~
kghose
Yes, it was interesting to see it labelled out like that. I liked to think of
the transform as a cross-correlation. You take a wave of a particular
frequency (f) and cross-correlate it with your signal s(t) across all time.
The fourier coefficients are the results of the cross-correlation, telling you
how much of each frequency you have in your signal.

------
6ren
> 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.

This _is_ 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.

------
SeanDav
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.

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...

Amazing what focus and hard work can achieve!

------
alter8
> he completed all 33 courses (...) in less than one year.

> That works out to around 1 course every 1.5 weeks

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.

~~~
klibertp
"you see the pieces meshing together"

...or not. Depends on many things, among which the teachers are very
important. If your lecturer thinks that his class is the most important and
the rest of your courses are trash then you won't see "pieces meshing
together". Been there, done that.

------
infinitesimal
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.

------
nnq
...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 :)

------
wbhart
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_ "mastered" elementary linear algebra in about 10 days.

------
gbeeson
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.

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

~~~
Evbn
It is a marketing scam for a self-help eBook.

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

------
frozenport
I thought Linear Algebra was the easiest math-class?

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

