
How to Ace Calculus: The Art of Doing Well in Technical Courses - motxilo
http://calnewport.com/blog/2008/11/14/how-to-ace-calculus-the-art-of-doing-well-in-technical-courses/
======
jerf
This generalizes to formal computer science as a programmer, and is probably
one of the best ways to put it I've ever seen. If you understand computer
science as the formula, you'll think it's pretty useless. If you get the
concepts, you start seeing how it is useful everywhere.

I do not sit down and prove my designs and code or use lots of tricky
algorithms, but I use a lot of the insights and ways of thinking I picked up
from the computer science concepts, thinking about invariants and the
maintenance of them, etc. There's few things sadder than sitting through four
years of school and coming out seriously thinking that it's all useless
wankery against the importance of "REAL PROGRAMMING".

(I've also noticed/learned that when you do a good solid job of designing your
system with strong foundational concepts, the system will talk to you as you
try to design it. I just got back from talking to a coworker about a case
where I need to bypass my permissions system and temporarily become a
superuser in order to do this particular thing, and I realized that rather
than that being "the solution", that was actually my permission system telling
me that I was doing something wrong. Only after I realized that did I reflect
on it for a moment and realize the permission system was right and I was
trying to do something potentially dangerous. I had thought about the thing I
wanted to do but didn't fully consider how it might be exploited. We may still
do it, we may not, but either way, listening to the code taught me something
important about my system. You don't get these insights when you're too busy
with your REAL PROGRAMMING and turning out mushy, concept-less code. You just
write the flaw in and let your customers or hackers find it.)

------
RiderOfGiraffes
There's something that I think most people are missing, although many of you
will already know this.

People are saying that you need to develop the intuition, to develop the
visualization skills, to develop the sense of what's happening rather than
simply memorizing the formulas.

But to me, the visualization is not the point. To me, the sense of what's
happening based on the visualization is not the point. To me, the point is the
richness of understanding, the combination of many ways of thinking.

This doesn't come without effort.

The lunk-to item seems to suggest that by having the picture in mind one can
avoid all the tedium of remembering the epsilon-delta limit arguments and can
avoid the definition of lim_{e->0}(f(x+e)-f(x))/e and so on, but that's not
true. The point is that the formula is tied up with the image, not that one
subsumes the other.

Allegedly Euclid said King Ptolemy (in response to a request for an easier way
of learning mathematics) that "there is no Royal Road to geometry".[1]
Likewise there is no "Royal Road" to a mastery of calculus. Or indeed, to a
mastery of _any_ subject. That which can be mastered with little effort has
long been surpassed, and work is required to gain the depth and breadth
required to make these things easy.

But we do these things "not because they are easy, but because they are
hard."[2] They are of value, and developing the mastery is satisfying in its
own right, but also makes you a rare commodity.

[1]
[http://en.wikipedia.org/wiki/Royal_Road#Cultural_references_...](http://en.wikipedia.org/wiki/Royal_Road#Cultural_references_to_the_Royal_Road)

[2] <http://er.jsc.nasa.gov/seh/ricetalk.htm>

~~~
lacker
You can certainly avoid all the tedium of the epsilon-delta arguments while
mastering calculus. In fact there were over a hundred years between the
invention of calculus and the formalization of the epsilon-delta definition!
Calculus was not invented in the same way it is taught, and the early masters
of calculus were much more intuitive than rigorous.

For references, calculus was first published in 1684:

<http://en.wikipedia.org/wiki/History_of_calculus>

But the epsilon-delta definition wasn't formalized until 1817:

[http://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of_...](http://en.wikipedia.org/wiki/\(%CE%B5,_%CE%B4\)-definition_of_limit)

~~~
pfedor
You could perhaps try to learn calculus the way it was taught three hundred
years ago, but I'm pretty sure it would be more difficult, not easier, than
the standard presentation. In the early years calculus was considered black
magic only the smartest few could master, probably not unlike the way, say,
String Theory is seen today.

------
fuzzmeister
"They switch to a philosophy major."

This comment seems odd to me, as philosophy is another subject where a failure
to sit down and really think about the topics at hand will leave you
hopelessly lost, if the class is taught to any degree of rigor.

~~~
hdctambien
Both of the CS majors I knew that switched majors switched to Philosophy.

I never asked why. They both were having trouble in the Intro Algorithms class
when they bounced.

Maybe people that think they are CS/Math people and then find out that they
might not be CS/Math people find solace in Philosophy? Maybe what they liked
about CS/Math was the logic more than the algorithms.

~~~
Apocryphon
Perhaps that way they can still get into A.I., except without having to learn
LISP. I wonder if there are any ex-CS people who similarly go into
Linguistics.

~~~
bane
"I wonder if there are any ex-CS people who similarly go into Linguistics."

It's surprisingly common in my personal experience.

------
Paulomus
This rings true for me. Certainly the courses I struggled in were ones where I
had a hard time getting a grip on the concepts. The best lecturers helped by
teaching in a way that made the key concepts clear and showed how they
developed from previous concepts taught.

------
nubela
As a chinese who is surprisingly good with technical courses and math, and
also a computer science major who is somewhat not that bad with programming:

Learn to see patterns. Math is all about patterns. Get obsessive. I just got
out of an obssessive period (3 days, to the point I didn't wanna talk to
anyone) where I couldn't solve problems. Visualise problems in your head, put
the entire problem domain into your head, lie on the bed. Solve it YOURSELF.

Always, always, solve it yourself, and only ask when you have TRIED AND TRIED.
Then when you finally ask and get the solution, you'll remember it for life.

Pattern, and self-attempting. Practise makes perfect too.

~~~
motxilo
Free idiom upgrade: Perfect practice makes perfect.

~~~
RickHull
Or: Practice makes permanent. (Inject qualifiers as needed)

------
billswift
I have to disagree, at least somewhat. Insight, understanding the basic
concepts has always come fairly easily to me. The REALLY hard part is the
disciplined practice needed to be able to actually apply what you know to real
problems.

~~~
swah
But if you haven't seen how it maps into real problems, was there any insight
at all?

------
mlinsey
This is why I did terribly in quantum mechanics; I was never able to generate
a whiff of insight about anything that was going on. The thing is, I'm not
sure the top students really did either.

~~~
discreteevent
Like everything there is a balance. I think that if I could go back in time
and give myself one piece of advice it would be: Don't try to gain insight
into everything. There isn't time. Its fine to do this for fundamental things
like calculus but for laplace transforms, fourier series etc it can take too
long given the time that a college student has. If it is your profession then
that is a different story. Also for things that are sufficiently alien to our
everyday experience (like relativity) I think maybe you just have to accept it
and,work with it for a while and then possibly try and look for insight after
because initially we have no other intuition to relate it to.

~~~
3pt14159
I wholly disagree. I don't buy the OP's argument for _tactically how_ to learn
one bit, but knowing concepts for everything you "learned" is the only way to
do it. After first year, I almost never had to work hard, I could just derive
anything I needed, and yeah, this included using fourier series and laplace
transforms.

To me the key is to observe that concepts are important then observe that the
easiest way to learn concepts is to learn them at the optimal time of day (ie,
not 8 am during the class when you are tired, that is a waste of time), from
the optimal person (ie, not your prof that just wants to get back to
research), during the optimal time of the term (ie, not in the first week of
class, more like the week before the midterm and the 10 days before the
final). To achieve all of this just requires two skills: 1. Knowing how to
learn from a text book 2. Knowing when a textbook is crap and getting a better
one from internet review sites.

Learning from a text book is easy. Cover the page with a piece of paper and
read each line. When they come up with a problem that you don't have a
function for _derive it_ and bam, you've invented the formula for lateral-
torsional buckling of non-uniform crossectional beams you will never have
trouble with the concept again. If you get stuck (stuck to the point of it
hurting your ego, not "I'm sure I would get it if I had the time" stuck then
look at the formula (not the proof if you can avoid it). Try to prove it
again! If you still can't prove it, find the proof somewhere (hopefully the
text, but if not email your prof or the book author for the proof). I
corrected the same (otherwise super awesome) text book 3 times over the course
of two years. The author loved me because out of the 5 corrections he did 1
was from himself, 1 was an email that said "I think this is wrong" and the
other 3 were from me _proving_ that he was wrong.

Anyways got kind of long there, but don't waste time learning from other
people, just learn how to learn and damn well get the concepts otherwise
what's the point?

------
pkananen
Why is the tendency to introduct concepts, and often leave out the insight?
Why don't professors start with insight and generalize concepts?

~~~
jonsen
A good teacher is a master of _two_ subjects. The subject of study _and_
pedagogics. A PhD in the subject of study and maybe one course of pedagogics
will necessarily skew things a bit.

~~~
motxilo
I would dare to say +95% of the teachers I had up until college were downright
mediocre because they lacked one single paramount side of outstanding
teachers: p-a-s-s-i-o-n. Sometimes this alone has the potential to drive one's
pedagogical skills.

~~~
nlawalker
I think most teachers would say the same thing about the students they've
taught. It's pretty hard to impart insight and understanding to someone who
doesn't care and doesn't want to.

~~~
motxilo
A passionate teacher is likely going to instill greater interest in a large
number of students than that of one who does not convey the slightest amount
of passion, dooming the poor disciples to such an ordeal -including the
brightest ones.

------
Fargren
Understand the topic and practice a lot? Well, yeah. That is good as an
objective, but what's missing is a way to get to the point where you actually
understand the thing.

~~~
hammock
"Insight" as used by the author is just a fancy (and misused) word for
understanding. "To succeed you can't just memorize the formula; you have to
understand how it works." Is that really such an insightful thing to conclude?

The most valuable part of the article for me is where he points out that a lot
of hard-working but unintelligent students write copious notes without ever
doing the mental gymnastics to understand what it is they're writing down.

------
tokenadult
Cal Newport, the author of the submitted blog post, draws comments both here
on HN and on his own blog pointing out that deep understanding of a subject
doesn't necessarily equate to VISUAL thinking about a subject. There is a big
literature on "learning styles" and some attempts by some schoolteachers to
categorize children by what their preferred learning styles are. When I have
taken learning style questionnaires, and when I have asked my wife (a piano
performance major and private music teacher) about this, the answer on
learning styles is "all of the above." I personally think, based on my
observations of successful learners of a variety of subjects, that learning
styles are themselves learnable, and a learner with a deep knowledge of a
particular subject will know multiple representations of that subject. My wife
has had many piano performance courses, and also music theory and ear training
courses, and has learned visual representations of music both in the form of
standard musical notation and in the form of "music mapping,"

[http://www.amazon.com/Mapping-Music-Learning-Teachers-
Studen...](http://www.amazon.com/Mapping-Music-Learning-Teachers-
Students/dp/0895793970)

which she has found very helpful.

As for mathematics, the subject I teach now, I have always cherished visual
representations of mathematical concepts, for example those found in W. W.
Sawyer's book Vision in Elementary Mathematics

[http://www.amazon.com/Vision-Elementary-Mathematics-W-
Sawyer...](http://www.amazon.com/Vision-Elementary-Mathematics-W-
Sawyer/dp/048642555X)

[http://www.marco-
learningsystems.com/pages/sawyer/Vision_in_...](http://www.marco-
learningsystems.com/pages/sawyer/Vision_in_Elementary_Mathematics.pdf)

But other mathematicians who taught higher mathematics, for example Serge
Lang, recommended memorizing some patterns of multiplying polynomials by oral
recitation, just like reciting a poem.

[http://www.amazon.com/Basic-Mathematics-Serge-
Lang/dp/038796...](http://www.amazon.com/Basic-Mathematics-Serge-
Lang/dp/0387967877)

The acclaimed books on Calculus by Michael Spivak

[http://www.amazon.com/Calculus-4th-Michael-
Spivak/dp/0914098...](http://www.amazon.com/Calculus-4th-Michael-
Spivak/dp/0914098918/)

and Tom Apostol

[http://www.amazon.com/Calculus-Vol-One-Variable-
Introduction...](http://www.amazon.com/Calculus-Vol-One-Variable-Introduction-
Algebra/dp/0471000051/)

are acclaimed in large part because they use both well-chosen diagrams and
meticulously rewritten words to deepen a student's acquaintance with calculus,
related elementary calculus concepts to the more advanced concepts of real
analysis.

Chinese-language textbooks about elementary mathematics for advanced learners,
of which I have many at home, take care to introduce multiple representations
of all mathematical concepts. The brilliant book Knowing and Teaching
Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in
China and the United States by Liping Ma

[http://www.amazon.com/Knowing-Teaching-Elementary-
Mathematic...](http://www.amazon.com/Knowing-Teaching-Elementary-Mathematics-
Understanding/dp/0415873843/)

demonstrates with cogent examples just what a "profound understanding of
fundamental mathematics" means, and how few American teachers have that
understanding.

<http://www.aft.org/pdfs/americaneducator/fall1999/amed1.pdf>

<http://www.ams.org/notices/199908/rev-howe.pdf>

Elementary school teachers having a poor grasp of mathematics and thus not
helping their pupils prepare for more advanced study of mathematics continues
to be an ongoing problem in the United States.

<http://www.ams.org/notices/200502/fea-kenschaft.pdf>

In light of recent HN threads about Khan Academy,

<http://news.ycombinator.com/item?id=2348476>

<http://news.ycombinator.com/item?id=2350430>

I wonder what Khan Academy users who also have read the submitted blog post by
Cal Newport think about how well students using Khan Academy as a learning
tool can follow Newport's advice to gain insight into a subject. Is Khan
Academy enough, or does it need to be supplemented with something else?

~~~
Radix
I think you make an important point, but I'm not sure the post takes any
particular focus on visual representation other than a graph is generally an
easier way to intuit what a derivative is. His repeated use of the word
concept suggests insight for him requires a more general abstraction.

As an aside, why do or did people claim there is visual learning aside from
spatial learning? I don't experience visual and spatial imagination as
different things. (With reasoning about time always assumed.)

~~~
pfedor
_[...] other than a graph is generally an easier way to intuit what a
derivative is._

Even that is a matter of personal preference. I honestly believe it's easier
to get the concept of a derivative by linking it to instantenous velocity.

~~~
psykotic
There are many different ways of thinking about mathematical concepts like
derivatives. The more you know, the more deeply you know them, the better.

Here's a random example: Marsden and Weinstein define derivatives in their
out-of-print textbook Calculus Unlimited without limits. The tangent to a
graph at the point x is the boundary between two line pencils, one of lines
entering the epigraph at x, the other of lines leaving. There's no limit-
taking of chords. It's a simple and neat definition that connects with
classical notions of tangency.

In his essay On Proof and Progress in Mathematics, Thurston lists a dozen
other definitions or conceptions of derivatives in his personal arsenal, some
very sophisticated. But even those among his definitions that are elementary
and have roughly the same scope there is a difference in their psychological
affordances, and that can make all the difference.

------
Tycho
Good article but I don't think it's the whole story. Works well for maths,
probably, where in my experience everything is obfuscated by a) lack of any
practical real-life application for most people and b) non-verbal symbolic
representations of concepts.

With programming you very likely do want to apply the stuff you're learning to
'real life' problems, and you're going to be expressing all your efforts in a
programming language with familiar keywords (and just a few
symbols/operators). Here the problem is not intuiting what the purpose is -
it's easy to explain what Ajax calls are supposed to do, for instance, but
actually implementing them is quite bitty. You need to set up a sort of chain
of connections between multiple points, and not until you've learnt all the
details of this process, can you tuck it all away neatly under one abstraction
and free up brain cycles to deal with higher problems. I find more and more
that when I learn a new corner of programming, there's just inevitably going
to be a certain number of hours of faffing about learning the details before
it 'clicks.' You feel stupid for a week or so, the boom You Know Kung-Fu, like
it was easy all along.

Having said that some students just really struggle with basic concepts like
'variables' and need to make sure they intuitively grasp them. But that's
about _passing_ , not getting straight As.

------
ionfish

        [T]he students who struggle in technical courses are those who skip
        the insight-developing phase. They capture concepts in their notes
        and they study by reproducing their notes. Then, when they sit down
        for the exam and are faced with problems that apply the ideas in
        novel ways, they have no idea what to do. They panic. They do
        poorly. They proclaim that they are “not math people.” They switch
        to a philosophy major.
    

This may well be true, but if it is, these students are setting themselves up
for a fall: if they wish to be any good at all at philosophy then they will
need to cultivate precisely this skill. Much of philosophy consists of taking
a general set of tools (concepts) and applying them to different situations.
It isn't terribly fruitful approach unless one understands those concepts in
the first place.

------
btilly
What is described should be effective, but seems like too much work to me. I
prefer the advice I offered in <http://www.perlmonks.org/?node_id=70113>
instead. Besides, you get the benefit of looking like a really lazy genius.

------
Confusion
This article is devoid of interesting content. The central thesis is: "In
order to understand something, you need to develop _insight_. Well, duh. It
does not explain how to develop 'insight', it does not explain what
constitutes 'insight' and it does not even justify that the main example,
visual representation of a derivative, results in 'insight' and results in
people having an easier time grasping and using derivatives. As I am skeptical
the example actually grants that 'insight' (it may seem obvious to you and me,
but that is the trap we must avoid. The point is that it _isn't_ obvious to
many), this article did not add anything to my understanding of 'teaching' or
'understanding' at all.

------
tumanian
The article is correct in the insight part, however it misses(or doesn't state
explicitly) the fact that insight comes from the definition of the problem,
what you are trying to do. Every concept is a solution to a problem, and to
get the insight, my approach is to go through a list _What am I trying to
accomplish (this is where visualizing comes)_ What are the other ways of doing
it _How does this method work, and why_ Why is this method better then others
*Where will this method not work.

After this solving any problem in the problem set is a piece of cake. Reduce
the problem to subproblems, check applicability of the concept to the
subproblems, apply the method, enjoy the result.

------
crasshopper
Like most study tips, these do not suit me.

From what I know of Cal Newport, I expected data -- evidence that these
techniques work better than X. I thought Dr. Newport might even say X works
better for group A and Y works better for group B.

------
sb
That is exactly the reason why studying technical subjects is such a joy! Once
you grasp a concept, you don't need to learn anything by heart (which I suck
tremenduously at.)

------
gohat
This article is very helpful, but in all honesty, I think the formula is
slightly more valuable than the picture. They're both essential, of course,
but the formula is really the key to understanding the concept.

This is said, of course, from the vantage point of someone who has studied
calc with 3 variables and so on, so may not have the fresh perspective.

------
hessenwolf
Erm... how to do well in technical courses - understand it. Profundity
squared. Not.

Author is missing the fact that yes, you need to understand it, but you also
need to practice the exams. The further into your university years you get,
the more your basis in understanding becomes valuable, but you still need to
practice the exams.

------
ezyang
A friend and I like to put it this way: after you learn information, you need
to _compress_ it. Find a way to make it take up as little brain space as
possible. Then you're more likely to retain it, to understand it, to be able
to use it.

------
haploid
This is all well and good, until the day when you decide to learn something on
your own from a book, and encounter concepts defined in epsilon-delta form
that make zero sense to a mind trained on intuiting concepts from graphs.

Visualization will not get you beyond 3 dimensions, nor will it get you
understanding systems in terms of Lagrangians/Hamiltonians, nor will it give
you the ability to read texts geared toward actual mathematicians.

Speaking for myself, it was surprisingly difficult un-learning the "slope of a
tangent line" type of conceptualizing in order to understand math with
sufficient rigor to be able to actually read math texts correctly.

~~~
mturmon
Yes, the "develop a visual analog" approach will not be effective if you spend
all your time translating back and forth between the linguistic abstraction
(for all delta, there is a small enough epsilon such that...) and your visual
analog. For example, I just checked, and Baby Rudin
([http://www.amazon.com/Principles-Mathematical-Analysis-
Third...](http://www.amazon.com/Principles-Mathematical-Analysis-Third-
Walter/dp/007054235X) ) does not contain a single picture or line drawing.

Additionally, some things like ordinary algebraic manipulation are very well-
suited to linguistic abstractions ("multiply the polynomials, take the
derivative, put all terms involving _z_ on one side of the equation, apply the
quadratic formula"). Sometimes only the linguistic abstraction can give the
solution (e.g., "this problem is easy because the quadratic coefficient
cancels out, and the equation is in the form t^3 + c t = d").

It's also worth noting that manipulating the linguistic abstractions takes a
lot of insight and talent (e.g., knowing the perfect substitution of variable
to make an integral fall into a known form, or knowing which one of the four
error terms will be hard to control, and working on it first).

It's not wise to be over-committed to the visual approach.

~~~
wnewman
I have generally found it very helpful to spend a lot of time understanding
the behavior simple concrete cases, and understanding how a general
mathematical principle applies to them. The visualizable low-dimensional case
of analytic geometry is a particularly flexible concrete case for
understanding many principles of calculus and linear algebra. I appreciate
fairly well how it breaks down: I went on to path integrals and other quantum
stuff where the number of dimensions is much larger than three. But the
understanding from 1-3 dimensions was very helpful. More generally, one can
make a habit of thinking about how general principles apply to special
concrete cases that you understand. When trying to understand group
theoretical theorems, you can check how they apply to your favorite concrete
groups. When trying to understand conservation of angular momentum, or the
Bohr correspondence principle, you can cross-check your understanding of them
with what you know about the behavior of hydrogen atoms and balls rolling off
the edges of tables and so forth. And probably many readers here will have
naturally tried thinking about how a nontrivial algorithm would work on some
simple concrete data set.

The advantages of thinking this way seem to be a little like the advantages of
test-driven development: time spent understanding representative concrete
cases doesn't teach you everything, but it can eliminate many
misunderstandings very quickly.

