
On Becoming a Math Whiz: My Advice to a New MIT Student - da5e
http://calnewport.com/blog/2011/04/28/on-becoming-a-math-whiz-my-advice-to-a-new-mit-student/?utm_source=feedburner&utm_medium=feed&utm_campaign=Feed%3A+StudyHacks+%28Study+Hacks%29&utm_content=Google+Feedfetcher
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davidtgoldblatt
What I don't like about this sort of advice is that it's almost completely
non-actionable. "Don’t just sit and stare at it: think hard; until you’re
exhausted; then come back the next day and try again" is the piece of advice
to the stuck student. The distinction between "think hard" and "look at the
problem and wait for a flash of inspiration" isn't something that a beginning
math student will be able to see. A better set of advice for a stuck math
student might be:

\- Take a concrete case of the abstract (e.g. instead of trying to prove
cauchy-schwarz for inner product spaces, prove it for R^n, and generalize).

\- Strengthen your assumptions, then weaken them (prove something for finite
dimensional vector spaces, then weaken your conditions to include all vector
spaces).

\- Try to find a counter-example of what you're trying to prove, and figure
out why it's so difficult.

To me, posts like this (themed toward real analysis but parts of it generally
informative): [http://terrytao.wordpress.com/2010/10/21/245a-problem-
solvin...](http://terrytao.wordpress.com/2010/10/21/245a-problem-solving-
strategies/) are much more useful than posts that say "well, try hard".

~~~
billswift
I agree; _"Don’t just sit and stare at it: think hard; until you’re exhausted;
then come back the next day and try again."_ is bad advice. Don't just _think_
about it, whatever that means to you; just thinking tends to get people's
minds running in smaller and smaller circles-which is why going away and
coming back the next day helps. But thinking, and working, differently will
help you more and help you more quickly. Try diagramming the problem, try
working a simplified or more specific example, try generalizing the problem to
see if it fits something you learned in another context. There are many things
you can _do_ to work on a problem.

Polya's _How to Solve It_ is mostly simpler types of math, it is intended for
teacher training after all, but the general methods he demonstrates can often
help with much harder and more complicated problems.

Wickelgren's _How to Solve Mathematical Problems_ (originally titled, _How to
Solve Problems_ , the new title is more accurate) has both basic and more
advanced tactics for problem solving.

The best way I have found to use these types of books, after reading them
through quickly for an overview, is to stop and browse in one of them when you
get badly stumped on a problem. Then go back to the problem; if you still
can't make headway, stop and browse a bit more.

ADDED: Mathematics involves three distinct types of learning and work:
learning the mathematical theory, which I generally find fairly easy. Learning
and applying problem solving methods to apply theory to actual problems, which
is much harder. And doing the calculations to solve the problems once you have
worked out how to apply mathematics to the problem, which I find really,
really hard, fortunately this is the easiest aspect to automate (calculators
and Mathematica, for example).

~~~
joe_the_user
Good point.

I might not have been the best at math but I've sometimes been considered a
"math whiz" - I took the undergraduate math seminar at UCLA when I was a High
School senior.

I've always had the impression that what made people _bad_ at math is the
exactly the "grind" attitude - "focusing" on a problem only reduces your
creativity. In fact, whenever I took this attitude, I became bad too.

Playing with a given problem every way you can is good. Enjoying a problem and
finding it interesting is important. Grasping the concepts is good. Letting a
given problem go whenever you can't solve it is good - I think there's a place
below conscious awareness where problem solving can happen well.

But don't just grind on mathematics, that's poisonous.

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bodski
Aside from the learning the theory a lot of being good at math(s) is pattern
matching type stuff and the intuition that goes with it. I suppose this is
what the author is getting at, practice, practice....

 _"Young man, in mathematics you don't understand things. You just get used to
them."_ \- Jon von Neuman

One of the most reassuring quotes I know!

~~~
Entaroadun
How is that reassuring? If I'm learning something I don't want to just "get
used" to it. I want a fundamental understanding.

~~~
spinchange
It's reassuring precisely because it acknowledges that conundrum --
fundamental understanding often comes after a tough, rote (and uncertain)
'getting used to it.' -- so "stick with it," as it were.

~~~
Entaroadun
I agree with that, because thats what it takes to internalize something,
however the quote says understanding never comes. What I wanted to opine on is
the idea that you shouldn't be satisfied with not understanding something just
because someone tells you so.

~~~
bodski
You might be taking the quote too literally. I think what von Neumann was
getting at is that the abstract world of mathematics is essentially artificial
and alien to our natural sensibilities and that if you expect to 'understand'
something to its very core you may be misguided. The deeper you go the more
counter intuitive things can get.

It reassures me because it feels at times that the people around me who are so
'great' at mathematics are born naturals and that I may as well give up. For
me is a great 'leveller' to hear that even the greatest minds struggle with
these things, albeit to differing degrees.

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astrec
There's quite a nice quote on the subject from Terence Tao (Livingston, Sir
Ken; The Element; pp 100-101):

"I think the most important thing for developing an interest in mathematics is
to have the ability and the freedom to play with mathematics -- to set little
challenges for oneself, to devise little games, and so on. Having good mentors
was very important for me, because it gave me the chance to discuss these
sorts of mathematical recreations; the formal classroom environment is of
course best for learning theory and applications, and for appreciating the
subject as a whole, but it isn't a good place to learn how to experiment.

Perhaps one character trait which does help is the ability to focus, and
perhaps to be a little stubborn. If I learned something in class that I only
partly understood, I wasn't satisfied until I was able to work the whole thing
out; it would bother me that the explanation wasn't clicking together like it
should. So I'd often spend a lot of time on very simple things until I could
understand them backwards and forwards, which really helps when one then moves
on to more advanced parts of the subject.

I don't have any magical ability, I look at a problem, and it looks something
like one I've already done; I think maybe the idea that worked before will
work here. When nothing's working out then I think of a small trick that makes
it a little better, but still is not quite right. I play with the problem, and
after a while, I figure out what is going on. If I experiment enough, I get a
deeper understanding. It's not about being smart or even fast. It's like
climbing a cliff -- if you're very strong and quick and have a lot of rope, it
helps, but you need to devise a good route to get up there. Doing calculations
quickly and knowing a lot of facts are like a rock climber with strength,
quickness, and good tools; you still need a plan -- that's the hard part --
and you have to see the bigger picture."

------
benthumb
One of the most entertaining things I've read period, and it's by a
mathematician talking about his coming of age:

<http://www.xamuel.com/homeless-by-choice/>

~~~
orborde
This blog is amazing. Thanks for posting it.

~~~
benthumb
That's what I thought, too. Glad I'm not the only one who appreciates it...

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akdubya
Sounds like the Feynman algorithm:

1\. Write down the problem.

2\. Think real hard.

3\. Write down the solution.

~~~
lkozma
He had a more practical "algorithm" as well: "You have to keep a dozen of your
favorite problems constantly present in your mind, although by and large they
will lay in a dormant state. Every time you hear or read a new trick or a new
result, test it against each of your twelve problems to see whether it helps.
Every once in a while there will be a hit, and people will say, “How did he do
it? He must be a genius!”

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lkozma
Similar advice by Paul Halmos on reading maths: "Don't just read it; fight it!
Ask your own questions, look for your own examples, discover your own proofs.
Is the hypothesis necessary? Is the converse true? What happens in the
classical special case? What about the degenerate cases? Where does the proof
use the hypothesis?"

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xiaoma
To be impressive in a sea of impressive people like MIT, I don't think just
hard focused work is enough. It's also important to consider nurturing your
general cognitive abilities. Primarily, don't drink too much in college! For
an extra edge, get cardiovascular exercise regularly. It leads to higher
levels of neurogenesis.

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atakan_gurkan
from the article: "Junior graduate students think senior graduate students are
smarter, but they’re not: they simply have more practice.

Senior graduate students think junior professors are smarter, but they’re not:
they simply have more practice."

My M.S advisor is a really hard worker, so he has a lot of experience doing
research and is very good at it. For a while as an undergrad, I thought he was
not particularly smart, just hardworking. However, after I started working on
my first real problem as part of my thesis, I changed my mind. I realized that
to work hard you actually need to be smart (of course there are many other
factors related to time and money), otherwise it is very frustrating.

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sireat
More actionable advice would be to read George Polya's seminal work How to
Solve it.

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

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anonlawstudent
Sort of mirrors the 10,000 hours of practice theory.

~~~
vecter
In some sense, yes. If you read this post of his though:
[http://calnewport.com/blog/2008/11/25/case-study-how-i-
got-t...](http://calnewport.com/blog/2008/11/25/case-study-how-i-got-the-
highest-grade-in-my-discrete-math-class/) you get the sense that it's not only
about brute forcing hours, but also spending those hours wisely. In the case
of math, that was making sure he could understand and recreate every proof
perfectly. This was probably a lot more effective than spending those hours,
say, studying only problem sets or lecture notes.

~~~
kenjackson
The 10,000 hours theory though requires something called _Deliberate
Practice_. The theory notes the difference between simply doing something a
lot, versus doing something with the intent to get better, presumably with
some form of feedback.

~~~
Retric
The 10,000 hours theory relates more to the human lifespan than any specific
training method. If you can be really good at something at 25, then at most
you had ~15 years * 52 weeks per year * ~40 hours a week of practice = ~30,000
hours. Now, change that to _deliberate_ practice and your looking at around
10k hours. However, some things like go blow those numbers out of the water.
You can focus your life on the game a 8 and still be improving at 50.

~~~
kenjackson
Well the Dan Plan is looking to show that it is about training. But I should
also note that the theory isn't that you stop improving at 10,000 hours, but
that's how many hours it takes to be an expert.

Maybe Go is an outlier. I don't know enough about the game. But I'd be
surprised that someone who did 10k hours of deliberate practice wouldn't be
pretty good by most metrics of the Go community.

~~~
xiaoma
Go is likely similar to certain parts of language acquisition and musical
talent in that there's a critical period involved. If you wait until you're an
adult to start, not even 20k hours under a great teacher will give you real
mastery. AFIK, every top player started training as a child.

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chegra84
I just did all the questions in the Maths and Statistics book.(Guaranteed A+)

Avoid books that don't have practice exercises.

~~~
mg74
Which "Math and Statistics" book?

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da5e
I think what Newport is saying is that you have to work smart harder in order
to get smarter and more hard-working. It's after the hard work and learning
that the inspiration comes. The paradox is that if you want something to "come
to you" you have to go after it first.

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VB6_Foreverr
I am no math whiz but I did have one road to Damascus moment that helped me
feel less daunted. Proofs of theorems can be very intimidating. With a lot of
effort I might eventually be able to understand and reproduce a proof. However
I'd think that the mind that could come up with such a proof in the first
place must be orders of magnitude smarter as some of intermediate steps would
seem so unintuitive. As in what possessed him to try that route?

But what you have to remember that what you're seeing can be the result of
years of effort, trial and error that eventually gets tidied up into a
narrative that's analagous to sticking your arm into a haystack and picking
out the needle in one smooth action.

It would help people a lot if this was pointed out more by teachers I think.

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Maro
Being a math wiz really only pays off if you become a mathematician. Even a
physicist doesn't really have to be a math wiz. Eg. being good with people is
a much better thing to get good at and has a much wider payoff horizon.

~~~
jacobolus
(a) Why are these mutually exclusive? (b) “Math wizardry” in the context it’s
being used here means something like “able to think and focus in a deep and
prolonged way about the connections and patterns in numbers and structures”.
This is about as useful a skill as you can develop in a huge number of fields,
including most sciences and engineering disciplines, finance, architecture,
mechanical jobs like auto repair, many kinds of art and music projects, some
of the trickiest aspects of law and government, and so forth.

