

Ask HN: How can I discover math? - HiroshiSan

So I've been trying to teach myself K-12 math using www.khanacademy.org now after reading http://www.maa.org/devlin/LockhartsLament.pdf I feel like I should be discovering math, playing with numbers asking more questions. The problem is that from up to this point in my life all I have been taught was to memorize a formula, plug in some numbers, get an answer. I want to learn math well, I want to enjoy the beauty of it, but I don't know where to begin that journey or how to start.
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plinkplonk
Get the book "Thinking Mathematically" ([http://www.amazon.com/Thinking-
Mathematically-J-Mason/dp/020...](http://www.amazon.com/Thinking-
Mathematically-J-Mason/dp/0201102382) _non_ affiliate link) which exposes the
process of how mathematicians think, using simple problems. You can then adapt
the process to your preferred level of problems. Also learn proof technique
(Velleman's How to Prove it" is a good book for this) and doesn't make any
assumptions of mathematical knowledge.

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samratjp
Check out this previous post: <http://news.ycombinator.com/item?id=1449799>

And also, I'd highly recommend art of problem solving.
<http://www.artofproblemsolving.com/>

Honestly, the best way to discover math is the Hacker's way i.e. find problems
to work on and then start working on them even in you don't know how
initially. The very digging in process will lead to further discovery and
further learning. Now, be sure to check out art of problem solving forums,
people ask questions there and there are very good detailed explanations
available.

Be also sure to check out Street Fighter math
([http://ocw.mit.edu/courses/mathematics/18-098-street-
fightin...](http://ocw.mit.edu/courses/mathematics/18-098-street-fighting-
mathematics-january-iap-2008/)) and fermi questions (e.g:
[http://mathforum.org/workshops/sum96/interdisc/classicfermi....](http://mathforum.org/workshops/sum96/interdisc/classicfermi.html))

And remember to print out these questions and carry them with you along with
tons of scrap paper. Sometimes, it helps just to read a problem and let it
sink in over days...

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gjm11
Soviet textbooks and the Princeton Companion (recommended by others) are
likely to be pretty brutal for you. I suggest problem-solving as a better way
in.

Take a look, e.g., at Project Euler (<http://www.projecteuler.net/>); find
some books aimed at K-12 students doing mathematical contests (there are lots;
what's best depends on how much you already know) and online resources aimed
at the same audience (e.g., <http://amc.maa.org/>,
<http://www.mathcomp.leeds.ac.uk/>); there are other sources of not-too-
routine mathematical problems online, such as <http://nrich.maths.org/>.

This pretty much guarantees that you'll be _doing_ as well as _passively
absorbing_ (one of many problems with which is that it's easy to think you're
absorbing when actually you're not). There's a very wide range of levels of
difficulty. And it's likely to be fun, if mathematics really suits you at all.

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mian2zi3
"Discovering math", depending on what you mean, could be setting the bar
pretty high. Memorizing formula is not what mathematics is about, but you're
unlikely to get very far if you just "play with numbers". To learn mathematics
and to think mathematically, you still need a structured environment:
Lockhart's complaint is that modern (K-12) math teachers neither have that
knowledge nor the ability to provide that structure.

If I were you, I'd go to where that knowledge does get taught: in
undergraduate mathematics classes. That usually starts with some rigorous
calculus (I second the recommendation for Spivak), real analysis (which might
seem hopelessly unmotivated if you don't know/remember much calculus), linear
algebra or abstract algebra, or discrete math. Depends on what you're
interested.

Reading mathematics is hard. Google around for advice. Do lots of problems,
which generally means finding proofs. This would very well fit the definition
of "discovering mathematics".

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CaptainMorgan
You asked about beauty and history, with an idea on how to discover...
personally, I started with "The Math Explorer: A Journey Through the Beauty of
Mathematics": (<http://www.amazon.com/gp/product/1591021375>). That book got
me hooked and from there I went on to pre-cal studies at university,
eventually completing sequences in discrete structures, probability, up to and
including calculus III, a level I never imagined I'd get to. That book was
literally my starting point since I never took high school math seriously nor
was I any good at that level. I think it'll serve you well considering the
level at which you're at, as it did me. I've reread that book four times to
date and still enjoy it.

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jonp
One book I'd recommend is "1089 and all that"
([http://www.amazon.com/1089-All-That-Journey-
Mathematics/dp/0...](http://www.amazon.com/1089-All-That-Journey-
Mathematics/dp/0198516231/)).

I've bought copies for friends and family whose maths backgrounds range from
school to degree level and they've all enjoyed it.

It's not a textbook, more a conversational account of a few topics the author
has found interesting through his life from a young boy to an Oxford don. It's
very readable and the author's delight in maths and problem solving shine
through

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joubert
Get "The Princeton Companion to Mathematics" -
<http://press.princeton.edu/titles/8350.html>

~~~
omaranto
Seriously? Your reply to "I'm trying to teach myself K-12 math" is "find out
what mathematicians think about, instead"? My guess: you are trolling or have
never read the Princeton Companion.

~~~
joubert
I'm suggesting it because, whatever _practical_ tutorials the poster is
looking for, the Companion will make for fascinating reading.

When I was in school and plodding through physics and chemistry class, I was
wildly excited by other, more advanced books, which I devoured. Ditto for
biology.

I wish the Companion was out then, for I would have loved it (although, my
love for mathematics wasn't in any way diminished).

Sure, the book is _not_ going to teach you K-12 stuff, but it is almost like
standing on the edge of mountain, giving you a vista of the wonderful world of
math.

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fmstephe
It will only teach you a small bit of math (i.e. Trigonometry) but try
programming a little 2D rocket ship game with some physics and movement etc.
It can be fun and you will get a better feel for the maths involved. don't use
libraries. Use javascript with canvas - easy as pie.

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sbe
Find a problem that piques your interest, then learn the math that's relevant
to the problem. You can choose a problem from most any subject, like math,
physics, or economics.

Having a problem that piques your interest--something to work toward--is a
good motivator.

~~~
HiroshiSan
How would I go about finding said problem? Sorry if this seems stupid. There
is no real problem that piques my interest right now perhaps because I don't
really know the field all too well. I just want to enjoy learning math and get
better at solving problems.

~~~
andresmh
Actually, this is what the book "Who Is Fourier?: A Mathematical Adventure" is
about. It's the adventure of a group of families who wanted to learn more
about languages, which led them to sounds which led them to learn about
Fourier transforms. It's a lot of fun and it actually has real math that you
often get to learn in a signal processing class in college. While it is very
focused, this book and the approach will pique your interest in other topics.

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tmsh
Might be a nice time for GEB. There's even a K-12-ish course for it at MIT
OCW:

[http://ocw.mit.edu/high-school/courses/godel-escher-
bach/ind...](http://ocw.mit.edu/high-school/courses/godel-escher-
bach/index.htm)

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themullet
The one that made me truely see the beauty of maths: Hallucinogenic drugs -
due to the hallucinations basis in geometric shapes and patterns.

Not really useful for teaching yourself K-12 math however for the beauty of it
nothing better

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chipsy
I recommend seeking out translated Soviet math textbooks. They're dense and
unforgiving, but are conceptually correct. You could do far worse.

~~~
HiroshiSan
Do you have any specific textbooks I could take a look at?

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GeneralMaximus
Hijacking the thread here, but could someone quickly recommend a college-level
calculus book?

~~~
jimmyjim
Spivak's Calculus, [http://www.amazon.com/Calculus-4th-Michael-
Spivak/dp/0914098...](http://www.amazon.com/Calculus-4th-Michael-
Spivak/dp/0914098918/ref=ntt_at_ep_dpi_1) is a good bet. But it's very theory-
involved.

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yoshiks
let's start with "how to solve it" (g. polya), if you love to learn the
thinking of math.

