
Ask HN: learning Basic Math, Reading Math - justlearning
Lately after couple of posts including the latest http://news.ycombinator.com/item?id=666407 , I gathered the courage to ask this:<p>problem description :  I want to finish "Introductions to Algorithms" (Cormen et al) and the math is holding me back. I have been trying and am too impatient. Previous comments from (do you buy books for long term) and a couple more helped me understand that this is normal to spend couple of hours on a single paragraph/page. But sometimes I don't get it for the lack of basics (it's been a while away from academics). I know I should work hard at going back, but however back I go, it feels like I don't know it. the longer it takes, more depressing it is. Sometimes, I feel my whole life is going to be depressed in this pursuit.<p>My problem in one line: I want to READ Maths like I read english (when i read the word mountain- i visualize a mountain, when i read an equation i see greek letters with an equals sign... you get the idea?)<p>my self analysis - too impatient, want to code  the pseudo-code eagerly, hate (or lazy?) to take the pen/paper, look for solutions without trying hard = typing/reading the solution rather than understanding, have no big picture of implementing math - project to work on (people reading math that helps in their work vs me doing math with no implementation makes my brain go to sleep.)<p>I have been trying to get into about math for a while. I have done my research and read few excellent blogs (including from atleast one member of HN)  for know-hows.
Books I know I should have to cover the basics:<p><pre><code>  *Discrete Mathematics with Applications  by Susanna S. Epp
  *How to prove it?
  *Concrete Mathematics - Knuth 
</code></pre>
(I got along well with Knuth for the first chapter and after a while my lack of basics &#62;&#62; depression &#62;&#62; stop and do something else.)<p>Which books have helped you?  which one should I pick first? How can I achieve my goal?<p>I need books basic enough to teach me discrete math (to achieve my goal of finishing introduction to algorithms) and in the long run read math like english.<p>I am looking for your experiences - obstacles you faced... something like this [ http://news.ycombinator.com/item?id=666615 ; i can't relate to few points - eg: not qualified to decide if a book is badly written]- anything i can relate to, any anecdotes? perhaps..my sub conscious will probably hold back my depression!  Everyone of you have truly enchanting stories.
I know I am taking your time, but I HAVE to  do this.<p>disclaimer: i am not a comp sci student. &#60;cough&#62;my experience has been as an enterprise developer&#60;/cough&#62;. pls don't be hard on me.<p>...and thanks for your precious time
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yequalsx
I agree with RiderOfGiraffes. You must do. Reading alone won't do the trick
but I'm sure you know this. I have found that there are two types of
mathematicians.

1\. Can not read a theorem without knowing every little detail.

2\. Can read a theorem, accept that it is true without knowing every little
detail and being able to justify every little detail.

You sound like type 1. For type 1 person I recommend taking courses and asking
the teacher lots of questions. I am type 1 and it took me a long time to
really understand how to read a mathematics book. I learned this in graduate
school because my teachers mostly sucked at teaching.

Good luck.

~~~
yequalsx
Let me add that one way to become a type 2 person is to play a game with a
theorem. Say you encounter something you don't really understand. The book
says it is true but you don't know why it is true. Forget about why it is true
and see if you can do problems assuming it is true. Say to yourself, "Provided
this is true what must logically follow from it?" See if you can use the
result even though you don't know why it is true or where it came from.

~~~
justlearning
thanks, good to know (both about types) and to get over to type 2.

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vomjom
Frankly, CLRS goes over a lot of the basic knowledge you need to get through
the book. I don't think a discrete math textbook will help you if your main
goal is to get through CLRS; the math in that book isn't that dense. Like all
things, it just takes practice. Spend a lot of time on sections that are hard
to understand, and future sections will become a lot faster and easier.

Try the MIT algorithms video lectures: [http://ocw.mit.edu/OcwWeb/Electrical-
Engineering-and-Compute...](http://ocw.mit.edu/OcwWeb/Electrical-Engineering-
and-Computer-Science/6-046JFall-2005/VideoLectures/index.htm)

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Arun2009
This is a topic I'm really interested in, and I _do_ sympathize with your
situation. I have no advice for you, but here's my experience.

I hold a Masters in EE (control theory). I have never really found studying
Mathematics formidably intimidating, probably because I've been at it for some
time now.

But I totally suck at the following things:

\- Solving problems that require really inspired "crux" moves. Think IMO level
problems, lighter versions of Euler's solution to the Basel problem
(<http://en.wikipedia.org/wiki/Basel_problem>) etc.

\- Coming up with abstractions and new Mathematics. Think (again) Euler's
invention of graph theory or Cantor's invention of Set theory.

\- Recalling what I've learned just months after their immediate use is over.
For example, I had to cover quite a bit of math for my BEng. But if you ask me
now about real analysis, PDEs, stochastic processes or fourier transforms, I'd
be stumped.

I strongly suspect that this is because we haven't analyzed and understood the
problem of learning Mathematics well. To give you an example, 200 years ago
people were trying to get fit. But it was not until the invention of
systematic strength training with free weights and machines, etc., that we
could really do a reasonably good job of it efficiently. Early body builders
look like sticks compared to monsters of today.

I am hoping that such a revolution happens in education as well. Before we
learn, we must really learn how to learn. Our knowledge of that is very
limited now.

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michael_dorfman
First, regarding CLRS: have you looked at the videos of Leiserson & Demaine's
MIT course on the subject? Watching video lectures might get through where the
book doesn't.

I was going to recommend the Knuth book, so I'm trying to better understand
your problem. Where, in the first chapter, did your "lack of basics" manifest?
If you spell it out in a bit more detail, perhaps we can make more specific
recommendations.

Needless to say, it helps to make sure your expectations are reasonable.
"Reading Maths like you read English" is probably never going to happen--
every mathematician I know can knock through a novel in half the time it takes
to read the equivalent number of pages of a mathematics book, and that's in
the best case.

~~~
justlearning
"Where, in the first chapter, did your "lack of basics" manifest?" - in the
towers of hanoi, there are several scattered equations(though Knuth starts
with basics) with mathematical "syntax" that makes me impatient. What makes me
'hallucinated' is that even after I read the clear explanations of Knuth, I
can't relate to the math equation just below the para.

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psyklic
To start off, I recommend that you get more practical-oriented books which
contain lots of exercises. There are actually some books which try to explore
discrete math using C or Mathematica/Maple. Those might make the subject a bit
more interesting and concrete.

I've found that (at least with computer books), for me the mindless visual
quick-start texts are best to begin with. They make it more fun to explore the
material even though the books have less "meat" for your buck. Even though I
am advanced enough to sit down with a manual and understand it, these books
are much more motivating for me.

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RiderOfGiraffes
One person's opinion ...

You will never read math like English. I'm a Ph.D., I've had this discussion
with many other graduates and post-graduates. Math is different from English
(or any other natural language).

In my circles of math, programming and education there is even a phrase "Read
like Math" as opposed to "Read like a comic book". Math is its own language,
and it is dense. When interpreted into English a single equation can become
several paragraphs. Learning to read and visualize an equation is possible,
but always necessary.

Simple equations are easy, but if there's real content, it's hard. You need to
ask what you're trying to accomplish, really.

If you want to learn stuff, you need to do it. You can learn about it by
reading about it, but you can only learn to do it by doing it (under guidance
or with hints).

See also:

<http://news.ycombinator.com/item?id=672067>

[http://www.scientificblogging.com/carl_wieman/why_not_try_sc...](http://www.scientificblogging.com/carl_wieman/why_not_try_scientific_approach_science_education)

EDIT: added reference: <http://c2.com/cgi/wiki?MathForProgrammers>

Quoting from the this last:

    
    
        I'm not certain that reading like prose is a benefit. 
    
        When algebra was first invented, people didn't use
        variables, operators or other notation. They just
        wrote things out in natural language. Here is an
        example from Al-Khwarizmi's famous book: 
    
        If the instance be, 'ten and thing to be multiplied
        by thing less ten,' then this is the same as 'if it
        were said thing and ten by thing less ten. You say,
        therefore, thing multiplied by thing is a square
        positive; and ten by thing is ten things positive;
        and minus ten by thing is ten things negative. You
        now remove the positive by the negative, then there
        only remains a square. Minus ten multiplied by ten
        is a hundred, to be subtracted from the square. This,
        therefore, altogether, is a square less a hundred
        dirhems. 
    
        Compare this to
            (x+10)*(x-10) = x*x + 10*x - 10*x - 10*10
                          = x*x - 100 
    
        Now imagine trying to do even simple calculus like
        this. If there's a lot of essential complexity, I
        claim that natural language becomes much harder to
        understand than specialized notation, which is why
        notation exists in the first place. A good notation
        is understood on its own terms, not in terms of
        translation to some natural language statement, for
        instance, algebraic manipulations are done according
        to certain rules, without having to re-justify them
        each time, because of the underlying soundness of
        algebra.

~~~
justlearning
Thanks, How then could you make math as lucid as possible to understand.

"Learning to read and visualize an equation is possible" - any tips on how?

sorry for the late reply - i wake up to the chinese sun.

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papaf
Another online course that starts with the basics:
<http://math.ucsd.edu/~ebender/DiscreteText1/>

