Ask HN: learning Basic Math, Reading Math

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.

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.

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

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...

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.

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.

"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.

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.

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).

http://www.scientificblogging.com/carl_wieman/why_not_try_sc...

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.

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.

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