When ever I've run up against "impenetrable" math I often ask "So how would you use this?" and connecting it to the real world helped tremendously.
The problem isn't the notation per se, it's that teachers don't spend nearly enough time explaining the notation itself. It's a foreign language that they are so skilled at that they don't understand how unfamiliar it is to their students.
I was well into a physics major before I really stopped and carefully considered the many different notations used to represent derivatives (dx/dy, f'(x), y', y-dot, Dsub-yX, etc.) I realized that I had developed separate context-specific bodies of calculus knowledge/skill from different fields with different notations and approaches, and that these were all the same thing. Different techniques in different contexts were an artifact of different styles of notation, not differences in the actual math.
I can understand Italian to some extent as a side-effect of my study of Spanish. If I took an electronics class in Italian, I would understand some of the concepts and misunderstand others. Would my troubles to understand certain electronics concepts be due to trouble with electronics or trouble with Italian? Who knows? Both types of misunderstanding would compound each other.
If teachers spent more time carefully teaching this foreign math language before (and while, and after) using it to teach math, I'd guess a lot of students' "math problems" would magically disappear.
The first problem is that the notation is usually "the first symbol that popped into some random genius's head 200 years ago". And once the notation is set, it's set, no matter how poor it is, or how many other places it's already in use etc etc. Then, as you note, sometimes there are multiple notations. Ugh.
The second problem is closely related to the first. Mathematical notation is write-optimised. This makes sense because of the long history. But that doesn't change that write-optimised languages are harder to read and understand, even for experts, than read-optimised languages.
In a programming context if you today reduce all your variable names to single latin letters and all your function names to single greek letters, you will be widely mocked and reviled. In maths it's Just How Things Are Done.
I guess what I'm saying is: the curse of mathematical notation is pen and paper. The boundaries of QWERTY liberated (almost all) programming languages from the curse.
Serious question. What do you think is the general impression of APL programmers? =)
I already compromised by adding "today" and "almost all" as qualifiers.
I actually really like mathematical notation. Once you get used to it, the terseness can make things a lot clearer than natural language.
(If you've ever tried to read old mathematical articles / books that are a few centuries old, you'll understand the power of mathematical notation. Check out "God Created The Integers" or "On the Shoulders Of Giants" by Stephen Hawking if you're curious: these are two books that provide excerpts of highly influential works from earlier mathematicians and physicists, respectively)
When given two numbers, if one wishes to find the quantities that give 0 when the second of the numbers is added to the product of the quantity and the first number and the quantity multiplied by itself, one should negate the first number and then either add or subtract the square root of the sum of the square of the first number minus four times the second number, and divide this summation by two.
That's just the quadratic formula in disguise:
Let b and c be real [or complex] numbers, then
x^2 + b x + c = 0
x = (-b +- sqrt(b^2 - 4c)) / 2
(Saying "let's do maths without the symbols" is a little like saying "let's do programming without special languages"... it is very very hard to make it work.)
As wonderful as it is, mathematics needs notation, and lots of it. You can express incredibly complex ideas in mathematics, totally unambiguously, through a collection of symbols. Not to mention that they're universally recognised.
The reality is that mathematics is 100% about thought. You'll struggle to put together the concepts in your mind long before the notation is the real issue. Once you have a clear picture of the abstract space you can use the notation you've learned to communicate the world you've created to others. What could be more wonderful?
[1] http://aleph0.clarku.edu/~djoyce/java/elements/bookVI/propVI...
Suppose we wish to make a rectangle with a given area and perimeter. This is an interesting problem! Does the number of possible answers depend on the specific area and perimeter? Certainly! It all comes down to thinking about squares, since squares maximize the area given a fixed perimeter. If the area of a square with the given perimeter is LESS than the desired area, then there's no way we can make such a rectangle. If the area of the square is equal to the desired area, then that's our only answer! Now, how about if our square's area is larger than the desired area? We'll get two possible different lengths of a given side of the rectangle - one representing the rectangle's width, and the other representing it's height. Or we could also think of them as two different rectangles - a tall one, and its rotation by a quarter turn (which makes it wide). By symmetry, we know that the difference between the square's side length and the shorter side will be the same as the difference between the square's side length and the longer side. How large is that difference? Exactly enough to diminish our shape's area from the square's area to the desired area. And that difference in length is simply the square root of the difference between the square's area and the desired area!
It would have been better with pictures :). Anyway, the quadratic formula is probably the greatest mistake in all of mathematics education. Somehow we use the word "quadratic" and even the phrase "complete the square," but never have I ever seen someone draw the said square!
While I think notation is often great for expressing ideas concisely and precisely, I think an excess of notation not a good way to communicate concepts. Nobody should memorize the quadratic formula! We should understand instead how to think about areas and lengths, and then we solve the problems in quadrature that we want.
FWIW, an animation of the quadratic formula/completing the square: http://en.wikipedia.org/wiki/File:Completing_the_square.gif
Let x, b, c \epsilon C:
x^2 + b x + c = 0 => x = (-b +- sqrt(b^2 - 4c)) / 2
Example of the difference: http://i.imgur.com/gwAqirx.png
generated by code:
\textbackslash{}epsilon: $\epsilon$ \\
Example: $x \epsilon \mathbb{R}$
\textbackslash{}in: $\in$ \\
Example: $x \in \mathbb{R}$
Computer science is also essentially about notation and vocabulary, but we have to make our notation understandable to the computer, which is a much higher standard than what mathematicians have to adhere to.
We are in a field that demands a much higher level of rigor than mathematicians are accustomed to, as much as they'd hate to hear it.
Coq notation, lisp-style notation, even python-style notation- anything would be better.
That said, it does bring me back to a math class where a professor, after realizing he needed to introduce a subscript, to a subscript, to a subscript of something that had both a subscript and superscript already, made a comment along the lines of "please excuse my poor notation."
Dr. Vesley blew them all away. There was notation, but the real clincher was his absolute clarity of communication. He covered a lot of material, but the pace never felt rushed. In fact, it was so calm and so clear it was refreshing, more like meditation by a babbling brook. I wasn't the only one to feel that way -- the whole class seemed to have a similar experience.
Related to the story in TFA, a friend of mine with a towering math background said a few years ago, "I remember when math was easy -- back when I had time." Math that we've learned and mastered is "easy", but new areas of math can require a LOT of mental energy to gain traction in.
My largest college regret was blowing off linear algebra- it was an annoying class taught in an annoying way (handwritten homework showing your work for each step of matrix multiplication, no proofs, ect). I blew it off because there were no applications of it in anything I cared about.
A semester later, it showed up somewhere in every single advanced computer science class. Really wish there had been a proof-based linear algebra class that showed up later in the curriculum so by the time we reached it we knew it had value.
With a basic course in linear algebra (such as Gilbert Strang's on MIT OpenCourseWare) and potentially some intro calculus you should fly through that course.
But following up on your comment, the difficulty of math is that it is build on foundations. If you miss something because you were distracted, that hole is going to be an impediment over and over and it will create more holes until it is very difficult to make progress.
see this video, starting at 8mins: http://www.infoq.com/presentations/Expression-of-Ideas