The revelation he has had looks trivial to me. Graphs as equations, equivalence of expressions or matrix as linear operators are all something I learned and understood from the very beginning.

I wonder if it has something to do with the textbooks. As a Chinese I often found American textbooks on mathematics so softcore. They have so many analogies, so many "real world" examples that masquerade the true mathematical meaning of the concept. Many of Chinese people argue that these are the reason that Americans are more creative, but I cannot help but wonder maybe the lack of rigor underlies some of problems with American math education.

Or maybe I am just the exception. Maybe other Chinese struggle with math just the same.

I think that maths at school it still too focused on creating engineers (the non software kind) and much of the material is aimed at producing people who can do calculus.

I'd prefer for there to be more focus on proofs, logic, geometry, number theory, graph theory, and cryptography etc. I think these concepts would give people a more rounded understanding of mathematics and prevent their experience of mathematics being one of pain.

Cool, i also had an "aha" moment while reading:

“It was only after grad school that I learned (from Lockhart’s book Mathematician’s Lament) to consider natural numbers as stones that can be arranged in various patterns that illustrate the different properties of a number. For example, evens are piles of stones that can be arranged into two equal rows, and square numbers have just the right number of stones to make a square! It’s really fun thinking about various operations in this way, and there are some beautiful proofs based on this technique. For example, why the sum of the odd numbers 1 + 3 + 5… Is always a square.”

I should hang out with maths teachers more often!

Two good ways to represent plane curves are parametric form:

(x(t), y(t))

or implicit form:

f(x, y) = 0

Parametric form is naturally associated with one point of view of what a plane curve is: the set of points traced out as a parameter is swept over its domain. Implicit form is naturally associated with a different point of view: the set of points that satisfy a certain relation.

Functions of the form

y = f(x)

can be easily re-expressed in parametric form:

(t, f(t))

or implicit form

y - f(x) = 0

so both the parametric viewpoint and the implicit viewpoint are equally valid and useful ways of understanding the graph of a function. You could rephrase the author's insight as saying that he had always understood graphs of functions parametrically, but later learned to also understand them implicitly.

Depending on the application, it may be more convenient to have a parametric representation of a curve, or an implicit representation of a curve. For example, it's easy to find a point on a parametric curve, but hard to test if a point is on a parametric curve; on the contrary, it is hard to find a point on an implicit curve, and easy to test if a point is on an implicit curve. If your curve is the graph of a function, it is easy to convert back and forth between these forms, but in general, converting from one form to the other may be quite hard.

For me, the relationship between implicit and parametric representations is a piece of math that I didn't really learn until long after I thought I had already learned math.

I somehow understood the same thing(a graph is a collection of points that makes a function true..) a year ago through Gilbert Strang Big picture of derivatives: https://www.youtube.com/watch?v=T_I-CUOc_bk

It looks like the blog is down at the moment. Here's google's cached version of the post in the meantime:

http://webcache.googleusercontent.com/search?q=cache:7SWRaHW...

I really didn't get the article. I mean, I've got that we have some mapping f: R -> R, or equivalently f: (x, y) \in RxR, so we could represent it by drawing on 2d plane, but no further...

Author claimed, that he had been missing something essential regarding that stuff, but I wonder what exactly does he talk about?

Is "secondary math" university mathematics? How can anyone think he's "learned" that?

What comes after Nonlinear Differential Equations, psy-op nigger?