http://www.psychologytoday.com/blog/everybody-is-stupid-except-you/201211/us-math-achievement-how-bad-is-it

Alright so this post here is a response to this here article in the post title, and it was kind of, pretty much, mostly, exactly what I was talking about in my last post. Only this article has the advantage of putting technical terms to what I was talking about. 

Conceptual learning, or a fundamental understanding of the material on hand, is infinitely more important than learning a procedure to solve math problems. I believe that this is because procedure leaves no room for critical thinking, which is exactly the skill that a) you need to do math and b) the skill that math improves. The article also says that we need to let the students struggle to understand the concepts on hand and I completely agree with that as well. I think that understanding through self-discovery is the only kind of understanding that really sticks with people. It becomes a concrete part of their logic and an integral piece of the puzzle that math represents 

Another interesting fact pointed out by the article is that the gap between American students and students from other developed countries actually widens as the students advance to higher education. This shows that one main cause of the problem is the school system. We come out as little math robots and as soon as something is outside the scope of our understanding we totally shut down.

From the article:  the two features of instruction that predict good math outcomes are

1. Being explicit about the conceptual structure, and interconnected-ness, of mathematics

1. Allowing students to struggle to understand mathematical concepts.

By trying to skip this struggle by instead teaching procedure the students are deprived of the basic knowledge that they need to learn. The problem is that when this happens over and over through school, though you can technically keep up through memorization, you end up not having the foundation that you need to apply that knowledge. 

Math beyond school is subtle, and until you understand the concepts there is no way to really pick up on the trends that someone who is good at math might be able to pick up on plainly. And, as stated in the article, none of the concepts represented up to a high school level are really all that advanced. People can learn it, the question, as it always seems to be, can you teach it?