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?

My Angst with Teachers.

Alright, now I'm an adult. There's a library 300 feet from me with proof of everything I could possibly need in mathematics. So there's no excuse for me not understanding any math taught in class. Just, you know, laziness. But at one point I was a wee child and all I could really do was ask questions. I was always good at math, excelled even, but a lot of the time I had questions. Questions (although I couldn't yet express it so clearly) a young me would consider fundamental to learning. Why does the math work?
I remember a particularly unpleasant teacher in 8th grade who would just point to the math over and over and say "see! It works" and I think we were talking about the Pythagorean theory.

I guess what I'm saying is that if we are all going to be teaching children math we have to be prepared to have our explanations challenged. That is the nature of children, or at least some children. You have to be prepared to explain something nine different ways because these kids are all different. Luckily for the kind of stuff that we'll be teaching there really is a chance to do that. I mean, this is the best chance to really sink in proper math skills in students. So we have to teach math without stressing it. We have to make it enjoyable, because imagine how much easier the hard math classes could have been if someone had really taken the time to make sure you really knew, really understood, how math worked. There is so much cool stuff you can teach kids, and most teachers do a lot of really creative and neat things with English or History, and Science too. But i remember math was always a very serious time, and that was okay for me, but a lot of kids struggled in that. I mean, seriously, a lot of adults struggle trying to pay attention in classes they paid hundreds of dollars to go to, but Greg, who spent all his money at the Cafeteria during lunch on ice cream, the nine-year-old is somehow to be expected to pay attention to the most boring thing he could possibly imagine. Seems like we need to improve the teaching more than anything else.

And furthermore, I'd like to say that we, as teachers of the future work force, may want to stress occupations that use serious math. Because those people exist, and they make a lot of money and wouldn't it be cool if Sally decided she was going to be good at math because when she was 4 she decided that she wanted to be an engineer. I'm just saying, the stigma against math is there and it starts early. But, we have to plan ahead for the future of these children. What jobs are there going to be 20 years from now? I'd put money that a lot of those jobs are going to be pretty heavy on the math and science.

In Conclusions, I really think that the only thing holding kids back from learning mathematics well are the teachers and their methods.