Building Math Talent
My friend, Marvin Levine, taught statistics to psychology majors. He said that there were two types of students. One type thought the class was easy, and they received an A. The second type thought the class was very difficult, they worked very hard at it, and they got a C.
Well, there you have it: The Haves and the havenots of math. If you one of the lucky ones, math is easy and you do well; if you are one of the disadvantaged, you have to work really hard just to avoid a failing grade.
Why the difference? This is my theory. Students walk into a math class with some basic understanding of mathematics. (This is their mental models, to use the terminology from my theory section.) The class presents something new. The students then have to somehow get from their current understanding to the new information.
Most students erect a temporary bridge. This is memorization  they memorize the procedure for solving the problem. The difficulty of this is influenced by how smart they are and how often they have built this same bridge in the past. But one of the biggest factors is simply how big of bridge they have to build. If the student walks into the class with a big understanding of math, they just need to build a small bridge. If the student walks into the class with a small understanding of math, they need to build a large bridge.
It is possible for students to build a permanent bridge  to add to their understanding of mathematics. Obviously, this is the better longrun choice  the student acquires a larger understanding of mathematics, making subsequent classes easier and improving the student's grades in those subsequent classes.
This is probably not the best shortrun choice. It is difficult to build understanding, and it takes a lot of time. This option is open only to the students who come into the class with good understanding and need only to build a small bridge. Students with a small understanding would not have the time to understand all of the mathematics in the gulf between what they understand and what they need to do on the math test.
Also, the students with small understanding probably do not have the slightest idea how to build permanent understanding. The style of teaching encourages just memorization.
So, the havenots start the class with a small understanding of mathematics. They have to work very hard, to build a bridge between what they know and what they have to do on the test. And even then, they often don't do well on the test (though that probably depends on the test). Meanwhile, they lack the time and knowledge needed to increase their understanding. So their understanding does not grow.
Meanwhile, the Haves understand most of what is happening in class and just have to build a small bridge. They have the skill and time to build a bridge of understanding, which increases their understanding of math. They walk into the next class with an even better understanding of math, increasing the gap between the Haves and the Havenots.
Is there an Innate Math Skill?
I know, I left out of this picture the idea that some people simply have more math skill than others. This is probably true. It is almost certainly true that some people enjoy math more than others.
But..... How do we know? Maybe some people, who seem to lack math skill, simply missed out on some of the fundamental facts of math. Their starting foundation of understanding simply wasn't as good as other people's. Then they didn't do as well at math, they didn't understand as much, and so they didn't enjoy as much. Eventually, the whole thing snowballed and they never developed any math skills.
You might think that everyone knows the foundations. I don't think so. There are two ways to look at this. First, I start out teaching the foundations. I teach foundations to second graders. I teach foundations to sixth graders. If I was working with mathchallenged adults, I would start at the foundations. Most people do not have good foundations.
Or, look at what the foundations are. The nature of numbers is a very deep topic. Zero is deep. Representing numbers in base10 is not simple  it took a while for that even to be invented. So anyone can have better foundations. Of course, you don't have to have a miledeep understanding of these to do well in math. But now we are back to the practical side of things  many or most students do not have good foundations.
The toughest is the foundations that are so basic that you can't even think of them. One is, I think, my concept of "transformer". This is a relatively easy concept to teach, but in my experience, without any teaching fifth graders tend to be weak on this concept (with girls doing worse than boys in a class where there was normally no difference). I cannot imagine doing math without the conceptualizing a function or formula as a transformer.
Or, I have packaging problems. Maybe it is just me, but I cannot imagine doing even addition or subtraction without the concept of packaging.
So, let us not diss the idea that the major source of difference between mathtalented children and mathuntalented children is the difference in the amount they understand of the foundations.
Or, my 10yearold daughter, who tends to be math adept, thought that 2 + 3 = 5 was a convention, and on other planets they might have a different convention. Of course, a different planet will have different names for these numbers, but she was thinking that once the names were given, the next decision would be what 2 + 3 should equal. There is a problem in her foundation!
Being FarSighted
One implication of this is that when you use the insight method of teaching math, you are not necessarily going to have students who are immediately successful. If the student has a short bridge to build for the material on a test, you can teach understanding; if there is a huge bridge to be built, you can build a permanent structure.
But, the insight method is farsighted. You are building students who will do well in their math classes and find them easy.
