To what extent does math ability improve with age?

I just completed tutoring for an eleventh grade student for a math class called advanced algebra. Firstly, I have some qualms about the material in the course. It jumped around from one topic to the next in two week increments, with no continuity between topics. It shifted from linear equations to combinatorics to logarithms to conic sections.

The other problem was that the student was just not getting it. He needed rules for everything and had no feel for the subject. For example, to find the square of the sqare root of x, he applied the rule of squaring the x term to get square root of x squared in order to get x as the answer. By definition the square root of x is a number that when squared equals x. The answer should be immediate.

I tried to show that the distance formula between two points in the plane was just an application of the Pythagorean Equation. I told him that the formua for midpoint made intuitive sense, since you just took the average of the x coordinates and the average of the y coordinates. I tried over and over to get him to be able to apply the two basic identities for (x+y)(x-y) and (x+y)^2. Nothing registered.

The student did not strike me as dumb. He made an effort and remained attentive. I am wondering if he were given the material in another year or two, if it would come more easily.