My mother is a math teacher. She teaches children at what is likely the most important stage of their mathematical development: as per state mandate, she is building a foundation for the algebraic thinking which will be so vital to the rest of their mathematical career. If her seventh graders succeed, they have a chance of understanding algebra and trigonometry. If they struggle with the material, when they continue on to higher levels of mathematics, they will likely down and make little effort to save themselves because they “just don't get math.”

The students who perform well don't get math either. The state has failed them. After seven years of required math lessons, they still think math consists of multiplication tables, the dreaded negative numbers, and the Cartesian coordinate plane which is only useful if you are selling hot dogs and hamburgers at the state fair. Their teachers are unable to help them really comprehend what they are learning because they have to plow ahead and cover the rest of the standards before the Benchmark Exam. We're not leaving children behind, we're dragging them ahead whether they're ready or not.

A child can walk out of my mother's class with an A and a mortal fear of fractions. They can pass the Exam with a fundamentally flawed concept of the definition of a function. They can learn everything the state requires and still carry the misconception that the variable names x and y are intrinsically different. Unfortunately, with few exceptions the students whom the state has deemed worthy have only a functional knowledge of the processes used to solve specific kinds of problems rather than a deeper understanding of the way the processes work: they know what to do but not why. It's magic.

The magic isn't quite where you think it is though. The magic is what the kids are missing. It is the rules that make the processes go. Descarte wasn't puzzling over how to make exactly $200 selling three times as many hot dogs as hamburgers when, by divine inspiration, he was blessed with the concept of rectangular coordinates. Newton and Leibniz didn't simultaneously pull the concept of a derivative out of thin air. The Pythagorean Theorem was not a gift from Apollo, and the pyramids weren't built by aliens. All of these results were derived from a basic set of rules.

Chances are that in a few years kids will have forgotten the processes. It has only been two years since I took trig, but I couldn't quote you the Law of Sines or any of the trigonometric identities. However, I could derive them with a few triangles and a couple of minutes because I know the rules and some definitions. With a sound knowledge of the relevant definitions and a little bit of direction, students can derive the processes the state thinks are important and have a better understanding of how they function than if they had memorized everything.