I'm not against the concrete approach to learning math, especially in the early grades. I think the building blocks, the scales, and the plastic pizzas can all be useful for understanding basic calculations.
But math is more than a system for sharing pizzas equally. It is also an abstract language with a consistent logic. And this is where constructivist math curricula fail miserably.
What I saw of Trailblazers was that they never took that next step to the abstract level. If the kids could divide a pizza, they were done. All too often they were asked to write about what they did in words, in a misguided attempt to cross disciplines.
But the language of math, like any language, must be learned on its own terms. Kids need to become fluent in the language of math by manipulating numbers and symbols.
One of the great benefits of math education is in teaching logic. A person who can construct a decent proof is on the way to clear thinking in any field.
Extra Credit for Overachievers: A man named Gödel proved that any logical system must be either inconsistent or incomplete. If you were going to design a system of arithmetic, which would you rather have it, inconsistent or incomplete? Which one is the system that we actually use, inconsistent or incomplete?
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