or: Why aren't our kids taught math properly?
Remember your first algebra class? No, you've probably repressed it. If you were like most kids you may have approached that first class with some misgivings. You had seen some samples of algebraic expressions, and they looked forbidding. Completely different from the familiar arithmetic; hardly looked like math at all. Expressions full of mysterious letters! You might have heard that the letters stood for "unknowns", which didn't help calm the butterflies in your stomach.
The teaching of algebra is an example of math teaching that often starts off on the wrong foot. With a little forethought it can be otherwise.
This "cold shower" introduction to "unknowns" in algebra has long puzzled and discomfited students, and has given the appearance of a chasm between arithmetic and algebra. What the students haven't realized, because their teachers haven't told them this, is that they've been working with "unknowns" since their first day of arithmetic. The elements of algebra flow naturally and seamlessly from familiar arithmetic – if the students are properly taught.
The point is that the simplest problems of arithmetic all involve an "unknown", namely the answer! The "sum" in addition, the "difference" in subtraction, the "product" in multiplication, and the "quotient" in division are all unknowns. This concept, that you are to find a third – unknown – quantity when given two other quantities and a relationship, is (or should be) elementary; it is what is in fact going on when a child does an addition problem. But it isn't taught that way. The child is not taught in terms of logically locating the missing element in a sequence when given other elements and a rule, though this is what he is doing. Instead, most children are still taught arithmetic in a mechanical and formulary way that does not prepare them well for moving on.
So, an elementary arithmetic problem like 2 + 3 = ? is of course the same as 2 + 3 = x; a child can easily learn the logic of this substitution. Place the unknown in a different position in the sequence and we have 3 = x - 2, which looks very much like "algebra". If children are taught from the first that what they do in arithmetic is solving expressions for an unknown quantity, I suspect that much of the age-old fear of algebra will evanesce. Here's hoping that the educational establishment, which must know all this already, will grasp the importance of a good mathematical foundation for all children, and will improve the teaching of the conceptual foundations, and not only in algebra.