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Math is our means of quantifying things, and serves as an essential tool of science. Math allows us to identify quantitative relationships between one parameter and another parameter, such as the relationship between the speed of an airplane and the lift generated by its wings, or the relationship between the amount of electric current through a wire and the heat generated. Math is ruthlessly logical, and teaching students math inadvertently teaches them to think logically.
The chronological development of math generally follows the logical development by which it should be taught. To illustrate this, consider, in pattern, how a student could learn geometry inductively. After a child learns how to count, add, subtract, multiply, divide, and measure things, he is ready for geometry. The first step is to introduce a one-inch square as a basic unit of area, analogous to a one-inch line as a basic unit of length. Next, show that a, say, 2" by 3" rectangle is made up of six unit squares, analogous to a multiplication table, and show that the area of any rectangle is obviously length times width.
The next logical step is to demonstrate that a right angle triangle (RAT) can be generated by cutting a rectangle in half across its diagonal, and that the area of any RAT is obviously "half base times height." Next, show that any non-RAT triangle can be divided into two RATs to calculate area. Furthermore, any polygon, such as a hexagon, can ultimately be broken up into right angle triangles to calculate its area. Next, show how the formulas for areas of rectangles and triangles can be used to prove the Pythagorian Theorem, which allows one to calculate the length of one side of a RAT given the other two sides.
At this point the student can be shown how the great Archimedes discovered the basic equations for a circle, calculated "pi", and came close to inventing calculus by drawing regular polygons inside and outside of circles, and breaking them up into RATs. Then the student will be taught to measure the angles of a triangle and will discover that the sum of all the angles always adds up to 180 degrees. Next, show how to prove that this has to be so by the principles of geometry already acquired. From here it's on to trigonometry, and so on. Taught this way, the student will grasp each step clearly and even learn to anticipate each new step--an invaluable skill. It is this basic inductive approach that will be used to teach all the principles of mathematics.
Math requires a lot of practice in order to automatize all the basic operations. In solving a complex problem, one doesn't want to waste time performing the simple mathematical operations needed to solve it. As Ayn Rand wrote: "All learning consists of automatizing one's knowledge in order to leave one's mind free to pursue further knowledge." (The Romantic Manifesto, p.36)
An important type of math problem to practice on is word problems. Words stand for concepts. Problems in the real world are first identified conceptually and then translated into mathematical terms and operations. The student must be proficient at translating word problems as such. |
