I often tell my students that math is about critical thinking not computation. This bemuses some and amuses others. The idea is not to just learn computation and memorize formulas but to learn how to apply this knowledge to solve problems. The universal language we use to solve problems via science or engineering is math.
The very concept that math is a language is lost on most, which is why we don't teach math like a language. If we taught English the way we teach math nobody would speak English. Computation just helps us "do the math" or get the answers required to solve problems. Somehow this seems to have been lost in translation by those who have tried to set up a system for teaching the mathematics. It becomes obvious that system doesn't work very well.
The focus is on computation, but not on how to use it. Just because you are good at computation does not mean you are good at math. Many students have found this out the hard way. You have to be able to apply it.
Any teacher who has taught math has heard the question "Why do we have to learn this stuff?" or "When are we ever going to use this stuff?" especially if the subject is algebra. My answer would be because it teaches you how to think critically. Critical thinking is helpful in all areas of life. Except we don't teach it that way. We have gotten lost in computation.
Quotes like "The human mind has never invented a labor-saving machine equal to algebra." and "Calculus is the most powerful weapon of thought yet devised by the wit of man." make no sense to the average student because math at some point stops making sense and becomes meaningless. Students memorize formulae long enough to spit it out for a test and then forget it. Algebra and calculus are the most failed subjects in the USA, and yet these are the very subjects students need to understand in order to become powerful problem solvers.
In the past all the greatest philosophers were also great mathematicians: Plato, Aristotle, Socrates, Confucius even Einstein, but all the great philosophers of antiquity were also great mathematicians because the mathematics is used for critical thinking. This is also why Universities still require "higher math" for degrees in Law or Medicine where I often hear students wonder aloud what the point of taking math they'll never use is. They miss the point that it's not the math and computation but the critical thinking that will make lawyers better at debate and doctors better at problem solving in their field. The mathematics has evolved to solve problems. The history of math is the history of men using numbers for problem solving.
Students can not use their computational skills (if they have them) to solve story problems. Two stage problems where students have to find one answer before they can solve another are missed by the vast majority. With concept based teaching, understanding the concepts makes math easier and even fun because once you understand the concepts you know what to do and the computation simply helps you do it. Once the concept is understood any question just becomes a variation on a theme and the computation is easy to apply.
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Math really can be fun and easy.
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