It has long been said that you don't actually understand algebra until you learn calculus and don't really grasp calculus until you learn differential equations. The experiences of Data-Dog and cognisonance would seem to confirm the first part of that observation. Much of the problem in math education today stems from the fact that many students are not exposed to different ways of thinking about and solving problems. For me, it started with drills. First were the drills in addition and substraction. Then came learning multiplication tables and applying them to learning not just how to multiply but to do division as well. Drills did more than teach me how to do mental arithmetic. I used them to discern relationships and principles that might have escaped my notice otherwise.
I don't intend to turn this thread into a mathematics class, but I want to follow through on Scott77's idea of what to share. High school algebra revolves around solving for unknown quantities. You are given the clues and your job is to solve the mystery, in this case the mystery being the unknown quantity. The equations given to the student will have one or more unknowns which are represented by the variables. Think of yourself as a sleuth with enough information or clues in your hand to unravel the mystery.
Geometry is the gateway to what mathematics is truly all about. For in geometry, the student is introduced to logic and how to use it to reach valid conclusions. Definitions, axioms/postulates and theorems are the building blocks to obtaining critical thinking skills and the ability to use reason. The introduction to the relatively simple theorems in high school geometry prepares the student to take on the concepts met in calculus and beyond because it is the ability to reason which will enable the student to see that mathematics is not about learning "formulas", manipulating numbers, and using them to solve problems, but to thinking both analytically and synthetically so that inferences and conclusions can be drawn which are not initially self-evident.
I strove to teach my students thinking ability in my math classes. I was hampered by the fact that many were weak in the four basic operations of arithmetic and that weakness was what frustrated them when it came to learning about fractions, decimals and numerical relationships. My students were amazed at my ability to work with these things without ever using a calculator or computer. The key, I would say to them, was to use the best computer ever devised: the human brain. I would express confidence in their native ability to learn and use mathematics saying that if I could do it, anybody could. Thinking ability lifted the fog in their minds about things like the relationship between fractions and decimals, what an irrational number really was, and how to distinguish among real, imaginary and complex numbers.
Let me also say this. Don't be deceived by the arithmetic and algebra you learned in elementary, middle and high school. What you were being taught in those early years was merely how to use these two branches of mathematics. You weren't taught how they really work. True arithmetic's and true algebra's workings lie on the other side of calculus and cannot be understood until calculus is mastered first. I don't say this to discourage anybody, but simply to put matters in their proper perspective. To those of you who are now studying calculus, I say hang in there. One piece of advice I will give when it comes to tackling the algebraic aspects of calculus is one I emphasize again and again to my students: TAKE YOUR TIME!!! Many mistakes can be eliminated by not being in a hurry to solve a problem. Also, look over the test when you get it and do the easy problems first. That will leave you a lot of time to pore over the more challenging ones and you won't feel yourself under as much pressure.
Quendi
