In this work, I will often contrast problem solving and descriptive mathematics. In doing so, I do not wish to imply that there a dichotomy between the two skills. Problem solving and description complement each other. To solve a problem, you must first be able to describe it. To create a high quality mathematical description, you must be able to solve problems.
The two skills work hand in hand, and it is often difficult to determine when problem solving stops and description kicks in. Take for example the very simple question of simplifying the expression 1 + 1. What exactly do we mean by "simplifying" an expression?
To "simplify" an expression, we want to find the most elegant way to write the expression that includes the pertinent information contained in the original expression. As we all know, 1+1=2. The number 2 contains the same information as 1+1. It is shorter and more concise. Solving the "problem" 1+1 is simply the process of trying to find a more elegant way to describe a system with two things.
Solving problems is largely a matter of finding the best description of the problem. Conversely, you need to have strong problem solving skills to understand basic mathematical descriptions. My telling you that 2x is the derivative of x2 means nothing until you have gone through the effort to learn Calculus. The best way to learn the meaning of this term involves a fair amount of tedious drill work.
My goal in creating this work on descriptive mathematics is not to circumvent the teaching of problem solving skills, but to emphasize the development of descriptive skills in conjunction with problem solving skills. The best way to do this is traditional methods, like drills, but with an emphasis on taking learning one step forward, and asking the student to communicate ideas with the knowledge they gained from the drill.
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