"Why?"

One of the most fearsome elements in a student's arsenal is the question, "Why?" The power of this interrogation to discover is formidable. Its power to irritate is also great. It is, however, a question that every student should ask themselves whenever something seems wrong. It is not a bad thing for a student to ask, "Why am I learning this?" It's a great opportunity to tell them that it's not pointless. They open up to the possibility of its being used, and the teacher supplies the use.

It is therefore advantageous for a teacher of mathematics to know the utility of each item in a curriculum. It also behooves such a teacher to understand the value of the lessons in mathematics as a whole. While I cannot cover all of the applications of each segment of school math, I can try to address some of the benefits of the classes as a whole.

Mathematics can teach you how to think. This does not mean that everyone will learn to think by doing math, and it certainly doesn't mean that in order to learn to think, you must do math. It is merely one of many subject areas where prolonged exposure develops logic, reason, and memory. Students who would think well using the math model might discover that they have difficulty thinking with other models. It would be a disservice to these students to not permit them the mathematical mind.

Aside from fundamental cognition, mathematics teaches a student to fluidly apply tools to a variety of situations. A fairly elementary problem in ballistics would be nightmarish and impossible without a functional grasp of the concepts of arithmetic, algebra, and (sometimes) calculus. One of the profound mysteries behind mathematics is that it relates to our world at all. Teaching a student to tie the mathematical tools to the real-world concepts that they can sense and recognize is invaluable and I can scarcely believe that anyone gets by without this skill.

Consider a simple problem of paying in change for something. You can count how much you have. The number you get is clearly representative of the value of the money you have. You can physically move the appropriate amount of money for the transaction, counting it out, and then you can again physically count the money you have to get the new number that represents the value. This last step is, of course, simplified dramatically if a student can draw the connection between doling out change and the arithmetic process of subtraction. However, that is not something that should be expected to come naturally to a student. If you really dig down into that, there's no proof that it should be related at all except for experiential coincidence. That's the kind of innovation mathematics can teach you, and you only get more tools as you continue advancing in it.

In fact, once someone is fully armed with school mathematics (arithmetic, geometry, algebra, trigonometry, and a little calculus), one can accomplish staggering feats of computation. The difficult part is actually translating the applicable data from the world into the recognizable forms with which these tools are used. You can take a demand curve from economics and with calculus you can easily discover what price you should charge to get you the maximum amount of money. No one's going to do any better than that! The difficulty is in obtaining that function where you want to apply the tool.

I've always said that mathematics is all about being as lazy as possible while giving up no ground. You cover all your tracks, but take as few steps as possible to get from one place to another. The fact is, school mathematics trivializes a great deal of work that would be required if one could not apply its tools to situations in the real world. This leaves more time for the fun stuff. We all want more time for the fun stuff.