What is Algebra, and What Is It Good For?
Consider the following: If you triple a certain number and increase that value by forty the result is the same as if you decrease the product of the number and ten by sixteen. It’s not that the reader can’t read the words in the sentence, and it’s not even that the meaning of the words is unknown to the reader, it’s just that the English language is very cumbersome, and often ambiguous when we are trying to quantify a situation. It becomes obvious that in order to make certain relationships clear it is advantageous to introduce symbolism, and grammar, punctuation, and a system of rules that are both precise and concise; algebra allows us to do just that. Consider: 3x + 40 = 10x – 16. This algebraic sentence (equation) says exactly the same thing as the English statement except that it is now in a form that is much easier to deal with, but only if the reader has the knowledge of the grammar, punctuation, and rules that govern the language of algebra. Thus, algebra forms the first of many intellectual instruments whose aim is to clarify the quantitative aspects of the world. In any real or contrived situation that is expressed in everyday language it is extremely important to be able to introduce enough symbolism to let what is truly relevant emerge and be grasped. The next step is to find ways to manipulate the mathematical notation in order to come up with a solution (a replacement value for x that makes the sentence true). Generally, the more convoluted the problem the more the symbolism is needed. Having a working knowledge of the language of algebra is an essential skill for anyone who hopes to be able to solve real problems in the world. The vocabulary seems so foreign, and the notation so bizarre in appearance, that the innumerate feel as though they have no chance of understanding algebra, and that those who do understand it are like aliens from another planet. The reality is that almost anyone can learn to use the language of algebra to quantify the world. The most difficult part of learning the language of algebra is getting past the boring repetition of computation and the rules of the language that are necessary to be able to understand algebra. Hanging around long enough to get to the application of algebra as a tool for solving interesting problems is rewarding and worthwhile. Part of the problem with understanding the language of algebra is its level of precision. Our every day English can always be punctuated with the nebulous words and phrases like “uh”, “you know”, and “like that”, or maybe even “whatever”. By comparison algebra seems almost pedantic in its precision. In ordinary language the definition of the even numbers might be, “ two, four, six, and all numbers like that”. Algebra, on the other hand, requires the definition; “any number that when divided by two leaves no remainder”. A great deal of what students struggle with in algebra is in the category of grammar: The correct way to write and manipulate precise expressions in the language of algebra. There are so many symbols that stand for words and phrases that students often feel that they are learning a foreign language; that may not be far from reality. Algebraic notation and the rules that govern the manipulation of that notation are boring, much like diagramming sentences in English. The difference is that the precision that dominates the language of algebra gives it power and makes it relatively free from the kinds of disputes that plague other areas of human endeavor. Algebra, then, is the language of all higher levels of mathematics, and it is the language of real world problem solving. Just as it is essential to have a solid foundation in arithmetic skills before tackling algebra it is essential to have a solid foundation in algebra skills before taking higher-level mathematics courses. Before expecting to be able to accurately quantify real world situations that defy attempts to apply less powerful problem solving tools a working knowledge of algebra is a must. 2 years of Algebra 1 is significantly better than the traditional 1 year that is dedicated to this course. |