Until recent history, mathematics had not been taught to the general population. Only those who were rich, powerful, and/or politically connected were given the opportunity to study math beyond basic counting operations. Many of my junior high students are excited about the prospects of returning to this situation. I have the opportunity to teach remedial math and math study skills courses for a local university. Many of the college students with whom I am involved are going back to school after many years in the work force.
Most of them experience a high degree of math anxiety because they have forgotten much of the algebra they learned in school. They’ve forgotten it because they don’t use algebra in their daily lives. In fact, many college students are quite successful in their various programs of study and yet struggle to pass their general algebra requirements. And almost everyone breaks into a cold sweat at the mere mention of the words “story problems”. Given the high anxiety level associated with the subject and the fact that so much of what we learn in algebra is not used by the general population, why is it so important that we teach it?
Be honest! When was the last time you needed to factor a polynomial or to find the asymptotes in a rational expression. Unless you must use these ideas in your work, your answer is probably “huh? ” Don’t get me wrong, I think there are compelling reasons to teach algebra to the general population. The first reason, of course, is utility. We use much of the algebra we’ve learned every day. For example, the ordering properties of our real number system are the basis for almost all of our comparisons–deciding which cereal is cheaper, alphabetizing lists, etc, etc, etc.
Negative numbers are useful in balancing our checkbooks–bummer! Of course, we add, subtract, multiply, and divide practically every day. I could go on and on. Most of the problem solving we do has a mathematical basis. Even “he loves me, he loves me not” is a simple mathematical progression–i. e. 1, -1, 1, -1,… Another reason we should learn algebra is to enhance our pattern recognition skills. Pattern recognition is an important problem solving skill. If I can make a problem match a similar previously solved pattern, then the current problem is solved. This is a powerful tool.
So powerful, in fact, that a mathematicians work is more involved with determining whether a solution exists than actually finding the solution. Three guys go to a scientific conference and check into a hotel into separate rooms. One is a physicist; one is an engineer; one is a mathematician. During the night, the baseboard heaters in each room come on for the first time in weeks. The accumulated dust causes a fire which spreads to the drapes. The physicist awakes smelling smoke, sees the flames and the extinguisher by the door. He grabs it, and uses it to put the fire out.
The engineer awakes smelling smoke, sees the flames and the extinguisher by the door. He grabs it, checks to see that it is charged enough to extinguish the fire and then uses it to put the fire out. The mathematician awakes smelling smoke, sees the flames and the extinguisher by the door. He is satisfied that a solution exists and goes back to sleep. We teach many patterns in math classes. Some have great utility. For example, every linear equation in one variable can be solved using the same pattern. e. g. 3 = 6x – 9 and 13 = 5x + 3 can both be solved by applying the additive and then multiplicative properties of equality.
So, in general, y = mx + b, where ‘m’ does not equal 0, is solved by adding ‘-b’ to both sides of the equation and then multiplying both sides of the equation by the reciprocal of ‘m’ giving x = (y – b)/m as the solution. It is as simple as following a recipe to bake your favorite cake–only altitude, barometric pressure, and relative humidity do not affect the outcome. Here’s another type of problem on which we spend a lot of time in a high school algebra class–factor a quadratic of the form ax^2 + bx +c, where ‘a’ does not equal 0. If a=1, there is a very strong pattern here.
The factors look like (x + g) (x + h) where g and h, along with their signs, multiply to ‘c’ and add to ‘b’. There are no exceptions. Indeed, g and h could be rational, irrational, or complex numbers and, in many cases, difficult to find. If ‘a’ does not equal 1, then the factoring pattern is much more complex. So along comes a knight in shining armor called the quadratic formula, which will calculate the factors of any and every quadratic. (i. e. it returns -g and -h in the above example). With such a powerful ally why would I attempt to factor any quadratic?
With the exception of a math class, when was the last time you ever needed to factor a quadratic in your life? Be honest! Then why would I, as a teacher, make my students spend countless hours learning to factor quadratics? The answer is pattern recognition. Discovering, learning, and using patterns is an act of logical reasoning. If the pattern is logical, it will produce similar results every time it is used and our logical reasoning skills are enhanced. Indeed, mathematics is full of logical patterns. The question is not whether the mathematical pattern is logical, it’s whether we see the logic therein.
The more we learn, the more we enhance our logical reasoning and problem solving skills. There is no better subject than math with which to teach thinking skills. So what is algebra really? Why do we teach it to everyone? To answer these questions, we need to make comparisons to the components of a language. A language has vocabulary and grammar–a set of rules which creates structure. A language is also made up of patterns. Remember diagramming sentences in English class? That was to help you learn the correct patterns of the grammar. In algebra we have a vocabulary where each word or symbol has very precise meaning.
And we have very precise rules which create the structure. Indeed, math is an almost perfect language. Each rule has few, if any, exceptions. In any other language, grammar rules have many exceptions. (e. g. in order to make verbs past tense we just add ‘ed’; except for those 137 exceptions which we learn one at a time. ) Do you think it would be good to study a foreign language, even though you may never visit that country which uses it? Are your own language skills better understood by having made that effort? Are there other rewards? The answer to these questions is a resounding yes. Think about thinking.
What language do you think in? When you realize that a thought exists, or you communicate it to yourself somehow, doesn’t it take form in your own language? Those who speak English think in English; those who speak Japanese think in Japanese; those who speak Spanish think in Spanish, and so on. Those who have become fluent in a second language have experienced thinking in that language and have an excellent realization of what I am trying to communicate here. Thinking and language are inseparably integrated. Twenty years ago, I was fluent in Japanese and I thought in Japanese, but over time I have forgotten it.
Algebraically speaking, mathematics is a language. If it is not used over a period of time, it is forgotten. But enhancement of our critical thinking skills may remain for a longer period of time. Preparatory algebra and algebra should be taught with this language idea in mind. Special care should be given to proper use of vocabulary–including signs and symbols. Rules of structure (grammar) should be carefully taught. We also have an avenue to teach problem solving. Solving story problems becomes a translation process. We are literally translating from English to math.
When you begin to learn a foreign language, you start by learning some vocabulary and grammar and then you translate simple phrases like “This is a pen. ” This process forces you to think systematically and logically, following the rules of grammar and choosing correct vocabulary. It is the same with algebra and story problems. A number increased by sixteen is two. ‘A number’ translates to a variable, say n. ‘Increased by’ becomes + and ‘is’ translates to =. The English sentence becomes the mathematical equation n + 16 = 2 and the solution to the problem is the solution to the linear equation above which yields n = -14.
By giving students lists of words which always translate into the various operations of +, -, x, and so on, the problem solving process becomes far less intimidating. For example, ‘plus’ always translates to + (add); ‘decreased by’ always translates to – (subtract); ‘per’ always means (divide), and so on. As we get better at the language, the sentences we translate become more complex. By increasing the complexity as learning occurs, our thinking skills are continually enhanced. In order to learn any language we must learn basic concepts and then practice. The only way to learn math is to do math.
Those who are unwilling to put forth an effort to work many practice problems struggle continually with math. Their anxiety levels rise and they begin to avoid the subject whenever possible. Learning algebra is not easy, but it is worth the effort for everyone. For several thousand years, mathematics has been the chosen medium to teach critical thinking skills. When we study math, our mental abilities, which are increased by learning language, are enhanced in a logical, useful manner. We must have the common sense to continue requiring the study of algebra for every diploma or degree we offer in education.