Not “Why Algebra?” But “What Algebra?”

The struggling student’s question, “Why do I have to study algebra?” has escalated to the New York Times.

My response:

Of course you need to take algebra – some algebra.  Every effective modern person needs to know certain mathematical concepts- the ideas of a variable and linearity and negative numbers to name a few.  In fact you need to learn all of the mathematics required to study fundamental statistical reasoning – knowledge essential for all informed citizens.  You can check my list in this post – Pre-Statistics.

To be honest though, some of the stuff in your algebra text is there for historical reasons or worse because we teachers of algebra grew up with it and are good at it and think that way.  For example all the material about roots is mostly a waste of time if you know about rational exponents and are good at operations with fractions.

Here is what could happen to you if you go on from algebra to precalculus and calculus classes   You might see more and more problems that require a series of tedious algebra steps.  A fair part of your test grades will be based, inadvertently, on your algebra skills rather than your new knowledge of functions, limits, differentiation and integration. This is because such questions are easier to grade (and make up).  And in the back of our instructor heads we still want to be sure you have good algebra skills which of course you need.  In practice, of course, we ourselves would use a computer algebra system something like Wolfram Alpha for a long algebra problem.

In addition the patterns that you will see in later math courses are not emphasized enough in algebra classes.  Consider the highly useful, $a^2-b^2=(a-b)(a+b)$ and $(a-b)(a+b)=a^2-b^2$ where $a,b$ are symbols that stand for any real number or expression that stands for a real number.  Notice that I wrote the identity both ways because it is used both ways.  (Idea: Why not teach multiplication and factorization at the same time?)  Anyway the similarities between $7^2-3^2, s^2-t^2, (x-1)^2-(x+2)^2, \cos^2-\sin^2, c^4-d^4,$ and $r^2-R^2$ should be evident. Other useful patterns abound.

Often we also teach you the long way to do algebra problems because we think it will help you remember a particular concept.  It is always good to know why, but in my experience most students forget about the “why” and just concentrate on the algorithm.  So we would do better to separate the why question from the algorithm.  For instance, “criss-cross applesauce” works really well for fractions with relatively prime denominators so we should explain why it works and then teach it.

This post is starting to ramble.  My point – some algebra concepts are essential for an educated person.  However the standard algebra curriculum could be pruned and reorganized and certain patterns should be emphasized more.