Over at Pharyngula PZ had a post about Tucker Carlson and some guy mocking algebra, describing it as just inserting letters into numerical calculations for no apparent reason, and thinking that we could well do without it. It always amuses me when people take some deep subject and dismiss it as absurd simply because they are ignorant of it or they do not see any sense in it. What kind of mind must they have to so casually dismiss a venerable field of study that has been around for over a thousand years, developed by some of the finest minds, and is a bedrock of mathematics and science?

But I don’t want to waste any time on Carlson and his idiot friend but instead want to look at what algebra is about.

In learning mathematics, the first major stumbling block is fractions. Children find it hard to wrap their minds around fractions, how to add, subtract, divide and multiply them, and even locating them on the number line.

The next stumbling block for students the world over is algebra because it is their first experience with abstract mathematical manipulations. Indeed it is algebra that is the gateway to real mathematics. Up until that point students were doing arithmetic, which is what manipulating specific numbers is, and not mathematics, Mathematics is not about working with numbers (although they do appear) but about relationships between quantities and involve proofs and theorems to get widely applicable general results, all done symbolically using letters and other characters.

I can well empathize with the difficulties that students experience when they first encounter algebra because of my own struggles with it. In my own case, the problem was made even more acute because when I was twelve, I had to be hospitalized for a year, missing school for the entire time from the middle of sixth grade to the middle of the seventh. When I left hospital, the new school that I enrolled in was willing to place me in the middle of seventh grade so that I didn’t lose a year. But I had to catch up on all the missed work and while I was quickly able to do so for most subjects, algebra was a major stumbling block. Because my classmates had started learning algebra at the beginning of seventh grade, I had already missed six months of it and so had to catch up on my own. It is bad enough starting algebra along with others, but this was much worse. I recall struggling with the mysterious ‘x‘ that the problems in the textbook kept constantly requiring me to solve for.

My teacher had little interest in assisting the new student catch up. While my father tried his best to help me, I just did not understand the point of it all until one day it dawned on me what algebra was all about, at least at that grade level. It is relatively easy to go forward in numerical calculations, where you are given the initial information in the form of a number and asked to perform manipulations on it (multiplication, division, and the like) to work your way to the numerical conclusion. So, for example, if I give you a starting number and ask you to perform a set of operations such as square it, then add five, then divide by 7, then take the square root, and so on, you will be able to tell me the final result fairly easily (if you have a calculator). That is all arithmetic, manipulating numbers.

But sometimes you have the numerical conclusion but do not know the initial numerical value that led to it. i.e., suppose I did not tell you the starting number but only told you the long sequence of operations and gave you just the final resulting number and asked you obtain the starting number. It is much harder to be given the numerical conclusion, and the operations that had been done on the initial number to arrive at it, and go backwards to find that initial number.

At some point it dawned on me that one could take the latter kind of difficult (backwards) problem and turn into the former, easier (forwards) kind. The way to do this was to introduce the mysterious x to represent the initial unknown number and then go forwards just as before but with x instead of numbers and obtain the final result in terms of x. One then solves that equation to obtain x.

Generalizing is straightforward. If one starts out with more than one unknown quantity (say two), that one needs to find, then one introduces two letters x and y, and proceed similarly. As an example, if I give you two numbers 8.7 and 3.5 and ask you to find the sum and difference of the two numbers, it would be easy to arrive at 12.2 and 5.2. That is arithmetic. But if I told you that two unknown numbers add up to 14.6 and their difference is 3.4 and asked you to find them, that is much harder using just arithmetic. Using algebra, we go forward by starting with x andy and write down two equations x + y = 14.6 and x – y = 3.2. These can be solved easily to give x = 8.9 and y = 5.7.

That is how algebra emerges naturally. All the machinery of algebra that one learns is to enable one to solve a wide variety of such problems. Once I grasped that that was what algebra was all about, I quickly caught up with the rest of the class and algebra became one of my favorite subjects, that I love to this day.

I am not saying that knowing algebra is essential for everyone. I am sure that plenty of people go through life quite comfortably without using it. But for those who are fortunate enough to grasp and appreciate its intrinsic beauty, it provides a great deal of aesthetic satisfaction as well as being highly useful.