On The Topic of Multiplication Tables
During a routine evening visit with my Mom last week, the topic of multiplications tables was broached. She expressed some fear that upcoming generations, now surrounded by these devices that took on the simple mathematics operations that used to by done by hand or in the heads of children, will not learn mathematics the same way she had. Whether this is true or false is not something I am qualified to answer; it has been years since I was in grade school. Instead, the question of whether or not we should be, or ever should have been, teaching math in the form of the multiplication table at all, is what has consumed my late night thoughts.
It has been many years since I sat in a classroom filled with blocks, big colored numbers on the wall, with other students so young our biggest problems were likely to be whether or not we were going to pee our pants or if we had forgotten our lunch money at home. But I recall one thing quite distinctly from that time period; a melody that I will likely never forget. The cassette recording of an upbeat voice saying "1 times 1 is 1! 2 times 1 is 2! 3 times 1 is 3!" all the way to "12 times 12 is 144!". I remember my first grade teaching dancing around the room, entreating us all to sing along, several times each day.
Having been through my fair share of education, both for things technical and things that require more scholastic study, I'm starting to wonder why we did this. It is very rare that I am in a position where I have no access to a machine of some sort that can do this level of mathematics for me; after all, the most basic calculator can do this several times fast and more accurately than me, and cost a whopping 98 cents. The argument made by my Mother was that we don't want to forget how to do math; we don't want the machine to do it for us, and for us to forget and be lost without them. But when we were singing that song, the teaching dancing, were we really learning how to do math? It seems to be we were memorizing simple facts, making a small index of correct and incorrect answers that we could pull at a moments notice.
Admittedly, there could be some value to having memorized the answers for the most common questions; it would save the two seconds it would take to plug it into the calculator, and make it easier to guess at simple patterns. But can we justify making an entire year of math education focused on such a menial task? Surely we would memorize the most common answers not through rote, but through application?
Learning by Discovery
One of my favorite ways to learn is to discover the things I need to know; the process of starting with a base number of facts, then, with guidance, discovering for the "first time" the techniques and methods that I can use to solve the problem. I spent countless hours of my high school years learning all about two-dimensional transformations; rotating raster images, detecting collisions between boxes, circles, triangles, even so far as discovering that to create realistic physics, I needed not only a speed at which things moved, but a speed at which the speed of things changed. Although my answers were sometimes ridiculous and complex, they only helped to give me more appreciated for the better solutions when I finally saw them. It was not a chore to memorize them; I memorized them without trying because I desperately wanted to know.
And so it should be why we memorize things in math; unless strictly necessarily, the memorization should not be required. Instead, we should encourage children to explore their environment, to encounter problems, and find solutions to their problems. So instead of reading off the multiplication table until they can recite it in their sleep, we have them figure out how many pizzas they need to feed the class.
Math teaches thinking, not facts
When one engages in a high-level mathematics course, often times the tests are open book, open notes, and open Internet. You are allowed to have access to all the knowledge that humanity has to offer, and yet, when taking that test, you will swear to the uselessness of it. This is the nature of math; the goal is not to memorize which buttons on the computer lead to a specific result, but why those buttons behave as they do.
This is where a large amount of primary education fails to accurately represent math. Until the end of middle school or high school, math is more of a game of memorizing the answers than learning how to come up with the answers. Even if it is memorizing a process, such as long division, it is still just that; memorizing the process. Ask the student why the process works. Ask them how they would apply it to a different numbering system, and they will have no idea.
If the students learned by discovery, instead of merely being given the answer, one would hope they have some inkling as to why the process works, and how to memorize it. Why would they have this inkling? Because noone gave them the answer; they had to observe that when they count from 0 to 9, the next number is one ten with zero ones; maybe they observe that to split that number into equal parts, they need to first find the largest number that when multiplied by another is still less than the original, and that this step needs to be repeated until they have the answer, or maybe the observe that if they instead count from 0 to 1, they can easily divide by 2 by shifting the digits. After struggling with this for some time, when the teacher comes along and says "Here is one way you can do it", they will appreciate that all the more. More importantly, they would have exercised the part of their brain responsible for thinking in a logical manner; and in doing so, prepared themselves for rediscovering calculus, physics, even social systems.