"I am my thoughts. If they exist in her, Buffy contains everything that is me and she becomes me. I cease to exist. Huh." -Oz, Buffy The Vampire Slayer
The Three 'R's represent an old idea on the fundamentals of education are, Reading, wRiting, and aRithmetic. While meandering through East Lansing, I fell to pondering why I still consider mathematics education relevant in a world where computers are increasingly able to perform the calculations that students are called upon to make. While answering this question, I also hit upon why I consider my dual background in math and philosophy a natural pairing, and an overall theory of education that can be explained by the Three 'R's.
To me, the natural product of education is not knowledge, but rather thought. In a society shaken to its foundations by the advent of the Internet, knowledge has become an increasingly available resource. On the other hand, thought, along with love, remain the cornerstones of that which is best in humanity. Thought can be further broken down into a three step process in which we must repeatedly engage to truly fulfill our calling as thinking beings. First, we obtain new knowledge, which I symbolize by Reading. Next, we synthesize the new knowledge with our existing knowledge and thought structure, aRithmatic. Finally, to complete the cycle we must make our new thoughts socially available so others to may obtain and respond to them, which is wRiting.
We can certainly specialize in one of these categories. Authors, advertisers, and my beloved educators focus on the presentation and sharing of information, all of which would fall into the wRiting category. Scientists and historians, for example, attempt to better acquire knowledge, and as such are Readers. Finally, people, such as mathematicians and philosophers, who seek to find structures in the raw data of life are practicing aRithmatic.
Although we may specialize, to fulfill our yearnings toward humanity we must complete the full cycle, although not necessarily completely alone. Thus, the mathematician need both examine previous works for background and present her or his own conclusions with at least a modicum of communicative savoir faire. Similarly, scientists need to synthesize their data, or find a mathematician to do so, then present their findings and educators must first educate themselves, then make their own separate peace with their subject material, before finally preparing it for classroom consumption.
To focus in on the middle step, the importance of aRithmatic lies in this. Somewhere between gaining knowledge and presenting our conclusions, we must add our own immeasurably valuable contribution and think for ourselves! Ideally this is the task for which we are made ready by our mathematics education. The true task of the mathematician is, once very basic ground rules are asserted, to go forth and discern what else must necessarily follow and why. Although this path has been blazed by mathematicians long dead, there is no reason that a student cannot walk it anew as they too are exposed to the shinning structure of mathematical knowledge. All too often mathematics serves the opposite purpose, and students are led to blindly memorize unmotivated methods, encouraging an arcane practice of "mathematics" which greatly resembles the showmanship of a magician, wherein things appear and disappear in somewhat predictable patterns but without respect for an underlying sense or reason.
My word choice here is not mere hyperbole, I often say to my students verbatim, "math is not magic," to emphasize that they ought know why the processes in which they are participating work. If one simply considers math to be the manipulation of numbers and symbols via memorized methods with the desire to produce some correct end value, then the questions, "why do I need to learn this?" and, "why can't I just use my calculator?" make perfect sense, furthermore, they have no good refutations. However, if one takes math to be a process by which one understands a set of rules and manipulates them to obtain logical, but not at all obvious, conclusions, then these questions make no sense, as well they shouldn't. The best computers we have are merely calculators, not true understanders.
This is why math, and more so philosophy, are so important to me. They are, perhaps even more than dancing, an essential expression of who I am and what it is to be human. Pure, magnificent, creation of new thoughts, MY thoughts, out of the read works of those who precede me, which I then write and send forth into the world that I may be known, and through being known create the possibility of fellowship and love. Thus, I implore educators to teach a comprehension, rather than calculation, based math. The math itself isn't important, it could be replaced by philosophy, law, debate, or any other subject based in the recombination of information. What is important is that we claim our birthright, the ability to create thought, without which not only may we never be human, we may never be loved.