There are three kinds of people in this word, it has been said. Those that think math is a waste of time beyond learning to count, add, subtract, multiply, and divide and thus is primarily useful for consumerism. Those that think that math is amazing because it is extremely useful for the construction of bridges, building, airplanes, cars, and more. These are the engineers. And, finally, those that think math is beautiful for its own sake. Each group is smaller than the previous.
A person’s approach to teaching math will differ based on which group he is in.
Group One teachers—we will call them Math Consumers—focus on the usefulness of math for everyday life. They are primarily concerned with how it can be helpful and are more likely to express their own frustrations with the subject.
Group Two teachers—we will call them Math Practicioners— focus on the “amazing” ability of math to be black and white, right or wrong. They emphasize the answer and how “amazing” it is that you will always get the same answer for the same problem no matter who does it, so long as they do it correctly. Both Tom in Montana and Susie in Vermont will always conclude that 375/25 = 15, every time. It is the answer that is of the utmost importance, and Group Two teachers (and students) are more likely to skip steps, combine steps, and conflate steps along the way because the steps don’t matter, the answer does.
Group Three teachers—we will call them Math Contemplators—focus on the “beauty” of math to reveal new ideas, thoughts, patterns, and meaning to us. They tend never to skip, combine, or conflate steps because the beauty of math is found in every step. It isn’t the end that is important, but the journey from beginning to end that is.
In Morris Kline’s book, Mathematics for the Nonmathematician, he describes two views of math in its history, which I will summarize as the Greek and the Roman views. The Greeks pursued math as Math Contemplators. They sought out the beauty of math, and they looked at math as a means of ordering the soul through the perception of new ideas, thoughts, patterns and meaning. The Greeks were so into math as contemplation that, the story goes, Archimedes died when Roman soldiers told him to halt and he, so caught up in his math problem, didn’t hear them or respond to their command. They killed him. Valerius Maximus writes:
Archimedes was drawing diagrams with mind and eyes fixed on the ground, a soldier who had broken into the house in quest of loot with sword drawn over his head asked him who he was. Too much absorbed in tracking down his objective, Archimedes could not give his name but said, protecting the dust with his hands, “I beg you, don’t disturb this,” and was slaughtered as neglectful of the victor’s command; with his blood he confused the lines of his art. So it fell out that he was first granted his life and then stripped of it by reason of the same pursuit.
Another Greek mathematician is purported to have constructed a machine to confirm his mathematical theory, then, after his theory was confirmed, he destroyed the machine so that it couldn’t be put to practical use, thereby reducing the value of math to the utilitarian.
The Romans pursued math as Math Practicioners. They used math to order the world around them, constructing roads, building, aqueducts, and weapons. They cared so little for the beauty and contemplative nature of math that Alfred North Whitehead would write, “No Roman ever lost his life because he was absorbed in the contemplation of a mathematical diagram!”
The Greeks, we might say, sought math as a means to heal and order the human soul and mind. As Socrates would argue in The Republic, well-ordered souls will lead to a just and well-ordered city. The Romans, we might say (if we are trying to see them in the best possible light), sought math as a means to a well-ordered physical world. They might have argued that a well-ordered city would lead to a well-ordered soul. Or, as the old saying goes, “Cleanliness is next to godliness.”
We could take the time to argue through the merits of the two views, but I’d like to consider just one thing: the consequences. Whatever the Romans might have intended by their approach to math, what followed it was a view of math that went from using math to order nature to one that used math to manipulate nature. When math is reduced to its practical and utilitarian purposes, you get questions from students such as, “What is this for?” or “Why do I have to learn that?” And if math is only for practical purposes, like ordering or manipulating nature, then why do our students need to study math beyond, say, Algebra? What is it for? Why do they have to learn that? That, I believe, is how two views of math led to three kinds of math people. Once the Roman view was accepted as the dominant view, people began dividing between the people who actually wanted to use math for its usefulness and people who saw it as useful but thought it better to leave its use to the other group.
The beauty of approaching math as a Contemplator is that you don’t have to be a master of Calculus or Astrophysics or Quantum Mechanics to be a Math Contemplator. I can be as well-versed in math as a Math Consumer and still approach the teaching and learning of it as a Math Contemplator. I simply have to choose to contemplate math because it is beautiful — every step, every process, every procedure, every law, every definition. I suspect, moreover, that if I am willing to approach them as such, that will be the first step in beginning to actually see them as such.