What is the nature of mathematics? This is something I’ve thought about on occasion, and it seems to me that it is an important question if I’m going to teach “math” to someone else. In my early years, I was taught that math was a trick that you do. You memorize all the answers, and then when a teacher quizzes you, you spit them out.
And then, as the elementary math progressed, I found myself feeling lost. Math didn’t “connect” in my brain. It was extremely difficult, and also frustrating for my father, who had to help me learn it anytime I had math homework. And then, one day, math did connect. It was like magic, really, for I don’t remember trying any harder than I did any other previous day. It was just that, all of a sudden, I could understand it.
I was ten-years-old, if I remember correctly, which is about the exact age that the brain is appropriately myelinated for the understanding of mathematics anyhow. Coincidence? I think not.
But enough about me. I am actually writing about math in regard to my own children. I daily fight the feeling that they are “behind” in math. I fight it because I don’t think that math is approached correctly within our culture, which is what leads to the vast majority of the population being practically “illiterate” when it comes to arithmetic. I also fight it because I have never read any actual evidence that says that children who delay math in the early years are actually behind their peers when all is said and done.
But mostly, I delay it because I believe in mastery rather than getting through the material. This is a hard position to take in a culture that prefers children to pass tests and keep up with their peers, no matter what.
You see, I haven’t yet encouraged my son to memorize even his addition math facts. And now I’ll explain why.
Bear with me. I am sorting this out while I write.
First and foremost, I chose to skip math entirely during kindergarten because he was struggling with a tic disorder, and I had read a few research studies connecting early math with such disorders. I spent his kindergarten year nourishing his mind on stories {both true and fairie} and reinforcing his phonics.
When first grade came around, we began math at a low key. He did one page per day, about four days per week. We use Math Mammoth because it is mastery-based rather than spiral, and also it is affordable. To be honest, I would have skipped it entirely again, but the State requires all private schools to teach math beginning in first grade, and so I complied.
Being that this was first grade, we started with addition, and he did fine. The next step was subtraction, and he had trouble with the transition. Because I believe that the primary cause of math struggles at early ages is due to physical brain immaturity, I immediately dropped subtraction.
I interrupt this anecdotal evidence with this:
If we present a child with learning tasks prior to the myelination of the areas needed to handle these tasks, we may be forcing the child, in its efforts to perform, to use less appropriate neural networks. By asking the learner to perform before the appropriate area is developed, we may be causing the failure and frustration seen in many children today.
We now return to our regularly scheduled blog post.
Like I said, we dropped subtraction. Instead, we worked on some other aspects of life that are typically considered to be part of the realm of mathematics. We learned to tell time to the half-hour. We learned to count coin money. We touched on some basic functions of shapes {sounds more impressive if you call it geometry}. And then I eased back into literal number work by focusing on place value for about three weeks.
When we returned to subtraction, which was actually the beginning of this year {second grade}, he flew through it. He begged to do many pages a day, so I gave him free reign. One day, he did six pages. He was enjoying playing with the numbers.
The big test came when we tried switching back and forth between operations. We’d begin with two groups of balls–say, 4 yellow balls and 3 blue balls–and we’d combine the groups {an act of addition} and then we’d divide them back up {an act of subtraction}. Over and over I was pressing for one thing: an understanding of the nature of numbers. Do you understand that numbers are quantitative in nature? was the question asked by every exercise.
It is only now, at almost seven and a half, that I can answer in the affirmative. So, in a couple weeks {when he finishes his current section}, we’ll spend some time memorizing the addition and subtraction facts. My goal was that memorization be preceded by understanding, so that he was never tempted to view it like his mother once did, as an impressive trick that pleases teachers.
So what is the proper view of math, exactly? Well, I think we can start with this quote from The Paideia of God:
Numbers do not exist on their own. If I add one apple to another apple the result is that I have two apples. By the same token, if I add a green apple to a red apple I get exactly the same result–two apples. Numbers are only adjectives, descriptive of those things that exist in the world God made. They have names, and because we intend to call them by their names, we are nominalists. These adjectives do not stand alone in some realm of the Forms or in any realm or dimension like the realm of the Forms. Using one as an abstract noun is fine, as long as we do not forget ourselves and begin thinking of it as having its own freestanding reality.
I would say that teachers generally teach math in a way which assumes numbers have their own “freestanding reality” and if the concept of quantity is introduced through the use of manipulatives, it is only in order to reinforce number work in general, not out of a real recognition that numbers by their very nature are quantitative.
As a number of us have been debating the definition of classical education, I got a chance to reread CiRCE’s official definitions for the seven liberal arts. I had never considered before that the quadrivium is so…connected. CiRCE says that arithmetic tells us how numbers behave, while geometry tells us how shapes behave, while music tell us how numbers behave in relation to each other {there’s that broad harmony concept again}, while astronomy tells us how shapes behave when they are moving.
So arithmetic tells us “how numbers behave.” This is the nature of math. But numbers don’t exist in their own world, being rather helpful adjectives with which we are able to signify quantities.
So we aren’t learning math, but rather math is teaching us about the world, explaining reality to us. In this way, math is our tutor. If this is so, it is important not to rush it. Our memory verse last week was Proverbs 19:20:
Listen to counsel and accept discipline,
That you may be wise the rest of your days.
If math is counseling us on how quantities interact, we must listen. Some children, will “hear” quickly. Others will take more time. My thoughts on math now are not that math should necessarily be slow, or necessarily any speed at all, but that velocity is not the point. In all things, we are seeking wisdom. Wisdom necessitates understanding. And understanding takes some of us longer than others.
When we who teach hear of some child, somewhere who is doing multiplication at the age of six, we should not consider this to have any bearing on our own students whatsoever, for our aim is nothing less than that each child might understand.
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