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

likethe 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*.

## 3 Comments

Okay, you girls have both given me something to think about here.

Mystie, I liked this:

If math is concrete, then we should teach it concretely, and young students are capable of learning concrete concepts.I hadn’t completely made that connection! It is an interesting thought actually: we treat a concrete subject abstractly and lose our children’s ability to learn it in the process.

Rachel, I liked this:

I don’t necessarily require that my children fully comprehend all of the Scripture verses I ask them to memorize. I figure that when they are ready for those concepts, the verses will be there to come to mind.I hadn’t thought about it that way, and I will have to consider that some more. With my oldest, I think I am on the right track because he often uses memorization as a trick to get around thinking. He is a good thinker, but often tempted to be lazy so that he can look “smart.”

However, I totally see your point, and I also see how I could be inclined to think of it that way when it comes to my second child, who uses memorized material more like what you described–as something that comes to mind when she is ready for it.

I remember having those moments, too – where I had struggled with a mathematical concept for months, and then suddenly it just “clicked.”

I’m not sure I agree with requiring mastery before memorization, though. Children’s brains are best suited for memorization before they’re well-prepared for understanding math, so it seems that would waste all of those best memorization years.

For the same reason, I don’t necessarily require that my children fully comprehend all of the Scripture verses I ask them to memorize. I figure that when they are ready for those concepts, the verses will be there to come to mind.

Excellent as always, Brandy.

I’ve never been on board with “delayed” anything, though part of it is semantic. Sure, if comparing work load or topics covered, our school might look “delayed.” But I’m not delaying, I’m just doing something pretty much completely different. 🙂

I’m not going to purposefully hold a student back from something he can do, which is what “delayed” implies to me: not even starting. However, I do know that is not usually what is meant.

I have LOVED Math-U-See for all these reasons you pointed out; the teacher and writer of the program understood all this. Still, although the author goes to great lengths at several points to stress there’s no “grade level” and there’s no “do a page a day” mentality, because there are pages and lessons it is still easy to slip into. I realized when we got halfway through learning (comprehending how it works, but not memorizing) the addition facts, “mastery” to the author also INCLUDED memorizing the facts. And you aren’t supposed to move from one lesson to another until mastery. And we can’t progress to the next lesson because the way he teaches the next set of addition facts relies upon quick recall of the previous addition facts.

I realized before last term that Hans thought of number order in the same way as alphabetical order: the order has no bearing on the particular unit. So when adding +1, when I said “Well, what’s the number after __?” He had no clue why that was relevant. But, a week after doing the problems with a number line (and not the blocks), it clicked. Now my tact is one I picked up from our phonics program, and I made the connection after listening to the CIRCE talk that math is another language. Our phonics program intro said good readers do sight read, but the sight reading comes after decoding the same word at least 100 times. So, just repeat reading the same word lists until they are sight reading the words. So, in math now we’re just doing addition facts (on paper with manipulatives that are based on place value) until he has “decoded” them enough times that he just knows them. I have no idea how long it will take, but we aren’t in a race, and I have noted progress.

That’s a great quote about numbers being adjectives (say I, as the grammar and not math person)! I had totally forgotten that, although I’ve read the book several times. 🙂 That’s why it seems to make more sense to me to use manipulatives and show numbers are concrete rather than waiting until they can handle abstraction and then teach math abstractly. If math is concrete, then we should teach it concretely, and young students are capable of learning concrete concepts.