Zen & the Art of Unschooling Math
Photo © Rachel Gathercole
I was measuring rice for dinner when my eight-year-old son, Saul, bounded into the kitchen. “Do 400 quarters and 400 more quarters make 200 dollars?” he asked me, waiting eagerly for a verdict. I had to think about that a moment. And my four-year-old Sadie, who loves to help cook and didn’t want to wait, took the liberty of adding the water to the pot herself (two cups water per one cup rice). Finally I agreed that yes, it does make $200. Satisfied, Saul dashed off to finish whatever he was doing and I was left with the question that had become my constant companion: How did he know that?
After all, my kids have never had a math lesson in their entire lives.
The very next day, while paying the bills, I overheard them finishing a game of Mancala. “I won by four!” said Sadie, to which Saul cheerfully replied, “And I won by negative four!” (Later he asserted that he had in fact won by four minus eight. A more decisive-sounding “victory,” perhaps?)
Here’s what was puzzling me: I never teach my kids any math, yet they continue to mysteriously acquire more and more advanced mathematical skills. It is as if they are having secret math classes somewhere in the far reaches of their minds when I am not looking, while they appear simply to be drawing pictures or playing with friends or eating their lunch. Even though I knew this self-directed learning would happen, and always does, I became fascinated with observing my children’s intriguing and often amusing shows of mathematical understanding. I found myself on something of a quest. I was driven to find the answers to two questions: How were they learning all this math, and why?
I have long been aware that self-directed, delight-driven learning works for my kids, having spent nine years as an unschooler watching them learn everything they need to know and more through this method. Though I am available to them, my self-taught children have no curriculum or lessons except the ones that emerge from within them in the form of their own interests and curiosities.
Yet, even to a life learner, math can seem different from other, more transparent, areas of self-directed learning. When a baby listens to adults and imitates their sounds, it is plain to see how this contributes to his learning language. When a child shows interest in the Middle Ages and reads books about it, there is little doubt that this can lead to her learning some history. Math learning is somehow more elusive. Though it does happen, the process seems uniquely cryptic and this can make the idea of natural math learning appear just plain unrealistic.
But as I watched my children, I discovered that math, like language, is all around us, and that the children are absolutely driven to use it.
The first day, for example, I overheard them playing a game of Monopoly.
“I owe you forty dollars, but I don’t have two
twenties,” said Sadie. “I could give you four tens.”
"When freed from the traditional pencil-and-paper, rote
method, children come up with surprisingly varied, creative, brilliant,
and eye-opening ways of understanding numbers and their interaction."
“Or eight fives,” said Saul. “That would work, too.” He thought a moment, and then continued. “Or you could give me forty ones! But I don’t think there are that many in Monopoly.”
The second day, in an effort to stop Sadie from fiddling with a $300 camera, Saul tried to explain to her in four-year-old terms how much it would cost to replace it. “You’d have to pay a hundred dollars plus a hundred dollars plus a hundred dollars. And you know how much a hundred dollars is? A thousand cents. No—ten thousand cents.”
It turns out that learning to manipulate numbers is actually very important to the kids. They are inwardly driven to do it for their own purposes. They want to compare their archery scores, triple the cookie recipe, negotiate a higher allowance and spend their birthday money. They want to express hugeness, run odd-job businesses and lemonade stands, build awesome Lego structures and get their fair share of the pancakes.
“I’m six-and-a-half,” one says. The other: “I’m six-and- three-quarters!” (“Well, I’m six-and-seven- eighths!”)
“You can have half of my cookie.” “Can I have two-thirds, instead?”
“How much more money do I need to buy that action figure I want?”
It is not that I, The Homeschooling Mother, am using these convenient real-life experiences as motivations or opportunities for my kids to learn. It is they who want to learn the math, because they want to play the game, be the scorekeeper, calculate their batting average or save money to buy a special Father’s Day present. They want to learn the math and they will use any method necessary to do it.
If they ask me a question (“How much do I need to buy four houses on Boardwalk?”) I simply answer it. If I were to make a lesson out of each question, they would quickly come to think of the game or activity (and asking me questions) as annoying and not worth the bother and it would be unnecessary besides. They learn the math anyway and rarely ask the same question twice. (If they do, I answer it again. The way I see it, that’s my job.)
Of course, oftentimes I glimpse their mathematical exercise without being privy at all to their inner motivations. They come to me with questions I could not have predicted, which emerge from them at seemingly random times. “What do two twelves make?” “Is five a quarter of twenty?” “Does negative twelve plus twenty make eight?” Where these questions come from I don’t know. I simply answer them and the answers are soon absorbed.
Even when we see it happening, though, it is sometimes difficult to imagine how children could be learning math that is not taught to them. The ways they acquire this knowledge remain mysterious and fascinating. Back when Saul was six, we were listening to music together one evening when he suddenly turned to me and asked, “Is two twenties forty?” Though baffled that he knew any multiplication at all, I simply answered that yes, it was, and left it at that. Later, though, I asked him: “Saul, how did you figure out that two twenties is forty?” He replied: “Because I’ve been pondering it for about a year and trying to figure it out for a month.” This gave me pause; still, I persisted. “But how did you figure it out?” I asked, until finally he answered (with a sigh), “I figured out that half of twenty is ten, so two tens must be twenty, and four tens must be forty, and stuff like that. I detected it. I detective-worked it.”
"It seems math just isn’t as elusive as I’d
imagined. I once thought a grown person would surely teach my
kids the ins and outs of manipulating numbers; now I see that I
couldn’t keep up with them if I tried."
When Saul, still at age six, announced quite out of the blue at the breakfast table that four threes was twelve, I asked him how he knew that. He again said that he had figured it out, and when I probed further as to how, it turned out to my utter surprise that he had used rhythm. In his mind he had made four beats with three numbers in each beat (ONE two three, FOUR five six, SEVEN eight nine, TEN eleven twelve), just as simple as that. At a time well before multiplication would have been taught to him on a traditional schedule, he had invented, of his own accord, a means of multiplying rhythmically, and now could easily produce multiples and products of two, three, four and more.
Much more recently, I overheard Sadie, at age four, counting by twos, fives and tens. How did she learn that? I inquired. The answer: she had heard her brother counting his money.
It seems math just isn’t as elusive as I’d imagined. I once thought a grown person would surely teach my kids the ins and outs of manipulating numbers; now I see that I couldn’t keep up with them if I tried.
But that’s all very haphazard, some will say. What about systematically learning the facts and concepts in order, and what about the harder stuff? The above do sound like isolated arithmetic facts when looked at from a traditional (memorization) perspective. However, when a child figures out these patterns on his or her own, without outside help, suddenly they are not memorized facts but observations, parts of a complete puzzle forming piece by piece in the child’s mind. Each mathematical observation leads to the next, rendering systematic memorization and drilling wholly unnecessary.
|"I think the idea of allowing kids to learn math naturally, on their own, is intimidating because many of us, as adults, harbor deep inside our own discomfort with numbers."|
It can be scary to look at this way. I think the idea of allowing kids to learn math naturally, on their own, is intimidating because many of us, as adults, harbor deep inside our own discomfort with numbers. We believe that math is inherently difficult, abstract, and unappealing. As the late, great educator John Holt pointed out, we learned to manipulate symbols before we understood the concrete meaning behind this shorthand. Conversely, given the chance to observe first, at their own pace, how numerical patterns work in real life, kids can then easily pick up the shorthand, which may even come as a welcome way to more easily express already-understood concepts.
At age six, Saul stumbled upon a worksheet and asked what it was. I explained that it was a sheet of math problems, and, never having done any written math before, he asked what the symbols meant. When I told him that “+” means add the numbers together and “–” means take the second number away from the first and “=” means “equals,” he nodded thoughtfully, and then, to my surprise, easily filled out the worksheet with mostly correct answers.
I am not concerned about my kids’ systematically memorizing arithmetic facts, because their ability and striving to figure out the seemingly random ones they do demonstrates an inner understanding of the very concepts that make these facts true. Some skills and comprehension naturally come sooner, some later. The fact that they have developed a grasp of these ideas on their own affords confidence that they will continue to acquire more complex mathematical skills and theory on their own as well.
After all, the “harder” stuff is out there for the learning just like the basics. Left to their own devices, children get into surprisingly complicated concepts, perhaps because they do not know that math, “higher” or otherwise, is “supposed” to be difficult. One night, lying in bed trying to go to sleep, Saul (then seven) suddenly lifted his head groggily. “If an hour is a really long time, would a day be a long time with 24 reallys?” I had to admit that, in a pre-algebra sort of way, yes, it would.
A year later, when he turned eight-and-a-quarter (an important occasion), he announced for our information that he was actually “six-and-nine-quarters.” Shortly afterward, he changed his age to “six-minus-a-quarter and ten quarters.” Why he preferred these complex, alternative fractional terms I don’t know. But they did add up to his “true” age: eight-and-a-quarter.
In the end, the answer to my quest is perplexingly simple. It seems that the mysterious source of my children’s math education is not so mysterious at all. They simply figure the math out on their own. And they do it because (strange though it may seem to the conventionally-schooled like me) it is inherently interesting and useful to them. Their lives are filled with compelling reasons to understand and use math.
|"They are learning that it’s useful, why it’s useful, when it’s useful and how to be comfortable with it. And they are finding out that they can do it themselves."|
Does it matter that these skills are coming to them in a seemingly random, disorganized fashion, or as I prefer to think of it, on a need-to-know basis? I don’t think so. Or rather, I do. I think it’s important. They are learning the math that they want to know, when and because they want to know it. They are learning that it’s useful, why it’s useful, when it’s useful and how to be comfortable with it. And they are finding out that they can do it themselves. It is not someone else’s list of facts for them to memorize. Math is theirs. Theirs to work with, theirs to play with, theirs to use as they see fit.
As a parent, my main educational goal for my kids is not to fill their heads with information on traditional subjects (though I have no doubt they will acquire such knowledge along the way). I want them to learn to think, critically and creatively, and to know how to learn whatever they want or need to know using the resources available to them. Figuring out the world of numbers at their own pace, for their own purposes, is part of this. It is not only the math itself that is important; the process of extracting these patterns from the world and putting them to use is just as valuable.
Sure, when they come to me and want to know how long the third side of their triangular treehouse needs to be, I’ll probably step up and tell them about the Pythagorean Theorem – because that’s how that information is determined. When they want to calculate the probability of rolling a nine on their next turn – or some other more difficult question – I’ll accompany them to go look it up, or to ask someone who knows. And if they want to learn calculus so they can go to engineering school someday, well then we’ll sign them up for a class. They’ll be ready.
Unless, of course, they figure it out on their own,
which, at this rate, they just might. After all, they’ll probably
understand things I myself could never imagine.
In fact, you might say they already do. Shortly after Saul’s seventh birthday he made the following cryptic announcement: “Twenty-nine is twenty-ten, if you count twenty as twenty-one.” This one left me reeling for a while, trying to wrap my mind around it. But on some level, I knew that what he said was true, in a way that, in the end, perhaps only the free mind of a child could fully understand.
Rachel Gathercole is a freelance writer and the proud mother of two delightfully autodidactic children. She is utterly fascinated with children and motherhood, and can’t help looking on in awe at the incredible, inscrutable learning process that daily unfolds before her eyes.