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Zen & the Art of Unschooling Math

"When freed from the traditional pencilandpaper, rote method, children come up with surprisingly varied, creative, brilliant, and eyeopening ways of understanding numbers and their interaction." 
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.”
“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 fouryearold 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 oddjob businesses and lemonade stands, build awesome Lego structures and get their fair share of the pancakes.
“I’m sixandahalf,” one says. The other: “I’m sixand threequarters!” (“Well, I’m sixandseven eighths!”)
“You can have half of my cookie.” “Can I have twothirds, 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 reallife 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 detectiveworked it.”
When freed from the traditional pencilandpaper, rote method, children come up with surprisingly varied, creative, brilliant, and eyeopening ways of understanding numbers and their interaction. Math suddenly becomes not a drudgery, but a fascinating and useful aspect of reality that can be learned in so many different creative and practical ways that perhaps only a child herself could show them to us.
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.
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 alreadyunderstood 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.
"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." 
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 prealgebra sort of way, yes, it would.
A year later, when he turned eightandaquarter (an important occasion), he announced for our information that he was actually “sixandninequarters.” Shortly afterward, he changed his age to “sixminusaquarter and ten quarters.” Why he preferred these complex, alternative fractional terms I don’t know. But they did add up to his “true” age: eightandaquarter.
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 conventionallyschooled like me) it is inherently interesting and useful to them. Their lives are filled with compelling reasons to understand and use math.
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 needtoknow 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: “Twentynine is twentyten, if you count twenty as twentyone.” 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. This essay also appears in the book Life Learning: Lessons from the Educational Frontier.