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One and a Quarter Pizzas:

Written by
Holly Rebekah Graff is an unschooling mom and former public school science teacher. She believes that every child deserves the freedom, time, and support necessary to pursue her passions and construct her own rich understanding of the world. She has worked with a diverse group of students, urban and rural, prekindergarten through high school, in a variety of settings from crowded urban classrooms to intimate groups of homeschoolers. She currently teaches science classes for homeschoolers at her home in the Catskills of New York. She blogs at Unschool Days about the schoolfree lifestyle. This essay also appears in the book Playing With Math: Stories from Math Circles, Homeschoolers, and Passionate Teachers, edited by Sue VanHattum (Delta Stream Media, 2015). 
My daughter, Luna, like most babies, began to grasp the concept of quantity the moment she discovered the delight of placing her foot in her mouth. If she tired of the one, there was always the other. But somehow there was never a third. Our bodies, in all their beautiful symmetry, are mathematical! And as Luna discovered her own body, she couldn’t help but discover math. Like all of us, she was immersed in mathematical concepts from the moment of her birth. So I never thought it unusual that Luna should love math.
Now she’s eight and tends to eat erratically, generally having little interest in food, but then becoming absolutely ravenous every second or third day. This was a ravenous day, and Luna was busy wolfing down slice after slice of spinach pizza with her friend, Maya.
Meanwhile, Maya and I were remarking on Luna’s enormous appetite and amusing ourselves by tallying her slices as she ate them. The conversation went something like this:
Me: You’ve eaten more than one whole pizza!
Luna: How many slices are in one pizza?
Me: Eight.
Luna: And how many more than eight have I eaten?
Me: Two more.
Luna: So eight and two…
Maya: That’s ten slices!
Luna (after some thought): That’s one and a quarter pizzas!
Me: How did you figure that out?
Luna: Well, if there are eight slices in a whole pizza, then there are four slices in a half, and two slices is a quarter because a quarter is half of a half.
“For some time, Luna had been collecting her experiences with numbers, quantity, fractions, and probably a dozen other ideas, turning them over in her mind, processing them. And on that day, within that context, she fit them all together to make a new and more sophisticated connection.” 
Luna’s learning is not tied to any externally imposed schedule. She is free to mull things over, to process ideas for as long as she desires. There are no deadlines to be met and no evidence required of her learning. I simply trust her to be curious and motivated, and to learn at her own pace. After all, millions of years of evolution have honed the human brain into a fine, responsive instrument for the purpose of seeking out knowledge. And just as Luna learned to walk and talk organically, without force or coercion, I believe that she will learn everything else she needs to know – including math – as a result of her natural interaction with the world. This postulate of unschooling, that children can and should be trusted with their own education, can be difficult to explain to others. But I think breastfeeding provides a great analogy.
Like unschooling, breastfeeding involves a lot of trust. It’s so common to see new mothers examining baby bottles after a feeding, noting the volume of milk consumed and expressing some degree of either surprise or dismay at their observations. They tend to feed their children according to a prescribed schedule, usually suggested by a pediatrician. In contrast, a breastfeeding mom has no quantifiable evidence of her child’s nourishment. She trusts her child to communicate when she is hungry and to drink until she is full.
As an unschooling mom, I am similarly lacking in any quantifiable evidence of Luna’s academic progress. I have no test scores and no report cards, no ounces of milk to cite as proof of her intellectual nourishment. Much of what she’s taking in, like breast milk, is invisible to my eyes. But I trust her to drink of the knowledge for which she thirsts, and I trust her to do so until, for the time being, she’s had her fill.
Indeed, unschoolers believe that the most profound learning often takes place silently and invisibly, in between activities and away from prying eyes. It is here that all those pieces of information, having been shaved from active experience, are pulled inward to jostle against one another in various combinations and arrangements until gradually, or sometimes suddenly, a new understanding emerges. This takes time, and time is one of the greatest gifts of the unschooling life.
Luna often experiences such moments of revelation, discovering key math concepts over time through her own thought and exploration. While reading a recipe, for example, she recently came upon the measurement, 1.5 cups. She’d seen the decimal point used in this way before and through experience understood that this was just another way of writing 1½. But on this particular occasion, a new connection was made.
Luna said, “Oh, so is that because the five could go up to ten, and then ten would mean another whole cup?” When I told her yes, that she was absolutely right and that’s why we call it the base10 system, she said, “I knew that because five is one half of ten, and if 0.5 means one half, then if the five went all the way up to ten then that must mean one whole.”
The joy that I felt upon hearing my daughter construct her own understanding of the decimal system was akin to the joy one feels upon watching her toddler take her very first step. For some time, Luna had been collecting her experiences with numbers, quantity, fractions, and probably a dozen other ideas, turning them over in her mind, processing them. And on that day, within that context, she fit them all together to make a new and more sophisticated connection. This idea would not have had nearly the same significance if I had simply told Luna that it was so.
“The most profound learning often takes place silently and invisibly, in between activities and away from prying eyes. It is here that all those pieces of information, having been shaved from active experience, are pulled inward to jostle against one another in various combinations and arrangements until gradually, or sometimes suddenly, a new understanding emerges.” 
Visual Models
A few days later Luna told me that she’d thought of a good way to explain how zero is used as a placeholder in the base10 system. “You can think of it like a vending machine that sells drinks or something,” she said. “Even if one of the spots runs out of bottles, there’s still a place there. Zero is holding the place until it gets more.” When Luna has been mulling over a concept in her head, she often comes up with interesting visual models to aid her understanding.
This is another hallmark of unschooling. Children are not restricted to the models and formulas taught by others. They are free to create their own. This allows for more flexibility and creativity when approaching a problem and also allows children to draw on their own strengths and preferred learning modalities. Luna is a strongly visual learner in general, so it comes as no surprise that she would often describe her understanding of math in terms of a visual model like the vending machine.
As a visual thinker, Luna loves to create art and has always been very interested in symmetry. She often incorporates symmetry into her artwork and invents symmetrical patterns. When she first began adding two numbers together, the problems that she was able to figure out immediately, without pausing to count in her head, were the following: 2+2, 3+3, 4+4, and 5+5. Other problems, those involving two different numbers such as 3+4 or 6+3, required more time and thought, and she didn’t always solve them correctly. I believe that her strong ability to double numbers is an extension of her symmetrical visualization skill. I find that fascinating, and it is a gift that will serve her well in the development of her mathematical ability. If Luna were in school, her teacher would probably focus on the problems that had given her trouble rather than on this very obvious and exciting pattern in her work. One of the benefits of unschooling is that Luna will always be regarded as an individual. I will always have the time and ability to try to understand her. I may not always succeed. But because I know my child so well, I have the ability to recognize her strengths and help her build on them.
Alternative Algorithms
Given so much freedom, Luna often devises original strategies when
performing calculations in her head. How’s this for an alternative
algorithm? During an extended stay in Paris last year, Luna asked how
many days we had left before going home, and I said six. Then she asked,
“How many hours is that?” So I said, “Well, there are 24 hours in a day,
and…” Before I could do the mental math and provide her an answer she
began it on her own. I happened to be writing at the time, so I jotted
down exactly what she said, “Well, 24 is almost 25, and a quarter is 25
cents, and there are four quarters in a dollar, so four 25’s is 100. So
if I had four 24s then it would be taking away 4 from 100. So… 100, 99,
98, 97, 96. 96 is four 24’s. Four, five, six… So we need two more. So we
need half a dollar now. So half a dollar’s 50. Take away 2 from 50. So
that’s 48. So add 48 to 96.”
Here things got a little tricky because 48 and 96 weren’t so easy to add
in her head. But after trying it a couple of different ways, she decided
to make the 96 into 100, add the 48 and get 148, and then subtract the
extra 4, giving her a final answer of 144 hours left in Paris.
At the time, Luna would have been in second grade. By this point in traditional schooling, there is much emphasis placed on memorizing “basic facts.” A second grader is expected to know basic addition and subtraction facts and to have begun memorization of the multiplication tables, often rattling them off by rote. “Two times two is four, three times two is six, four times two is eight…”
Luna hasn’t memorized her “basic facts.” She doesn’t know the multiplication tables by heart. She still even counts on her fingers a lot of the time when solving some basic addition problems. But when in her daily life Luna encounters a math problem that she’s interested in solving for one reason or another, she always finds a way to do it. Always.
More importantly, it never seems to occur to her that for any reason she couldn’t do it. While a problem like 24 hours times 6 days might be completely uncharted territory for her, she approaches it with confidence. She makes use of what she knows to extrapolate into what she doesn’t. Her problem solving process is always creative and fascinating. She doesn’t always solve problems correctly, and sometimes her process is so meandering that she forgets what she was trying to solve in the first place! But I’m not concerned about these things. I see her playing with math, manipulating it, using it, and owning it.
Luna is much more interested in working problems in her head than working them out on paper. She prefers to talk the problems through, describing the steps as she performs them in her head. I think that’s great. I was always good at math, but because I was taught one “correct” way to work out problems in school, I’ve always felt very dependent on pencil and paper. If there’s none available, I’ll even use a finger to “write” invisible numbers on my palm so I can keep track of their positions as I borrow and carry. It was only in grad school, when I had to do more advanced math, that I finally started to break away from these schoolimposed restrictions and started to feel comfortable manipulating numbers in a way that suited me.
Because Luna was never told that there’s a “right way” and a “wrong way” to work a problem, she’s already comfortable juggling numbers around in her head, pretending they’re 25s instead of 24s because that’s what works for her, and then later taking away the extra ones she gave them earlier. She is in command of those numbers. She calls the shots. As I said earlier, math is a tool. Like a hammer. You can wag it menacingly over a child’s head. Or you can put it in her hand and watch her swing it.