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 . . .

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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.