Two of our members are gearing up to take the AMC 10 (American Mathematics Competition) next week. This is an elective math exam offered annually by the Mathematical Association of America for kids in the tenth grade. The math involved is far beyond the usual standard for the grade and involves not only considerable mathematical knowledge but also a high degree of intricate problem solving.

A typical (popsicle-headache-inducing) question looks like this: *A solid cube has side length 3 inches. A 2-inch by 2-inch square hole is cut into the center of each face. The edges of each cut are parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid?*

*A)7 B)8 C)10 D)12 E)15*

Within the non-traditional education community math can be an especially polarizing subject. For some parents, it epitomizes the irrelevance of traditional schooling (“Why does one have to learn algebra?!” many homeschoolers ask). Yet for other parents it represents a central anxiety about removing their kids from school (“But how will my kids ever learn math?” is a question perspective parents might ask places like Macomber). So how do two kids that have never set foot in a traditional school learn not only the skills of math but also develop a genuine enthusiasm for the subject? Yesterday, while I was writing this in the ski lodge at Wachussett Mountain, one of the kids in question ran back in to get his gloves - so I asked him: “How *did* you learn math?” The kid looked at me for a moment unblinkingly (his black facemask and helmet highlighted the inscrutable look parents will recognize in their teenagers). “I didn’t!” he said suddenly. And before I could finish questioning him, he ran out to join his friends on the slopes.

So what did he mean by this? I think partly this was typical teenage insouciance, and partly he meant that he didn’t *learn* math because he was never really *taught* it. In fact, as I later discovered, both these kids have never really been taught math. So how did they learn so much about math if they were never taught it? Probably in the same vague and difficult- to- describe way in which they learned to ski. They were certainly never forced to learn it. And it was never presented to them through a conscious program of study designed to make a difficult and forbidding subject more easily approachable. There were also no obstacles placed in their way and they were never subjected to the constant threat of evaluation to make sure they were learning at an arbitrary standard along an arbitrary schedule.

Their parents and home life also played an important role. They may have some natural or inherited aptitude (though neither strikes me as a *savant *or a prodigy) and they both grew up in families where one or both parents had positive feelings toward the subject. Probably there was an expectation that the kids would enjoy math (as a family of skiers might expect a kid to enjoy skiing) but neither was drilled from a young age. Their parents also enrolled them in supportive environments like the Macomber Center where math was treated the same as any interest (like skiing or Pokemon or punk rock music) and in so doing it was never coded as “nerdy” or “boring” compared to other supposedly “cool” or “fun” subjects.

It is also fair to point out that although they have never been *taught *math, they have nevertheless put far more work into math than they would let on. For the past two years, for instance, they have been using Khan Academy online together to study precalculus and calculus on a roughly weekly schedule. Dan helped them with this earlier on, but as their interest progressed they started working on it alone in the music room. It was their interest and excitement in calculus that got them to do math in a more formal setting and in preparing for calculus they learned a ton of background mathematics to fill in the gaps in what they knew. This is how it is possible to learn a subject without being taught it in a self-directed environment!