In what was to be one of many *Dear God, what did my students even LEARN in the first dozen math classes they took?* moments last year, I had a student come to my office for help with a linear equation he couldn’t solve. “I don’t know why I can’t get this one,” he said, and showed me one of the homework questions: *Solve for x: x/4=(x-3)/3+1*.

“Show me what you did,” I instructed, and he obediently went through the steps, reversing order of operations, just as he’d been taught. “And I got x=0,” he reported.

“Okay,” I said. “What’s the problem?”

“x can’t equal zero,” he explained ^{*}. Then he paused, and said, “Can it?”

“Why not?” I replied. “Zero is a number, isn’t it? And when you plug it into the original equation, it checks.”

“I guess,” he said, but I could tell he wasn’t convinced.

My student didn’t understand the concept of zero. Alex, on the other hand, understands the concept of zero:

“Alex has a zero-like concept; it’s not identical to ours, but he repeatedly showed us that he understands an absence of quantity.”

That’s Alex the African Grey parrot. And, I mean, I used to have pet parrots, and they were actually really, really smart – smarter than anyone expected – but god*damn* that there are *humans* that age who’ve taken a dozen years’ worth of math classes who don’t seem to have attained that level mathematically.

^{*} And now, my attempt to make sense of this absurdity, which I’ve encountered more than once: I suspect that somewhere along the way, my students were told “You’re not allowed to divide by zero in an equation”, which they processed as “blah blah CAN’T blah blah ZERO blah blah EQUATION”, like in the Far Side cartoon.

Makes perfect sense now, doesn’t it? This sort of thing explains a variety of student errors, such as how an offhand mention of “you can’t use the quadratic formula to solve an equation if there’s no x^{2} term” got applied as “you can’t solve an equation if there’s no x^{2} term” by a college student who didn’t understand why her explanation of *x-3=0, therefore, no solution* didn’t get the full credit that IT SO CLEARLY DESERVED.