I find myself lying to my students. A lot. I suppose it doesn't much bother me on a moral level. For one thing, my conscience is perhaps less muscular than it ought to be. For another, I'm generally pretty open with my kids. They know, for instance, that I'm divorced. That I'm quitting smoking for my 30th birthday. That for several years I was professionally violent. I go out of my way to let them know that, within reason, I won't shy away from their curiosity. Still, I lie.
Some of the lies are self-serving. (I will totally have this graded by tomorrow.) Some are self-preserving (Tell your mom it must have ended up in my spam folder.) Some of them aren't really even told, more like nurtured from the soil of the teenage imagination. (I have served time in prison. I am the father of the health teacher's new baby.) But the lies that keep me up at night are mathematical. (You can't take the square root of a negative number. Functions approach their asymptotes, but never touch them.) And, in this last category, I'm not sure what to do about it.
A while back I wrote a post on the equals sign in geometry. One of the difficult concepts for new geometry students is congruence, which looks an awful lot like equality, but isn't quite. Equality can either be used a relation between numbers, or between geometric objects that are identical as sets. Objects with the same fundamental characteristics (e.g. polygons with the same corresponding side lengths and angle measures), on the other hand, are said to be congruent. The standard high-school-level method for keeping the two concepts straight is to say something like, "Numbers are equal, geometric objects are congruent." Full stop.
But this is a lie. Things are more complicated than that. In fact, congruence is a weaker condition than equality (geometric equality implies congruence, but the converse is not true). So this equality/congruence lie was one I was just bound and determined not to tell. Of course (and many of you can see this coming), things went disastrously. A few kids internalized the message, but most were so paralyzed by the nuance that they fell back on a self-constructed numbers:equal, objects:congruent shortcut, which is true in all but the pickiest of circumstances.
Christopher Danielson tried to warn me:
"What does this expanded version of "equal to" ... get you that this simpler distinction doesn't?"
The answer, of course, is nothing. It gets me nothing, really, except the satisfaction of not having heaped one more lie onto the steaming pile of lies that another teacher, in a faraway time and place, is going to have to un-lie. But at what cost?
So this is my struggle. How do I determine the absolutely necessary lies from the lazy ones? How do I lie with the bare minimum frequency and intensity? My kids are certainly to the point where I can un-lie the domain of the square root function, but they certainly weren't when their first teachers fibbed it with such good intentions. Most of my kids will never obtain the skill set to get at a mathematically rigorous definition of an asymptote, so I'm stuck with: a line that a function approaches, but does not reach, except in certain situations where a function can indeed cross the line, so long as it behaves like an asymptote at some point...eventually. But for a few of them, this is another lie to be untold as they move toward calculus.
I have no idea how to avoid lying, but I'm going to try to at least be more explicit about lying. I'm going to try to help kids mentally bookmark my pretty little lies in the hopes that the reptilian parts of their brains will expect to be compensated with genuine truth at some point...eventually.