Listen to and learn from your students; they hold the key to deeper understanding.
My precalculus class yesterday was exploring graphical behavior of rational functions. We started with which the group easily handled with their understanding of even and odd vertical asymptotes (VAs) from the previous class.
The curve approaches on the right, the VA at is odd so the rational function’s graph “passes through infinity” there, and “bounces off infinity” at .
We hadn’t explored what happened when rational functions had variable expressions in their numerators, and they had never seen holes in these curves, so I had no idea what they would do when I asked for the graphical behavior of at . As a student, I was taught to perform algebraic simplifications, but I thankfully remained silent.
My students were initially bothered by the form of the function at , and various clusters were working toward different solutions when student NC declared, “at , the denominator has the dominant exponent, so there still is a VA at that point. It might be a different kind of VA, but it’s a VA.” In an instant, everyone in the room understood what was happening. Changing the degree of the numerator’s factor in later examples ultimately led to holes on and off the x-axis, but NC’s dominance argument yielded far deeper and lasting understanding than my plans for an algebraic approach ever would have.
OK, I admit that you can get to the same place by doing algebraic simplifications, but my deliberate silence allowed my students to develop their own subtly different understanding that exceeded what I had planned to offer. After years of being the information dispenser in my classes, I’m learning the uncomfortable lesson that it’s often better to set them up and then shut up.