When My Students are More Clever than I am

This happens often enough to keep me excited in my job. I love the feeling when I learn from my students. Nearly everyday I learn important things about people, about human interactions, about kindness and community. What happened today was a great example of when I learn some math from my kiddos. I blogged yesterday about helping a boy with a L’Hopital’s Rule assignment. What I did not mention was that there was a problem I could not solve. I’m not good enough with typesetting here on wordpress so I’ll do my best here. We were trying to evaluate the right hand limit of (x – 1) * tan (pi*x/2) as x approaches 1. Two clues made me think of L’Hopital. The first is that the student told me they were working on L’Hopital’s Rule. the second was the indeterminate form of 0 *  negative infinity.

We looked at a graph to convince ourselves that there is a limit and then we tried to make this a quotient to fit the rule. My instinct with trig functions is to avoid the cotangent, cosecant, and secant functions simply because I feel more comfortable visualizing the basic cosine, since, and tangent. So I made the product a quotient by dividing the tangent expression by 1/(x-1) creating an indeterminate form of negative infinity divided by positive infinity. A quick ratio of derivatives yielded something even word and the second time around was even scarier. We walked away from the problem – in part because I was trying to do three other things at the same time and in part because he was exhausted from thinning his way through the other examples. I asked one of my best BC students to consider it overnight and had some hope that he would. 

This morning at 8 I posed this question to my quiet BC class of 7 and they pounced on it. One student advised that I rewrite the tangent function as a quotient of sine divided by cosine so that L’Hopital is immediately satisfied. Another advised that I rewrite tangent as a cotangent so that I have a ratio as well. In each case, my students saw a way to rewrite a product in terms of 0 / 0 rather than my way of infinity / infinity. Neither format is lovely but we all seemed to agree that 0 / 0 seems less scary. One round of derivatives on each idea led to the conclusion that the limit was – 2 / pi. Geogebra agreed.

What strikes me is that, despite me showing an undesirable approach AND me asking them to recall something from the past, my morning class was perfectly willing to be flexible and to TRY something. Two sound ideas in about 90 seconds and we were off. I was so delighted to start my way this day.

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