Implicit Differentiation
Yesterday I had a really rough Calc class. The subject was implicit differentiation, which is where you take a derivative of a function that is not strictly defined in terms of y. For example, xy + y = 2 is implicitly defined. Traditionally, it is a very difficult concept to teach, and I saw it coming so I prepared as well as I could.
I should probably interject here with a word on lesson plans. I rarely use them in the classical way they are defined by education theorists, because within five minutes of the class starting the best laid plans will be in ruin. A good teacher is not one who can determine the duration and sequence of every single problem to be done during a particular class, meticulously mapped out. A good teacher is someone who can effectively respond to student questions; he/she can take a seemingly tangential point, explore it, and then rope the investigation back into the larger discussion.
So in order to prepare, I mapped out the possible questions the students might have; I explored what they are weak on (chain rule in particular), and tried to think of innovative alternative explanations for when they inevitably will get stuck.
The class started, and they encountered some of the difficulty I had predicted, but a brave student said something I wasn't prepared for and that put me back on my heels:
"Mr. Brescher, I don't think [we learned/you taught] power rule right. This just doesn't make any sense"
I was just floored by the comment, because I am very personally invested in the amount of learning that goes on in my classroom. I paused for a second, and responded as best I could:
"Well, I know we only spent one or two days on it, but I have used it several times every class since then. If I was at all deficient in teaching it then, there have been ample opportunities over the last two weeks to ask questions and pick it up."
In my mind this was an unsatisfactory answer that just popped out to avoid an uncomfortable silence. I might just have to go back and reteach all of those rules.
I should probably interject here with a word on lesson plans. I rarely use them in the classical way they are defined by education theorists, because within five minutes of the class starting the best laid plans will be in ruin. A good teacher is not one who can determine the duration and sequence of every single problem to be done during a particular class, meticulously mapped out. A good teacher is someone who can effectively respond to student questions; he/she can take a seemingly tangential point, explore it, and then rope the investigation back into the larger discussion.
So in order to prepare, I mapped out the possible questions the students might have; I explored what they are weak on (chain rule in particular), and tried to think of innovative alternative explanations for when they inevitably will get stuck.
The class started, and they encountered some of the difficulty I had predicted, but a brave student said something I wasn't prepared for and that put me back on my heels:
"Mr. Brescher, I don't think [we learned/you taught] power rule right. This just doesn't make any sense"
I was just floored by the comment, because I am very personally invested in the amount of learning that goes on in my classroom. I paused for a second, and responded as best I could:
"Well, I know we only spent one or two days on it, but I have used it several times every class since then. If I was at all deficient in teaching it then, there have been ample opportunities over the last two weeks to ask questions and pick it up."
In my mind this was an unsatisfactory answer that just popped out to avoid an uncomfortable silence. I might just have to go back and reteach all of those rules.
Labels: Calculus
posted by Kyle at
4:33 AM
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