Teaching Repertoire: How I Teach Calculus (Part I: How I Lecture)
When I teach calculus, each day falls into one of two categories: lecture or problem solving.
Both types of days start out the same way: I will hand back graded papers, I will answer any homework questions that have been emailed to me ahead of time, then I will make the announcements. In addition to upcoming deadlines, I will give the students my impression of how the class is going, and how the current topic fits into the larger unit we are studying.
On a lecture day, I will then transition into what I'm going to say. Typically I try to start with a problem that is a natural consequence of previous work but that is difficult, unpleasant, or impossible to solve with the techniques we already know. For example, when I taught the chain rule, I started with an example of underdamped simple harmonic motion. The previous class had been on derivatives of trigonometric functions, and one of the examples had been modeling simple harmonic motion by y = Asint. I started the lecture by reminding the class of this example, and I explained why this was a pretty crude model of what happened in the real world. Then I provided an example of a function that modeled underdamped SHM (without deriving it), and sketched a graph, making an intuitive argument as to why this was a better model. And then I said that, unfortunately, our class does not yet know how to take derivatives of this type of function (I pointed to the arguments of the exponential and trigonometric functions, explaining that "we have only learned how to take derivatives when the argument is our variable -- not when it is a function"), so we could not explore this model or compare the acceleration function to the position function.
Then I admitted that this was a fairly complicated function for us to be starting with, and I suggested that we should start with something simpler. So I wrote on the board f(x) = \sqrt{x^2 + 1} and said that we would find its derivative. This is one of my standard techniques: interesting problems are too hard to start with, so I typically have to start somewhere simpler. I reminded my students (in an almost offhand comment) that "this can not be simplified," and I pointed out why our previous techniques for taking derivatives would not apply here: we could do it if this [points at x^2 + 1] was a variable, but here it is a function. I connect this to the previous example: we are having trouble taking derivatives of functions applied to functions.
This has probably taken me longer to type than it took me to do in class.
No matter what I'm lecturing on, this is always the set-up for teaching a calculational technique: Plausible example of why we'd want to know this followed by simple example where I can demonstrate it.
You will note that I have not yet said to them the words "chain rule" nor have I written the formula for the chain rule on the board. If I think that the symbolic form of a formula is confusing, I will not state it at the beginning. Remember the inverted pyramid of journalism? I have adapted that to my class: catchy beginning followed by the most essential and accessible parts of the material. When students get confused, they are more likely to stop paying attention for the rest of class.
Next, I took the derivative of the function, showing them the steps. As I went through the steps I referred to the "inner function" like we do when talking about the chain rule. But I still haven't said "chain rule."
Quickly I admitted that this was probably novel and confusing for them and that I wanted to show them some more examples. So I wrote three more fairly simple chain rule problems on the board and then slowly and carefully took the derivatives, explaining each step.
I pointed out to my students the patterns in all the examples I had done. I summarized these by (finally) writing the chain rule on the board, explaining it in the same words I had used in the examples and connecting those words to the symbols.
Then I wrote a few more easy chain rule examples on the board and asked my students to work them in their notebooks. (This has now become a mixed-mode lesson: lecture + directed practice.) Once most of the students had made some progress but before my weakest students got too upset, I went over those problems slowly and carefully.
Most of the rest of a lesson continues similarly: I demonstrate one or more examples of a situation (in this case, what if you need to use the chain rule more than once), then I have them work a few problems in their notebooks, then I go over the problems. Only near the end of class, once I am pretty sure that everyone can do the technique, will I prove or justify it.
Both types of days start out the same way: I will hand back graded papers, I will answer any homework questions that have been emailed to me ahead of time, then I will make the announcements. In addition to upcoming deadlines, I will give the students my impression of how the class is going, and how the current topic fits into the larger unit we are studying.
On a lecture day, I will then transition into what I'm going to say. Typically I try to start with a problem that is a natural consequence of previous work but that is difficult, unpleasant, or impossible to solve with the techniques we already know. For example, when I taught the chain rule, I started with an example of underdamped simple harmonic motion. The previous class had been on derivatives of trigonometric functions, and one of the examples had been modeling simple harmonic motion by y = Asint. I started the lecture by reminding the class of this example, and I explained why this was a pretty crude model of what happened in the real world. Then I provided an example of a function that modeled underdamped SHM (without deriving it), and sketched a graph, making an intuitive argument as to why this was a better model. And then I said that, unfortunately, our class does not yet know how to take derivatives of this type of function (I pointed to the arguments of the exponential and trigonometric functions, explaining that "we have only learned how to take derivatives when the argument is our variable -- not when it is a function"), so we could not explore this model or compare the acceleration function to the position function.
Then I admitted that this was a fairly complicated function for us to be starting with, and I suggested that we should start with something simpler. So I wrote on the board f(x) = \sqrt{x^2 + 1} and said that we would find its derivative. This is one of my standard techniques: interesting problems are too hard to start with, so I typically have to start somewhere simpler. I reminded my students (in an almost offhand comment) that "this can not be simplified," and I pointed out why our previous techniques for taking derivatives would not apply here: we could do it if this [points at x^2 + 1] was a variable, but here it is a function. I connect this to the previous example: we are having trouble taking derivatives of functions applied to functions.
This has probably taken me longer to type than it took me to do in class.
No matter what I'm lecturing on, this is always the set-up for teaching a calculational technique: Plausible example of why we'd want to know this followed by simple example where I can demonstrate it.
You will note that I have not yet said to them the words "chain rule" nor have I written the formula for the chain rule on the board. If I think that the symbolic form of a formula is confusing, I will not state it at the beginning. Remember the inverted pyramid of journalism? I have adapted that to my class: catchy beginning followed by the most essential and accessible parts of the material. When students get confused, they are more likely to stop paying attention for the rest of class.
Next, I took the derivative of the function, showing them the steps. As I went through the steps I referred to the "inner function" like we do when talking about the chain rule. But I still haven't said "chain rule."
Quickly I admitted that this was probably novel and confusing for them and that I wanted to show them some more examples. So I wrote three more fairly simple chain rule problems on the board and then slowly and carefully took the derivatives, explaining each step.
I pointed out to my students the patterns in all the examples I had done. I summarized these by (finally) writing the chain rule on the board, explaining it in the same words I had used in the examples and connecting those words to the symbols.
Then I wrote a few more easy chain rule examples on the board and asked my students to work them in their notebooks. (This has now become a mixed-mode lesson: lecture + directed practice.) Once most of the students had made some progress but before my weakest students got too upset, I went over those problems slowly and carefully.
Most of the rest of a lesson continues similarly: I demonstrate one or more examples of a situation (in this case, what if you need to use the chain rule more than once), then I have them work a few problems in their notebooks, then I go over the problems. Only near the end of class, once I am pretty sure that everyone can do the technique, will I prove or justify it.
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