And I Don't Even Teach Calculus Any More
So today The Topologist asked me how I prove the ratio test in Real Calculus 2. I responded that I tell the students that if the ratio is less than 1, then you can make an argument about the relative sizes of the terms and then use that to compare the series to a convergent geometric series. I don't say more than a few sentences, and I don't write anything down.
He was horrified.
Especially since the ratio test is my favorite series convergence test. How could I not prove to the students how it works!?
Proving the ratio test doesn't rise to the level of a proof that I'm willing to spend class time on. I ask myself if a proof is going to help most of the students understand more, be something that the students can follow but possibly not improve their overall understanding of calculus, disappear almost immediately from their memory (as if it never happened) with no lasting impact (either positive or negative), or make the students even more confused.
Anything that serves only to enhance confusion needs to be pretty important (like how I outline the argument behind the Fundamental Theorem of Calculus) in order to be worth class time. As the sequences and series are chock full of confusion, I find no need to add inessential extra confusion.
Oh, and The Topologist harbors this quaint notion that if you say something in class once that almost all of the students present will hear what you've said. For example, he feels that with the correct preamble before proving the ratio test that students will not regurgitate back mutated parts of what was written on the board during the proof as their attempted answer to the most confusing homework problem. (Clearly the answer to the most confusing homework problem is very closely related to the most confusing thing that happened in class. Since they are both very confusing, they must be the same thing!)
I also know people who start the definite integrals section with an example (like integrating something linear or quadratic over an interval) and setting up the Riemann sum, evaluating the limit, and using induction to show that the series adds up to what they say it does. The only way that you could get me to do that is if my class were over-enrolled (or had too many people who were taking Calc 2 despite failing Calc 1), and then I'd do that calculation every day until the "change your schedule" deadline. Heck, I might even drag out some complex analysis to find the total area under the bell curve. But I wouldn't do it under normal circumstances.
Hardly matters what I would do, though, as I'm not teaching calculus any more.
And in teaching tangents: Biology continues to be awesome. I need to nominate the awesome Russian dude for a teaching award. Yesterday he gave a 50-minute lecture with PowerPoint. Do you know how many slides he had? Three. Fucking brilliant. If you do work in an area similar to what he does and he's giving a talk at a conference, you need to go.
He was horrified.
Especially since the ratio test is my favorite series convergence test. How could I not prove to the students how it works!?
Proving the ratio test doesn't rise to the level of a proof that I'm willing to spend class time on. I ask myself if a proof is going to help most of the students understand more, be something that the students can follow but possibly not improve their overall understanding of calculus, disappear almost immediately from their memory (as if it never happened) with no lasting impact (either positive or negative), or make the students even more confused.
Anything that serves only to enhance confusion needs to be pretty important (like how I outline the argument behind the Fundamental Theorem of Calculus) in order to be worth class time. As the sequences and series are chock full of confusion, I find no need to add inessential extra confusion.
Oh, and The Topologist harbors this quaint notion that if you say something in class once that almost all of the students present will hear what you've said. For example, he feels that with the correct preamble before proving the ratio test that students will not regurgitate back mutated parts of what was written on the board during the proof as their attempted answer to the most confusing homework problem. (Clearly the answer to the most confusing homework problem is very closely related to the most confusing thing that happened in class. Since they are both very confusing, they must be the same thing!)
I also know people who start the definite integrals section with an example (like integrating something linear or quadratic over an interval) and setting up the Riemann sum, evaluating the limit, and using induction to show that the series adds up to what they say it does. The only way that you could get me to do that is if my class were over-enrolled (or had too many people who were taking Calc 2 despite failing Calc 1), and then I'd do that calculation every day until the "change your schedule" deadline. Heck, I might even drag out some complex analysis to find the total area under the bell curve. But I wouldn't do it under normal circumstances.
Hardly matters what I would do, though, as I'm not teaching calculus any more.
And in teaching tangents: Biology continues to be awesome. I need to nominate the awesome Russian dude for a teaching award. Yesterday he gave a 50-minute lecture with PowerPoint. Do you know how many slides he had? Three. Fucking brilliant. If you do work in an area similar to what he does and he's giving a talk at a conference, you need to go.
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