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Let's say I'm teaching an undergraduate Calculus 1 course using Stewart's Calculus textbook. Should I spend a fair amount of time in lecture doing examples that are taken straight from the textbook chapters?

I have been doing this, because often the examples are quite good. This book is highly polished, and user-tested over many years, and so I'm unlikely to be able to routinely think of better examples to teach the material.

On the other hand, my lectures might then sometimes feel a bit redundant. A student who has already read the relevant textbook sections might think, "I've seen these examples already, so why do I need to be here?"

What is your opinion about this? My current viewpoint is that it is useful to go over textbook examples in class, because I can't assume the students will have read the book in advance. (And some students find it easier to understand things when explained in a lecture than by purely reading.) I'd like to see what other people think about this issue.

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    $\begingroup$ I think we should teach from the heart. That said, you should probably remember many students don't read the textbook so your examples are still new to them. $\endgroup$ – James S. Cook Aug 26 '18 at 17:10
  • $\begingroup$ I can't assume the students will have read the book in advance Sure you can, depending on how you run your class. Your current practices send a loud and clear message to your students that you're assuming they will not read the book. If you want your students to read the book, change your practices. For example, give a 5-minute easy multiple-choice quiz on the reading at the beginning of each class for which there was assigned reading. $\endgroup$ – Ben Crowell Nov 4 at 14:08
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Sometimes Stewart picks exactly the example I would have picked, even if I hadn't been prepping from his book. It is usually when the natural example is relatively simple. I wouldn't worry too much if you reuse a few of them -- as you say, they probably didn't read the book and if they did, who knows if they understood it -- but it is good to diversify.

You can do things like covering homework questions which you aren't going to assign, picking a second calculus text for your reference, mining the textbooks of other STEM fields, and keeping a log of good examples you find incidentally to keep things from being verbatim Stewart's explanation.

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An approach I sometimes use is to call students' attention to an example from the book without presenting it in detail, and then present a variation on it.

For example, if I were teaching optimization with constraints, I might say something like this:

In your textbook, which as always I assume you have read before class today, there is an example of minimizing the surface area of a cylindrical can with a given volume. You should make sure you read and understand that example -- it's Example 3 on p. 426. I'm going to do a variation on that problem: let's take a can with a given surface area and try to maximize the volume. Essentially I'm going to exchange the roles of the constraint function and the objective function. Let's see how it looks:

or...

Let's take a second look at Example 3 in the textbook, which (as always) I assume you have read before class. The text solves this problem by using the constraint equation to solve for one variable in terms of the other; when you plug that in to the equation for the surface area you end up with an equation of one variable. But you can also solve this problem by using implicit differentiation. Let's see how that method looks:

This serves multiple purposes:

  • It reminds students that they are expected to read the textbook and that you will hold them accountable for what they read; at the same time,
  • It makes connections between what is covered in class and what is covered in the textbook (most important, in my opinion) without devoting class time to something that should be done outside of class.

One way to think about this is that the examples done in class should be homomorphic to the ones in the textbook, and sometimes even isomorphic, but they should very rarely be identical in all respects.

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  1. My advice is to follow the book very closely in topics, methods, notation, and explication.* But I would not literally use the same examples unless you feel time pressed in preparation. Do an analogous problem but slightly different numbers or formula.

  2. You could also look at the examples and the homework problems and at least do a sanity check. Is there a "missing example" for doing the problems? Or if there are too many examples to work in class (along with time for drill), put your brain to work on selecting those you think more important.

*The desire by instructors to be creative ends up doing more harm than benefit. I believe it comes from a sort of romantic view of the university lecturer rather than a sound pedagogical view of how humans learn (at all levels and ages).

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    $\begingroup$ I wonder if textbooks are driven by good pedagogy or instead the desire to grab as large of the market as possible. $\endgroup$ – James S. Cook Aug 26 '18 at 17:11
  • $\begingroup$ JSC: Much too much of that. Agreed. All that said, instructors doing their own thing are 98% worse. I have had to endure being the test class for instructors trying to move their lectures to books. There are very, very few uni professors who have the anal level of diligence needed to be good textbook writers. $\endgroup$ – guest Aug 26 '18 at 17:18
  • $\begingroup$ If everybody followed this advise, there would be no progress in the way we teach mathematics. Oh, wait... $\endgroup$ – Peter Saveliev Aug 28 '18 at 19:10
  • $\begingroup$ @PeterSaveliev well, we need to give up these romantic ideas about teaching and being creative. All that matters is course evaluations and our ability to give excellent customer service. I hate this idea with every fiber of my being. If the teacher/student relationship cannot be customized to reflect both the student AND teacher then why on earth would any reasonably intelligent person go into university teaching? $\endgroup$ – James S. Cook Aug 29 '18 at 0:51
  • $\begingroup$ @James S. Cook In spite of the rigidity of the curricula, there is always some room for experimentation. $\endgroup$ – Peter Saveliev Aug 30 '18 at 18:43
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When I choose to closely follow the book, I use exercises as examples. For me, it's more work but also more freedom. Meanwhile, the students are exposed to both a polished presentation in the book and an informal discussion during the lecture.

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