From Andrew Gelman, an interesting pedagogical suggestion:

**The split screen.** One of the instructors was using the board in a clean and organized way, and this got me thinking of a new idea (not really new, but new to me) of using the blackboard as a split screen. Divide the board in half with a vertical line. 2 sticks of chalk: the instructor works on the left side of the board, the student on the right. On the top of each half of the split screen is a problem to work out. The two problems are similar but not identical. The instructor works out the solution on the left side while the student uses this as a template to solve the problem on the right.

Has anyone tried anything like this? It sounds rather elegant to me.

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I’m skeptical about the value of this — my experience teaching calculus is that what many students learn in high school is how to apply “problem templates” (either from the text or that the instructor worked in class) to their own problems without much understanding. Once they get to college, especially in vector calculus, they are required to have some grasp of the underlying concepts and some students really struggle with this transition.

This semester I was teaching vector calculus with the WebAssign online homework system. It offered something similar to the above. It would let you “practice another version” of a problem, which basically meant various constants were changed. For that other problem, WebAssign would also show you the various steps to solve it. I definitely had students who leaned on this feature too much and were able to complete the HW just fine but then did miserably on the exams. Indeed, when I teach this course next time, I think I will turn off this feature, at least for some of the problems.

My experience has been that by the time students get to vector calc most of the awful ones have been weeded out by then. The bigger stumbling block is usually sequences and series, which is unlike anything they’ve seen before and extremely unlikely to have been taught in high school. Students can’t fall back on a ‘template’ quite as easily as they can with the integration/differentiation material that comes before it, and don’t have the problem solving skills to so much as recognize what type of technique you need to solve them.