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What to do when a majority of students have insufficient mastery of the prerequisite material?

Course Queries Syllabus Queries

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( 3 months ago )

I'm currently teaching a second-semester calculus course in which a significant percentage of my students (over half) either tested into the course just out of high school, took a much easier first semester at a nearby community college, or simply got through the preceding course by the skin of their teeth. For many of them, it has been over a year since their previous math course.

It seems as though there is almost no recollection of limits, differential calculus, or the basics of integration that were included in the first semester material. They seem to be able to keep up with the new topics as long as they are independent of the previous coursework, but whenever I ask them to recall old material, things start to get dramatic. For example, a recent lecture on numerical integration (e.g. Simpson's rule) contained a discussion of error bounds, which involved maximizing a function on an interval. This quickly dissolved into a review of finding extrema while I responded to student questions, and by the time I was done, we had no more time to talk about integration formulas.

I feel like the natural thing to do would be to simply start over, i.e. give a comprehensive review of the entire first semester as quickly as possible before moving on. However, this is a summer course, so it is accelerated, and there is no time for this. Things are already behind schedule.

How can I facilitate a reasonable understanding of the curriculum under these conditions?

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