I wrote the title for this post, and then hit "publish", accidentally posting it without any content. I thought about leaving it that way, since it neatly sums up my view of the topic!
OK, that's not true. "Typical" extra credit at Tiny U consists of giving students points for a short reflection paper on a presentation/lecture/movie viewing/concert/etc that is related to the class subject. This type of extra credit has two positive outcomes: 1) it encourages students to take advantage of cultural opportunities around campus, which is particularly useful at a liberal arts college; and 2) it makes students happier, since they think they're improving their grade, even if the points involved are minuscule. (My students tend to be bad at math, and even worse at assessing their risk of doing poorly in the class, whatever "poorly" means to then.)
Note I didn't include "helps students fully understand and apply class concepts" as one of the positive outcomes. In my experience, this type of extra credit very seldom does so.
I've moved away from that model of extra credit, and now, in most of my classes, I have a limited number of pre-fabricated extra credit activities. They are worksheets, self-testing quizzes, fill-in-the-blank study guides, and other material that is useful for studying for the exams. For example, for the primate section of my physical anthropology class, the extra credit activity was to fill in a chart with information about the physical and social adaptations of all the major primate groups. The activities are available on the class webpage from the beginning of the semester, and they are due on the same day as the exam to which they relate. I frequently take exam questions from those extra credit assignments. In other words, the extra credit helps them study for the exam.
Not only does the extra credit directly relate to class concepts, and sometimes help students apply those concepts to previously unfamiliar case studies, I get a lot fewer questions about extra credit opportunities. Students know up-front how many opportunities there are and when they are due, since they are listed directly in the syllabus. I don't have students constantly asking me "will there be more extra credit opportunities coming up?", especially at the very end of the semester when the poorest students suddenly realize that skipping class has left them with gaping holes in their grade and poor test scores.
I've been very happy with this type of extra credit. It's the only kind I'll use from now on.
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