Can I offer bonus opportunities without being unfair to high-scoring students?

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Gamma
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I am teaching a college level physics course. In one of the recent tests, some of my students did not perform well. These students requested additional work in order to raise their scores. How can I give additional work while being fair to the ones who scored well? There were couple of students who scored 100% in the same test and I don't want those students to have scores more than 100. What are some ways to boost their scores?

Note: In the past, when the entire class performed badly in a quiz, I allowed them to re-do the test and averaged the two scores.


Thanks.

Gamma
 
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Tough call- personally, I do not give out bonus points/extra credit work (even though I have been asked to do so by students).

As long as the students are all treated equally- meaning everyone has the same opportunity to get bonus points, there's no real problem (IMO)- but since you set a 'cap' of 100%, you have set up a situation where some students are treated differently than others.
 
I only give two bonus opportunities all term (a pre-class and post-class survey about learning in science), and these are available to all students. I give 9 test equivalents (a mini exam with one weight, 2 midterms each with 2 weights, and a final worth 4 weights) and drop the lowest 2 equivalents (to allow a missed or poor midterm). I'm a little concerned about this last test -- I hope I don't have a bimodal distribution... that's the most difficult case, and what you suggest you have.

A redo (or second new test on the same material, at a given time outside class), with an averaged score, does, as you suggest, keep a cap on scores... and the high scoring students can then choose to participate in the redo or not (in our case, enjoy the sun and warm weather outside as the end of the term nears). It seems to me to be a good idea if you feel up to it. I don't think high scoring students will hold it against you, and I'm sure the low scoring students might appreciate the chance to improve scores.