Type #1: Explain
Have students take a test on their own (don’t grade the tests). Once they’re done, give students the key and have them grade their own.
Ask students to make corrections and then create a video of their explanations of the three (or however many) most difficult problems for them. They may or may not be questions they initially got incorrect. Students should submit their initial test, corrections, and video. Grade is given on accuracy of grading, accuracy and cleanliness of corrections, and appropriate video explanations. Grade is not based on initial number of questions missed as this would encourage nefarious means.
Type #2: Create
Have students create their own test, key, and solutions. Give detailed directions on number and type of questions.
For multiple choice questions, have students explain why the distractors are good. For free response questions, give the students the questions but have them create the stem and the answer key.