I am not an instructor but for the online classes I have taken, all of them were administered through Blackboard. Blackboard allows the instructor to set a time limit on exams/quizzes. While taking the exam a small box appears on the right-hand side of the screen showing a countdown from the maximum time allowed by the instructor. It will also give a 1-minute warning when you are about to run out of time.
Once the exam/quiz is complete, it will state the amount of time taken to complete it at the top of the screen. I imagine this is the information you are looking for. Unfortunately I think that Blackboard is not meant to be used for just one class. It is an online course system that the college or university you work for would need to invest in. Still, it is a great system and something worth checking into.
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