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In the USA most mathematics undergraduate classes seem to base the final grade on the following items
I believe that there can be a lot of value in oral exams, so my question is: What is a concrete model for implementing oral exams in an undergraduate class? How could the format look like?
I can imagine that this will look differently for different classes, so specifically I would be interested in beginning calculus.
What would your purpose be for doing oral exams?
My purpose is to allow students to explain something they bombed on a test.
I allow students to retake tests in Caclulus I. On the first test, I ask them to explain the definition of the derivative, comparing it to slope. On the retake, I ask the question a different way. I can't ask that question any more ways. So if they want a second retake on that, they must come to me and explain. I have only had one or two students do this so far.
The problem with oral exams is that they (almost surely) cannot be given simultaneously to a large group (unlike written exams). If the questions are the same and are not known in advance, then those taking the exams earlier are at a disadvantage, because those taking them later might get additional information about the questions from those who took it earlier.
Another disadvantage of oral exams over written exams is that they would take a longer time to administer. Thus the examiner will be forced to make the exam short (shorter than a corresponding written exam), that is, to use fewer questions and cover a smaller scope of topics.
One way to do tackle these two disadvantages is to have a large set of questions announced beforehand, where the actual oral exam consists of a random sample of that set. Everyone has the same information about the questions, the exam would take a shorter time to administer, and the student needs to understand or to be skilled in a wide range of topics to have a good chance of passing the test.
I would also suggest that your oral exam consist of questions that test for understanding and for skill. For example, if the topic is, say, limits, then example tasks that assess understanding could be to give a precise definition of a limit, to give some examples of practical applications that use the concept of limits, or to determine whether a given statement about limits is true or not. An example task that assesses skill would be to find the limit of a given specific function or sequence.
For each homework, a small sample of answers is selected "at random," and the students are required to explain what they handed in. The oral exam grade is then the grade for the homework.
The rationale is that it is impossible to give oral exams to a group of 60 students in a reasonable timescale, and doing it in a fair way. And for homework I'm not interested really if they did the work, but if they understand how it is done. By hanging the threat of an exam over their heads, I hope to get them to at least look at what they turn in.
