This post was written by Jeb Collins.
Welcome back to the nuts and bolts of mastery-based testing. This post is part 2 about the logistics of actually implementing this assessment method in the classroom. If you haven’t read part 1, Katie does a great job explaining about how to prepare before the semester and what happens during the first week. In this post I will be talking about what happens once the semester starts: how to create the multiple tests and quizzes that are written throughout the semester, how to grade those tests and quizzes, and how the final grade is calculated. I’ll be writing about my experiences and choices in how this is implemented, with the huge caveat that there are many ways of implementing this method, and mine is simply one of them. I’m sure you’ll see many others in future posts.
Choosing Exam Problems
Once the mastery topics are chosen, then the individual problems must be chosen to appropriately test those topics. There are a couple of things to keep in mind when choosing the problems. First, one of the main differences in exams from a points-based approach is that you need multiple versions of each question. Therefore, you want to choose questions that can be easily modified. I have found that such questions tend to come in two forms. The first is something like a differentiation or integration question, where the question can be changed completely by a small change such as 
I also choose slightly harder problems for my mastery-based tests. I have a couple of reasons for this. First, I don’t give a cumulative final, so once a student has mastered a topic they are never tested on it again. So I want to be sure that students who obtain mastery on a topic have truly mastered the material. Also, each test has only four new problems on it, and therefore is slightly shorter than a typical points-based test.
Finally, something to consider when writing questions is their length. Mastery questions will often have multiple parts to them, since most topics cannot be tested by one problem. However, this can lead to trouble if the question is too long, since failure on one part can lead to a loss of mastery on the whole topic. I have one example from Calculus 2 where I wrote an integration question that included both trig substitution and partial fractions. I noticed that often students would get one part right and another wrong, and therefore wouldn’t get mastery on the topic. The next semester I decided to break both of them up into two separate topics, and it worked much better. I had more total mastery topics, but the students were better able to demonstrate their ability on the different integration techniques.
Writing Exams
Something I didn’t realize when I started using master-based tested was that writing the tests can be considerably different from a points-based method. In a points-based method, each test written is the same length and covers distinct topics. In mastery-based testing, each test is longer than the last, covering new material in addition to different versions of problems from the previous tests. In a points-based method, the quizzes are uniform; that is, each student takes the same quiz. In mastery-based testing, I may have to create as many as 18 different quizzes for one day, as each student may wish to attempt a different problem. All of this adds up to needing a different method to create these tests and quizzes in an efficient manner.
What I like to do is create a library of problems for each mastery topic. I will create a different file for each topic and store all the versions of my problem for that topic in the file. I usually create about 3-10 problems for each topic, creating less versions for the later topics since they will be used less often. This is a good thing to do during the summer or winter break, when I have some extra free time. With this library in hand, writing the exams or quizzes becomes a matter of copy and paste, and can actually be done very quickly.
Quizzes
In mastery-based testing, quizzes are an opportunity for students to obtain mastery on a single topic between the exams. Since failure is not only expected on the exams but is considered beneficial for the students learning, I want to give my students as many opportunities as possible to demonstrate to me they have learned from their failure on previous tests. As the semester goes on, the exams get longer and more stressful for the students, and quizzes are an opportunity for students to demonstrate mastery on a single question. This obviously increases their grade, but it also leaves fewer questions for them to attempt on the next test, which reduces stress. For these reasons I actually find students asking for more quizzes than I have time to give.
When I give quizzes, I let each student choose which mastery questions they would like to attempt on the quiz. This only makes sense since each student will have mastered different topics on the previous exams. For conservation reasons, I don’t print out quizzes with all the questions on them. I have the students email me by the day before the test to let me know which problem they want to attempt. This is usually the biggest problem because students forget about the email, and are then unable to take the quiz. So I remind them often for about a week before the quiz to send me a quick, one-line email, and I also print out extra quizzes . Mostly, by the time the second quiz happens, all the students who want to take the quiz remember to email me. I always give the quiz at the end of the class period. This is because there are some people who will not take it, either because they have mastered all the topics up to that point, or because they are simply not prepared by the time the quiz comes around. I am fine with this, and just let the students leave early.
Grade Distribution
I use the following grade distribution in my mastery-based classes:
- Tests: 80%
- Homework: 20%
I choose to have the majority of the student’s grade decided by the mastery score. My main reason for doing this is because I design the exam questions such that if they show mastery of them, they understand the material of the course. Another reason is because the mastery exams are where growth mindset is emphasized in the course, I want that mindset to be emphasized for the students.
One aspect of mastery-based testing that I have found to vary widely implementers of this testing method is how the average test score is calculated. The way I calculate the test average is to subtract 5% for every question not mastered. This of course means that you can master no questions and still get a positive exam average, but since this would result in an F under any reasonable grading scale, it really doesn’t matter. However, this does make it somewhat difficult for students to understand how to calculate their grade as the semester progresses. To help with this, I provide the following table on my syllabus:
This helps the students get an idea at the beginning of the semester of how many questions they need to master to obtain the grade they want. More importantly, it helps them determine their grade during the semester. This table also allows me to easily emphasize the importance of doing homework, as the students can easily see that doing poorly on homework requires them to do very well on exams to compensate.
Hopefully these posts have given a good broad overview of how a mastery-based class could be run. As I mentioned above, this is only one way in which the class could be set up. There are really as many implementations of this method as there are instructors using it, but these basic guidelines should give a good starting place for those of you thinking about trying out mastery-based testing.
Mastery-Based Testing in Undergraduate Mathematics 
[…] before the course begins — how should you write the syllabus and explain MBT to your students? Part 2 explains the specifics of writing exams and grading that you will be dealing with once the course […]
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