Hi everyone, Amanda Harsy and Jessica OShaughnessy here to add a few thoughts on a variation of Mastery-Based Testing. Both of us have been using Mastery-based Testing in our courses over the last two years. Jessica uses MBT in Calculus I, Calculus II, and Introductory Statistics, while Amanda uses it in her Calculus II, III, and Real Analysis courses.

We usually have 16-18 concepts in our Mastery Exams and students can add to their exam grade by mastering these concepts. Now we have a choice: should we treat all the concepts as equal? That is, can students master any of the concepts to build their grade? In some courses, it may make sense to let the students choose.

But what if you feel that not all concepts are created equally? That is, are there some concepts you really think students should have grasped after taking a class?

Both of us use a slight modification in our mastery-based grading to address this belief. For example, since Calculus II is a sequential course, there are certain concepts we think all Calculus II students should master in order to be successful in Calculus III. We want students to successfully be able to differentiate transcendental functions, calculate area, and use integration by parts. In order to enforce these concepts, we have introduced “Core Concepts.” In Amanda’s Calculus II course, for example, these concepts include differentiation and integration of transcendental functions, L’Hopital’s Rule, Advanced Integration Techniques, Calculating Area, Calculating Volumes, and Interval of Convergence for Series. Students must master these 7 concepts in order to earn at least a C for their exam grade. Similarly, Jessica breaks her Calculus I classes into 16 topics and 7 core/required concepts. The students must master these 7 before any other questions count. If they do only these required concepts, they will receive a 70% as their test grade.

We both have a love/hate relationship with core concepts, which we tell students are “required” concepts. We’ll start with the love:

- First of all, it forces students to have “mastered” important concepts they will need to understand to be successful in the subsequent course. As much as we love related rates, our students can probably manage Calculus II without fully understanding the topic. However, the chain rule is absolutely critical. Previously, they could get a C by half understanding the chain rule and half understanding related rates. This way, they know they cannot move on to calculus II without a full grasp on the chain rule (and other required topics). They are better prepared to build on these topics.
- It also helps students focus on particular topics for each exam. For our weaker students, we can direct them to these concepts to get started. We want them to focus on the calculus concepts that they really need.

The cons (we will use cons instead of hate since we think hate is really too strong):

- This method for calculating the mastery exam grade is a little more complicated than treating all concepts equally. For example, it requires students to keep working on some of the more complicated integration techniques until they master them.
- What about when a student masters 14/16 concepts, but one of the two they didn’t master is a required concept? Like limits? Should they be allowed to pass the course when they understand so much of the material? Granted, we think this is an extreme case, but we all know these extreme cases happen so it is important to think about how to handle these situations. Here is one way that Jessica has dealt with these cases.
- “Occasionally I will have students with several required topics going into the final exam. This stresses the students because they feel like they will fail the course if they do not master those required topics. Although I do not tell them, I grade these in a different way on the final. Typically, they master or do not master. However, I write up those that are “close”. If they are close but have not fully mastered a question, then I will count it toward their passing grade, but I will not add it as an extra “mastered” question. For example, 7 mastered questions is a 70% for my test. If they mastered 7 questions: 6 required concepts and 1 non-required, this means they would have 1 required concept unmastered. If they are “close” I will go ahead and give them a 70% for the test grade. I will allow up to two that are “close”. My definition of close may be a few algebraic mistakes or some minor conceptual mistakes. Students who have no idea how to do the chain rule do not obtain a ‘close.’”-Jessica

- What should we do when students don’t try the required concepts? Unfortunately, we both have had students who haven’t attempted the core concepts. Then they complain that they had 7 questions mastered and they don’t understand why they received a D in the course. It is extremely important to emphasize that some concepts are required and to spell this out carefully in the syllabus. Of course, there will always be students who do not pay attention to this, but at least we have done our best.

Overall, we both really like core concepts. This way, we can make our derivative questions hard enough that students will really show us they have mastered the concept without the ability to skip that concept because it is “too hard”. It allows students to be better prepared for their sequential courses and encourages them to review their old material. If you worry that students will skip some important ideas that won’t necessarily show up in other concepts because of the mastery-based system, this may be a good variation for you!