By Emily McMillon and George Nasr (University of Nebraska-Lincoln)

We—Emily McMillon and George Nasr—are graduate students at the University of Nebraska-Lincoln. We implemented mastery based testing for two sections of a course on geometry for pre-service elementary teachers during the Spring 2020 semester, and found that our students

• looked over mistakes on assessments to improve their understanding,
• felt less stress and testing anxiety,
• experienced increased confidence in mathematics and greater growth mindset,
• viewed exams as an opportunity to show knowledge, and
• reflected on the purpose of assessment in student learning.

In this post, we will discuss what led us to try mastery based testing for this student population, how we implemented mastery based testing in our courses, and some student survey responses.

Motivation

We first heard about Mastery Grading late in 2019 when Austin Mohr gave a talk at UNL on the topic. At the time, we were both teaching mathematics courses for pre-service elementary teachers. Hearing about Mastery Grading, we both, independently, thought this type of grading would be excellent for pre-service teachers. Hence, with permission from our department, we decided to implement mastery grading in two sections of the same course in the Spring 2020 semester.

Before describing what exactly Mastery Grading is, we would like to discuss some general learning goals we find valuable in a course for future elementary school teachers. Our first goal is to guarantee that our students fully understand most of the course concepts upon leaving the class. We feel that it is particularly crucial that students in an education program fully understand concepts, given that they are responsible for being able to articulate similar concepts to their future students.

A second goal is to encourage students to revisit and reflect on their previous work and mistakes. It is particularly imperative that future teachers understand that mathematical ability can be improved upon, as studies have shown that elementary teachers pass on their views of mathematics to their students.

Our third goal is to broaden the scope of students’ understanding of the purpose of assessments beyond a numeric score. As future teachers, it is important that they are at the very least aware of different styles of assessment, and, ideally, critically assess different styles of assessment to determine which is ideal for their own future students.

Overall, we believe it is important that elementary education mathematics classes are designed in a way that encourages future teachers to continue working on concepts until they have demonstrated understanding. We want our students leaving these classes feeling confident that they have truly mastered the concepts that they may one day teach for themselves. We also want assessments to be seen as low-stakes opportunities for students to show us the progress they have made, while also incentivizing them to look back at their mistakes and try to understand what it is they have yet to learn. We believe this can be accomplished with Mastery Grading.

Mastery Grading

Mastery Grading is a grading scheme by which students are expected to show complete understanding of course objectives. This is done by offering multiple opportunities throughout the semester to reattempt course objectives for all or nothing credit. There is no penalty for students taking longer to master a course objective. The goal is for students to eventually show that they understand the material, not for students to necessarily demonstrate complete understanding of material the first time it is assessed.

There are many variations on mastery based grading; our implementation as described below is but one example. Many additional resources are available online. We found the following blog very helpful and so pass it along to the interested reader: https://mbtmath.wordpress.com/.

We believe that Mastery Grading helps achieve the three goals we mentioned in the preceding section on our motivation. Mastery is designed to encourage students to revisit concepts to receive full credit for learning them. In a point-based class, students can earn partial credit for partially learning something and then may never have to revisit that concept again. In this way, students will ideally leave this course with a robust understanding of the course content.
Another feature of mastery is that it only rewards students points for a problem once they have shown full understanding of the underlying concept. This incentivizes learning from mistakes and has the potential to help students cultivate a growth mindset toward mathematics. We also feel that mastery provides students with another perspective on how to run a class and assign grades.

Course Structure

Usual Course Structure

Geometry Matters is a required course for most elementary education majors at UNL. The course covers geometry and measurement and follows chapters 10-14 of Sybilla Beckmann’s textbook Mathematics for Elementary Teachers. This course is part of a three-course sequence that covers chapters 1-14 of the aforementioned textbook. The first course in the sequence is Math Matters, which must be taken prior to Geometry Matters, covers chapters 1-7 in the textbook. The other course in the sequence, Math Modeling, covers chapters 8-10 and can be taken at any point.

The course is taught by faculty, lecturers, and advanced graduate teaching assistants, depending upon instructor availability in any given semester. The course grade is usually determined by some combination of assessment scores, homework scores, and written project scores (so-called “Habits of Mind” problems). Students tend to do well in the course — in the last six years, pass rates have ranged from 79% to 100%, with most semesters having over 90% of students pass the course. Hence, grades and pass rates were not a reason we decided to implement Mastery Grading.

How We Applied Mastery Grading

We divided the course content into 18 Learning Outcomes. Student grades were based 60% on mastery of these outcomes, with homework and project problems making up the remaining 40%. Grading for individual Learning Outcomes was for all or nothing credit, and homework and project problems were graded with a traditional points-based system. Our original plan was to test outcomes 1–7 on the first assessment, 1–13 on the second assessment, and 1–18 on the third assessment. The final would not cover new material but would be a final opportunity to master previously not mastered outcomes. In addition, we planned to offer occasional opportunities to take one to two outcomes as “quizzes” in class as the opportunities arose.

Modifications (Moving Online)

As these courses were taught during the Spring 2020 semester, we were forced to move the courses online in March of 2020. We chose to make some modifications to the course assessment structure to better work in the online, asynchronous format required by our university.
Before the move to online, we had given the first assessment as well as two mastery quizzes. The second assessment had to be taken online. We decided to eliminate the third assessment and instead replace it with weekly mastery quizzes that would each test a single new concept and offer an opportunity for students to reattempt up to two learning outcomes they had not yet mastered. Recall that quizzes were made up of exam-level problems—the only difference between these and exams was the quantity of problems. The final exam remained as previously scheduled, albeit online.

An Example of a Learning Outcome and Student Work

The following is a description of one of our 18 Learning Outcomes assessing areas of polygons other than rectangles, which spans sections 12.3 and 12.4 of our textbook.

1. Be able to determine the area of triangles and parallelograms in various ways, including by making reference to the moving and additivity principles of area.
2. Be able to use the area formulas for triangles and parallelograms to determine areas and solve problems.

That is, to earn points for this learning outcome, students would have to show mastery of both parts A and B.

Here is a sample two-part problem assessing this learning outcome, along with work from a student that did not master the concept on their first try. Students knew they may use the formulas for the areas of standard shapes such as rectangles, triangles, and parallelograms. Students also knew they were expected to express reasoning for their conclusions, and to substantiate their reasoning with ideas such as principles of area.

On part (a), the student was very close and would have earned most points for this part, but we would have liked the student to say that you can form a rectangle out of two triangles of equal area, and hence, half of the area of the rectangle is the area of either triangle. One can infer from the dashed lines the student drew on the triangle provided that they are thinking about this as two triangles forming a rectangle, but being explicit in their explanation was critical for us to ensure their understanding.

However, the work on part (b) is what really led us to feel it was critical to have the student spend more time reviewing this outcome. The point of this part was for the student to recognize the shaded region could be decomposed into a triangle and parallelogram, and that adding the area of these shapes would yield the area of the original region. The student’s attempt still showed some understanding of how one can decompose and move regions in an effort to figure out the area, which showed a desire to use principles of area. However, if one carefully checks, it is not possible to fit both triangles the student creates in the specified regions, and this is critical for the student’s method to work. (Though coincidently, they do get the right answer for the area.)

Recalling that each learning outcome was worth 15 points. We would say the student would have earned around 8 points on this problem had this been graded with points. However, due to the mastery grading system, this student had a second chance to demonstrate their understanding. Below, we show a second version of a problem for this learning outcome and the student’s response.

We recognize that part (a) has some unconventional notational choices, but we feel it is clear the student showed comprehension of the underlying concept being assessed. On part (b), we see clear improvement from the first attempt in the student’s ability to find the area of a large shape by decomposing it into smaller, familiar regions. This student earned certification of mastery on this attempt.

Student Performance

We would like to give a brief overview of how our students did. By the end of the semester, 37 of our 42 students had mastered at least 17 of our 18 learning outcomes, and no student mastered fewer than 14 learning outcomes. Many concepts were mastered by students on their first attempt, and the majority of students needed at most two attempts to master a concept.

Survey Results

At the end of the semester, we surveyed our students on their experiences in the course. There was no concrete incentive to complete the survey, but 41 out of our 42 students completed the form.

This survey consisted of two parts — a series of Likert questions, and a series of open-response questions.

Likert Questions

We asked students to respond to the following three statements on a scale of 1 to 5, with 1 meaning “strongly disagree” and 5 meaning “strongly agree”.

1. I feel like mastery grading allowed me to demonstrate my understanding of the course content.
2. Mastery based grading influenced me to look at exams and try to understand my mistakes.
3. This course has made me more confident in my ability to learn math.

Below are the results.

We noticed two students who wrote overwhelmingly positive things in the next part of the survey but responded with “strongly disagree” to these questions, so we infer that these responses to the Likert survey were the opposite of their intended responses.

Free Response Questions

We also wanted to give students a chance to share, in their own words, how their experiences with mastery grading impacted their experiences. We asked our students a few questions regarding their experiences with mastery grading. We also asked them to compare these assessments with points-based assessments they’d had in Math 300 (a prerequisite course also about mathematics for future elementary teachers) and to reflect on how these experiences would impact their future teaching.

Working through the responses, we found several themes that were shared among many students, which we now discuss, categorized into expected and unexpected results.

Results We Expected

Content Understanding: As instructors, we noticed that the work being turned in by our students was of exceptionally high quality as compared to previous semesters of mathematics courses we had taught for future teachers. A few students remarked on their personal feelings that they had taken more away from the course than they might have under a points-based system.

• “Even though we didn’t have a final, I think I would have been able to pass a final easily because I actually remember the learning outcomes. This is probably due to doing the homework and actually caring to learn what I did wrong and how I can fix it. In the past, I just did the word for an ‘A’ and didn’t really bother to learn it.”
• “I felt less pressure to cram studying and to be perfect. I felt like I studied to actually understand the material.”
• “[Mastery grading] made me care more about my learning rather than stressing over a test score. I was more willing to put in the work and less motivated to use shortcuts.”

Learning from Mistakes: We found that mastery grading encouraged our students to look back at their mistakes on their exams. Of our survey respondents, 16 mentioned learning from mistakes as a positive takeaway of the course in their open survey responses. Many commented that they would have never looked back at mistakes they made on exams in other classes. The following two quotes are representative of the types of responses in this category. Some students mentioned specific learning outcomes they learned best, while others gave more general responses indicating that looking back at their mistakes benefited their learning.

• “I felt I learned how to do [Learning Outcome 5] the best during this course. I learned this because I failed the first time and I had to go back and figure out what I was doing wrong.”
• “I had multiple chances to show that I could master concepts and could prove that I can learn from mistakes and better myself in math.”

Math Confidence and Growth Mindset: As one may expect from our third Likert question, several students indicated feeling more confident in mathematics. Students mentioned how they were able to learn content they did not think they would have been able to learn at the start of the semester. We also found some encouraging comments about students’ development of their growth mindset. In total, 8 respondents explicitly mentioned math confidence or an increased growth mindset in their responses. A representative comment is:

• “It isn’t the end of the world if you don’t pass on the first try, you just have to keep trying.”

One of the interesting results was that some students even commented on growth mindset oriented toward their future students, as in the comment that follows.

• “I will always tell my students to keep trying and they will get it eventually, sometimes it just takes more time and effort!”

Results We Didn’t Expect

Stress and Anxiety: 15 students indicated in their responses that exams felt a lot less stressful since they could redo their mistakes. Several of our students admitted to struggling with testing anxiety and said that this grading scheme gave them some relief to that. It should be noted that several students commented that at the beginning of the semester, the “all or nothing” nature of these exams seemed daunting. However, all these students said that things improved once they became more familiar with the grading scheme and started passing outcomes.

Exams as Opportunities: Mastery grading also affected how at least 6 respondents felt about exams. In particular, they felt that exams were an opportunity to show their knowledge and understanding as opposed to a hurdle to be overcome. The following quotes represent these responses.

• “I knew that my instructor was looking for key factors that indicated I knew the material [on assessments].”
• “[Mastery] was based on creating a genuine understanding of the content. I feel that traditional math assessments can sometimes be more discouraging with trick questions, and this is more transparent with concrete goals and objectives.”

In our experience, students sometimes view mathematics exams as being antagonistic, unfair, or that our goal as instructors is to trick them or make them get a lower grade. To us, these responses show that students saw this grading scheme as being friendlier and more conducive to allowing them to demonstrate their understanding.

Student Learning: Our future teachers also demonstrated an immense capacity to think about their future students. It appears mastery grading encouraged some to think more carefully about what their concrete goals are as instructors, such as with the following student.

• “[Mastery] has helped me realize that as a teacher, I want to always ask myself, `What do I really want them to know? How do I want them to show it?’ … There are outside factors that may have messed them up in the moment, but the mastery of that skill is what I should be looking for.”

They also showed an ability to preemptively empathize with their potential students. In particular, many students who admitted to struggling with testing and/or math anxiety commented on wanting to try mastery based grading with their own students as a way to alleviate their students’ testing anxiety.

• “Sometimes students learn at a different pace, [and it’s] not fair to give one shot at something [where] if they don’t do well they can’t redeem themselves.

Other students perhaps did not struggle with testing anxiety, but still saw the importance of giving students multiple opportunities to demonstrate their knowledge.

• “I would want my students who struggle with math exams [to] benefit as much as they can like I did!”

Challenges: It is important for us to acknowledge that not all feedback we received was positive. There were two common themes among those that found issues with Mastery Grading. First, students did not enjoy having to redo outcomes when they thought they misunderstood only a small portion of the outcome or made only a small error. Second, students still wished they could get some partial credit for the ideas for which they did demonstrate a good understanding. A small number of students commented that having multiple chances led them to care less about any individual assessment, and so they studied less. We also noted a trend that students who had taken more “traditional” math courses, i.e., calculus sequences courses, seemed more frustrated by having to retake outcomes when they made fairly small errors.

Discussion

We believe we accomplished two of our three main goals. Students seemed to be successful in understanding to course content. In addition, students appeared encouraged to learn the content and felt motivated to understand their mistakes. We even saw that students felt more confident with mathematics and demonstrated a growth mindset. However, we are less confident that we broadened the scope of students’ understanding of the purpose of assessments beyond a numeric score, although based on some student comments, it appears that our students at least started thinking about this.

There were a few less expected results that we were delighted to see. Students generally reported feeling less anxious about exams since they knew they would have multiple opportunities to show what they know. Students felt assessments gave them the chance to accurately show their knowledge. They also reflected on their experiences with mastery and how it might inform their future teaching.

We feel that future teachers were the most amenable to this style of grading as they themselves tend to value the opportunity to grow and learn.

More to this point, we have already seen evidence of how mastery has affected their future experiences. We highlight one piece of evidence here. During Fall 2020, the semester after we implemented mastery grading, some of our students took another math for future teachers class with our colleague, Kelsey Quigley. At one point during the semester, Kelsey offered an opportunity for her students to earn points back on their first exam. As she discussed logistical considerations with them, a student suggested that the way they should earn points back would be to redo the problems they individually did not do well on, as opposed to the problems the class did not do well on overall. They said, “It’s like you’re mastering the concepts you missed versus going back and doing the ones that you understood.” Kelsey had the impression that this student gained this perspective through their experiences with mastery based testing in our course.

The challenges mentioned in the previous section are important to address. While many of the challenges presented by the students are inherent to mastery grading, we feel that there are a few things that instructors can do to address the issues.

1. Have regular conversations with your students. Mastery is likely to be new, so having these conversations can help them understand how to feel. Be transparent. Tell them why you’re doing this.
2. Positive feedback may compensate instead of partial credit. While it is not the same as getting points, you at least send the message that you recognize the good work they did.
3. You can discuss how not getting partial credit doesn’t mean you didn’t do anything good, and how it doesn’t hurt to continue to practice skills you already understand.
   What we described is not the only way to approach mastery. Some implementations have multiple “Levels” of mastery, so in that sense students earn partial credit.

In conclusion, we found mastery grading to be a rewarding experience both for us as instructors as well as for our students. This testing style felt like a perfect fit for pre-service teachers, and we would encourage any instructors of pre-service elementary teachers to consider giving mastery based grading a try in their courses.

Acknowledgements

We wish to thank Allan Donsig and Michelle Homp for backing our desire to teach this class using mastery-based testing, Wendy Smith for her help in designing our study and methods of data collection, and Yvonne Lai for her helpful feedback and guidance in writing this article. Finally, we would like to thank Austin Mohr for introducing us to this testing method and inspiring us to try it ourselves.