An Apprentice in Mastery (Part 3)

Now that I explained the components of my mastery-based system to you in my last post and to the students on the first day of the semester, I want to share some thoughts about how this all played out in practice.

Changing the language and the narrative. This probably isn’t surprising, but it is interesting – students quickly adapted their language to match the language of the grading system. “Checkboxes” or just “checks” became the predominant vocabulary along with “tokens” and “understanding.” Early on, students knew what was expected to achieve mastery of an objective and most of their language matched my goal of improving student mindset (and aligned with the mindset language I regularly used in the classroom).

Homework Assignments. The initial plan was for these assignments to be a great opportunity to earn feedback towards your mastery of objectives and not impact your final grade, but because they were de-emphasized in the grades and only grading for completion, they slid further down my own list of priorities, especially with two other preps this semester. I moved these responsibilities to my TA for the class and I am also more convinced now than before that it is okay for homework to be optional if students are able to demonstrate mastery in the end and that I should provide them with practice problems but not force them to turn in the problems. The shift in language to “practice problems” might have a positive effect as well, especially if there are ways for students to quickly self-check their answers/progress (such as through an online “quiz”).

Writing Assignments. Unsurprisingly, the quality of these assignments decreased since students just needed to efficiently answer the questions to earn a procedural objective rather than need a more comprehensive write-up as a percentage of their grade. With that being said, I still believe that the content in these assignments brings the important political context to my course, so my plan is to adjust and combine these assignments to mini-research projects with more corresponding checkboxes to increase the quality and emphasize the importance of the political context. My initial thought is to create one of these per unit (four in total) while keeping the Redistricting project as one of the four assignments.

Grading. I can only remember two small instances of students contesting any grading and both were cases of them arguing that they demonstrated enough mastery to be awarded a check on an objective from an exam question. As you might imagine, the grading process itself was certainly quicker than in previous semesters. While there are some cases that are borderline in terms of determining whether a student’s reasoning sufficiently demonstrates mastery of an objective, many are clear-cut and the lack of any partial credit accelerates the grading process.

“What’s my current grade in the class?” It’s a question we get all the time, and one that isn’t usually too hard to answer – it’s whatever the grade formula gives you based on the assignments and tests so far. Unfortunately, this question was more difficult for me to answer (other than in the extreme cases) using this system because it was so dependent on your performance on each upcoming objective and unit and because most of the students still had tokens in the bank until the end of the semester. As long as I can find a nice way to do this in Google Sheets, my current plan is to implement a “progress bar” that automatically shows your progress towards final letter grades based on how many objectives you have mastered at any point.

Token system. A few students chose the strategy of retaking objectives shortly after exams because they were fresher in their minds, but the vast majority saved all of their tokens until Finals week as I allowed them to use the tokens at any point before the final. I always talked about token strategies with the whole class and with individuals to guide them in ways that I felt made the most sense for each student. Common strategies I emphasized were (i) save them until the end, (ii) retake procedures first (based on final grade implications), and (iii) retake objectives from the same unit (less content to review if taking the final).

Up until Finals week, I had no issues managing token use as not too many were used and I would generally write out a couple of questions or have an oral exam/conversation for students to demonstrate their understanding of whatever objective(s) they were coming in to retake. But with a class of 30 students and roughly 100 tokens collectively still to use (some students didn’t need any or needed less than 5 because they had already achieved all of the objectives), Finals week was overwhelming to say the least. My “solution” was to create a sign-up so that no more than 4-5 students would be in the office at once, but I ultimately needed 12 hours (4 hours each on 3 straight days) to complete all the retakes. This was by far the most inefficient element of the system and my plan is to create a digital repository of questions (rather than a mental one) to draw from for retakes and essentially just print out quizzes to make this process go (hopefully) smoother than this first time.

A smaller change in the system is to have a 1-1 ratio instead of a 1-2 ratio of tokens to objectives that can be retaken. I still think that retaking up to 10 objectives “feels right” and although it is arbitrary to an extent, again my goal was to balance initial motivation with growth mindset and opportunities to learn at your own pace over time so my plan is to give 10 tokens this time around. This also has the effect of slightly decreasing the impact of any “bonus tokens” which I will probably add a couple more opportunities to earn as well to hopefully increase motivation on assignments.

Overall Objectives. To further streamline the final grade conversion, I believe that it is more sensible to convert holistic aspects such as attendance and participation from final grading categories to “overall objectives” that would carry the same weight as a procedural or conceptual objective. Placing a small amount of these overall objectives on the spreadsheet would allow every component of the class to be visible (thus “tangible”) to the students.

Student Reflections. At the end of the semester, I gave a survey to the students about their thoughts and experiences on various components of the mastery-based grading system and are some pieces of feedback that best capture the class as a whole:

  • “clear visual of what I accomplished and what I needed to work on”
  • “less stressed because I know I can redo checks”
  • “focused more on understanding instead of memorizing material”

The transparency of the system is what students talked about most, and this is encouraging as one reason to use this system is to keep everyone closer to the content of the course. Lowering test anxiety is also a critical feature as I always try my best to not only be responsive and reactive, but proactive to the mental health concerns of students. And setting up the two categories of procedural and conceptual objectives was done to emphasize the importance of conceptual understanding.

Students also offered suggestions for improvements to the system such as additional tokens and partial credit opportunities, but no single suggestion appeared more than twice. I should also mention that 20 of the 24 students that responded to the survey (from a class of 30) said that they liked or really liked (the top 2 choices from a 4-point Likert scale) the use of mastery-based grading.

Stay tuned for more updates as I implement changes to the system this semester based on my experiences. Thanks for reading and I hope this makes you think more about your own course objectives and assessment. From one reflective practitioner to another.

An Apprentice in Mastery (Part 2)

For my second post, I want to describe for you the mastery-based system that I used for my first attempt, but before I do that, let me give you a quick overview of the course in which it was used.

Course: Math 114 – “Math and Politics”

This is a general education course and one that I taught in each of my previous two semesters at Trinity. The course content and assessments have evolved through the semesters, with the course covering the following four units (some associated topics):

  • Voting Theory (unweighted voting methods, fairness criteria, weighted voting)
  • Apportionment (methods, history, paradoxes, voting power)
  • Redistricting (overview, efficiency gap, compactness)
  • Game Theory (Nash equilibrium, simultaneous and sequential games, mixed strategies)

Before implementing the mastery-based system, assessment in this course consisted of unit exams, regular homework assignments, written responses to articles relating the topics to politics, and a group project.

Keeping in mind my goals from the last post of procedures vs. concepts, simple and transparent, and including a retake system, here’s the grading system I used in Fall 2019:

Learning Objectives

I translated content goals from having taught the course before into explicit learning objectives in which students must display mastery. Each unit had their own set of objectives with half of them categorized as “Procedural” and the other half categorized as “Conceptual” objectives based on the associated level of thinking. There was a total of 76 objectives split almost equally across the four units:

  • Voting Theory – 10 Procedural, 10 Conceptual
  • Apportionment – 10 Procedural, 10 Conceptual
  • Redistricting – 10 Procedural, 10 Conceptual
  • Game Theory – 8 Procedural, 8 Conceptual


These objectives were assessed through in-class quizzes and exams. For the first two units, two quizzes were given only covering procedural objectives and the exam covered all of the conceptual objectives. Part of the motivation for this was having 50 minute class periods and not wanting students to use any of that time going through any potentially tedious (procedural) calculations. For the last two units, the quizzes and exams covered a mix of objectives, but this was a result of the way the content was organized as opposed to an intentional shift in structure.

The regular homework assignments were maintained but could not count towards demonstrating mastery. They were graded simply on based on completion and were used as an opportunity for students to gain practice and feedback on their progress towards the objectives rather than as a higher-stakes portion of their grade based on correctness. See below for how these assignments contributed to their final letter grade.

The written responses were maintained but streamlined for students to only answer questions that I viewed as the most crucial to get out of the article and to briefly reflect on the political context. The main point of each article (one per unit) was included as one of the procedural objectives and successful completion of the assignment was enough for students to demonstrate mastery of that objective.

The group project was also maintained and streamlined compared to the previous semesters. The project was for students to create their own new congressional district map for a state using an online web application and to write a brief report on the history of redistricting in that state and to show the efficiency gap and compactness calculations of their proposal which they had learned in that unit. The main goals of the project were translated into 3 of the 10 conceptual objectives for that unit.

Spreadsheet and Checkboxes

To ensure that students had easy access to their objectives at all times, I created a Google spreadsheet with all objectives separated by unit and separated by procedural/conceptual. Each objective had a checkbox next to it that I would mark when students achieved that objective and I set up conditional formatting to highlight the objective green when it was mastered.

Logistical note: I created one master spreadsheet file with a tab for each student and then a separate file for each student that was shared with them and automatically synced to their tab on the master file. This is why I had to use Google Sheets because Excel Online doesn’t support syncing from another file. Below is a screenshot of the spreadsheet (click to enlarge).


To address retakes and support student’s control over their learning, I created a digital token system and gave each student 5 tokens to use for the semester. Each token could be used to retake up to 2 objectives or to submit a homework assignment later than the deadline. There was also an opportunity to earn an extra token for an excellent group project report. Students could retake objectives at any time during the semester and could retake any objective multiple times if needed. Using a token to retake objectives consisted of coming to office hours and me providing them with new questions related to the objectives for them to demonstrate mastery. As you may have noticed, tracking their tokens was also a part of each individual’s spreadsheet.

As you might imagine, many of these tokens were used at the end of the semester as students saved them to see how many objectives they would need to retake. More on this and other reflections coming in the next post!

Final Grades

To convert a student’s mastery of objectives into a final grade, I created and used the conversion table below which factored in the number of objectives checked off in the procedural and conceptual categories separately as well as the number of assignments completed and their attendance and participation in the class. You can see the table below taken from the course syllabus (click to enlarge) where your final grade would be the first row in which you met all four categories. For example, if you completed all assignments, attended and participated in class, mastered 35 procedural objectives, and mastered 29 conceptual objectives, your final grade would be a C+ as you would not have achieved every column in any of the rows above that grade.

Part 3 will be some of my reflections on how various components of this system actually worked and thinking about some changes for next semester!