Revisiting Grading

Fifteen months later I’m back writing again and sharing my grading journey. If you are curious about the whole evolution, take a look at my earlier posts with this tag.

First, I’m sure you could imagine why I haven’t written in a while but I finally feel that I’ve moved past the overwhelming sense of burnout from this past year and am ready to continue my quest of revolutionizing the college math classroom. For the purpose of documenting progress and continued steps and changes I do want to share how I handled assessment over the last year.

To provide some context, all of the classes in which I used mastery grading last year were in a hybrid format (some students in classroom, some joining via Zoom) and naturally, the format of the class impacted the way I handled assessments as well. I continued to use and modify the grading system in the same Math and Politics class described in the previous posts and I also incorporated the system in a Statistics course which I taught twice last year. Starting from Spring 2020 (described in the last post), here’s an overview of the main features of my grading systems over the last year which I will follow with more thoughts.

Spring 2020: Math and Politics – 80 total objectives (31 procedural, 34 conceptual, 15 assignments/class)

Summer/Fall 2020: Math and Politics – 25 total objectives (20 content, 5 assignments/class)

January Term 2021: Statistical Data Analysis – 50 “Mastery Points” (34 from content, 16 from assignments/project components/class)

Spring 2021: Statistical Data Analysis – small content rearrangement but same overall structure as above
Math and Politics – 50 “Mastery Points” (40 from content, 10 from assignments/class)

You’ll notice a couple of big changes here. The first relates to the grain size of my objectives; ultimately it was too many and too clunky to deal with. I didn’t change the content of the course at all, I simply consolidated the objectives into broader categories and tested them on individual quizzes rather than collective exams. The second is this notion of “Mastery Points” which refers to my shift from a binary system (you either did or did not meet the objective) to a three-tier system of “Mastery/Intermediate/Beginner” which corresponded to 2, 1, or 0 “Mastery Points”, respectively.

Consolidating Objectives

For my Math and Politics class, I consolidated the longer list of 65 content objectives into 20 content categories which was easy to do given my familiarity with the course material. For Statistics, I started using this approach which ended with 17 content categories. The process was a bit different in that the nature of the course content is more cumulative and so some of the categories were really just demonstrating a conceptual understanding of a wider range of prior content because this is something that can get lost when placing (and assessing) every topic in isolation. In my opinion, breaking down a semester-long course into 15-25 content categories or objectives is the way to go as it will feel (much) more manageable and it is perfectly fine if an objective or category encompasses multiple related ideas or sub-topics. Click below to see my categories and descriptions for each course.

Math and Politics Objectives

Statistics Objectives

Content Quizzes

Admittedly, my approach here was shaped by the move to online teaching. Starting in Summer 2020, in every course I had Moodle (our LMS) quizzes for each of the content categories as my main form of content assessment. For each of the categories, I had a pair of corresponding Moodle quizzes – one labeled “Practice” and one labeled “Mastery.”

The Practice quiz would be available immediately after covering the material in class and contained similar types of problems that they be expected to see on the Mastery quiz. The practice questions would be a mix of multiple choice, drag and drop definitions, fill in the blank, and numerical entry and with all of these questions, students could check the correctness of their answers in the system immediately. I intentionally set up the questions so that if a student’s answer was incorrect, the system would not show the correct answer as I encouraged students to make use of office hours and work together when they were stuck on a problem. These quizzes were optional in the sense that they didn’t “count” towards demonstrating understanding but of course highly recommended as the place to practice without being graded or penalized for mistakes, aka learning.

The Mastery quiz was scheduled ahead of time and once students opened the quiz, the quiz would be on a timer (length varied between 30 and 60 minutes) and all questions appeared as one “essay question.” In other words, there were no multiple choice or simple numerical entry questions as students had to type or write and scan all their work and reasoning to demonstrate understanding of the content. In my Math and Politics class, these quizzes happened during class time and in Statistics, these happened outside of class time (generally available for a day or two) due to the amount of content covered and class time needed, and in both structures students were allowed and encouraged to use their notes. With that being said, I’m not going to pretend that academic dishonesty wasn’t an issue because in both formats it certainly was. This concern warrants its own post but for now I’ll just say that it existed as it does with most classes and types of assessments.

I can report back that students enjoyed this quiz format because it allowed them to grow from the Practice quizzes and ask questions during office hours and then the smaller chunks of content on each quiz (as opposed to a traditional midterm) took some anxiety and pressure away from the assessments, along with the existence of tokens allowing them to retake and improve their scores. On that note, I’ll add that students had five digital tokens that could be used to retake quizzes or submit assignments late and there were a couple of instances in which students could earn an additional token through exceptional work and effort.

Two Tiers to Three Tiers

Why? In my binary system, I noticed many instances of a tricky middle ground and I wanted space to acknowledge growth and progress among students whose understanding was not quite there yet. I will add that reading and listening to experiences of other professors also helped – here’s a shoutout to the Mastery Grading community.

In practice, I still experienced instances of a tricky middle ground now between a 1 (“making progress”) and 2 (“mastery”) and some student feedback revealed they thought it was too difficult to get a 2 which highlights to me the importance of adding more metacognitive work because I don’t think it’s a case of me being “too tough” or setting “too high a bar” but rather students not necessarily knowing what they don’t know.

Using the tiers for quizzes along with different categories (project, assignments, class engagement, etc.) allowed me to put everything on the same scale of “Mastery points” which I personally like because I think it is easier to track for students and is simple to put together a translation of a final grade (see images below for final grade translation and what the shared Google sheet looked like in Statistics). It also allows you to think of the relative weight of a project or other assignments in the overall scheme (yes I know this feels closer to traditional grading, but keep in mind the goal is still working towards objectives) and give you the flexibility to grade other components using a rubric or whatever metric you prefer.

See for my Google Sheets templates and instructions.

What’s Next?

First and foremost – the name. The community is moving away from the “mastery” term for good reason – language matters and it’s best to disassociate from anything that brings up the notion of slavery and if you resist that move you should probably ask yourself why. With that being said, I don’t exactly have a replacement term yet; “standards-based” sounds dry to me to be honest.

I will also be implementing this system in other courses, including undergraduate probability (math major course) which I will be teaching for the first time this Fall. I’ve always held that this system is best when teaching a course that you’ve taught before because you are more familiar with the content objectives but I am confident enough in my abilities at this point to extract appropriate objectives to set up and then execute the system.

Ungrading. This is a big shift from standards-based grading that I will also be implementing this semester in my redistricting course. The motivation: there is a small but growing movement in higher education to remove grades (other than final letter grade) altogether to address the existing power dynamic of grading and increase students’ metacognition and accountability and hopefully move them away from the “path of least resistance.” (“What’s the least I can do to get an A?”) The idea: students will co-construct (first week of semester) what constitutes an A, B, C, and D and grade themselves at the end of the semester. I am not planning to change my existing assignments and projects, but I will build in more regular opportunities for reflection and metacognition for them to track their own learning and progress. I do have to reserve the right to change their final grade if I do not agree with their given grade, and this can go in both directions as research shows that female students are more prone to undervalue themselves and their work in self-assessment.

I will update my progress on both fronts in my continued journey through assessment. And I don’t plan to lose sight of the bigger picture – for me, re-envisioning the mathematical experience for students means creating humanized spaces in the three key areas of (a) content, (b) pedagogy, and (c) assessment. Let’s do this.

Mastery Grading: System Updates

My first three posts were about my motivation, initial design, and first implementation of my mastery-based grading system and now I want to talk about the purposeful changes I made last semester and how the system was impacted by the shift to remote learning.

I taught the same course (Math and Politics – a general education course, see this post for more information and my initial mastery grading setup) so I was able to make direct changes to various components of the mastery system based on the initial experience and my reflections (see this post).


I made a few minor changes to specific objectives (consolidating, eliminating, rewording, etc.) but a more significant change was converting assignments to objectives to (hopefully) give them more value by identifying the main learning goals I sought from the assignments and placing them alongside the other content goals for each of the units. The assignments included completing practice problems (students chose which ones to write up from a larger set) and writing assignments similar to those I had assigned in the past.

As mentioned in my previous reflection, I also added a category of “overall objectives” which included attendance, participation, and four brief reflections that were not tied to a specific unit in the course but that I view as valuable in a general education math course (the relationship between math and privilege, productive struggle, growth mindset language, and a final reflection on the course).

This time around I had a total of 80 course objectives:

  • Voting Theory – 7 procedural, 10 conceptual, 3 from assignments (20 total)
  • Apportionment – 7 procedural, 9 conceptual, 3 from assignments (19 total)
  • Redistricting – 9 procedural, 7 conceptual, 3 from assignments (19 total)
  • Game Theory – 8 procedural, 8 conceptual (16 total)
  • Overall Objectives – attendance, participation, 4 reflections (6 total)

Final Grade Conversion

Rather than using a “bundle” approach as last semester, I decided to simplify the process by having a total number of objectives mastered correspond to a final letter grade. This makes it much clearer to students where they stand at all times and does not distinguish (grade-wise) between the procedural and conceptual objectives. After going through the spreadsheets from last semester I didn’t think it was necessary to separate them in order to translate to a final grade but I still labeled them in their respective categories to identify the different types of (mathematical) thinking that correspond to the objectives.

The updated grade system was as follows:
80 (out of 80) objectives mastered = A
77-79 = A-
74-76 = B+

59-61 = C-

50-52 = D-
< 50 = F


I gave each student 10 (digital) tokens for the semester where each token could be used to retake any objective, submit an assignment late, or resubmit an assignment. In an attempt to motivate higher quality assignments (beyond the minimum needed to satisfy the objectives), I also awarded an additional token to students that submitted excellent work.


This looked similar to last semester with a couple of changes. I added a “progress bar” on the top with boundaries for the different letter grades and a display of their total number of objectives mastered and the corresponding final grade. I also added checkboxes and formatting to the “token bank” to make it easier for myself and add another visual element for students. (Click to enlarge the screenshot below.)

Remote Learning

Given the circumstances, the transition to remote learning amidst the pandemic actually happened at a good time in the course because I had just given the exam on the second unit (it was right before our scheduled Spring Break) meaning I could start fresh on the remaining two units in a new format. For the first two units, I had constructed guided note packets that we would work through in class and quizzes and exams were used to assess the objectives.

For this class, I decided to go the asynchronous route and make videos that went through the notes and examples and then created assignments that corresponded to the videos and the objectives. I also wanted to give students as much flexibility as possible with all the challenges they faced with remote learning and various life situations. As a result, I turned the content into weekly modules where I would post some videos (myself narrating and working through problems on my tablet) and an assignment on Monday which would be due the following Sunday. With 6 weeks of remote learning in the semester and two full units remaining, this worked out nicely in terms of having 3 modules for each unit. (Notice if you look closely at the spreadsheet screenshot above, there are no “procedural” and “conceptual” labels for the last two units as the objectives are organized by the weekly modules instead.) I did not give any quizzes or exams and mastery of objectives was completely based on grading their assignments and the questions that corresponded to the objectives. In terms of accessibility I had virtual office hours each week at set times where students could discuss that week’s assignment with myself and each other.

Another outcome of remote learning is that I did not give a final exam which I did last semester as a “final opportunity” to earn any missing objectives and instead I scheduled individual virtual meetings with students for them to use their tokens. Given the circumstances and stress I did not give them new problems to demonstrate mastery and I simply had them go through their old quizzes, exams, and/or assignments and explain to me why they got certain problems wrong and how the problem would be corrected. The students knew this ahead of time so that they could adequately prepare and this token use is not a structure I would use in “standard” semesters but my human response was to ease some of the expectations in the trying times.

As an anecdote, I’ll add that the final distribution of grades was the most bimodal that I’ve ever had as about 60% of the students got A’s and the other 40% received C’s. I attribute this to students having the ability to change grades to pass/fail and some students with less favorable learning conditions had no problem taking this route to satisfy their general education math requirement with no negative impact on their GPA while others were motivated to earn an A that would boost their GPA.

Student Feedback

I just wanted to share a few things that students shared about their experience with mastery-based grading this semester:

  • “Tokens were great to fall back on but they were limited so it pushed me to study”
  • “The system allowed mistakes to be made and actually learned from”
  • “It was nice to study for understanding rather than for a grade”

Reaction to the grading system this semester was unanimously positive which is extremely encouraging and only continues to motivate me to extend such a system to other classes as well. Thanks for reading and I will continue to share my experiences in math education.

If you’re interested in trying mastery-based grading and want a spreadsheet template, I’m sharing mine with instructions as well:

Spring Heartbreak 2020

Friday, March 13th, 2020. That was the last day of in-person classes this semester.

Tuesday, September 8th, 2020. That’s (for now) the first day of classes for the upcoming Fall semester.

Yes, that’s almost six months (!) without classes in the traditional sense. Now don’t get me wrong, every decision being made is for the greater good and I’m absolutely on board with doing my part to slow down the spread of this pandemic while we wait for more testing and medical advances. With that being said, this hurts. Every single person is affected by all of this in one way or another. It’s about all of us and I simply want to share my feelings to add an example of the myriad of ways that we are impacted and to hopefully shed light on what others might be experiencing.

Here’s the thing: Teaching saved my life. It gave me a purpose, a platform, a passion, and a voice that I haven’t been able to find anywhere else. No matter what stages or levels of anxiety and depression I’ve experienced in the past decade, teaching has been something that I’ve always looked forward to. The classroom is my home. It’s where I feel most confident, empowered, valued, proud, and happy.

And now that place with those feelings and all of the incredible people has been taken away for the next 6-7 weeks. Yes, teaching will attempt to continue in a digital space (more on that shortly), but the teaching that I know is gone. It would be gone for four months during the summer anyway and my past experience knows that this is a problem. Even outside of a pandemic, my day-to-day life is one that isn’t too far off from a self-quarantine (feel free to insert the “finally all of your preparation pays off!” joke) and during the semester this is perfectly fine – occupied by teaching prep, take care of my apartment and body (questionable), and have the classroom to look forward to most days. It’s the between semester breaks where I sometimes find myself too isolated and where depression wants to challenge me to a battle. Some days I win, and some days it wins and I’m left with negative motivation to even leave the couch.

So naturally I’m worried that adding two more months to the already long summer break might have a negative impact on my mental health. Although the move to online classes for the remainder of the semester wasn’t made official until this week, as things progressed last week most of us knew that this was inevitable and there was an aspect of finality to classes. I have an incredibly special Calc 2 class this semester that feels like a family of sorts. The absolute last thing I was going to do in class last Friday was teach calculus. We spent the time and space with each other practicing Tik Tok dances (it’s what college kids do) and then I shared some thoughts similar to what you’re reading here. And naturally I cried. In a “I’m acknowledging the value that all of you incredible people have” sort of way. And to have that feeling reciprocated throughout the room meant everything and it’s exactly why I feel that value and confidence in that space.

Now what? It’s my responsibility to do the best job I can at delivering the remaining content online for my two classes over the rest of the semester. All of the different thoughts and resources shared by faculty all over the country this week have led me to what I believe is a reasonable approach which is to create (short) videos introducing content and explaining key concepts for students to watch on their own time and then use videoconferencing (live and recorded) to practice examples and be able to answer questions. For many of us, this is our first experience with online classes and a week of transition time certainly isn’t enough to figure everything out, but all teaching requires flexibility and ability to adapt in different ways.

I can’t stress how important that flexibility and ability to adapt is in this current situation. It puts academics in its place and to expect all of the same things from students that you normally would is simply inhumane. We’re people first and I shared my feelings as honestly a mild example of how we are impacted. When you build connections you realize there are so many students whose lives are completely different away from a campus community. Additional demands and pressures on their space and time, environments that may not be conducive to work or focus, taking away social groups and activities such as clubs and sports, limited access to resources, the list goes on.

So please, whether you are in academics or not, act with kindness. Act in equity. Act with empathy. Act in love. Certainly good habits to have in general, and if it takes a pandemic to remind us of them then so be it. #SpreadLoveAlways

Know Your Platform

It hasn’t been that long since the tragedy that took the life of Kobe Bryant, his daughter Gianna, and seven others, but I’m still in shock. This one really hurt and was the first time I was impacted by the loss of someone I’ve never met.

Growing up, Kobe was my favorite player and I rooted for the Lakers simply because of him. Obviously, he gained worldwide fame for his abilities on the court, but I’m not here to talk about any of his stats, championships, or other basketball accomplishments.

There are plenty of celebrities – athletes, actors, musicians, etc. – who have become famous and commanded large followings, whether these followings are deserved or not. Our culture has a strange relationship with fame and in the social media age it has become even easier to acquire some level of fame for various good or questionable reasons. And in my quantitative mind, I think it’s reasonable to claim that the impact of our words is proportional to our “fame” or level of following.

And that’s why it hurt so much – it’s all about what Kobe had done post- “Mamba out” with his incredible amount of acquired fame. That leads me to my main message: Know. Your. Platform. Every single one of us has a platform in some form, so it’s a matter of acknowledging and recognizing your audience and reach and what you can do with your platform. Think about the people and places in your life, both physical and digital. This is your immediate audience and reach. Now think about all of those people and the people and places in their lives. This is the opportunity for your reach to spread.

My platform consists largely of college classrooms, young minds, and anything I stand for and represent in those spaces. Those of you that know me know that one of these things is trying to be an advocate for women’s sports and women’s basketball in particular. There’s the Kobe connection. He took his immense platform and following and used his voice to promote women’s basketball and empower young women and grow the game. And I’ll certainly never forget seeing him and Gianna wearing UConn gear in Gampel on Katie Lou and Napheesa’s Senior Day last season.

I could go on and talk more about his children’s book or Oscar or academy, but I’d rather leave you with an opportunity to reflect. What’s your platform? What do you stand for, advocate for, and represent? Your voice matters.

Much love.

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!

An Apprentice in Mastery (Part 1)

In addition to creating a new course this past semester, I also took on another creative teaching endeavor by implementing mastery-based grading in one of my courses for the first time. I’m going to write a short series of posts talking about my approach and my experiences:

  1. Motivation guiding my design (this post)
  2. Implementation of my system
  3. Reflections and moving forward

If you’re reading this, I’m not assuming you have any background in what mastery-based grading is, but I don’t want to spend too much time explaining the various components because they will mostly be revealed through my reasoning and approach. However, it is important that I would describe the key tenets of such a system as (a) assessment and grades are based on the understanding (“mastery”) of learning objectives rather than a traditional numerical scale, and (b) there should be a way for students to have additional attempts to demonstrate mastery.

So why did I want to do something like this in the first place?

  • Giving grades more meaning. When you give a final grade in a course, how does it correspond to what the student actually has accomplished? For example, in a traditional grading system a B could translate to (i) getting everything correct the entire semester but missing a few assignments, (ii) getting B’s on the vast majority of assessments, or (iii) having a wide variety of grades that average to a B in the end. A mastery-based approach treats a course like a license or certification in that you have to demonstrate certain skills to earn the license. Grades then translate to how many of the skills you have demonstrated mastery in and no matter the grade, you will have a tangible list of objectives that you have learned as a result of taking the class.
  • Student mindset. It is always important to attend to students’ mental dispositions and a mastery-based system (in theory) has the ability to make positive impacts on the mindset and anxieties of our students. It is one thing to preach the value of a growth mindset and it is another to align your practices with this value. Allowing students additional opportunities to demonstrate their understanding of learning objectives embodies the growth mindset philosophy of encouraging productive struggle and that learning (especially at the level of mastery) can take different amounts of time for different types of learners. In addition, such a system would hopefully reduce test anxiety where students would know that it is okay to not understand everything at the time of an exam and that they won’t be penalized or judged as inferior for attempting some of the objectives at a later time (and hopefully with additional motivation to achieve mastery).
  • Discourage academic dishonesty. When reading perspectives of others who have utilized mastery-based grading, you probably won’t see this as a reason to employ the system, but I try to be as transparent as possible about my own thought process. In math, homework traditionally counts for 15-30% of your final grade and in my opinion, this only encourages the academic dishonesty that is rampant today (this could be a whole blog post on its own). From a student’s perspective, imagine sitting down the night before a homework assignment is due and not knowing how to do half of it. If the homework is graded for correctness, you would ask a friend for the answers or Google them too, wouldn’t you? For the record, I’m not faulting the student at all for doing this, especially given their stress associated with courses, extra-curriculars, and social life. So academic dishonesty is probably the wrong term, but why use a system that encourages a strategy that is not conducive to making progress towards actually understanding course content? I think it is fair to say that we would like final grades to be (as much as possible) associated with a student’s level of understanding and not the understanding of their friends or people online. We should be encouraging the use of these resources as a means of developing understanding rather than having students feel anxious because they are “cheating” on their homework.
  • Instructor time? In reading perspectives of others, “saving the instructor time” is a commonly cited reason for implementing a mastery-based system, but in my opinion, this isn’t a reason to make a pedagogical change. Yes, as professors we have a lot on our plates and would like to be more efficient, but I would never prioritize this over doing something because I believe it will benefit students in one way or another. In reality, any changes we make will take a lot of time initially and take less time as we gain experience and fine-tune our approaches. I will be expanding on this in one of my next posts, but for now I will say that my implementation this semester added time on the front end in determining objectives, took less time to grade during the semester, and added time in administering retakes.

With these motivations in mind, here were the initial principles I had in mind for designing the structure of my mastery-based grading system:

  • Separating procedural and conceptual thinking. I believe that being explicit about the cognitive demand level we expect from our students is important and I wanted to honor the way in which the mathematics education community differentiates procedural (following steps) and conceptual (displaying reasoning / non-routine problems) thinking.
  • Simple and transparent. The last thing students want is a complicated grading system, especially when it is different from what they are used to, so I wanted to make sure that it was easy to understand how their understanding of objectives would translate to a final grade and that they could see (online) at all times which objectives they have (and have not) mastered.
  • Retake system. This is a key feature of mastery-based grading, so I knew I wanted to incorporate a retake system with a personal goal of trying to find the “sweet spot” of not having too few opportunities for retakes (wouldn’t alleviate the test anxiety issue) and not having too many opportunities for retakes (could take away initial motivation to learn).

I also wanted to figure out a way to incorporate my existing assignments into the system, either as a category or as their own objectives. This is a good time to mention that I did this in a course I had taught the previous two semesters and I think it is much easier to go through the challenging process of identifying learning objectives if you have taught the course before so that you are closer to the content and student outcomes.

In my next post, I will talk about the course, the specifics of the mastery-based system that I created and used, and how the system played out during the semester. Thanks for reading!

The Journey of a New Course

As yet another semester comes to a close, this one was even more special than usual because my vision of creating a new class finally came to life. Considering I assigned my students a final reflection assignment, I thought I’d share my thoughts and experiences as well.

First some background if you haven’t been following along: the goal of creating an entire class on redistricting that authentically uses math in the context of a relevant (and often contentious) political issue was almost two years in the making. In my last semester at UConn (Spring 2018), Adam Giambrone (now teaching at Elmira) and I tried the “let’s throw some things [content] at the wall [students] and see what sticks” approach to teaching this topic which for my money is an underrated approach to education – you don’t always need to have a plan! (More on that later.)

Anyway, upon being hired at Trinity, I was connected to Jack Dougherty who made me quickly realize that my goal of creating such a class was far from a dream, in fact it was exactly the sort of thing the institution wanted and that it had the potential to be brought to life even more through community / government partners. I was also fortunate enough to have the support of my department (Mathematics) who easily could have asked for something more concrete for a new course offering.

Through the help of Jack and Megan Hartline (who both do an amazing job organizing and supporting Trinity’s Community Learning Initiative) I was put in touch with various local professionals who had some connection to redistricting or politics and voting in general. This led me to a government employee (who would prefer to remain anonymous) who is simply incredible and whose knowledge and passion further inspired my own. Our first meeting is when my vision started to feel real for the first time.

Ok so I had a government partner supporting my vision and the course was quickly approved by the school – with the Political Science and Public Policy & Law departments even counting it towards their major! – and my passion for the topic and for education were always there, but creating a class from scratch isn’t easy. To be honest, up until the beginning of this semester I had nothing prepared beyond a list of potential topics, an idea for the final project (having them redistrict Connecticut as authentically as possible), and some materials I developed and used for a 2-3 week unit on the topic in my general education “Math and Politics” class.

Back to not having a plan…here’s the thing: It wasn’t that I didn’t know what to do and it certainly wasn’t laziness. For me part of education is being responsive to your audience and I wanted to gauge the mathematical and political science background of my students and use that to guide my instruction. This was especially important because this was my first time teaching a subject area outside of math or education and I had no prior experiences to rely on to somewhat estimate the level of the students.

As you might imagine, this “approach” led to a lot of late nights making lecture slides, handouts, readings, etc. I always joked with the students that twice a week I felt like one of them working on a class presentation assignment. Creating a course organically and in real-time is not something I would recommend (especially when you have two other preps in the same semester) but it did keep me very close to the content and to current events. On that note, thanks to the state of North Carolina for providing excellent content for this class!

What ended up happening is that effectively all of my course content was front-loaded into the first eight weeks (out of 14) of the semester. At that point I think both myself and the students (we affected each other) sort of “ran out of steam” when it came to content. There were definitely a few other related topics I could have covered, but I’m confident that I covered all of the crucial content and in a way that the students successfully grasped.

At the same time, I was communicating with various stakeholders to create and schedule the authentic learning experiences that I believed were absolutely essential for the course. Managing these logistics takes time and naturally forced these experiences into the second half of the semester, but I think this worked out because students went into the experiences with a strong foundational knowledge of redistricting policy and the associated mathematics.

What experiences am I referring to? In addition to two excellent guest speakers in the first half of the semester, there were three experiences I was able to put together and add (in my and the students’ opinion) incredible value to the course. First, I’m extremely thankful for the Caliper Corporation that generously gave my class a license to the software program Maptitude for Redistricting, which is the actual program used by a majority of the states. Thanks to Cheryl Cape and others in Information Technology, we were able to put together a “Redistricting Lab” in the library for students to grind away and act as independent redistricting commissions to propose a new State Senate map for Connecticut. There were certainly challenges and a learning curve for all of us in operating the software, but it was more than worth it for the authentic experience and the ability to empower students as some of the most qualified individuals for future redistricting in any state.

Second, I can’t thank State Representative (and Chief of Staff to Trinity’s President) Jason Rojas enough for leading a behind-the-scenes tour of the State Capitol and Legislative Office Building. While the focus was not directly on redistricting, so many students reflected with incredible positive feedback about the experience and how it humanized the people working in the government. And it’s not every day you get a chance encounter with the Governor!

Third, I was hoping that students could get an even more personalized meeting with elected officials involved in the redistricting process both to add anecdotes that I or any reading wouldn’t be able to provide and to humanize the process as well. Fortunately with Jason’s help, I was able to have smaller groups of students meet with two former Democratic Speakers of the House (Chris Donovan and Brendan Sharkey) and a former and current Republican Minority Leader of the House (Themis Klarides and Larry Cafero). I am thankful for each of them giving some of their time to our class and I was personally involved in two of the four meetings and I know I learned so much more from them than in researching my lecture slides.

I want to clarify that this class isn’t about me. Anyone can have an idea or vision for a new course but without interested and engaged students it’s simply not possible. This is a good time to mention that registering for this new class was by permission only and that the initial interest of my 22 students (from an initial cap of 19) actually did carryover throughout the semester. I can only imagine our collective attitudes toward education if all of our classes were this way.

With that being said, I do want to share a couple of personal anecdotes. First, I don’t have any formal background in Political Science (I completed half of a minor as an undergraduate!) and I never pretended that I did. I joked that I was a “Math professor masquerading as a Political Science professor” and it was no secret that the advanced Political Science majors were more well-spoken about the field in general than I was. And that’s ok! There’s something to be said about learning together and using our unique knowledge as individuals to advance our collective knowledge. As for my own knowledge, I certainly noticed a shift as the semester progressed. Thanks to the students, the guest speakers, the legislators, and hours of research in preparing lecture slides and activities, I noticed myself transition from someone who is knowledgeable and interested in redistricting to someone who is an expert in redistricting (particularly in the context of Connecticut).

My final takeaway: In looking back, I’m proud of what I was able to provide my students both in terms of content and experiences in my first attempt at doing something like this. It’s really amazing to think back to an initial vision and realize everything that contributed to its reality. Thanks to my experiences and my students’ input, my head has been spinning at all of the tweaks I can make to improve this class when (not if) I teach this class again. I firmly believe that this type of class represents something we should do more often in education, but clearly it’s not easy. Most importantly, we can’t do this alone. It takes (a lot of) communication and collaboration with others at your institution and the right people in your community. It takes others to trust and support your vision. It takes the right students. It takes the Supreme Court making a key ruling two months before the semester (just kidding, but it does help). It takes a belief that math truly matters. And finally, it takes a commitment to the philosophy that the sharing of perspectives and experiences are at the core of education.

For more information about the course content, see my page about the course.