Interrogating Grading Practices in Mathematics Methods

Creating Threshold Opportunities for Examining Institutional Norms
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AssessmentMathematics MethodsRelational Teacher Education
Calls for interrogation of practice urge educators to consider ways to dismantle traditional assessment systems. Additionally, leaders in teacher preparation research urge investigations that build understanding of knowledge and practices of teacher educators, including knowledge of how teacher educators develop practices such as grading and assessment to support the learning of prospective teachers. As mathematics teacher educators, we responded to these calls by engaging in long-term self-study of grading practices in a mathematics methods course where prospective teachers are encouraged to build autonomy for their own pedagogical growth. Data (including journals, class audio recordings, final evaluation conference transcripts, and the transcript of a whole group interview with 10 PTs focused on grading practices) were drawn from the second term of ungrading along with research documentation. In addition, the authors recorded every research team meeting and kept collaborative meeting notes. We describe how exploring impacts of this revised assessment system presented a threshold opportunity to rethink the use of traditional institutional structures such as grading rubrics. Results of this study reveal that removing one institutional structure produced an unintentional focus on a different structure and point to the need for close consideration of institutional structures that impact the work of teacher educators.


Calls for interrogation of practice urge educators to consider ways to disassemble traditional assessment systems (Brubaker, 2010; Buhagiar, 2007; Rojstaczer & Healy, 2012). Additionally, leaders in teacher preparation research urge investigations that build an understanding of knowledge and practices of teacher educators (e.g., Beswick & Goos, 2018; Krainer et al., 2021; Vanassche & Berry, 2020), including knowledge of how teacher educators develop practices such as grading and assessment to support the learning of prospective teachers (PTs). As mathematics teacher educators, we responded to these calls by engaging in long-term self-study of grading practices in a mathematics methods course where prospective teachers are encouraged to build autonomy for their own pedagogical growth. We describe how exploring impacts of this revised assessment system presented a threshold opportunity to rethink the use of traditional institutional structures such as grading rubrics. Results of this study point to the need for close consideration of institutional structures to inform the work of teacher educators.


This self-study of teacher education practices took place in a mathematics methods course taught at a four-year institution in the southeastern United States. Alyson, the course instructor and first author, engaged in relational teacher education (Kitchen, 2005) from a constructivist teaching paradigm (Steffe & D’Ambrosio, 1995) and actively worked toward creating mathematics methods courses in which PTs’ experiences and reflections played a vital role in the development of PTs’ pedagogical practices.

Background Literature

Despite discontent with traditional grading systems (e.g., Blum, 2020), there is little scholarly inquiry (Lee & Mewborn, 2009) into course grading systems. Two studies specifically focused on methods for determining course grades and impacts of alternative approaches to this process. McClam and Sevier’s (2010) self-study reported on efforts to change to a student-assigned grading system in a multi-section methods course and the resistance met among faculty and students as power relations changed. Brubaker (2010, 2012, 2015) reported on his implementation of individualized grading contracts developed by students in an effort to shape a democratic classroom environment. He reported this practice shifted the focus “from grading to learning” (p. 264) and changed the “intellectual atmosphere of the class” (p. 265) to one of collaboration and problem-solving around relevant ideas.

Experiencing a living contradiction (Whitehead, 1989) between her assessment practices and theoretical perspective, Alyson investigated how assessment practice changes could help resolve this misalignment (Lischka et al., 2020). Her initial research led to questions about interactions between grades and PTs’ ability to take up authority for their pedagogical growth. Authority in classrooms is situated in and informed by constructs such as “values, norms; school ethos and policy; . . . [and] institutional features of schooling” (Pace & Hemmings, 2006, p. 1) where one such feature is the grading system. Scholars have argued that grading is often used to maintain authority or control (Kohn, 1993; Pace & Hemmings, 2006) rather than to inform teaching and learning. In mathematics education, alternatives to traditional grading have been emphasized in the form of performance assessments evaluated by rubrics (e.g., Bush & Greer, 1999; Thompson & Senk, 1998), including recent efforts that urged involvement of learners in helping to create rubrics (e.g., Liljedahl, 2020). Rubrics have traditionally been used in grading to support efficiency and consistency (Perlman, 2003; Tierney & Simon, 2004), often touted as a way to “justify to parents and others the grades that they assign to students” (Andrade, 2000, p. 13). Though rubrics can be confining and limit the ways feedback is given to learners (Andrade, 2000; Wolf & Stevens, 2007), they also serve as supports to make explicit the cultural assumptions and expectations that may be hidden from students of color, multilingual/multicultural students, and first-generation learners (Delpit, 1988; Wolf & Stevens, 2007). As such, assessment systems and constructs such as rubrics provide structure to courses that support learners lacking fluency in the dominant culture of learning environments. Concerns over alternative assessment cited possible ways in which removing grading structures might disadvantage some learners and increase equity gaps (Talbert, 2022; Supiano, 2022).

Classroom Context of the Study

The methods course in which this study occurred was the second of two mathematics-specific methods courses in an undergraduate teacher preparation program. This course occured at the penultimate term of the program, with only the full-time practicum experience following. The overall program model included classroom-based experiences and exposure to inquiry-based teaching methods throughout the four-year sequence of courses. By the time PTs took this course, they had developed understandings of how learners should engage in inquiry-based mathematics classrooms and had examined mathematical tasks that allow for inquiry-based instruction. In this course, PTs focused on the mathematics teaching practices (National Council of Teachers of Mathematics [NCTM], 2014) required to engage learners in inquiry-based learning. Major products of the course included two multi-day unit plans, teaching rehearsals, and a letter-writing exchange with secondary mathematics learners. The primary goal of this course was for PTs to develop personal pedagogical practices that encompass research-based teaching practices. Because of the placement of this course in the program of study, PTs were not under any grade point average requirements to gain entry to the education program, although they had to maintain a C average in coursework to graduate.

This teacher preparation program served all PTs at this institution seeking secondary teaching licensure in agriculture, science, and mathematics. As a step toward retention of PTs in these critical areas, program faculty regularly implemented community-building activities that resulted in strong peer relationships that were both supportive and critical. PTs in this program learned from and with each other as they grew their pedagogical practices.

Over the two years prior to this study, Alyson transitioned this methods course from one in which grades were determined by the instructor through numbered scores assigned using rubrics to a course where PTs assigned their own grade through reflection on their work individually and through peer review. For all course assignments, Alyson provided rubrics that describe the intended product and then gave feedback on all submitted work either in writing or during in-person conferences. At the midpoint and end of the course, PTs assigned themselves a course grade based on a rubric description that emphasized reflection, improvement, and growth. During in-person evaluation conferences at the end of the course, PTs assigned a grade and produced evidence of meeting the grade criteria using submitted work from the course. Alyson recorded the course grade PTs stated at the end of these conferences.

Positionality of the Researcher

Alyson transitioned to instruction in teacher preparation at the university level during her doctoral program after completing 15 years of teaching secondary mathematics. Prior to this, she served as a mathematics department chair and system-level professional development leader. Having taken up constructivist teaching practices (Steffe & D’Ambrosio, 1995) in her classrooms, Alyson transferred these practices and structures to her work with PTs. Alyson’s perspective led her to develop instructional activities in which she could build models of PTs’ practices and “study the mathematics teaching and learning constructions of teacher-learners [PTs] and interact with teacher-learners in a learning space whose design is based, at least in part, on a working knowledge of teacher-learners’ concepts of mathematics teaching and learning” (Kastberg, 2014, p. 352). Alyson’s methods practice, therefore, focused on valuing PTs’ autonomy and viewing PTs’ existing models of teaching as rationally developed from their own experiences.

Taking up constructivist teaching practices requires that instructors come to know the PTs with whom they work. Alyson drew from Kitchen’s (2005) relational teacher education, described as “a reciprocal approach to enabling teacher growth that builds from the realization that we know in relationship to others” (p. 17), to make sense of what it meant to know PTs. Blending ideas from constructivist teaching and relational teacher education, Alyson’s methods course activities provided opportunities for reflection and interactive communication among PTs and for the class to corroborate ideas about teaching and learning, thus building PTs’ knowledge of teaching. With this perspective, Alyson engaged in self-study of her teacher education practice with a goal of aligning her practice with her theoretical perspective.


Alyson has engaged in collaborative self-study of her methods practice since 2013. That collaboration (with colleagues who are not authors on this study) has supported Alyson in examining her feedback (e.g., Kastberg et al., 2018, 2020) and discussion practices (e.g., Kastberg et al., 2019) in methods, while also unearthing other living contradictions (Whitehead, 1989) within her practice. Through investigation of her feedback practices, Alyson was challenged by the mismatch between her theoretical perspective and assessment practices. In 2018, Alyson began the process of restructuring her methods course evaluation system to better align with constructivist and relational perspectives. From 2018 to 2020, Alyson changed her course evaluation system and gathered data to support inspection of impacts of the change on her PTs and her own practice. Results of analysis during these years pointed toward potential impacts on PTs’ autonomy for their pedagogical growth (Lischka et al., 2020) but left Alyson with questions regarding her own assessment practices. In this study, Alyson and her critical friends retrospectively examined research documentation from analysis of data focused on Alyson and her interactions with PTs gathered from 2018 to 2022 in order to answer the question: In what ways does changing a course evaluation system through ungrading (Blum, 2020) impact the instructor’s other assessment practices in a constructivist and relational mathematics methods course?


Alyson, the course instructor, collaborated with Natasha and Jennifer, second and third authors, as critical friends (Schuck & Russell, 2005). Natasha, formerly a graduate student of Alyson’s, has collaborated with Alyson on a multi-year professional development project that produced several co-authored research and practitioner publications. Jennifer, a graduate student of Alyson’s and a high school mathematics teacher, was an observer in Alyson’s fall 2019 methods course as well as an active participant in research meetings. All three authors have grown to trust each other and openly share struggles and successes in teaching with each other. We engaged in self-study of teacher education practice that is improvement aimed (LaBoskey, 2007) and characterized by openness, collaboration, and reframing (Samaras & Freese, 2009). Through examination of artifacts of teaching and research documents, we aimed to build understanding of teacher-educator practices.

Data for this study were drawn from the second term (Fall 2019) in which Alyson engaged in an alternative assessment structure along with research documentation beginning in summer 2019. During Fall 2019, ten PTs participated in regular journaling exchanges with Alyson as part of course assignments, engaged in peer- and self-evaluations related to course assignments, and defended their final course grade in an evaluation conference. Data included: PT journals, class audio recordings, final evaluation conference transcripts, instructor journal, and the transcript of a whole group interview with PTs focused on grading practices (conducted by Jennifer). In addition, the authors recorded every research team meeting and kept collaborative meeting notes. Research progress on this data was disrupted by the Covid-19 pandemic, which resulted in data from three years of research team analytical discussions and notes.

The researchers analyzed data in multiple phases that moved from examination of PT artifacts to coding research team meeting notes and transcripts. From January 2020 through early Fall 2021 (with Covid-19 interruptions) analysis began with qualitative coding of PTs’ artifacts during which Natasha openly coded selected PT data, identifying initial themes of interest. Codes were reviewed by Alyson and Jennifer with any disagreements resolved. Next, all authors participated in analytical dialogues (Guilfoyle et al., 2004) to determine emerging themes the team wished to explore, which produced an interest in understanding authority roles related to assessment. Next, Natasha applied emergent codes related to authority to multiple data sources including: each PT’s journal and final conference transcript, selected class recordings, the instructor journal, and the whole group interview. Alyson and Jennifer reviewed this layer of coding with discussion to reconcile any disagreements and then sorted within codes to determine emergent themes (Saldana, 2016). During this phase of analysis, we recorded all analytical conversations.

In Fall 2021, the research team was prompted by external reviewers of a manuscript in progress to revisit conversations around the use of rubrics in assessment. From December 2021 through Spring 2022, the team reviewed collaborative research notes to find evidence of discussion related to rubrics across the entire study. We transcribed meetings in which these discussions were found and coded them for the ways rubrics were discussed. This retrospective analysis, compiled in chronological order, offered an opportunity for “the researcher – me, now, investigating the archives and artifacts left by the informant – me, then” (Ham & Kane, 2004, p. 114) in order to gain perspective on how Alyson’s practice had changed after implementing an alternative approach to assessment.

Trustworthiness of this study was supported by the use of multiple forms of data and through multiple authenticities (Grant & Lincoln, 2021). Ontological authenticity, or knowledge of self revealed, was developed by the sharing of story that demonstrated how Alyson had “come to understand their own tacit positions and recognize how they “own” them” (Grant & Lincoln, 2021, p. 5). Throughout the following findings section, we tell the story of how Alyson uncovered her tacit understandings about the role of rubrics in course assessment and confronted the way in which those understandings conflicted with her theoretical perspective on methods instruction. Educative authenticity, which “connects strongly to the premise of qualitative inquiry that information, data, and interpretations do not belong solely to those who hold power and/or money but rather are the right of the stakeholders to have and to work with” (Grant & Lincoln, 2021, p. 5), was evidenced by the foundational data representing conversations among Alyson and her PTs as found in course artifacts. In the next section, we provide a chronological narrative using evidence from data sources to describe how self-study provided opportunities for Alyson to both align her assessment practice more fully with her theoretical perspective and to come to understand the role of rubrics in her practice more fully.


In fall of 2018, Alyson first initiated changes to her course assessment system in mathematics methods and gathered data for a previous study. Her syllabus from 2018 shows that each major assignment was accompanied by a rubric, and the overall course grade was determined by Alyson based on the number of assignments on which PTs met expectations according to the rubric. Through a prior study, Alyson and Natasha explored the tensions between institutional expectations on grading and relational teacher education. When summarizing the results of this earlier research, Alyson wrote:

If I state on the syllabus what I want to do in grading and carry it out, then I am supported by the administration. So, the tension for feedback/assessment lies in the faculty member’s hands. I need to be comfortable identifying the pieces of the cultural practices of teaching that stand in opposition to my beliefs about teaching and learning and stand against them. (Meeting Notes, 6/5/2019)

However, the practice of using rubrics, a cultural practice of teaching, had not yet been revealed as a living contradiction (Whitehead, 1989) for Alyson.

In July 2019, Alyson and Natasha were writing about the prior study and, in search of supporting literature, found Biggs (1996) which suggested the formation of an evaluation rubric based on performance growth. This reading, along with study findings, urged Alyson to again restructure course assessment for Fall 2019. Wanting PTs to take more ownership for their assessment, Alyson crafted a rubric for course evaluation that would require PTs to assess their own growth throughout the term. She planned to implement additional self- and peer-assessments aligned with the new rubric. Alyson noted that rubrics were essential “in order for students to be able to defend the grade they feel they have earned” (Meeting Notes, 8/15/19). Alyson said during a meeting:

You wrote here that there were times that I was hesitant to use the rubric or felt constrained by the categories? And I think that is something to think about, because I'm still thinking about grading as me being the one assigning the grade. And so, the rubric is still this measure that I'm using, but I don't think you get rid of the rubric. I think that's what sets the standard. (Meeting Transcript, 8/5/2019)

At this time, Alyson still used rubrics in the traditional sense, a tool used for establishing grades consistently, however, this is the first time she referred to rubrics in a different way, as that of standard setting.

At the start of the 2019 term, Alyson provided the following rubric description for earning an A in the course:

The teacher is able to reflect on their own teaching using the language and content from the course; evaluate the decisions made about lesson planning and assessment; improve their practice; formulate a personal theory of mathematics teaching that demonstrates research-based practices; and generate new approaches to teaching based on content and principles discussed in class. (Methods Course Syllabus, Fall 2019 – Fall 2022)

Throughout the fall 2019 term, Alyson collected data from the ten PTs in her class and continued thinking about the usefulness of the current course grading system. Studying the data from the previous year pushed the new inquiry toward consideration of the ways in which this new evaluation system supported PTs to develop autonomy or authority for their own pedagogical growth.

Analysis of the fall 2019 data started in January 2020, with a focus on understanding constructs related to power, authority, and autonomy. With interruptions due to Covid-19, analysis was slow but revealed the use of rubrics as an important element in the shifting of authority from instructor to PT (Atlas-ti Memo, August 2020) through the first phase of coding. In analytical discussions reflecting on the possible meanings of this theme, Alyson struggled with discomfort. Natasha said, “there's this separate thing going on between talking about the rubric and how the rubric is used, versus the students taking up authority” (Meeting Transcript, 8/10/20). Later, Alyson summarized by saying:

I think there are some cases where I use the rubric as a way to take the authority back from the students. But then there are other ways where I see the rubric as a way to kind of give them the authority to make some of these decisions for themselves. So, there's give and take. (Meeting Transcript, 8/10/20)

As Alyson planned for another term of her methods course, this time during a pandemic, the team opted for no additional changes to the evaluation system but continued to consider the ways in which Alyson supported PTs’ use of rubrics in self-evaluation.

During fall 2020, the team struggled to keep up with pandemic teaching requirements, presentations at virtual conferences, and moving data analysis forward. In spring 2021, the team completed phase two of coding and analysis of the 2019 data. In analytical discussion, rubrics surfaced again in the statements that PTs made in journals and class discussions. Alyson, explained her thoughts and questions:

I've been thinking about that rubric one. At first, I don't know what I feel about the rubric taking such a big role. I'm feeling better about that now. And I may change my mind once I get into it. But they're [PTs] going to get into their own classroom, and they're not going to have somebody standing there saying, This is good, this is bad. They need . . . some kind of standard to which they're comparing themselves, so that they can stop and think about, is this practice that I'm doing a good practice? . . . And so, I almost think that shifting them to thinking about what is the standard is not really a bad thing? (Meeting Transcript, 4/29/21)

These ideas prodded Alyson to think more carefully about how she presented rubrics and standards to her PTs during methods classes. She said “if I want to shift the power to the students [PTs] but the rubric is still sitting there . . . I need to think about how to talk about that with them, in better ways” (Meeting Transcript, 5/20/21).

In fall 2021, as the team continued to write about this study, external reviewers of a manuscript in progress urged them to consider how Alyson’s thinking about the use of rubrics had changed. Alyson, Natasha, and Jennifer recognized that the delays in writing caused by the pandemic provided an opportunity to examine evidence of how Alyson’s thinking changed over time as recorded in meeting notes, transcripts, and analysis logs. They proceeded with a search of this data for discussion of rubrics and organized those conversations and transcripts as a tool for reflection. Reflecting on these changes, Alyson realized that part of her struggle related to the role of rubrics as tools for standardization (Meeting Notes, 4/20/22), which was not her goal as a constructivist teacher. She stated, “My current thought on rubrics is that they are guideposts for students but not the totality of grading (in any course) and that I provide them for PTs to have knowledge of what to aim for” (Meeting Transcript, 4/27/22). Rather than being the solitary tool used to determine grades in consistent ways, Alyson now viewed rubrics as a tool to both communicate best practices and support PTs in evaluating their own pedagogical growth. Moving forward, Alyson plans to attend to the ways in which she discusses rubrics and standards with her PTs, with a clear understanding of the ways in which she intends to use rubrics in supporting their pedagogical growth.


At the beginning of Alyson’s alternative assessment journey, she relied on prior experiences in mathematics teaching and designed an assessment system that lacked numerical grades but included rubrics describing expectations for success. As found in the literature (Bush & Greer, 1999; Perlman, 2003; Thompson & Senk, 1998), rubrics were a familiar tool that clarified expectations and supported her ability to justify earned grades for students. She removed numerical rules but was still using rubrics in a traditional sense. Although questions of authority and autonomy led the direction of the analysis, questions about the use of rubrics surfaced in discussions of the data. PTs’ discussion of rubrics found in data caused Alyson to reframe ideas about rubrics to thinking more closely aligned with her goals. Through self-study and critical friend conversations, Alyson recognized this incongruence, reframed her thinking, and is working toward communicating about rubrics and grading in new ways.

Ungrading (Blum, 2020) is a growing trend among educators at all levels. However, research that examines these practices is not keeping pace; grading practices merit scholarly inquiry (Supiano, 2022; Talbert, 2022). This self-study begins to provide research that informs how alternative assessment, as a form of ungrading, is enacted and the ways in which it impacts both learners and instructors. Grading practices provide necessary structures for all learners to find success (Delpit, 1988; Wolf & Stevens, 2007) so the removal of these structures may have unintended consequences for learners. Though this study did not find unintended consequences, it revealed that removing one grading structure (numerical grades) produced an unintentional focus on another structure (rubrics). The ways in which traditional structures such as rubrics and grading operate on authority and autonomy for learning warrant closer inspection (Pace & Hemmings, 2006).

Alyson learned about her own assessment practice as the authors examined data representing the voices of Alyson’s PTs alongside Alyson, Jennifer, and Natasha’s wonderings about the impacts of changing assessment systems. This research provides an example of “teacher educator knowledge as that which is enacted in practice, while engaged in one’s professional activities, as constantly evolving and developing from experience, and as situated in a particular context” (Vanasssche & Berry, 2020, p. 188). Through this example, the field of teacher preparation can “focus on critically analyzing the (normative) beliefs and assumptions about teaching that underpin practice” (Vanassche & Berry, 2020, p. 207). Assessment practices are not often critically examined yet have the potential to influence the ways in which prospective teachers develop as educators. Additional research is needed to better understand the ways in which grading, or ungrading, shapes learning.


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Alyson E. Lischka

Middle Tennessee State University

Natasha Gerstenschlager

Western Kentucky University

Jennifer Webster

Middle Tennessee State University

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