By Elise Lockwood, Contributing Editor, Oregon State University
On many occasions when I grade my students’ proofs, or when I read their solution to a particularly interesting problem, I am surprised by something I read. Sometimes I am surprised because I am disappointed with a given argument or a hand-wavy proof, but often I am surprised because I am impressed by a clever insight or an eloquent way of expressing an argument. Indeed, there have been occasions when I have learned something through the experience of grading my students’ work. Also, seeing the sheer variety of solution strategies that my students offer helps me to appreciate various mathematical approaches and makes me more attuned to their respective mathematical ways of thinking.
In this post I will discuss an activity that I call peer grading, by which I mean having students provide formative, written feedback on their classmates’ assignments. This involves giving students the opportunity to engage with and analyze work that their classmates have done. Peer grading has been used by other teachers (see the references at the end of this post), and my personal reflections on the value of engaging in the process of grading have convinced me that students can similarly benefit from grading other students’ work.
Practical notes – so what does this look like?
I have experimented some with peer grading, but I still have more to fine tune in terms of the practical details. There are several possibilities for how it might best work, and I outline some guiding practical principles:
1) Be transparent about the process. Before my students begin the peer grading process, we have explicit classroom discussions about the purpose of the activity and what is expected in terms of the nature and the amount of feedback they should give. I understand that learning to give valuable, constructive feedback may take time, and that there will be a range of quality, especially initially. I give students a handout with some examples of what helpful feedback looks like (probing or challenging questions, thoughtful comments with reasons and justifications, catching serious mathematical errors) and what unhelpful feedback looks like (suggestions for changes without explanation or justification, positive comments with no reasoning, missing mathematical mistakes), in order to exemplify an effective critical analysis.
I also check over the peer graders’ comments to ensure that they are putting effort into the process, and this is a small part of their grade. I will typically offer comments and use a check/check plus/check minus system to give feedback to the peer graders (so far this has been sufficient), but one could also come up with a more rigorous feedback system. Also, the peer grader’s comments should have no actual bearing on the other students’ grades, as I view the activity of peer grading as a formative assessment, not a summative one.
2) Tailor assignments for peer grading. Not every assignment is appropriate for peer grading, nor should students necessarily grade most or all of the assignments your students complete. I designate a handful of assignments that are explicitly geared toward peer grading. One such assignment might consists of a few proofs (perhaps in the midst of or at the end of a unit on proving) or some carefully selected counting problems that have the potential for being particularly illustrative of important ideas. I intentionally pick tasks that make a nice “peer grading assignment” so as to make the process efficient and effective.
3) Facilitate variety. I think there is benefit to letting students see a variety of proofs or problems from a variety of their peers. It is not necessarily helpful if a student grades only one other student’s proofs – rather, the power in the exercise is in being able to see what a handful of other students did. There are many options for how to implement this – a student could grade few problems from many students, or many problems from few students, or several problems from several students. I prefer the third option, which I feel gives students the most opportunity to see a variety of responses. Practically, this may mean that students take shifts in peer grading. In a class of 30, I may have six students grade one peer grading assignment (which would consist of a few problems), so they each grade five classmates’ work. Then, in the next assignment, six different students would do the grading. Over the course of the term, students would get to see a variety of problems, proofs, and topics from many of their peers.
Different topics, different courses.
There are a number of different types of courses for which this could be particularly useful. One type of course is proof-based, upper division courses for math majors. These would include courses like discrete mathematics, introductory analysis/advanced calculus, and abstract algebra. It is in courses like these that students are make formative steps in their development as proof writers, and they are also encountering advanced, challenging topics. Students are learning important ideas both about what is convincing (to themselves, to a professor, or to the mathematical community) and also what a proof must include to be convincing and also rigorous. In these courses, students encounter difficult topics and must learn, in a relatively short amount of time, how to solve increasingly abstract problems and how to write proofs that are clear, concise, mathematically correct, convincing, and rigorous.
Another type of course for which I think peer grading would be particularly useful is in preservice teacher courses, whether geared toward elementary, middle, or high school teachers. Using peer grading in such courses not only gains the affordances outlined below, but there is also the pedagogical benefit that future teachers could uniquely appreciate. Peer grading simulates the exact kind of activity teachers will eventually need to do. Thus, gaining experience making sense of, interpreting, and assessing student’s arguments is great practice for what will become a prominent part of their jobs. They have to think hard about what other people are thinking, which is an essential part of being a teacher. Sometimes we try to simulate this kind of activity by providing examples of hypothetical student work that we ask preservice teachers to evaluate, but peer grading could offer a more authentic kind of experience.
In addition, peer grading might also be particularly beneficial for students who will go on to do technical work in industry (such as data scientists, engineers, actuaries, etc.). Experiences with peer grading could help them hone critical analysis skills that such work will require.
What might be gained from allowing students to see and evaluate their peer’s proofs?
I offer a few specific benefits that students might experience if they interact meaningfully with other students’ problem solutions and proofs.
1) Improved conceptual understanding. Students can gain mathematical insights from their classmates. They may learn a more elegant or clean way of formulating an argument (perhaps demonstrating an efficient total-minus-bad solution to a counting problem), or they may actually be introduced to a brand new mathematical technique or idea (such as a multiplicative expression for the sum of the first n natural numbers). Additionally, the student who is grading may have key mathematical ideas reinforced through the process reading through many different proofs or solutions to the same problem. That is, ideas that might have been only partially developed for students as they wrote a proof may now be solidified through revisiting and evaluating many different formulations of the same argument. The point is that mathematical ideas can be strengthened and even developed by immersing oneself in others’ mathematical work.
2) Enhanced mathematical communication. In addition to actually learning some mathematics, students can learn valuable lessons about communicating mathematically. By seeing what others say and how they say it, students can appreciate how they themselves communicate. Perhaps they thought they were being clear, but then they read another more articulate response and realize how to say something more clearly. Seeing a variety of proofs or problem solutions can highlight the distinction between an effective bit of communication and a poorly articulated argument.
3) Renewed empathy for others and increased confidence. Finally, the experience of grading can help students better understand and appreciate their own struggles and success. For students toward the bottom of the class, they benefit from the mathematical insights and seeing well-formulated proofs that their classmates produce. For students at the top, they can gain empathy for their classmates, appreciating that others might actually be struggling with the material. Also, in my experience, students can tend to lack confidence about their own abilities, and many students are hesitant to raise questions for fear of appearing unintelligent or lost (this is especially true for female math students). However, it is often the case that many students in a class will find an idea challenging, and the activity of peer grading could help to bring this reality to light. Students can realize that others and can be empowered by it – not in a “I can’t believe how stupid so and so is” kind of way, but rather in a “Wow, so it seems that other people are struggling with these topics, too” kind of way. This realization can, in a sense, level the playing field, and can perhaps even boost confidence among students who need it.
I have framed these benefits in terms of how they will help the grader, but note that the student whose work is being graded also stands to benefit from peer feedback. They may realize that ideas they thought they were communicating clearly are actually not easily understood, or they might come to learn particular mathematical topics that they need to work on.
Freeman, S. & Parks, J. W. (2010). How accurate is peer grading? CBE Life Sciences Education, 9(4), 482-488. http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2995766/
Peer Assessment. Retrieved from http://ctl.utexas.edu/teaching/assess-learning/feedback/peer-assessment.
Self and Peer Assessment Resources. Retrieved from
Sivan, A. (2000). The implementation of peer assessment: an action research approach. Assessment in Education, 7, 2: 193-213.