Mathematical Communication is a developing collection of resources for engaging students in writing and speaking about mathematics, whether for the purpose of learning mathematics or of learning to communicate as mathematicians.

Feedback and assessment for writing

Most of this page focuses on “learning to write”; i.e., on assessing how well students communicate mathematics. Near the bottom of the page are sections on “writing to learn” and “writing to assess”: some writing assignments are designed primarily to help students to learn mathematics, while others are designed primarily to help teachers assess student understanding of mathematics. These call for different feedback and assessment strategies than “learning to write.”

Commenting strategies

As you give students feedback on their writing, you might consider commenting on mathematical correctness, clarity, flow and organization, and other general principles of communicating mathematics. A balance must be found among verbose commenting, a reasonable time investment, and what’s most helpful for students. Some suggestions:

  • Consider your purpose for commenting and craft your comments to achieve that purpose.
  • To gain a sense of the effect of comments on students, see the video “Beyond the Red Ink: Teachers’ Comments through Students’ Eyes.” Although these students are first-year humanities students, experience suggests that upper-level mathematics students are not far different.
  • Consider meeting with each student to provide feedback. Although scheduling meetings can be a hassle, feedback given in person is often more efficient and richer than feedback given in writing, and you’ll get to understand the students better as writers by hearing their responses to your comments.
  • Point out what the student does well. Positive feedback is remarkably effective.
  • Remind yourself to consider the text at all scales: it’s easy to overlook large issues when you’re focused on details.
  • Consider how much the student can learn in the available time: there’s no need to list all types of problems in a paper. Decide what the student most needs to learn at this point and consider giving only those comments. [When planning an assignment, consider scheduling multiple revisions so large-scale issues may be addressed during the first revision and smaller-scale issues may be addressed during later revisions.]
  • If you feel you must provide many comments, you can help students recognize the relative importance of the various comments by drawing attention to the most important ones in a summary note.
  • Less important issues can be de-emphasized through the use of coded comments (see this example of comment codes from MIT’s communication-intensive mathematics subjects and this example of comment codes from an engineering class at MIT).
  • Do not mark each instance of a recurring problem: instead tell the student how to identify the problem for him/herself.
  • If you’re distracted by an error until you mark it, very briefly mark errors on a scratch paper or in a temporary document, providing only enough information for yourself. Then when you’ve finished reading, decide which comments to give the student.
  • If the writing will not be revised, then limit yourself to general comments that apply to future writing.
  • Suggest that the student keep an editing checklist of mistakes he or she often makes.
  • The student is the author, so when you draw attention to a problem, avoid dictating a solution unless necessary.
  • To avoid comments being taken personally, refer to the text, not to the student: “This paragraph could be more concise.” not “You don’t write concisely enough.” To write helpful comments, consider these dimensions of commenting.
  • Reading papers is time consuming and occasionally frustrating. Take care to remain professional.

For further explanation of the above suggestions, see these two pages about Commenting.

In addition to or instead of giving individualized feedback, you may choose to give feedback to the class as a whole. Example: Guidance for revising an algorithm assignment.

Choosing grading criteria

Identifying and prioritizing grading criteria before grading is important to prevent unintentional, subconscious bias, even in graders who consider themselves objective, as found by this study of hiring decisions based on criteria prioritized before/after learning about an applicant:
Uhlmann and Cohen, “Constructed Criteria: Redefining Merit to Justify Discrimination,” Psychological Science, Vol 16, No 6, pp. 474-480, 2005.

Guidance for how to create a rubric is provided on the MAA Mathematical Communication page “How can I objectively grade something as subjective as communication?”

For a detailed explanation of this strategy, see this grading handbook, which includes sample lists of characteristics of good mathematical writing and sample rubrics, as well as guidance for grading drafts, etc.

Grading for small writing assignments

One strategy is to assign two grades to each small writing assignment: one for mathematical content and one for quality of exposition. To speed up grading, you may want to assign an overall exposition grade rather than one for each problem on the assignment (if there are multiple). Make sure your students are aware that effective communication is being evaluated and will contribute to their grade. If possible, give them a rubric or a sample of what you are looking for.

Example: Janet Preston’s rubric for week-long projects like‘s problems of the week.

Example: 18.310C rubric for grading heapsort algorithm

If you give students a rubric before the assignment is due, they are likely to write to the rubric. If you are new to teaching mathematical writing, consider not providing a rubric for the first writing assignment, so you have the opportunity to see how students write on their own. You may then design future rubrics to emphasize whichever writing characteristics you most want students to emphasize.

Grading for large projects

The grading of the final project should reflect all of its aspects: proposal, any intermediate drafts, peer-review, and final product. It may be helpful to the students to see the grading rubric you will use; many of them will not have written a long piece of mathematics before.

Here is an example of a rubric for grading a draft. Two grades are assigned: the grade that counts is based on effort/completeness, but students are also given a temporary “advisory grade” based on the rubric for the final paper. This strategy rewards effort while allowing room for improvement and giving information about how much improvement is needed.

For the final paper, you may want to grade the following:

  • Correctness of mathematics
  • Clarity of exposition
  • Flow and organization of paper
  • Other general principles of communicating math that you consider to be most important
  • Extent to which feedback was incorporated. (Some students may be disinclined to incorporate feedback unless doing so is built into the grading.)

A sample rubric for the final paper is included with the rubric for the draft (above). A rubric can be designed not only as an assessment tool but also as a teaching tool, as explained and illustrated in this blog post. Another option is to use a more detailed grading grid; however, writing mathematics well is complicated, and it may not be possible to create a precise grid that sufficiently captures the diversity of strengths and weaknesses of students’ writing.

Encouraging students to do more than superficial revision

Sometimes students revise only superficially when major revisions are needed. To help students understand the extent of revision needed, and to encourage them to make significant revisions when those are required, consider the following strategies:

  • If major revision is needed, give only high-level comments. Giving detailed comments will cause students to focus on details and may cause them to lose sight of the big picture, thus preventing major revision.
  • Explicitly tell students which sections of the paper ought to be “rewritten” rather than “revised” and what’s meant by those terms.
  • Avoid scheduling revision at the end of the term when students are pressed for time.
  • Meet with students individually to give them a chance to ask questions about feedback. The one-on-one attention also makes clear to students that you consider the quality of writing to be important.
  • If you have a writing workshop in which you show students drafts of your own writing, choose drafts that exhibit major revision. By modeling major revisions yourself, you’ll help students to realize that they may also need to make major revisions.
  • Have students assess the quality of their own writing and submit a revision plan.
  • Require students to indicate which changes they made and explain why changes were not made.
  • Include in the grade the extent to which instructor and peer comments were addressed.

Literature on assessing writing

  • Patrick Bahls’ 2012 book Student Writing in the Quantitative Disciplines: A Guide for College Faculty contains a chapter on Assessing and Responding to Student Writing.
  • Crannell, A., “Assessing Expository Mathematics: Grading Journals, Essays, and Other Vagaries,” Assessment Practices in Undergraduate Mathematics, MAA Notes #49.
  • Emenaker, C.E., “Assessing Modeling Projects in Calculus and Precalculus: Two Approaches,” Assessment Practices in Undergraduate Mathematics, MAA Notes #49.
  • Crannell, A., “How to grade 300 math essays and survive to tell the tale,” PRIMUS 4 (3), 1994.
  • Houston, S.K., et al., Developing Rating Scales for Undergraduate Mathematics Projects, University of Ulster, 1994.
  • Dennis, K. “Assessing Written and Oral Communication of Senior Projects,” Supporting Assessment in Undergraduate Mathematics, The Mathematical Association of America, 2006, pp. 177-181.
    Contains rubrics for presenting and writing, with recommendations.

Assessing “writing-to-learn” assignments

If the primary purpose of a writing assignment is to help students to better learn the mathematics, then providing feedback on the process is likely to be more important than grading writing quality.

Using writing to assess understanding of mathematics

  • Morgan, C., Writing Mathematically: The Discourse of Investigation, Falmer Press, 1998.
    From the Google Books review: “…[using] written language to serve as ‘evidence’ of their mathematical activity and achievement,… raises two important questions. Firstly, does this writing accurately present children’s mathematical activity and ability? Secondly, do math teachers have sufficient linguistic awareness to support their students in developing skills and knowledge necessary for writing effectively in their subject area?”
  • See also the general resources for assessing understanding of mathematics, listed on the Assessment page.

General resources and research (not specific to mathematics)

  • How can I handle responding to drafts? The WAC Clearinghouse
    These pages provide strategies for

    • focusing your commenting energies
    • using a grading sheet (including sample grading sheets)
    • evaluating writing to learn assignments
  • Beyond the Red Ink: Teachers’ Comments through Students’ Eyes” Conversations with Bunker Hill Community College Students, Bedford/St. Martin’s
  • John C. Bean’s Engaging Ideas: The Professor’s Guide to Integrating Writing, Critical Thinking, and Active Learning in the Classroom includes a clear and helpful chapter on Reading, Commenting On, and Grading Student Writing.
  • Nancy Sommers’ Responding to Student Writing Bedford, St. Martins, 2013. (publisher’s page)


  • Bracey, E., “The Will to Revise: Commenting, Revision, and Motivation in College Students“, Xchanges, Issue 9.1, 2013.
    Bracey’s research supports some prior results by indicating that the revisions of students who use a writing center focus on those aspects of writing that are emphasized most in their professor’s comments: e.g., if the professor comments only on grammatical issues, the students focus only on grammatical issues and resist writing center assistance on clarity, coherence, etc. This 2013 article begins with a helpful review of research on the factors that motivate student revision.
  • Taylor, S. S., “‘I Really Don’t Know What He Meant By That’: How Well Do Engineering Students Understand Teachers’ Comments On Their Writing?Technical Communication Quarterly, Vol. 20, No. 2, pp. 139-166, 2011.
    Teachers and engineering students from one institution were interviewed about the teachers’ comments on the students’ papers. Students recognized the focus of about three quarters of teachers’ comments but understood the reasons for only half of the comments. The results are broken down by type of comment and discipline of teacher. Recommendations are suggested, including that teachers provide more explicit explanations for comments. This 2011 article begins with a helpful review of research on response and on student reception of response.
  • Summers, N., C. Rutz, and H. Tinberg. “Re-Visions: Rethinking Nancy Sommers’s ‘Responding to Student Writing.’ 1982.” CCC Vol 58, No 2 (2006) pp. 246-266.
    This collection of three essays summarizes some past and current (as of 2006) research and thinking about responding to student writing. In the first essay, “Across the Drafts,” Nancy Sommers summerizes some of the findings from the Harvard Study of Undergraduate Writing, which followed the writing of and feedback received by 400 students over their four years at Harvard.
  • Anson, Chris M. “Response Styles and Ways of Knowing.”  Writing and Response Theory, Practice, and Research. Ed. Chris Anson. Urbana: National Council of Teachers,1989. 332-366.
  •  Grading Writing: the Art and Science–and why computers can’t do it, by Doug Hesse.
  • Uhlmann and Cohen, “Constructed Criteria: Redefining Merit to Justify Discrimination,” Psychological Science, Vol 16, No 6, pp. 474-480, 2005.
  • See also the resources on this site’s general page about assessment.

Please suggest key research to add to this list.

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