math writing assignment

Assignments on writing

Examples of short assignments, term papers, designing assignments that enable students to write well.

Writing well requires mastery of writing principles at a variety of different scales, from the sentence and paragraph scale (e.g., ordering information within sentences so content flows logically ) to the section and paper scale (e.g., larger-scale structure ). To simplify teaching, you can begin the term with shorter assignments to address the smaller-scale issues so you can more easily focus on the larger-scale issues when you assign longer assignments later in the term. At all scales, students best learn to communicate as mathematicians if the assignments are as authentic as possible: if the genre and rhetorical context are as similar as possible to those encountered by mathematicians.

Many of the following ideas are currently implemented in M.I.T.’s communication-intensive offerings of Real Analysis and Principles of Applied Mathematics .

• Require that at least one question on each problem set be typed up and written in the style of an expository paper (rather than the usually terse and sometimes scattered style of a homework solution).
• Assign short exposition tasks such as summarizing the proof of a theorem done in class or filling in the gaps in an explanation given briefly in class.
• To help students learn LaTeX or how to use equation editors, have an assignment requiring at least basic math formatting due early in the semester so students aren’t required to learn it as they’re researching and writing their term papers. Begin with simple math formatting exercises, building to more complex: e.g., see the assignments for M.I.T.’s Real Analysis recitations 1 (text with math) , 2 (table and figure) and 13 (slides containing a figure with LaTeXed labels) .
• Begin with communicating simple arguments, building to more complex (e.g., having students explain the heapsort algorithm and then revise the explanation based on feedback provides a rich opportunity for teaching about writing clear definitions, giving conceptual explanations as well as rigorous details, and presenting information in an order that is helpful to readers.) See the sequence of assignments from M.I.T.’s Principles of Applied Mathematics .
• Have students revise part of a concise textbook such as Rudin’s, Principles of Mathematical Analysis in the style of a more-thorough lecture note.
• Before an exam, have students formulate and submit to you a list of 2+ questions they have about the material. Students have a hard time formulating precise questions, yet this is an important communication and learning skill. Some students may feel they understand the course material, so permit questions that go beyond the scope of the course. You can use the questions to focus a review session. More detail about this assignment is given in this lesson plan from M.I.T.’s communication-intensive offering of Real Analysis.

The following books, articles, and websites contain short writing assignments.

• Stephen Maurer’s Undergraduate Guide to Writing Mathematics has an extensive appendix of writing exercises designed to target various aspects of writing mathematics.
• Writing Projects for Mathematics Courses: Crushed Clowns, Cars, and Coffee to Go , by A Crannell et al . [link goes to MAA review] This 119 page book from the MAA contains “writing projects suitable for use in a wide range of undergraduate mathematics courses, from a survey of mathematics to differential equations.” Each prompt is written in the form of an (often amusing) letter from someone who needs help with a “real-world” problem that requires math expertise. Students must solve the problem and write a letter of response. On his website, Tommy Ratliff (one of the co-authors) gives a brief account of using such projects in his calculus course.
• Annalisa Crannell’s Writing in Mathematics website has writing assignment for Calculus I, II, and III as well as links to colleagues’ websites that have further writing assignments.
• Quantitative Writing from Pedagogy in Action, the SERC Portal for Educators, has many examples of short and long writing assignments based on “ill-structured problems,” which are “open-ended, ambiguous, data-rich problems requiring the thinker to understand principles and concepts rather than simply applying formulae. Assignments ask students to produce a claim with supporting reasons and evidence rather than ‘the answer.'”
• The Nuts and Bolts of Proofs by Antonella Cupillari includes exercises for an introductory proof-writing course. Proof topics include calculus and linear algebra.
• Platt, M. L.. (1993). Short essay topics for calculus. PRIMUS: Problems, Resources, and Issues in Mathematics Undergraduate Studies 03.1 , 42-46.

Additional information about journal-writing assignments and other writing-to-learn assignments can be found on the page about using writing to help students learn math .

For each assignment, indicate your expectations about audience and length, so students know how much explanation to include. An appropriate audience is often other students in the class who are unfamiliar with the specific topic of the assignment, or other math majors not in the class.

Term papers enable students to pursue areas of their own interest and so can be among the most rewarding assignments for students. To help students succeed, give students guidance for choosing a sufficiently focused topic, for finding helpful sources, and for using sources appropriately. See this assignment for proposing a term paper topic , from M.I.T.’s Principles of Applied Mathematics –it includes guidance for how to choose a good paper topic.

One of the (interesting) challenges of assigning a term paper is generating a list of possible paper topics. Ideally, each topic should have well-defined scope and have at least two or three available resources accessible to students in the course. You may want to emphasize to the students that they are not expected to do original mathematics research. However, the paper must be their own — they cannot paraphrase and closely follow a published survey paper.

One of your institution’s librarians may be happy to collaborate with you to show students how to find useful sources.

To provide students with an authentic rhetorical context for their term papers, consider showing them samples of expository papers and suggesting that they write for a journal that publishes expository papers (e.g., The American Mathematical Monthly , Math Horizons , Mathematics Magazine , and The College Mathematics Journal .

Don’t assign a term paper unless a variety of topics exist at an appropriate level. For example, a term paper may not be appropriate for an introductory class in analysis.

Be aware that plagiarism may be an issue particularly in large classes on subjects for which a wealth of material is available online. In such classes, you may find it to be helpful to tightly specify the paper topics or to supply a specific slant to the papers (e.g., apply such-and-such method to an application of your choice). Vary the assignments from year to year. These precautions may be less important in small classes.

In some classes (e.g., applied mathematics classes), it may be necessary to carefully guide students to choose topics that contain sufficient mathematical content. For that reason, using caution when approving unfamiliar topics.

A poorly focused assignment will leave students confused about what is expected of them and is likely to result in poor writing. Students are likely to write their best if the assignment is interesting and if students are told (or are able to confidently identify for themselves) the following:

• educational objectives of the assignment
• audience knowledge and interest, and author’s relationship to the audience
• purpose of the text to be written (e.g., to convince, to entertain mathematically, to teach, to spark interest)
• content to be addressed
• details of the genre ( proof ? research paper? funding proposal?)
• how the writing will be graded
• an effective writing process (you can provide support by assigning intermediate due dates or revision )

The following resources explain these points and give further guidance for designing effective assignments:

• Bahls, P., Student Writing in the Quantitative Disciplines: A Guide for College Faculty , Jossy-Bass 2012, pp. 36-46, contains sections on structuring writing assignments (includes sample prompts), sequencing assignments throughout a course, and sequencing writing from course to course.

General resources (not specific to mathematics)

• How can I avoid getting lousy student writing?
• What makes a good writing assignment?
• The webpage Integrating Writing and Speaking Into Your Subject , provided by MIT’s Writing Across the Curriculum, has several subpages about writing assignments.
• Creating Writing Assignments , MIT’s Writing Center

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• Prof. Haynes Miller
• Dr. Nat Stapleton
• Saul Glasman

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Learning resource types, project laboratory in mathematics.

Next: Revision and Feedback »

In this section, Prof. Haynes Miller and Susan Ruff describe the criteria for good mathematical writing and the components of the writing workshop .

A central goal of the course is to teach students how to write effective, journal-style mathematics papers. Papers are a key way in which mathematicians share research findings and learn about others’ work. For each research project, each student group writes and revises a paper in the style of a professional mathematics journal paper. These research projects are perfect for helping students to learn to write as mathematicians because the students write about the new mathematics that they discover. They own it, they are committed to it, and they put a lot of effort into writing well.

Criteria for Good Writing

In the course, we help students learn to write papers that communicate clearly, follow the conventions of mathematics papers, and are mathematically engaging.

Communicating clearly is challenging for students because doing so requires writing precisely and correctly as well as anticipating readers’ needs. Although students have read textbooks and watched lectures that are worded precisely, they are often unaware of the care with which each word or piece of notation was chosen. So when students must choose the words and notation themselves, the task can be surprisingly challenging. Writing precisely is even more challenging when students write about insights they’re still developing. Even students who do a good job of writing precisely may have a different difficulty: providing sufficient groundwork for readers. When students are deeply focused on the details of their research, it can be hard for them to imagine what the reading experience may be like for someone new to that research. We can help students to communicate clearly by pointing out places within the draft at which readers may be confused by imprecise wording or by missing context.

For most students, the conventions of mathematics papers are unfamiliar because they have not read—much less written—mathematics journal papers before. The students’ first drafts often build upon their knowledge of more familiar genres: humanities papers and mathematics textbooks and lecture notes. So the text is often more verbose or explanatory than a typical paper in a mathematics journal. To help students learn the conventions of journal papers, including appropriate concision, we provide samples and individualized feedback.

Finally, a common student preconception is that mathematical writing is dry and formal, so we encourage students to write in a way that is mathematically engaging. In Spring 2013, for example, one student had to be persuaded that he did not have to use the passive voice. In reality, effective mathematics writing should be efficient and correct, but it should also provide motivation, communicate intuition, and stimulate interest.

To summarize, instruction and feedback in the course address many different aspects of successful writing:

• Precision and correctness: e.g., mathematical terminology and notation should be used correctly.
• Audience awareness: e.g., ideas should be introduced with appropriate preparation and motivation.
• Genre conventions: e.g., in most mathematics papers, the paper’s conclusion is stated in the introduction rather than in a final section titled “Conclusion.”
• Style: e.g., writing should stimulate interest.
• Other aspects of effective writing, as needed.

To help students learn to write effective mathematics papers, we provide various resources, a writing workshop, and individualized feedback on drafts.

Writing Resources

Various resources are provided to help students learn effective mathematical writing.

The following prize-winning journal article was annotated to point out various conventions and strategies of mathematical writing. (Courtesy of Mathematical Association of America. Courtesy of a Creative Commons BY-NC-SA license.)

An Annotated Journal Article (PDF)

This document introduces the structure of a paper and provides a miscellany of common mistakes to avoid.

Notes on Writing Mathematics (PDF)

LaTeX Resources

The following PDF, TeX, and Beamer samples guide students to present their work using LaTeX, a high-quality typesetting system designed for the production of technical and scientific documentation. The content in the PDF and TeX documents highlights the structure of a generic student paper.

Sample PDF Document created by pdfLaTeX (PDF)

Sample TeX Document (TEX)

Beamer template (TEX)

The following resources are provided to help students learn and use LaTeX.

LaTeX-Project. “ Obtaining LaTeX .” August 28, 2009.

Downes, Michael. “Short Math Guide for LaTeX.” (PDF) American Mathematical Society . Version 1.09. March 22, 2002.

Oetiker, Tobias, Hubert Partl, et al. “The Not So Short Introduction to LaTeX 2ε.” (PDF) Version 5.01. April 06, 2011.

Reckdahl, Keith. “Using Imported Graphics in LaTeX and pdfLaTeX.” (PDF) Version 3.0.1. January 12, 2006.

Writing Workshop

Each semester there is a writing workshop, led by the lead instructor, which features examples to stimulate discussion about how to write well. In Spring 2013, Haynes ran this workshop during the third class session and used the following slide deck, which was developed by Prof. Paul Seidel and modified with the help of Prof. Tom Mrowka and Prof. Richard Stanley.

The 18.821 Project Report (PDF)

This workshop was held before students had begun to think about the writing component of the course, and it seemed as if the students had to be reminded of the lessons of the workshop when they actually wrote their papers. In future semesters, we plan to offer the writing workshop closer to the time that students are drafting their first paper. We may also focus the examples used in the workshop on the few most important points rather than a broad coverage.

• Download video

This video features the writing workshop from Spring 2013 and includes instruction from Haynes as well as excerpts of the class discussion.

« Previous: Writing | Next: Sample Student Papers »

In this section, Prof. Haynes Miller and Susan Ruff describe how students receive feedback on their writing and what is expected from students during the revision process.

Feedback and revision are critical to students’ development as mathematical writers in the course. For each project, each student team is required to write a first draft, meet with course instructors for a debriefing meeting, make revisions, and submit a final draft. This process provides an opportunity for a mid-project check-in about the students’ writing as well as their research, and it pushes them to produce a stronger final draft than what most could have managed on their own.

In the best situations, a team’s first draft represents the students’ best efforts but is still somewhat rough; we give them lots of feedback for reworking their paper, and their final draft is substantially clearer and more rigorous, well-motivated, and technically precise. In our experience, each subsequent paper is typically better than the one before.

Instructor Feedback on Writing

After a team submits its first draft, the team’s mentor for that project, and sometimes Haynes and sometimes Susan, reads the paper and crafts feedback. First drafts typically have plenty of room for improvement. We try not to overwhelm students with a huge number of comments; commenting on everything often leads to students getting lost in the details and unable to distinguish the most important points from more trivial points. Instead, we draw attention to the most important things for the students to improve. We try to craft constructive comments so that, rather than being discouraged, students will be inspired to revise. Sometimes a second round of revision is necessary. This whole process is quite like the refereeing process for journal articles.

Debriefing Meetings

Students receive feedback on their draft at a team debriefing meeting, which usually occurs several days after the first draft is submitted. Sharing feedback via the debriefing meeting provides two key advantages:

• Clarity and emphasis via discussion. Speaking face-to-face allows us to emphasize the most important feedback; to ask students questions and understand the intentions behind their writing; and to have some back-and-forth to make sure that students understand the feedback.
• Efficiency. Reading papers and commenting on papers takes a long time. The debriefings allow us to convey some of the feedback efficiently in person rather than on paper.

Most students take the debriefing sessions very seriously. They do not see our feedback on their work beforehand, and they are naturally curious and may be somewhat anxious, especially the first time. The face-to-face interaction always helps to frame suggestions in a constructive manner, and students almost never respond defensively. They generally listen attentively and make a sincere effort to respond to our critiques.

Immediately after the debriefings, we scan the marked-up papers and send an electronic copy to the team members. The final draft is typically due a week after the debriefing, giving students time to think about research extensions of their work and to improve their writing.

Self- and Peer-Editing

One of the things we look for in papers for the course is consistency of voice and notation among sections written by different team members. We encourage the students to help each other revise.

« Previous: Revision and Feedback

To illustrate the writing and revision process for the student papers, two sample projects are presented below.

Sample Paper 1: The Dynamics of Successive Differences Over ℤ and ℝ

This project developed from the project description for Number Squares (PDF) . To view the practice presentation and final presentation from this team of students, see the Sample Student Presentations page.

The student work is courtesy of Yida Gao, Matt Redmond, and Zach Steward. Used with permission.

• First Draft of Sample Paper 1 (PDF)
• First Draft of Sample Paper 1 with Comments from Susan Ruff (PDF - 2.5MB)
• First Draft of Sample Paper 1 with Comments from Prof. Haynes Miller (PDF - 3.6MB)
• Additional Comments on Sample Paper 1 from Prof. Haynes Miller (PDF)
• Final Version of Sample Paper 1 (PDF)

Debriefing for First Draft of Sample Paper 1

This video features the debriefing meeting for the first draft of Sample Paper 1. The student team first presents their findings, and then the course instructors offer feedback and discuss the mathematics and the writing for the project.

Sample Paper 2: Tossing a Coin

This project developed from the project description for Tossing a Coin (PDF) .

The student work is courtesy of Jean Manuel Nater, Peter Wear, and Michael Cohen. Used with permission.

• First Draft of Sample Paper 2 (PDF)
• First Draft of Sample Paper 2 with Comments from Susan Ruff (PDF - 1.7MB)
• First Draft of Sample Paper 2 with Comments from Prof. Haynes Miller (PDF - 2.7MB)
• Additional Comments on Sample Paper 2 from Prof. Haynes Miller (PDF)
• Final Version of Sample Paper 2 (PDF)

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