Principles of Applied Mathematics

This lecture-based course at M.I.T. was originally developed by Daniel Kleitman. It was transformed into the form published here (2011) by Peter Shor, Michel Goemans, Neil Olver, Olivier Bernardi, and Susan Ruff. One lecture is devoted to writing topics (information order and connectivity), but most writing instruction comes in the form of online resources and feedback on writing assignments.

Enrollment is typically about 50; thus feedback on students’ writing must be supplied by TAs. One of the greatest communication-related challenges for the course is training and supervising these TAs to ensure the quality of the writing feedback students receive.

Writing Assignments

Most problem sets include a short (1-2pp) writing assignment and many also ask students to revise an earlier writing assignment after they’re received feedback. In the second half of the term, a longer (3-5pp) writing assignment is given; this paper is also revised.  The topics for the assignments vary from term to term. Examples of assignment types follow.

  • Assignment: Prove a given statement. (1-2 pages)
    Notes: Many students have not written proofs before, and proof-writing is an important focus of the class.
    Communication objectives: proof writing, precision, rigor, guiding text
    Handout: formatting a proof and using guiding text
    Rubric for short writing assignments
  • Assignment: Describe an algorithm and calculate its running time. (2-3 pages)
    Notes:The algorithm is complex enough that students must introduce and explain sub-algorithms. Students are faced with several structural challenges:

    • Should the algorithm be presented as a sequence of steps or should the high-level approach be given first with successively greater detail as needed?
    • Should the running time for each piece of the algorithm be given when that piece is explained, or should the entire running time calculation be delayed to the end of the paper?
    • Should sub-algorithms be presented first, within the context of the larger algorithm, or at the end?
    • Should necessary definitions be stated before the algorithm is presented, or should new terms be introduced in context?

    Some guidance for handling these challenges is provided to students.
    Communication objectives: structure; defining terms; context vs. detail

  • Assignment: Explain a topic in discrete applied mathematics to two different audiences. (2-3 pages)
    Notes: Students are presented with a situation in which someone asks them a question: they must do a rhetorical analysis of the context and then answer the question appropriately. Students choose two contexts/questions from three or more that are all related to the same topic. Thus students must adjust their explanation of the topic based on the context. Because some contexts call for an oral response, students are given the option of writing a dialogue.
    Communication objectives: rhetorical analysis of context (including audience knowledge and purpose), designing communication appropriately given the context (including level of detail).
  • Assignment: Analyze a linear program and its dual (5 pages)
    Notes: Students choose from among several suggested linear programs.
    Communication objectives: structure, focus, cohesion, and smaller scale issues as needed.

The following assignments are no longer used:

  • Assignment: Term paper on a topic of the student’s choice (10+ pages)
    Notes: Guidance for choosing a term paper topic was provided.
    This assignment is no longer used in this class. This assignment is likely to be more valuable in smaller classes in which students can be individually encouraged to use the assignment to improve their understanding of the class material.
    Communication objectives: structure, focus, cohesion, and smaller-scale issues as needed.
    Rubrics for draft and for final paper
  • Assignment: Peer critique (1-2 pages)
    Notes: Due to scheduling constraints, peer critique was conducted online rather than face-to-face. Students report that online critique of shorter writing assignments isn’t very helpful, so this assignment has not been used since the term paper assignment was suspended.
    Handout: Peer review guidance plus rubric

TA training and supervision

The TAs include one graduate student and about 3 undergraduates. The undergraduates are chosen based on demonstrated ability with writing and peer critique from communication-intensive math classes that they’ve taken as students.

The TAs are trained to comment on writing by the lead instructor(s) for the course and by a lecturer from M.I.T.’s Writing Across the Curriculum (WAC). Training includes the following:

  • Shortly after the first papers of the term are received, the entire team meets to discuss how to use a rubric to grade the papers. The same rubric is used throughout the term.
  • For each assignment, a few of the papers assigned to each TA are commented on as well by the WAC lecturer. This redundancy enables the TAs to learn from the lecturer’s comments and enables the lecturer to assess the TA’s comments and to provide guidance as needed. All of the WAC lecturer’s comments are made available to all TAs.
  • For the first several assignments, the team meets after completing the grading to ensure that all TAs are grading consistently. TAs come prepared with their grades and brief reasons for assigning each grade. Each TA identifies their best and worst papers. These papers are circulated among the teammembers to ensure that grades are consistent at the ends of the grading scale. If needed, papers at other grade boundaries in the middle of the scale are circulated as well, with grades being adjusted as needed to ensure consistency. These meetings typically require an hour a week, and often include discussion of which writing characteristics are most important and how to write effective comments.
  • Once the staff have conducted enough joint norming sessions to agree on the desired writing characteristics, the weekly meetings may be suspended. Instead, for each assignment, the course instructor quickly grades three papers (one strong, one weak, one average) as soon as the assignment is due. The resulting grades serve as anchors for the other graders. Throughout the term, all graders are reminded to give honest praise in addition to constructive feedback; as the term progresses, graders are encouraged to recognize improvement.

The following rubrics and guidance have been provided:

  • Rubric for short writing assignments (Spring 1010) For the sake of simplicity and consistency, this rubric is used for both the first draft and the revision.
  • Guidance for assigning grades to drafts and revisions when the draft receives a grade based only on effort (this rubric is not currently used)
  •  Guidance for grading the term paper draft (Spring 2009) Students were assigned a grade for the draft that was based on effort/completeness as well as an advisory grade for the quality of the writing; the writing grade was revised when students revised their papers.
  • Guidance for identifying and responding to suspected plagiarism
  • A common issue is how to grade and provide feedback on the writing of second-language learners (those who are in the process of learning English). The following guidance is given to TAs in this class:
    • Becoming fluent in a language takes years, so have realistic expectations for both yourself as commenter and for the student as learner. Developing an “ear” for article, number, and prepositions can take particularly long for some second-language learners, so comment on such grammatical issues only once and then ignore them. If you like, you may underline such grammatical issues if doing so enables you to read past them–once the author sees the underlining, he or she is likely to be able to identify the grammatical problem and correct it. Feel free to suggest that the student ask a friend to do such underlining. You could also suggest using a grammar checker, but note that grammar checkers are not foolproof.
    • Comment and grade primarily on issues that affect clarity (e.g., word choice, use of guiding text, structure).
    • Some structural styles are cultural, so it may be helpful to point out the structures commonly used in the American academic mathematics writing culture.
    • If a student makes a recurring mistake of grammar or word choice, note it once and suggest that the student add it to an “editing checklist” or to a list of words and phrases to learn. [This comment applies to all learners of “mathematical English.”]

Instructor observations

The term paper assignment was suspended in part because the lead instructor(s) felt that students were not learning the mathematical content of the course well enough and that, with such a large class, it was difficult to ensure that each individual chose a sufficiently challenging term-paper topic. To address these problems, the term paper was replaced with more shorter assignments targeted to specific concepts within the course.

To further help students to learn the course material, a proposal has been made (and approved) to add a recitation to the course. The proposed recitation will serve two purposes:

  • to further enable students to reinforce their understanding of the core content by discussing it in recitation
  • to provide opportunities for more small-group instruction in and discussion of communication topics.

The proposed recitations have been loosely modeled on the recitations for M.I.T.’s communication-intensive offering of Real Analysis. Most of the writing assignments and TA training will remain as described above, but a few assignments will be added, and the credits for the course will be increased.

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