specifications in analysis

Earlier this week, I wrote about expectations for my analysis class this fall (which also apply broadly to upper-level math classes) and some things I learned about specs grading this summer. In this post, I’ll share the specifications I have created for analysis. (I have taught real analysis before, and last time I tried a standards-based approach. Frankly, that basically turned into a point system, albeit a simplified one, which is why I’m trying something completely different this time.)

The rest of the post is taken verbatim from (the current draft of) my syllabus.

---

Effective learning requires effective methods of assessment. The assessments should relate as directly as possible to the expectations of the class, and they should provide both feedback on how to improve and opportunities to demonstrate improvement as the semester progresses. In my experience, “traditional” grading schemes based on assigning points or percentages to individual tasks do not serve these functions well. Therefore, this course adopts specifications grading*, in which grades are tied to specific outcomes. This is likely to be different from grading policies in other classes you have taken, so feel free to ask me questions or let me know if you have concerns. I hope that this system will make clear the connections between the expectations stated in the previous section and the ways you will be assessed.

Overall grading. At the end of the semester, I am required to submit to the university a letter grade reflecting your achievement in this class. That grade will be determined on the basis of a set of specifications in four areas: (1) class participation, (2) written proofs, (3) exams, and (4) synthesizing activities. Each of these areas will receive a simple grade of A, B, C, D, or F. The following sections describe how these grades will be determined. Your final grade will depend on your performance in all four areas, according to the following table.

Final grade	based on individual grades of
A	all As, or 3 As and 1 B
A–	two As and two Bs
B+	one A and three Bs
B	all Bs, or 3 Bs and 1 C
B–	two Bs and two Cs
C+	one B and three Cs
C	all Cs, or 3 Cs and 1 D
D–	two Cs and two Ds

I will use my discretion to assign a final letter grade to other combinations of individual letter grades.

Class participation. Attendance at every class meeting is required. Most weeks, we will alternate days between discussing reading assignments and presenting solutions to exercises. The end of this syllabus has a schedule of what we will be doing in class each day (with allowance for adjustments, as needed).

Reading. In order to participate effectively on discussion days, you will need to read the textbook before coming to class. Each reading assignment is about 10 pages. The textbook attempts to be very accessible, but that does not mean it is easy. We will be working with ideas that stretch reason and imagination. You should be prepared to spend at least 1–2 hours on each reading assignment; rereading pages, paragraphs, or sentences; working out examples; and writing questions or comments in the margins or on separate paper. You should be especially mindful of definitions. These are not always set apart from the text, so pay attention when new vocabulary is introduced. Start working on a list of definitions and theorems from the start of the semester. The chapter summaries can be an aid in this process.

Collaborating. On days with a reading assignment, you will work in small groups to discuss the material. I will assign these groups at the start of each week. You should bring your own questions and thoughts to these discussions. If there is extra time, you can also discuss the current set of exercises.

Presenting. On the remaining days, you will take turns presenting solutions to exercises distributed previously. The solution you present does not necessarily need to be entirely correct, but it should show evidence of a serious effort. You should also be prepared to answer questions from me or other students. To maintain balance, no one will be allowed to present more than once every two weeks, unless every student in the class has already presented during that time period. In exceptional cases, some of these verbal presentations may be made to me outside of class (no more than one per student).

To earn a	you must do the following
D	attend at least 75% of class meetings present at least one proof in class
C	attend at least 85% of class meetings and contribute to discussions present at least three proofs in class
B	attend at least 90% of class meetings and contribute to discussions present at least four proofs in class
A	attend all class meetings (2 unexcused absences allowed) and contribute to discussions present at least five proofs in class

Written proofs. Over the semester, you will develop a portfolio of work that you have submitted for formal assessment. Most of your contributions will be proofs. Each week I will indicate one or more exercises whose solutions could be submitted to your portfolio. You may discuss your work with other students in the class, to have them check whether it meets the standards of the class and give you feedback. A proof for the portfolio is due the Monday after it is assigned. These proofs must be typed using LaTeX, Google docs, Microsoft Word, or another system.

When you submit a written proof for your portfolio, I will judge whether it is Successful, Quasi-successful, or Unsuccessful (see the earlier section on “Proofs” under “Expectations” for details about these ratings), and mark it correspondingly with one of S/Q/U. Proofs marked Q or U will not be counted towards your grade. However, proofs can be resubmitted at the cost of one or two of your allotted tokens; see section on “Tokens” below.

To earn a	your portfolio must contain
D	at least four successful proofs
C	at least six successful proofs
B	at least eight successful proofs
A	at least ten successful proofs

Exams. There will be two midterm exams and a final exam. Each one will have a take-home portion and an in-class portion. [Dates and times, listed in syllabus, omitted here.]

The take-home portions will consist of two or three proofs that you are to complete on your own, without consulting other students. (You may discuss your work with me before turning in the exam, although I might not answer questions directly.) These will be judged as successful, partially successful, or unsuccessful, like the proofs in your portfolio. They cannot be resubmitted after grading, however.

The in-class portions will test your mastery of definitions and the statements of theorems. You will need to be able to state both definitions and theorems properly. You will also be asked to recognize and provide examples of situations or objects where a definition or theorem does or does not apply.

To earn a	you must do the following
D	correctly answer 60% of in-class test questions write at least two successful proofs on take-home exams
C	correctly answer 75% of in-class test questions write at least three successful proofs and one quasi-successful proof on take-home exams
B	correctly answer 85% of in-class test questions write at least four successful proofs and two quasi-successful proofs on take-home exams
A	correctly answer 95% of in-class test questions, write six successful proofs on take-home exams

Synthesis. To master the ideas of the class, you must spend time synthesizing the material for yourself. The items in this graded section will be added to your portfolio, to complement the proofs. All materials in this section must be typed using LaTeX, Google docs, Microsoft Word, or another system.

List of definitions and theorems. It should be clear at this point that being able to produce accurate statements of definitions and theorems is essential to success in this class. To encourage you to practice these, I am requiring you to create a list of these statements for the entire course. Your list should be organized in some way that makes sense to you—e.g., alphabetically or chronologically.

The textbook can be used as a reference, as can the internet, but how do you quickly recall what definitions we’ve used and how they're related? How do you find the phrasing of a theorem that’s become most familiar? This list should help you in these situations. More importantly, creating it will help you review and organize the material in your own mind.

I will verify your progress on these lists at each in-class exam.

Papers. Twice during the semester, once in the first half and once in the second half, I will provide a list of topics that we have been discussing, from which you can choose to base a paper on. These will be due approximately two weeks after the midterm exams.

There is a third paper that can be completed at any point in the semester on a topic of your choosing, but you must get the topic approved by me before Thanksgiving.

These papers will for the most part be expository, meaning they will present previously known mathematical results (not original research). Here are the requirements for a paper to be acceptable:

• It should have 1500–4500 words.
• It should use correct grammar, spelling, notation, and vocabulary.
• It should be organized into paragraphs and, if you wish, sections.
• It should cover the topic clearly and reasonably thoroughly, with an intended audience of other math students (who may be assumed to have studied as much analysis as you).
• It should contain a proof of at least one major result.
• The writing should be original to you. Of course, small pieces like definitions may be taken directly from another source, but apart from these the paper should be your own work.
• Citations are generally not necessary in expository mathematical writing, except for the following: a statement of theorem that you are not proving, a peculiar formulation of a concept/definition, or a creative idea (e.g., an uncommon metaphor or illustration) from another source.
• You may choose to follow the style of our textbook, or a more formally structured math textbook, or something more journalistic or creative, as long as the previous criteria are met.

Papers that do not meet these criteria will be considered unsatisfactory and will not count towards your grade. An unsatisfactory paper can be revised and resubmitted at the cost of three tokens.

To earn a	you must do the following
D	create a list of definitions and theorems to include in your portfolio
C	create a list of definitions and theorems to include in your portfolio write a paper on one of the topics provided
B	create a list of definitions and theorems to include in your portfolio write two papers on the topics provided, one during each half of the semester
A	create a list of definitions and theorems to include in your portfolio write two papers on the topics provided, one during each half of the semester write a third paper on a topic of your own choosing related to the class

Tokens. You start out the semester with seven (7) virtual “tokens,” which can be used in various ways:

• One token allows you to resubmit a written proof initially judged to be quasi-successful (must be used within one week of initial grading).
• Two tokens allow you to resubmit a written proof initially judged to be unsuccessful (must be used within one week of initial grading).
• Three tokens allow you to resubmit an unsatisfactory paper (must be used within one week of receiving paper back).
• One token gives you a 48 hour extension past the due date for a paper.

Unused tokens may be exchanged for a prize at the end of the semester. [maybe?!?]

*Based on Linda Nilson’s book Specifications Grading: Restoring Rigor, Motivating Students, and Saving Faculty Time.