In this section, Prof. Haynes Miller and Susan Ruff describe their approach to grading and the development of grading rubrics for the course.
In Spring 2013, the three project papers and the presentation were weighted equally in students’ final grades.
|Project 1 paper||25%|
|Project 2 paper||25%|
|Project 3 paper||25%|
As part of the presentation grade, students were expected to attend all of their classmates’ presentations and submit peer comments.
The grades for papers were broken down as follows:
|Mathematical vision (This intentionally vague category captures how well students tackled the mathematics in the context of their background and abilities. While subject to some vagaries, this criterion is important to avoid preconceptions about the “right” approach to a project.)||50%|
|Process (This reflects how students progressed and whether they met milestones.)||10%|
|First draft (This primarily serves to reflect how complete the first draft was.)||10%|
Each team was given a team grade for each project. Students were also asked to declare which part of the writing they did, and each received separate grades on that as well. It did sometimes happen that members from the same team received different grades.
The students in the course span a wide spectrum in terms of background and experience in mathematics, so grades cannot be based solely on mathematical performance. But all students can be expected to carry out a research project at some level and write it up coherently. We aim to grade relative to the starting point of each team and what we believe they are capable of achieving. We also take into account improvement over the course of the term, both in research technique and in writing.
Evaluating project papers is quite challenging, and the grading process for the course is still a work in progress. A main challenge is that the process is somewhat subjective; there are many factors that contribute to the quality of a paper. Grading is not simply a process of checking whether answers are correct.
Each paper is read by the group’s mentor for that project, and about 40% of them are also read by Susan. Each reader completes a scoring sheet, and all scores are then averaged.
In Spring 2013, for the first project, we held a grade norming meeting during which we discussed grades for all nine papers. Reviewing nine papers in depth as a group is quite an involved process, but it is important that grades be consistent across instructors. It helped us grade consistently and also gave us an opportunity to collectively identify teams that particularly needed attention.
Grades are typically lower at the beginning of the semester and better at the end because students get better at both research and writing. In Spring 2013, we were surprised by the lack of complaints about grading. This is striking because as math people, we’re used to having clearer grading situations. It’s encouraging that we were able to grade projects in a way that students perceived as fair.
Development of Rubrics
Grading rubrics are useful for maintaining standards and consistency. They are especially useful in a course like Project Laboratory in Mathematics, in which a substantial portion of the grade is based on assessment of performance in an area (communication, in this case) that is outside the experience of many people involved in the course. Ideally grading rubrics should be consistent from one term to the next, but for various reasons this has been hard to achieve. Below, we share the rubrics from Spring 2013 and Fall 2013.
The writing rubric we used in Spring 2013 was derived from a form used by Haynes the previous time he taught the course, in Spring 2007. We attempted to break down the components of a paper as finely as possible in order to help us assess all the aspects of a successful piece of work. Our rubric identified characteristics ranging from broad (e.g., Are the different parts of the paper in place? Do they perform their function correctly?) to fine (e.g., Are mathematical terms and symbols used correctly?) We were not happy with this rubric because we generally found that each component was more or less okay in isolation, and total grades were fairly high, but the grades did not necessarily reflect the overall quality of the papers.
A key challenge in grading presentations is that there are many different good ways to present mathematics. For example, using slides and using the blackboard are both valid and common ways for mathematicians to communicate their ideas. We developed a checklist, posted it on the course website, and used it as a framework for assigning grades. Staff usually completed the student response sheets, which were published with the others. After each presentation, all instructors sent their thoughts on the presentation and proposed grades to Haynes, and he compiled that feedback to generate a final presentation grade.
Select student response sheets are available on the Practice and Feedback page.
In Fall 2013, Prof. Larry Guth, who was the lead instructor, and Susan developed new rubrics for the course. These rubrics were built upon an extensive list of characteristics of effective mathematics communication derived from the Spring 2013 course and earlier versions of the subject, with ideas emerging from a workshop conducted by Susan, Joel Lewis (co-instructor in Spring 2011, now at the University of Minnesota) and Mia Minnes (UCSD) at the 2013 Joint Mathematics Meetings.
Fall 2013 Paper Rubric (PDF) (Courtesy of Susan Ruff and Prof. Larry Guth. Used with permission.)
Fall 2013 Presentation Rubric (PDF) (Courtesy of Susan Ruff and Prof. Larry Guth. Used with permission.)