Mathematics students often have a reasonable idea of how to solve a problem but then struggle to write it down. They may also have an exaggerated idea of how clear their work is to others. This initiative encourages students to think critically about their mathematical writing by peer-marking a question in the fourth year combinatorics course.
The rationale for the peer-marking exercise is explained to students in advance. This motivates them to do the exercise and gets them thinking about mathematical writing.
The problems in the course are often challenging to solve, but usually fairly easy to understand. The chosen question (Question 2 on Sheet 3) has an easy start, accessible to all students, and ends with an optional part, intended to stretch the more advanced students.
The exercise was run in an informal way with little administrative overhead. I give students about four days to do the marking, then collect the marked work and check it overnight, before returning it to the student. I have never needed to make any changes to the peer-marking, but have occasionally added extra comments of my own.
A detailed marking scheme is issued. I also stress the importance of being polite and positive when giving feedback. Peer-markers are anonymous, but are identified to me by using the final digits of their student numbers.
One potential issue is a peer-marker not taking their job seriously, or writing a comment likely to cause offense. Neither has happened in four years of running the exercise. My experience is that peer-markers put considerable effort into their marking, far more than the 15 minutes I suggest.
Peer-marking is not done much in mathematics and may represent a substantial cultural shift for some students. I encourage anyone who has concerns to talk to me informally after a lecture or in an office-hour. In practice no-one has even taken this up: it may be that having access to a detailed marking scheme helps assuage any fears.
Students who do not submit work are not used as peer-markers. Submission rates for the peer-marked question have been very good (typically all but one or two in a class of 30).
Having a mixed ability group seems to be a good thing for this exercise. When checking the peer-marked work, one comment I found was `Wow! I wish I wrote as clearly as you'. This, and other similar comments, clearly show that the exercise can increase students' appreciation of the value of well-written mathematics.
Analysis suggests that students put in more effort into the peer- marked question and usually do better at it than the other questions on the sheet. I take advantage of this by setting a problem related to the peer-marked question in the final exam for the course.
Students have several times commented to me that they had only appreciated errors in their own thinking after they saw the same error exhibited on the script they marked. They also see the difficulty in reading overly terse or poorly explained work.
Informal feedback suggests that students find the peer-marking exercise useful and interesting. I think it helps to create a good atmosphere in the classroom, and shows students that their views and judgements are valued.
Writing the marking scheme is time-consuming, but only has to be done once. I now find the exercise saves me a small amount of marking time each year.
I have several times had students remark to me, after doing the exercise, that they had a much better sense of the difficulty of giving useful feedback on mathematical work. This seems to be a welcome side-effect of the exercise.
Scaling the exercise to bigger groups might require a formal approach. I think it is probably better suited to a 3rd or 4th year course where the students can be expected to be more mature, and there is more scope to set questions requiring extended mathematical writing.