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By having students write a letter to a friend about your course, you can get useful information both on what the students have learned and on what they thought of the course. Dear Professor Math,
Thank you for your interest in my use of topic letters in lower division mathematics courses. I am sending you more information on these assignments in letter format so that you will have some sense of their style. Let me begin by outlining for you their Background and Purpose.
For a number of years, I have been using student writing both to develop and evaluate conceptual understanding in my lower division courses. The topic letters are one such assignment. In these letters, students are asked to write about mathematics to someone they know; for instance, I might ask them to describe the relation between various concepts (e.g., what is the relation between exponential and logarithmic functions?), the relation of a concept to previously studied concepts (e.g., how does knowledge of the logarithmic function extend our calculus repertoire?), or the relation of some concept to its motivating problem (e.g., how does statistics address the problem of prediction?). Because these types of questions are conceptual in nature, I have used topic letters primarily in my calculus and liberal arts courses, although there are ideas in more skills-related courses like college algebra for which a letter could also be used.
I am fortunate that class sizes at USC average 25-45, so that evaluating the letters is manageable. Initially, my objective with the topic letters was solely to evaluate student conceptual understanding. I have found, however, that they also provide an evaluation of how well I've done at designing course experiences to develop conceptual insight and understanding. Unlike traditional anonymous student questionnaires which focus almost exclusively on teaching mechanics like "punctuality," the topic letters tell me something about the quality of the learning environment I tried to create, my success in creating it, and its success in building conceptual understanding. An end-of-semester "Letter to a Friend" addressing the course as a whole was therefore a natural extension of the topic letter assignment.
My Method is illustrated in these partial instructions for an End-of-Semester letter in first semester calculus:
You have a friend who foolishly decided to attend another university in the east. (Like Harvard has anything on USC.) This friend has heard that calculus is a requirement for many disciplines, and is considering a course in calculus next semester to keep all major options open. Knowing that you are just completing this class, your friend has written to you asking for your insights.
Having never heard much about what calculus really is, this person would like to know what kinds of problems calculus can solve, what its most important ideas are, and what, if anything, is interesting or exciting about the subject.
Write a letter to your friend giving your answers and insights into these questions......
The audience for the assignment is set very deliberately. I emphasize to students that they must write to someone who has never studied the mathematics in question, but who has the same working knowledge that they had at the start of the topic, or the start of the course. Without such an audience, students are less inclined to describe their personal understanding of the mathematics, and more inclined to try to impress me with the use of formal terms that they may or may not understand...and I will probably let them.
I do assign a grade to the letters, based primarily on the insight and mathematical correctness displayed. I have found that it is important to stress that quality of insight (and not quantity of facts) is the key feature of a strong letter. Students are instructed to write no more or less than is needed to convey the ideas clearly. To allow them as much freedom as possible in how they do this, I do not set page limits or other formatting requirements. (Most letters fall into the 3-5 typed page range.)
Students who submit weak topic letters are encouraged to revise them, and I provide each student with specific written suggestions on areas that could be improved. Revisions of the end-of-semester letters are not permitted since they are collected during the last week of class. Since my main purpose with the end-of-semester letter is to get a more meaningful course evaluation, I simply give all students full credit as long as they seem to have made a serious effort with it.
Findings of note include the fact that the grading itself can take some time.....10-15 minutes per letter since I read each one twice (first for comments and a preliminary grade, then again to ensure consistency and to assign the actual grade). Most of this time goes to writing comments; in larger classes, this lead me to require fewer letters than I might otherwise, with four being the maximum I would consider for any class.
The time is worth it. First, the students receive an excellent opportunity to synthesize the material. Most students find this useful (although a few question the place of writing in a mathematics class), and quite a few students go out of their way to be creative. I've gotten interesting letters to a variety of real and imagined friends, family members, pets, plus the occasional politician and movie star, which makes the letters fun for me to read. I also enjoy the letters that take a more straight-forward approach to the assignment, since even those students use their own voices in the composition, something which rarely happens in other assignments.
The value of the topic letters for course evaluation became especially clear to me when I first began using topic letters in my liberal arts mathematics course about two years ago. I had been using them for some time before that in other courses, and knew that the assignment sheet needed to warn them (loudly) that working examples was not the point, and encourage them (strongly) to convey the "big picture" in a way meaningful to their "friend." Still, most students included numerous examples in their letters, and not much else. After reflecting on this, I realized that much of our class time focused on just that: working examples, and not much else. In fact, several of the letters came straight from those class examples. Naturally, we looked at conceptual explanations in class (as did the book), but these were not included in the letters. Was this just because we spent more time on the examples? And why was I spending so much more time on examples? With further reflection, I realized that virtually all my homework, quiz and test questions were focusing on mechanics; the letters, I thought, would address the conceptual side of the course. But with such a heavy emphasis on mechanics in the course grade, I had (somewhat unwittingly) emphasized examples in class in order to prepare students to earn those points, and students responded to this in their letters.
My own Use of Findings came through various modifi-cations to the course. After some initial experimentation, I replaced in-class quizzes based on mechanics with take-home quizzes that require students to discuss and apply concepts.. I've also developed a set of "course questions" which I use on the final exam, after giving them (verbatim) to the students at the beginning of the semester and returning to them during the semester as we cover the relevant material. Along with these changes in the course assessment structure, I modified instruction through the use of more discussions centering on the course questions and concepts, and less emphasis on examples of mechanics in lectures. These modifications are now having the desired effect; the topic letters are much better, as are students' responses to the conceptual questions on various tests and quizzes, and their mastery of mechanics has not suffered.
The information I gain from the end-of-semester letters also helps me to fine-tune course instruction and the overall syllabus, as well as instruction in specific topic areas. To get course evaluation information from these letters, I watch for comments that indicate the value of the various activities we did during the semester (such as advice to their friend concerning important things to do), an indication of the emphasis placed on concepts versus skills (such as a list of rules with no concepts mentioned), or an indication of a concept that may have been under (or over) emphasized (such as no mention of the Fundamental Theorem in a calculus class).
Some Success Factors to consider have already been mentioned; the time concern is one which I would reiterate. With the topic letters, there is also the difficulty of sorting course evaluation information from student assessment information; for example, when I get a letter emphasizing examples and algorithms instead of concepts (as I still do), it may not be due to the class experiences provided. The student may not have understood my expectations or the concepts, or may simply have taken the easy way out. This means that I must look at the collection of letters, as well as the individual letters, to get a "course" reading. All these factors make it essential to set the objectives of the letters carefully, so that both you and the students gain from the time investment.
Another concern I have is the fact that the letters are not anonymous, which may limit their validity as a course evaluation tool. On the other hand, some of the things students put in their letters suggest they are not overly-concerned with anonymity. In some of these, I suspect the student is going for shock value (with tales of drinking parties and other such escapades). As far as possible, I sort out the mathematical and course information in these, without reacting to their provocative side. A more distressing type of "no-holds barred" letter I have received comes from students who did not do well in the course, or whose confidence was shaken along the way, and who tell their friend very clearly how this has affected them. Although I've never read any anger in these, they are very difficult to read and comment on. Despite that (or perhaps because of that), this type of letter has given me good information about the course, and what kinds of experiences are discouraging to students.
I hope you too will find the letters valuable, and that
you'll share your experiences with me. Best wishes...J.B.
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