Student write expository papers in an honors, non-science majors' calculus course to integrate the major ideas they're studying.
Background and Purpose
Millersville University is one of fourteen institutions in the State System of Higher Education in Pennsylvania. It has an enrollment of approximately 6,300 undergraduate students, about 280 of whom participate in the University Honors Program. Each student who plans to graduate in the Honors Program must complete a stringent set of requirements, including a course in calculus. For honors students seeking degrees in business or one of the sciences, this particular requirement is automatically met because calculus is required for these majors. However, most honors students major in the humanities, fine arts, social sciences, or education. To accommodate these students, the mathematics department designed a six-credit, two semester sequence, Honors Applied Calculus. There are usually between 25 and 30 students in each course of the sequence.
At Millersville University, courses are not approved as honors courses unless they include a well-defined writing component. At its inception, this requirement was not enforced for Honors Applied Calculus because no one at any level in the university associated writing assignments with undergraduate mathematics courses. The lack of an honors writing component troubled me, so I attended workshops such as "Writing Across the Curriculum" in order to obtain ideas about incorporating writing into the two courses. As a result, not only was I able to find ways to add writing assignments to Honors Applied Calculus, but I also decided to replace the usual in-class test on applications of calculus with a portfolio of applications. Writing would be an integral part of the portfolio because the students would be required to preface each type of application with a paragraph explaining how calculus was used to solve the problems in that section.
I use, on a regular basis, four topics for writing assignments in these courses. They are 1) a discussion of the three basic types of discontinuity; 2) a discussion of how the concept of limit is used to obtain instantaneous velocity from average velocity; 3) an analysis of a graph representing a real world situation; and 4) a discussion of how Riemann sums are used to approximate area under a curve. For the Riemann sum paper, I specify the audience that the students should address; usually it is someone who has had minimal exposure to mathematics such as a younger sibling. In the first semester, I assign one of the first two topics during the first half of the semester and the graph analysis paper near the end of the semester. In the second semester, the portfolio of calculus applications is due by spring break, whereas the Riemann sum paper is assigned near the end of the semester. I always allow at least three weeks for students to complete each writing assignment.
Typical instructions for the graph analysis paper are:
Find, in a publication, a graph of a function that represents a real world situation, or make up your own graph of a function that could represent a real world situation. The graph you use for this assignment must exhibit at least six aspects from the list of graph aspects that we compiled in class. You are to make a chart indicating each aspect of the graph, where it occurs, and what it indicates about the function or its derivatives. Then you are to write a narrative in which you explain the connections between each graph aspect and the real world situation it represents. The final paper that you submit must include the graph, the chart, and the narrative.
I believe that this assignment is the definitive assessment tool for the conceptual material that we cover in the first semester. By transferring what they have learned in the course to a situation outside of the course, the students clarify for themselves the significance of such things as extrema, discontinuities, and inflection points. It is important to realize that in this assignment, I have no interest in evaluating computational ability. Rather, I am assessing students' critical thinking, analytical thinking, and interpretative skills.
When I grade writing assignments, my primary concern is accuracy and my secondary concern is organization. No one is required to be creative or original, but most students voluntarily incorporate both attributes in the work they submit. The portfolio project, on the other hand, involves problem solving and computation in addition to writing. Each student includes two or three problems, chosen from a list provided by me, for each type of application (Newton's Method, optimization, differentials, and related rates). Each section of the portfolio corresponds to one of the types of applications and includes the brief written explanation mentioned earlier, followed by the statement of each problem to be solved in the section and its detailed solution, including relevant diagrams. I evaluate the portfolio primarily on correctness of solutions and secondarily on writing and organization.
Students have reacted favorably to the writing assignments. On the Honors Program course evaluation questionnaire, many students claim that writing about mathematical ideas is a greater intellectual challenge and accomplishment than doing calculations or solving problems. They also have claimed that writing clarifies concepts for them. So far, no student has indicated that he or she would prefer to have tests in place of the writing assignments. However, students have indicated that putting together the portfolio is very time consuming and several have remarked that they would have preferred a test in its place. There is also universal agreement that the writing assignments and portfolio do comprise a more than adequate honors component for the sequence.
It is interesting to note that on the graph analysis assignment, approximately the same number of students choose to make up their graphs as choose to use graphs from publications. In the latter case, many students use graphs related to their majors. Some examples of situations for which students have created their own graphs are:
1) a graph of Joe Function's trip through the land of Graphia where a removable discontinuity is represented by a trap;
2) a graph of Anakin Skywalker/Darth Vader's relationship with the force where the x-intercepts represent his changes from the "good side" to the "dark side" and back again;
3) a graph of profits for Computer Totes, Inc., where a fire in the factory created a jump discontinuity. The narrative written by the student in this paper was in the form of newspaper articles.
Some examples of graphs students found in publications are:
1) a graph of power levels in thermal megawatts of the Chernobyl nuclear power plant accident where there is a vertical asymptote at 1:23 a.m. on April 16, 1986;
2) a graph of tourism in Israel from 1989 to 1994 with an absolute minimum occurring during the Persian Gulf War;
3) a graph of the unemployment percentage where it is increasing at an increasing rate between 1989 and 1992.
The assignment on Riemann sums has also produced some very innovative work. One student prepared a little book made of construction paper and yarn, entitled "Riemann Sums for Kids." A French major, writing in French, explained how one could amuse oneself on a rainy day by using Riemann sums to approximate the area of an oval rug. Another student wrote a guide for parents to use when helping their children with homework involving Riemann sums. An elementary education major prepared a lesson plan on the subject that could be used in a fifth grade class. Several students submitted poems in which they explained the Riemann sum process.
The portfolio produces fairly standard results because it consists of more "traditional" problems. However, there have been some creative presentations, such as prefacing each section of the portfolio by a pertinent cartoon.
Use of Findings
When I first started using writing assignments in Honors Applied Calculus, I only counted them collectively as 10% of the final course grade. At that time, I did not value the writing as much as I did the in-class computational tests. However, I soon realized that students devoted more thought, time, and effort to the writing assignments than they did to the tests. I also realized that I learned more, through their writing, about how my students thought about and related to mathematics than I did from their tests. As a result, I now give equal weight to writing assignments, the portfolio, tests, and homework when determining the final course grades in the course.
Furthermore, when I began to use writing assignments, I did not feel comfortable evaluating them. This situation resolved itself once I realized that the students, writing about mathematics for the first time, were just as unsure of what they were doing as I was. Now, I encourage the students to turn in rough drafts of their work the weekend before the Friday due date. This provides me with the opportunity to make suggestions and corrections as needed without the pressure of assigning final grades right away. Sometimes I will assign provisional grades so students will have an idea of the quality of their preliminary work.
I also have learned about curriculum from these projects. For example, based on results from the writing assignment on types of discontinuity, I am convinced that the most effective way to introduce limits is within the context of continuity and discontinuity. This is the approach I now use in all my calculus courses.
It would be difficult to imagine better conditions under which to use writing in a mathematics course than those that exist in the honors calculus sequence. The students are among the most academically motivated in the university. They are majoring in areas where good writing is highly valued. The pace of the sequence is determined by the needs of the students and not by a rigid syllabus that must be covered in a specific time frame. As a result, there is ample class time to discuss topics after they have been introduced and developed. For example, in the first semester, we spend twice as much time on the relationship between functions and their derivatives as is typically scheduled in other calculus courses.
Even though I initially introduced the writing
assignments to fill a void, I now view them as an indispensable part
of learning calculus. Consequently, I am looking for ways
to incorporate writing in all my calculus courses.