Creating successful calculus writing assignments

Primus: Problems, Resources, and Issues in Mathematics Undergraduate Studies, Jun 2002 by Green, Kris H

ABSTRACT: I discuss three different writing assignments that I have used in my calculus courses. These assignments are introduced with a discussion of purpose and audience. Defining these qualities of an assignment will ensure that writing assignments are more successful. The assignments discussed and explored here represent three different purposes: personal, informational, and a blend of the two. The audiences for these assignments are diverse and force the students to incorporate particular modes of writing that demonstrate much of their thinking. Assessment of student learning as a result of these assignments is discussed. A fourth writing assignment is developed from a basic idea in order to illustrate the use of purpose and audience to create meaningful assignments.

KEYWORDS: Writing, calculus, audience, purpose, meaningful assessment.

1 INTRODUCTION

We have been compiling evidence on The Exponential family for years. They [the members of this family] have contributed to many events, from drug overdoses, increases in debt, the polar ice caps melting, and even the hole in the ozone layer. These all depend on an initial amount, a rate, and an amount of time.

This excerpt was part of one student's response to the second writing assignment below. In the excerpt, the student has described a wide variety of applications for exponential functions. We did not discuss these applications in class, and they are not explicitly stated in the text. She has either made connections for herself or gone to other resources for these applications. More likely, she has combined these two approaches. The student has furthermore generalized from these separate examples to identify the qualities of an exponential function - initial amount, rate, and time - using her own language. While the excerpt does not demonstrate an ability to solve problems that involve exponential functions, the student has built a framework in her mind so that future problems involving exponential functions can be more readily assimilated. In fact, since this assignment was completed before rates of change and derivatives were discussed, the student has demonstrated that her mind is ripe for a new understanding of "rate" that will apply to more general settings. At the same time, the writing demonstrates that there is a critical "missed connection" for this student. She has stated that all exponential functions depend on three quantities, but she did not explain how these functions depend on those quantities. In responding to her work, I can call her attention to this missed connection by asking, "How do exponentials depend on those quantities?" Thus, her writing gives a thorough peek into the way she thinks about these functions. This type of assignment clearly exhibits the conceptual understanding of the students.

This is in stark contrast to the more in-depth, application-type problems that require several computational skills to be used in order to determine a specific answer. The problem with these assignments as a measure of conceptual understanding is that it is difficult to distinguish between the students who understand the concepts but cannot perform all of the separate skills in the problem, and those students who have all the skills but do not possess a "big picture" of the problem to tie the skills together or to aid them in deciding which of the many possible skills are appropriate. It can be argued that writing assignments muddy the waters of assessment further since the ability to write is essential to express one's understanding. While I agree that this is a potential danger, Assignment III below and the concluding remarks discuss a multi-step, shared (between student and instructor) model for writing assignments that can mitigate this difficulty while helping students to improve their writing skills.

Writing assignments provide students with opportunities to demonstrate conceptual understanding at various levels. First of all, as the excerpt above shows, a good writing assignment provides an unparalleled way to look at what the student knows and does not know. Writing assignments provide an alternative learning style through the verbal/linguistic intelligence that allows a broader range of students to participate in the course to a higher degree. (By forcing our students to write more often, we are forcing them to 1. develop stronger reading skills for collecting, analyzing and synthesizing information, 2. develop the ability to proofread their own work, and 3. develop the ability to write within the discipline of mathematics.) The process of writing helps students to scaffold their learning by forcing them to collect and organize their thoughts on a subject. This last reason deals with metacognition: thinking about the way you think in order to create a single framework in which to place new ideas and skills.

Below I describe three writing assignments that have been successful in my calculus courses at St. John Fisher College. A brief overview of two important writing characteristics of the assignments, purpose and audience, is given. Discussion of each assignment concludes with notes on how I grade the assignments and how the assignment can be used to assess the learning of the students.