Increasing the Dialogue About Calculus with a Questionnaire

A practical survey is provided here which might be found useful for departments looking to initiate discussion about goals and expectations in a Calculus course.

Background and Purpose

Many mathematics departments across the country are considering change in the teaching of calculus. In our work on the Calculus Consortium Based at Harvard Evaluation and Documentation Project (CCH EDP) [2], we developed a set of questionnaires. Items from these questionnaires can provide mathematics departments with a vehicle for beginning discussions about possible change in the teaching of calculus or for assessing changes already made. Throughout the project we were constantly reminded of the wide variation in participants' perspectives and local situations. The implementation of reform-based calculus should be different at each institution because implementation patterns reflect local student needs, faculty attitudes and beliefs, and characteristics unique to institutions.

Method

The CCH EDP began in the fall of 1994. In this project, we sought (a) to investigate faculty perceptions of student learning and faculty attitudes and beliefs towards calculus reform and (b) to examine and describe the evolution of efforts to reform the teaching of calculus in the context of CCH Curriculum Project materials ([3]). As part of the CCH EDP, we surveyed 406 faculty members, who were using or had used CCH Curriculum Project materials, at 112 institutions of higher education. In the surveys, we asked about types of courses, types of students using the materials, faculty interpretation and use of the materials, pedagogical approaches, the extent of technology use, institution and department reaction to the use of the materials, factors influencing the initiation of reform efforts, and faculty attitudes and beliefs.

Findings

The questionnaire (see Appendix I) contained in this article represents a subset of the original CCH EDP survey items. The items were chosen because we found, based on responses to CCH EDP survey, that they revealed significant information about how faculty interpret and implement reform-based calculus. Many of the questionnaire items were based on goals for reform in calculus instruction established at the Tulane Conference in 1986 [1].

Use of Findings

Our experience indicates that casual and formal discussions within a department are critical in assessing a calculus program. There are several ways you might use the questionnaire. You could use the questions as the basis for calculus-instructor or full-department discussions without the faculty completing the survey beforehand. You may decide to modify the items to develop a questionnaire more suited to your own situation or to include fewer or more comment-type questions. On the other hand, you might ask faculty members to anonymously complete and return the questionnaire prior to a meeting. You could collate and analyze the responses and distribute an analysis of the quantitative responses and a listing of the comments. The responses to each set of items could serve as the foundation for more formal calculus-instructor or department discussions. To get a discussion rolling, you might ask the following useful discussion questions:

• How would you characterize your department's approach to calculus instruction? What are the more important features, the less important?
• Is using technology the basis for your efforts? Why or why not?
• Do your efforts encourage student engagement with calculus? How do you know?
• What role do the following pedagogical methods play in your approach: cooperative groups, projects, student presentations, using concrete materials to teach calculus? What solid evidence do you have that these techniques work or that the lecture method works?
• What does "rigor" mean to you, and what importance do you give "rigor" in the teaching of calculus?

Success Factors

Continued use of this questionnaire will refine the issues that are important to you as a faculty and the process could initiate bringing about changes in the questions you ask. The questionnaire provides a format for meetings in which the faculty, and possible student representation, can come together periodically to discuss the results. In this way, your department can create a beginning for reforming the learning environment. Above all, your faculty and students will become more sensitized to the goals of the department and the issues surrounding the teaching of calculus.

References

[1] Douglas, R., ed. Toward a Lean and Lively Calculus: Report of the conference/workshop to develop curriculum and teaching methods for calculus at the college level (MAA Notes Series No. 6), Mathematical Association of America, Washington, DC, 1986.

[2] Ferrini-Mundy, J., CCH Evaluation and Documentation Project, University of New Hampshire, Durham, NH, 1994.

[3] Hughes-Hallett, D., Gleason, A.M., Flath, D.E., Gordon, S.P., Lomen, D.O., Lovelock, D., McCallum, W.G., Osgood, B.G., Pasquale, A., Tecosky-Feldman, J., Thrash, J.B., Thrash, K.R., & Tucker, T.W. Calculus, John Wiley & Sons, Inc., New York, 1992.

Appendix

CALCULUS QUESTIONNAIRE

Please answer all questions and comment as often as you wish.

1. CONTENT:

1A. Please indicate the level of emphasis that you feel should be placed on the following topics in a first-year calculus course.

		Amount of emphasis
		little or none			heavy
a.	Preliminaries (functions, absolute value, etc.)	1	2	3	4	5
b.	Limits (lengthy treatment, rate "heavy")	1	2	3	4	5
c.	Derivative as a rate, slope of tangent line, etc.	1	2	3	4	5
d.	Using definition to find derivative	1	2	3	4	5
e.	Techniques of differentiation	1	2	3	4	5
f.	Applications of the derivative (max/min, related rates, etc.)	1	2	3	4	5
g.	Fundamental Theorem of Calculus (lengthy treatment, rate "heavy")	1	2	3	4	5
h.	The definite integral as area (lengthy treatment, rate, "heavy")	1	2	3	4	5
i.	Techniques of integration	1	2	3	4	5
j.	Applications of integration (arc length, volumes of solids, surface area, etc.)	1	2	3	4	5
k.	Applications of exponential/logarithmic functions (growth, decay, etc.)	1	2	3	4	5
l.	Solving differential equations	1	2	3	4	5
m.	Applications of differential equations	1	2	3	4	5
n.	Series (lengthy treatment, rate "heavy")	1	2	3	4	5
o.	Series — techniques to determine convergence	1	2	3	4	5
p.	Taylor series (lengthy treatment, rate "heavy")	1	2	3	4	5
q.	Applications of Taylor Series	1	2	3	4	5
r.	Parametrizations, Vectors	1	2	3	4	5

1B. Please add any other topics that you feel should be included, and any other remarks concerning content. Technology will be addressed separately.

2. THEORETICAL METHODOLOGY

2A. Please indicate the level of emphasis that you feel should be placed on the following methods in a first-year calculus course. It might be helpful to think about how much emphasis you place on these items in your assessment of students. Technology will be addressed separately.

		Amount of emphasis
		little or none			heavy
a.	Formal definitions	1	2	3	4	5
b.	Statements of theorems, counterexamples, etc.	1	2	3	4	5
c.	Proofs of significant theorems	1	2	3	4	5
d.	Historical themes in mathematics	1	2	3	4	5
e.	Writing assignments	1	2	3	4	5
f.	Student practice of routine procedures	1	2	3	4	5
g.	Applications of real world problems	1	2	3	4	5
h.	The analysis and solution of non-routine problems	1	2	3	4	5

2B. Please state your own definition of the phrase "mathematical rigor."

2C. Based on your definition, please describe what level is appropriate in first-year calculus and how you would assess (grade, for example) at that level.

3. TECHNOLOGY FOR LEARNING:

3A. Please state your feelings about the role of technology in the classroom, in promoting or hindering learning in a first year calculus course.

3B. To what extent do you feel technology should be integrated throughout the course versus for special projects, if at all.

3C. Please select the response that best represents your views about the ideal use of calculators or computers in the classroom.

		Amount of emphasis
		little or none			heavy
a.	Calculators for numerical purposes	1	2	3	4	5
b.	Calculators for graphing purposes	1	2	3	4	5
c.	Calculators for symbolic manipulation	1	2	3	4	5
d.	Computer courseware (Maple, Mathematica, etc.)	1	2	3	4	5
e.	Modifying existing programs/Programming	1	2	3	4	5
f.	Spreadsheets or tables	1	2	3	4	5

4. CLASSROOM TEACHING APPROACHES

4A. Please respond: In an ideal calculus course, how frequently would your students use the following instructional systems?

		not frequent			very frequent
a.	Use lecture notes as basis for learning	1	2	3	4	5
b.	Participate in a specially designed calculus laboratory	1	2	3	4	5
c.	Use concrete materials/equipment to explore calculus ideas	1	2	3	4	5
d.	Work in small groups on mathematics problems	1	2	3	4	5
e.	Work in small groups on projects that take several class meetings to complete	1	2	3	4	5
f.	Practice calculus procedures in the classroom	1	2	3	4	5
g.	Make conjectures, explore more than one possible method to solve a calculus problem	1	2	3	4	5

4B. Please add any additional comments. Perhaps you would like to address the phrase "ideal classroom." What types of support might your department in an ideal world provide you with to help you accomplish your teaching goals?

5. STUDENT ASSESSMENT/ EVALUATION

5A. In your calculus course, what importance to course grade do you assign to each of the following items? (1) Please rate on the scale. (2) Please circle the methods of assessment that you would like to discuss.

		unimportant			very important
a.	Quizzes, tests, or examinations that measure individual mastery of content material	1	2	3	4	5
b.	A final examination that measures individual mastery of content material	1	2	3	4	5
c.	Individual tests of mastery of content material	1	2	3	4	5
d.	Small group tests of mastery of content material	1	2	3	4	5
e.	Lab reports — individual grades	1	2	3	4	5
f.	Lab reports — group grades	1	2	3	4	5
g.	Quizzes, tests, or examinations of material learned in labs	1	2	3	4	5
h.	Homework exercises — individual grades	1	2	3	4	5
i.	Homework exercises — group grades	1	2	3	4	5
j.	Projects — individual grades	1	2	3	4	5
k.	Projects — group grades	1	2	3	4	5
l.	Journals	1	2	3	4	5
m.	Class participation	1	2	3	4	5
n.	Portfolios	1	2	3	4	5
o.	Other: Please describe:	1	2	3	4	5

6. PERSPECTIVES ON CALCULUS REFORM

6A. How aware are you of "calculus reform" issues and efforts? Please explain.

6B. What do you find encouraging about the directions of calculus reform?

6C. What are your concerns about the directions of calculus reform?

7. PERSPECTIVES ON THE CURRENT TEXT: Please write your comments about the text in use in your department? Do you want to change the current text? Please explain.

8. PERSPECTIVES ON THE CURRENT STUDENTS: It is important to share perspectives about the students we teach. Please give your impressions of the latest Calculus class you have taught, and share any comments from the students that you would like to pass on to members of the department.