MAA Calculus Study: Good Teaching

One of the primary goals of the MAA Calculus Study, Characteristics of Successful Programs in College Calculus (NSF #0910240), has been to identify the factors that are highly correlated with an improvement in student attitudes from the start to the end of the calculus course: confidence in mathematical ability, enjoyment of mathematics, and desire to continue the study of mathematics. To this end, Phil Sadler and Gerhard Sonnert of the Science Education Department within the Harvard-Smithsonian Center for Astrophysics constructed a hierarchical linear model from our survey responses to identify these factors. The factors reside at three levels: institutional, classroom, and individual student. Not surprisingly, most of the variation in student attitudes can be explained by student background, but there are influences at the institutional and classroom level. We have been particularly interested in what happens at the classroom level where there is the greatest opportunity for improvement.

Sadler and Sonnert ran a factor analysis of the classroom-level variables, clumping those responses that were highly correlated. They discovered that the responses broke into three distinct clusters, which we are labeling “technology,” “progressive teaching,” and “good teaching” because these seem to describe the characteristics of the instruction. By far, the most important of these in terms of high correlation with improved attitudes is “good teaching.” Listed below are the 21 student-reported characteristics of instruction that are highly correlated with each other and highly correlated with improvements in student attitudes, characteristics that collectively we are calling “good teaching”:

My calculus instructor:

1. Asked questions to determine if I understood what was being discussed.
2. Listened carefully to my questions and comments.
3. Discussed applications of calculus.
4. Allowed time for me to understand difficult ideas.
5. Helped me become a better problem solver.
6. Encouraged students to enroll in Calculus II.
7. Acted as if I was capable of understanding the key ideas of calculus.
8. Made me feel comfortable asking questions during class.
9. Encouraged students to seek help during office hours.
10. Presented more than one method for solving problems.
11. Made class interesting.
12. Provided explanations that were understandable.
13. Was available to make appointments outside of office hours, if needed.

My calculus instructor did not:

14. Discourage me from wanting to continue taking calculus.
15. Make students feel nervous during class.

My instructor often or very often:

16. Showed how to work specific problems.
17. Asked questions.
18. Prepared extra material to help students understand calculus concepts or procedures.

In addition:

19. My calculus exams were a good assessment of what I learned.
20. My exams were fairly graded.
21. My homework was fairly graded.

The good news is that most calculus instructors rated highly on most of these characteristics. This good news needs to be tempered by two facts: Instructors could and in many cases did elect not to participate even though other instructors at their institution were involved in the study, and these responses were all collected at the end of the term. They reflect the opinions of the students who had successfully navigated this course, predominantly students who were earning an A or a B in the course (roughly 40% A, 40% B, 20% C).

It is interesting and informative to see how students at different types of institutions rated their instructors on these criteria. We followed CBMS in categorizing post-secondary institutions by the highest mathematics degree offered at that institution. I am using “research” to designate universities that offer a PhD in Mathematics (predominantly large state flagship universities), “masters” if the highest degree is a master’s (predominantly public comprehensive universities), “undergrad” if it is a bachelor’s degree (predominantly private liberal arts colleges), and “two-year” if it is an associate’s degree (predominantly community and technical colleges). As shown in the graphs at the end of this article, instructors at research universities got the lowest ratings on every characteristic except “showed how to work specific problems.” For most of these characteristics, instructors at undergraduate colleges were the next lowest, then masters universities, and most of the time instructors at two-year colleges received the highest ratings. 

There were a few notable exceptions. Instructors at undergraduate colleges received the highest ratings in some of the areas where one would expect them to be strong:

• Acted as if I was capable of understanding the key ideas of calculus.
• Encouraged students to seek help during office hours.
• Was available to make appointments outside of office hours, if needed.
• Did not make students feel nervous during class.

Masters universities scored highest in often or very often showing how to work specific problems, and just barely edged out two-year colleges in “listened carefully” and “my exams were fairly graded.”

There are a number of possible explanations for the weaknesses of research universities and the strengths of two-year colleges. One is class size. The largest classes are found at the research universities where average class size is 53, the smallest at two-year colleges where the average is 21. However, average class size at masters universities is larger than at undergraduate colleges, so class size cannot be the only explanatory variable. Some of the discrepancies between institution types may be explained by student expectations. This is because SAT scores and high school mathematics GPA are highest for research universities, then undergraduate colleges, then masters universities, and lowest for two-year colleges. Better students may have higher expectations of their instructors, or they may be more discouraged by encountering difficulties in this course. The differences may also have something to do with age and thus maturity of the students. The youngest students are at research universities, the oldest at two-year colleges. They also may be related to the relatively large number of instructors at research universities who teach calculus but have little or no interest in teaching this course, as opposed to two-year colleges where the interest is very high (see my November column, MAA Calculus Study: The Instructors). Nevertheless, it is discouraging that students at research universities seem to be getting calculus instruction that has a worse effect on student attitudes than instruction at other types of institutions.

Figure 1: Instructor Characteristics 1–5.

Figure 2: Instructor Characteristics 6–10.

Figure 3: Instructor Characteristics 11–15.

Figure 4: Instructor Characteristics 16–18.

Figure 5: Instructor Characteristics 19–21.

The MAA national study of calculus, Characteristics of Successful Programs in College Calculus, is funded by NSF grant no. 0910240. The opinions expressed in this column do not necessarily reflect those of the National Science Foundation.