Title: Redesigning the calculus sequence at a research university:

issues, implementation, and objectives

Author(s): Harvey B. Keynes and Andrea M. Olson

Publication type: journal article

Online: link (full-text for subscribers only)

Authors' abstract. The paper discusses the progress and challenges of a new reformed calculus sequence for science, engineering, and mathematics students developed by the Institute of Technology Centre for Educational Programs and School of Mathematics, University of Minnesota. The main objective of the Initiative is to enable undergraduates to better learn calculus and the critical thinking skills necessary to apply it in a variety of science and engineering problems. Changes in content and pedagogy are emphasized, including instructional teamwork and student-centred learning, involving students working cooperatively in small groups and exploring mathematical ideas using appropriate technologies. Achievement and retention of Initiative students are compared with a control group from the standard calculus sequence. Student attitudes about the usefulness of the Initiative's curriculum, pedagogy, and its influence on learning are discussed. Future implications including new uses of distributed learning are also addressed.

The University of Minnesota performed an, in principle interesting, experiment on calculus teaching. The usual way calculus is taught at a research university is as follows: the students have three hours of lectures a week by a professor and one hour of discussion led by a teaching assistant (usually a graduate student). One can of course investigate whether this is the optimal mix or not. In the experiment the University of Minnesota traded one hour of lectures a week for two hours of discussion a week (and renamed the discussion 'workshop'). This is from the student point of view. From the university point of view the situation is somewhat different since there are multiple workshop sessions for one lecture session. The typical situation is depicted in the following tables.

traditional | #students | #hours | total# |

lectures | 100 | 3 | 3 |

workshops | 25 | 1 | 4 |

experimental | #students | #hours | total# |

lectures | 100 | 2 | 2 |

workshops | 25 | 3 | 12 |

So from the university point of view 1 hour of lectures is replaced by 8 hours of workshops. Even though teaching assistants are cheaper than professors, this will mean an increase in cost. The authors of the study indeed write

Even after implementing all reasonable economies, there is an incremental cost difference of 20-25% over the standard calculus sequence

This is probably an underestimate since in the actual experiment the professor was also supposed to help during some of the workshops and the teaching assistants were supposed to be present during the lectures. The indicated cost increase does seem to be somewhat realistic if this team-teaching is abolished and only the trading of lectures for workshops is considered. Apart from the increased cost there is another problem that the authors mention: staffing. The experimental condition requires three times as many teaching assistants. One of the ways in which this was addressed in the experiment was by employing high-school teachers and undergraduates as teaching assistants. This of course raises all kinds of issues.

Lets look at the results: does trading an hour of lectures for two hours of workshops actually lead to better results? The authors claim that it does, but I'm not convinced. Comparisons were made between the experimental classes and the traditional classes. Students were not randomly assigned to one of the two conditions. It is justifiable to not do this, but then one should be very careful in making comparisons. The authors look at the average calculus grade point average: this is 3.27 for the experimental condition and 2.85 for the traditional condition. They also looked at how many students took a second year of calculus: this was 77% in the experimental condition and 56% in the traditional condition. The authors also gave partially the same questions on the final exam for both conditions. In the experimental condition 76% of the responses to these seven common questions was correct whereas in the traditional condition only 60% was correct. So this is all clearly in favor of the (more expensive) experimental condition. Now comes something strange. The authors compare the grade point average of the students in all their upper division courses. In the experimental condition 43% had a GPA of 3.5 or higher and only 23% had a GPA less than 3.0. For the traditional condition these percentages are 15% and 58%, respectively. From this the authors conclude that the experimental condition provides students with strong mathematical skills necessary for success in future courses. The more obvious interpretation of this difference is that the students in the experimental condition were just smarter. Remember that students were not randomly assigned! And it becomes even stranger. Like all mathematics departments at research universities the University of Minnesota has a mathematics placement test that is administered to all incoming students. This can serve as a pre-test to determine whether the two groups are comparable and to statistically adjust scores if they aren't. This is however not done. The only reason that I can think of that this is not done is that the placement scores are similar to the upper division GPA scores (which shows that this difference is indeed due to the fact that the students in the experimental condition are smarter) and that if one adjusts for this, then the experimental condition turns out not to outperform the traditional condition. At this point it is good to say that the authors of the article are involved in the experiment and therefore have much to loose if the experiment is deemed a failure.

We can do some ballpark statistics on the above information on GPAs. We assume that students with a GPA of more than 3.5 on average have a GPA of 3.75, those with a GPA in between 3.0 and 3.5 have on average a GPA of 3.25 and those with a GPA of less than 3.0 have on average a GPA of 2.5. Then the average GPA of students in the experimental condition is 3.29 and that of the students in the traditional condition is 2.89. Both of these figures are extremely close to the average calculus GPA for these conditions. So the calculus grades seem to fit in perfectly with the grades in all other courses.

Based on the information on the website of the University of Minnesota something can be said about the aftermath of this experiment. Both the (now no longer) experimental and the traditional condition still exist at the University of Minnesota. The traditional sequence seems to attract twice as many students.