After doing midterm semester surveys and observations in several calculus courses, I compiled a list of suggested resources and strategies for improving student performance in calculus. I also posted this to the POD list along with a little more background on controversies surrounding math reform (aka the ‘math wars’) over the past 100 years.
Seven Characteristics of Successful Calculus Programs
There was an important national study of calculus programs conducted recently. “Our survey revealed that Calculus I, as taught in our colleges and universities, is extremely efficient at lowering student confidence, enjoyment of mathematics, and desire to continue in a field that requires further mathematics.”
However, here is a summary of what the best calculus programs are doing:
MAA Calculus Study: Seven Characteristics of Successful Calculus Programs
The list includes:
- Regular use of local data to guide curricular and structural modifications.
- Attention to the effectiveness of placement procedures.
- Coordination of instruction, including the building of communities of practice.
- Construction of challenging and engaging courses.
- Use of student-centered pedagogies and active-learning strategies. [A recent meta-analysis confirmed yet again the advantages of active learning over traditional lecture courses.]
- Effective training of graduate teaching assistants.
- Proactive student support services, including the fostering of student academic and social integration.
Another big study was on Inquiry-Based Learning in College Mathematics. Not only did students learn more in inquiry-based learning math courses than traditional courses, but there were dramatic gains for low achieving students, pre-service math teachers, and female students. Here are some articles on that:
Teaching Calculus in Engineering or Other Real-World Contexts
Another thing that has been shown to increase student learning and retention in college math courses is to use problem-based learning (real-world examples) and teaching the math in context. These help show the students WHY they are learning the concepts and how they apply in the real world. It would also involve students working in groups sometimes on real-world problems. Learning how to teach with inquiry learning or problem-based learning would require faculty training and incentives. although the first option below (creating an engineering math class students take before calculus) would require no changes at all from math faculty (instead, engineering faculty teach an engineering math course that precedes calculus).
Here also are some articles on problem-based learning in calculus:
At the University of British Columbia they switched their calculus course over to peer instruction (using clickers, think-pair-share, small group work, etc.) and showed some positive results. Peer instruction is a well tested strategy that works, but it would require faculty be trained. Warren Code led this study:
Online Homework Systems
One strategy that can both boost student performance and save some time for the instructors/TAs is to use an online homework system (or adaptive learning system).
For example there are tools like Webassign
, or MyOpenMath
(which is open source). These tools give the students instant feedback when they are working on problems – problems which are matched up with the problems in their textbook.
Calculus Concept Inventory
I might recommend giving the Calculus Concept Inventory to students at the beginning and end of the course (pre and post) to measure their conceptual understanding of calculus – both what they understand conceptually coming into the class, and what gaps they may still have at the end of the course. This is the most direct way to measure what students learn.
Here is more info about the Calculus Concept Inventory
Contact the author Jerome Epstein for a copy.
Tutor Training, TA Training, and Communication
Hopefully all math tutors, TAs, and anyone who comes into contact with students undergoes rigorous training on how to help students and teach students. It’s also important for the tutors and TAs to be in communication and in sync with the instructor and what’s going on in the class.
Here’s an example of a CRLA certified tutor training program at USF:
There are several books on tutor and TA training, and the CIRTL network I believe has some courses to prepare grad students and TAs: http://www.cirtl.net/
Incorporating Matlab or Programming
Some calculus courses are starting to incorporate programming into the course, such as using Matlab. I don’t know that this would increase student performance or retention, necessarily, but it may connect the course better to engineering and to the tools that engineers and scientists and other professionals use. Here are a couple of books about using Matlab in Calculus, for example:
Animations, Graphs, Online Tutorials
Search online for animations or diagrams or tutorials or videos related to calculus.. These are the kind of supplemental resources I would point students to on a tutoring site, too. Several students I have surveyed have indicated that they use supplemental resources in the course to help them learn and understand the material, and I’ve seen research showing that the students who do the best are often the ones who find outside resources to learn from (can’t remember reference offhand).
Research on Technology in Calculus
Here are a couple of articles/case studies on integrating technology into calculus by a couple of folks well known for their work in math education research:
When looking for innovative techniques in undergraduate math teaching, I first search the proceedings of the RUME conference (research on undergraduate math education):