Section 3.1 Evolution of the process for integrating mathematics instruction into the junior-level physics courses
The original design of the Paradigms in Physics curriculum in 1996 had initially integrated some mathematics instruction within the new courses. This had been inspired by an early collaboration between the PI and one of the co-PIs. They had attempted to coordinate content in their respective upper-division physics courses in mathematical methods and electricity and magnetism, as described by the PI under ``Early collaboration'' in Part III, Section 4.1 of this study.
The content was synergistic...I would teach mathematics in math methods just prior to when they needed it in his course. We were just beginning here to play with both other pedagogical strategies and coordinating content. Up until that time, there was a teaching committee that would tell faculty through course descriptions what content needed to go in these courses, but people were left up to their own to figure out how to do that.
Coming to agreement on the desired order of the physics content as well as of the mathematics needed had absorbed many meetings and consultations with the other faculty, both during the original design process and the current re-envisioning of the program. A key aspect was re-ordering the sequence of courses based on building mathematical as well as physics expertise. The PI explained:
You have to have the courses sequenced so that there is a natural build up of the mathematics content; if you just jump around in terms of the mathematics, it's not going to work. I think there are a number of such sequences that can work but you need to think about do you need to do this piece of math before you do that piece of math and if they're in two different math bits, they have to go in some particular order.
The re-envisioning process had involved making multiple complex judgments based on how the courses would be increasing the students' mathematical sophistication.
Although some mathematics instruction had been integrated into several of the junior-level courses, the mathematical methods course had survived as a separate physics course throughout the two decades of development of the Paradigms in Physics program. As part of the Paradigms 2.0 redesign, however, the decision had been made to eliminate the separate mathematical methods course from the required physics major. Keeping the mathematics at a level that is necessary but manageable can be critical in retaining students in the major whose aspirations are achievable without learning the most complex mathematical techniques available. The PI reflected on the motivation for eliminating the separate mathematical methods course at this time:
I think it's that I have always tried to teach a math methods class that would help students bound for graduate school be prepared for graduate school and I think we just recognized that we have now too many students who are not bound for graduate school and they don't need all that math. We were trying so hard to make room in the curriculum for students to take some selection of advanced physics courses, so the decision was to encourage the graduate-school-bound students to take the graduate math methods course and to cut some of the math methods material out of what is required for everybody.
This decision reflects the challenge of maximizing outcomes for students by considering what would be the best use of instructional time for students who were heading to industry or other employment that would not require some of the advanced mathematical techniques taught in the separate course while still meeting the needs of those students headed for graduate study.
The PI described the evolution of integrating mathematics into all of the junior-level physics courses as follows:
In the original implementation of the Paradigms (courses), we had two extra weeks because we had 3 three-week things and ten-week terms and...we took an extra week originally in the fall too where we were teaching some math methods ‘just-in-time’. The fall one disappeared but we started the spring one, like ten years in, we started using the spring one to do math methods for Energy and Entropy (an introduction to thermodynamics) and the winter one was always an extra week for Spins (an introduction to quantum mechanics). So when we did Paradigms 2.0, the paradigms (courses) that had this extra week associated with them, it just seemed to work really well, so with Paradigms 2.0, we decided to try it with all of them.
The PI was responsible for undertaking this major change in embedding mathematics instruction within all the junior-year physics courses during the 2016-2017 academic year.