A Study of the Development of the Paradigms in Physics Program

Maintaining a commitment to generalizing the mathematics.

As a consequence of teaching the math bits this year, the PI has become more aware of the importance of teaching mathematics language and generalized mathematical approaches as integral parts of helping students to learn how to think like a physicist.

When I originally taught math methods before the paradigms (courses), it was very very math-y, and when I started teaching the paradigms, we were putting some of the math methods just-in-time as part of paradigm courses, but I think even I fell into the trap of integrating it a little too much, and doing the math only in the one physics context and in the one set of notations which is relevant to that course... I felt like some of the making it more abstract was the responsibility of the math methods course, but now that we've gotten rid of the math methods course, I'm putting some of that responsibility back into the math bits. I really want the students to see the math in its most context independent form so that they can see where they're using the same math both in this course and in some other course. I feel like the math bits have a real strong responsibility for teaching the students the math language as well as the physics language.

This is a commitment that may not be shared by some of the course instructors:

...my colleagues look at what I'm doing in the math methods and the first thing they want to do, is cut back on the time that I have because they want to move to interesting applications and things, and they really see this as an encroachment on the physics...

However, the students' responses indicate they still need such instruction:

I'm becoming convinced by the response of students that it's the right thing... I try to meet very specific mathematics and physics needs that they have, and when I do particularly the little short white board activities or something and I walk out and I see how much they're struggling to respond even to these what should be short easy prompts, it just makes me understand that they still are at that stage of learning how to do those things and that each of those things is difficult and needs its time and space for them to wrap their brains around it.

Such differing views about the importance of separate math instruction within physics courses suggest that the faculty member embedding mathematics instruction within upper-level physics courses may require considerable personal influence within a department as well as knowledge of mathematical ways of speaking and thinking in various physics contexts.

Scheduling the math bits within a course.

The previous version of integrating mathematics instruction into the junior-level Paradigms in Physics courses had primarily involved starting a term with a week of mathematics instruction before starting the physics course. This new plan, however, opened up other possible arrangements. In an interview midway in the academic year, the PI noted:

The challenges are that it's basically a team teaching situation so I have to work closely with whomever is teaching the course and I have to get them to agree, we have to agree together about when it's going to happen. Scheduling has been the biggest challenge because we're beginning to realize that it is not necessarily best to put this material first...

The PI initiated the new arrangements during fall term with a close colleague with whom “sprinkling” the math bits throughout the course “just-in-time” for the next topic in physics turned out to be problematic:

The first paradigm it was (a colleague) and me teaching. We have team taught before, and we're both pretty flexible in the classroom about when things happen, so we tried a version where I would do a day here and a day there and two days somewhere else, but we split it up all over and sprinkled it around; even as flexible as both of us are, we both got pretty frustrated so after that experience I decided I would at most split it up into two pieces.

Another possibility was putting the mathematics instruction into the middle of the physics course rather than at the beginning. The PI noted that this was welcomed by a faculty member who was teaching a Paradigms in Physics course that involved some integrated labs:

Now I'm working on doing one with (another colleague), and he actually wants me to do it in a middle week because his course has labs in it and he wants the students to have some time to write up the labs while they're doing math bits, so we'll see how that works. It's an interesting challenge because when we've talked about what it is that I should do, he's already doing it, but in a physical context, and since he wants me to do it in the middle, he will already have done (some mathematical methods) in a particular physical context, and then I'll come along and generalize it so I'm really curious to see how that is going to work.

By the interview during spring quarter, the PI had formed a firm point of view on how to schedule the math bits for most of the courses:

Figuring out when in the schedule the math methods is going to be, it's becoming pretty clear that for most of the courses, doing it first is not the best choice, because I think giving the students a little bit of context for how they want to do it, why they want to know these things, helps.

The PI offered an example:

So (another colleague) chose to do just two hours of physics first, and then had me do a chunk, and I think that worked particularly well, it's probably different for the different courses, knowing exactly when it should be.

Reflecting upon the learning experiences encountered this academic year, the PI summarized:

...with the first one... I chopped it up into lots of little pieces and that did not work well, but with most of them now I'm chopping it up into two pieces, one that comes early on and then one that comes a little later with more advanced materials, that seems to be a nice amount, so it's either three contact hours or four contact hours at a time.

This is the recommendation now made for others contemplating integrating mathematics instruction into upper-level physics courses.

Planning collaboratively what should be taught by whom, when, and how.

Like other team-teaching situations, negotiating what should be taught as well as by whom, when, and how was challenging. With so many changes underway, the PI needed to be aware of how the other faculty were revising the junior-level physics courses in order to plan appropriately for which mathematics to present. Some courses had had mathematics instruction embedded previously, others had not; the PI was more familiar with the content of some courses than with others, and some of the content was being shifted between courses as they were being revised. In the interview during spring quarter, the PI reflected on discussions of what to teach:

The major challenge across all of them (the 2016-2017 Paradigms in Physics courses) has been actually coordinating with the other faculty members; they have had, some of them have had very firm opinions about what needs to go in, which haven't always been aligned with what I think needs to happen.

The PI and a colleague differed, for example, on the nature of a topic, with the course instructor thinking of the topic as physics and insisting on teaching it, whereas the PI thought of this topic as an important component of the mathematics methods needing to be taught separately:

(This colleague) is solving (a type of) equations very specifically...in the context of (a topic) with the physics language about (a central concept)...and it was interesting to me that he insisted on teaching that himself first, whereas I thought that was math methods, and he just still can't think about that as math bits at all. He wanted me to do other math bits, so when I tried to say I even thought of those (types of equations) as math bits, he so much didn't think about that as math bits, that when I tried to do the generic...equations, that's the piece he didn't want to be in the course, but it was in the math methods course and I still want it to be somewhere in the curriculum, I feel strongly it needs to be somewhere in the curriculum in that language.

As a senior member of the department, the PI had undertaken numerous negotiations with both long-time and new colleagues that required both being accommodating and not when deciding on when the math bits sessions would occur and what they would entail:

You have to be both willing to be flexible and stick to your guns when working with colleagues.

Such negotiations can become complex and it is important to keep a clear sense of what the most essential aspects are.

The PI also reflected on how hard the planning process could be:

It's hard to fit into the flow of some other course. It's hard to make a flow for the math bits on their own; the math bits that are relevant to any given course tend to be lots of little separate things that don't particularly go together...

For one course, the PI described the collaborative planning process as follows:

We outlined what might be the math bits for the course; so this was (two colleagues) and I and we x'd out the things that were really physics-y and then there were also little isolated things and we put them back into the physics portion of the course and then we grouped together the things that took more than one day to talk about and said that those will be the math bits for this course.

Such collaborative planning, however, was sometimes problematic. Faculty travel, family commitments, research priorities, even just a person's comfort with flexibility, are typical issues that can interfere with one's ability to coordinate one's own teaching with others.

Assigning homework.

As with other team-teaching situations, establishing effective routines for sharing homework assignments also can be challenging. The PI commented on both structural and pragmatic issues encountered:

There are typical team teaching challenges like trying to figure out how much homework the math bits can give in other regular homework assignments and getting those homework assignments to the person who is designing the course; also trying to decide whether or not there should be a specifically math bits question on the final exam... It's really hard for me to get the homework to people and everybody has a different format for their homework...whether it's Word or LaTeX, part of my homework archive or not my homework archive, and who is going to write up the solutions, just how people give the homework problem to the students, and whether they give it to them in a pdf, just every case is different, so at some stage, we might want to think about that, so that's been a persistent annoying thing.

The paradigms courses have had a shared homework policy in terms of when assignments are due, Wednesday and Friday, but not a shared format nor process for integrating two instructors' plans for homework into one assignment.

Respecting timing issues.

Whenever one is teaching it can be difficult to estimate ahead of class just how much time one is going to need for the tasks at hand, especially if one is using interactive engagement strategies. As the PI noted, this is particularly challenging in a team- teaching situation:

...as soon as you start mixing it (mathematics instruction) into the course, then whatever agreement you have if either the other professor or if I run over, it impacts the other person's plans, so there are typical team-teaching challenges... The bigger issue is that it's hard in some cases to get it into just one week, that I need more time...

Balancing the time available with the time needed within a course was a process being worked out during this implementation year.

Responding to colleagues' requests.

A collaboration frequently involves a mutual exchange of perspectives and goals, out of which may come ideas for something new. The effort needed to develop and implement that new thing, however, can change what happens.

Another thing that was a surprise was how much all the faculty didn't know what exactly was going to be in their courses; I guess this was a challenge more than a surprise, essentially all of them have pulled the rug out from under me, in terms of telling me at the last minute that they want me to do something different from what I thought I was doing and what I thought we had agreed on. All of those requests for changes have been positive and important, but it's taken a lot of, it's stretched even my ability to do things on the fly. It's definitely been a challenging experience. And I think for them as well to incorporate me into what they're doing and I hope that they will in the long run find it positive.

The PI also described the unfortunate outcome of such a collaboration for a course where the math had been taught previously during the week before the physics course started:

...that was a course where I had done the math bits before, it (the course) had had a week attached to it before, and so (the course instructor) thought what I was going to do was very similar to what I had done the previous year but we were going to try and put it in the middle of the course and then I didn't do what I said I would do; he got me interested in something else and I tried to do that and then I ran out of time to do the thing that he thought was important and he just got really (unhappy).

Such interactions can be unsettling although not unexpected when new approaches are under development.