*A question about the end results of professional development.*

A colleague (NB) and I were discussing a question about limits, in particular some cases where local improvement does not necessarily result in global convergence. And, as often is the case, my thoughts turned to the same question, but related to teaching:

If I (or any teacher) make changes (hopefully, improvements) to my teaching year after year as the result of formal or informal professional development, will my teaching converge to some limit which represents ‘excellence’?

For fun, and related to other conversations, I thought I’d try to frame this question more precisely (a.k.a. mathematically) and then discuss how we might think about the answer.

Let be the set of all the teaching skills I have in year . This represents both the content, technical and meta- knowledge I have that’s relevant. It is my teacher’s toolbox.

Then, each year there is an update: where I throw out the old () and add in the new ().

Although I will teach (and live) for a finite number of years, I’m now interested in . Now for a series of questions:

- Does the limit exist? For an individual teacher, probably. For all teachers, probably not.
- Does the limit represent ‘excellent teaching’? First we have to put a measure on a teacher. So given a set of teaching skills, and a particular class of students , we imagine some value which quantifies the quality of the teaching/learning. What is ? Probably not something one can (or should
^{1}) articulate, so we’ll leave it alone for now. We can then use some combination of these values over a set of classes, e.g. like an expected value, to produce a single value for the set , called . So back to our original question, which is now is where $Q_m$ is the maximum or an excellent-level value for ? No. Even if , it is unlikely that the limit reaches the maximum. - So how should professional development go so as to increase the likelihood (and the rate of convergence) that the limit will be reached at an ‘excellent’ level? The challenge is that there are many possible growth paths and although it is always possible to get better, only focusing on improving the end product will leave foundational weaknesses that keep one from excellence. So (a) allow and take time to reflect and review teaching at the basest levels, (b) create the opportunity to view and discuss alternates across disciplines, looking for commonalities that lead to excellence, and (c) avoid quick, technical fixes as a substitute for addressing fundamental weaknesses.

Does this really tell us much about good/excellent teaching? Probably not, but it does help a bit with thinking about where we need to focus our attention.

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^{1}One could argue that establishing, or the trying to establish, a universal is bad both because there is too much variety in what one should be measuring (e.g. in details and timeframe), and it is possible that an explicit would be vulnerable to gaming, thus leading to poorer teaching.