# Best Ever

A confession and example of a teaching transformation.

A Confession: As much as I talk about teaching and ideas about ‘good’ teaching both here and with colleagues, I don’t really practice what I preach as much as I could or should. I’d say I’m a good teacher, but in the very traditional sense, i.e. in a way that my colleagues and students would recognize as good teaching. After teaching for many years (30!) I realized that I could keep refining my technique but that it wouldn’t really increase the impact I had on student learning. So I tried to figure out what I could change, and that lead me back to the very beginning, and here’s a result:

An Example: For Fall 2010 I was assigned to teach a new course, a course for current high school and community college faculty that wanted or needed graduate work in mathematics. The topic was Analysis, a.k.a. Calculus. From what I knew of the course and what I had learned in my search for changes to my teaching, I realized that this course was both a good place to try these new things and a course that really required a new approach to teaching.

First point: If you ever get the chance to teach teachers, do it.  If you give them the respect they deserve, they are the most understanding, accepting, and encouraging students to teach. Most of my students worked all day and then took time to spend 3 hours of their evening learning about math, because they wanted to.

Second point: This course transformation I’m about to describe, breaks all the rules that I usually give to someone interested in changing his or her teaching. (1) I had never taught the course, nor anything like it, before, (2) I changed everything that I had ever done in teaching rather than just a part, and (3) all I really only had a small set of good ideas that I had never tried, to go on.

Details: The main idea was that the students were going to do all the work, including choosing what work to do. The only parts I set were that they had to document it all online and that they had to ‘make sense’ of Calculus as a whole. And I told them so, a little bit at a time. I didn’t answer any question that I could avoid answering.I didn’t evaluate their work, but made them evaluate it themselves. I did teach some to the whole group, as needed, on topics usually prompted by student questions or my or student interests. Most of the teaching was done one to one or with small groups.

Results: It was the best course ever. Between the students and the way the class ran, it was the most enjoyable teaching I had ever done. For the final evaluation (i.e. grades), I had each student review and reflect on the body of work they had done. The work the students had done was very diverse, each organized according to the students needs and interests. The reflections showed appropriately deep learning, greater understanding, new insights, and many other wonderful things. Sure, some students were frustrated by the course structure, and some may have done less than they could have, but I would again risk that to see the kind of growth and excitement that the course encouraged in most of the students.

Conclusion: I believe, as I wrote a couple of days ago, that development as a teacher requires not just incremental improvements, but an occasional review of the foundation. This experience, where I risked disaster while trying to create true and deep learning, only confirms this belief, and encourages me (and hopefully you, the reader) to reflect, plan some core changes, and then take the plunge.

Good luck and good teaching!

# MOOCs, LMSs, Publishers, et. al.

In response to the not-so-surprising announcement that Blackboard is joining the MOOC fray.

I just read (here) that Blackboard has decided that they’ve been on the sideline long enough and will be offering support for MOOCs.  This should come as no surprise to anyone who’s paid any attention to the changes going on in higher education.  The latest indicators were the agreements that Coursera signed with various universities to provide the Coursera software for use in the university’s courses. With a MOOC selling itself as a learning management system (LMS) (or as a publisher), it wouldn’t take long for the LMSs (like Blackboard) to find a way to get involved with MOOCs.  It probably won’t be long before all the different service parties try to gain a foothold in each other’s areas. We already have text-book publishers already involved in building their own mini-LMSs and MOOC providers acting as text-book publishers where their course materials (videos, quizzes, etc.) are used as the source for a traditional course. Will it be long before an LMS starts providing their own material?

As Blackboard has many detractors, I’m sure the web will soon be full of reasons why Blackboard supporting MOOCs is a bad idea, but I’d like to offer a reason why this (and all the other conglomerations of learning services) might turn out to be good. It could cause the ‘7-11 phenomenon’1 to occur for all sorts of smaller, focused learner service providers. Some companies like Piazza, that focuses on discussion boards with nice support for mathematics, or Canvas, that offers an alternative LMS experience, or any of the open-source LMS, MOOC, Publisher, … could find more success as professors seek specific solutions to instructional problems and don’t want or need the expanse of the current LMS+MOOC+Publisher entities.  Of course the 7-11 analogy is weakened by the fact that universities will be heavily investing in these L+M+Ps and thus will encourage/force instructors to use those resources rather than the smaller focused ones. It is also possible that the L+M+Ps will just buy up and absorb (or eliminate) these smaller entities.

The learning service landscape is not even close to being settled, so stay tuned to many more changes.

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1When I was in college many years ago, I took a mandatory course in political science (really a US government course), and one day we were talking about the different political parties.  The discussion came around to wondering why there were 3rd parties, when the two major parties were dominating all the action.  The professor’s response was that it was the ‘7-11 phenomenon’: as in cities where bigger and bigger grocery stores are opened that provide all sorts of services, it can actually make it more profitable to open up small convenience stores in between the big stores, rather than drive them out of business.

# Limit of Professional Development

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 $T_k$ be the set of all the teaching skills I have in year $k$. 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: $T_{k+1} = (T_k \backslash O_k) \cup N_k$ where I throw out the old ($O_k$) and add in the new ($N_k$).

Although I will teach (and live) for a finite number of years, I’m now interested in $\lim_{k \to \infty} T_k = \cap_k \cup_{j \ge k} T_j$.  Now for a series of questions:

1. Does the limit exist? For an individual teacher, probably. For all teachers, probably not.
2. Does the limit represent ‘excellent teaching’? First we have to put a measure on a teacher.  So given a set of teaching skills, $T$ and a particular class of students $S$, we imagine some value $Q(T,S)$ which quantifies the quality of the teaching/learning.  What is $Q$?  Probably not something one can (or should1) 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 $T$, called $Q(T)$. So back to our original question, which is now is $\lim_{k \to \infty} Q(T_k) = Q_m$ where $Q_m$ is the maximum or an excellent-level value for $Q$? No.  Even if $Q(T_{k+1})>Q(T_k)$, it is unlikely that the limit reaches the maximum.
3. 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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1One could argue that establishing, or the trying to establish, a universal $Q$ 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 $Q$ would be vulnerable to gaming, thus leading to poorer teaching.

# What I learned about teaching from Everything and More: A Compact History of Infinity

How a popular math book taught me some things about teaching.

I’m reading David Foster Wallace’s Everything and More: A Compact History of Infinity.

It’s a well-written and enjoyable history of the infinity from the Greeks to Cantor’s set theory. He doesn’t hide the math, so you need a bit of a background of math, at least Calculus and probably a first course in Analysis.  Also, he has a distinctive1 writing style, which I very much enjoy, but it takes some devotion to keep up with it.  Once I finish, I hope to post a more complete review. But from a teaching standpoint, he’s reminded me of three things:

(1) Have One Theme and Point it Out: Early in the book, Wallace establishes the basic idea from Zeno’s and Aristotle’s writings that he carries through the rest of the book.  In each part of the book, representing a different time of history, he relates the issues of the time back to the original ideas. It’s not just that he relates it, but that he relates it by recognizing that we the readers will be seeing how the arguments he constructs and the problems he faces are very similar to ones he presented earlier. He both respects the intelligence of the reader, and includes the reader into the various issues of the times.

(2) Be an Enthusiast: Wallace is not a mathematician, but he is a fan.  He honestly enjoys mathematics, and he can’t help to share his enjoyment through his writing. His enthusiasm is contagious. Also, he obviously has more to say, but is able to keep and progress the main parts of his arguments in the primary part of the text, relegating the extra bits to footnotes (labeled IYI = if you’re interested) and other asides. This provides for a rich (or as rich as you want) exploration of the topic.

(3) Have a Story with Interesting Characters: The book is a story of an idea, but it is also much about the people struggling with that idea.  Those people, however, are not the only (nor the most interesting) characters.  Early on he introduces Mr. G, his high school math teacher.  Wallace uses Mr. G, and the other characters, to make the writing more personal and to provide an opening for some non-standard mathematics and off-topic references.

Altogether these make the book (and teaching) more engaging as it keeps the intent clear, and respects and provides opportunities for the students.

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1If you are aware of his style, you will recognize this as a major understatement2.

2And you’ll recognize these footnotes as a (pale) tribute to his style.