Reflecting on Practice

Once we're done the morning of math (with a brief coffee break) the teachers all get back together for an hour of math education pedagogy. Like the mathematics we cover, each year is something a little different. For example, in previous years we've focused on Lesson Design (with Drs. Nicole Bannister & Gail Burrill), Teaching through Problem Solving or Learning the Open-Ended Approach (with Dr. Akihiko Takahashi).
This year the organizers tried something a little different; they tapped six of the returning participants to look at Questioning in the Classroom from the practicing teachers' perspective. As one of those teachers leading the professional development it was a considerable challenge to not only meet the expectations of the participants and the organizers but also our own expectations -- my colleagues are amongst the premier educators in the States (National Board certified, AP consulants, you name it). We began with a working weekend in Denver in the spring, pulling together resources and a timeline -- our biggest fight was avoiding putting too much in. And then, when actually talking about pedagogy with professional teachers there is a huge struggle against anecdotes; everyone wants to share their stories. In discussing Questioning we want to move beyond what we do now and move towards something better. And so we start with what the research said.
This blog post is only to set the scene for a series of posts; I will go into this at greater depths in the future but our motivation was the results of the 1999 TIMSS video study -- James Hiebert presented the results to us in 2003 at PCMI and it was the most astonishing moment I've had in a lecture in a long time and it has been the prime motivator in my teaching ever since:

Almost all (ed: statistically 100%) of the problems in the U.S. that start out as making connections tasks are transformed, in a variety of ways. Often a teacher steps in and does the work for the students-sees students struggling, gives a hint that takes away the problematic nature of the lesson, and tells students how to solve it. These are not incompetent or poorly intentioned teachers but simply teachers who have picked up very well an American way of teaching mathematics. One of the cultural agreements we have made in this country, with ourselves as teachers and with students, is that it is the teacher's job to tell students how to do the problem and how to get the right answer-that it is not fair to allow students to struggle or be confused.

In other words: we are far too nice. So, for the past six years I have worked hard not to be nice and tried to persuade colleagues near and far to cowboy up¹. I've presented on this at OAME directly and in any other presentation that I've done I've pressed the point. It was encouraging to see Dan Meyer come to a similar conclusion in his presentation to open source programmers (yes, the context is a bit bizarre but makes sense if you follow his blog). Be sure you should watch the video.
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¹I include "cowboy up" only because I had to explain the phrase to Gail this year :)

Graph is created from data produced in the TIMSS video study and is from here: http://www.mathforum.com/pcmi/hstp/sum2009/reading/Hiebert_Improving_Math_Teaching_2004b.pdf

The 830 at PCMI

PCMI is a 3 week program; each day from about 830 to 1040 we have what can best be described as a math class. But it's unlike any math class most people have ever had.
Each day starts with its own problem set designed by the class' organizers, folks from the Education Development Center and Harvey Mudd College. The problem set is well structured, beginning with a simple idea or concept and then continually developing in both depth and breadth, although this may be obvious only several days later. The questions are also in categories: Important (things you'll need to know for upcoming days), Neat and Tough (can be really tough! Clay Prize tough!) -- we aim to get through at least the important stuff in our morning together.
The classroom is composed of 12 tables of 5-6 people each (we do have guests from the other programs) and as a table we tend to worth through things together; there's a table sandbox monitor who is there to ensure that the teachers exercise all those collaborative skills they try to encourage with their students. Not only that, but we never tell people ideas, we create a situation in which they can they discover it themselves. This is not easy and like any skill takes practice and continual reinforcement. It is at the heart of the whole morning class (indeed, of PCMI) and the mathematics could almost be the motivation for appreciating this whole process. It's why I call them "organizers" above and not teachers -- it's not instruction as you know it.
The math is very accessible and very deep - low threshold, high ceiling - and it is too easy to look at it only superficially. Teachers will occasionally race through the questions to get them done (remind you of any students?) and will miss out on the complexity of the mathematics. I remember my first year doing the same thing.
As one of the participants said "I've taken courses in number theory but never understood prime numbers until now." This has been true for every topic I've encountered at PCMI -- teachers seldom get the chance to think deeply of simple things that Al Cuoco of EDC, and one of the course's authors, encourages.
If you visit PCMI @ the Math Forum you can click on Class Notes to read over the problem sets from previous years. Or, to get a very insufficient glimpse of the questions, the MAA has a book of Al's work Mathematical Connections that includes material we've looked at during PCMI. It's condensed (remember, we get three weeks) and doesn't have the same level of personalization that our questions set have -- the authors adapt the problem sets from day-to-day to build off of our ideas, suggestions, questions & comments.

Paperless?

California's recent announcement that they are moving to e-textbooks will mean a lot more resources for 1:1 schools. Right now, using a tablet computer means either having a CD copy of the textbook (now a departmental requirement for our texts and fortunately most Ontario publishers have agreed) or several hours spent at the photocopier, scanning the questions in. Some publishers copy-protect their CDs but in the age of snipping tools, it's a lost cause. I understand they're concerned with sales but a quick check of class lists will ensure they're selling what they should.
Since my students have tablets, I use a OneNote file each day for their work: I get to pull questions from the textbook and sequence them the way I want. I can also make different levels of homework depending on the students -- this is particularly nice and, since the students don't necessarily see each other's OneNotes, they don't know who has what. I also put the answers from the text at the bottom of the OneNote for their reference. With OneNote, of course, I can also add in links to resources for the questions, my only little running commentary (either helpful hints & tips or notes about the phrasing of the question, where to find other questions like this and so on. Images, videos and applets can also be incorporated. It's this kind of environment I'm hoping that California will come up with.
I know that many of the math teachers don't do this; it's another little bit of work each day. I just find it inefficient to ask the student to copy the question from the textbook (since an answer in isolation is useless in review) and then flip to the back of the book for the answer. Not to mention most desks don't accomodate a math textbook and a tablet computer (and a soft drink, chips, ipod, etc).
Some teachers do it for the whole unit; I find that a little wishful thinking. So many good questions & thoughts arise from class that I like to tip them in either the same day or the next day -- and it's not just the math stuff I put in, either. Current events, humourous things from them... it all adds a little bit to the work.
If you're a math or science teacher, OneNote is likely only effective if you have a tablet (or a plug-in tablet as I used to use). For other subjects a laptop or netbook would be sufficient.