Working Groups

Before I start dealing with reflecting on the content of the classes, I've got two more aspects of PCMI to mention.
The first is the most productive: The working group. Each of the teachers is assigned a working group in a topic of secondary mathematics for the afternoon (Wednesdays off) in which they, typically in groups, will produce a product useful to classroom teachers.
As the person in charge of a group this can be very challenging: these are all energetic, enthusiastic and talented teachers -- who all teach in very different classrooms. So what may be appropriate for one school system could fail utterly in another, not just in terms of content but departmental expectations, school standards, etc. As the working group leader I have to steer these folks towards a consensus: a project that is meaningful to them, useful to others, and able to be accomplished in three weeks. Most of the time this takes the form of a lesson plan or activity that is refined throughout the three weeks -- I find that too limiting and I'll discuss what we did in a later post. I will say my group this year (go Discrete Math!) took on a huge challenge and did an amazing job; I was overwhelmed with how they took on their responsibilities and always questioned "how can we do this better, or different?"
We also spend some time looking at different problems in the mathematical area and we're always fortunate to have 200 world-class mathematicians running around in the corridor (well, they don't run so much as shuffle) to snag for a few hours. It's funny when you speak to them at lunch and then after lunch realize WHO they really are. They typically stride the mathematical world like colossus and you've asked them if they liked the carrot cake! :) We were lucky enough to have Joe Malkevitch (yes, THAT Joe Malkevitch) spend almost two hours discussing problems with us -- starting with the Art Gallery problem and then seeing where that took us. That is how lucky we are at PCMI!
The other group of activities I have to mention are the cross-program ones. This is a huge umbrella and can cover things like I mentioned below, James Heibert discussing the TIMSS Video Study results, at least two Clay Scholars every year discussing their work (with us! High school teachers!), Gov. Huntsman (at the time) speaking of math at the state/national level, and even Tom Garrity explaing how "Functions describe the world". The level and content varies so greatly, an exhaustive list would be its own (rather dull) blog post. Suffice it to say, it's the kind of opportunity you would have to hang around Harvard for, for at least a few years.

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.

Three things...

I managed to sign myself up for a How to write a better blog online course. Because, dear reader, this blog isn't just for you... no, this is to teach me how to be a better writer and a better reflecter (I'll bet that's not even the right use of the word... but I'm going to pull a you can do anything on the internet, grammar and spelling don't count)
So my task today to improve said blog is to provide a list. Totally open-ended. The rest of the 10,000 participants in this online course are mostly marketers, trying to sell something (not necessarily material but also opinion). That's not my goal so my list then is this, right off the cuff. I have to get this done because I have planning to do for tomorrow. I want to use Google Sketchup in my MPM1D Geometry class and that will take a little time.

Three things that will make me a better teacher:

1. Reflection. Reflection. Reflection. Reflection on what I am teaching, how I am teaching it, how it was received, how it can be improved. The issue, of course, is time. But, as is constantly mentioned, if you find it important, you'll make time.
2. Patience. As has been previously noted, I'm not particularly patient. Surprisingly, that has no effect in the classroom... I'll quite happily sit with a student to go over mathematics for hours. It's what I love to discuss so I have no problem spending the time or effort. What I am impatient with is bureaucracy. Stupid rules. Rules that are there only to make things fit into neat little forms. I will be a better teacher when I get over the fact that I can't change this. Stop tilting a windmills and do what I can.
   
3. Be more of a out-front leader. Previously, I've opted for the sit-back-and-lead-from-behind. Doesn't work. A decade has taught me a lot. Those who push and get themselves out there (and not always in a bad, back-stabbing, conniving way -- which, unfortunately does seem successful for some -- but in an open and sharing fashion) are those that are leading nowadays. Waiting for someone to notice what I'm doing is useless. I have to publish. I have to share.
   

So that's my list. I'm sure my students would have a completely different one. Hmm... I think I'll make up a Google Form and ask them.
Oh... and I figure a blog posting is better with pictures.