We were reminded this morning that tomorrow is the halfway point of PCMI! How can this be? Time has begun to take on that quality of seeming to speed up when you want to slow it down. This post may be a little shorter because (a) I don’t want to spend time writing when I should be experiencing and (b) I only want to write what might be meaningful to say. I also want to point out that anything I write here reflects only my experience at PCMI, and how I am processing all the activities in which I am participating and everything that I am learning. Just a caveat to my readers.
We were put into new groups in morning problem-solving, which I have just reminded myself is called Developing Mathematics. At my new table, I met someone who teaches in the same building in which I used to teach in Prospect Heights, Brooklyn (the building houses 4 schools), and the stories this teacher has heard recently about the principal I and most of my colleagues ran away from in 2011 continue to substantiate everything I remember as driving me out of the school. Luckily, he teaches in the most functional school in the building. Although hearing about the continued outrageous behavior of this woman is somewhat validating, there are 400 students who are being educated under her misguidance, which is sad and unconscionable on the part of those who supervise her. The work pace at my new table is quite different – I am sitting with a group of people who appear to work much faster than I do, and with more on-going discussion. My discomfort is a challenge to me – how can I produce good work and participate in this group in the most meaningful way possible? And how can this discomfort illuminate my understanding of student behavior and performance when I want them to work together in groups I have selected? One of the many wonderful features of this conference is the many layers of learning that are presented to us. The actual problem set is terrific – we are exploring complex numbers as rotations, an idea I have never completely understood until now, and the skill with which the progression of problems has been designed takes our understanding deeper as we work through them.
Meghan’s Eye View of the Crew
Our Reflections on Practice session was a little shorter today because of the group photos being taken. – Nonetheless, it was very fruitful with takeaways large and small. We shared new ways to collect evidence of student understanding – Plickers, Desmos Teacher Tools, Jeopardy – and how some of those tools provide data from more open-ended questions than others. Plickers does not require technology except for the teacher’s phone, but it is collecting answers to multiple choice questions. One teacher suggested a tweak on the speed-dating activity using whiteboards (another suggested call the activity Meet and Greet to avoid discomfort with the ‘speed-dating’ title) for sharing solutions to a range of problems, and yet another teacher pointed out that good old fashioned eavesdropping on student conversations about their work could provide meaningful formative assessment as well. We looked at the activity “Two Truths and a Lie” as applied to mathematics: students are directed to write two mathematical truths and a lie in any order; the lie must be well written, and in a way “which would challenge your last year’s teacher”. As we reviewed some examples in our groups, we saw how rich discussion could result from this activity. We finished the session by learning about a “Hinge point” question: it is a carefully crafted check for understanding midway through a lesson to see if students grasp the central concept, need to have it briefly clarified, or need the teacher to start all over again. This idea, written about by Dylan Wiliam in his wonderful book, Embedded Formative Assessment, is simple yet powerful. As teachers, we check for understanding throughout our lesson, but how intentional are we each day when we do this? By examining this practice, and crafting our hinge questions (and potential responses to the results) while planning our lessons, we can do way more than pay lip service to the idea of regular formative assessment.
The features of good hinge questions as described by Wiliam are:
- They are concise; students should be able to answer them in under 2 minutes, and teachers need to decide on a response to the results in 30 seconds.
- The questions can’t be answered correctly for wrong reason; common errors and misconceptions should be visible.
- Teachers should be able to see responses from every student using some method of data collection.
As Cal Armstrong put it, the goal of the Hinge Point question is a quick read of the classroom, “taking the temperature, not doing an MRI.”
[so much for a briefer post]
I finally feel on track in our professional development group, and managed to craft a solid framework of a draft for our portion of the course. My partner and I are still working on adding activities to our lesson, but the structure exists – essential questions, learning outcomes, and content. I am relieved; it felts as if we were floundering for a few days, but I suppose that was part of the process of allowing our ideas to coalesce.
The rest of the day was filled with enrichment: a visit to the gym, a Cross Program Pizza and Problem-Solving session in which the High School Math Camp participants gave us a big run for our money, and a Line Dancing Workshop run by David Nacin, a professor at William Patterson University and participant in the Undergraduate Faculty program here at PCMI. Who knows – maybe there will be a talent show at the last night in which we can perform our group tango!
And shameless promotion of my super-talented child (or a minute of video entertainment for my readers):