## A Residue of Professional Development

So, the session I wrote about a few days ago (you can find that post here ) continues to pay dividends. Yesterday my Precalc Honors kiddos had a test. Today we were to begin discussing vectors. I had what felt like a pretty clever idea this morning. I started off by posting this image (stolen from the opening evening problem that Amy and Allyson shared with us )

I got a quick question back asking if the dots were equidistant. I confirmed and then my students began to quietly count. I encouraged them – as I always do – to chat with each other and I was hearing things about medium sized squares, big squares, etc. I suggested that some more formal classification might be helpful. A couple of kids quickly concluded that there are 30 squares to be formed. This is a correct answer under certain restrictions. unfortunately, these restrictions were not placed on the question. A student named Max said 40 out loud, then said 50. This shook up the crowd a bit and people began to dig in. However, they were hesitant to debate Max because he has a reputation (well deserved) for being pretty on point with questions like this one. SPOILER ALERT: I AM ABOUT TO UNVEIL OUR SOLUTION. IF YOU WOULD LIKE TO AVOID THAT AND THINK ABOUT IT FOR AWHILE FIRST, COME BACK LATER.

Still with me? Good, happy to have you. I went to the board and drew a square of side length sqrt(2) and got two great reactions right away. One person called this a diamond but acknowledged it is also a square. Another said we should redefine squares to avoid this. I then stepped out of the way to encourage discussion about sizes of diamonds that could be formed. We had a list on one side of the diagram listing number and size of ‘squares’ and developed a list on the other side of the number of, and size of, the different diamonds. We had some great debates about the parameters here. We decided that the only diamonds had size lengths of sqrt2, sqrt5, sqrt8, and sqrt10. We were unsatisfied with the seeming lack of a clear pattern here. You will see in the picture below how I tried to impose a little bit of order on the counting by making sure that I identified groups of diamonds or squares in numbered sets that were all perfect square integers in their count. What you will also see in the picture (coming soon, I promise!) is that I pivoted the conversation soon to vectors. My Precalc Honors kiddos took a test yesterday and we are prepared to start a new chapter on vectors. I did not particularly advertise that this was the next topic, but it felt like I could pivot in that direction. Many of the kids in this class took Geometry at our upper school with a text I wrote. In that text, I intentionally introduce some vector language early in the year. When I got to school today, I did not intend to pivot from this diagram straight into talking about vectors, but when we were discussing diamonds of length sqrt5 I realized that it was meaningful to distinguish between a horizontal change of 2 with a vertical change of 1 versus a horizontal change of 1 and a vertical change of 2. Time for the photo now and then a little more explanation.

So, in the photo above, a bit of glare there unfortunately, you see a green side of delta x = 1 and delta y = 2. I drew an arrowhead and one student muttered ‘vectors!’ It felt like such a natural trigger and frame to discuss vector notation. Almost instantly kids were discussing magnitude, direction, remembering notation, etc. Man, it was a great way to start the day!

I ended up sharing this problem with a couple of other classes during the day and each time I confessed that my partner and I only found 45 squares and were VERY confident of our answer. Each class figured out where we had gone wrong and they seemed pretty proud that we worked through this all together.

Another opportunity here to thanks Amy and Allyson for the great PD session and I know that I will be pulling some other tricks out of the bag of tools that they provided for us last week.

## Meaningful Professional Development

Back in November we were having conversations at our school about improving our ability to place new students in our curriculum. Every year we have a wave of brand new students who have to move down from our original placement suggestions and it is always a frustrating thing for them and for us. So, I did what I do and I reached out to a number of department chairs at schools like ours seeking advice. One of the people I reached out to was Amy Hand. She is the math chair at Packer Collegiate Institute in Brooklyn, NY. In addition to sharing some wisdom, she also mentioned a workshop she was putting together. Here is the flyer she sent me –

I was immediately intrigued and I went to my admin to pitch the idea. We decided that we could afford to send a handful of folks together there and we ended up inviting our two Geometry teachers and one of our middle school teachers to go with me. I have been on the inquiry bandwagon for awhile but I knew that I could learn some new wrinkles to add to my game. I was excited to bring along three colleagues to listen to someone other than me pitch this idea. I also knew that the power of being in a room together working side by side with colleagues is always a powerful thing. Well, we returned Friday afternoon and I have had a couple of days to let some of the ideas sink in – as well as a couple of days to get caught back up with my life here. I am happy to spend a little time here telling you about what a wonderful experience this was. I feel that there is some positive energy that can help move our department forward in examining how to open up our classroom culture to encourage more open inquiry from and for our students.

Amy and Allyson Rohrbach, a colleague of hers from Packer, put together a really meaningful and packed two days for us. We started with a short introductory session on Wednesday night. This seemed like an unusual idea, but it was a great way to start. We had what seemed like a completely innocent problem on our tables. It was a problem from Brilliant that involved counting squares. I wish I had the image of this problem, perhaps I can find it soon to share. What seemed like a completely innocent problem instead became a lovely conversation about counting procedures with different folks going to poster paper at the front of the room to share their strategies. We got to know each other over snacks and beverages and discussing math ideas, Amy and Allyson framed our work efficiently that night and I think that we all left the room that evening energized for our work the next day. Thursday was the heavy lifting day but even that was paced really well and Any and Allyson kept us shifting gears so we did not feel like we were sitting with one idea or one problem for too long. There was plenty of space to explore and I don’t think that any of us felt rushed. One of the problems we worked on is one that Amy and Sam Shah worked on together at Packer and Sam blogged about that problem here I know that the next time I am teaching Precalculus I will be framing our discussion through this problem. I feel certain that I read Sam’s post when it came out, but working side by side with folks in this environment brought the problem and the pedagogy behind it to life in SUCH a meaningful way.

Again, we had GREAT conversations discussing/debating/explaining ourselves. I certainly have fun listening to my students debate like this but there is a different level of fun when I can get lost in the math myself to this degree. It is also energizing to hear from and share with people that I have never met before. There is such a sense of open curiosity in a carefully designed environment like the one that Amy and Allyson helped create.

I walked away from this event convinced that I will have an ongoing exchange of ideas with Amy and Allyson and I am already discussing a school visit to bring some of my colleagues who did not make it to this event. You can follow Amy and Allyson over on twitter where they are @MathSenseLLC. you can also check out their next workshop which is described here They will be on the west coast for this trip. If this is more in your neighborhood and you want to be recharged in your commitment to inquiry driven education or if you want to be nudged in this direction, I cannot recommend this highly enough.