Professional Growth in a Connected Age

I’ve been teaching for a long time now. It makes me feel old when I realize I am in my 27th year in the classroom now. I joke with my students that I have been teaching longer than any of them have been alive. When I started teaching the predominant models of professional development were the inservice days at school where the school administrators decided how we needed to grow, the weekend workshops or summer workshops that I would scramble to find funding for, or the one or two day workshops that would cause me to miss school. It’s a different world now. I know I’m preaching to the choir if you are even reading this but this world of twitter, of blogs (both writing them AND reading them), of online simulcast workshops, or improv EdCamps, the list goes on. In this day in teaching I am fully convinced that if you want feedback and you want connections to help you think about your craft and to expand your toolbox – if you really want it – there is an ocean of resources at your fingertips. Literally (since I’m typing this right now!) at your fingertips. Not all of it fits everyone. I know that I am still wrestling with the timing and pace of my twitter feed, but I think I’m getting better at it and I KNOW I’m growing as a result of it. I spent a long time reading blogs, then commenting on blogs before I felt confident enough to launch my own. I have two kids at home so I know how tight time can be, but I also know that the past two Saturdays (that I blogged about separately here and here ) where I spent a combined 16 hours out of the house were worth the time and effort. Luckily Mrs. Dardy is kind and flexible and supportive of this pursuit.

I’ve been thinking quite a bit about this, about how different my life as a teacher is in the past few years. I’ve been in regular communication with one of my former colleagues, Gayle Allen. Gayle (@GAllenTC) hired me seven years ago when my family left Florida and we landed in New jersey for a while. Gayle is a remarkable, energetic thinker and was a great boss. She and I have been engaged in a long conversation about professional growth and one of the results of this conversation is an article that got posted today over at a website called Getting Smart. I know that I am not unique in this journey, but I also know that there are still many of our colleagues who have not taken this plunge. Some because they are not interested in doing so, some because they don’t know where to start. I’m pleased to be able to give shout outs of thanks to Dan Meyer (@ddmeyer) and to Sam Shah (@samjshah) through that article and I’m pleased to be connected again (even if we are nearly 3000 miles from each other) with Gayle.

This summer will see a trip to OK to take part in TwitterMathCamp. This would not have happened if Tina Cardone (@crstn85) had not reached out to me and asked me to join in on the fun. This summer will see me finish an in-house Geometry text for our students. This project would never have happened without the encouragement and advice of Jennifer Silverman (@jensilvermath). This summer will see me work on plans to help a brand new teacher in our high school take the leap from teaching Algebra I in the middle school to teaching Honors Precalculus for the first time. All of these experiences will help me grow as a professional. 27 years at it now and I feel like I still have an awful lot to learn. I hope to be smarter this time next week about this craft than I am right now.

Circle – Square Problem (Stolen from Dan Meyer’s site)

We are only three (well, really two and a half) days from spring break. Spring itself comes late in NE PA but spring break comes early for us. We are in the thicket of infinite series in Calc BC and I wanted to take the last three days on a tour of an old AP Free Response section. I wanted to send them on their break with the full realization that they know almost everything that they’ll need to know in May. So, yesterday I ran off copies of the two calculator questions from an AP test along with the grading rubric. Figured it would be a good working day while I listened in and roamed a bit. Then at 6 this morning I read Dan Meyer’s newest blog post and immediately decided to scrap my plans. He presented the following problem

Given an arbitrary point P on a line segment AB, let AP form the perimeter of a square and PB form the circumference of a circle. Find P such that the area of the square and circle are equal.

and, as usual, presented a challenge to his readers.

What can you do with this? How can you improve the task?

I’m not sure how much I improved the task, but I did make some decisions about what to do with it. The first decision I made was to hand my students some blank paper and simply read the question to them. I read it carefully twice. I did not want to visualize it for them, I wanted to see what they would do with this. I was surprised that most of them did their best to jot down the exact words that I said.  (This was in my quiet, small morning group – I am just getting ready to meet with my afternoon group) Almost all of them proceeded to draw a line segment and label a point P pretty near the middle. One student identified the two partitions as x and y. Very algebraic! My more engaged (and much larger) second class responded almost unanimously by drawing a segment. They hardly bothered with the words at all. One student had a segment with a square imposed on one side and a circle imposed on the other. Nice.

Both classes were comfortable (being BC Calculus kids, I expected them to be) with some general statements of the area either interns of a variable or in terms of segment notation. Both classes decided that it would be nice to have a default length for the segment AB and they each suggested 1 as the length. This was interesting, I went immediately to 100 as a nice default value. I was thinking in terms of a percentage of length. My first class let me get away with that and we reached a solution to the quadratic pretty quickly. My second class was insistent that 1 was a better choice. Of course, the critical balance value is found regardless. By the time the afternoon class had met the hive mind was working full bore on this problem. I shared Dan Anderson’s terrific Desmos demonstration and I also showed the the Math Hombre’s GeoGebra sketch of the problem.

As per Dan’s suggestion on twitter, I asked my students to try and optimize the situation. Both classes were very quick to conclude that the largest area is the trivial case of the entire segment being used to make the circle. When we graphed the function that represented the sum of the areas it looked tantalizingly close to the equal area point being the minimum. Sadly, this was not to be. We also discussed another scenario proposed on twitter and that was to have different regular shapes, not just a circle and a square.

I was proud of how open my students were to exploring this open-ended question. I was impressed that they were such careful listeners when I presented the problem to them verbally. Many of them are not note-takers, but they were willing to dive in and play on paper with this problem.

I am left with a some ideas/issues to ponder.

  1. I need to figure out a way to find time at the beginning of next year to give my students the opportunity to familiarize themselves with Desmos and GeoGebra. I wonder how I can structure this appropriately. When they see the dynamic visualizations the conversation opens up.
  2. By my standards, I think I did a pretty good job of getting out of the way today. I still started too many of the conversations, but I let the questions percolate from them. I need to find some sort of mantra or something to remind myself to be more quiet and take more time to let the questions arise.
  3. As with many class conversations, the pace was dictated by a few students who are more eager to share ideas and ask questions. I need to work on respecting this while also creating a buffer for those who take a few more moments to ponder.

So happy I scrapped my plan of canned AP FR questions today. I hope that the students are happy about this as well.