Month: March 2023

I am flattered and excited to be part of a pretty cool event happening next month in California. A little history here to explain this. In 1995, and again in 2008, I was fortunate enough to attend the Ajna S. Greer Annual Conference on Secondary School Mathematics, Science and Technology at Philips Exeter Academy. Looking at those dates, I think I need to make my way back again before too long!

Each of my summers there were fantastic in a number of ways, but in terms of todays post the particular fantastic thing to point out is that I met Carmel Schettino there. My memory is that we met the first time I was there, she may remember otherwise, but the important thing is that I met her. Carmel is a thoughtful educator who is committed to Problem Based Learning in the math classroom. You can read up on some of her work here ( https://www.carmelschettino.org/ ) She currently works at a pretty amazing school called The Avenues School (you can see what they are about here – https://www.avenues.org/ ) and I have had the pleasure of visiting their New York City campus and taking part in a summer workshop there a few years ago (pre-COVID) and Carmel and I have been in touch off and on for quite some time now. I was VERY flattered when Carmel reached out to me in January to ask if I would be a presenter for what she calls a Jam Session at an upcoming conference she was organizing. I jumped at the opportunity and was able to work it out with my administration and I am totally happy to share that I will be in San Jose, CA at the newest campus of The Avenues School for what is being called a PBL Math Teaching Summit at The Avenues Silicon Valley campus from April 27 through April 29.

While Carmel has worked at schools where there is a more intensive focus on problem based learning, she knows that I have been working at schools that have a more traditional view of their curriculum. However, even at the three schools I have worked at in the time I have known Carmel, I have been able to regularly carve out some space in our curriculum for problem based learning. I have stolen ideas (and some problems!) from a number of sources over the years. I imagine that anyone reading this is familiar with the Exeter Math Curriculum and their public problem sets (you can find them here – https://www.exeter.edu/mathproblems ), I have also used math competitions that I have found on the internet, I borrow liberally from the bookshelf of texts I have in my room, I have borrowed from The Park School curriculum, and a number of other schools that have been generous in sharing their work out on the web. The proposal I put together for Carmel for this event is that I am going to speak out the social learning aspect that I see my problem sets creating in my school. I am going to concentrate on the school where I currently work. I have been here since the fall of 2010 and in four of my courses here I have made take home problem sets a part of the curriculum and a part of how I assess my students’ progress. I live and work at my school. We are a PK – PG day and boarding school and I lived for six years in one of our student dorms. In my time here I have regularly seen groups of students gathered working on my problem sets. In the dorms, in the cafeteria, in the library, after school in my classroom during our extra help conference time. I routinely see students gathered and working bouncing ideas off of each other. The real joy is that they often ignore me, there are times when even though I am in the room with them, they simply pick each others’ brains. Sometimes students who aren’t in my class come by because they hear an interesting conversation. Sometimes students who had been in my class the year before drop in to share their memories or some tips to help move the conversation forward. I have been reaching out to a number of our alum to ask them to reflect on their experiences and I have been receiving some terrific replies from folks and I am excited to share in depth when I am in CA in April. I will be setting up a shared drive with resources and I will be sharing some of the particularly valuable reflections that my students’ have shared. Mostly though I am looking forward to spending some time in the company of other math teachers who are able to share their experiences in this approach to teaching and meeting a number of math teachers who are looking to make the leap and expand their toolbox.

I will be sharing my experiences after the conference in early May when I get back and get settled in. If you are interested in more information about this conference, there is contact information on Carmel’s website and you can reach out to her that way.

As always, if you want to reach out you can comment here or find me over on twitter where I am @mrdardy

Our school has a two week spring break that happens at a time in the year that does not feel much like spring in our part of the world. My Precalculus Honors class is due to start a chapter on systems and we will learn to work with matrices. However, for the intro to the chapter I knew I wanted to revisit old skills in a new, hopefully more interesting, way. I had the germ of an idea last Thursday (IIRC) and I sent out a tweet to my math team.

I thought that this would be a fun sort of open way to get my students thinking again about systems. As the #MTBOS is inclined to do, I immediately got helpful feedback. Among the suggestions were to make sure that the line was not horizontal or vertical as this would be WAY too easy. Noted! Nice suggestion.

A favorite early suggestion came from a teacher I had not interacted with before on twitter (at least I do not remember any extended interaction)

This got my mind working at a much deeper level. I had kind of thought of just throwing out a quick challenge question. This suggestion, and the resulting conversation, got me thinking along the lines of a more organized activity leading to this challenge question. A few more exchanges with Pete and with a couple of other twitter pals and I landed here with a great suggestion from Karen.

After a few more tweets back and forth, and with school looming finally, I crafted an activity (you can find the finalized product here )and shared this link out to three twitter pals who were pretty actively engaged in this conversation. I was so touched by the fact that all three of the folks I shared the doc with took the time to look it over and share some ideas/questions/insights into how I constructed this. Years ago when I first started blogging then took the leap into twitter this kind of collaboration and sharing of ideas was exactly why I took that leap. I feel remarkably fortunate to have found this community of open, curious, and generous minds.

I decided to jump right in on day one back and present this activity to my small Precalculus Honors class. I have seven students in this class and they normally work in two small groups. One group of three boys usually work together and another group of four students (two boys and two girls) usually share their ideas together. I specifically asked them to start this work on their own and I spent the first ten minutes or so of class just sitting quietly and trying to spy on the work that they were doing. I have small whiteboards at our big Harkness-style table and most students do their thinking on those whiteboards rather than in a notebook. After a little while I began taking notes on our GoodNotes app and asked the students question by question to share their ideas. You can find our class notes on the hyperlink in the previous sentence.

Next, I asked them to sketch their graphs and identify how many intersections there were. They were asked to find the coordinates of the solutions to their system. I suspect that few of them recognized right away that the point (3, 4) had to be one of their solutions, but that is fine. Not too much time or energy was wasted here. Now comes the payoff for the activity.

As soon as Ava shared that she remembered that a circle and a line that intersect once has this special name from Geometry (with a little prompting from me) her neighbor Abby immediately remembered the perpendicular slope notion. Once Abby said this out loud a couple of her classmates nodded in recognition that they, too, knew this at some point. Now we were off to the races. The link I shared above to the GoodNotes file has a number fo Geogebra sketches for the circles and lines that my students developed. We had a great conversation about whether it was easier to think first of the circle or the line first (we seemed to agree that the line as a starting point was more efficient) and everyone was pretty engaged in the conversation. I am pretty thrilled about how well this went and how my students were able to dissect this in our 45 minute class today. I want to thank Pete K (@PAKalenik9), Karen Campe (@KArenCampe), and Benjamin Leis (@benjamin_leis) for their thoughts and this activity was so much richer because of their contributions.