Category: Carmel Schettino

Updating Some Old Ideas

Years ago (more than I care to remember!) I was fortunate enough to attend the Ajna S Greer conference at Philips Exeter. I met Carmel Schettino there ( https://www.carmelschettino.org/ ) and I was introduced to the Exeter math problem sets ( https://exeter.edu/mathproblems/ ) and my mind was kind of blown. I worked at a school that was more traditional in its approach but I had some freedom to introduce some of the ideas I came home with. My solution to the situation was that I decided in my AP and Honors classes I would write open problem sets. I typicall gave my students about 8 – 10 days to let these problems marinate, to collaborate with each other, to come and pick my brain. I understood that many students did not think about it the whole time, but many did at least dip their toes into thinking about the problems. When I moved to my current school in 2010 I saw this in a really rich way. We are a day and boarding school and I lived in a boys’ dorm for 6 years and still live on campus. I see impromptu study groups, I see kids in the dining hall working (sometimes!), they gather in the dorms, in the libraries, they came to our after school conference times, and I saw some real collaboration. Over the past three years or so, likely a post COVID issue in part and also an AI issue in part, I have witnessed less collaboration than I was used to. This past summer I spent some time thinking about how to modify or whether to abandon the idea of problem sets as a meaningful part of our life together.

I decided that modifying was the solution I wanted. I came up a solution that I was optimistic about, but I was also worried about it. I bounced it off of a few colleagues, tweaking it along the way. Our school has a rotating schedule and once every seven days we have a 90 minute ‘long block’ and that ended up being the key to this situation. Here is where I landed:

  1. Problem sets are to be done in class during a long block
  2. The first 25 minutes are spent quietly working individually. No internet, just notes, calculators, brains working together. In this 25 minutes your job is to write down notes, ideas about these problems. You might not finish any problem, but you should have ideas about each of them.
  3. Turn in those notes and I make a copy of them. This portion of the class is worth 5 points for a participation grade.
  4. We take a 10 minute break while I make photocopies.
  5. I return the notes and for the next 55 minutes you work as a single group turning in a single solution set. That work is worth 25 points.
  6. One person is assigned to be the scribe and that person largely stays out of the debates. The next time around it has to be a different scribe.

My AP Calculus BC class and my AP Precalculus class each get 5 problem sets per grading period (we are on a trimester schedule) and we have already done 3 each so I have a sample size that supports my thesis. Let me tell you what I have witnessed.

Kids have been up to my four whiteboards debating and presenting their arguments. I have seen kids move from group to group grabbing ideas and defending their own. I have seen kids win their class over with a compelling argument. I had one student, new to our school, look me in the eyes on our first problem set day and tell me ‘this is the best math class I’ve ever had’ Not every student is as engaged as I wish and not everyone is comfortable debating yet. What I am convinced of is that more students are more engaged than where this project was two to three years ago. I am overhearing fantastic conversations and I am getting positive feedback from the students.

Below is a link to the problem sets I have written this fall. They are combinations of problems from various texts, from the Exeter problem sets ( https://exeter.edu/mathproblems/ ), from other competition math sets I have gathered over the years, from twitter conversations, etc. These are not intended to focus on the topics of conversation in our classes, they are intended to generate fun debates, to prompt remembering interesting ideas from the past, and to introduce a sense of play in my classes. I set up a folder with the problem sets I have already written this year ( https://drive.google.com/drive/folders/1AYRHRxL4AUufZuT9DY97XcVq3e-78Q21?usp=share_link )

I hope that google link works! If you want to have a conversation about any of this you can start up a chat in the comments or reach out to me over on my math twitter account @mrdardy

Empowered Problem Solving / Empowered Teachers

Not too long ago according to my calendar, but a long time ago now according to how the pace of school life moves, I finished an online workshop run by Robert Kaplinsky. The workshop, in six modules, was called Empowered Problem Solving. The modules were released on a weekly basis and were centered on videos of a workshop that Robert ran. These videos were accompanied by some outside reading in the form of blogposts and some PDFs. There were question prompts to encourage lively conversations on a message board, and there was quick support through emails from Robert and others working with him in the one or two cases early in the course when questions popped up about navigating the interface that they had set up. I did not recognize the names of folks on the message board there but I came to develop a sense of kinship through our conversations over the course of almost two months. Several themes emerged, of course, and it was interesting to go back through message boards from earlier lessons to see how my thinking was moving/growing and how the conversations deepened over that time. Looking back now, a few weeks after the course ‘ended’ [we still have access online for at least another month to revisit ideas and to help deepen our understanding/comfort with the ideas of the course] at a folder I created with documents that Robert organized for us, I realize that it will probably be out extended Christmas break when I can really digest and inject some of the habits of mind that are encouraged in the course. It made me think of my journey in grappling with/enacting/understanding the principles of inquiry and open-ended problem based lessons in the math classroom. I was forutunate to have had a Master’s Degree class in 1987 (before my teaching career began) called Mathematical Problem Solving. My grad school advisor, Prof Mary Grace Kantowski earned her Ph.D. in 1974 and her dissertation was Processes Involved in Mathematical Problem Solving, so I got a dose of this working with her and taking her class. I entered the high school classroom in the fall of 1987 and I have been honing, adapting, striving, to really figure out how to incorporate something more meaningful than practice exercises with my students. I was further energized by my first visit to the Anja Greer Conference at Phillips Exeter (I know it was between 2001 and 2005 but I cannot remember for sure what year it was) when I met Carmel Schettino and learned from her about problem solving in the math classroom and I am certain that this was my first exposure to the Exeter problem sets . The conference was mind-blowing and I was fortunate enough to attend one other time since then. Carmel’s work and advice energized me further and I started writing my own modest problem sets. Later, I wrote my own Geometry text that our school used for five years and in the process of that, I wrote HW for the course in the form of smaller problem sets. I have been fortunate enough to attend a summer think tank styled workshop that Carmel ran. I went with three colleagues to a workshop run by some folks from Packer Collegiate Institute in Brooklyn last year. I visited the Peddie School in New Jersey with three colleagues and we saw what they had done with their curriculum. Our school was visited by a member of the math department from Saint Andrew’s School in Delaware and he shared what they have done with their curriculum. All of these experiences led me to want to enroll in Robert’s online classroom and it was well worth my time and energy and the school’s investment of professional development funding. Conversations are happening in our school about the direction we want to go for our students and the visits and workshops last year helped prompt these conversations. The ideas and resources from Robert Kaplinsky’s workshop will be immensely helpful in moving this conversations forward.

All of this is a long winded way of me saying thank you to Robert, to Carmel, to the folks at Peddie who welcomed us, to Eric Finch from St. Andrew’s in Delaware, to my advisor Prof Kantowski. All of these voices throughout my career seem to be pointing the way to a more meaningful way of teaching and learning mathematics. Robert will be running his workshop again in February and March and I encourage you to take part. Whether you are just beginning to grapple with the ideas of running your classroom as a place of open inquiry and driven by problems (rather than exercises – a distinction that Prof Kantowski often discussed) or if you have been working with these ideas for years and are looking to be re-energized or more organized, this will be a great experience for you.