Professional Development: Doing mathematics

This may sound like a no-brainer, but I find it’s really important to engage elementary and middle school math teachers in doing mathematics during professional development. The experience of doing math in a different way than the way it was learned is critical before we talk about how to teach it in a different way than we learned.

It’s hard for me to figure out which tasks to use. I try to find activities that allow teachers to explore a particular mathematical concept in a different way than they may have learned it when they were students. For example, I recently worked with a group of teachers who knew the formula for surface area and volume of a cylinder but never had a chance to unpack why or how it works.

My goal is that through experiencing math this way, teachers will see a benefit to this way of learning–that when we have the experience of seeing why a formula works or how it works, we have a different experience, which leads to a different type of understanding.

I’ve been trying to think about how I select and modify tasks I use with teachers. It’s similar to how I select tasks for student in some ways and different in others. Here’s what I have so far:

  1. The task starts by asking for a prediction or estimate of the final answer: This gives everyone an entry point, builds on what they already know, and it allows me to assess where the group is. It’s not always necessary but it helps.
  2. Knowing the formula is not enough to complete the task: Because teachers have often memorized formulas or procedures, I need to make sure the task can’t be answered solely by knowing the procedure. I want a task that creates a need to unpack the formula or procedure they already know or apply it in a new way.
  3. I can provoke an interesting discussion based on a common misconception. Sometimes someone in the group has the misconception. Sometimes I will bring it up. Either way, this type of discussion allows me to have a conversation about how important it is to anticipate misconception.
  4. They have to prove WHY something is true. This allows me to set norms about what it means to convince or prove in math. I can create the shared understanding that we don’t just want to prove something works but that we want to explain why it works.
  5. It can be used with students with minor tweaks. At the end of the day, I want teachers to walk away with something they can try out in their classes

I’ll share some of the tasks I’ve modified next time, but I’m curious what else you would add to this list. How do you choose tasks to use in professional development?

12 thoughts on “Professional Development: Doing mathematics

  1. rpdent

    Thank you for make this explicit! I struggle with the same and spend a LOT of time trying to get it right. I’m seeing a lot of value in Fosnot’s landscapes for teachers to articulate where they are at, and what’s a good next step to focus on. Although it’s not necessarily giving direct support in choosing mathematical tasks, it helps to identify what you do with them and what you want to get out.

    Can’t wait for the next post sharing tasks!

  2. tjzager

    I love this list, Nicora. So helpful!

    I find that I like to start the very first session with warm-up tasks that lead to discussions about different ways to see or solve a problem. Especially when working with a new group of teachers where the rapport is young, I want to get everyone talking math, soon. I find an open strategy share that leads to some interesting math can be just the ticket.

    I’m thinking about a conversation one group had about Michael Fenton’s peaches picture, which I can’t seem to post here but will tweet you. When teachers shared how they each saw it, a few great things happened:

    – Everyone was doing math, early.
    – The approachability of this picture helped reduce the math PD jitters. Everyone feels like they can start this task.
    – They started to catch on to how fun it is to listen to other strategies.
    – I began unraveling their ideas about what it means to teach math (stand at the board and talk stuff).
    – I was able to model facilitating a math discussion around a shared experience we could refer back to. We did, too.
    – And, really great math came up! Once I had their strategies, I asked them to compare two of the solutions: (2 x 3) x 4 + 3 and 2 x (3 x 4) + 3. How can these both be true? The teachers had seen them very different, visually. So, before we knew it, we were deep in the associative property, which is a great place to be for K-6 teachers.

    I didn’t have your organized list before I picked this task, but it is one I’ve found works really well, so I’m curious about how it fits in with your thinking!

    To be sure, I love all the criteria you’ve posted and definitely use tasks that meet them during other parts of PD. I just wanted to add open tasks that lead to strategy shares as great warm-ups. I have a file of similar pictures, like Chris Hunter’s Othello boards and some of the other #arraychat pictures. I really like that we quickly dispense with, “What’s the answer?” and move toward “How did you count?” or “How did you see them?” as our first step together.


    1. Nicora Placa Post author

      I’m glad you brought this up. I’m in the process of planning and a PD session and I’m trying to think carefully about how I plan each part. I think you are right about the importance of the warm up task. I love your idea of using the open tasks with the pictures and then having everyone share out their strategies. It really sets the tone for the session and is something teachers can easily incorporate into their classes. I usually start with a warm-up task that models one of the strategies from here: for similar reasons–they are accessible to everyone, allow for multiple solutions, and can lead to a nice conversation about how we would use these in different level classes.

      I think the warm-up serves a different purpose in some ways than the more in-depth math tasks I was talking about in the post so now I need to think about what qualities I look for in warm-up tasks.

      1. tjzager

        Yes! I was thinking about it more later. You have a great list going for criteria of tasks we use once we’re nice and warm. The warm-up task is a little different, so has different criteria, with a lot of overlap. I really appreciate you making me think about both!

  3. mathmindsblog

    I could not agree more Nicora! Having teachers DO math is so incredibly important, especially, as you pointed out, if it is not necessarily how they learned it, or only the formula/procedural way they continue to think around particular ideas. Engaging in the WHY is invaluable for teachers in understanding student thinking.

    As Tracy said, I like to start any professional development with some type of warm up to get everyone talking before engaging in a task. Whether it be Talking Points or a Number Talk, it gets the participants talking and establishes talking about math as an important focus in our work together. I think your lists works wonderfully with either of those activities!

    One thing I do try to also be aware of in PD is the comfort level of the teachers. I do not want anyone shutting down or disengaging due to discomfort with the content. If it is a situation in which I think teachers may be uncomfortable not knowing how to engage in a task, I ask them to think about how their students would engage in the work and compare that to how they are thinking about it. It makes it safer until a community is established and relates to the work in their classroom.

    I look forward to continuing this conversation a lot and reading A LOT more blogs about it! Thanks for getting the ball rolling on this and I can’t wait to see the tasks you have coming!


    1. Nicora Placa Post author

      Hi Kristin–That’s such an important point you bring up about comfort level. I think I need to add something about engaging all teachers to the list. And on the other side, what do we do with teachers who aren’t challenged by the task or finish early? I need to think more about how to differentiate for different level teachers.

      I’m so glad we are all having this conversation because I feel like so much of what I do with PD is done in isolation. I really want to change that because I know I’ll learn more by forcing myself to write about what I do,from getting feedback and from hearing about what you all do.

      1. mathmindsblog

        Hey there! I have found comfort level is such a huge engagement factor that I need to always prepare for now. It can make the biggest difference in conversations! As far as the ones that are finished quickly, I ask them to think about how their students may solve it and how can they make connections between the various strategies. I also ask them to think about questions they would ask students who are “stuck” and what they would ask students who are finished early. I push them to differentiate their questioning. I think Michael Pershan’s shadowcon talk is great for this discussion!
        I am glad for this discussion as well since I am not planning PD for this summer! You will not be able to get away from my questions now!!!

  4. goldenoj

    I like your list. I’m not sure if it’s a part of the task, but I try to have every problem end with a reflection. The framework Dave Coffey introduced me to to help reflect is ‘what? So what? or Now what?’ They write, then they share at the table, and maybe or maybe not whole group depending on what I eavesdrop.

    Sometimes for task, I like to start with something pretty pedestrian and model a shift.

    I like starting with predictions. It does so much good, but one of the things it does is help make sure people understand the problem by considering or hearing what a solution looks like. But sometimes I start with a schema question that is more like a preassessment.

    It was neat reading this and Hedge’s new post close together.
    Her experience helped remind me that it doesn’t have to be new or fancy (especially not to me), it needs to be solid or significant. Real math is engaging. (Especially compared with typical school math.)

    1. Nicora Placa Post author

      I’m glad you brought up the reflection piece. I like the “what? So what? Now what?” prompt. I’m trying to figure out what other questions force us to reflect on our activity. I find that if I leave the reflection too open-ended, there is a risk that teachers don’t focus on the learning I’m trying to foster. Maybe that means I need to rethink the task or maybe I need some more specific reflection prompts. I’m not sure.

  5. Pingback: PD: A Math Task for Teachers - Bridging The Gap

  6. Pingback: A Round Up of Posts on Professional Development – Reflections in the Plane