Teaching Demonstration

This video is a brief demonstration, about 15 minutes, of some teaching I did at the 2015 New Jersey Association of Mathematics Teacher Educator meeting.


Unfortunately, the video tracking is not great so much of the annotation I was doing of the participant ideas is not easy to see as I am doing it.

The instructional activity itself is called Contemplate, then Calculate and was developed by Grace Kelemanik and Amy Lucenta. The slides, script, resources, and references are available here.


Things I would do differently:

  • Set up the projector screen in advance of starting the activity so I don’t have to fiddle with it during the strategy sharing.
  • Not wing the recording of noticings and strategies because I ran out of time to prepare before this talk, but take the time to make a template for collecting data.
  • Record the initial noticings of participants about the problem as they were happening.
  • Bring my own markers so that I can ensure I have access to more than one colour when recording the student ideas.


Things I decided to do or not to do somewhat deliberately:

  • I did not focus on student to student discourse during the full group portions mostly in the interest of time. My meta-objective for this activity was to share the overall structure in a somewhat limited amount of time.
  • I did not enforce participants writing using the prompts mostly because I knew I had little to no relationships with participants and I wanted to make sure no one felt alienated during this portion of the talk.
  • I did make sure that when I was recording student strategies that I tried not to impose, as much as I could, much interpretation of those strategies. One of the participants actually came up after-ward and said she really noticed that I was making an effort to write down representations of what participants were saying rather than filling in too many of the gaps based on my understanding of the problem.
  • I also focused on having participants share multiple strategies to solving this problem rather than attempting to focus participants on seeing one particular strategy.
  • I decided to summarize participant ideas at the end rather than take the time to have them share out to the room, mostly in the interest of time.


What else about what I did do you have questions or feedback about?




  • Grace Kelemanik wrote:

    David, thank you for inviting us into your learning. I too find the more I prepare to lead Contemplate then Calculate– i.e. anticipate potential noticings, anticipate shortcuts that might grow out of the various noticings, think through the structure underlying the shortcuts and play around with how I might record those approaches in a way that highlights the structural thinking — the better able I am to, as Max Ray would say, “listen 2 students”. And it certainly helps to have several copies of the task pre-recorded so I can annotate without needing to rewrite the entire task:) The biggest challenge our residents (and teachers with whom we work) have when facilitating is keeping the focus on the structural thinking, i.e. through the questions they ask, the instructions they give and the ways in which they record student thinking. So often we see the structural thinking “left on the table” at the end of the enactment. So, rehearsing for this focus on structure helps a great deal. But it typically is not the first thing our residents rehearse for when learning to lead C then C. We often come back to it once residents have enacted the routine a few times an gotten comfortable with the flow.

    A question I have is when you asked the group to reflect on the decisions you made and what kinds of knowledge you might need to “teach the way I did”, what did you mean by “the way I did”?

  • David Wees wrote:

    Hi Grace,

    While I think it is hard to see what I am thinking, I saw my purpose while I was doing this as;

    1. Treating the members of the room as sense-makers,
    2. Trying to figure out myself what sense my students are making.

    Both of these seem somewhat different than most teaching I see but after watching this video, I can see that it is difficult to see that these were two thoughts in my head at the time!

    #1 I think I was reasonably successful at. A participant even noticed that I was careful to not fill in gaps in what the participants were saying.

    However, the insufficient preparation I had done as well as the challenges the space offered (it was challenging to actually circulate around and behind people and I didn’t have much in the way of supplies with me) made #2 hard.

    In particular, the pre-recording the questions to add to was something I considered, so as to make it easier to annotate work. Unfortunately I was not able to bring chart paper with me (without hauling an hour and a half on the train 🙁 ) and I never had access to the room alone to hide some pre-work on the board behind the projectors.

    I also assumed that the room would have more than one color pen which was false and so I know that in order to do this activity again I need to:

    1. Bring chart paper with me with the some copies of the question pre-written.
    2. Bring my own different colored markers.
    3. Prepare better, as you have noted.

    I think also that I may need to extend the way I understand structure. I feel like if you were to write down a lot of examples of structure on a piece of paper and I were to do the same, that your piece of paper would have more examples than mine and that we might have some disagreement in what definition our examples suggest structure is.

  • Thank you so much for sharing this! It was so fun to watch and see someone else work through this IA (this specific example, btw, has become one of my favorite dinner party conversation starters when people ask what I do for work– major thanks to Grace for that!).

    I totally hear you on the prep/materials… I’ve started traveling with my own kit so I know I have everything in multiple colors: chart markers, Expo markers, post-its, Flipcam, tripod 🙂

    Some of the things I noticed were ones you highlighted as well– the nonjudgmental restating and recording (as you said, it’s hard to see, but I can tell you’re being strategic about spacing and arrows) and the multiple strategies in particular. I also appreciated your framing for the audience– that this is math they know, that they know how to talk to each other– which immediately positions them as competent even though you don’t have a prior relationship with them.

    I also noticed that the first participant to share a noticing said something like “the obvious thing is that they’re all addition/subtraction and positive/negative numbers”. I thought your explicit connection between the two things she said nicely called attention to a key *structural* component of this problem. I think I might have added something like “that may have been obvious to you, but someone else may not have seen that right away; let’s hear what was obvious to someone else” to remind participants that there really are different ways of seeing and thinking about mathematical expressions, and those differences are not just valid but also useful (because they could lead to different strategies and different structural observations later), and to mitigate the possibility that someone else in the room might be thinking “oh, I didn’t see that, I must not be smart/paying attention the right way; maybe that’s what I’m supposed to be doing.” I think this is a subtle way of chipping away at the mindsets that math is about smartness or quickness or right answers.

    Another thing I’ve seen Grace do that I have borrowed is to ask, after people have shared their strategies, which of the initial observations they used. This gives purpose to the initial noticing section, helps students draw connections between their observations and their ideas (and supports the value of group learning if they used someone else’s observation), and provides an opportunity to emphasize particular structural elements (we’re allowed to group or rearrange because…).

    Thanks again for posting this– I hope there’s more to come 🙂

  • David Wees wrote:


    I like your two observations. Is it fair to summarize like so?

    1. Pay attention to the ways people share things to see how they will interact with other people’s beliefs about performance in mathematics.
    2. Ask people to share what structure they used to solve the problem so that everyone thinks about structure as opposed to just the solution to this problem.

    I will share more videos like this in the future. I realized the other day when Grace K was coaching a group of us that it was the very first time I had gotten live feedback on my teaching.

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