PCMI #3: Talking and Listening

We were introduced to a protocol called “Talking Points” by Elizabeth Statmore (@cheesemonkeysf). Participants take turns agreeing, disagreeing, or expressing uncertainty about a list of statements. Each response must have a reason, and no commenting is allowed. The process occurs a second time with the same statements, and final opinions are tallied during the third round.

The protocol got me thinking, ‘What does it take for students to talk and listen to one another?’ No single reflection can do this question justice, so I’ll just go through two thoughts, one about talking and one about listening.

1: Students won’t speak if they’re rarely given the chance, and even if given the chance, they’ll rarely take it unless they think their thoughts are worthwhile. That’s why Elizabeth Statmore’s Talking Points and Ilana Horn’s ideas about competence have been so valuable in helping us think critically about our practice. Student talk requires space and intention. Protocols like Talking Points give students the opportunity to speak and a framework for structuring their thoughts, while surfacing our students’ competencies (like being willing to take risks or highlighting the strengths of others) make it more likely that our students will exercise them through dialogue. Recent posts by Dylan Kane (who makes a compelling distinction between getting students to feel competence and getting to students to recognize their competence) and Deb Barnum (who makes a great point that such protocols can “keep our conversations grounded in evidence”) resonate with me in these respects.

2: Listening to someone else’s ideas about math is different from listening to someone else’s ideas about other subjects. Because of the high stakes people feel about their own math abilities, it’s hard to accept an idea that’s different from your own because it may mean your idea was incorrect or incomplete. To paraphrase one teachers’ comment: you’re not going to listen to someone else’s thought if that would mean yours is wrong.

I wonder if this makes listening to mathematical ideas less about hearing what another person says and more about being willing to be influenced by them. How someone sees mathematics is closely tied to this issue. If math is mainly about being right and wrong, a person’s own thoughts are more likely to be a barrier from those of others. As teachers who want our students to become not only better talkers but also better listeners, we have an obligation to normalize for our students the idea that doing math requires making mistakes and revising our thinking. This is a necessary step toward ensuring that receiving feedback doesn’t get in the way of recognizing ones own competence. But this is also multi-dimensional work, and so helping our students form a healthy understanding of math is merely one aspect of designing a classroom that takes into account how students position one another and themselves. But it’s a step in the right direction.

PCMI Reflection #2

Today’s reflecting on practice focused on using student-generated questions to make math more meaningful for students. In addition to great conversations and an interesting activity on analyzing data sets, two concrete resources were shared by members of our group:

Screen Shot 2018-07-05 at 3.49.36 PM


“Meaningful” is a really interesting word, especially when it comes to math education. I think for a lot of teachers, the term means “real world applications”. But I don’t think that has to be the case.

I like the idea of asking students to create their own questions because their questions can still be meaningful to them without the questions having to do with real world situations. I think having students engage with questions that they’ve created centers the students while they experience math. Other group members also mentioned how asking students to ask their own questions creates ownership, improves metacognitive skills, gives students voice, and creates new opportunities for students to show competence in math class.

To me, the opposite end of the spectrum is the case (far too common in classrooms, probably even my own from time to time) where adults impose “real-world/fake work” activities on children which they think children will be interested in.

In The Having of Wonderful Ideas, Eleanor Duckworth has this wonderful* quote:

First, the right question at the right time can move children to peaks in their thinking that result in significant steps forward and real intellectual excitement; and, second, although it is almost impossible for an adult to know exactly the right time to ask a specific question of a specific child – especially for a teacher who is concerned with 30 or more children – children can raise the right question for themselves if the setting is right. Once the right question is raised, they are moved to tax themselves to the fullest to find an answer.

I’m particularly interested by Duckworth’s use of the word “setting”. That is, it may not be enough to simply ask students to ask their own questions. Rather, the setting must be structured for students to ask good questions. My students know when they’ve made something they’re proud of and when they’ve made something just because I told them to. So I wonder if the motivation and meaningfulness from asking a question comes not from the question itself, but rather from how much they’ve found their own question to be worthwhile (and also truly theirs).

Unfortunately, we didn’t have enough time to discuss how we might approach the work of asking students to create their own questions. But instead of seeing “create your own question” as a simple pedagogical trick, I do believe it should be done as part of a larger project to change the way students view mathematics and their mathematical selves in ones classroom.

*did you see what I did there?!

PCMI Reflection #1

I have the privilege to be currently attending the 2018 Teacher Leadership Program (TLP) at the Park City Mathematics Institute (PCMI). The TLP is a 3-week conference of math teachers doing math together, reflecting on their teaching practice, and working in small groups to achieve certain professional development goals.

The theme of this year’s reflecting on practice is “Mathematics and Motivation”. Using Ilana Horn’s book Motivated, the facilitators will be helping us develop a framework for thinking about motivating and engaging our students.

On the first day of reflecting on practice, we were asked to contribute one word (ONE word, c’mon!) that described a “good math student.” Here’s the wordle of the results:

(do you see the “NotAJerk”?)

Having noticed that curious was the most common term, we launched into discussions and activities centered around the potential noticings and wonderings that our students might have about mathematical concepts such as linear equations.

By the end of the session, we were asked to reflect on the following question:


This image has stuck with me since and is the main reason I wrote today’s post. For me, it raises the question:

How have I been constructing competence in my classroom?

For a better sense of this question, I should mention the work of Melissa Gresalfi. In this article, Gresalfi argues that competence, rather than being a set of skills particular to an individual student, should be seen as the relationship between what a student does and the opportunities for learning that he or she was given based on the classroom context. In this sense, competence is constructed by the tasks and participation structures designed by the teacher and the way they are realized by the students. This is important because how competence is defined in a particular classroom determines what kind of learning can take place.

For example, competence could be constructed in a classroom as correctly carrying out procedures established by the teacher. But this is not the only form of competence that could be constructed. It could also mean experimenting, making mistakes, and learning from those mistakes. In both cases, the classroom sets forth an image for success, and that image determines the path that students must take to get there.

Looking back at my own classroom from this year, I believe that competence was defined for my students as following procedures, reasoning about them, creating original mathematical methods, justifying them, and applying known methods to new situations. Today’s closing question raises the idea that curiosity can (and probably should) be added as a component of competence in my classroom. I praise students for solving problems with originality and justifying their reasoning. However, I should also (but don’t yet) consider whether my students do so out of intellectual urgency or whether they are simply complying with classroom norms – i.e. “I used a different method because I was supposed to.”

I would also like to start holding my students accountable for being curious about other people’s reasoning and sense-making. In discussing accountability, Gresalfi points out that teachers should not only think about for what students are accountable (what students must do) but also to whom students are accountable (who students must convince). In many classrooms, including my own, I am the only one to whom students must ultimately justify their work. Sure, I tell them to discuss their reasoning in pairs or in groups, but in the end I am the only one to whom my students feel accountable. In this situation, students in the lower-right-hand corner of the graph above continue to be rewarded while students in the upper-left-hand corner fail to have their competence of seeking and sharing ideas acknowledged.

Here’s my final thought for now: How does one go about construcing an open view of competence that includes curiosity? Probably, the first step is to critically examine the practices and rituals of ones own classroom. In Engaging Children: Igniting a Drive for Deeper Learning, Ellin Oliver Keene asks the reader to consider when your students speak up in class. Is it only to answer a question posed by the teacher? Or do they also speak up to ask questions, challenge one another’s thinking, and pose areas for further exploration? The nature of students’ unprompted discourse may be an excellent starting point for examining the ways that competence has been constructed in the classroom.