Reflections on the Problem-Based Learning (Symmetry Mini-Unit)

This is the last post in the series on experimenting with problem-based learning in Claire’s fourth grade class.

The other posts in the series are:

• Getting Started with Problem Based Learning in Grade 4 (Monday)
I addressed some of my motivations formalizing the idea of problem-based learning with Claire. I also shared insights into our first planning session — what Claire said seemed to be important about this math content (symmetry), what Claire said has been challenging for students in the past, and where she posted to take students this year
• Problem Based Lesson #1: Launching Symmetry in Grade 4 (Tuesday)
I shared experiences from the first lesson in our symmetry mini-unit. This includes some of our facilitation strategies, and also many different ideas that students introduced. I examined some ways in which Claire and I leveraged multiple mathematical competencies through this work.
• Problem Based Lesson #2: Making Connections with Symmetry in Grade 4 (Wednesday)
You know those “orphan” standards — the ones that seem disconnected from the rest of the curriculum? In this post, I explored how we connected a seemingly-orphan standard to more central work of the grade level, and how students made connections between these different ideas.
• Problem Based Lesson #3: Learning From One Another in Grade 4 (Thursday)
In the third lesson in this mini-unit, I explored both going deeper with the mathematics and challenging spaces of marginality in the classroom. We examine student work samples, and the process that Claire and I went through to choose which student ideas would anchor our class discussion.

Reflections on Lesson Structure

The lesson format we used is helpful — it’s predictable, and centers student thinking — but it’s not the only way to engage students in mathematical work and sensemaking. For example, the small group instruction during math workshop that Kassia Wedekind describes in Math Exchanges (Stenhouse, 2010) . These math exchanges have the potential to fit into a lesson with an entirely different format, but that focus on that same launch of a task, time for students to mess around with the mathematics, and learning from one another. There are also lessons in other programs, like TERC’s Investigations 3, that follow similar patterns of engagement with math and one another.

By lesson two, students had fall into the pattern of this lesson structure. Launch, work, discuss.. launch, work, discuss. It takes time to build up classroom norms around the function of the lesson components, like how to have a classroom discussion that really pushes thinking forward, but Claire and her students did an admirable job.

Reflections on Lesson Planning

The free content illustrations from Illustrative Math were a great starting place to find tasks worthy of discussion. Of course, I’ve found great tasks all over. I love using NRich, for example. (It’s one of my favorite math sites on the web!) I have also borrowed tasks from other curricula, like the aforementioned Investigations 3.

In our particular case, I was grateful to find tasks from IM built on one another beautifully. Because these tasks formed a coherent mini-unit, Claire and I were able to focus on enacting the tasks rather than writing them. I wrote about this in Cultivating a Listener’s Mindset: A Case for Curriculum (January 18, 2018).

Because Claire is a hurricane of a teacher, we were able to work not just on internalizing the lesson structure — an enormous task unto itself — but also on issues of equity and access with things like the five principles for equity-based instruction. These principles from from The Impact of Identity in K-8 Mathematics: Rethinking Equity-Based Practices by Julia Aguirre, Karen Mayfield-Ingram, Danny Martin (NCTM 2013).

• Going deep with mathematics
• Leveraging multiple mathematical competencies
• Affirming mathematics learners’ identities
• Challenging spaces of marginality
• Drawing on multiple resources of knowledge

Many of these principles felt natural to Claire. Going deep with the mathematics is her impulse; she’s someone that likes to stop me in the staff room to show me photos from the notice/wonder routine her students did that morning, and discuss the connections students made. She celebrates the different ways that they approach problems. She thinks about issues of status in the classroom.

While I knew that these principles were all, in varying degrees, endemic to her practice, I believe it is also valuable to make deliberate space for them. I know that she likes to challenge spaces of marginality. How will we do it in this lesson?

While we sometimes planned for these principles in advance, at other times they came about more organically, and we reflected. (Sometimes we reflected in person, and sometimes I reflected on my own via this blog.) These reflections helped Claire and I flesh out what it might look like to affirm mathematics learners’ identities. (Honestly, Claire and I haven’t revisited these principles in a while, either… these posts were originally written last May, and we came back to school in September to some new challenges.)

Reflections on the Math

One of the things I am most proud of regarding this mini-unit is that we were able to address exactly what Claire identified as a previous issue: that students only had experience with folding to find lines of symmetry, and that the lessons on symmetry felt disconnected from other bodies of mathematical knowledge.

Symmetry has connections to all sorts of things. We were able to think about line length, and line relationships, and angles, and even the equal sign/balancing equations. All good curricula embeds these ‘minor’ standards in the context of the major work of the grade. Making connections improves the experiences of sensemaking.

Where We Are Now

Fast forward several months. Claire is now piloting the a problem based curriculum — the upcoming IM K-5 Math — in her classroom. Every day, she uses the same structure that we worked on in this series: warm-up, activities (launch –> problem solve –> discussion), and a synthesis and cool down/exit ticket. This structure is second nature to her and her students.

In the first week of the school year, Claire and I worried over whether her new crop of fourth graders would connect to the problem based experience like her previous class had. We co-taught a few of these lessons to a roomful of silent, blinking children. Would this be what it would be like all year? Lesson after lesson felt like it was flopping. Failing. Could we even find evidence of students learning? Making progress?

Then, two or three weeks into the school year, things started to gel. Students were talking — all of them! Everyone had something to say during the notice and wonder launch. They offered different perspectives during the number talks and activities. Claire and I were thrilled! This lesson structure seems to work for yet another group. This means that Claire gets to focus on developing her pedagogy. What questions will she ask to help students dig deeper? How will she select student work to share? How will she confer with students during work time? These moves are so much more important to student learning and growth than “how will I manage the materials so that students know where to go and what to do.”

The other day, Claire stopped me in the faculty room. “Look at this!” She grinned, while pulling out her cell phone to reveal a photo from that morning’s math lesson. “Do you see what Oscar said? And check out what Ella said! Her comment connects back to what we did yesterday!” She’s focused on the students.

Tracy Zager says that if you get hooked on student thinking, the rest will follow. I think that’s true. The rest might include instructional format, or a set of frameworks to use. The rest might include specific mathematical tasks and practices.

…and the rest definitely focuses on what students have to say, and how we can help them move forward.