Lesson Study Cycle 1
How do we support students in developing their agency as mathematical and scientific thinkers?
In this first cycle of lesson study, our group investigated different classroom activities and routines that would support students in developing agency and authority as learners of math and science.
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Essential Question: How do we support students in developing their agency as mathematical and scientific thinkers?
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Content Understanding Goal: Students will understand that because ratios represent similar quantities scaled to a set amount, they can represent a ratio graphically as a diagram and as a math sentence. For example, they will understand that (xxx:y) is equivalent to 6:2. Students will be able to apply this understanding to represent ratios in real life both as a diagram and math sentence. For example, students will be able to craft recipes using pictures that represent recipes in real life.
Key Research
Competence is a construct. Classrooms can co-construct a definition of competence that centers around asking questions, taking risks, and making sense of problems. Classroom activities and routines should emphasize students' accountability to each other for making sense.
Gresalfi, M., Martin, T., Hand, V., & Greeno, J. (2009). Constructing Competence: An Analysis of Student Participation in the Activity Systems of Mathematics Classrooms. Educational Studies in Mathematics, 70(1), 49–70. https://doi.org/10.1007/s10649-008-9141-5
Students should own ideas, author their own debates, and take responsibility for processing each others' ideas. Pedagogical moves that support this are "appropriating" students' ideas to elevate them in whole-class discussion; extended wait time; and maintaining neutrality.
Solomon, Y., Hough, S., & Gough, S. (2021). The Role of Appropriation in Guided Reinvention: Establishing and Preserving Devolved Authority with Low-Attaining Students. Educational Studies in Mathematics, 106(2), 171–188. https://doi.org/10.1007/s10649-020-09998-5
Care and autonomy support from their teacher is a major predictor of intrinsic student motivation. However, it isn't enough to care--the students have to perceive that you care.
Bieg, S., Backes, S., & Mittag, W. (2011). The role of intrinsic motivation for teaching, teachers’ care and autonomy support in students’ self-determined motivation. Journal for Educational Research Online / Journal Für Bildungsforschung Online, 3(1), 122–140.
Research Base
Plan - Do - Study - Act
"Snapback"
Students must address the student who spoke before them with a question, agreement, acknowledgment, etc.
Neutrality
Neutral teacher feedback pushes students to process and evaluate their classmates' thinking for themselves.
The Price for One
Students will understand that because ratios represent similar quantities scaled to a set amount, they can represent a ratio graphically as a diagram and as a math sentence. For example, they will understand that (xxx:y) is equivalent to 6:2. Students will be able to apply this understanding to represent ratios in real life both as a diagram and math sentence. For example, students will be able to craft recipes using pictures that represent recipes in real life.
Focal Student 1 is an English multilingual learner who dreams of being a doctor or a veterinarian. She has a large extended family that she feels very close with. She benefits from language supports like sentence frames and discourse protocols, and from visual mathematics like diagrams.
Focal Student 2 is an aspiring artist who is proud of his dad's career in graphic design. He is a passionate communicator, and he often does his classwork quickly and with many mistakes, so he benefits from group and class discussions to challenge common assumptions and misconceptions.
Focal Student 1 initially struggled to access this activity. Our host teacher began the work time by checking in with her to ensure she understood the directions for the activity, and that she had access to translation software should she need it. She drew a number line, struggled to label it, then erased it and waited for the whole-class discussion to reproduce another students' double number line solution. Her partner at her table group was absent for much of the work period, so she communicated very little with her table group during this task.
Focal Student 1
Focal Student 2 did not draw a diagram, table, or double number line to visualize the ratio. He recorded his answers in the sentence frames on his page, then erased and modified his answers during the whole class discussion when other students presented strong reasoning in support of a different answer. It is unclear which strategy he was persuaded by.
Focal Student 2
The Task
Given a rate, students will represent the rate using a diagram, ratio table, or double number line (of their choosing). Using their visual, they will try to find the unit rate.
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This task emphasizes student agency and authority through group work and whole-class discussion, in which students engage in mathematical discourse about ratio reasoning and compare different visual strategies. The teacher should remain neutral through the discussion and instead emphasize students' responsibility to one another to craft sensical arguments.
Reflection
This first cycle of lesson study was a valuable opportunity to research, reflect on, and implement some classroom routines to support student agency. However, as a brand new teacher, especially having started at my new school midway through the semester, every classroom routine felt new and experimental. It feels hard to say whether the PDSA Cycle routines were driving greater student agency and authority when I was implementing so many structures that were new to my students, including class discussions, open-ended tasks, and group roles. That said, I believe the PDSA routines are important, and they are research-supported, so I plan to continue using them. The "snapback" can support students in meeting expectations that students listen to each other and are accountable to one another for making sensible arguments. Additionally, I strive to practice greater neutrality while facilitating class discussions.
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As for the lesson study lesson itself, our results were mixed. In particular, one change that was meant to provide language scaffolding and accessibility actually intimidated students and decreased safety around the task. In an effort to provide language support, our host teacher introduced sentence frames for students to present their answers. The students had only ever seen sentence frames like this on tests where they were graded for accuracy, and in a debrief discussion, many students agreed that they thought their warm-ups were being graded like tests. As a result, they were less willing to experiment with different types of visual reasoning, and they overall felt nervous. Having strange adults in the room (us observing teachers) also impacted their sense of psychological safety. Going into the next cycle, we are going to plan more intentionally for safety in data collection, and introduce any new routines that are not the focal point of the lesson study well before the spotlight lesson, so that they do not interfere with data collection.
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Going into next cycle, I have a few potential areas of interest. I feel like one area of growth for me is my board work--how can I structure the visual presentation of information to help students learn and support them in literacy development (by taking good notes)? Additionally, I've been wondering about the balance of group work and individual work, and I'd be interested in researching about the organization, timing, and support for these different kinds of mathematical work. I'm also looking at shifting more authority to students by implementing a student-led "daily discourse," and I'm wondering how to implement that most effectively, especially with my sixth grade class.