Our Lesson
The goal of this lesson was to provide students with real-world context for their work with trig ratios and right triangles. In doing so, we sought to help students reimagine who it is that, to borrow from the phrasing of a student question, "just invented math." The process was twofold: on the one hand, we wanted students to understand that ancient peoples (like the Egyptians) were discovering math long before these Europeans. In addition, however, we wanted to leverage that new historical understanding into an understanding of students themselves as mathematicians.
As we planned this lesson, it became clear that it would take more than one class period. As a result, Javier (the host teacher of this cycle) taught a pre-lesson in which students uncovered assumptions about who it is that "created math," began to think about how right angles apply to real world scenarios, and co-created a list of the six possible ratios of a triangle's sides.
In the second lesson (during which our team observed focus students in person), student learned about how Egyptians discovered cotangent in order to measure the pyramids, and began to see themselves as mathematicians. The latter realization was rooted in their own discoveries of the various ratios between a triangle's sides from the first lesson. The Egyptians' discovery was also tied to students' own experiences when using the pyramid-shaped building located in Miramar as an example for their own use of cotangent and tangent.
As we planned this lesson, it became clear that it would take more than one class period. As a result, Javier (the host teacher of this cycle) taught a pre-lesson in which students uncovered assumptions about who it is that "created math," began to think about how right angles apply to real world scenarios, and co-created a list of the six possible ratios of a triangle's sides.
In the second lesson (during which our team observed focus students in person), student learned about how Egyptians discovered cotangent in order to measure the pyramids, and began to see themselves as mathematicians. The latter realization was rooted in their own discoveries of the various ratios between a triangle's sides from the first lesson. The Egyptians' discovery was also tied to students' own experiences when using the pyramid-shaped building located in Miramar as an example for their own use of cotangent and tangent.
Our Goals
As we planned this lesson, the following goals served as our team's "North Stars"
Equity Goal:
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Our ResearchThis lesson was also informed by our team's research on the following theory of action (a sort hypothesis for change in our teaching practice).
Theory of Action:
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Our Students
Focus Student 1
Teacher Notes
Observation Notes (Observation done pre-lesson study)
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Focus Student 2
Teacher Notes
Observation Notes (Observation done pre-lesson study)
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Day One:
Pre-Work
The Lesson
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Participation
My focus student's participation was tracked with an emphasis not only on verbal participation, but also on "quiet" participation.
Focus Student 1
There were a couple of particularly telling moments in the data collected on Focus Student 1's participation.
- While FS1 largely participated nonverbally, she did participate every time she was addressed, and clearly tracked the teacher and referred to the board regularly throughout the lesson. In a longer study of her participatory patterns, I would love to see how continuing to bring her voice into the space might help her gain confidence to ask more than one question voluntarily and bravely (as she did in this lesson).
- My second noticing was less positive. At the end of the lesson, FS1 wrote down a guess for what information they would need to solve the problem on her paper. After hearing two of her peers give the same answer, she erased what she had written and, when asked by the teacher if she had any other ideas, said, "I agree." In a longer study, I would love to see how we might create structures that uplift diverse answers and give students like FS1 more confidence to disagree publicly.
Final Thoughts
At the end of this lesson study cycle (and at the end of this year of lesson study overall) I find myself with four main takeaways:
All of this, combined with my research, has left me feeling invigorated for the future of my classroom. I'm excited to make quiet participation public and bring new social-emotional group roles into my classroom. I'm excited to continue digging into my teaching practice and unearth new ways to make our classroom community a place in which care and critique go hand in hand.
- Make everyone's voice heard. Throughout the lesson, Javier offered opportunities for all students' voices to be heard. As we make students' quiet thinking and participation public, it is also meaningful for each student to hear their own voice fill the classroom.
- Cite your sources (the students!). Making participation and thinking public is critical, and it doesn't need to be something extra. In fact, our lessons are always based on student thinking from the previous lessons—why not simply build this into our teaching transparently?
- Be a little silly. There was a moment during the lesson where Javier asked students who invented math the day before. (The answer was each and every one of them.) As they pointed to each other, he turned to a student and, referencing their noticings of the demographics of mathematicians from the previous day, asked, "Are you an old white dude?" Laughter ensued, but it also drove home a simple idea: math is not just for "old white dudes!"
- Work with others. This lesson study cycle in particular has left me excited for collaboration in the future. It has of course reinforced the connections between history and math, but it has also pushed me to think about math as an opportunity for deep thinking, and figuring out how to (as Dr. Taylor said in my interview with him) "do a lot with a little information."
All of this, combined with my research, has left me feeling invigorated for the future of my classroom. I'm excited to make quiet participation public and bring new social-emotional group roles into my classroom. I'm excited to continue digging into my teaching practice and unearth new ways to make our classroom community a place in which care and critique go hand in hand.