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HTH/MAIC Public Lesson Study #4 - 9th Grade - Lines of Best Fit

Updated: Jun 22, 2020

Over the 2018-2020 school years teacher teams from the Mathematical Agency Improvement Community (MAIC) conducted research lessons culminating (so far) in four public lesson study events. This post documents and memorializes the fourth lesson of the series – a 9th grade lesson on lines of best fit.

Why Lesson Study?

“Knowledge for teaching is embodied in instruction and is spread and refined as teachers watch and discuss practice” ~ Lewis & Hurd, 2011

In order to improve student learning, teachers need opportunities to observe student thinking and discuss teaching. Lesson study provides a structure for teachers to collaborate around pressing problems of practice and to engage in cycles of inquiry to explore the intersection of pedagogy and student learning. Engaging in lesson study supports educators in:

  • Developing pedagogical skills necessary to enact ambitious instruction in classrooms

  • Building the mathematical knowledge necessary to teach for student understanding

  • Crafting a communal vision of ambitious student-centered teaching for equity

What is a Research Lesson?

Lesson study can take many forms – quick lesson study cycles lasting 1-2 days to test a particular practice, or longer more in depth research lessons that explore more complex problems of practice. This post memorializes a 9th grade research lesson on lines of best fit.

A research lesson is driven by a pressing problem of practice such as "how can we develop students' ability to engage in rich mathematical discussions, including listening to, building on, and challenging each other's ideas?" To explore the problem, the research team develops a mathematical understanding goal, an equity related teaching goal, and a theory of action to guide their inquiry. The team then designs a research lesson by researching best practices, exploring the standards and situating the lesson within the common core conceptual progressions, choosing focal students and anticipating student thinking.

During a research lesson, a teaching team introduces students to new learning, and tests out a new lesson structure, pedagogical move, or task, and collects data to determine how it impacted student learning and the development of an equitable learning environment. The lesson is observed by two outside individuals: a content commentator, and an equity commentator. The commentators provide feedback to the team about the degree to which the lesson achieved the research team’s content and equity goals.

The Research Team

The cross-grade level research team included:

  • Cate Challen, 9th grade, High Tech High Media Arts

  • Timmy Ro, 10th/11th grade, High Tech High Media Arts

  • Mele Sato, 12th grade, High Tech High Media Arts

The team met for one hour a week for eight weeks to research and design the research lesson.

Public Lesson Study Event

The Pre-Brief

This lesson took place Nov 6th, 2019 at High Tech High Media Arts. The following videos document portions of the pre-brief, the research lesson, and the post-lesson debrief, including commentary from the equity and content experts.

The Research Question & Theory of Action

The team developed a research question based on observations of student behavior and content knowledge from their contexts. How can we help students use the ideas of others to connect mathematical concepts in order to become adaptive mathematical experts?

The research team shares their research question and theory of action.

And also developed a theory of action to guide their inquiry cycles:

If we as teachers design cognitively demanding, open-ended tasks that warrant multiple mathematical strategies and perspectives, in a classroom where student discourse is the primary mode of learning, then students will be expected to make connections across different ideas and concepts, resulting in flexible mathematical thinking.

The Context

Next the team shared the context for the lesson. What students were exploring in their current unit.

The research team shares the context for the problem.

Do the Math!

We have found it helpful to have the audience members attempt the math problem themselves before watching the lesson. This supports audience engagement and understanding as students work through the task. We encourage you to try it too

The math task!

The Mathematical Understanding Goal & Anticipating Student Thinking

Learning from a previous research lesson highlighted the importance of having a concrete mathematical understanding goal. The specificity of the content understanding goal has an impact on what can be learned from the research lesson. The more specific the content understanding goal, the easier it is to determine whether students meet the content goal for the lesson. A common misconception among educators is that a standard is a mathematical understanding goal, the truth is that one standard is often made up of many different, critically important understandings that need to be "chewed" on multiple times for students to develop a robust schema/understanding for the concept. In this case the team was focused on the second part of this mathematical understanding goal: Students will interpret and compare lines of best fit, including making sense of the slope and intercept of a linear trend-line in the context of data.

The research team also anticipated how their focus students might approach the task and share the connections they hoped to spark during the lesson.

The team shares the mathematical understanding goal and the anticipated student thinking for the lesson.

The Equity Goal

The team also generated an equity goal for their lesson. Equity goals relate to the effectiveness of the classroom learning culture and patterns of student participation.

The team's equity goal: Students will listen to, value, and build off of each other’s ideas.

Mathematical Practices

The Common Core outlines key mathematical practices that students should be developing to build their mathematical identities.

The research team outlined the practices focused on during this research lesson.

Data Collection & Assessing Student Thinking

For data collection, each team member (other than the presenting teacher) closely observed the thinking of one of their focus students throughout the lesson. By tracking how their focus student experienced the problem and what factors contributed to their learning. What did they think about during the independent work time? How did questions/ideas from peers shape their thinking? Any "a-ha" moments? These observational data are examined along with student work samples by the team during their debrief session.

The Research Lesson & Observation

Warm-up & Task Launch

The lesson followed a notice & wonder warm-up followed by a launch, explore, discuss' lesson structure.

The notice and wonder warm-up with Cate's line of best fit.

The lesson task launch.


During the explore phase of the lesson the students worked independently to make sense of the problem, followed by working as a group to share ideas and make additional headway on the problem.

The explore phase of the lesson.


During the discuss phase of the lesson, groups are asked to present their thinking and the students engage in a whole class discussion about the different strategies.

The groups share their thinking and discuss the different strategies.

The research team also included a 'Go Meta" portion of the lesson where students were asked to reflect on what they think Google is doing to create its line of best fit, and what questions do they still have.

The class closing and reflection discussion.

The Debrief

After the lesson, the presenting teacher has the opportunity to reflect first on how the lesson went. Afterward, the research team engages in a discussion about how well they met their mathematical and equity goals and what they learned about their research question and theory of action based on the observations and data collected from focal students. In this public lesson Cate reflected publicly and the team debriefed in a follow-up meeting, capturing their reflections in the memorialization document.

A research lesson debrief also includes commentary by an outside expert content commentator and equity commentator. In this lesson Dr. Elizabeth (Libby) Butler from San Diego Unified School District served as the content commentator and Bryan Meyer from Escondido Unified School District served as the equity commentator.

Cate's Reflection

As the presenting teacher, Cate reflects on how the lesson went and possible next steps.

Cate reflects after the lesson.

Key Learnings for the Team

The team identified three key learnings about teaching and learning from the research lesson:

  • The importance of all students participating when sharing the group's thinking

  • The need to incorporate more opportunities for language learners to make sense of the mathematics in their native or preferred languages

  • A new wondering the team had was how to push students to build on their initial response to "something you understood or a question you have" without inducing anxiety?

Dr. Elizabeth Butler's Content Commentary

Dr. Butler shares her comments about the mathematical content and possible next steps to promote student learning.

Expert content commentator, Dr. Butler.

Bryan Meyer's Equity Commentary

Bryan Meyer comments on the team's equity goal and poses a thought provoking question to the audience.

Equity commentator Bryan Meyer.


Daisy Sharrock works at the Center for Research on Equity and Innovation at the High Tech High Graduate School of Education, and is part of a Student-Centered Learning Research Collaborative-sponsored research team that is currently engaged in the following study: Leveraging the Power of Improvement Networks to Spread Lesson Study. Read more about their current study here. We are grateful to JFF, KnowledgeWorks’, and the Student-Centered Learning Research Collaborative and its funders for their support. Learn more at

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