Robin Fies stands at the blackboard in a small auditorium at Watertown High School in Massachusetts as 16 teenagers pass through the door and take their seats. In her 30 years of teaching high school mathematics, Fies has seen just about every classroom configuration possible, from lectures and rows to small groups to the open classroom. But today her precalculus class looks different than it ever has before. In addition to her usual group of high school juniors, she also welcomes 30 peers—colleagues from the Watertown High mathematics department as well as math teachers from four other high schools in the region and researchers from EDC and Boston University.
The lesson today is on combinatorics, or the study of combinations, permutations, and probability. Students begin the 58-minute class by working together to determine how many different possibilities exist when appointing a president, secretary, and business manager from a group of five students. They go on to figure the number of possibilities for a committee of three people to be chosen from a group of five students when none has a distinct title. As the guest observers look on and take notes, the students work through nearly a dozen such problems. They construct their own model of Pascal’s Triangle and consider how it can be used as a tool for solving additional combinatorics problems. Finally, they move from these paper-and-pencil approaches to derive a method for solving combination and permutation problems when the numbers involved are very large. The lesson ends with the students using the method they derived themselves to solve the following problem:
In the Massachusetts Megabucks lottery game, there are 42 numbers from which the player chooses any 6. The state then selects six of the numbers at random. The winning ticket(s) for the grand prize are those on which the player’s six numbers match the state’s six numbers. From the 42 numbers, how many groups of 6 are possible? [Answer: 5,245,786]
This lesson is part of EDC’s innovative professional development project, Lesson Study Communities in Secondary Mathematics. Based on the lesson study model popular in Japanese schools, the project has the dual aim of improving student learning in mathematics while enhancing teacher knowledge and skills. With funding from the National Science Foundation, EDC educators are working with 20 middle and high school mathematics departments in Eastern Massachusetts to test and refine the model for use in American secondary schools.
In a typical cycle of lesson study, a team of teachers meets on a regular basis to identify specific learning goals for their students, both for subject-matter content and for more general learning skills. The team then works together to craft a lesson that addresses those specified goals. “A team will often start with the question, ‘What’s something that’s hard for students to learn?’” explains EDC Project Director Jane Gorman. “Then they choose a topic that kids have trouble learning and try to figure out how to teach it better.”
For the team from Watertown High, the general learning goal was for students to become more independent problem solvers, part of the school’s larger mission. The specific content goals included differentiating among types of combinatorics problems (combination, permutation, or multiplication counting principle); using mathematical tools, like Pascal’s Triangle, to better understand combinatorics concepts; and using mathematical reasoning to solve combinatorics applications without relying on formulas. These goals also align with several outlined in the statewide mathematics standards.
“Lesson study gives teachers a chance to really dig into a subject together and think about what it means to learn that subject,” says Gorman. “High school teachers have a very long list of what has to get covered in their classes and they don’t have much opportunity to sit down with one another and say, ‘Okay, today I’m teaching algebraic equations, so what do we really want the students to understand about that?’ The structure and culture of high schools tend to isolate teachers from one another. They have few opportunities for extended discussions with colleagues about teaching and learning. Lesson study gives them that opportunity.”
Once a lesson has been developed, one member of the team teaches it to a group of students while the others observe. The group then reconvenes to share observations, evaluate the lesson, reflect on student learning, and develop a revised lesson plan to be shared with other teachers. Sometimes the cycle culminates in a regional open house, as was the case with the Watertown High combinatorics lesson, where teachers from many schools come to observe the lesson and participate in colloquia led by the lesson study group.
“We have teachers in our project—both new and veteran—who say they rarely have the opportunity to go into other classrooms and observe other teachers teaching,” says Gorman. “So they’re very excited by the chance not just to talk about what should work, or learn the theory, or get a new curriculum handed to them, but to actually see someone teach what they also teach and then talk about it.” One of the goals of the project is to learn whether the lesson study method holds promise for altering the culture of professional isolation among teachers in American high schools.
The emphasis of a lesson study observation is on student understanding rather than teacher performance. While observing, teachers are asked to direct their attention to the students in order to collect “data” on what students are learning—how they are solving problems, responding to questions, interacting with materials, relating to peers. This data is used to inform the discussion that follows.
classroom observation component offers “nonevaluative” feedback, says
Gorman. “Lesson study is built on the assumption that collecting
concrete evidence is an inseparable part of understanding and gradually
improving instruction,” she explains. “Dialogue among teachers is
traditionally tilted toward observation about what could happen, what
might happen, what should happen, with far less attention to what did
happen.” The lesson study approach also moves the conversation away
from a subjective evaluation of the teacher and toward something more
research-based and constructive—how well are students achieving
specified learning goals?
“Teachers are always observing students learning during the course of their day, but naturally they can’t do it to the extent that a group of observers can,” says EDC’s June Mark. “When I attend a research lesson, I focus on one group of students and watch how they grapple with the mathematical task at hand. The lesson study process allows for lots of eyes to gather data, but it also brings a lot of experienced teachers together to consider what the data means for planning lessons and teaching challenging content. The public aspect of the lesson also helps teachers become more articulate and deliberate about their own thinking and work,” she continues. “It gives them an opportunity to be explicit about what they do and why they do it. It also allows them to open up their work to examination and analysis.”
Teachers learning together
While an important goal of lesson study is improved student learning, the process of co-constructing lesson plans gives teachers an opportunity to enhance their knowledge of mathematics and to sharpen their teaching skills. Each of the lesson study teams in the project has a “coach” to help guide it through the process. Teams may also seek out additional expertise to help them with the mathematics itself. At Watertown High, the team worked with Glen Hall, a mathematician from Boston University, who is collaborating with the high school on another EDC project called Focus on Mathematics.
“The teachers decided to use this lesson study cycle as an opportunity to learn about combinatorics more deeply themselves,” says Mark. They decided to kick off their new cycle by inviting Hall and Al Cuoco, a mathematician from EDC, to meet with the group. “Al and Glen worked with the Watertown team on the important mathematical ideas behind combinatorics,” says Mark. “They helped the team draw relationships among combinations, Pascal’s Triangle, and binomial theorem.” In addition, Hall regularly participates in the department’s monthly meetings, offering a mathematical perspective on their ongoing work. After the recent open house, Hall also led a session on combinatorics for the participating teachers.
“Working together like this is the best professional development I can provide my staff,” says Chuck Garabedian, coordinator for mathematics at Watertown High. “Part of my job as department coordinator involves walking around and observing the faculty at work. A couple of the teachers here are as good as you’ll ever see, and I get the opportunity to learn by watching them. I want all of the faculty to have the opportunity to watch what they do. Lesson study gives us that opportunity.”
Garabedian arranged for the department to use its monthly in-service early release day for lesson study as well as some paid professional development time after school.
The value of lesson study is not lost on Kari Brookshire, a second-year math teacher at Watertown. “There’s so much knowledge among teachers here. This is my chance to learn from them—how to organize a lesson, how to approach a problem,” she says.
The Watertown team’s five-day unit, the product of the department’s joint effort, will now be available to all the teachers participating in the larger lesson study community. “It’s a great resource for teachers,” says veteran teacher Karen Trenholm. “A single person would have to spend many hours putting a lesson like this together.”
Along with these practical appli¬cations, the project also has a research agenda. Throughout, the project has focused on three key research questions: (1) How does lesson study affect the prevailing, and often counterproductive, norms of privacy among teachers that are common in U.S. schools? (2) How does lesson study serve as a form of teachers’ professional development? and (3) How does lesson study meet the particular needs of U.S. secondary school mathematics departments? The research protocol includes baseline and follow-up surveys to all participants, interviews with 10 teachers, observation of the full range of lesson study activities, and case studies of three lesson study teams. The EDC team will produce a final report addressing the core research questions and the collection of case studies.
Originally published on September 1, 2005