Insights Into Algebra 1: Teaching for Learning
Mathematical Modeling Teaching Strategies: Lesson Study
When the Third International Mathematics and Science Study (TIMSS) was conducted in 1996, more than 60 percent of American math and science teachers who participated in the study reported that they never had the opportunity to observe or be observed by another colleague. In contrast, Japanese teachers observe colleagues in classrooms all the time.
Derived from the Japanese word jugyokenkyu, the term “lesson study” refers to the process by which a group of teachers cooperatively plan lessons and examine how students respond to these lessons. Lesson study is the favored form of professional development among teachers in Japan, and it is an emerging form of professional development in many parts of the United States.
What It Is
According to James Stigler and James Hiebert, authors of The Teaching Gap, the Japanese technique of lesson study is the most powerful method for changing classroom practice. During the process, teachers work collaboratively to:
- Identify an instructional problem to solve or goal to attain.
- Plan a lesson – including what activities to use, the order of activities, and perhaps even the questions to ask.
- Teach or observe the lesson.
- Discuss the lesson as it was taught – what worked well and what could improve.
- Revise the lesson.
- Teach the lesson again, if necessary, possibly using a different teacher in the group.
- Share the results of the discussions and observations with colleagues.
Listen to what Orlando Pajon has to say about the process of lesson study:
|Listen to audio clip of teacher
Transcript from Orlando Pajon
This lesson has been possible because of the collaboration of the math teachers here in our department, especially the ones that are working with the SIMMS curriculum. We do meet on a regular basis, and we plan lessons together. It’s been two years since we started the lesson studies, and it’s been very successful. During the whole working day that we have for professional development, for example, we gather together, we pick a topic, and we develop a lesson …Then we combine everybody’s ideas, everybody’s experiences, in order to get a very powerful and perfectible lesson. So this is not a lesson of my own; this is a lesson that has been created by a group of teachers. And if something goes wrong – which is supposed to happen, because it’s supposed to be perfectible – it’s not my fault. It’s everybody’s fault. We can always go back, and we can always revise it. And every time we revise it, for sure we are going to get a better lesson than the one we had the previous time.
Lesson study is not about making a perfect lesson. Rather, it is a formal process that allows teachers to work together, observe one another’s classrooms, examine their practices, discuss ideas, and learn from each other’s successes in order to become more effective educators. The following elements of lesson study are taken from the online article “Improving Student Learning One-Lesson-at-a-Time” by Jenny Sue Flannagan and M. Gail Derrick in ENC Focus:
Focusing the Lesson
Lesson study usually focuses on a broad, school-wide goal such as “independent thinking” or “love of learning.” The teachers help determine these broad goals and choose the specific topic of the lesson. The topic often relates to concepts that the teachers have observed are difficult for their students to understand.
Planning the Lesson
The teachers research the topic of the lesson, reading books and articles about the target concept. They collaborate to develop the lesson plan and present a draft to their peers for feedback.
Teaching the Lesson
One teacher from the team presents the lesson in his or her classroom. The other teachers observe the lesson very closely, taking notes on what the students and the teacher are doing and saying. They may document the lesson through video, photographs, audiotapes, and student work.
Reflecting and Evaluating
The group meets to discuss the lesson and their observations. The teacher who presented the lesson speaks first, outlining how he or she thinks the lesson went and identifying problems. The other teachers contribute their own observations and suggestions.
Revising the Lesson
Based on problems identified in the first presentation, the group revises the lesson. Changes are usually based on student misunderstandings that the teachers noticed during their observation. The group may meet several times to improve the lesson and prepare for a second implementation, although sometimes the teachers may decide not to re-teach the concept [or lesson].
Teaching the Revised Lesson
A teacher presents the lesson to a different group of students. The same person may teach the lesson a second time, or a different teacher may try it out. Often, all the teachers in the school observe the revised lesson.
Reflecting and Evaluating
The entire faculty participates in the second debriefing session, which may cover more general issues of learning and instruction. An outside expert working with the lesson-study group, such as the subject-area coordinator, also participates in the debriefing.
Teachers share the lessons they develop through this process, creating a bank of meticulously crafted lessons to draw upon for the future. The teachers may publish a report about their study, including the teachers’ reflections and a summary of group discussions. In addition, teachers from outside the school may observe teachers presenting the lesson.
Read what David C. Webb has to say about the process of lesson study:
Transcript from David C. Webb
What you may not notice in this video is that teachers have worked extensively to select good problems that can be used to launch activities and to engage students in modeling. The simulations would be an example of the activities the teachers might have selected … This planning is done beforehand by a group of teachers that commits to using this lesson with their students. One or several teachers will use this lesson with their students, and they’ll observe the lesson or the lesson will be videotaped, and the teacher will come back and reflect on the lesson as it was done with a particular group of students. And they’ll look for ways to improve upon it.
Lesson study differs from the planning that teachers already do in that it involves an observation component not generally associated with writing lesson plans. In addition, it may include the testing of new ideas, discussion of beliefs about learning, and reflection on what was observed in the classroom. Through lesson study, teachers learn from their practice, and they engage in developing and sharing professional knowledge.
Lesson study also differs from other collaborative activities in which teachers participate. Most importantly, it requires teachers to focus on a specific goal, namely improving their own understanding of student thinking. In addition, lesson study requires the interaction of teachers within the classroom during observations. This differs from traditional forms of professional development – college courses, all staff meetings, workshops, and conferences – because it allows teachers to discuss lessons in a tangible setting, involving students, rather than discussing instruction in the abstract, without students.
Read what teacher James Salazar has to say about bringing lesson study to a school:
Transcript from James Salazar
To begin a lesson study effort, it may be best to start with a small group of interested and dedicated volunteers. Requiring mandatory participation of all teachers in your school or department from the beginning may be overwhelming – both to you and to them – and it will likely lead to frustration and, ultimately, failure. Instead, have only a few teachers pilot a lesson study program. This can be a learning experience, and the pilot group can iron out many of the kinks before attempting a large scale program.
Be sure to inform the other teachers in the school as to what the lesson study program is and why it is happening. Keeping open lines of communication with other teachers, important school personnel, and the community will make it clear that the lesson study is aimed at improving student achievement. Time spent capturing the interest of school board members, district administrators, and parents will help when the program grows large enough to require additional funds for district wide implementation. Administrative support is crucial; at Bel Air High School, the participating teachers are released from the classroom for one full day per week to conduct lesson studies. Finally, let the students know what is happening. If they know that other teachers are observing their class for the purpose of improving their education, they will be receptive to the process and appreciate the effort.
Listen to what David C. Webb has to say about the culture of a school engaged in lesson study:
|Listen to audio clip of teacher educator
David C. Webb
Transcript from David C. Webb
What really struck me was the way teachers engaged in reflecting on the lessons they were observing in the video and the way they were open to discuss different ways to run the lesson in the future, or at least to ask a different assessment question or to use different numbers. Sometimes these changes that are offered are very subtle, but their impact on what happens in the classroom can be significant. For instance, when they choose to begin a pattern from a number 5, well, 5 is just as good a number as any – except if you are trying to halve, or do something where a pattern that initiates with an even number might come a little easier … Eventually, you might want students to work with messier numbers, but when it’s an initial investigation, you don’t want the arithmetic to override the mathematical reasoning. As teachers reflected on the assessment and the lesson, you got a sense that this was a department that took great care in what they were doing. Orlando was willing to share this with students – maybe to give them a heads-up as to why teachers may be coming and going in the classroom – but even more so that the teachers care about student learning and the teachers care about improving what they are doing.
List the ways in which teachers at your school share with one another their knowledge about teaching.
The Benefits of Lesson Study
According to the Iowa Association of School Boards, “The small amount of staff development that focuses on teachers’ instructional knowledge and skills is often not sufficiently rigorous or sustained to produce lasting on-the-job changes.” Recent research supports that assessment, suggesting that instructional change only happens when professional development is long term, sustained, and intellectually demanding. In addition, the content of professional development activities must focus on the material that teachers teach and on strategies for teaching it. Professional growth is most likely to occur when teachers learn together, assist one another with lesson planning, focus on improving the quality of student work, and work collaboratively to resolve the daily challenges of teaching and learning.
Read what David C. Webb has to say about the benefits of lesson study:
Transcript from David C. Webb
Most teachers, especially in middle school and high school, are somewhat isolated. They do not have an opportunity to engage their colleagues in improving lessons or discussing what another perspective might be on the same topic, or other strategies that could have been used with the same lesson that other teachers have found to be successful. Very simply, lesson study is about collaboration that offers teachers a chance to reflect on designing lessons, doing the lessons, how the lessons were executed, either observing it in the class or on video. Through that reflection, there’s this opportunity for teacher inquiry into what makes a good lesson. Did we reach the objectives? Are students learning the mathematics we want them to learn? How can we approach this differently? By having teachers collaborate in lesson study, it also allows an opportunity for the different perspectives that teachers bring to a department to come to the fore – and that’s critically important, because as you are working out your own lessons, you bring only one perspective, and [you] may not be able to capture other ways to turn the lesson on its head, to make the lesson more accessible to students, to make some subtle changes to the way the problem is designed, to bring about more impact and greater student achievement.
Many teachers believe that lesson study may not be worthwhile because the limited amount of time they can devote to it will only allow them to produce a few lesson plans a year. Teachers often ask, “With so much to teach, is it really wise to devote so much time to just a sliver of content?” The answer is yes. Lesson study is not meant to serve as a venture in curriculum writing, and it is not expected that new lessons will be written for every day of the school year. As stated by Clea Fernandez and Sonal Chokshi in a 2002 article on lesson study for Phi Delta Kappan magazine, “Lesson study is not a vehicle for creating a library of tried and tested lessons for teachers to borrow from a shelf and import into their own classrooms. It is a process for creating deep and grounded reflection about the complex activities of teaching that can then be shared and discussed with other members of the profession.” By writing, observing, and revising just one lesson, the teachers involved in lesson study will be able to share their knowledge with one another.
Lesson study provides a forum in which teachers can bring forth all the good ideas they’ve learned at workshops and conferences or developed on their own. In addition, lesson study may be the mechanism for opening new lines of communication between colleagues and may serve as the starting point for further discussions. Although a lesson study group may only formally plan and revise one lesson a year, the ideas brought forth throughout the process may help with the planning and delivery of other lessons.
Read what David C. Webb has to say about lesson study in mathematics:
Transcript from David C. Webb
In any lesson that a teacher designs, there’s a limited amount of time that teachers have to engage students in developing new mathematics or revisiting topics, and good problems can reap great rewards for a teacher … Orlando and his colleagues put a great deal of thought and care into selecting good problems that will engage students to contrast linear and non-linear patterns … [This lesson is] a nice example of why good problems are well worth discussing with colleagues and why it’s difficult to construct good problems or even select good problems on your own. The various experiences that teachers bring [to] a math department really gives great cause to engage in lesson study or even just to collaborate to plan lessons. It’s much easier to find good problems when you are working with colleagues.
As noted by lesson study expert Patsy Wang Iverson, lesson study can serve as the basis for district wide staff development.
Lesson study should not be viewed as just another program to be added to a potpourri of professional development opportunities; neither should it be viewed as a replacement activity. Rather, lesson study can become the foundation of site-based professional development, which treats teachers as professionals and builds their capacity for lifelong learning. It can help integrate the variety of professional development activities, courses, conferences, and workshops that teachers attend individually or in groups, and which, up until now, have not provided much synergy. Lesson study can serve as the means by which teachers can share the knowledge they gain from their various professional development experiences and channel them into the collaborative research lessons they develop and then teach while assessing student learning and understanding. (From “Why Lesson Study?” Lesson Study Conference presentation, 2002)
Explain how a lesson-study group might foster dialogue about instruction among the teachers at your school.
Workshop 1 Variables and Patterns of Change
In Part I, Janel Green introduces a swimming pool problem as a context to help her students understand and make connections between words and symbols as used in algebraic situations. In Part II, Jenny Novak's students work with manipulatives and algebra to develop an understanding of the equivalence transformations used to solve linear equations.
Workshop 2 Linear Functions and Inequalities
In Part I, Tom Reardon uses a phone bill to help his students deepen their understanding of linear functions and how to apply them. In Part II, Janel Green's hot dog vending scheme is a vehicle to help her students learn how to solve linear equations and inequalities using three methods: tables, graphs, and algebra.
Workshop 3 Systems of Equations and Inequalities
In Part I, Jenny Novak's students compare the speed at which they write with their right hands with the speed at which they write with their left hands. This activity enables them to explore the different types of solutions possible in systems of linear equations, and the meaning of the solutions. In Part II, Patricia Valdez's students model a real-world business situation using systems of linear inequalities.
Workshop 5 Properties
In Part I, Tom Reardon's students come to understand the process of factoring quadratic expressions by using algebra tiles, graphing, and symbolic manipulation. In Part II, Sarah Wallick's students conduct coin-tossing and die-rolling experiments and use the data to write basic recursive equations and compare them to explicit equations.
Workshop 6 Exponential Functions
In Part I, Orlando Pajon uses a population growth simulation to introduce students to exponential growth and develop the conceptual understanding underlying the principles of exponential functions. In Part II, a scenario from Alice in Wonderland helps Mike Melville's students develop a definition of a negative exponent and understand the reasoning behind the division property of exponents with like bases.
Workshop 7 Direct and Inverse Variation
In Part I, Peggy Lynn's students simulate oil spills on land and investigate the relationship between the volume and the area of the spill to develop an understanding of direct variation. In Part II, they develop the concept of inverse variation by examining the relationship of the depth and surface area of a constant volume of water that is transferred to cylinders of different sizes.
Workshop 8 Mathematical Modeling
This workshop presents two capstone lessons that demonstrate mathematical modeling activities in Algebra 1. In both lessons, the students first build a physical model and use it to collect data and then generate a mathematical model of the situation they've explored. In Part I, Sarah Wallick's students use a pulley system to explore the effects of one rotating object on another and develop the concept of transmission factor. In Part II, Orlando Pajon's students conduct a series of experiments, determine the pattern by which each set of data changes over time, and model each set of data with a linear function or an exponential function.