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Influence of Lesson Study on Teachers’ Mathematics Pedagogy

  • Jo Clay Olson
  • Paul White
  • Len Sparrow
Chapter

Abstract

The experiences of five U. S. elementary teachers who engaged in lesson study to investigate the teaching and learning of mathematics were quite different. Three teachers accepted the challenges highlighted by the lesson study process. They reflected on and changed their mathematics teaching practice in fundamental ways. The other two teachers rejected the challenges and maintained their traditional pedagogy for teaching mathematics. System-wide requirements (e.g., state testing) constrained the development of these two teachers while the ability to personalize insights and critically reflect became catalysts for professional growth of the three who changed.

Keywords

Social Capital Professional Development School District Cognitive Demand Mathematical Thinking 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Lesson study enables teachers to use their desire to understand their students’ learning on a journey of self-discovery. In lesson study, teachers set their own goals, create a research lesson, investigate student learning, and interpret students’ responses from their own perspectives and knowledge. This process provides teachers with opportunities to collaborate, reflect, and analyze student responses. Thus, lesson study encourages teachers to develop a continuous practice of inquiry in which they investigate student thinking and refine their own practice.

Teachers have multiple opportunities to gain insights about teaching and learning while participating in lesson study. These insights are seen as valuable by teachers (Fernandez and Yoshida 2004; Lewis et al. 2004; Southwell and White 2004; Wiburg and Brown 2007). But, do teachers integrate these insights into their daily practices? Specifically, two questions were posed:
  1. 1.

    In what ways do teachers integrate insights into classroom practice gained from lesson study professional development?

     
  2. 2.

    What aspects of lesson study promote teacher reflection, collaboration, and change in classroom practice?

     

Ten teachers joined one of three lesson study groups during the beginning of the school year. Two groups completed their lesson study and one group abandoned it while planning their research lesson. From this context, it was possible to contrast the learning of teachers who completed a research lesson with those that did not and identify aspects of lesson study that supported teachers’ professional development. In this paper, reasons that led the one team to abandon lesson study are outlined. In addition, factors that both encourage and discourage teachers’ involvement in the lesson study process are noted.

Context of Lesson Study

Over 200 elementary teachers and administrators in a Midwestern school district sought support from the mathematics education faculty in a large university as they phased in Investigations in Number Data, and Space ([Investigations], Akers et al. 1997), a reform elementary mathematics curriculum that was aligned with both state and national standards. The goal of the collaboration between mathematics educators and the school district was to increase student achievement in mathematics by improving the district’s K-5 mathematics program and grade-to-grade coordination through a three-year grant (PRIME).

The teachers participated in PRIME by attending one-week summer institutes, monthly professional development sessions during the school year, and coaching sessions in their classrooms to discuss pedagogy. The teachers requested that the leadership team create a framework for continued professional development that supported mathematics reform after PRIME. A Leadership Institute was created in response to these concerns and all the teachers participating in the project were invited to submit an application to join.

The Leadership Institute provided teachers with the skills needed to facilitate discussions about the teaching and learning of mathematics and promote educational reform. Ten teachers representing different elementary schools volunteered to participate in the Leadership Institute and also to facilitate professional development sessions for teachers in their school.

The institute was broken into three segments. Segment 1 was a summer institute which met for one week in the summer. Teachers became familiar with research on leadership, learning, and different models of professional development. The teachers selected lesson study as a professional development model after watching the video, Can You Lift a 100 kilograms (Lewis 2000). They believed that lesson study would allow them to investigate student cognition in a meaningful way and serve as a model that would engage their colleagues. Three lesson study teams formed along grade level bands. Teachers engaged in lesson study during segment 2. Monthly meetings were held after school for six months in which teachers planned, taught, and reflected on a research lesson. Segment 3 spanned the next 12 months. It focused on developing teachers’ skills, planning professional development, and facilitating discussions with peers.

Theoretical Framework

One characteristic of lesson study is teacher collaboration. Within a collaborative group, individuals interact with each other through words, gestures, and written symbols. Symbolic interactionism is a perspective from which researchers can interpret how interactions between people create meaning through observations, experiences, and words (Kuwabara and Yamaguchi 2007). Knowledge grows out of interactions between people and it is the interplay between personal and social meanings that creates new ideas. This knowledge is dynamic and changes as individuals create and reconstruct it. The character of verbal language changes as people assimilate new ideas (Kumpulainen and Mutanen 2000).

From the perspective of symbolic interaction, examining the teachers’ discourse provides a window into their thoughts. Boden (1990) and Kumpulainen and Mutanen (2000) found that changes in classroom discourse indicated a shift in a teacher’s interpretation of reform mathematics. The discourse among teachers during the Leadership Institute and between each teacher and her students during classroom observations were analyzed for new patterns of interaction. Thus, changes in the teacher’s discourse were interpreted as a change in pedagogy.

Methods

To describe how lesson study influenced teachers’ practices, a research design using case study was created (Merriam 1998). Data collected included (a) video recordings of instruction, (b) audio recording of instruction and lesson study meetings, (c) interviews, (d) teacher reflections, and (e) field notes of classroom observations and Leadership Institute meetings. Transcriptions were made of the recordings. Data were analyzed using a qualitative program (winMAX, Kuckartz 1998) that allowed hierarchical coding to characterize teachers’ practices before beginning, during, and after lesson study. Data analysis was compressed using conceptual and time-ordered matrices to identify patterns that indicated change of practice (Miles and Huberman 1994). These changes were characterized and compared to lesson study activities to describe how the lesson study activities influenced teachers’ growth. Observations continued for an additional 12 months to determine whether changes in teachers’ practices were sustained.

The purpose of each research lesson was to create opportunities in which teachers could ask students questions that would uncover misconceptions and identify the point when a problem-solving strategy led students to an erroneous solution. Two frameworks, namely the tasks’ cognitive demand (Stein et al. 2000) and secondly question types (Driscoll 1999), were used to create problems and hypothetical questions for the research lesson that would engage students in the exploration of mathematical ideas and the articulation of their thinking.

Case Study Teachers

Five elementary teachers (Table 1) who were members of the Leadership Institute volunteered to participate in a case study. The other five teachers in the Leadership Institute participated as team members and an analysis of their practices was not investigated.
Table 1

Case study participants in the Leadership Institute

Participant

Prior experience

Characteristics

Place Value lesson study (three teachers in the team)

Pat grade 1

1st and 4th grade for 12 years

· Passionate about deepening her mathematical content knowledge

· Enjoyed discussions about mathematical thinking

· Principal was surprised at her interest in math and leadership

Paula grade 2

1st and 2nd grade for 16 years

· Limited to procedural knowledge

· Hoped to gain insights about children’s problem-solving approaches

· Principal designated her as a model for literacy instruction

Count-Back-Change lesson study (three teachers in the team)

Courtney grade 3

K and 3rd grade for 30 years

· Limited mathematical understanding

· Wanted to ask new kinds of questions

· Principal provided limited support

Subtraction Regrouping lesson study ® Curriculum Guide (four teachers in the team)

Rachel grade 4

4th grade for 15 years

· Conceptual and procedural knowledge was built from science applications

· Wanted to expand her expertise in math

· Principal gave her extensive praise for her dedication and teaching

Rose grade 4

3rd and 4th grade for 20 years

· Procedural knowledge was connected to different representations

· Wanted to meet the wide range of students’ abilities in mathematics

· Principal recognized her as an outstanding teacher

· Granted release time and rarely taught mathematics

Findings and Discussion

Two phases are used to describe how lesson study influenced teachers’ pedagogy. First, the initial lesson study experience for the three teams of teachers is described. Second, the lesson study experience of the two teams who completed their research lesson is outlined followed by a description of the group who abandoned their research lesson. These descriptions provide the context for contrasting the pedagogy of teachers who completed their research lesson with those who did not. The changes are characterized through illustrative examples and then linked to specific aspects of lesson study that supported teachers’ professional development.

Initial Lesson Study Experience

The three lesson study teams began their lesson studies simultaneously at the beginning of the new school year with the goal of uncovering students’ conceptions about a mathematical idea to help teachers develop a practice of inquiry. This required teachers to shift their focus from step-by-step teaching actions to the priority of student thinking. The teachers were familiar with professional development in which they met to discuss mathematical concepts that students struggled with and then planned instruction that would support student learning. The lesson study experience went beyond their previous experiences as a catalyst for teachers to become coresearchers with the facilitator to investigate student thinking.

At the second lesson study meeting, teachers were confronted with a choice: becoming a coresearcher or being a recipient of research. Two of the lesson study teams decided to investigate students’ mathematical thinking as coresearchers and the other team decided to create a 4th-grade curriculum plan. Reasons for these choices are discussed after a description of the two teams that completed their research lesson.

Completing the Research Lesson

Three months later, a teacher in the Place Value (1st/2nd grade) and Count-Back-Change (3rd grade) volunteered to teach the research lesson. The problem that the 1st/2nd-grade team used was, “There are 36 cupcakes. We want to put 10 cupcakes in each box. How many boxes would we need to fill? How many extra cupcakes?” This problem allowed them to investigate how young students thought about place value. The problem that the 3rd-grade team asked was, “Use a count-back-change method to find the change for a snack that costs 65 ¢ if you give the clerk one dollar.” Their goal was to investigate how students made change, but they immediately began instructing students how to use the count-back-change strategy.

During the research lesson, the teachers moved around the classroom to observe what students were doing at their desks and asked the students questions to find out why they used a particular approach to solve the problem. The interaction here was a deliberate strategy to help teachers gather more close-up evidence of student thinking, providing a necessary experience enabling teachers to focus on student learning during a lesson observation. After observing the lesson and interacting with students, the teachers met with the facilitator for debriefing.

The Place Value and Count-Back-Change teams discussed the difficulties that students had using manipulatives to solve the problems designed for the research lesson. The facilitator recognized that it was difficult for the teachers to talk about students’ learning and misconceptions without immediately creating an intervention to remedy their limited understanding. After the initial debriefing, she selected a portion of the videotapes and transcribed the interactions for each team to prompt deeper reflection. The teachers’ in each team watched their videotape and classified the types of questions that they asked. Through this analysis they realized that many of their questions were “leading” the students toward a preferred solution strategy rather than uncovering students’ conceptions related to the content. As a result, the Place Value team decided to revise and reteach the lesson to further investigate student thinking. The Count-Back-Change team shifted their goal from investigating student thinking to designing a lesson to teach a particular counting procedure. Thus, reteaching of the lesson was not considered worthwhile by the team because the purpose of the lesson had changed.

This change of goals created a set of challenges and discomfort for the Count-Back-Change team. The inconsistency between wanting to let students use their own strategy and believing in the usefulness of a preferred procedure created tension. This clash between new knowledge and long-term beliefs is felt by many teachers as they adapt mathematics reform recommendations into their pedagogy. The Count-Back-Change team wanted to follow the reform recommendations, but they found themselves using the pedagogy of lecture, telling, and showing students a particular procedure that seemed disconnected to students and their ways of thinking.

The Count-Back-Change team struggled to resolve their new knowledge with a set of beliefs built on a history of experiences. The teachers preserved the procedure that did not make sense to students. Lesson study helped them notice the tension and created an opportunity to examine it. The 3rd-grade teachers discovered a conflict between knowing and believing. In spite of not resolving the conflict, the experience prompted Courtney to change her pedagogy by adopting an inquiry stance while working with her students.

Outside Japan, lesson study frequently takes different forms, revising and reteaching a lesson is an example of this variation. In this study, the choice of whether to repeat the lesson was left to the two teams. There are multiply ways that lesson study can be enacted to meet the specific needs of teachers and professional development goals.

Abandoning the Research Lesson

The 4th-grade teachers wondered why so many of their students did not remember how to regroup when subtracting and formed a team to investigate regrouping. In contrast to the other two lesson study teams, these teachers were ready to teach a lesson on regrouping at the end of the first lesson study meeting. The facilitator intervened and suggested they give a problem (Fig. 1) to their students and gather students’ responses. The teachers discussed the problem and anticipated that many students would say that problem A was bigger than B. The facilitator asked why they thought that and the teachers responded that students would subtract the tens without regrouping.
Fig. 1

Suggested problem for 4th-grade students to consider

One vocal teacher (Rose) became quiet and withdrawn while the other teachers talked about why students might subtract the tens without consideration of the units. The withdrawal of Rose signaled that she did not value the discussion, and investigating student thinking was a diversion from her goal: design a lesson to teach regrouping. The session ended with three other teachers agreeing to pose the problem to their students and collect their responses for the next meeting.

The next lesson study planning meeting began with Rose forging ahead with planning the regrouping lesson. The facilitator again intervened and asked how students responded to the problem. Rose spoke first, “We didn’t ask them. We knew what would happen and we didn’t have time.” The facilitator reemphasized the importance of taking advantage of this collaborative time to investigate why students forgot the procedure. The team finished planning their lesson to teach regrouping by the end of the session.

These 4th-grade teachers felt pressure to “get the kids ready for the state test” and creating a lesson to reveal students’ conceptions was moving too slowly. They sought quick solutions to improve students’ test scores on the mandated test. Investigating students’ conceptions was valued if it could be done quickly and a remedy was available to “fix it.” Simply put, lesson study took too much time, distracted them from “getting through the curriculum,” and managing the three sets of materials that they had to work with (Investigations, textbook, and supplemental activities). They abandoned lesson study and began organizing the materials into a curriculum guide.

Two months later, the facilitator noticed that many of the activities placed in the guide by the 4th-grade teachers could be described as “drill and practice.” She asked the team to analyze the tasks using the task’s cognitive demand (Stein et al. 2000) and determine whether each task met the projects’ objective, to develop the mathematical thinking of students, before including the activity in the curriculum plan. The cognitive demand of tasks characterizes problems in four ways: Memorized (reproducing previously learned facts, rules, or definitions); Procedures Without Connections (using a specific procedure without explanation); Procedures with Connection (linking procedures with conceptual ideas); Doing Mathematics (analyzing the problem and constraint to create solution strategies from which mathematical conversations emerge). Tasks that rely on memorized information have the lowest level of cognitive demand while tasks that require students to do mathematics have the highest level of cognitive demand.

To clarify the levels of cognitive demand, a page from the textbook was selected by the teachers. The facilitator made the mathematical concepts in the lesson explicit and classified it as a procedure without connections. Problem solving was not evident and it did not represent the kind of activities that met the goals for the professional development project. The facilitator encouraged the 4th-grade team to use the cognitive demand framework as a filter before including any activity in their curriculum guide.

After a short discussion, the team characterized the next activity as a task with high cognitive demand even though it was similar to the lesson just discussed. Rose explained, “Students [can be asked to] describe a real life situation for which a given mathematical equation could be used to solve the problem.” Rachel continued, “This type of task has high cognitive demand because there are multiple answers and it requires complex and non-algorithmic thinking.” Rose added, “It develops students’ mathematical thinking because it required students to solve their own problem.” The teachers decided that all of the textbook pages could be classified as high cognitive demand by asking students to create a situation for one of the exercises. Thus, the team avoided the opportunituy to critically consider the type of learning opportunity created by the lessons.

Teachers’ Pedagogies

The five teachers who began lesson study were dedicated and thoughtful about instructional decisions and students in their classrooms performed well on standardized tests. They asked questions that led students to expected responses and their discourse followed a traditional IRE pattern in which the teacher initiates a question, listens to a response, and evaluates the response (Cazden 2001). Thus, the teachers’ pedagogies were similar (Table 2).
Table 2

Initial pedagogies of the case study teachers

Teacher

Questioning

Discourse pattern

Task’s cognitive demand

LS participation

1st/2nd grade: Place Value lesson study

Pat

Leading1

Traditional IRE

Procedures with and without connections

Fully

Paula

Leading

Traditional IRE

Procedures with and without connections

Fully

3rd grade: Count-Back-Change lesson study

Courtney

Leading

Traditional IRE

Procedures with connections

Fully

4th grade: Regrouping lesson study ® Curriculum Guide

Rachel

Leading

Traditional IRE

Procedures with and without connections

Withdrew

Rose

Leading

Traditional IRE

Procedures with and without connections

Withdrew

1 Questions that led students to an expected response

The pedagogies of Pat (Place Value lesson study) and Rachel (Regrouping/Curriculum Guide team) are now described to provide a more detailed look at how lesson study and the ensuing collaboration can result in two totally different professional learning experiences.

Initial Pedagogies of Pat and Rachel

Prior to beginning lesson study, Pat and Rachel had many similar practices. They mainly designed lessons to help students learn procedures which resulted in teaching that required a low cognitive demand. Both were confident teachers and espoused similar beliefs not necessarily reflected in what happened in the classroom. Pat stated, “I present situations to students, question them, and let them explore.” Rachel claimed she pushed her students “beyond finding an answer and stopping. We try to find if there are other possible answers, why or why not.”

Pat’s Initial Pedagogy

Pat’s 1st-grade mathematics instruction was divided into two segments, calendar math and class instruction. At the beginning of the project, Pat’s pedagogy was characterized by the following three qualities: (a) questions were asked to help learn procedures for describing the passage of time during a calendar math, (b) questions were asked for helping students understand the procedure to solve problems during class instruction, and (c) students were given time to respond to questions.

During calendar math, a child put one straw to a jar to represent each school day. When ten straws were collected in the jar marked ONES, they were bundled with a rubber band and placed in a container marked TENS. The following excerpt from October is typical of Pat’s calendar math and provides an illustrative example of her questioning.

153: Pat

How many days have we been in school?

154: Jeffrey

Thirty-nine.

155: Pat

Thirty-nine, very good. How many ones do we have?

156: Ellen

Nine.

157: Pat

How many ones are in 39?

159: Brandon

Nine.

160: Pat

Nine. How many tens do we have? Chris

161: Chris

Ten.

162: Pat

What was my question?

163: Chris

Three.

164: Pat

What was my question?

165: Chris

(Pause 4 seconds) How many tens is there?

166: Pat

Yes, how many tens do we have?

167: Chris

Three.

Pat began mathematics lessons with calendar math and she asked the same set of questions (lines 153, 155, 157, 160). The cognitive demand was low as students were focused on reproducing an expected response. In this example, Pat elevated the task to Procedures with Connections when she asked the student to restate the question (line 164) before she accepted his response (line 167). Throughout this episode and other interactions, Pat maintained control of the classroom discourse using the traditional IRE discourse pattern.

Rachel’s Initial Pedagogy

Rachel’s 4th-grade mathematics instruction was divided into two segments, warm up and problem solving. At the start, Rachel’s pedagogy was characterized by three qualities: (a) questions were asked to make connections between representations and mathematical symbols, (b) questions during class instruction helped students create and practice definitions, and (c) students were given time to respond and self-correct themselves. She typically presented a task to elicit students’ prior knowledge at the beginning of each mathematics lesson. The following excerpt illustrates both her introduction and the interactions that she typically had with students.

Rachel held up several rectangles that represented halves of a square. Students taped two rectangles together to form a square that represented one whole. As they taped the rectangles together, Rachel asked a series of questions that led the students to a correct symbolic representation.

28: Rachel

How many halves are here (pointing at one rectangle)?

29: Marie

One.

30: Rachel

And here (pointing a rectangle next to the first rectangle)?

31: Marie

One.

32: Rachel

How many altogether?

33: Marie

Two.

34: Rachel

Can you write 2 halves? (Pause 5 seconds) The whole is divided

35

into how many parts? (Pause 4 seconds) Who can help?

36: Steven

Two.

37: Rachel

Two, that’s the denominator. And what is the numerator?

38: Mark

Two?

39: Rachel

That’s right. Can you write that (to student 1). That’s right, put a 2

40

for the denominator and a 2 for the numerator.

Rachel had selected an activity that provided an opportunity for her to assess students’ knowledge and review definitions or procedures. The cognitive demand was low as students were focused on reproducing an expected response (lines 29, 30, 32). When Marie was unable to write a symbolic representation of halves, Rachel asked another student to answer her question (lines 35 and 37). Rachel then led Marie to a correct symbolic representation of two halves. Like Pat, Rachel maintained control of the classroom discourse using a traditional IRE discourse pattern.

Pedagogies After Lesson Study

The three case study teachers who completed their lesson study changed unique aspects of their pedagogy (Table 3). These changes were revealed in their interactions with students and colleagues.
Table 3

Pedagogies after lesson study

Teacher

Grade level

Classroom questioning

Discourse pattern

Task’s cognitive demand

Place Value lesson study

Pat

1

Elicit mathematical thinking1

Reform: Student Led Discussions2c (Initially teacher dominated)

Procedures with connections and doing math

Paula

2

Prompt reflection1

Reform: Accept Alternative Strategies2a

(Initially teacher reinforced specific strategies)

Procedures with connections

Count-Back-Change lesson study

Courtney

3

Elicit mathematical thinking

Reform: Community of Inquiry2b (Initially teacher had sole authority)

Procedures with connections and doing math

Regrouping lesson study ® Curriculum Guide

Rachel

4

Leading

Traditional

Procedures with and without connections

Rose

4

Leading

Traditional

Procedures with and without connections

1 Questions that elicit mathematical thinking encourage students to make observations about patterns and create conjectures. Questions that prompt reflection encourage students to reflect, justify, and extend their thinking (Driscoll 1999)

2 Three types of discourse indicate reform pedagogy. They include (a) Accepting Alternative Strategies where students present their solutions and provide a justification, (b) Community of Inquiry where students pose problems and initiate topics for investigation, and (c) Students Led Discussions (Cazden 2001)

In contrast, the facilitator did not discern changes in the 4th-grade teachers’ pedagogy who did not complete the lesson study process. Thus, only one excerpt from Pat’s classroom interaction is presented to serve as illustrative examples of the teachers’ evolving pedagogies.

Changes in Pat’s Pedagogy

Pat’s pedagogy changed in three ways. She shared responsibility for leading the class discussion with children, she elevated the cognitive demand of the task during instruction, and she wrote the entire instructional task on the board instead of a short version that displayed only the important information. An example of how she shared authority with a child is presented as an illustrative example of the changes she made. Pat drew a tic-tac-toe grid on the board to represent a portion of a hundreds chart and placed a 23 in the center box (Fig. 2).
Fig. 2

Tic-tac-toe grid drawn by Pat (bold) and Jasmine’s response

Pat gave the marker to Jasmine. Jasmine wrote the number 24 in the square to the right of the 23. She then turned and faced the class.

324: Jasmine

Do you have a question? Raise your hand.

325: Student 1

How do you know that?

326: Jasmine

I counted, after 23 comes 24.

332: Student 3

How much tens is there in 20?

333: Jasmine

Uhm, the amount…(pause 14 sec)

334: Pat

Did you understand her question? She asked you how many tens

335

there were in 20.

336: Jasmine

There’s 20.

337: Pat

How many did you say?

338: Jasmine

Two.

Jasmine assumed responsibility for the classroom discourse and asked the class if they had any questions (line 324). One student asked her to justify the placement of 24 (line 325) and another child asked her to think about the meaning of twenty (line 332). During the interaction between students, Pat assisted her students as they developed their questioning skills (lines 334). This classroom discourse pattern was a departure from the traditional pattern with the teacher maintaining control. Pat shared authority with students by allowing them to lead the discussion and pose questions to each other.

Summary of Teachers’ Pedagogy After Lesson Study

Detailed analysis of the three case study teachers who completed the lesson study process changed their pedagogies over 18 months. Pat, Paula, and Courtney asked more purposeful questions and posed contradictions based on students’ responses. Before lesson study, the three teachers maintained control of the classroom discourse and frequently asked students “Why?” without knowing what to do with the information that they gained. After completing their lesson studies, these three teachers asked purposeful questions and used the students’ responses to guide instructional decisions. Thus, we describe these changes as a practice of inquiry in which they asked questions, listened to students’ responses, analyzed those responses, and made instructional decisions.

In contrast, Rose and Rachel showed students procedures and attempted to help students understand them. Rachel continued to introduce a lesson by asking students to recall previously learned information and suggesting procedures to follow. The suggested procedures were further explored using manipulatives to concretely demonstrate the procedure. The classroom discourse remained traditional. Rachel asked questions that led students to expected responses and reinforced their explanations by repeating students’ responses. Rachel viewed her primary job as preparing students for the state mandated test by teaching them to replicate preferred procedures.

Linking Teacher Change to the Lesson Study

Examining the pedagogy of those who completed the lesson study allows a better understanding of what aspects of lesson study supported sustained changes in pedagogy. First, the experiences of the Place Value team are discussed and linked to changes in Pat’s and Paula’s pedagogy. Second, the experiences of the Count-Back-Change team are discussed and linked to changes in Courtney’s pedagogy.

Place Value Team

During the first lesson study meeting, the 1st-grade teachers theorized that students who understood place value would break a multi-digit number into groups of tens and ones and utilize the structure of the hundreds chart to solve problems. They indicated that most of their students did not break multi-digit numbers apart and relied on the inefficient method of counting by ones to solve problems. During the second meeting, the primary teachers discussed how pictures could be incorporated with the hundreds chart to develop place value understanding.

They selected a task that connected a picture with the place value and created the following two problems: “There are 36 cupcakes. We want to put 10 cupcakes in each box. How many boxes would we need to fill? How many extra cupcakes?” and “We have three boxes of cupcakes and 6 extra on a plate. Each box has 10 cupcakes. How many cupcakes do we have?” During the third meeting, these primary teachers discussed how teachers maintain or reduce the cognitive demand of tasks by the types of questions that might be asked. This discussion led to the creation of probing questions for the research lesson that would not reduce the cognitive demand of the problems.

The Place Value team finalized their research lesson during the fourth meeting. They decided to state the problem orally to the students and then Pat would write the important information on the board, “36 cupcakes” and “10 in a box.” After a lengthy discussion about whether students needed the shortened problem, Pat concluded, “I think I inadvertently lower[ed] the cognitive demand [of the problem when writing the short version on the board].” Her critical reflection created an opportunity for team members to discuss their classroom routines and assumptions that impacted a task’s enacted cognitive demand. During this discussion, the teachers shifted from planning a single lesson to sharing aspects of their own pedagogy that influenced students’ opportunities to learn. This shift of focus from the impersonal to personalized reflection on their own actions and beliefs can be thought of as critical reflection. It goes beyond reflection on actions to deeply considering how those actions reflect a set of beliefs and assumptions that may not in fact support student learning.

After this critical reflection, Pat maintained the cognitive demand of posed tasks by writing the full problem on the board instead of an abbreviated form. She posed tasks with higher cognitive demand during calendar math and established a new discourse pattern during a hundreds chart activity by encouraging students to lead a classroom discussion.

Paula shifted the focus of her reflections. She initially interpreted her students’ responses as correct or incorrect. After sharing aspects of how she also reduced the cognitive demand of tasks, Paula began to speculate about her students’ responses as an indication of their mathematical thinking. This was very difficult for her because she often did not know when her students responded with an important mathematical idea. She began to build her content knowledge by discussing students’ responses with other teachers.

Count-Back-Change Team

During their reflection of lesson study, the intermediate teachers in the Count-Back-Change team gained a new insight when they recognized that they could not explain why a student incorrectly added a dime to 95 ¢ to make a dollar. This realization led to a discussion of questions. Questions could be posed in such a way that revealed students’ thinking. More importantly, they realized the limitations of quickly determining whether a response was correct or incorrect without first considering the reasoning that the student used. Thus, they began to ask follow up questions that probed students’ thinking a bit more before providing the preferred procedure.

After the analysis of the research lesson, Courtney maintained the cognitive demand of tasks when students struggled. She began to wonder why students made particular responses and asked probing questions to help her understand what they were thinking. Courtney asked her students questions to uncover their intuitive approach and then used leading questions to help them solve the problem using their own strategy. This was a surprising change because Courtney had a limited understanding of mathematics. She seemed comfortable that her students might “know more math” through their intuitions than she did. Courtney expressed joy that she was learning from them. She reflected, “I know when they get the answer right, but I learn when I ask questions to help me understand what they did. We are learning together.”

Implications

This chapter describes the pedagogical changes that teachers made while engaged in lesson study. Before participating in lesson study, the teachers focused their attention on teaching methods and content delivery. They made cosmetic changes to individual tasks and lessons. Their questions led students to reaching a teacher-preferred procedure or strategy. The teachers’ reflections were general and focused on delivery of the curriculum.

The three case study teachers who completed their lesson studies became more curious about students’ thinking and understanding. They asked students questions to help them identify which mathematical ideas the students were applying and to understand the reasons for these choices. These teachers recognized that their actions influenced not only how students learned but also what they learned, suggesting a new insight. This insight was personal and led them to reexamine their beliefs about teaching and learning, leading to a practice of inquiry. The reflection was critical because there was some discomfort the first time they considered whether their actions supported their beliefs. Critical reflection appeared to be a catalyst for their professional growth and is typical in lesson study (Perry et al. 2002). Their insights translated into sustained change in their pedagogy, personal beliefs, and assumptions during the two years following completion of the lesson study.

In contrast, the teachers who abandoned the research lesson midstream did not change their pedagogy. They shied away from opportunities to reflect deeply on student learning and how their actions might influence opportunities for students to learn while designing the research lesson and working on their curriculum plan. Their beliefs and assumptions did not change and they continued to teach using a comfortable pedagogy of past practice.

Leadership in Lesson Study

The contrast between the teams that completed lesson study and the one that abandoned it is stark. Why might the same professional learning opportunity affect two teams of teachers so differently? One possibility is the contrasting group dynamics of the teachers who completed lesson study and those who abandoned it. How the teachers interacted with each other influenced the nature of their collaboration, decision-making process, and actions. Portes (1998) and Putnam (2000) hypothesized that individuals’ social capital impacts their interactions and the dynamics of the group. Social capital refers to connections within and between social networks. While there are a variety of definitions, social capital describes an individual’s level of influence derived from cultural factors such as education, social interactions, and network of relationships.

Shared Leadership

All the teachers who completed lesson study had a similar level of social capital. None of these teachers had a high-profile leadership role prior to the Leadership Institute and they were not identified as an outstanding teacher by their principal. They viewed each other as equals with different experiences from which they could learn. Thus, no single teacher in the team assumed leadership or dominated the decision-making process.

The needs of the teachers to learn and collaborate took priority in all of their actions. The teachers in both lesson study teams shared their classroom practices and the tensions between practice, knowledge, and beliefs. In doing so, they became vulnerable and found new strength from each other. Gaining strength from each other is characteristic of teams who face challenges together. Both lesson study teams voiced a need for new pedagogy because “students are not learning enough using the old methods. We can’t expect to get different results if we do the same thing.” With limited confidence in their initial pedagogy, the teachers who completed lesson study wanted to collaborate, learn, and share leadership with each other. They gained social capital through the process of collaboration and developed a strong network of colleagues for support. This ultimately garnered them increased influence in the school district.

Individual Leadership

In contrast, the team that did not complete their research lesson relied on a single leader for guidance. Fourth-grade was the first year that students were assessed in mathematics on a state mandated test and these test scores determined whether the school was labeled satisfactory or unsatisfactory. Thus, the teachers felt pressured to have the students ready for the test and the use of time was critical. Diversions that consumed valuable class time for work that could be accomplished more efficiently by teaching procedures was not valued. As a result, the lesson study team looked for guidance from a respected teacher.

This teacher had considerable social capital that was derived from three sources. First, she was highly respected by her principal and awarded special privileges after creating a school district writing program. Second, with confidence in her pedagogical content knowledge, she articulated content that needed to be retaught each year and anticipated students’ response to new content. Third, she became a 4th-grade spokesperson who questioned how reform recommendations were translated into pedagogy. The teacher used her social capital to influence her colleagues by encouraging them to abandon the lesson study process. This process was slow and required teachers to rethink teaching and student learning. Wanting a useful product (curriculum guide) that would minimize instructional planning during the school year, the leader used her social capital to influence her team.

While the team had opportunities to think more critically about how activities may promote students’ mathematical thinking, the teachers stayed focused on the curriculum guide. Critical thinking about how teachers create or limit learning opportunities is essential to develop a practice of inquiry that enhances students’ mathematical learning.

The Influence of School Administration

After the Leadership Institute ended, a new superintendent was hired who refocused the school district’s energy back to reading. School district support for mathematics education evaporated. The commitment of several teacher leaders kept the vision of teachers asking mathematical questions, listening to students, and planning instruction to develop students’ thinking alive within their classrooms, but they were hesitant to step forward. They felt silenced by the new administration.

The administration changed several times over the next six years. The teachers who developed a practice of inquiry created networks in mathematics and extended them to support literacy education during an administration that focused attention on literacy. Eventually, the teachers who completed lesson study found a voice in the school district’s mathematics instruction. Their opinions were sought when the school district revised their grade level mathematics goals and selected new supplemental materials and textbooks for the school district. In addition, they offered mathematics professional development courses for their colleagues using a workshop model.

The teachers who did not complete the lesson study process maintained their level of influence in writing and science education, but did not extend their influence to mathematics. This suggests that they maintained old networks but did not create new ones that extended their influence.

These findings indicate that lesson study supported teachers’ development in two spheres. First, participation in lesson study encouraged teachers to critically reflect on their own pedagogies and this helped them develop a practice of inquiry. Classroom practices were transformed after teachers discussed a new insight with a colleague. The moment of insight was sometimes uncomfortable as teachers realized that their actions did not necessarily portray spoken beliefs. Thus, it appears that pedagogical changes are supported by a social group to create new meaning and to personalize shared experiences through critical reflection. A supportive group, such as a lesson study team, appears to encourage teachers to consciously investigate their own practices.

Second, the lesson study teams allowed teachers to establish a new network of support throughout the school district. This network enabled them to survive an administration that subdued teacher leadership through top-down decision-making policies. After several administrative changes, the teachers who completed lesson study utilized their social capital to reestablish a leadership team that supported mathematics education. Thus, it appears that participation in lesson study developed teachers’ social capital through which they maintained relationships and hope for a future in which attention for mathematics education reemerged.

Creating a Successful Lesson Study

Lesson study provides a sound structure for professional learning and is certainly worth utilizing in school settings. But, there are factors which can limit its success. Teachers often work in isolation and do not share their classroom experiences. Limited venues for exchanging ideas, practices, and beliefs can bring discomfort to teachers when they try to break down the walls of isolation. Lesson study often builds a closely knit community in which teachers share experiences and vulnerabilities from which professional learning and pedagogical change can emerge.

Successful lesson study teams appear to function through collaboration in which all the teachers have a similar level of social capital. They are more likely to recognize that each person contributes to the process in different ways, but no single contribution is considered more valuable. The members of the lesson study teams that completed their research lesson appreciated the gift of time to invest in the process of thinking deeply about a lesson and student thinking. They also released themselves from external expectations to meet state mandated benchmarks so that they could modify pacing guides to work on specific content beyond the allotted time. They were open to suggestions from an outside facilitator even though at times she sometimes pushed them to think about uncomfortable things. They trusted each other and engaged in critical reflection believing that the risks they took would lead to important professional growth.

The current environment of high-stakes-mandated tests creates many tensions for teachers who want to investigate new pedagogies and consider how they influence students’ learning. These tests put pressure on teachers to prepare for the exam, and schools sometimes rely on quick fix methods to make sure that students meet grade level benchmarks. Time to investigate student thinking and confidence to reallocate time by deviating from pacing plans takes courage. School administration can support teachers’ professional growth and student learning by creating time in pacing guides and space in the curriculum. More research is needed to describe how lesson study and other types of professional development supports sustained teacher change and the impact of new pedagogies supported by professional development on student learning.

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Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Washington State UniversityPullmanUSA
  2. 2.Australian Catholic UniversitySydneyAustralia
  3. 3.Curtin UniversityPerthAustralia

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