Mathematics Lesson Study in Elementary Pre-service Teacher Preparation Rosemarie Michaels, Dominican University of California

Mathematics Lesson Study in Elementary Pre-service Teacher Preparation

Rosemarie Michaels Dominican University of California

Abstract

As elementary teachers are expected to understand and engage students in mandatory mathematical practices, teacher preparation programs are tasked with ensuring pre-service teachers begin their careers with the pedagogical knowledge and skills to be able to do so. A university lesson study program was collaboratively designed with elementary teachers to intentionally integrate mathematical pedagogical knowledge and pedagogy required by state and national standards. A mixed-method design was used to answer the research question: How does lesson study facilitate the development of elementary pre-service teachers’ pedagogical knowledge and pedagogy in mathematics? Findings indicate that lesson study has a positive impact on the development of pre-service teachers’ pedagogical knowledge and mathematics pedagogy. Lesson study programs that focus on mathematical pedagogical knowledge and pedagogy in collaboration with classroom teachers can serve as models for exemplary practice in elementary pre-service teacher preparation.

Keywords: Lesson study, elementary teacher preparation, mathematics pedagogy, pedagogical knowledge

Introduction

This study describes the effect of a university lesson study program on elementary preservice teachers’ mathematical pedagogical knowledge and pedagogy. This paper contains a review of literature on lesson study with pre-service teachers preparing to teach mathematics in elementary school, methods for implementing lesson study in collaboration with classroom teachers, and lesson study’s impact on the development of mathematical pedagogical knowledge,

pedagogical knowledge in general, and mathematics pedagogy in pre-service teachers.

Context and Background

A challenge for university teacher preparation programs is to prepare elementary preservice teachers to effectively teach mathematics when they are classroom teachers. In 2010, a majority of states in the United States adopted the Common Core State Standards in Mathematics (CCSS-Math) that specify eight Standards for Mathematical Practice (SMPs) for K-12 students (Common Core State Standards Initiative, 2020). SMPs describe the habits of mind that foster mathematical understanding and the skills needed to be successful in school and career (see Appendix A for list of SMPs). Although SMPs were designed for students, “they apply to all who do mathematics, including elementary teachers” (Max & Welder, 2020, p. 844). As classroom teachers struggle to implement the CCST-M and SMPs, teacher preparation programs are called upon to prepare teachers to understand and teach these standards (Max & Welder, 2020; Koestler, Felton, Bieda & Otten, 2013). Moreover, California’s Commission on Teacher Credentialing (2016) adopted new Teacher Performance Expectations (TPEs), which include SMPs within the subject-specific pedagogy for teaching mathematics. As the TPEs are a set of knowledge and skills that pre-service teachers need to demonstrate prior to earning a teaching credential, pre-service teachers are expected to understand and engage students in the SMPs, along with more overarching and subject-specific pedagogies. In 2017, the Association of Mathematics Teacher Educators (AMTE) released Standards for Preparing Teachers of Mathematics to provide universities with guidance in revising their teacher preparation programs to effectively prepare pre-service teachers to teach the CCSS-Math. Central to AMTE standards is learning to teach through mathematics methods courses that integrate the SMPs and provide foundations in mathematical pedagogical knowledge; and opportunities to learn in guided fieldwork settings through collaborative partnerships between universities and schools with a shared vision of high-quality mathematics instruction (AMTE, 2017). Incorporating lesson study in pre-service teacher preparation programs is a way to address both the AMTE standards and California’s TPEs. Lesson study in elementary preservice teacher preparation is a professional development tool wherein pre-service teachers plan,

observe, analyze, and reflect on lessons in collaboration with classroom teachers and professors (Chichibu, 2016; Michaels, 2015). A growing body of research indicates that collaborative lesson study programs that deliberately connect professors with classroom teachers; and teacher preparation coursework with classroom instruction, provide high-quality professional development opportunities and learning experiences for pre-service teachers (Burroughs & Luebeck, 2010; Cajkler, Wood, Norton, & Pedder, 2013; Michaels, 2015). Hence, the focus of the university mathematics lesson study program was revised in collaboration with classroom teachers to incorporate the pedagogical knowledge and mathematics pedagogy described in the TPEs (California’s Commission on Teacher Credentialing, 2016), while simultaneously addressing the AMTE standards (2017). Similarly, the mathematics methods course in the teacher preparation program was revised to embed the SMPs as a focus for understanding, learning, and teaching mathematics.

Purpose

The purpose of this study is three-fold: 1) describe the literature on lesson study with elementary pre-service teachers and their preparation to teach mathematics; 2) discuss the effects of lesson study on the development of pre-service teachers’ mathematical pedagogical knowledge, pedagogical knowledge in general, and mathematics pedagogy; and 3) examine implications for effectively preparing elementary pre-service teachers to teach mathematics. The following question guided this research: How does lesson study facilitate the development of elementary pre-service teachers’ pedagogical knowledge and pedagogy in mathematics?

Literature Review

Background

Traditionally, lesson study is a professional development practice that engages collaborative teams of classroom teachers in a cycle of planning, teaching, observing, analyzing, revising, and reteaching lessons with the goal of refining instruction to improve student learning (Fernandez & Yoshida, 2004; Lewis & Hurd, 2011). It is not surprising then, that the majority of lesson study research is conducted in K-12 schools with classroom teachers and their students

(Larssen et al., 2018). Due to the success of these inservice lesson study programs, professors are beginning to incorporate lesson study into pre-service teacher preparation coursework. As it is a recent practice, the research base is small; only 17% of peer-reviewed lesson study research involves pre-service teachers in university teacher preparation programs (Larssen et al., 2018). In addition, as professors adapt lesson study to their unique teacher preparation programs, there is little consistency in how lesson study is conducted or studied in pre-service teacher preparation (Cajkler et al., 2013; Larssen et al., 2018; Wood, 2018). Therefore, the following review of the literature serves to describe the varied, yet small, research base of teacher preparation programs using lesson study to assist in preparing pre-service teachers to teach mathematics in elementary school.

Mathematics Lesson Study in Elementary Pre-service Teacher Preparation

This literature review includes all lesson study research with pre-service teachers in university teacher preparation programs with a focus on learning to teach mathematics in elementary school. In most of these seven studies, professors conducted research with preservice teachers in their mathematics methods courses using varying forms of the traditional lesson study cycle, that is, collaboratively plan, teach, observe, analyze, reflect, revise, and reteach a mathematics lesson. Fieldwork in elementary schools was a requirement for all preservice teachers in the courses and typically the lesson studies took place at those school sites. Elementary classroom teachers were integral partners in three of the studies. Pothen and Murata (2007) designed a lesson study program for pre-service teachers in their mathematics methods courses, wherein, small teams of pre-service teachers collaborated to plan, observe, and then analyze a mathematics lesson. Pre-service teachers taught the mathematics lesson at their elementary school fieldwork site observed by all other team members. Debriefing sessions consisted of sharing observations, reflecting on teaching practice, and discussions on lessons’ strengths and improvements. Pothen and Murata (2007) collected data on pre-service teachers’ mathematical content and pedagogical knowledge through pre- and post-lesson study surveys and post-lesson study reflective essays. They observed the planning meetings and classroom lessons, but did not facilitate the sessions.

Pothen and Murata (2007) reported that pre-service teachers’ mathematical content and pedagogical knowledge improved, due to the lesson study experience, and provide a few examples of how this manifested. While planning the lesson, pre-service teachers realized they needed to relearn elementary mathematics concepts, in order to teach them. While collaborating with their peers on lesson planning, they came to deeper understandings of mathematical content. Pothen and Murata (2007) also found that pre-service teachers learned the value and use of assessment in elementary classrooms, specifically, the importance of assessing prior knowledge and students’ learning during instruction. The researchers contribute the characteristic aspects of lesson study for this result, that is, collaborative planning and debriefing mathematics lessons with peers provided pre-service teachers with opportunities to plan, discuss, and think deeply about mathematics content and pedagogy. Small teams of pre-service teachers in Myers (2013) mathematics methods course also collaborated to plan, teach, observe, and then analyze mathematics lessons. She studied preservice teachers’ reflective practice through small group and individual reflective essays of their lesson study experiences. She participated in each aspect of the lesson study process, providing advice and subject matter expertise for her pre-service teachers during lesson planning meetings and post-lesson debriefing sessions, but did not facilitate or guide the sessions. Myers (2013) was surprised to find an absence of critical reflection in her pre-service teachers’ individual and group essays. Myers (2013) also noticed a lack of curiosity and attention to the complexity of teaching and learning. She suggests that pre-service teachers need a knowledgeable and supportive facilitator to provide continuous guidance toward reflective practice throughout the lesson study process. Similarly, small teams of pre-service teachers in Parks’ (2008) mathematics methods course also collaborated to plan, teach, observe, and then analyze mathematics lessons. While Parks (2008) provided guidance during lesson planning sessions, she did not facilitate the lessons taught at fieldwork sites or debriefing sessions, which were held during a mathematics methods class meeting. The focus of this study was on the development of pre-service teachers’ mathematics content knowledge and equity lens through analyses of lesson study essays and audio-recordings of groups’ planning and analysis sessions. Mixed results were reported; some pre-service teachers developed mathematical content knowledge and equity lenses, but others

formed problematic beliefs about teaching and learning that do not support effective, equitable mathematics pedagogy. Parks (2008) recommends professors carefully structure and facilitate pre-service teachers’ lesson study experiences. Chassels and Melville (2009) studied the effects of lesson study on their elementary preservice teachers’ mathematical content knowledge, pedagogical knowledge, and reflective practice in their mathematics methods courses. Small teams of pre-service teachers not only collaborated to plan, observe, and then debrief a research lesson, they revised the lesson and a new team member retaught it at their fieldwork sites. Like Myers (2013), data were collected through group and individual reflective essays of their lesson study experiences. As in Pothen and Murata’s (2007) study, Chassels and Melville (2009) found that the lesson study experience increased pre-service teachers’ mathematical content and pedagogical knowledge. Specifically, pre-service teachers developed a deeper understanding of mathematics content and effective instructional strategies, demonstrating knowledge and skills in lesson analysis, reflection, giving and receiving constructive feedback, and confidence in their ability to plan lessons. However, Chassels and Melville (2009) also report that there were many challenges in implementing lesson study at elementary schools, including a shortage of time to plan and debrief lessons and an overall lack of support for collaboration at the school sites. They found that pre-service teachers need guidance in planning and lesson analysis from classroom teachers that host the lesson studies. Chessels and Melville (2009) recommend that professors provide classroom teachers with information about lesson study and guidance with their important roles in the cycle. The next three studies demonstrate how pre-service and classroom teachers can collaborate effectively in lesson study experiences. In Burroughs and Luebeck’s (2010) study, pre-service teachers observed a team of classroom teachers collaboratively plan a mathematics lesson. During the next mathematics methods class, professors and pre-service teachers discussed the lesson plan and provided classroom teachers with improvement ideas. Unfortunately, only two pre-service teachers were able to observe the actual lesson and participate in the subsequent debriefing session. Instead, a debriefing and analysis session was held with the team of classroom teachers, pre-service teachers, and professors later in the week. Burroughs and Luebeck (2010) collected data through reflective essays, video-recorded lesson

study sessions, and focus group interviews; they reported results for emergent themes. Preservice teachers demonstrated pedagogical knowledge by understanding the importance of indepth planning and assessment of students’ prior knowledge and learning during instruction. Pre-service teachers also demonstrated the ability to collaborate meaningfully with classroom teachers, and to reflect and critically analyze mathematics lessons and the complete lesson study cycle. Michaels (2015) also studied a collaborative lesson study cycle with pre-service and classroom teachers; however, research was conducted in a seminar class rather than a mathematics methods course. Similar to Burroughs and Luebeck’s (2010) research, classroom teachers planned a mathematics lesson, which was then analyzed by pre-service teachers during a class session on the university campus. All pre-service teachers observed the mathematics lesson at the school site and participated in the subsequent collaborative debriefing and analysis session, facilitated by the professor. Data was collected through post-lesson study surveys. Michaels (2015) reports that pre-service teachers’ pedagogical knowledge of mathematics instruction increased (i.e., using subject-specific pedagogical skills, assessing student learning, and making content accessible to students), due to the lesson study session. Lastly, Post and Varoz (2008) studied the effects of a complete lesson study cycle on teams of pre-service and classroom teachers, who collaborated to plan, observe, teach, debrief, revise, and reteach a mathematics lesson. After observing classroom teachers teach the first lesson, pre-service teachers taught the revised lesson. Professors facilitated the lesson study experiences, collecting written observations through the lesson planning and teaching sessions. Results indicate that pre-service and classroom teachers’ pedagogical knowledge of mathematics improved, due to the lesson study cycle. Pre-service teachers shifted their attention from the logistics of managing the lesson and students to demonstrating understandings of students’ mathematical thinking. Collaboratively planning lessons provided a format for pre-service teachers to learn about classroom teachers’ planning processes, while classroom teachers developed new strategies for mentoring pre-service teachers. Although the research base is small, overall results indicate that lesson study provides structured experiences for pre-service teachers to increase their understanding of mathematical pedagogical knowledge, content knowledge, and pedagogy. However, there is little consistency

or clarity on how these constructs are defined or studied in elementary pre-service teacher preparation. The current study defines mathematical pedagogical knowledge, pedagogical knowledge in general, and mathematics pedagogy using standards established by California’s Commission on Teacher Credentialing (2016) and Common Core State Standards in Mathematics (2020), that is, the TPEs and SMPs, standards for both pre-service teachers and K12 students. In addition, the methodology is clearly described and based on prior research. Therefore, this study contributes to the growing body of literature by systematically connecting mandated, clearly defined standards in mathematics pre-service teacher preparation to collaborative lesson study experiences in elementary classrooms.

Methodology

This study took place in a small, private university in western United States and an elementary school in the community. A mixed-method design was used to measure the extent to which lesson study develops pre-service teachers’ mathematics pedagogy, pedagogical knowledge in mathematics, and pedagogical knowledge, in general. Post-lesson study surveys and reflective essays were created and analyzed for this research.

Participants

Participants were 66 pre-service teachers in the elementary teacher preparation program at the university. Pre-service teachers were undergraduate (n = 31) and graduate students (n = 35) enrolled in one of five mathematics methods courses over a two-year period, taught by the researcher. All pre-service teachers were concurrently participating in fieldwork at elementary schools during the semester, as was required by all studies in the literature review.

Instruments

A post-lesson study survey was used to measure a lesson study session’s impact on preservice teachers’ mathematical pedagogical knowledge, pedagogical knowledge in general, and pedagogy. The researcher and professors in the department created the survey, which was piloted with pre-service teachers and revised as needed and suggested (see Appendix B for PostLesson Study Survey).

The post-lesson study survey measured pedagogical knowledge as defined by California’s Commission on Teacher Credentialing (2016) through the TPEs. TPEs were used as the basis for this research as pre-service teachers need to demonstrate their knowledge of the TPEs prior to earning a teaching credential. The post lesson study survey measured pre-service teachers’ pedagogical knowledge in mathematics in three areas: SMP 1: Makes sense of problems and persevere in solving them, SMP 4: Model with mathematics, and academic discourse. These focus areas were chosen as they are the elements that are traditionally difficult for pre-service teachers to comprehend; they are also standards that elementary students are expected to master (Common Core State Standards Initiative, 2020); and can be observed readily during classroom instruction. In addition, the post lesson study survey measured pre-service teachers’ pedagogical knowledge of four of the six general, overarching TPE domains: Assessing student learning, reflecting on teaching practices, maintaining effective environments for student learning, and engaging all students in learning, excluding two TPEs on lesson planning, which was not a focus of this study (California Commission on Teacher Credentialing, 2016). The researcher collaborated with classroom teachers to develop a post-lesson study reflective essay assignment, which was piloted with pre-service teachers and revised as recommended. The reflective essay assignment guided pre-service teachers to reflect on four areas: SMP 1: Make sense of problems and persevere in solving them; SMP 4: Model with mathematics; assessment of student learning; and their pedagogy, in addition to their own choices for professional reflection.

Procedures

Pre-service teachers in a mathematics methods course, the professor, and a classroom teacher collaborated on a lesson study cycle to plan, analyze, teach, observe, debrief, analyze and reflect on a mathematics lesson. The professor carefully structured and facilitated each aspect of the lesson study program, as recommended by previous researchers (Chessels & Melville, 2009; Myers, 2013; Parks, 2006). To maintain consistency, the same third grade classroom teacher taught the mathematics lesson study session for each class of pre-service teachers in this study and also participated in designing the reflective essay. The classroom teacher was chosen based on her expertise in the SMPs and knowledge of lesson study practice. She planned the

mathematics lessons to meet her students’ academic and social needs, while purposefully incorporating the TPEs of focus (SMP 1, SMP 4, and academic discourse). Similar to Burroughs and Luebeck’s (2010) study, during a mathematics methods class session, the professor and pre-service teachers analyzed the lesson plan, discussing the focus TPEs, student learning outcomes, and considerations for the upcoming classroom observation. Just prior to observing the mathematics lesson, the professor and pre-service teachers met with the classroom teacher at the school site to preview the up-coming lesson. Pre-service teachers asked questions, in order to fully understand the classroom teacher’s thinking and lesson planning process. While pre-service teachers observed the mathematics lesson, the professor guided the experience by highlighting important aspects of the lesson for pre-service teachers. Immediately following the lesson, the classroom teacher and pre-service teachers met to collaboratively debrief and analyze the lesson, facilitated by the professor, with a focus on mathematics pedagogy, pre-service teacher and student learning, strengths of the lesson, and improvement ideas. The classroom teacher started the debriefing session with a commentary on the lesson and reflections on student learning. Then, pre-service teachers asked questions and shared their perspectives and observations. The professor ensured pre-service teachers were actively and equitably participating and guided their understanding of the classroom teacher’s thinking and decision-making processes and elementary students’ learning. During the next mathematics methods class session, the professor led a concluding discussion about the mathematics lesson study session, focusing on pre-service teachers’ learning and their current perspectives on teaching mathematics. Hence, pre-service teachers’ pedagogical knowledge and pedagogy were constructed throughout the shared lesson study experience.

Data Analysis

Due to the small sample size, descriptive statistics were used to analyze the data collected from the survey. Means and percentages of pre-service teacher survey responses were calculated for the lesson study sessions. Reflective essays were analyzed and coded by focus areas on the essay and post lesson study survey; and then further analyzed for emergent themes. Data are represented in Tables 1-3.

Results/Findings

To investigate the impact of a lesson study program on pre-service teachers the following question was asked: How does lesson study facilitate the development of elementary pre-service teachers’ pedagogical knowledge and pedagogy in mathematics? Survey results were analyzed and the generated descriptive statistics are reported in Tables 1-3, including percentages and means for survey items. Each response was coded with a score of 4, 3, 2, or 1. The weight of 4 was given to the response that had the most positive outcome, that is, responses were coded: a great deal = 4, a lot = 3, a little = 2, nothing = 1. Lesson study reflective essays were analyzed quantitatively and results are reported simultaneously with survey responses. Results indicate that lesson study has a substantial impact on pre-service teachers’ pedagogical knowledge of mathematics (Table 1), pedagogical knowledge in general (Table 2), and mathematics pedagogy (Table 3). Reflective essays serve to elucidate how participation in lesson study facilitated this development.

Pedagogical Knowledge in Mathematics

Pre-service teachers learned “a great deal” about pedagogical knowledge in mathematics (Table 1), specifically SMP 1: Make sense of problems and persevere in solving them (M = 3.67); SMP 4: Model with mathematics (M = 3.76); and academic discourse in math (M = 3.48). Although a small percentage of pre-service teachers reported they learned “a little” about SMP 1 (6%) and academic discourse in math (11%), overall results indicate lesson study has a positive impact on pre-service teachers’ pedagogical knowledge in mathematics. Reflective essays support the survey results. Pre-service teachers consistently demonstrated their new pedagogical knowledge of the SMPs by describing the teacher’s pedagogy and how her pedagogy supported students in learning the content. For example, a preservice teacher explained SMP 1, “The teacher directed the students to write instructions on how to solve a word problem if they had to explain it to a kindergartener. This activity activated prior knowledge and made the students think about the concept that they were about to work on” (preservice teacher 1). Pre-service teachers also shared the rationale for teaching students modeling with mathematics (SMP 4), “Modeling helps students build genuine understanding through direct experience and

inspires curiosity about the topic” (pre-service teacher 2). Many pre-service teachers listed effective instructional strategies observed for each SMP and then included aspects of the listed strategies in their essays. In this example, a pre-service teacher saw a variety of ways that the teacher “encouraged persistence among her students by offering multi-step problems, checking for understanding, giving explicit encouraging words, and multiple opportunities to practice solving mathematical story problems” (pre-service teacher 3). Pre-service teachers related important realizations about mathematical pedagogical knowledge for problem solving, such as, “It’s okay for students not to understand how to solve the problem in one lesson. As long as they are making progress, it’s a success” (pre-service teacher 4). Lastly, pre-service teachers commented on their pedagogy, “I plan to use exercises like this when introducing new math concepts” (pre-service teacher 5). Pre-service teachers reported learning “a great deal” (59%) or “a lot” (30%) about academic discourse in mathematics during the lesson study experience. Reflective essays clarify what pre-service teachers learned about academic discourse, for example, “Students gain insight from one another by listening to their peers’ explanations and ideas, then try out the strategies that are presented for other real-world problems” (pre-service teacher 6). Pre-service teachers discussed how academic discourse can assist students in varying levels of concept understanding, “This discussion solidified understandings for some kids, and helped others learn from patterns that they did not independently notice (e.g., some models have different side lengths, but the same area)” (pre-service teacher 5). Others noticed how academic discourse provides the teacher with insights into students’ thinking. Pre-service teachers also observed the teacher’s roll during academic discourse, as the teacher “guided the conversation in the appropriate direction, which was a really useful strategy to learn” (pre-service teacher 7). It was common for pre-service teachers to state that they had not observed academic discourse in their fieldwork nor experienced it as elementary students, but were planning on using this strategy, “I will provide multiple opportunities for students to explain why they came up with their answers and to prove that their answers make sense utilizing reasoning and alternative strategies” (pre-service teacher 8).

Pedagogical Knowledge

Pre-service teachers’ pedagogical knowledge, in general, was positively impacted by the lesson study sessions (see Table 2). Specifically, pre-service teachers’ learned “a lot” about assessing student learning (M = 3.21), reflecting on teaching practice (M = 3.41), engaging all students in learning (M = 3.41), and maintaining effective environments for learning (M = 3.36). Even though no pre-service teachers reported learning “nothing” during the lesson study, small percentages of pre-service report learning “a little” about pedagogical knowledge. Indeed, 38% of pre-service teachers learned “a great deal” and 45% learned “a lot” about assessing student learning, while 17% learned only “a little.” Overall, assessing student learning scored the lowest of all elements of pedagogical knowledge, even though assessing student learning was a focus of the lesson study sessions. In reflective essays, pre-service teachers were consistently able to assess and discuss evidence of student learning using professional educator language as evidenced by statements such as, “The students appeared to understand the concept of solving a problem in different ways; they were able to follow through with providing an alternative method when asked to do so, which further indicted to me that they had already incorporated this flexibility into their math skill set” (pre-service teacher 9). Many pre-service teachers were able to suggest the next pedagogical step for the teacher, “Most of the students demonstrated that they could measure area by counting unit squares. They likely need further practice to generalize this idea, and to be able to use it with symbols rather than concrete materials” (pre-service teacher 10). However, a common practice for pre-service teachers was to discuss student assessment through the classroom teacher’s perspective, rather than their own, “The teacher was assessing the students by observing them and then asking students to share what they learned out loud. She assessed student understanding by seeing if the students were able to: practice multiplication and division and solve word problems” (pre-service teacher 11). And another pre-service teacher stated, “The design of the lesson made it easy to assess students’ understanding because they had to physically build the model. If they weren’t able to build it, the teacher could tell that they didn’t understand” (pre-service teacher 12). Pre-service teachers discussed the importance of assessing students throughout the lesson. A pre-service teacher stated that it “is a critical aspect of the teaching and learning

process, as it allows for educators to strategically evaluate the effectiveness of their teaching by measuring the extent to which students in the classroom are understanding the course material” (pre-service teacher 13). Progress monitoring was a new learning for many pre-service teachers, as they previously thought about assessment as something teachers do after the lesson, “I have always thought that assessing students should be a separate column in a lesson. After listening to” this teacher during the debriefing session, “I realized that she assesses as she goes. This is such a good idea and something that I will forever have engraved in my brain moving forward” (pre-service teacher 14). Other pre-service teachers stated they will use what they learned about assessment in their own classrooms, “I will plan to incorporate multiple means and opportunities for formative assessment during lessons to help to fine-tune while class lesson planning and ensure all individual students are progressing” (pre-service teacher 6). Lesson study impacted pre-service teachers’ pedagogical knowledge for reflecting on teaching practice: 51% learned “a great deal,” while 38% learned “a lot.” There was not a prompt on the essay to discuss reflective practices specifically, although the written assignment was an exercise in reflection in and of itself. Within the essays, pre-service teachers rarely discussed their own participation in reflective practice or what they learned through or about collaborative reflection during the debriefing session. Rather, pre-service teachers discussed reflection in general terms: “Reflection is what helps teachers improve upon teaching technique and procedure” (pre-service teacher 4) or shared the teacher’s reflections during the analysis & debriefing session, “We were privy to her immediate reflections on the lesson, which gave insight into her observation process” (pre-service teacher 2). Engaging all Students in Learning and Maintaining Effective Environments for Student Learning were not main focus areas of the lesson studies, however, survey results indicate that lesson study impacted both elements of pre-service teachers’ pedagogical knowledge “a lot” (M = 3.41 and 3.36, respectively) and the reflective essays support this finding. When pre-service teachers discussed these elements, they explained their importance, stated that the teacher’s pedagogy was effective, and provided specific examples observed in the lesson. Pre-service teachers also discussed what they learned from the teacher during the debriefing session, as exemplified in this comment, “I keep thinking about what she said about engaging kids because it’s the teacher’s job to keep the students engaged at all times. The more teachers make the

lesson engaging the more kids want to do the activity and want to learn (pre-service teacher 14). Pre-service teachers discussed the impact on their own pedagogy, “This lesson impacted my teaching practice by showing me how a teacher really can create a positive environment, engage a student in a lesson, and support all learners at the same time (pre-service teacher 2).

Pedagogy

Pre-service teachers learned “a great deal” about the pedagogy for teaching math (M = 3.59), are more confident in teaching math (M = 3.48) and will be able to teach math to students more successfully (M = 3.56) due to their participation in the lesson study experience (see Table 3). Throughout the reflective essays, pre-service teachers consistently discussed the lesson study’s impact on their own pedagogy and teaching practice, as noted previously for elements of pedagogical knowledge. A few students admitted that, “Teaching math has always been a scary thought for me, but learning strategies such as these ones make me feel much more confident” (pre-service teacher 11). Most explicitly stated that they learned a great deal from the lesson study experience and “will definitely use these strategies in my fieldwork and later in my own classroom” (pre-service teacher 6). Lastly, pre-service teachers saw the connection between theory to practice by noting that the lesson study experience afforded them the opportunity to “see in practice how to use all the different strategies we’ve been learning about in class” (preservice teacher 15).

Discussion

Summary of Findings

Lesson study integrated into a mathematics methods course through a partnership with classroom teachers has a significant impact on pre-service teachers’ pedagogical knowledge and mathematics pedagogy. Specifically, collaborative lesson study experiences developed preservice teachers mathematical pedagogical knowledge of SMP 1: Make sense of problems and persevere in solving them; SMP 4: Model with mathematics; and academic discourse; and preservice teachers’ pedagogical knowledge in general (i.e., assessment, reflection, student engagement, effective classroom environments). Pre-service teachers’ pedagogy was also

impacted, both their knowledge of mathematics pedagogy and their confidence and ability to successfully teach students mathematics.

Conclusions

In this study, a mathematics methods course was revised to integrate SMPs as a focus for understanding how to teach mathematics in elementary school. The embedded lesson study program was also revised in collaboration with classroom teachers to incorporate required pedagogical knowledge and pedagogy for pre-service teachers (California’s Commission on Teacher Credentialing, 2016) following recommended AMTE standards (2017) for teacher preparation programs. A key to the results may lie in the collaborative, reflective nature of lesson study in combination with a professor-classroom teacher partnership with a shared vision of high-quality mathematics instruction in elementary classrooms. A common focus on the required SMPs for both teacher preparation and elementary classrooms, modeled by the classroom teacher, and facilitated by the professor provided important links between theory and actual teaching practice for pre-service teachers. Facilitation of the entire lesson study process may have contributed to its success. In this study, the professor carefully structured and facilitated all aspects of lesson study: classroom teacher collaboration, lesson plan analysis during mathematics methods course, guidance during lesson observation, and leadership in the subsequent debriefing session. This validates previous researchers’ recommendations for guidance from a knowledgeable facilitator to improve preservice teachers’ experience and learning during the lesson study cycle (Chassels & Melville, 2009; Myers, 2013; Parks, 2008). Results of this study supports previous research with pre-service teachers in mathematics methods courses in other ways. Pre-service teachers learned the value and use of assessment in elementary classrooms, in particular the importance of continual assessment of students during instruction, corroborating results of Pothen and Murata (2007) and Burroughs and Luebeck (2010) research. However, it is important to note that assessing student learning was the lowest element of pre-service teachers’ learning. Although pre-service teachers demonstrated pedagogical knowledge and skill in assessing student learning, the tendency was to report how the teacher assessed the students, rather than taking ownership of their own assessment of

student learning. This is an interesting, yet not surprising finding, as pre-service teachers are in the process of becoming teachers and as such, do not yet see themselves in the professional educator roll in assessment of student learning.

Implications

This study contributes to the growing body of literature by systematically connecting required pedagogical knowledge and mathematics pedagogy standards to collaborative, facilitated lesson study experiences in elementary classrooms. When professors and classroom teachers are a collaborative team, pre-service teachers are provided opportunities to learn to teach through guided, structured experiences that link coursework theory to teaching practice in schools.

However, limitations of this study should be considered. Generalizing the results may be problematic due to the small sample size. This study also lacks information from the perspective of classroom teachers and professors. How does lesson study impact classroom teachers’ work with pre-service teachers and elementary students? How can researchers examine professors’ observations of pre-service teachers’ growth in pedagogical knowledge and pedagogy? Further, as the strength of this study is collaboration in authentic classroom experiences and consistency in studying clearly defined constructs (i.e., pedagogical knowledge and mathematical pedagogical knowledge), future researchers can build on this work, thereby continuing to clarify best practices in lesson study research in elementary teacher preparation in mathematics. For example, how can university lesson study programs further develop pre-service teachers’ pedagogical knowledge and confidence in assessment? Results of this research can also inform teacher preparation programs. As participation in fieldwork appears to be an established requirement during mathematics methods courses, university-school affiliations can be expanded to include guided lesson study experiences. In methods coursework that already integrates lesson study, professors can partner with elementary classroom teachers more closely to ensure pre-service teachers are involved in seamless, authentic, and structured experiences to acquire the pedagogical knowledge and skills required of them. Professional partnerships such as these are not a new idea; Darling-Hammond (2006) has encouraged this practice for over 15 years; however, partnerships are still not common. Lastly,

SMPs need to be embedded in mathematics methods courses to provide pre-service teachers with practice in developing the habits of mind that foster mathematical understanding, so they are prepared to facilitate the transfer of these knowledge and skills to their future students. A goal for teacher preparation programs is to prepare pre-service teachers to effectively teach mathematics. University lesson study programs that focus on mathematical pedagogical knowledge and pedagogy in collaboration with classroom teachers may serve as a model of exemplary practice in elementary pre-service teacher preparation.

Author Biography

Dr. Rosemarie Michaels is an Associate Professor of Education at Dominican University of California in San Rafael. She is chair of the Education Studies Teacher Preparation Program. An experienced classroom teacher, Rosemarie has taught in higher education for over 20 years and is dedicated to developing university-school partnerships, both locally and abroad. She was a visiting professor at the University of Tokyo, Japan in spring 2019, where she worked closely with professors and graduate students on professional development through lesson study. Rosemarie recently received the Francoise Lepage Award for Global Innovation in Global Education. Her professional interests include effective, equitable pedagogy in teacher education and K-12 classrooms, collaborative lesson study, and the 21st century skills.

References

Association of Mathematics Teacher Educators (2017). Standards for preparing teachers of mathematics. https://amte.net/standards Burroughs, E. A., & Luebeck, J. L. (2010). Pre-service teachers in mathematics lesson study. The Mathematics Enthusiast, 7(2), 391-400. Cajkler, W., Wood, P., Norton, J., & Pedder, D. (2013). Lesson study: Towards a collaborative approach to learning in initial teacher education? Cambridge Journal of Education, 43(4), 537-554.

California Commission on Teacher Credentialing (2016). California teacher performance expectations. https://www.ctc.ca.gov/docs/default-source/educator prep/standards/adopted-tpes-2016.pdf?sfvrsn=0 Chassels, C., & Melville, W. (2009). Collaborative, reflective, and iterative Japanese lesson study in an initial teacher education program: Benefits and challenges, Canadian Journal of Education, 32(4), 734-763. Chichibu, T. (2016). Impact on lesson study for initial teacher training in Japan: Focus on mentor roles and Kyouzai-Kenkyuu. International Journal for Lesson and Learning Studies, 5(2), 155-168. Common Core State Standards Initiative. (2020). Preparing America’s students for success. http://www.corestandards.org Darling-Hammond, L. (2006). Constructing 21st-century teacher education. Journal of Teacher Education, 57, 1-15. Fernandez, C., & Yoshida, M. (2004). Lesson study: A Japanese approach to improving mathematics teaching and learning. Lawrence Erlbaum Associates. Koestler, C., Felton, M., Bieda, K., & Otten, S. (2013). Connecting the NCTM process standards & the CCSS-M practices. National Council of Teachers of Mathematics. Larssen, D. L. S., Cajkler, W., Mosvold, R., Bjuland, R., Helgevold, N., Fauskanger, J., Wood, P., Baldry, F., Jakobsen, A., Bugge., H. E., Naesheim-Bjorkvik, G., and Norton, J. (2018). A literature review of lesson study in initial teacher education: Perspectives about learning and observation. International Journal for Lesson and Learning Studies, 7(1), 8 22.

Lewis, C. C., & Hurd, J. (2011). Lesson study step by step: How teacher learning communities improve instruction. Heinemann. Max, B., & Welder, R. M. (2020). Mathematics teacher educators’ addressing the common core standards for mathematical practice in content courses for prospective elementary teachers: A focus on critiquing the reasoning of others. The Mathematics Enthusiast, 17(2&3), 843-881.

Michaels, R. (2015). Bringing lesson study to teacher education: Simultaneously impacting pre-service and classroom teachers. Journal of Scholastic Inquiry: Education, 4(1), 46 73.

Myers, J. (2013). Creating reflective practitioners with pre-service lesson study. International Journal of Pedagogies and Learning, 8(1), 1-9. Parks, A.N. (2008). Messy learning: Pre-service teachers’ lesson-study conversations about mathematics and students. Teacher and Teacher Education, 24(5), 1200-1216. Post, G., & Varoz, S. (2008). Lesson study groups with prospective and practicing teachers. Teaching Children Mathematics, 14(8), 472-478. Pothen, B. E., & Murata, A. (2007). Transforming teachers’ knowledge: The role of lesson study in pre-service education. In Lamberg, T., & Weist, L.R. (Eds). Proceedings of the 29th

annual meeting of the North American Chapter for the Psychology of Mathematics Education, University of Nevada, Reno, Reno, NV, 231-238. Wood, K., (2018). The many faces of lesson study and learning study. International Journal of Pedagogies and Learning, 7(1), 2-7.

Table 1

Impact of Lesson Study on Mathematical Pedagogical Knowledge, in percentages

A great deal A lot A little Nothing M

SMP 1: Make sense of problems and persevere 73 21 6 - 3.67

SMP 4: Model with mathematics

76 24 - - 3.76

Academic Discourse in Math 59 30 11 - 3.48

Note. Focus areas are in boldface. (n = 66)

Table 2

Impact of Lesson Study on Pedagogical Knowledge, in percentages

TPEs

A great deal A lot A little Nothing M

Assessing student learning 38 45 17 - 3.21

Reflecting on teaching practices 51 38 11 - 3.41

Maintaining effective environments for student learning 52 33 15 - 3.36

Engaging all students in learning 54 32 14 - 3.41

Note. Focus TPE is in boldface. (n = 66)

Table 3

Impact of Lesson Study on Pedagogy, in percentages

A great deal A lot A little Not at all M

Pedagogy for teaching mathematics 62 35 3 - 3.59 I am more confident in teaching math 50 49 1 - 3.48 I will be able to teach math more successfully 56 44 - - 3.56

Note. (n = 66)

Appendix A

Standards for Mathematical Practice, California Commission on Teacher Credentialing (2016) 1. Make sense of problems and persevere in soloing them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics

5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure

8. Look for and express regularity in repeated reasoning

Appendix B Post-Lesson Study Survey

Check one space to indicate how much you learned during today’s Lesson Study.

A Great Deal A Lot A Little Nothing

1. MP 1: Make sense of problems and persevere in solving them

2. MP 4: Model with mathematics

3. Academic Discourse in Math

4. Reflecting on teaching practices

5. Maintaining effective environments for student learning

6. Engaging all students in learning

7. Assessing student learning

How did the lesson study further develop your pedagogy?

at All A Great Deal A Lot A Little Not

1. Pedagogy for teaching math

2. I will be able to teach math more successfully

3. I am more confident in teaching math