Preservice teachers' perceptions of a practice-focused lesson study

Abstract

Purpose

The purpose of this study was to explore middle and secondary mathematics preservice teachers' (PST) perceptions of the benefits, challenges and impacts of implementing lesson study.

Design/methodology/approach

This paper presents a single case study in the university context. Embedded units of analysis included a group of middle and a group of secondary mathematics PSTs, who were in the internship year of a teacher preparation program.

Findings

This paper presents several perceived benefits, challenges and impacts according to PSTs. PSTs indicated benefits included their focused student observations and their collaboration in lesson design. Perceived challenges included observing as a nonparticipant observer and aspects of planning. The different impacts occurred in three phases: planning, observing during teaching and reflection.

Research limitations/implications

First, the study participants included a small group of middle and secondary mathematics PSTs from one university in central Texas. Second, the PSTs in this study were not able to reteach the lesson, which may have influenced their perceptions.

Practical implications

Mathematics teacher educators could use insights from this study as they implement lesson study or other field experiences to support PSTs in their growth as mathematics teachers.

Originality/value

This paper examines PSTs' perceptions, which could benefit mathematics teacher educators as they consider how to introduce or implement lesson study with PSTs.

Citation

Shelton, R.N., Rogers, R.M. and Wilkerson, T.L. (2023), "Preservice teachers' perceptions of a practice-focused lesson study", International Journal for Lesson and Learning Studies, Vol. 12 No. 3, pp. 214-225. https://doi.org/10.1108/IJLLS-11-2022-0168

Introduction and purpose

The process of lesson study allows teachers to closely examine particular lessons to look for and address areas in a lesson for improvement (Lewis, 2000, 2002). In our teacher preparation program, middle and secondary mathematics preservice teachers (PSTs) engage in lesson study as a form of professional development. PSTs in this program engage in other forms of professional development including professional learning activities, action research and attending conferences and workshops. However, lesson study provides a unique opportunity for PSTs to engage in the development and improvement of a particular lesson with their peers that focuses on student mathematical thinking (Amador and Carter, 2018; Amador and Galindo, 2021).

The middle and secondary mathematics certification programs at our university began using lesson study with preservice mathematics teachers in 2010. We implemented lesson study with PSTs to increase their knowledge in assessing students' understanding and reflecting on their professional growth as a result of the experience. During this extended period, researchers discovered problematic constraints and consequently identified elements crucial to providing meaningful lesson study experiences for PSTs: (1) a research theme that identified qualities PSTs wanted their students to have, (2) observers focus on students' comments and actions and (3) design the lesson around mathematical practices rather than mathematical content.

In the spring semester of 2020, to explore preservice middle and secondary mathematics teachers' experiences with lesson study, we designed a research study around these elements: research theme, a focus on students' comments and actions and mathematical teaching practices. The purpose of this study was to explore middle and secondary mathematics PSTs' perceptions of the benefits, challenges and impacts of implementing lesson study. We selected a lesson study focused on the eight Mathematics Teaching Practices (MTPs) identified by the National Council of Teachers of Mathematics (NCTM, 2014). The MTPs support educators in their understanding and implementation of specific knowledge for teaching mathematics through instructional design and learning processes, and NCTM provides examples of teacher actions and student actions related to each MTP. We believed this focus would support PSTs because unexpected changes connected to school environments would not derail the lesson study and would provide a framework for examining student actions. As such, we designed a study using the following research questions as a guide:

1. What are middle and secondary mathematics preservice teachers' perceptions of the benefits of implementing lesson study?

2. What are middle and secondary mathematics preservice teachers' perceptions of the challenges of implementing lesson study?

3. What are middle and secondary mathematics preservice teachers' perceptions of the impacts of implementing lesson study?

In this article, we provide the details of the study, discuss the findings and offer implications for mathematics teacher educators implementing lesson study in their teacher preparation programs.

Literature review

Since the early implementation of lesson study in the United States in which lesson study consisted of identifying a research theme, planning a lesson, implementing the lesson, recording observations from the lesson, reflecting on the lesson and then modifying the lesson before teaching it again (Lewis, 2000, 2002), lesson study implementation has varied. However, a common cyclical format of planning, teaching, observing and reflecting has remained consistent (Amador and Galindo, 2021; Lamb and Aldous, 2016). The interest and popularity of the use of lesson study in teacher education programs continue to grow, and as a result, there have been many studies conducted on a variety of research topics (Ponte, 2017). Previous research focused on the planning phase (Cavanagh and Garvey, 2012; Ponte et al., 2015), while others focused on the reflection phase of the lesson study cycle (Lamb and Aldous, 2016; Ricks, 2011). Studies have also examined the impacts lesson study had on PSTs' content knowledge, specialized content knowledge and pedagogical content knowledge (Akerson et al., 2017; Leavy and Hourigan, 2018).

More recently, studies have focused on the use of lesson study to support PSTs to notice students' mathematical thinking (Amador and Galindo, 2021; Amador and Carter, 2018). Professional noticing consists of observing and interpreting events in the classroom (Amador and Carter, 2018) which specifically connects to one of the lesson study phases, observation. Researchers have found that student noticing can increase pedagogical knowledge by formulating engaging lesson plans that include components of effective teaching (Jacobs et al., 2010; Philipp et al., 2007). While research on implementing lesson study in teacher education programs continues (Angelini and Alvarez, 2018; Lamb and Ko, 2016; Ponte, 2017), additional research must identify conditions that make lesson study a positive learning experience for PSTs to notice students' mathematical thinking.

There are, however, difficulties and constraints related to engaging PSTs in lesson study (Carrier, 2011; Chassels and Melville, 2009; Kannellopoulou and Darra, 2019; Sims and Walsh, 2009). One problematic constraint to successful lesson study implementation with PSTs is time pressure (Kannellopoulou and Darra, 2019; Sims and Walsh, 2009). The constantly changing school environment, particularly unexpected changes in the schedules of schools, should be considered with this constraint. When unforeseen changes in the instructional day or unplanned school events occur, long-term, detailed plans prepared by PSTs implementing lesson study are significantly impacted. These unforeseen occurrences can be especially disruptive if the lesson study is focused primarily on content, as PSTs may lose opportunities to teach the particular lesson they have developed. Consequently, another constraint to successful lesson study experiences with PSTs is shifting their focus from the delivery of the lesson to a focus on student learning (Kannellopoulou and Darra, 2019). The lesson study is about the mathematical understanding of the students rather than about the teacher (Lamb, 2015). In addition to the difficulties and constraints around implementing lesson study with PSTs, it may also be important to consider PSTs' perceptions of lesson study.

Preservice teachers' perceptions of lesson study

In a previous study related to PSTs' perceptions of benefits and challenges related to lesson study, Chassels and Melville (2009) reported that benefits included a greater insight into student needs, teacher collaboration, using different teaching and learning approaches with colleagues providing observations for areas of potential improvement and engaging in conversation with mentor teachers to receive and give feedback. The researchers found that challenges included time to engage in lesson study, that some candidates preferred to work alone and other organizational concerns. Chassels and Melville reported that benefits included a greater insight into student needs, teacher collaboration, using different teaching and learning approaches with colleagues providing observations for areas of potential improvement and engaging in conversation with mentor teachers to receive and give feedback.

In 2011, Carrier, who conducted a study with preservice teacher as part of an elementary science methods coursework, indicated that PSTs valued working collaboratively with peers, team teaching and peer feedback. The PSTs in this study recognized their lessons improved through lesson study because they could share ideas with one another in planning lessons. Further, they shared that they learned from one another during the team teaching of the lesson study, particularly in improving their teaching practices in the moment. Some of the PSTs' perceptions of the difficulties in working with one another included time to collaborate, hurting one another's feelings and feeling intimidated by team teaching in front of a peer.

As part of a large phenomenological study, Elipane (2011) found that lesson study provided an important opportunity for critical reflection for a particular PST, both personally and as part of a group. In reflection time, it supported further learning about teaching effectively. When analyzing the responses from this PST when they shared their observations of the lesson, Elipane noted, “Ascertaining others' perspectives assisted the ST [PSTs] into being able to consider things that he wasn't able to discern in his personal observations and in the enactment of his decisions” (p. 311). This particular PST, which was the focus of Elipane's (2011) report, emphasized the importance of the lesson study and the reflection. Given the potential implications of PSTs' perceptions of lesson study impacting their engagement in the lesson study process, we developed a study to learn more about their perceptions of the benefits, challenges and impact of lesson study in our program. We share the details of this study in the following section.

Methodology

To examine PSTs' perceptions surrounding lesson study, we designed a single case study with two embedded units of analysis (Creswell and Poth, 2018). The embedded units of analysis included a group of middle and a group of secondary mathematics PSTs in the internship year of a teacher preparation program at our private university in central Texas. The following sections provide an overview of our teacher preparation program, the participants who were PSTs engaged in lesson study, data collection and data analysis.

Program overview

The teacher education program at our university is a four-year program that begins with general education courses and includes methods courses, practicums and seminars which are typically separated by grade band. In the final year of the program, most PSTs complete a full-year internship in professional development schools. The PSTs who are interns (student teachers) are typically in schools Monday through Thursday, and they have seminars on Friday mornings with university faculty. While these seminars are separated by the grade bands related to certification areas, the three researchers, who teach middle and secondary mathematics education courses at the university and are the authors of this article, decided to conduct lesson study with middle and secondary mathematics PSTs in the second semester of their intern year.

Participants, lesson study description and data collection

Participants included nine (one male, eight female) middle mathematics PSTs and three (female) secondary mathematics PSTs. We clustered the participants into three lesson study groups. There were two groups of middle PSTs and one group of secondary PSTs.

We introduced PSTs to the lesson study cycle in one of the Friday morning seminars. A lesson study cycle can be defined by the four phases of study, plan, teach and reflect (Lewis, 2002). We used Can You Lift 100 Kilograms? (The Lesson Study Group at Mills College, 2022) as a video case study to highlight the phases of lesson study and allow for discussion about the lesson study cycle. We used the video case study to focus on PSTs' noticing of students' mathematical understanding and how the instructional plan connected to mathematical teaching practices and the research theme. After introducing the PSTs to lesson study using the video case study and providing examples of research themes, we explained that PSTs would be in lesson study groups and that each group would develop a research theme for their lesson study. Then, the three groups of PSTs each developed a research theme by responding to guiding questions from Lewis's (2002) Developing a Research Theme Protocol. We provide the research themes in Table 1.

Each group of PSTs considered the flexibility of the scope and sequence of classroom curriculum to determine in which cooperating teacher's classroom they would present the lesson. Then, they used a version of the Lesson Study Proposal given by the Lesson Study Alliance (2018) to write their lesson plan based on the topic provided by the cooperating teacher and their group's research theme over a period of several weeks. We modified the Lesson Study Alliance (2018) template by removing items related to unit planning, particularly because we focused on the MTPs (NCTM, 2014), and we also removed a prompt related to what the PSTs should post on the classroom board. We provide an overview of the PSTs' lessons in Table 1. One researcher served as a content expert to support the PSTs during their planning. One PST from each group volunteered to teach the lesson after the research lesson was designed by all group members, and the other PSTs in the group along with the researchers conducted a focused observation using an open, simple observation form on which observers could document the time and their observations as they saw fit. From our years of experience, we found a simple observation form with time and open space to write and illustrate was most effective in capturing the sequence and details of the lesson and students' thinking. When we had more complex observation forms with specific categories, observers seemed more focused on fitting observations into the appropriate categories. As such, we opted for an open, simple observation form. We also reminded PSTs who observed the research lesson to focus on noticing the actions and comments of students, not the PST who was teaching the lesson.

After the PSTs completed the research lesson, they responded in writing to seven prompts (see Table 2). Each group along with the researchers gathered for a discussion about the PSTs' experiences in the lesson study. While there were several collected data sources, in this article, we focus on the PSTs' perceptions of the beliefs, challenges and impacts of the lesson study, as reflected in questions 3, 4 and 5 from the PSTs' written responses.

Data analysis

The three researchers analyzed the written reflection responses by first establishing a coding procedure with initial codes. We independently read the PSTs' responses to questions 3, 4 and 5 and identified initial codes. Next, we met to discuss the codes and came to a 100% agreement on the final coding scheme. We each independently coded the PSTs' responses to prompts 3, 4 and 5 based on the established codes, and we met to compare our coding and discuss the emerging themes. We also discussed the differences between middle and secondary PSTs as we identified themes that emerged based on each research question.

Ethical considerations

There were four ethical considerations for this research study. First, the PSTs were engaged in this lesson study as part of the required coursework in their teacher preparation program. We addressed this ethical consideration by emphasizing the importance of professional development and learning about lesson study rather than the grades assigned for the completion of the work. Second, we also encouraged each PST to contribute to the lesson study. One way that we did this was to avoid assigning which PST would teach the lesson until the research lesson was fully designed. Third, we recognize that the responses to the final lesson study debrief prompts were self-reported, and as a result, we indicated that these are PSTs' perceptions of the benefits, challenges and impacts of lesson study. Lastly, we use pseudonyms throughout to protect PSTs' confidentiality.

Findings

Examination of the data sources revealed several themes related to benefits, challenges and impacts. These findings and themes are shared in the following sections, which are organized according to the research questions.

RQ1.

Preservice teachers' perceptions of benefits of lesson study

PSTs described the benefits of implementing lesson study in two areas: observation and collaboration. PSTs saw the benefit of participating in lesson study as an opportunity to conduct very focused, intentional observations of students during the lesson implementation. These observations allowed them to capture details that they may not have otherwise seen as a part of regular instruction. For example, Sally noted, “Relating this to professional growth and my greatest benefit, it was really cool to, in a way, step out of my role as their teacher and just watch them be students and observe their interactions and discourse. I think it is important to see your students as your students, but it's also important to see them as little people! You have to know your students to really see how you can best serve them.” Other PSTs indicated the importance of observing in a classroom they had not been in before to see how the students reacted to the instruction and how it helped them grow confidence in their mathematical abilities along with observing student reactions, how they think and their conversations so they could see the impacts of the lesson on the students. Audry provided further evidence, as this was her regular classroom placement and knew the students, yet she did not teach the lesson. She described being able to hear outside observations of the lesson of her students as beneficial, stating, “It forced me to think back on my own practices and how I would adjust them moving forward.”

The second area of benefit related to collaboration was PSTs' collaboration in the planning process and the reflection process. They saw the benefit of sharing instructional ideas and of working together to improve lessons. Molly shared, “The greatest benefit from participating in lesson study was the practice of planning out the details of a lesson from start to finish, alongside other educators. We were able to collaborate and put our ideas together to create a lesson that had much intention and purpose toward a specific goal. This gave me more confidence to approach big goals with individual lessons as I move forward in my teaching experience.” The PSTs shared that through collaboration, you get reasonable ways to improve a lesson, ways to modify a lesson to engage students and it built their confidence in their understanding of mathematics. They saw these benefits for this particular lesson and how the benefits would or could impact their future lesson planning. Emma shared, “This was a result of collaboration amongst colleages [sic] which I believe is very powerful in creating a lesson that is most beneficial to students.” This collaborative approach also allowed the PSTs to see the importance of the research approach to studying student learning. Jackie noted, “The greatest benefit of the lesson study was learning a new way of working with other teachers to research students.”

The PSTs in this study saw benefits of lesson study in conducting observations of the students and in collaborating. They perceived their observations to be an opportunity to learn more about students and work with them in ways that differed from their internship, capturing small details by watching one small group of students for the duration of the lesson. They also perceived opportunities to work together as PSTs in small groups on a particular lesson to be a benefit of lesson study. They shared that this helped them learn from one another and build confidence.

RQ2.

Preservice teachers' perceptions of challenges of lesson study

PSTs described the challenges of implementing lesson study in two areas: planning and being a nonparticipant observer; however, our findings reflected differences related to whether they were middle or secondary school PSTs. For example, all three of the secondary PSTs said that planning the research lesson was the greatest challenge, and the majority of middle school PSTs discussed the structure of observing related to not being able to interact or talk with the students during the lesson. To illustrate the secondary PSTs' challenge of planning, Audry, who taught the lesson, shared, “The greatest challenge in participating in lesson study was making sure I kept everything focused on our research goal.” Another secondary preservice teacher talked about the challenge of aligning their lesson to fit the scope and sequence of the school and how it would change from day to day before they were to teach, so they had to change plans and be very flexible, but that it was very challenging to do so. Shanda explained, “It was a challenge to make our goals a reality,” and others shared the challenge of designing problems that were accessible to students and fit their goal of problem-solving.

The middle PSTs, however, perceived the challenges differently and found it challenging to serve as a nonparticipant observer. Jennifer, a middle school preservice teacher, indicated, “I believe the greatest challenge in participating in this study was having so many adults in the room, but only one could talk. There were instances where some of the observers had to restrain from saying something to the students because they saw the confusion that some of the students had.” Many wanted to talk with the students either to assist in their learning, to help them with following directions or to ask more about their thinking. Observers in lesson study are tasked with not interacting with the students to focus on the observation. Molly, the middle school preservice teacher whose class was where the lesson was taught even though she was not teaching the lesson, lamented, “The greatest challenge from participating in lesson study was not being able to interject as an observer. There were things that I noticed as an observer that the teacher could not see or was not thinking about.” In our teacher education program, we emphasize the Mathematics Teaching Practices (NCTM, 2014) along with the related teacher actions and student actions. For example, when teaching PSTs to ask purposeful questions (MTP 5), or facilitate meaningful discourse (MTP 4), we explain the importance of listening and responding to students to intentionally guide the conversation about the mathematics. Eliciting evidence of student thinking (MTP 8) also requires PSTs to frequently engage with and interact with PSTs. In the case of the lesson study, it was interesting to discover that, in light of preparing PSTs to adopt these practices, the request for them to be nonparticipant observers during lesson study was a challenge for them.

The PSTs in this study saw challenges of lesson study in planning and in being a nonparticipant observer, and these perceived challenges varied by grade band. The secondary PSTs shared about the challenges of staying focused on the research theme.

RQ3.

Preservice teachers' perceptions of impacts of lesson study

PSTs described the perceived impacts of implementing lesson study related to transferring to teaching practice in three distinct phases of the lesson study cycle. Impact occurred for some during the planning of the lesson, others experienced the greatest impacts during the lesson specifically related to observations they made during the research lesson and still others noted the greatest impacts at the end of the experience during the reflection particularly highlighting the importance of professional collaboration. Some of the PSTs identified an impact in just one of the phases, while others identified an impact in two or all three phases.

For example, related to the planning phase, Krista shared that she saw that it was important to have overarching goals in the lesson for the students, stating, “I am preparing students ultimately to be contributing members of society, and I need to keep that in mind as I am creating lessons. Collaboration and group work is something they will see in life, and preparing them to work well in a group is a life skill. In the future, I will structure lessons to have students start to work on and hone in skills that will prepare them in the real world.” Carrie added that she found value in taking time to “thoroughly plan out how we would assess student, help students, and challenge students.”

In reflecting on the lesson after it was taught, both Audry and Jackie described the impacts related to how it made them think about how student learning was affected. Jackie said she felt that experiencing lesson study impacted her professional growth by showing her different ways to assess students, see how well students were learning and see how students react in different ways. Further, Marla said it made her think about how to increase student collaboration in her classroom particularly related to purposeful seating students in different ways, and that using tasks was important to promote productive struggle and student collaboration. As Shanda considered the impacts engaging in lesson study had on her, she shared, “It was a lot of fun to do. It helped me to be more attentive to detail when planning lessons and encouraged me to reflect on why I choose to do certain things in lessons. The reflection allowed me to hear different observations and learn to look for those in the future.” Jennifer added that it gave her “some insight on anticipating student responses as well as anticipating parts of the lesson that students might struggle with.” Several mentioned how important collaboration was with their peers, with Emma stating, “Collaborating with my colleagues about the results of a lesson also taught me new ways to be more intentional in my instruction.” Thus, Emma also indicated how it would impact her future instruction.

Several PSTs also described the impacts related to their observations during the lesson. They felt it helped them understand the students much better and to be able to focus on student thinking. Jennifer explained, “This opened my eyes to some of the rich conversations as well as deep confusion students have when the teacher is not present.” It was also interesting that Molly, who was usually the PST teaching in the classroom for her internship, indicated that being an observer during this lesson gave her “the opportunity to see my own students from a different perspective while conducting purposeful observation.”

The PSTs in this study saw the impact of lesson study in three phases of the lesson study cycle, particularly as it related to transferring what they learned to their practice. When we asked this question, some PSTs focused on the planning, some on the observations they conducted during the teaching of the lesson, some on the reflection after the lesson and other PSTs focused on two or all three of these phases.

Discussion

The purpose of this study was to explore middle and secondary mathematics PSTs' perceptions of the benefits, challenges and impacts of implementing lesson study. The findings suggest that PSTs saw benefits in conducting focused, intentional observations of students and that PSTs valued collaborating with their peers in lesson study. Further, they pointed to the challenges of planning and of being a nonparticipant observer; interestingly, trends showed the challenges were different from PSTs in different certification grade bands. Lastly, PSTs reported their new understandings related to teaching would transfer to their practice in different ways. Some PSTs reflected on how lesson study would transfer to their future planning, some in observations and others in professional collaboration. It seems important to discuss the different perspectives of middle and secondary PSTs related to the challenges of lesson study.

In contrast with Chassels and Melville's (2009) study in which challenges included time to engage in lesson study, that some PSTs preferred to work alone and other organizational concerns, we found two primary challenges of planning and being a nonparticipant observer which varied by grade band. In the current study, middle PSTs focused on noticing and indicated that they wanted to interject in the lesson to engage in discussion with the students, while secondary PSTs focused on content knowledge and planning. The middle PSTs' focus on students seems to fall in line with findings shared by Amador and Weiland (2015), who indicated that using lesson study in PSTs' field experiences may support a focus on student mathematical thinking. In our study, middle PSTs focused on student mathematical thinking and wanted to further engage with students to learn more and to help them. However, secondary PSTs focused on planning, much like some of the participants in Leavy and Hourigan's (2018) study of elementary preservice teachers conducting lesson study focused on early number concepts. We suspect the PSTs' differing perspectives in our study could relate to PSTs' coursework, their previous field experiences or their research planning goal. In our teacher preparation program, the middle and secondary PSTs take many general education courses together, but their methods courses and internship-related courses are separated by grade band. It could be that this differing coursework or differing field experiences influenced the PSTs' perspectives on the challenges of lesson study.

Differing research goals may have also played a part in PSTs' perspectives of the benefits and challenges of lesson study in the current study. The middle PSTs groups' research goals were about student confidence and student collaboration. Interestingly, the middle PSTs wanted to interact to help students or address their confusion, but we instructed them that they had to be nonparticipant observers. Because the middle PST groups focused on research themes that were more socially focused when compared with the secondary PSTs who focused on problem solving and perseverance as their research theme, it seems plausible that this would reflect in their perceptions of the benefits, impacts and challenges of lesson study, as they wanted to talk with students. Further, because the secondary PSTs focused on a more content-related goal, it seems possible that this would be reflected in their perceptions of lesson study. In our discussions during data analysis, we wondered if secondary PSTs may not have been comfortable in other content areas outside of those that they had taught. For example, this lesson study experience was in a geometry course, but some PSTs had primarily had experiences in algebra or precalculus placements. Certainly, it would be important for mathematics teacher educators to consider how to address potential unfamiliarity with the content, particularly with PSTs in lesson study.

It may also be important to note that there were several different takeaways for impacts with some PSTs focused on planning, some on teaching and some on future practice. Further, some PSTs focused on two or all three of these areas. Again, because PSTs have had different experiences in their coursework, field placements and research themes, it seems reasonable that they would have different perspectives on the impact of lesson study. It is possible that while some PSTs made decisions about their current and future teacher related to their perceived impact of lesson study, some PSTs did not see how lesson study impacted or could impact their planning, teaching or future practice.

Implications for practice

Findings from this study have three implications for mathematics teacher educators using lesson study to engage PSTs in reflection about lesson design, instructional delivery, student learning and revising particular lessons. First, mathematics teacher educators should consider that the middle and secondary PSTs in this study indicated that the benefits of lesson study included the power of observing students in unique ways that may be different from observations while they are teaching in their internship and in a distinctive opportunity to collaborate with their peers in developing a lesson. When using lesson study during internship, this is particularly important for mathematics teacher educators to consider, given that the PSTs are in their internship and typically only work with cooperating teachers at this point. Writing lessons with other PSTs is a departure from writing lessons with cooperating teachers in PSTs' field experiences and from collaborating on lesson planning in other field experiences, and it could allow PSTs to work together to ensure the inclusion of techniques and strategies learned in university settings.

Second, mathematics teacher educators should consider that middle and secondary PSTs may see different challenges with lesson study based on their grade band. In this study, secondary PSTs indicated that planning the research lesson was the greatest challenge, while many of the middle school PSTs noted that it was difficult not to talk to the students during the lesson. This information could be useful in introducing PSTs to the purpose of lesson study and preparing them for their roles during the teaching of the lesson, particularly as they select their research theme. Because the middle PSTs shared the difficulties of not communicating with students during the lesson when they had a socially focused research theme, while secondary PST shared the difficulties of planning when they selected a research theme about problem solving and perseverance, it seems important to consider the practical implications of this. Mathematics teacher educators could have PSTs predict what they may find difficult related to the planning and teaching of the lesson when they consider their chosen research themes. This finding could also support course discussion about the importance of teacher observations in the classroom, as teachers must consider when and how to respond to students during mathematical tasks.

Third, mathematics teacher educators can reflect on the different areas of impact PSTs may see when engaging in lesson study and how PSTs may transfer what they learn into their practice. In this study, some PSTs reflected on the impact lesson study would have in three distinct phases, including on their future planning, others on their instructional delivery and others on their reflections with colleagues. Mathematics teacher educators may want to draw attention to each of these three areas when using lesson study with PSTs, highlighting the ways lesson study can support their planning and teaching. Mathematics teacher educators can consider discussing these phases with PSTs either before or after PSTs conduct the lesson study.

Limitations and further research

This study contained two limitations. First, our study participants included a small group of middle and secondary mathematics PSTs from one university in central Texas. Additional research could include PSTs in our program in the elementary grades, or we could expand to include additional PSTs from other university settings. This would allow us to further explore the benefits, challenges and impacts of lesson study according to mathematics PSTs. We could also include PSTs in other content areas.

A second limitation of this study was that PSTs were not able to reteach the lesson, which is typically the next step in the lesson study cycle (Lewis, 2000, 2002). The inability to reteach the lesson, which would have included adjustments to the lesson, may have influenced what PSTs believed to be the benefits, challenges and impacts. We are conducting additional research to gather more information as we explore critical elements of the lesson study cycle when facing potential constraints such as time and collaboration with school partners.

Conclusion

The Association of Mathematics Teacher Educators (2017) indicated that “Well-prepared beginning teachers of mathematics have foundations of pedagogical knowledge, effective and equitable mathematics teaching practices, and positive and productive dispositions toward teaching mathematics to support students' sense making, understanding and reasoning” (Standard C.2, p. 58). Lesson study is a powerful and authentic experience for providing that foundation for PSTs' professional growth as they develop their understanding and practices of teaching mathematics. We explored middle and secondary mathematics PSTs' perceptions of the benefits, challenges and impacts of implementing lesson study. We found that PSTs perceived benefits of lesson study in focused student observations and in lesson design. Second, PSTs in this study perceived the challenges of lesson study to be in the planning and in acting observing as a nonparticipant observer during the lesson. Third and finally, PSTs indicated impacts were in three phases: planning, observing during teaching and reflection. In alignment with previous studies that revealed PSTs' perceptions of lesson study (Carrier, 2011; Chassels and Melville, 2009), our findings indicated that PSTs perceive benefits and challenges to conducting lesson study. In our case, they saw those benefits in observations and in lesson design. However, they perceived challenges in planning and not being able to interact with the students they were observing. These study findings contribute to mathematics teacher educators' understanding of PSTs' perceptions of lesson study, which may be useful for those who are implementing lesson study in their teacher education programs.

Table 1

Research lessons overview

Groups (pseudonyms)	Research theme	Lesson topic	Lesson overview
Middle group 1: *Jennifer Emma Laura Sally	For students to develop confidence in their own abilities inside and outside of the classroom	Attributes of Quadratic Equations	• Basketball Free Throw Video Introduction • Informal Vocabulary Review • Small Groups Create Free Throw Videos and Describe Attributes • Whole Class Views Videos for Compare/Contrast • Whole Class Summarizes Attributes and Discusses Graphs
Middle group 2: *Krista Jackie Jason Molly Marla	For students to learn to work collaboratively in a group so that they will be cooperative members of society	Nets of Prisms and Pyramids	• Introduction with Transformers • Small Groups Explore and Identify Attributes of Nets • Partner Practice to Build Nets • PST Presentation of Nets with Name, Shape of Base, Lateral Faces • Exit Ticket Question
Secondary group: *Shanda Audry Carrie	Students will be able to problem-solve and persevere through mathematical questions when the answer is not readily available to them	Surface Area of 3-Dimensional Shapes	• PST Reviews Surface Area • Students work in Groups of 3 on 3 Surface Area Problems at Stations in the Hall • Turn and Talk to Share Something the Student Learned

Note(s): *Indicates the PST who taught the lesson

Source(s): Table created by the authors

Table 2

Final lesson study debrief

1. A) What did the team hope to observe during the lesson?

B) What was actually observed during the lesson?

2. What were the major points raised during the post-lesson discussion?

3. What impact did lesson study have on your professional growth?

4. What was the greatest benefit from participating in lesson study?

5. What was the greatest challenge in participating in lesson study?

6. Would you participate in lesson study again if given the opportunity? Why or why not?

7. Please share any additional comments and/or observations from your experience with lesson study

Source(s): Table created by the authors