Supporting collaborative inquiry skills through lesson study: Investigation of high school mathematics professionals

Abstract Lesson study provides collaborative platforms for enhancing mathematics teachers’ knowledge and pedagogical skills. Research on the collaborative learning of professional programs is well documented and specifies that it can support mathematics teachers in transferring their pedagogies, knowledge, and attitude about teaching mathematics to improve students’ achievements. This study examined the role of lesson study projects in developing collaborative inquiry skills among high school mathematics teachers in Saudi Arabia, as well as the impact of lesson study projects on students’ mathematical learning. This study implemented a scale with certain design features to evaluate the level of collaborative inquiry skills of two groups of high school teachers: Those who attend lesson study projects and those who never do. The study analysis uncovered that high school mathematics teachers who engage in lesson study professional development projects demonstrate higher collaborative inquiry skills than those who never engage in such professional development experiences. A significant finding of this study determined that lesson study experience had a positive impact on the improvement of students’ mathematics learning.


Abstract:
Lesson study provides collaborative platforms for enhancing mathematics teachers' knowledge and pedagogical skills. Research on the collaborative learning of professional programs is well documented and specifies that it can support mathematics teachers in transferring their pedagogies, knowledge, and attitude about teaching mathematics to improve students' achievements. This study examined the role of lesson study projects in developing collaborative inquiry skills among high school mathematics teachers in Saudi Arabia, as well as the impact of lesson study projects on students' mathematical learning. This study implemented a scale with certain design features to evaluate the level of collaborative inquiry skills of two groups of high school teachers: Those who attend lesson study projects and those who never do. The study analysis uncovered that high school mathematics teachers who engage in lesson study professional development projects demonstrate higher collaborative inquiry skills than those who never engage in such professional development experiences. A significant finding of ABOUT THE AUTHOR Maha Saad Alsaeed: Alsaeed, PhD (Ohio University), is a Saudi Arabian assistant professor of mathematical education. She has 16 years of academic experience. She has published internationally through well-known publishers, such as Taylor & Francis. She has attended various academic events in Saudi Arabia and the USA. Her research interests include mathematics, education, technology and professional development. Her recent administration activity involved the advancement of pre-service teaching practices. Currently, she is engaging in various research projects that are exploring the advancement of teaching mathematics through science, technology, engineering and mathematics (STEM) education. In addition, she has worked as an academic department head.

PUBLIC INTEREST STATEMENT
Human society and educational systems are going through major shift due to extensive daily innovations and technological integration. Collaborative learning in professional development educational setting has been well documented in literature to support mathematics teachers facing educational challenge, transferring their classroom practice, and improving students' achievement. This study examines the role of lesson study projects, a Japanese core of educational development, in developing collaborative inquiry skills among high school mathematics teachers in Saudi Arabia and its impact on students' mathematical learning. The results of the current research imply that lesson study experience improve mathematics teachers' collaborative skills and support their engagement of professional dialogue and instructional reflection within their community. A consequence implication for educational stakeholders and policymakers is the importance of integration of content-based and collaborative-based professional development programs to assist mathematics teachers in transforming their practice during the innovative and challenging era of education.

Introduction
Lesson study has been recognized internationally as a Japanese educational development platform with strong efficacy in terms of classroom practices and teaching transformation. It is a collaborative professional development model that allows teachers to work together on a particular topic to improve their pedagogy, knowledge, attitude, and classroom practices. Lesson study is practice-based, research-oriented, and content-oriented; it helps teachers improve their knowledge through continuous dialogue and collaboration (Huang & Shimizu, 2016).
In the late 1990s, the lesson study model was introduced and became well known in the United States after the release of the TIMSS video study ; it then spread and was adapted globally. In Saudi Arabia, the Ministry of Education is currently undertaking steps toward the development of mathematics teaching, including the enrichment of school professional development cultures (Al-Abdullatif & Alsaeed, 2019). Through the Saudi-Japan 2030 vision and partnership put together in 2017, both countries are aiming for "achieving the national socioeconomic visions of both countries by building on commonalities, activating complementarities and maximizing synergies" (Ministry of Economy and Planning in Saudi Arabia). Both countries are focusing their partnership growth on three partnership pillars: Diversity, innovation, and soft values. These pillars include the reinforcement and adaption of the Japanese lesson study professional development system.
Beyond political ambitions, lesson study has been used to establish sustainable professional development strategies throughout the school system of the Kingdom of Saudi Arabia. The movement was used to support school systems in building their own professional development structures by engaging all teachers, administrators, and school principals in this system and helping them transform their learning philosophies and pedagogical perspectives through continuous discussion and reflection. The Kingdom of Saudi Arabia is pushing the idea of apprenticeship teaching through a collaborative program that engages all school staff in building robust professional communities of practice (Ministry of Economy and Planning in Saudi Arabia). Through this community, teachers in Saudi Arabia have learned to work together to improve their pedagogical practices.
The meaning of teaching collaboration is crucial. It indicates that teachers work together, learn from each other, engage in critical dialogue regarding common purposes and pedagogical concerns, and support each other in addressing challenges and issues that reflect on their own practice (Robutti et al., 2016). Teachers who engage in collaborative inquiry with their community of practice are able to change their practice and improve their classroom environment in support of their students' learning (DeLuca et al., 2014). Whether collaboration occurs through online or face-to-face platforms, such environments can support a teacher's social and cultural interactions and empower their communication skills for the 21 st century (Corporan et al., 2020;Sumtsova et al., 2018).
Research on the collaborative learning of professional programs is well documented, and results have shown that the act of collaborative inquiry supports teachers of mathematics in shifting their pedagogies, knowledge, and attitudes about teaching mathematics, thus leading to improved student achievements (Darling-Hammond & Richardson, 2009;Lawrence & Chong, 2010).
The current study focuses on the collaborative inquiry (CI) structure of a professional development program in mathematics education. In addition to highlighting the role of these skills in lesson study programs, this study also particularly aimed to examine the effectiveness of lesson study projects in developing the CI skills of Saudi high school mathematics teachers. Therefore, the following guiding research questions were investigated: (1) )What is the impact of lesson study professional development program on developing CI skills among high school mathematics teachers?
(2) )Do variables such as gender and teachers' years of experience affect teachers' level of CIS when they are engaged in the lesson study professional development project?
(3) )To what extent do high school students' scores in the aptitude test improve as a result of their respective mathematics high school mathematics teachers' participation in the lesson study professional development project?
The present research documents several key contributions made to the fields of teachers' professional development innovations. First, lesson study has been used to engaged experienced teachers to improve their teaching routine practices (Ermeling & Graff-Ermeling, 2014;Takahashi & McDougal, 2016) and to support initial teachers to deepen their understanding of the curriculum and overcome new workload challenges (Chassels & Melville, 2009). However, investigation of the comparison between experienced and beginning teachers has not received much attention in term of lesson study research designs (Coenders & Verhoef, 2019). Thus, this present practical advantage advancing knowledge in this area. Additionally, gender engagement in the work environment has been investigated by previous research and has revealed variations in teachers' engagement with students, curriculum, and work load (Topchyan & Woehler, 2021).
For example, Klassen and Chiu (2010) indicated that female teachers had greater workload stress, greater classroom stress from student behaviors, and lower classroom management selfefficacy than their male colleagues. In term of CI, no studies have been found to investigate the differences between male and female teachers in their level of collaboration within the professional community of practice.
Moreover, this study provides an opportunity to advance our knowledge of how lesson study projects may impact students' mathematical learning. With this study, we aimed to increase the body of knowledge about professional growth in the context of lesson study models and to distinguish elements that contribute to the improvement of professional collaboration inquiries among mathematics teachers. This not only enables us to determine whether participating in lesson study programs leads to changes in teachers' collaborative skills, but also allows us to determine whether a relationship between these two constructs exists, as well as to examine possible explanations using the quantitative data.

Theoretical foundation and perspectives
Lesson study provides platforms for enhancing teachers' collaboration around their pedagogical concerns, inquiries, and learning. It supports teacher connections in collaborative learning by engaging them in discussions about learning gaps and everyday classroom problems.
A number of studies have indicated the significant impact of lesson study on transforming mathematics teaching (Chen & Yang, 2013;Stigler & Hiebert, 1999), leading to it becoming an effective professional development tool (Borko et al., 2010), improving teacher beliefs and knowledge of mathematics, supporting mathematics instruction (Lewis et al., 2012), and improving student learning (Perry & Lewis, 2011). In the current study, lesson study cycles through four professional development steps. .
• Investigation and goal setting: Communities of professionals study current student characteristics, knowledge, and skills, set long-and short-term goals for their students, identify learning gaps and difficulties, and then formulate their research theme. An example of a long-term goal is the improvement of students' problem-solving skills across different mathematics topics.
• Planning: Teachers collectively plan lessons to address these goals and fill the gaps in students' knowledge and skills. For instance, a group of teachers may plan some extra tasks and demand real-world problems to be addressed and implemented with their students.
• Research lesson: One team member teaches a lesson while the other members observe and collect data on student learning.
• Reflection: The team discusses the results from the research lesson, provide implications, write a report about the redesign of the lesson, and share these findings to the broader community of teachers. The shared findings allow team members to recognize gaps in their lesson plans and support them in refining their instructional practice (Fernandez & Yoshida, 2004;Lewis, 2002).
The action of collaboration is obvious in the lesson study cycle. Studies have shown that CI relies on two components: Working together in the teaching community and deepening the understanding of a specific pedagogical and subject matter discussion (Nelson & Slavit, 2008;Ontario Literacy and Numeracy Secretariat, 2010). In addition, CI encompasses multiple attributes, as Nelson and Slavit (2008) indicated: • Practitioners sharing a vision of high-quality learning and teaching through interaction; • Through inquiry groups, teachers share the same needs, goals, and norms; • Teachers work together to design the inquiry focus; • Teachers are supported by experts in the design and implementation phases of teaching practices; • There is an action-oriented phase in the local context, and then teachers transfer their knowledge to a broader context.
The current study focused on the four important characteristics of CI: Teachers as learners; a shared vision; inductive, deductive, and abductive reasoning; and the cyclical process. These four characteristics and sub-skills are embedded in the four stages of the classroom professional development cycle: Investigation, Planning, Research Lesson, and Reflection. The following studies in the literature ground the rationalizations behind using the four characteristics to describe and develop the scale for measuring CI skills for high school mathematics teachers.
Teachers are their own learners through collaborative inquiry: The CI cycle involves connecting personal experiences and self-reflection to the entire professional group's learning and actions (DeLuca et al., 2014). CI skills position teachers to learn about their students' learning and thinking processes. Teachers can improve their planning, instruction practices, and assessment models using the data they generate from constructive dialogue with other teachers. The knowledge teachers have about pedagogy and content is fixed, but can be continually refined based on what is happening in their classroom. The term flexibility describes how teachers can use inquiry to improve and change their way of teaching by interacting with other teachers based on the learning environments they encounter (Ontario Literacy and Numeracy Secretariat, 2010) Teachers develop a shared vision of learning and teaching through collaborative inquiry: Through the CI process, practitioners work collectively to develop a shared vision of high-quality teaching and learning. Advancement of teaching practices does not happen without reconstructing the inquiry groups' beliefs, attitudes, and goals regarding effective teaching and learning (Nelson & Slavit, 2008;Putnam & Borko, 2000). Engaging with a professional community helps teachers shift their ways of thinking about non-traditional practices and evolving shared ambitions. As Putnam and Borko (2000) pointed out, that inquiry community has always been a resisted system because of the powerful support systems that are resistant to fundamental change and development. The beginning of each change happens when teachers move forward and rethink the way they view learning and teaching.
Collaborative inquiry drives inductive, deductive, and abductive reasoning: Through CI, teachers engage in pedagogical and curriculum reasoning which, according to Shulman (1987), is a set of processes that help teachers shift their understanding about their own pedagogical knowledge of teaching from initial understanding to new comprehension (Pella, 2015). Three kinds of pedagogical and curriculum reasoning are involved in a CI professional development program; the discourse between practitioners can be inductive, deductive, or abductive (Ontario Literacy and Numeracy Secretariat, 2010). Inductive reasoning occurs when teachers work together to review classroom instances and identify convergent themes. Teachers then work inductively to draw conclusions regarding these themes. For example, teachers in an inquiry group may test various ideas by reviewing evidence from students' homework, drawing conclusions about what students know and what is misunderstood. Deductive reasoning occurs when an inquiry group tests a practice that is widely known to be effective with their students. For example, teachers might include rich classroom discourse with their students and may work with other teachers to discuss how they are doing when they apply that practice with their students. Abductive reasoning is a mixture of the other two types of reasoning, helping teachers to form and test different pedagogical hypotheses at the same time. For example, teachers may begin with a hypothesis that if students use technology in learning, they will attain better grades. Then, teachers can test that assumption by reviewing the grades of students who are using technology. The team of teachers may also need to review the body of literature to verify their assumption. The conclusion reached from the teachers' actions in this regard is their best decision based on the given knowledge and evidence (Rapanta, 2018).
Teacher collaborative inquiry is a cyclical process: Through CI, teachers engage in a cyclical process of knowledge and pedagogical generation (Nelson & Slavit, 2008;Ontario Literacy and Numeracy Secretariat, 2010). CI often involves the cyclical process of professional learning. Teachers work in groups to identify challenges in the curriculum or in the pedagogy, work together to set a plan for a teaching experiment, scaffold the experimental process, and make improvements in the district and school . DeLuca et al. (2014) reviewed 42 studies of CI professional development and demonstrated that these programs commonly engage teachers in a process that varies from three stages to a maximum of 11 stages. These stages share three designing features: Dialogical sharing, taking action, and reflection.

Participants
The current study investigated the impact of lesson study projects on the development of CI skills among Saudi high school mathematics teachers from a small city in the Eastern province of Saudi Arabia. Teachers in Saudi Arabia are annually requested to voluntarily participate in local lesson study projects that are administrated within the Saudi professional development system for teachers. Once a group of teachers from a specific content area, and usually from the same school, have accepted the project, they meet regularly and collaboratively create their group goal or learning gap to be researched and then work together to achieve this goal through the lesson study professional development cycle, as addressed by lesson study literature (i.e., Fernandez & Yoshida, 2004;Lewis, 2002). Typically, the study group aims are reached within the same year, which is the standard duration of the lesson study program in Saudi Arabia. In their final meeting, the teachers typically present their successful lesson study to their lesson study group, or sometimes give a joint presentation with mathematics teachers from other school districts.
The targeted population in this study was high school mathematics teachers in Saudi Arabia and their high school students. To undertake the process of the investigation, the researcher first obtained ethics approval from the Ministry of Education in Saudi Arabia. To obtain the research sample, the researcher worked with some high school administrators to locate mathematics teachers who either participated or not in lesson study projects. The researcher then contacted the intended teachers to respond to the scale of the study. Finally, these participants assisted in reaching their high school students to fill out a survey in a link sent to them during school time with the assistance of their teachers. The schools' administration and researcher confirmed to students that their responses were completely voluntary and that revealing their grades and information would be used solely for research purposes.
Based on the previous process, the final research sample consisted of 142 high school mathematics and 262 respected 12th grade high school students from a small city in the Eastern province of Saudi Arabia. The details of the research sample is presented in Table 1.

Data collection and instruments
This study applied the collaborative inquiry scale (CIS) to examine the level of CI skills the high school mathematics teachers acquired. The CIS, which was prepared by the researcher, consisted of 33 phrases measuring the extent to which mathematics teachers utilized CI skills and worked collectively to improve their teaching of mathematics within their professional community. The scale consists of four dimensions: Investigation, Planning, Research Lesson, and Reflection. Participants were asked to rate their level of CI skills on a five-point Likert scale, as follows: 1 = Not at all, 2 = To some extent, 3 = To a moderate extent, 4 = To a great extent and 5 = To a very great extent.
The choice to use these four dimensions was based on two important reasons. First, typical mathematics teachers practice these four instructional planning events or processes. Second, these four phases are the planning phases that usually exist in most standard lesson study professional programs (DeLuca et al., 2014). Thus, a teacher may be able to answer the items of the CI scale and self-rate their level of collaborative inquiry skills, regardless of whether they had or had never engaged in lesson study professional programs. The items presented in the four dimensions demonstrate the participants' level of CI experiences and skills, as well as how they perceived themselves depending on whether they had ever engaged in lesson study professional development programs.
The dimension Investigation refers to how, through their professional community, teachers examine and negotiate their initial understanding of the real meaning of effective teaching and learning of mathematics. As indicated by Nelson and Slavit (2008) and DeLuca et al. (2014), professional development with collaboration inquiry fosters ongoing dialogue among teachers from the starting point to the end of the professional development cycle. The dialogue may involve many topics that concern the teaching of mathematics, such as standards of mathematics education, new trends in mathematics education, and movement of mathematics education reform. Thus, this dimension explores teachers' current knowledge, beliefs, attitudes, pedagogy, and their vision of effective teaching and learning of mathematics. An example item of this factor is presented in item I1: "I was inspired and motivated by my peers to improve my mathematics instructional practices." This item, as illustrated in figure 1, is related to the CI trait and is a part of the CI skill: Teacher as a learner and developer of a shared vision. The items in the Investigation construct describe the initial formulation of teachers' beliefs and attitude as they communicate and engage with their community of professionals.
The second dimension of the CIS examines teachers' level of inquiry collaboration while they interact with mathematics topics through ongoing lesson planning (Butler & Schnellert, 2012;Nelson & Slavit, 2008). The dimension Planning refers to the phase where, after generating and discussing the meaning of effective teaching and learning of mathematics, the teaching professional community digs deeply into examining specific and challenging mathematics lessons and topics. In this phase, the teachers select a challenging lesson or learning concept to explore in detail and formulate a shared goal regarding how they might treat the topic in their classrooms.
An example item of this factor is presented in item P2: "I collaborate with my colleagues to set up a goal about the intended level at which we wish our students' quantitative skills to be." This item, as illustrated in Figure 1, is related to the CI trait and is part of the CI skills cyclical processes, since goal setting may evolve over time and take more than a single step. The item also describes how teachers set up goals through their community to improve the current mathematics performance of their students. Similarly, in the Research Lesson dimension, teachers may engage in implementing previously discussed lessons regarding specific mathematics topics, while other teachers observe the implementation and collect data. This dimension evaluates teachers' levels of collaboration as they take action, enact new learning strategies, and adapt innovative classroom tools (Butler & Schnellert, 2012;Nelson & Slavit, 2008).
The Reflection dimension in the CIS refers to the stage when the teachers think about their own practices and use the data they have generated from interacting with other teachers through constructive dialogue to improve their teaching practice. The items in the fourth dimension bring together all four CI characteristics and traits to describe the level of collaborative inquiry skills that teachers exercise within their community. Figures 1 and 2 demonstrate the design features of the CIS model, as well as the interrelations between the items in the scale and the four characteristics of the CI skills. The CIS in its current structure sought to allow teachers to self-observe their CI level through their instructional planning cycle.
To measure mathematics learning among high school students, this study used data from the results of the quantitative section of the general aptitude test (GAT) to measure the growth of high school students' mathematical knowledge as a result of their teachers' engagement in lesson study projects. The GAT, usually given four times during high school, is one of the aptitude tests employed as a prerequisite for entering universities and obtaining jobs in Saudi Arabia. The test consists of two sections: The verbal section and the quantitative section. The quantitative section scores were offered willingly by high school students through the electronic survey sent to their schools.

Item generation and content validity
The present study applied the CIS scale to examine the level of CI skills that high school mathematics teachers acquired. According to Warner (2007), content validity is crucial phase in scale design process to ensure whether test items represent all theoretical dimensions or content area that must be tested and such information should be verified by experts to decide whether content coverage is completed. As previously discussed, the CIS scale was developed by the author using the theoretical model presented earlier linking four dimensions to one construct termed collaborative inquiry skills with four dimensions.
The content validity of CIS scale was evaluated to determine whether all important aspects were covered, identified, and essential; items that were not desirable in specific construct domains were excluded. Thus, the final version of the scale was tested using a panel of experts in the field of mathematics education. The panel contained mathematics education professors, as well as mathematics teachers, who worked with the researcher to examine the clarity, content, and applicability of usage of items with intended mathematics teachers. Afterword, a pilot study was conducted with 60 male and female high school mathematics teachers, who were randomly chosen from the main sample of the study, to calculate reliability coefficients. The results of the analysis are provided in the following paragraphs.

Confirmatory factor analysis (CFA)
Following the content validity process, CFA was performed to test the structure validity of Collaborative Inquiry Skills (CIS) and to test how the observed variables are related to one latent variable (CIS) using Lisrel 8.8 software. To assess the scale goodness-of-fit, the following fit indices were reported: normed chi-square (χ 2 /df), the goodness-of-fit index (GFI), adjusted goodness-of-fit index (AGFI), normed fit index (NFI), comparative fit index (CFI), relative fit index (RFI), root mean square error of approximation (RMSEA), and root mean square residual (RMR).
The results of the CFA, using the maximum-likelihood estimation method, indicate that the fourfactor model provided an adequate fit to the data. In addition, the standardized factor loadings of items for CIS varied between 8 and 97 (figure 3 and Table 3). For the current study, the CFA model for the CIS scale has significant indicators of good fit, and the value of expected cross-validation index (ECVI) for the default model is less than the ECVI for the saturated model. The rest of values reported in (Table 2) of the indicators fell within the ideal range for each indicator, which indicates a good model fit.

Reliability
To investigate the reliability of the CIS items the internal consistency was determined by calculating the correlation coefficients between the item scores and total scores for the dimension being    (Table 4). All of the correlation coefficients were high and statistically significant (p < 0.01). Furthermore, all Cronbach's alpha and Spearman-Brown split-half tests coefficient for all dimensions were high ranging between 8 to 9. The values are presented in Table 5 which reflect a good degree of reliability (Field, 2013) In addition, the corrected item total correlation of the CIS items was calculated using correlation coefficients between the score for each item and the total score for the dimension when one item was deleted from the total score of the dimension. The results indicate that all Cronbach alpha when one item is deleted are less than Cronbach alpha for the dimension when all items are presented, which indicates that all items contribute positively to internal consistency of the scale (see Table 6).

Data analysis and results
This study used a quasi-experimental design method using t-test data analysis, the eta-squared test, and one-way ANOVA to answer the research questions. The research design used two groups of subjects from the population: One encompassing teachers who participated in lesson study professional development projects and one with teachers who never participated in such projects. The study also used two groups of high school students. The results for the target aim of the study are presented in the following sections.

The impact of the lesson study on developing CI skills
To measure the impact of lesson study projects on the CI skills of teachers, the researcher used t-tests for two independent samples, examining the differences between high school mathematics teachers who participated in lesson study projects and those who did not. The results of this comparison are presented in Table 7, which also presents the eta-squared results (η 2 ). The etasquared test was used to evaluate the magnitude of the effectiveness of the lesson study projects in developing CI skills among high school mathematics teachers, where the value of η 2 was calculated from the following equation: ; where T 2 ¼ Squared of T test and df ¼ degree of freedom: According to Cohen (1988): If η 2 = 0.01, there is a small effect; if η 2 = 0.059, there is a medium effect; if η 2 = 0.138, there is a large effect;  if η 2 = 0.232, there is a very large effect. Table 7 shows the following consequences: • There was a significant difference (p < 0.01) between the mean scores of mathematics teachers who participated and those who did not participate in lesson study projects for all of the sub-dimensions. The total score for the CIS was higher for teachers who participated in lesson study projects.
• The eta-squared value of 0.166 indicates that the lesson study projects had a large effect in developing the set of skills in the Investigation dimension. The eta-squared value also indicates that lesson study projects can explain 16.6% of the variation in the Investigation set of skills. That is, 16.6% of the variation in the Investigation dimension score is predictable from the attendance and engagement of the lesson study professional development program.
• The eta-squared value of 0.159 indicates that the lesson study projects had a large effect in developing the set of skills in the Reflection dimension. The value of eta-squared also indicates that lesson study projects explained 15.9% of the variation in scores for Reflection skills. A large amount of the variation was explained by the lesson study projects. That is, 15.9% of the variation in the Reflection dimension score is predictable from the attendance and engagement of the lesson study professional development program.
• The eta-squared value of 0.115 indicates that lesson study projects had an average effect on the development of the set of skills for the Planning dimension. The value of eta-squared indicates that lesson study projects explained 11.5% of the variation in scores for the skills in the Planning dimension.
• The eta-squared value of 0.094 indicates that the lesson study projects had an average effect on the development of the set of skills for the Research Lesson dimension. This value of etasquared indicates that 9.4% of the variance in the dimension Research Lesson is predictable by the engagement and attendance of the lesson study professional development program.
• The eta-squared value of 0.137 indicates that lesson study projects had an average effect on the development of the total score for all four dimensions of the CIS. This value of eta-squared indicates that lesson study projects explain 13.7% of the variation in the total scores for the entire CIS, which is an average level of variation explained by lesson study projects. That is, 13.7% of the variation in the CI total score is predictable from the attendance and engagement of the lesson study professional development program. To conclude this section, the statistical analysis revealed a link between lesson study professional development projects and CI skills. The mean scores of high school mathematics teachers participating in the lesson study projects for all sub-skills and the total CIS scores were significantly higher when compared to the scores of high school teachers who did not participate in lesson study projects. Additionally, lesson study projects had a large effect on the development of the set of skills presented in the Investigation and Reflection dimensions. Lesson study projects had an average effect on the development of the set of skills in the Planning and Research Lesson dimensions. The lesson study projects also had an average-sized effect on explaining the variation in total CIS score.

CI skills and the impact of high school mathematics teachers' gender and years of experience
To measure the differences in the levels of CI skills among male and female high school mathematics teachers who had engaged in lesson study projects, a t-test was conducted for the two independent samples. The results are presented in Table 8.
The results shown in Table 8 reveal that there were no significant differences for any of the subdimensions of the CIS between male and female mathematics teachers who participated in lesson study projects. In other words, there was a convergence between the mean scores of male and female mathematics teachers who participated in lesson study projects in all sub-dimensions of the CIS and in the overall total score. This study also explored the impact of years of experience on the level of CI skills among high school mathematics teachers who participated in the lesson study program. One-way ANOVA was implemented to explore whether CI skills differed based on the number of years of experience. In addition, a follow-up least significant difference (LSD) test was used to further identify the direction of the mean difference. The result of this comparison is presented in Table 9.
The results presented in Table 9 show that there were no significant differences in the CIS skills presented in the Investigation dimension due to years of experience. The results also reveal that there were significant differences in the CIS skills for the Planning and Reflection dimensions (p < 0.05 and 0.01, respectively) due to years of experience. The data also showed that the total CIS score was statistically significant due to different years of experience of the participating teachers. Using the LSD test, the source of these variances is revealed and is presented in Table 10.
The results in Table 10 show that significant differences in the mean scores for skills in the Planning dimension were better for teachers with more years of experience. In the Reflection dimension, the results were similar, with teachers who had more than 10 years of experience having significantly higher mean scores than those with 5-10 years of experience. The mean scores of high school mathematics teachers who participated in lesson study projects with more than 10 years of experience were also significantly higher than those with less than five years of experience in the overall total CIS score.

Students' mathematical learning and CI skills
To measure the impact of lesson study projects on students' mathematical learning, an independent samples t-test was used to identify changes in the high school students' quantitative scores among students whose teachers participated in lesson study projects and those whose teachers did not participate in lesson study projects. Independent samples t-test is used to compare two means when those means have come from different groups of entities ( (Field, 2013). In this study, both groups are independent and comparable in numerous characteristics such as age, grade level, schools' socioeconomic background, schools' geographic area. The analysis was performed to compare two different GAT scores (quantitative section) of two groups of high school students and the results are presented in Table 11.  Table 11 analyzes the impact of lesson study on high school students' mathematics learning. The data from the independent sample t-tests determined that the first mean GAT scores were significantly higher (p < 0.01) for high school students who had teachers participating in lesson study projects than they were for students whose teachers did not participate in lesson study projects. However, there was no significant difference for the second GAT scores.
Lesson study provides a platform to enhance teachers' collaboration skills around their pedagogical concerns, inquiry, and learning. It supports teachers connecting in collaborative learning by engaging them in discussions about learning gaps and everyday classroom problems. The aim of this study was to examine the role of local lesson study projects in developing CI skills among Saudi high school mathematics teachers and their influence on their students' mathematics performance. This study implemented the CIS model to evaluate the level of CI skills among mathematics high school teachers, studying several variables.

Conclusions and implications
The current study stressed that collective professional development programs experience emphasize mathematics teachers' shared values and vision by focusing on student learning and common difficulties, take an inquiry stance on sharing experiences and expertise, demonstrating a willingness to experiment with alternative strategies and engage in reflective dialogue. As noted by Robutti et al. (2016), these aspects are essential areas of collaboration among mathematics teachers to empower and support them to address teaching issues and challenges.
The current results of the study revealed promising influences of lesson study projects on mathematics teachers' levels of practice, collaboration, and ability to actively engage and commit to their community. Furthermore, these projects offer an opportunity for math teachers to learn and be empowered by them. The CIS scale provides a method to assess teachers' levels at which they productively communicate and engage in professional dialogue.
The consequences of the study demonstrate that a large amount of the variation in the Investigation and Reflection dimensions can be predicted by teachers' attendance and engagement in the lesson study professional development program, as these factors showed a large effect on the professional development project. The findings elucidate the power of lesson study in initiating professional discourse about teachers' current mathematical knowledge, along with their beliefs, attitudes, pedagogy, and how they formulate positive attitudes toward  mathematics. The findings also imply that lesson study experience can strengthen teachers' self-reflection and support teachers in thinking over their own practices using the data they generate from interacting with other teachers. This could in turn improve their pedagogy and knowledge about teaching mathematics. Based on the design features of the scale, the teachers in the reflection phase tended to practice all four collaborative characteristics: being learners of their own, producing a shared vision with their colleagues, practicing reasoning about their teaching, and engaging in the cyclical processes of teaching change. Lesson study supports teachers' reflections of their own practice, which may contribute to positive changes in their teaching. However, this aspect of the findings needs to be re-tested through qualitative explorations to identify how lesson study can contribute to the initiation of teachers' professional dialogue and self-reflection, as well as to examine whether reflection practice allows teachers to shift their respective teaching practices.
A remarkable finding from this study was the influence of lesson study experience on improving the quantitative GAT scores of students. This finding is in accordance with other studies that indicated the impact of the lesson study approach on changes in teaching and students' achievements (Chen & Yang, 2013;Lewis et al., 2012;Stigler & Hiebert, 1999). Although further analysis needs to be undertaken, and the small-scale nature of this research has been acknowledged as a limitation of the study, the current study offers valuable insights for educational leaders in Saudi Arabia and internationally, emphasizing the importance of developing more content-based professional learning communities for teachers. In addition, this study recognizes that other factors, such as peer support, use of technology, tutoring, innovative teaching methods, and many other factors, may interfere with the increase in mathematics achievements in high schools.
Research in mathematics education has highlighted the importance of integrating content-based collaborative professional programs to support students' learning of specific concepts or mathematical topics (Appelgate et al., 2020;Chen & Yang, 2013;Lewis et al., 2012;Stigler & Hiebert, 1999). The findings of this study suggest significant implications for policymakers, school leaders, researchers, and teachers. It illustrates how lesson study projects provide a platform for teachers to engage in discourse about fundamentals in their careers, such as their students' learning of mathematics, and to improve their attitude about teaching mathematics. For professional development models to be successful and supportive of instructional change, the platform should be embedded into the school system. This requires that school leaders appreciate this kind of professional program and to provide a conducive environment and convenient school timetable for teachers to engage in a collaborative learning community (Akiba & Wilkinson, 2016). This means that important modifications to school structures and leaders' rules would be necessary before adapting the professional development models in schools in Saudi Arabia and internationally outside of the context of Japan (Akiba & Wilkinson, 2016). Leaders would be responsible for encouraging teachers' initiatives in school lesson studies and the valuation of their success and empowerment. Another implication emerged from the characteristics of efficient professional development programs discussed in the current study is that teachers demanded to: engage in collaborative learning community to self-improve their pedagogy, interact with knowledgeable peers when planning their lessons, share and observe data on students' learning of mathematics, and engage in deep thinking about their own pedagogy and teaching habits. These aspects of collaboration should be part of teachers' regular practice to support students' learning and their pedagogical development.
More investigations are needed to identify the design features of content-based, collaborative, and inquiry-based learning activities that influence the nature of teaching discourses in ways that enhance learning opportunities. Specifically, longitudinal and large-scale studies are needed to identify the magnitude of the impact of collaborative skills on students' learning. Given the recent increase in innovative educational resources, instructors are required to have the collaborative skills necessary to face these challenges. Future studies are needed to investigate how lesson studies and other collaborative school-based professional models can be implemented and stand out in a world full of technological innovations and open collective applications.

Funding
The author received no direct funding for this research.

Disclosure statement
No potential conflict of interest was reported by the author(s).

Citation information
Cite this article as: Supporting collaborative inquiry skills through lesson study: Investigation of high school mathematics professionals, Maha Saad Alsaeed, Cogent Education (2022), 9: 2064406.