Mathematics coordinators as school team PD leaders before and during COVID-19

This study explored a professional development (PD) program preparing mathematics coordinators to lead effective PD in their schools. Due to the COVID-19 pandemic emerging in the midst of this study, the PD program and school team PD were performed online instead of face-to-face. We explored the PD program and school team PD design before and during COVID-19 period, and the coordinators' expertise-based PD leadership professional identity (LPI). Findings revealed structural stability of the PD program design, and support of coordinators’ LPI. Coordinators enacted a structural transition of effective PD design into the school team PD, demonstrating their LPI in practice.

School settings can serve as a venue for supporting and challenging teachers' PD through authentic workplace experiences, particularly expanding opportunities for collaboration within teaching teams and formation of professional learning communities (PLCs) (Borko et al., 2014;Darling-Hammond et al., 2017;Hawley & Valli, 2000). Moreover, applying a PLC structure in teaching teams in different school settings, implementing PD teaching materials, and imparting systemic support, all combined, can contribute to the scalability and sustainability of teachers' PD (Borko et al., 2010;Cobb & Jackson, 2011;Hawley & Valli, 2000;Prediger et al., 2019a;Roesken-Winter et al., 2021). Yet, in order to maintain teachers' effective PD in schools at scale, there is a critical need to prepare a cadre of PD team leaders (Baker, 2021;Borko et al., 2014;Fennell et al., 2013;Jackson et al., 2015;Loucks-Horsley et al., 2012;Marrongelle et al., 2013;Prediger et al., 2019a). In the local Israeli context, mathematics coordinators can be utilized to lead their teaching teams' PD in alignment with their role definition. In a reform applied in Israel in 2016, the role of the coordinator was defined as a "subject matter leader and managerial facilitator of the professional team" (The Israeli Ministry of Education, 2016). Israeli coordinators have the dual responsibility of teaching students as well as leading the teaching team. As part of their leadership role, they are responsible for the planning, followup and assessment of the discipline-based teaching and learning aspects, hand in hand with their teams; they need to lead weekly team meetings for teamwork and consultation; but it is not explicitly stated that they should lead team PD (The Israeli Ministry of Education, 2016). In their role as PD leaders, in addition to strong teaching skills, they need to strengthen their instructional expertise to be able to facilitate adult learners in a PD program (Borko et al., 2014;Loucks-Horsley et al., 2012). The study adds a new dimension to the coordinators' current teaching and leadership role, where they are expected to extend their skills and reconstruct their perception of themselves as professionals. Beijaard et al. (2000) claim that teachers construct their professional identity (PI) throughout their careers through self-perception of their subject matter, as well as their didactic and pedagogical expertise. PI constructs self-efficacy and motivation to engage in educational change and implement new practices (Beijaard et al., 2000). PI is formed through teachers' active engagement in the improvement of their practices (Beijaard et al., 2004). Our focus on the professional development of novice PD leaders led us to earlier research regarding extended expertise for our conceptualization of PD leadership PI (hereinafter, LPI). LPI is constructed through PD leaders' self-percived expertise, subject matter and didactic expertise facilitating adult learners' PD, pedagogical expertise supporting the team during change processes, and leading a school-based PLC (Baker, 2021;Borko et al., 2014;Loucks-Horsley et al., 2012).
Further research is needed about a PD program design that best supports the development of PD leaders' expertise to apply effective PD in school settings (Baker, 2021;Borko et al., 2014Borko et al., , 2021Tauber et al., 2021). This study addresses this knowledge gap by exploring how a coordinators' PD program design was targeted to support their LPI in terms of subject matter, didactic and pedagogical expertise. Additionally, a more in-depth exploration of school team PD implementation by leaders is necessary to understand how to scale-up and sustain effective PD design in school settings (Baker, 2021;Borko et al., 2014). We address this by exploring the school team PD design implemented by the coordinators, as well as their LPI in practice.
The study was conducted during the 2019e2020 academic year, in which COVID-19 erupted. The resulting social restrictions changed immediately the PD program sessions and the school team PD sessions from F2F to online. Leaders and teams were required to adjust to online teaching and learning practices, maintain new forms of collaboration, and were in need of ongoing professional support (Abedini et al., 2021;Bakker et al., 2021;Darling-Hammond & Hyler, 2020;Tsybulsky & Levin, 2019). Acknowledging the challenges faced by school teams, team leaders, PD coaches and PD designers during this time (Cutri et al., 2020;Darling-Hammond & Hyler, 2020;Eddy et al., 2021;Hodges et al., 2020), we compare the PD program design and the support of the coordinators' LPI, as well as the coordinators' applied team PD design and their LPI in practice, before and during COVID-19 period.

Theoretical background
Below we describe the theoretical model of effective PD, further elaborating the strategies supporting the scalability of a PD program. We then describe the model of teachers' PI and conceptualize the coordinators' LPI. We conclude by briefly describing how COVID-19 period affected the current research context.

Effective professional development and scale up strategies
Effective PD includes content features situated in practice and focused on students' learning, and process features modelling teaching strategies enacting active learning and the facilitaion of teachers' PLCs (Borko et al., 2010;Desimone, 2009;Guskey, 2002Guskey, , 2003Hawley & Valli, 2000;Polly & Hannafin, 2011). There is a concensous about the effective PD features that manifest both content and process aspects, leading to desirable changes in teachers' practices and consequently improve students' outcomes (Avalos, 2011;Borko et al., 2010;Darling-Hammond et al., 2017;Guskey, 2003). Effective PD is content focused and aligned with specific curriculum-based teaching practices and strategies; involves active learning through teachers' engagement as learners who have experience and professional interest and integrate reflective discussions regarding learning experiences that are essential for class implementation; supports collaboration by encouraging teachers to work together, exchange ideas and perceptions, and develop teamwork skills; involves practice of teaching models and modelling of instruction to help teachers plan future implementation; provide expert support to model strong teaching practices and facilitate changes in classroom teaching practices; involves feedback and reflection about teaching practices; and has sustained duration by allocating enough time to adjust teaching methods by practicing implementation and holding reflective discussions (Darling-Hammond et al., 2017).
The community-based strategy is applied by forming teachers' PLCs, which maintain supportive structural conditions (place, time schedule) in which teachers experience collaborative learning and sharing of practices aiming to enhance their professional skills and their students' learning (Hord, 2009;Stoll et al., 2006).
According to the Israeli education reform, coordinators should conduct weekly teamwork meetings (The Israeli Ministry of Education, 2016). This enabled us to apply the community-based strategy by utilizing these meetings to foster PLC structures in different school settings, guided by coordinators as PD leaders.
The material-based strategy is applied by introducing teachers to curriculum-based teaching assignments designed to stimulate their application of practices that enhance students' thinking and inquiry (Guskey, 2002;Heck et al., 2019;Prediger et al., 2019a). Such teaching materials were introduced in the PD program, ready to be implemented in schools.
The systemic support strategy considers the school context through management and leadership levels as means of support for teachers during practice change (Prediger et al., 2019a). We applied this strategy, as coordinators leading their school team PD, supported by the school management and the Ministry of Education as official PD training.
However, implementing effective PD design at scale does not guarantee the quality of PD in schools, nor teachers' practice change, since it needs to be further supported by skilled PD leaders (Baker, 2021;Darling-Hammond et al., 2017;Guskey, 2003). Baker (2021), as well as Tauber et al. (2021), identified PD features that support PD leaders, revealing that their participance in collegial reflective learning, professional discussion, and learning of practical teaching methods was realized in their improved expertise, self-efficacy and motivation to implement effective PD in their teams. Beijaard et al. (2000) define teachers' professional identity (PI) as the way teachers perceive themselves as professionals, based on their teaching expertise; Subject matter expertise reflects a deep understanding of the discipline itself, the relations among its' different concepts, and the skills to integrate these aspects in teaching. Didactic expertise reflects the skills to plan, apply and assess lessons in a manner that can influence students' thinking, facilitate their learning, and increase their responsibility for the learning process. Pedagogical expertise focuses on the social, moral, and ethical aspects of teaching, and the skills to acknowledge students' opinions, thoughts, personal problems, and dilemmas. Beijaard et al. (2000) define teachers' professional identity (PI) as a dynamic, lifelong process of constant self-interpretation regarding one's teaching practices and perceptions, as well as adaptation to changes in the professional environment (Beijaard et al., 2000).

Conceptualizing the LPI of PD leaders of mathematics teachers' teams
In a subsequent paper, Beijaard et al. (2004) emphasized that teachers form their PI through continuous present and future selfview of themselves as professionals, and engagement as active learners in individual and collaborative learning experiences (Beijaard et al., 2004).
Based on prior research of school team PD leaders' skills (Borko et al., 2014;Loucks-Horsley et al., 2012) we propose extending the Beijaard et al. (2000) PI model to school team PD leadership expertise. We conceptualize the LPI as the way PD leaders perceive themselves as professionals, based on their PD leadership expertise; subject matter expertise consists of extensive mathematical knowledge as well as the skills to facilitate adult learners in mathematical solution strategies; didactic expertise includes the skills to exemplify teaching practices and facilitate adult learners in inquiry of students' solutions; and pedagogical expertise is the skills to lead a PLC environment for sharing and reflection, as well as to support teams during practice change (Borko et al., 2014;Loucks-Horsley et al., 2012).
Relying on the theoretical background regarding effective PD, PD scale up and mathematics teaching PD leaders' LPI, we explored the coordinators' PD program design, and the way it was targeted to support their LPI. Further, we explored the school team PD design, as it was implemented by coordinators, and their LPI in practice.
During the sudden outbreak of COVID-19, the PD program and school team PD sessions transitioned immediately from F2F to online Zoom videoconferences. This challenged the coordinators who needed to assist teams in adjusting to new practices during online teaching, in facilitating new forms of collaboration, and in maintaining the online school team PD.
Comparing and complementing of the two periods' data expanded the study perspective regarding the PD program structural design and the structural transition of effective PD features to school teams' PD, as well as the support in coordinators' LPI before and during a crisis.
1.4. We addressed the following research questions RQ1: Which effective PD features are exemplified in the coordinators' PD program and how (if at all) were they targeted to support the coordinators' LPI, before and during the COVID-19 period?
RQ2: Which effective PD features were applied by the coordinators in school team PD, and how (if at all) were they associated with the coordinators' LPI, before and during the COVID-19 pandemic period?

The Israeli context for this study
In 2016, a reform in high school mathematics education, called Oz Letmura, was fully applied in all Israeli schools. One of the main changes introduced was to require each Israeli teacher to receive 60 h of teaching training within the school setting, which is embodied in their salaries. Israeli educational coordinators serve both as teachers and as leaders of teams of teachers. While coordinators are not formally required to lead their teams' PD, it is acceptable to bring in external instructors for this purpose. Teaching teams' weekly sessions with the coordinator are meant for planning, developing teaching materials, data analysis, team guidance, and monitoring and evaluating work processes. Using these weekly team meetings for collaborative collegial learning in PLC environments, this study suggests assisting coordinators in becoming PD leaders of their teams.

The PD program
During the 2019-20 academic year, a PD program for mathematics coordinators in middle and high schools was offered at one of the universities in Israel. The program was designed to develop mathematical-didactic and leadership practices and prepare the coordinators to lead team PD in their school teams. The PD sessions were focused on mathematical-didactic curriculum-based materials, ready for implementation in teams and classrooms; and on leadership activities designed to develop the coordinators' interpersonal skills as PD leaders and their understanding of the challenges the teams would encounter when changing their practices.
The PD program was conducted by three highly professional coaches, of whom two, Anna and Betty (these, and all other names used in this article, are pseudonyms), led the mathematicaldidactic aspect, with Benny heading the leadership aspect. The coaches have vast experience in school-based teaching and management roles, as well as teachers' PLC leadership.
The second author led the PD program planning and objectives along with the coaches. She also communicated with them regularly before each session to become updated with their pre-planned activities. The first author was present in all the PD sessions and provided a general reflection regarding the activities and coordinators' participation, which was considered by the second author and the coaches. This paper explores the program sessions between November 2019 and June 2020, including the transition from F2F to online sessions in March 2020 due to COVID-19. Further, we explore the school team PD sessions that were scheduled to start in February 2020, after three months of preparing the coordinators for their role as PD leaders and continue until the end of the academic year.
The PD program was approved by the Israeli Ministry of Education as 60 h of professional, in-service training for the coordinators. To complete the program, the coordinators had to actively participate in the program sessions and report on leading two team PD sessions, for their choice. The original program included 11 F2F sessions (4 academic hours each), and 16 academic hours of asynchronous sessions. The asynchronous sessions were dedicated for planning the team meetings or for designing of a teaching unit that would reflect collaborative teamwork, as a final project. However, due to the COVID-19 pandemic, the first six sessions were F2F (4 academic hours each) as planned, and starting from March 2020, seven sessions were held online via the Zoom application (2 academic hours each). The online sessions were shortened to suit pedagogical recommendations and enable parents of young children to participate. Accordingly, the number of asynchronous hours rose to 22, in an effort to provide the coordinators with more time to prepare for leading the online team PD sessions.
The teachers received credit for 30 h of professional, in-service training as part of their obligation to complete 60 h of training in the school setting. To complete the 30-h program, they had to actively participate in the team sessions and submit a collaborative design for a teaching unit as a team's final project. The school team PD structural design was at the discretion of the coordinators, with the premise that their experience with collaborative and reflective learning of curriculum-based teaching materials will lead them to apply these in their teams within a PLC structure. Prior approval for the study was received from the university's Institutional Review Board (#IRB-2019-003).

Participants
Participants were coordinators and teachers from different schools in Israel (Table 1).
A typical PD session structure included a mathematical-didactic activity in small groups for the solution of mathematical assignments, followed by two plenary sessions in which reflective discussions were held about mathematical-didactic and leadership aspects (Fig. 1).
The mathematical-didactic activities focused on reflecting on the curriculum-based assignments and their solutions, in order to allow for their immediate application in teams and classrooms. For example, in the Findings section, we describe the geometry multiple solution assignment that can be introduced to middle and high school students and solved in different mathematical ways.
All the six F2F sessions followed the typical structure, as well as three of the online sessions. The other four online sessions focused only on leadership reflective discussions, aimed at enhancing collegial sharing time during the COVID-19 challenging period. Further, the online sessions also focused on technological mathematics teaching applications and online teaching strategies to support the transition to online teaching and online team PD leadership. For example, in the Findings section, we mention the use of an online graphic web application to present the calculus question manipulation inquiry.
PD materials were shared via an online learning platform (Moodle), and coordinators communicated in a WhatsApp group between PD sessions.

Research tools
The research tools included observations of the PD program aimed at detecting its structural design and the support of coordinators' LPI. Observations of school team PD and reflective questionnaires for coordinators and teachers, all aimed at tracing the school team PD structural design and the coordinators' LPI in practice. Finally, we used a quantitative pre-and post-self-report questionnaire aimed at assessing the coordinators' perceptions regarding their LPI.

Observations
The first author of this study conducted observations of all the PD program sessions and several school team PD sessions.
Since the PD program design mostly exhibited the typical structure outline (Table 2), we compared two typically structured sessions with, one before and one during COVID-19.
We begin by analyzing the 5th F2F session in February 2020, to provide the perspective of a session that was held after a substantial period of familiarization with the PD program and during the initial school PD implementation period. We analyzed a school team PD session that was held F2F during February 2020, after the 5th PD session.
Further, we analyze the 8th PD session, which was held online on March 2020 during the first month of the online transition due to the COVID-19 outbreak, which allowed us to explore the PD design through the adaptation to the online format. The 8th PD session had a typical structure, with the exception of two leadership reflective discussions instead of one.
We then analyze one online school team PD session, held during March 2020, after the 8th PD session.

The reflective questionnaire
This questionnaire used an open-ended question that asked the coordinators and the teachers in their teams to elaborate on two practices that had changed in their teamwork since the coordinators joined the PD program and began leading team PD. The coordinators were asked to respond to the reflective questionnaire twice before the COVID-19 pandemic in December 2019 and February 2020; once after the outbreak ein March 2020eApril 2020; and once towards the end of the PD program in MayeJune 2020. The teachers were asked to fill in the reflective questionnaire about a month after the coordinators were formally scheduled to implement team PD in their schools, i.e., in March 2020, and again in MayeJune 2020. As the team sessions moved online in March 2020, the teachers' reflective questionnaires refer only to the online period, while the coordinators' reflective questionnaires refer to both periods.

LPI self-reported questionnaire
The coordinators' pre-and post-LPI self-reported questionnaire was based on the questionnaire designed by Beijaard et al. (2000) to assess teachers' perceptions of PI, and was adapted for the current study to explore the coordinators' perceptions of LPI. The questionnaire included 29 closed-ended items on a 1e5 Likert scale (1-incorrect, 5 -highly correct). The first 8 items assessed the coordinators' perceptions regarding the PD program support of their LPI. For example, the next item assessed the subject matter expertise of the coordinators: "I see the value of spending time and effort preparing content for my team sessions in the same topics we dealt with in the PD program". The next 21 items examined the coordinators' perceptions of their LPI in practice, as they lead their school team PD. For example, the next two items assessed the didactic and pedagogical expertise of the coordinators (respectively): "In team meetings, we discuss didactic issues such as dealing with classroom heterogeneity and students' common misconceptions"; "In team meetings, I lead a respectful, professional discussion, accepting different points of view and learning from mistakes". The questionnaire was given twice e at the beginning of the PD program in November 2019, and towards the end of the program in June 2020. Content validity and inter-rater reliability for the adapted questions were analyzed by three mathematics education experts and reached a high level of agreement (k ¼ 0.90).

Data analysis
Observations: All the selected observations were transcribed verbatim and analyzed using qualitative, directed content analysis (Hsieh & Shannon, 2005). Each observation was divided into the segments that comprise the typical structure (Fig. 1). Each segment was read thoroughly and coded based on effective PD features (Darling-Hammond et al., 2017). Each segment was then associated with one of the coordinators' main LPI (Beijaard et al., 2000;Borko et al., 2014;Loucks-Horsley et al., 2012); as the findings of the PD program observations were classified as support in LPI, and the school team PD observations as LPI in practice.
The reflective questionnaire: We used directed content analysis (Hsieh & Shannon, 2005) to code the responses given by the coordinators and the teachers regarding the coordinators' practice change as school team PD leaders. We began by reading each response thoroughly, looking for examples for practice change. The practices identified were then divided into segments, each representing one change. Next, we used the effective PD program model to associate each segment with one effective PD feature, and the LPI concept for associating it with one coordinators' LPI in practice. Table 3 presents the analysis method. In the left column, the identified practice change segments are highlighted in color.
Finally, we grouped the segments that reflect each of the effective PD features and LPI (separately) and calculated the percentage of each group of segments of the total coded segments. For O. Cohen-Nissan and Z. Kohen Teaching and Teacher Education 121 (2023) 103921 comparison goals between the two periods investigated (for the coordinators' data) and the two groups of participants, we grouped the coordinators' reflections separately for each period -F2F and online -and compared the periods to one another. We then compared the coordinators' responses with teachers' responses during the online period and combined the two to compare them with the coordinators' responses for the F2F period. LPI questionnaire: The coded responses to the quantitative questionnaires provided a descriptive view of the coordinators' perceived LPI, before and after participating in the PD program. We used ANOVA with repeated measures to compare the various types of LPI, using post-hoc tests with Bonferroni criterion for significance and dependent T-test analyses, with respect to a) coordinators' perceptions of the PD program support of LPI, and b) coordinators' perceptions of their LPI in practice. Table 4 summarizes the research tools, data collection and data analysis.

Findings
Addressing the first RQ, we analyzed the PD program observations to exemplify their structural design and detect the manner in which it is associated with support of coordinators' LPI before and during the COVID-19 period. To answer the second RQ we analyzed the observed school team PD sessions, as well as coordinators and teachers' reflections, in an attempt to exhibit the effective PD design applied in the school team PD and the coordinators' LPI in practice, before and during the COVID-19 period. Finally, we supplemented both RQ answers with the LPI questionnaire results of coordinators' perceptions regarding the program's support of the LPI in practice.

Qualitative observations
We present the two pairs of observations-based findings e the 5th F2F and 8th online PD program sessions and their paired school team PD sessions, relying on their typical structure activities (Fig. 1). We describe the effective PD features and LPI by highlighting them in italics and provide local summaries associating the sessions' main activities with the coordinators' LPI. Finally, we compare the paired F2F observations with the paired online observations regarding the effective PD design and the association with the coordinators' LPI.

Observation of the 5th coordinators' F2F PD program session
The 5th session was held F2F in February 2020, in a typical structure, starting with the mathematical-didactical activity that was content focused on a curriculum-based geometry assignment (Fig. 2).
In the small group collaborative learning activity coordinators collaborated as active learners, consulting and suggesting diverse mathematical solutions, while coaches supported as experts, facilitating the teams' motivation, and advising solution strategies.
This activity included mathematical inquiry and solutions, targeting to strengthen the coordinators' subject matter expertise, preparing them to implement PD materials within their team.
In the following mathematical-didactic reflective discussion, the coaches were observed supporting as experts and providing didactic highlights, while the coordinators collaborated as active Coordinator "These days, our team session is conducted differently. There is a higher-level mathematical discussion, and I feel we have gone through a process of knowing ourselves better as a team" "There is a higher-level mathematical discussion"

Content focus
Didactic "I feel we have gone through a process of knowing ourselves better as a team" Feedback & Reflection

Pedagogic
Coordinator "We started to solve mathematics questions in the team sessions and felt the learning process of our students" "We started to solve mathematics questions in the team session" Active learning Subject matter "… and felt the learning process of our students" Feedback & Reflection

Didactic
Teacher "The coordinator presents interesting math assignments from the PD program, encouraging us to apply it in class" "The coordinator presents interesting math assignments from the PD program …"

Content focus
Subject matter learners presenting their solutions. The coaches enacted group feedback and reflection, introducing the advantages of an applied teaching model that presents a mathematic assignment with multiple solutions.
The following excerpt of the discussion illustrates the way the coaches addressed the multiple solution teaching model's adaptability for different curriculum contexts, and its alignment with the Israeli education system's objectives: Coach Anna: "Presenting multiple solutions to one question enhances students' understanding and encourages flexible thinking".
Coordinator Ela: "I agree, when I teach middle school students, even though I present a full formal geometry proof, I try to be flexible with my students and accept various solutions".
Coach Betty: "This is great, you are in line with the objective of the Israeli educational system to make geometry more accessible, as it is regarded as a barrier to higher level mathematics learning".
Coach Anna: "Please notice that you found several geometric solutions as well as solutions from different mathematical branches: geometry, trigonometry, vectors; this assignment fits both junior-high and high school curricula".
This mathematical-didactic discussion focused on rationalizing the value of the teaching model for facilitating student learning, targeting to support the coordinators' didactic expertise and prepare them to lead didactic-oriented discussions with their teams.
Coach Benny opened the leadership activity by facilitating the groups' reflection, referring to the need to support the team's practice change by saying: "People perceive their practice change differently, incorporating fears and hopes, but leaders, like entrepreneurs, should be able to transcend this, embrace the opportunity and manage the team's fears". Coach Benny supported the team as an expert, heard each opinion and created a safe environment for discussion. Coordinators collaborated and reflected their thoughts, like Niv, who said: "Changes in our work routines are positive, as we learn about the team's strengths, commitment, and cooperation". This activity focused on the coordinators' pedagogic expertise, preparing them to lead a PLC environment and support the team's practice change.

Observation of a F2F school team PD session that followed the 5th F2F PD program session
The session was held after the 5th PD session, in a typical structure, yet with a different order of activities. When coordinator Limor asked the teachers to share one thing they have learned about their students during the semester, she engaged them in a collaborative and reflective discussion. Limor supported them as an expert, listening and creating a respectful environment exhibiting her pedagogic expertise.
The mathematical-didactic activity content focus was on the solution of the geometric assignment (Fig. 2). Limor led the small group collaborative learning activity, where coordinators collaborated and solved the assignment as active learners, while she was observed motivating them to work together. She supported them as an expert, suggesting mathematical strategies to solve the assignment, demonstrating her subject matter expertise.
In the following mathematical-didactic discussion, teachers collaborated as active learners, presenting their solutions. Limor supported the team as an expert, facilitating their feedback and reflection about the didactical advantages of the multiple solution teaching model.

Demonstrating her didactical expertise
The paired sessions comparison is presented visually in Fig. 3. The highlighted cells indicate whether a PD feature was exemplified in the session, and the LPI with which it is associated.

Observation of the 8th coordinators' online PD program session
The 8th session was held online on March 2020, using the Zoom videoconference application.
It had a typical structure, with slight changes. Due to transition to online teaching in schools during COVID-19 period, the coaches added another leadership activity at the beginning of the session, in which each coordinator presented a humoristic viral picture and reflected their feelings and challenges during COVID-19. The coordinators collaborated in discussing their professional difficulties, while the coaches supported each sharing as experts and created a safe and respectful environment for reflection. This activity was targeted to support the coordinators' pedagogical expertise, preparing them to lead a PLC environment and support their teams through practice change during COVID-19 period.
The mathematical-didactic activity's content focus was on the solution of a calculus question sent to the group before the PD session (Fig. 4).
To prepare for the small group collaborative learning activity, coach Betty reviewed the solution to the question and shared some students' misconceptions regarding it. She introduced the teaching model, that is "a question manipulation", by saying: "I've manipulated the question, changing it to a parametric function which represents various functions, f ðxÞ ¼ x 2 À4 2xþc , and you are asked to analyze it according to sections A-E". The coordinators were then divided into five virtual breakout rooms for 5 min to solve this task. We managed to observe only two out of the five Zoom breakout rooms, and in both the coordinators were engaged in the content focused mathematical inquiry as active learners. This activity was targeted to support the subject matter expertise and prepare the coordinators to implement mathematical assignments in an online school team PD.
In the following mathematical-didactic discussion, the coordinators collaborated as active learners and reflected upon their inquiry in the breakout rooms. One teacher suggested an algebraic solution, and another presented a visual solution using the online Desmos graphing calculator. Coach Betty supported the coordinators as an expert, discussing these two perspectives by presenting both with the algebraic solution and a graphical solution on the GeoGebra online application. Betty then discussed the teaching model advantages: "By manipulating the question we are developing our didactic skills; we learn to upgrade the teaching materials to promote our students' thinking". This mathematicaldidactic discussion focused on development of the didactic expertise of the coordinators, preparing them to lead an online, didactical-oriented team discussion.
Finally, Benny led the leadership activity, discussing challenges posed by COVID-19, including team uncertainty, social distance, etc.

O. Cohen-Nissan and Z. Kohen
Teaching and Teacher Education 121 (2023) 103921 The coordinators collaboration and reflection were observed in the Zoom chat room, such as Gila who wrote that "juggling everything is complicated, especially maintaining a regular schedule with students", or Shirly who wrote that "these (PD) meetings are an opportunity to change the current home routine". Benny supported the team as an expert, saying: "Team collaboration supports our adaptation to online teaching challenges". Coordinator Ilana added: "Our team collaboration is improving, and our experiences here (in the PD sessions) and with the team have accelerated our adjustment to this challenging period". Clearly, Benny created a respectful, safe environment to share and reflect. This activity was targeted to support the coordinators' pedagogic expertise, preparing them to lead a PLC environment and support their team's practice change during COVID-19 period.
3.1.5. Observation of an online school team PD session that followed the 8th PD program session The session was held using the Zoom videoconference application and had a typical structure, yet it started with the leadership activity. Coordinator Gila supported the team as an expert, encouraging teachers to collaborate and reflect their feelings regarding the COVID-19 challenges through pictures, demonstrating her pedagogic expertise.
The mathematical-didactic activity combined the collaborative learning in small groups with a follow-up didactic discussion, both of which were content focused on the PD program's final assignment of preparing a teaching unit. Coordinator Gila and the team solved a percentage problem they prepared for their 9th grade students as active learners. Gila supported as expert suggesting a solution strategy and demonstrated her subject matter expertise. In the discussion, Gila supported the team as an expert by facilitating their collaboration and active learning, asking them to reflect on a teaching model she had suggested in order to scaffold students' thinking using tables, and demonstrated her didactic expertise.
The paired session's comparison is presented visually in Fig. 5.

Reflective questionnaire
The frequencies of PD leadership practice change retrieved from coordinators and teachers' reports, and coded to effective PD features, can be seen in Fig. 6. Each column represents the frequency of segments that were coded for the various effective PD features, so that their total reaches 100% of the coded segments. For example, the coordinators' report in the F2F period included 60 coded segments of practice change, of which 33% were associated with the expert support effective PD feature. It should be noted that the coordinators' reflected practice change for the F2F period were initially comprised of 66 responses, of which 13 responses (19.7%) from the beginning of the PD program (December 2019eJanuary 2020) indicated no practice change. These 13 responses were excluded since they could not be coded for the effective PD features, leaving 53 responses that were coded into 60 segments.
The coordinators described their practice change in the F2F period as associated with their support of the team as experts and their leading of content focused and collaborative school team PD  (33%, 24%, 20% respectively). They also mentioned facilitating the team feedback & reflection, and barely addressed applying the teaching model and active learning in school team PD (13%, 7%, 3% respectively).
The online period included 59 responses, of which 4 responses indicated no practice change. These were 7% of the total responses (compared to the 19.7% of "no change" responses during the F2F period). We excluded these 4 responses, leaving 55 responses that were coded into 62 segments. Findings indicate that the coordinators' reports on their practice change included more content focused experiences in the online period than in the F2F period (48% vs. 24%). Furthermore, in the online period they reported providing less expert support than in the F2F period (15% vs. 33%). Through both periods they reported similar facilitation of collaboration (19% vs. 20%) and reflection (13%), whereas the teaching model and active learning experiences remained insignificant (5% vs. 7%).
Comparing teachers' and coordinators' reports during the online period provides a wider perspective of the school team PD. Teachers reported more intense experiences of teaching models (34% vs. 5%), and hardly addressed content focus meetings (5% vs. 48%). They also mentioned more intense experiences of feedback and reflection (30% vs. 13%) and similar experiences of teamwork collaboration (16% vs. 19%), expert support (14% vs. 15%) and active learning (1% vs. null).
Figs. 7 and 8 present the coordinators' reports of their practice change in the F2F and online period (respectively), and Fig. 9 presents the teachers' reports on PD leadership practice change during the online period. Each cell represents the frequency of segments that were coded for both effective PD feature and LPI in practice, out of total number of coded segments (excluding the no-change reports). For example, in Figs. 7 and 10% of the 60 coded segments were assigned simultaneously to the expert support PD feature and the didactic LPI. These 10% accumulate with other frequencies in the expert support row, reaching a total of 33% that represent this effective PD feature, as well as in the didactic column reaching a total of 40% of segments, representing the didactic LPI.
Figs. 7e9 connect visually the reports of PD leadership practice change in school team PD during both periods, with different types of LPI and application of effective PD features at varying intensities. There were less reports of subject matter LPI in both periods, mainly through the application of content focused team PD. The F2F period included more reports of pedagogic LPI than didactic LPI (53% vs. 40%) and vice versa in the online period (35% vs. 49% and 17% vs. 79%).
The reports of didactic expertise in both periods referred to the application of most of the effective PD features. The coordinators reported that they demonstrated didactic LPI in both periods by applying content focused school team PD sessions (17%, 32%), whereas teachers reported experiencing didactic LPI with fewer content focused PD experiences (2% vs. 32%) and more teaching model and feedback and reflection experiences (34% and 20%). While the coordinators described content focused sessions through application of the PD assignments, the teachers described the use of the teaching models with these assignments.
In the online reports, both the coordinators and teachers described less demonstrations of pedagogic expertise as expressed by fewer implementations of expert support and team collaboration (11% and 3% vs. 23%, and 11% and 3% vs. 17%, respectively). Furthermore, all the reports included a relatively similar percentage of association with pedagogical expertise through feedback and reflection experiences.

O. Cohen-Nissan and Z. Kohen
Teaching and Teacher Education 121 (2023) 103921 3.1.7. Quantitative perceptions of the coordinators' LPI The pre-and post-perspectives provided by the LPI questionnaire associated the changes in the coordinators' perceptions with their exposure to the PD program. Findings indicate that before the PD program and the school team PD implementation, no significant interaction between type of measurement (subject matter, didactic, and pedagogic) and type of environment (PD program, vs. school team PD) was revealed, F (2,27) ¼ 1.38, p > .05, h2 ¼ 0.087.
Yet, after the PD program and the school team PD implementation, a significant interaction was found, F (2,27) ¼ 4.22, p < .05, h2 ¼ 0.238. The coordinators perceived the emphasis placed on pedagogical expertise in the PD program as similar to their level of pedagogical expertise in leading team PD, both before and after the PD program and school team PD implementation. Further, they perceived their didactical expertise as lower compared to the didactical professional expertise that was emphasized in the PD program, both before and after the PD program and the school team PD implementation. These results suggest that the coordinators maintained consistent perceptions regarding their pedagogical and didactic expertise as team PD leaders. Despite viewing the didactical expertise as an essential component of the PD program, the coordinators did not alter their perceptions regarding their didactical expertise as school PD leaders.
However, with respect to the subject matter expertise, while before the PD program and school team PD implementation, the coordinators perceived the emphasis placed on this expertise in the PD program as similar to the expertise they enacted in practice in leading team PD, after the PD program they perceived their subject matter expertise as lower compared to the emphasis placed on this expertise in the PD program. Table 5 presents the T-values, means and standard deviations of the coordinators' perceptions towards the various types of LPI.

Discussion
Below we discuss the results for the first RQ, which rely on the   observations of two (F2F and online) PD program sessions which outline the program's structural design and identify how it targeted to support the coordinators' LPI. We then discuss the results of the second RQ, taking a holistic view of school team PD design and the coordinators' LPI in practice from both selected observations, as well as the coordinators and teachers' reflective questionnaire data.
The results of the LPI questionnaire are also used to compare the coordinators' perceptions of the PD program support of their LPI and their LPI in practice.

The PD program effective PD design and the support of LPI
Our findings presented the PD program's common design of typically structured sessions, which included all the effective PD program features, targeting to support the coordinators' subject matter, didactic and pedagogical LPI (Beijaard et al., 2000;Darling-Hammond et al., 2017;Loucks-Horsley et al., 2012). Each of the PD sessions activities was mainly oriented to develop a particular LPI expertise through experiences of a group of effective PD features. Below we conceptualize these groups as effective PD feature clusters based on their joint support of the coordinators' LPI.
The small collaborative group learning consisted of curriculumbased learning experiences and the use of mathematical solution strategies, targeting to support the coordinators' subject matter expertise. The F2F activity, was longer, and the coaches were available in the room opposed to the shorter online breakout rooms activity. Accordingly, the F2F activity presented a richer cluster of effective PD features of content focus, active learning, expert support, and collaboration, as opposed to mainly content focus and active learning in the online activity.
Content focus and active learning recuring appearance in both periods indicates their vitality in PD leaders' preparation. PD leaders need to expand their specialized knowledge of mathematics as team PD leaders, they need to experience curriculumbased teaching materials and actively practice solution strategies that assist their facilitation of their team learning (Borko et al., 2014(Borko et al., ,2021Eddy et al., 2021;Loucks-Horsley et al., 2012;Prediger et al., 2019a;Smith et al., 2020).
The mathematical-didactic reflective plenary discussion activity during both periods exemplified the coaches' expert support while facilitating a collaborative collegial reflection, with a focus on promoting students learning and rationalizing new teaching methods. Targeting to support the coordinators' didactical expertise, it incorporated all the effective PD features. PD experiences that engage in reflective didactical discussions that are focused on students' thinking and on instructional practices are essential to prepare PD leaders to orchestrate high quality discussions within their teams (Borko et al., 2014;Loucks-Horsley et al., 2012;Smith et al., 2020). Expert support and group collaboration are vital during leaders' PD, the first construct a future vision of strong professional skills to lead didactical discussions and the second models the facilitation of team collaboration (Borko et al., 2014;Eddy et al., 2021;Polly & Hannafin, 2011). Earlier research revealed that providing novice PD leaders with experiences of didactical oriented feedback and reflection prepares them to conduct profound didactical discussions in their teams (Baker, 2021;Darling-Hammond et al., 2017;Eddy et al., 2021;Loucks-Horsley et al., 2012;Smith et al., 2020).
The teaching model feature was applied by the PD coaches' demonstration of didactical PD leadership skills and designated teaching strategies such as the multiple solution strategy and the question manipulation method. Teaching model feature encourage peer consultation and bridges between the PD participants' learning process and their implementation in practice (Borko et al., 2021;Darling-Hammond et al., 2017;Eddy et al., 2021).
The leadership plenary reflective discussions in both F2F and online sessions exemplified the coaches' expert support of a collaborative community environment. Coordinators could share and reflect on their strengths and weaknesses as novice PD leaders, and discuss strategies to support school teams during the school PD.
This PD features' cluster of expert support, collaboration, feedback, and reflection was targeted to support of the coordinators' pedagogic expertise by preparing them to lead a PLC environment and support their team during practice change.
Earlier research advocate of the need to enhance novice PD leaders skills to lead safe and respecting learning communities and be prepared to support the team members through practice change implementations (Borko et al., 2014;Loucks-Horsley et al., 2012). Furthermore, prior research conducted during COVID-19 outbreak regarding online PD programs for leading teachers revealed that a continuous collaborative and reflective collegial experiences which subsequently supported the application of new leadership practices, contributed to leaders' professional growth (Eddy et al., 2021). Naturally, a PD design should consider the fact that educators tend to implement instructional practice change that is congruent with their professional perceptions (Hawley & Valli, 2000;Polly & Hannafin, 2011).

LPI construction and implementation of school team effective PD design
The coordinators implemented most of the effective PD features in their school team sessions. We revealed similar effective PD feature clusters in the observed F2F and online school team PD implementations, which demonstrated the coordinators' LPI in practice and were in line with the PD program LPI support. However, the coordinators and teams' joint online reports expanded our perspective regarding the online school team PD implementations. Despite the comprehensive support of the PD program in the coordinators' LPI, they applied PD feature clusters that demonstrated LPI in practice at varying intensities. Below we discuss the various feature clusters implemented by the coordinators in the school team PD which demonstrated their LPI in practice, while presenting the disparities in applying these clusters during the F2F and online periods.
The coordinators implemented the cluster of expert support and team collaboration, demonstrating pedagogic and didactic LPI in alignment with the program support. The intensity of this cluster regarding the demonstrated pedagogic expertise decreased during the online period, showing a consistently low level of didactic expertise. The low level of these features in relation to the didactic expertise in both periods aligns with earlier research which indicates challenges that novice PD leaders experience in facilitating collaborative adult learning (Baker, 2021;Borko et al., 2014;Knowles, 1970). Further, the decrease in the pedagogical aspect of expert support and facilitaion of team collaboration in the online period, as well as the steady low intensity of both in the didactic aspect, reflect the struggle of teacher leaders to support their teams throughout the rapid change of teaching practices, and the challenges in facilitating online collaboration during COVID-19 period (Eddy et al., 2021).
The coordinators' F2F reports present their application of the content focus and active learning cluster, demonstrating didactic expertise, while the content focus feature demonstrated subject matter expertise as well. The content focus feature intensity increased during the online period regarding the subject matter and didactic expertise, when the coordinators implemented the PD program curriculum-based materials. Such implementations can support a rapid enactment of school teams' PD in different contexts and PD formats (Boles et al., 2020;Borko et al., 2014;Heck et al., 2019;Marrongelle et al., 2013;Smith et al., 2020). Further, the emergent transition to online teaching challenged the teams with new teaching strategies, which encouraged the novice PD leaders to implement the PD curriculum-based teaching materials (Bakker et al., 2021;Darling-Hammond & Hyler, 2020;Eddy et al., 2021). The intensity of active learning was consistently low during both periods regarding didactic and subject matter expertise. This exposed the challenges that novice PD leaders experience in facilitating active learning in their team PD (Boles et al., 2020;Borko et al., 2014;Eddy et al., 2021;Knowles, 1970;Smith et al., 2020).
The coordinators' reports included the feedback and reflection feature regarding the pedagogic expertise, while teachers' reports complemented this perspective by relating it to coordinators' didactic expertise. It seems that from the coordinators' view, applying feedback and reflection using pedagogical skills creates safe environments in which teams can share and exchange professional practices, and cope with practice change (Hawley & Valli, 2000). From the teachers' view, applying feedback and reflection presents leaders' high-level instructional skills supporting adult learning, and can thus be perceived as didactic expertise (Borko et al., 2014(Borko et al., ,2021Eddy et al., 2021;Loucks-Horsley et al., 2012).
Both coordinators and teachers reported teaching model implementations which demonstrated the coordinators' didactic expertise, while teachers' reports were more intense. Indeed, leaders that apply PD teaching materials and methods exemplify strong instructional skills (Borko et al., 2021;Darling-Hammond et al., 2017;Loucks-Horsley et al., 2012).
Finally, the significance of sustained duration was discussed in earlier research as an effective feature that affords teachers the time to adjust their teaching methods through practice implementation and reflection (Borko et al., 2010;Darling-Hammond et al., 2017;Eddy et al., 2021;Hawley & Valli, 2000;Heck et al., 2019;Polly & Hannafin, 2011;Smith et al., 2020). The PD program duration of one academic year allowed for consistent, cohesive application of effective PD structural design in both periods.
Coordinators' implementation of effective PD design, while demonstrating their LPI, imply their self-efficacy and motivation to engage in educational change and implement new practices as novice PD leaders throughout both periods. Yet, despite their demonstrated LPI in practice, according to the LPI questionnaires, their perceived LPI showed doubts in their subject matter and didactic expertise, and confidence in their pedagogical expertise. Studies found that PD leaders are more challenged with attaining subject matter and didactic expertise to support adult learners' PD, since that requires practice time, expert consultation experiences, and substantial opportunities for collegial feedback, while skills for leading a sharing learning community are more easily acquired (Borko et al., 2014(Borko et al., , 2021Jackson et al., 2015).

Structural stability, structural transition, and effective PD implementation in scale
Comparing the PD program and the school team PD structural design, broadened our perspective of the suggested integrated PD model for leaders and their teams (Fig. 10). With similar effective PD feature clusters and their repetitive orientation to support the construction of coordinators' LPI in both the F2F and online typically structured sessions, we conceptualized the structural stability of the PD program design. This manifested in the coordinators' implementations of similar effective feature clusters in different schools throughout both periods, exemplifying LPI in practice. We conceptualize this phase as evidence for the PD program scalability through structural transition of effective PD feature clusters, related to the continuous construction of coordinators' LPI in practice.

Research limitations
Although our data offers a vast perspective of school teams' PD implementations based on reflective questionnaires collected from both the coordinators and their team teachers, the number of school teams' PD observations was limited. More observations on school teams' PD implementations could have clarified the strengths and challenges of the coordinators as school PD leaders. Furthermore, we did not examine the school teams' sessions before the PD program started, with the aim to identify both the PD design and the coordinators' LPI. Yet, the reflections which indicated practice change, exposed the practices that were not applied before.
Another limitation is the use of teachers' PI model to define PD leaders' PI which can be criticized as incompatible. We suggest linking the coordinators' current role as teachers and leaders of teachers with their new role as instructors of their team PD. As novice PD leaders, they need to extend their perceptions of themselves as professionals. Our conceptualization of LPI is based on teachers' PI, using the original model's expertise, i.e., subject matter, didactic and pedagogic, and extending them to suit adult instruction in a PD program. We base this extension on earlier research elaborating the skills of PD leaders as profound instructional skills which exceed the skills required to teach students to suit the facilitation of adult learners (Borko et al., 2014;Loucks-Horsley et al., 2012).
According to Beijaard et al. (2000), teachers' PI effects their selfefficacy and motivation to implement new practices, therefore suits our view of coordinators' implementation of a new role. Further, Beijaard and his colleagues' (2004) updated explanations describe PI as the present and future vision of professionality, and as being formed through constructivist learning experiences. This fits our inquiry of the effect of the PD program and the applied school team PD on the coordinators' LPI.
Finally, our use of pre-and post-self-reported questionnaire may have limited our view of LPI. First, before the PD program coordinators were unfamiliar with LPI concepts; second' after their participation they may have only referred to their recent experiences during the challenging Covid-19 period. However, the observations and reflective questionnaire findings provide a comprehensive view of coordinators' LPI construction.

Conclusions and contribution
The current study suggests an integrated PD model for mathematics coordinators and their teams that showed structural stability throughout the COVID-19 challenge, along with evidence for structural transition to school team PD. The model emphasizes clusters of effective PD features targeted to support coordinators' LPI. Structurally wise, before and during COVID-19, the PD supported the coordinators' didactic expertise by applying a cluster of all the effective PD features (Darling-Hammond et al., 2017); the subject matter was mainly supported by the cluster of content focus and active learning; and the pedagogical expertise by the expert support, collaboration and feedback and reflection cluster.
Through the profound PD support during both periods, coordinators applied similar effective PD feature clusters which they experienced in the PD program. They perceived themselves as less professionally skilled compared to the PD program support in both didactic (before and after the PD program) and subject matter (after the PD program) expertise; and they were confident in their pedagogic expertise. Use of the LPI model enabled a transparent view of their naturally evolving self-perception of the new role, before and after applying new practices, which were vivid evidence of their engagement.
Comparing the two periods showed that COVID-19 period included more intense implementation of the clusters related to didactic expertise and less intense implementation of the clusters related to pedagogical expertise. The differences in intensities of implementation, as well as the low self-perception of the coordinators in respect to their pedagogical expertise, could be the result of the challenges that coordinators faced as novice PD leaders, as well as the challenges of the COVID-19 period.
The study offers a theoretical contribution to the interconnection between the role of PD leaders and scaling-up teachers' effective PD design. The mathematics coordinators' subject matter, didactic and pedagogic LPI were evident in practice, as they implemented the PD teaching materials in their teams (materialbased), demonstrated practices of learning community environments (community-based), and supported their teams through practice change challenges (systemic support), particularly in the shade of COVID-19 (Boles et al., 2020;Maass et al., 2019;Prediger et al., 2019a;Prediger et al., 2019b;Roesken-Winter et al., 2021).
A further theoretical contribution is the integration of the effective PD model and the PI model, as well as the extension of the last as defined for teachers, to include mathematics team PD leadership expertise e LPI (Borko et al., 2014;Darling-Hammond et al., 2017;Loucks-Horsley et al., 2012).
The study's methodological contribution is its dual-content analysis of both the effective PD and PI models within the reflections and objective observations. Further, the reflections in the coordinators' self-reports and teachers' reports regarding their coordinators provided a bi-directional perspective which broadened our view of the nature of school PD implemetaions.
Finally, from a practical viewpoint, we propose a PD model for mathematics coordinators which preserves the structural stability of effective PD design targeted to support the construction of their LPI during a challenging academic year. They experienced a meaningful learning process that constructed their expertise as team PD leaders and was manifested in their transition of the PD structural design to their team, thus facilitating the scalability of an effective PD program. We encourage more large-scale research on scaling up effective school-based team PD, which will ultimately lead to improved teaching and consequently better student learning.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Data availability
The authors do not have permission to share data.