Improving Instructional Leadership Through the Development of Leadership Content Knowledge

Abstract

Instructional leadership is integral to improving mathematics teaching in secondary schools. However, administrators often lack sufficient content knowledge in mathematics to be effective in this role. This study examined the impact of professional development focused on developing leadership content knowledge in algebra. Data included written assessments, case discussions, and interviews from 10 principals. Analysis identified shifts in principals’ algebra content knowledge and their frames for interpreting algebra instruction. Principles improved their connections between mathematical representations and shifted from using frames highlighting teacher characteristics toward using frames highlighting teacher and student thinking. Implications for leadership professional development design are discussed.

Keywords

leadership curriculum and instruction instructional leadership mathematics education leadership content knowledge

Systemic change in mathematics teaching has been a persistent challenge in American schools. In the Common Core State Standards era, districts face changes to curriculum, pedagogy, professional development, and assessment that reflect a vision of ambitious teaching (Common Core State Standards Initiative, 2010). Implementing this vision is frequently cast as a problem for which the solution involves enhancing teachers’ mathematical knowledge for teaching (Ball, Thames, & Phelps, 2008). Although teacher knowledge is important, administrators play a critical part in systemic change given their role in the professional lives of teachers (Boyd et al., 2011; Hoekstra & Korthagen, 2011; Tickle, Chang, & Kim, 2011). Studies of systemic change in mathematics identify strong instructional leadership as integral to success, particularly in changing policy contexts (Kaufman & Stein, 2010; Stein, Silver, & Smith, 1998). Often, however, administrators focus on the form of an initiative rather than the substance and philosophy (Spillane, 2000), making it difficult to provide teachers with critical support for meaningful change (November, Alexander, & van Wyk, 2010; Stevens, 2004).

Administrative support is especially critical at the secondary level. Although student-centered mathematics curricula have traction in the elementary grades (Remillard, Herbel-Eisenmann, & Lloyd, 2009), high school implementations have been less successful. The Curricular Options in Mathematics Programs for All Secondary Students (COMPASS) Center (St. John, Fuller, Houghton, Tambe, & Evans, 2005) identified six factors that characterize successful secondary math reform efforts. Three of these factors—multi-level leadership, programmatic vision, and professional supports—implicate administrators in central, active roles. Administrative support and a deep understanding of the goals of student-centered teaching were hallmarks of the successful cases. Little is known, however, about what specific knowledge of mathematics teaching and learning might be useful to administrators, and principals in particular, in supporting systemic change. The study reported in this article took place in the context of a content-focused professional development (PD) project for principals and teacher-leaders aiming to bring about systemic change in the teaching and learning of algebra. This article explores the impact of the PD program on the leadership content knowledge (LCK) of principals (Stein & Nelson, 2003) related to the teaching and learning of algebra.

Leadership Content Knowledge: What Knowledge Do Principals Need About Content?

Principals have an important role in supporting teachers to effect systemic instructional change. This support can take multiple forms, from classroom-based observation and evaluation to identifying PD opportunities and leading systemic change. To provide this support, principals need knowledge of a district’s systemic structures, of the teachers with which they work, and some fluency with the content that teachers teach. A principal’s ability to participate in and promote teacher learning, particularly about content, has a significant impact on student learning (Robinson, Lloyd, & Rowe, 2008). There has been a paucity of work, however, related to what knowledge of school content principals might need. This issue is particularly salient at the secondary level, where subject matter is more complex than at the elementary level and principals are less likely to have a multi-subject background (Stein & Nelson, 2003). Hiebert, Morris, Berk, and Jansen (2007) argue that it is impossible for teachers to be experts in every content area they teach. Such an expectation is even less realistic for principals, particularly when administrative certification programs rarely focus on content knowledge in ways that teacher certification programs do.

Stein and Nelson (2003) use the construct of LCK to describe the knowledge principals need to identify good teaching, to support its development both when it is and is not present, and to set conditions for continuous improvement. They describe three specific types of knowledge as a part of LCK: knowledge of teaching and learning at the classroom level (about the subject and how students learn it), at the school level (about teaching and teacher learning), and at the district level (how to guide the learning of teachers within a system). They contend that to support systemic change, principals must engage in discourse around teaching and learning at each of these levels. Stein and Nelson suggest that PD aiming to develop these three levels of content knowledge might take a postholing approach, in which principals engage in deep exploration of a slice of content from the perspective of a learner, a teacher, and a principal. They posit that this approach would afford transit across the levels and support principals in engaging in all facets of instructional leadership. For example, a principal with some knowledge of algebra content could use that knowledge to have discussions with teachers about key features of mathematics teaching and learning, to make informed decisions that support teacher learning through district-wide math reform, and to communicate about systemic change in mathematics to the community.

Postholing echoes two current theories for supporting systemic change. The first is the notion that to effectively support teachers, instructional leaders need to understand research-based best teaching practices that support meaningful student learning (Robinson, 2006; Stein & Spillane, 2005). The second posits that focusing on student engagement in mathematical explanations through the analysis of student work can support the development of content knowledge (Kazemi et al., 2011). This focus on student work is important for two reasons: It is an activity that principals might engage in with teachers in the context of a classroom observation, and it can be incorporated into PD for principals. As such, the strategy of investigating a slice of content as a learner, teacher, and principal is likely to foster changes in leadership practices that support systemic change in mathematics teaching.

The postholing construct can serve as a design framework for PD activities at each of the three levels. In the context of mathematics, knowledge at the classroom level involves solving rich mathematical tasks that teachers might pose to students. For example, principals might engage in patterning tasks that afford multiple solution paths, discuss their approaches, and make connections between mathematical representations. Knowledge at the school level involves engagement in teacher learning tasks, including the analysis of a narrative or video case of a teacher engaging in a task with students, discussing and analyzing teacher questions, or evaluating and categorizing students’ mathematical work. Knowledge at the district level zooms out to consider the ways in which teacher learning and systemic change might be supported district-wide. In the context of algebra, this might include engagement in curriculum adoption processes or the use of teacher evaluation tools that target research-based student-centered teaching practices. The postholing construct offers a framework that, together with research on supporting lasting systemic change, can undergird the design of a PD experience.

Engaging administrators in PD designed to develop LCK does not guarantee systemic change. This knowledge needs to be integrated into the knowledge, beliefs, and practices of both the individuals in the school system and the system itself. Rather than adding new knowledge or practices to the current canon, PD that leads to meaningful change must be transformative in nature (Thompson & Zeuli, 1999). Transformative PD moves beyond specific curriculum materials or activities toward general principles that link teaching and learning. Such experiences in mathematics focus on content knowledge, how students make sense of content, and considering how to move student thinking forward (Smith, 2001). A postholing approach to PD underscores the importance of transformative change. If principles about teaching and learning mathematics are not generalized, the impact of the PD might be restricted to the content discussed therein (such as algebra). To support transformative change, the PD design must challenge principals’ existing thinking rather than adding on new practices. In this study, we describe the impact of a PD project using a postholing design on the LCK of principals. Specifically, we focus on the first two levels of LCK—the classroom and school levels.

The Professional Development Project

The PD project described in this study was a 2-year partnership between a large research university in the Midwest United States and six nearby districts ranging from small-city to distant-rural. The project’s goals were to support principals and teacher-leaders in designing, enacting, and supporting systemic reform of algebra instruction in their districts. The project consisted of three phases: a content-focused study group for principals and teacher-leaders related to algebra, the development of site-based algebra improvement plans for each district, and the implementation of those plans by teacher-leaders and administrators. Phase I (September 2008-June 2009) focused on LCK at the classroom and school levels and Phases II (August 2009) and III (August 2009-July 2010) focused on LCK at the school and district levels (see Figure 1). This article describes outcomes related to principal learning in Phase I.

Figure 1.

Phases of the professional development project.

Consistent with a postholing approach (Stein & Nelson, 2003), the project’s focus was on algebra as the study of patterns and functions. Algebra was of particular local interest given recent state policy mandating four mathematics credits including Algebra II for a high school diploma. This policy resulted in a greater focus on algebra teaching, more (and often less-qualified) teachers teaching algebra, higher failure rates in Algebra I, and a more diverse student population learning algebra—a finding mirrored in national reports (Loveless, 2008). Curriculum developers and researchers have identified the idea of function as a critical and unifying feature in the study of algebra in secondary school (e.g., Chazan, 2000; Coxford et al., 1997). Research has shown limitations to teacher knowledge of function, specifically with respect to identifying the key aspects of function, providing examples and non-examples, and making connections between representations (Even, 1993; Stein, Baxter, & Leinhardt, 1990). These research findings coupled with local policy shifts suggested algebra as the postholing focus for the project.

Phase I Professional Development Design

The project study group involved principals and teacher-leaders. The two groups engaged in the same PD activities separate from one another, with the principals beginning their sessions 3 months prior to the teacher-leaders. Principals observed the teacher-leaders engaging in the PD in the final sessions, providing opportunities to compare their teacher-leaders’ interactions in the PD with their own. The design of the principal study group targeted knowledge at the classroom and school levels. Knowledge at the classroom level involved learning about the content of algebra as the study of patterns and functions. Knowledge at the school level encompassed learning about the ways in which algebra teachers can plan, teach, and reflect to promote rich student learning outcomes.

The PD made use of practice-based materials (Smith, 2001) composed of mathematical tasks, narrative and video accounts of classroom practice, and student work authentic to secondary classrooms. The 30-plus hr of PD included four units, each including work on a mathematical task, the analysis of a narrative or video case of a teacher teaching the task, and the analysis of student work and/or teacher planning artifacts related to the same task. Cases serve as examples of practice that can be analyzed from a variety of perspectives and from which general principles about teaching and learning can emerge (Markovits & Smith, 2008). The cases used in this project were dilemma-based, in that they raised critical questions about mathematics teaching and learning and featured teachers implementing ambitious instruction using high cognitive demand tasks. We drew cases from three published volumes (Boaler & Humphreys, 2005; Smith, Silver, & Stein, 2005; Stein, Smith, Henningsen, & Silver, 2009). Each case discussion began with a prompt asking principals to consider particular aspects of the teacher’s practice and implications for student learning (see Table 1).

Table 1.

Case Discussion Key Ideas and Focus Questions.

Case discussion	Key ideas	Focus questions
Catherine Evans and David Young	The way a teacher implements a task can affect students’ opportunities to learn.	What are the similarities and differences in Catherine’s and David’s classes?
Monique Butler	Teaching a procedure first, then addressing conceptual connections, has limitations.	What did Monique Butler want her students to learn?
		What did her students actually learn?
Ed Taylor	Making observations, noticing patterns, and building toward generalization can be a powerful learning experience for students.	Identify pedagogical moves.
		Which of the Five Practices is each move connected to?
		How did the move support student learning?
Edith Hart	Using and connecting multiple mathematical representations can provide rich learning experiences. Teacher questioning facilitates these connections.	Identify the mathematical ideas students learned.
		How did Edith Hart support student learning?

Each case addressed an aspect of algebra as the study of patterns and functions: linear relationships and comparing linear and quadratic relationships, factoring quadratics, solving systems of equations, and interpreting qualitative graphs. Each unit began with principals solving and discussing the mathematical task used with students in that case, followed by reading or watching the case and analyzing the ways in which teaching practice supported or inhibited students’ mathematical learning. Units closed with activities extending further into classroom practice, such as analyzing student work, teacher questioning, assessment, or planning. Ten administrators from six districts participated in the PD. Participants varied in their administrative experience (1-25 years) and their primary content teaching area (see Table 2; all names are pseudonyms). Three of 10 principals had some experience teaching mathematics. The group met for eight after-school sessions totaling over 30 hr of work.

Table 2.

Participant Background.

	Leadership (years)	Teaching (years)	Education
Garrett	High School asst. principal (1)	High School math (11)	Secondary math
Nancy	Middle School asst. principal (1)	Elementary (15)Middle School math (3)	Elementary education, math minor
Wade	Elementary/Middle principal (6)	Elementary (6)	Elementary education, math minor
Cassandra	High School principal (12)	Elementary (12)	Special education, math minor
Douglas	High School principal (5)	High School science (9)	Secondary science
Steven	High School principal (11)	High School social studies (30)	Secondary social studies, accounting
Erin	Middle School principal (8)	High School social studies (11)	Secondary social studies
Mona	High School principal (3)	High School language arts (7)	Secondary English
Chad	Middle School principal (5)	High School language arts (6)	Secondary English
Ed	Middle School principal (13)	Middle/High School social studies and Spanish (6)	Secondary social studies, Spanish

Method

The purpose of this study is to explore the impact of a PD program on principals’ LCK at the classroom and school levels. To address this purpose, data collection included video recording of all PD sessions, pre- and post-PD assessments of principals’ knowledge of algebra and their analysis of a short lesson transcript, and semi-structured interviews conducted at the end of Phase I. The PD was led by the first, fourth, and fifth authors, faculty members in the university’s teacher education department.

To consider the ways in which principals’ LCK at the classroom and school levels changed, we selected two data sources from the PD sessions. For knowledge at the classroom level, we were interested in data that showed principals’ abilities to solve mathematical tasks involving algebra and specifically to understand the nature of and use functions. To understand shifts in knowledge of function, we analyzed items from a written assessment designed to assess principals’ LCK at the classroom level, administered at the beginning and end of Phase I. This assessment included items used in previous research to evaluate teachers’ mathematical knowledge related to functions (Steele & Hillen, 2012; Steele, Hillen, & Smith, 2013). Items asked principals to provide a definition of function and examples and non-examples of functions, and to solve problems using functions. These items measured three important aspects of knowledge of function at the classroom level: the ability to define function, the ability to identify examples and non-examples of function, and the ability to create and make connections between multiple representations of functions (Cooney, Beckmann, & Lloyd, 2011).

To investigate LCK at the school level, we analyzed transcripts of principals’ discussions of the narrative and video cases in the PD sessions. Although these case discussions do not equate to observing principals’ practices in evaluating teaching, these discussions helped us to understand the features of a classroom that a principal might be likely to attend to during observations and discussions with teachers. In fact, we argue that these cases afforded a unique lens into what principals noticed about instruction because they were not burdened by existing observation tools or rubrics used in their individual districts.

Data Analysis

We looked for change in LCK at two levels. With respect to knowledge at the classroom level, we expected to see changes in the ways in which principals engaged with the mathematics through both their talk and written work in solving mathematical tasks. Knowledge at the school level was measured by shifts in the ways in which principals talked about algebra teaching and learning. Specifically, we wanted to understand how principals framed their arguments about algebra teaching and learning during analyses of classroom practice. Following Copland (2000), the concept of frame refers to the terminology and conceptual apparatus in which one couches ideas. For example, after observing a classroom, a principal may frame what she notices as being related to teacher characteristics, such as the teacher’s experience or pedagogical style, or as related to teacher thinking, such as the nature of questioning. This frame analysis quantifies the ways in which principals appropriated PD ideas and integrated them with their own perspectives.

Knowledge at the classroom level

The written assessment measured three aspects of knowledge at the classroom level, including the ability to define a function, identify examples and non-examples of functions, and make use of and connect multiple representations of functions. Two researchers coded the items with an interrater reliability of 85% or better. A correct definition of function met two criteria (Even, 1993): explicitly included univalence (each input has one and only one output) and did not explicitly rule out arbitrariness (functions need not be numerical or a written rule). Definitions that did not include univalence or made erroneous statements (e.g., must be linear relationships, must involve numbers or variables) were coded incorrect. Definitions were coded inconclusive if there was not enough information to evaluate the univalence criterion. Such definitions included correct statements (e.g., passes the vertical line test) that provided no further explanation regarding why the statements implied a relationship was a function. Examples and non-examples of functions were coded correct, incorrect, or inconclusive based on the same mathematical criteria. The pentagon pattern task asked principals to generalize a geometric pattern and to explain how their generalization related to the visual diagram. Responses were coded using a rubric shown in Table 3. This rubric, developed and validated in previous research (Steele et al., 2013), captured the degree to which principals connected the visual representation to their generalization. The criteria in the rubric represent a progression from a procedural, numerically based understanding of the functional relationship to a well-connected understanding of function (change of three for each train with two constant sides) across symbolic, tabular, and visual representations.

Table 3.

Representations Rubric for Pentagon Task.

Score	Description	Example
4	Explanation fully connected to visual patternGeneralization evident (verbally or symbolically)All aspects of generalization explained accurately with respect to the visual pattern	For each pentagon on the end of the train you count 4 sides, so that is always 4 × 2 = 8. There are two fewer pentagons in the middle of the train than the train number itself, and each one contributes exterior 3 sides to the perimeter: 8 + 3(n − 2), where n is the train number.
3	Explanation partially connected to visual patternGeneralization evidentAt least one aspect of generalization explained accurately with respect to the visual patternRemaining of generalization explained incorrectly, vaguely, or not explained	n(5) − (n − 1)(2)n is the train number, multiply this number by 5 then subtract one less than the total number times 2.From the visual representation, we see that 2 pentagons share one side. This shared side is on the inside and is not included in the perimeter.
2	Weak explanation partially connected to visual patternGeneralization evidentAt least one aspect of generalization explained, but the explanation is incorrect or vague	(3x) + 2When a new train is added, only three sides of that train are actually added. The (3x) is 3 sides from before multiplied by the train number.
1	No connection to visual patternGeneralization evidentElements of generalization explained but not connected to the visual pattern	3n + 2Multiply the number of trains by 3 and then add 2. I know this works because it fits my pattern. My description is independent of the visual representation.
0	No explanation

We analyzed the PD discussions involving mathematical tasks to triangulate data on connections between representations. We identified connections principals made between representations in each of the transcripts of the mathematical task discussions. For each discussion, we examined the extent to which principals were pressed to make connections by the facilitator as compared with making these connections on their own.

Knowledge at the school level

We also investigated how principals took up the key constructs in the PD and how these key constructs were situated into broader frames related to the teaching and learning of mathematics. The oral and written case analyses provided principals with opportunities to analyze teaching practice. We coded transcripts of four case discussions and the pre- and post-PD written case analysis to identify the frames principals used to describe algebra teaching and learning. To code a comment within a particular frame, we first identified the construct (e.g., teacher experience, questioning, motivation) principals noticed in the case, and then situated that construct within a frame (e.g., as teacher characteristics, student thinking). The research team used a constant comparative method (Strauss & Corbin, 1990) to identify the constructs principals used in analyzing cases of algebra teaching. Two researchers named constructs and proposed them to the team, with disagreements resolved through discussion.

Consistent with the notion that strong school leaders look beyond surface-level features of teaching, we identified four dominant frames for the constructs principals noticed in discussing the cases. Two frames represented the notion of characteristics: static aspects of teachers and students such as years of experience, certification, and prior mathematics achievement. Two contrasting frames captured teacher and student thinking: the dynamic work of engaging in the teaching and learning of mathematics, capturing ideas such as questioning, planning practices, and student discussion. Figure 2 shows the frames (represented as boxes) and constructs (text inside the boxes) identified in the data. The box in the center of the thinking column shows the constructs at the intersection of teaching and learning introduced in the PD.¹ The lower right box identifies other constructs not related to characteristics or thinking.

Figure 2.

Frames and constructs identified in case discussions.

Shifts in the frames principals used over time to analyze the cases were taken as evidence of changes in knowledge at the school level. For each case discussion, we identified the constructs principals discussed and situated them within the relevant frame. We then described shifts in frame use across the four case discussions.

Results

The purpose of this study is to explore the impact of a PD program on principals’ LCK at the classroom and school levels. At the classroom level, we note changes in principals’ abilities to define, represent, and provide examples of function, and to draw connections across representations. At the school level, we describe shifts in the frames used to analyze cases of algebra teaching. We then trace one principal’s learning across the PD.

Changes in Algebraic Content Knowledge

Three items in the written assessment measured principals’ knowledge of the definition of function and their ability to create examples and non-examples. The first item asked principals to define function. On the pre-test, no principal provided an accurate definition of function meeting the criteria of univalence and arbitrariness. Three made no attempt, one allowed for arbitrary functions but violated univalence, and five both ruled out arbitrary functions and violated univalence. For example, one principal’s initial response stated that “the value of an unknown quantity in an algebraic equation/solution ➔ impact it will have in the outcome solution.” This definition does not include univalence and rules out arbitrary functions. The post-test showed significant improvement: five of the eight² responses were entirely correct, with one omission. The same principal noted above provided this definition, which includes univalence and does not rule out arbitrary functions: “A function is a relationship for which there is one and only one value (solution).”

When identifying examples and non-examples of function on the pre-test, two of nine principals gave a correct example (five omissions) and none provided a non-example. On the post-test, six out of eight provided an accurate example and six an accurate non-example. (None of the examples or non-examples was drawn directly from the PD.) Shifts were also evident in representational use. All pre-test responses used only symbolic representations, whereas post-test responses included three graphs, two tables, and four responses with multiple representations. These results suggest principals’ conception of functions was more robust after the PD.

Principals were also asked to describe a linear function expressed as the perimeter of a pattern containing pentagons (Figure 3), extend the pattern, and write a general description connected to the visual pattern. Table 4 summarizes the pre-test and post-test rubric scores for the eight principals completing both items.

Figure 3.

The pentagon task.

Table 4.

Representation Scores for Pentagon Task.

Principal	Pre-test	Post-test
Erin	0	3
Chad	1	4
Garrett	2	4
Steven	4	4
Cassandra	1	1
Wade	1	2
Nancy	0	4
Ed	1	2

Principals made stronger connections between the visual pattern and the generalization after the PD. Two of eight made some connection to the visual representation in the pre-test, whereas seven of eight made a connection in the post-assessment (score 2 or higher). Moreover, half of the principals connected every aspect of their generalization to the pattern (score 4).

These shifts in representational use were also evident in PD discussions of math tasks. In early sessions, the facilitator had to ask specific questions to prompt representational connections. For example, in the first discussion, the facilitator asked principals to relate the visual pattern to the symbolic formula. In the first two discussions, 50% and 80% of connections between representations were prompted by facilitator questions. In two later task discussions, 14% and 20% of the connections were prompted by facilitator questions.

Together, these data demonstrate that after the PD, principals were better able to define and provide examples of functions, and to describe functions using multiple representations. These changes in algebra knowledge at the classroom level resonated with the PD goals, so in that sense, one might not consider these changes to be surprising. Given that the content of function is learned by first-year high school students, we argue that this learning is significant. Understanding the mathematics content is also critical for making sense of mathematics teaching; as such, the results with respect to changes in principals’ analyses of cases of teaching algebra in part rest on their ability to make sense of the underlying mathematics.

Changes in Knowledge of Teaching Algebra

To examine changes in leadership content knowledge at the school level, we identified constructs and frames used by principals as they analyzed cases of algebra teaching. Broadly stated, principals began to use the PD-introduced constructs in concert with, rather than replacing, the constructs they originally brought to the initial case discussion. This integration allowed principals to focus on mathematics-specific issues in discussing the cases. Furthermore, through the use of PD-introduced constructs, the principals shifted from static frames of teacher and student characteristics toward dynamic frames related to teacher and student thinking. To illuminate the shift, we compare the initial case discussion with three subsequent case discussions.

Initial Frames for Discussing Cases of Algebra Teaching

The first case discussion analyzed the Case of Catherine Evans and David Young. The goal was to introduce the Mathematical Tasks Framework (Stein et al., 2009), a construct situated in the thinking frame. Despite this goal, principals overwhelmingly framed their comments in the teacher characteristics and general frames. In describing similarities and differences between the two classes in the case, principals focused on teaching experience, teacher confidence or personality traits, and educational background (teacher characteristic frame). For example, after several comments were made about the differences in teaching experience between Catherine and David, Steven discussed teacher confidence:

[Teachers now] don’t have the same confidence in what they are doing [as in the past]. They are doing what they think they were told to do rather than taking a learning situation and making it part of me and pull all of it together, for the learning of the students that are there, with the confidence that what I will do will work.

Although most constructs related to the teacher characteristic frame, one construct was discussed in the teacher thinking frame—modifying instruction. Steven noted that Catherine modified her teaching but did not “reduce what she expected of the students.” Erin added,

One difference [between the two cases] that Steven brought up, that I think is one of the most important, is that Catherine modified her instruction . . . whereas David did not necessarily modify. He was busy getting through his delivery and his instruction. He did a lot of great things, but he didn’t rewrite things necessarily.

Modifying instruction represents a teacher thinking frame but it was used superficially. Principals discussed whether the teachers modified instruction but not the specific modifications or implications for student learning. Steven’s comment connects Catherine’s modifications to her expectations, but does not elaborate on how the modifications maintained expectations.

Principals also identified some constructs situated in a general frame, such as traditional instruction, understanding, and student involvement. Principals used traditional instruction to compare Catherine’s and David’s teaching styles without describing specific teaching practices. Principals discussed “understanding” with respect to how teachers checked for whether students understood, but did not describe the nature of the understanding. For instance, Steven identified Catherine’s goal as having students “conceptually understand the concepts that she is trying to teach,” but did not elaborate what conceptual understanding meant. Principals mentioned student involvement, but they did not articulate its meaning or importance.

Subsequent Frames for Discussing Cases of Algebra Teaching

In subsequent case discussions, principals’ analytical frames shifted to foregrounding student thinking and the interactions between student and teacher thinking. The discussions show principals connecting general constructs to thinking frames by articulating meaning for ideas such as understanding and involvement, allowing for explicit connections to the PD constructs.

The second case illustrated a teacher (Monique Butler) introducing a concept procedurally on a previous day in response to local testing demands, then trying (unsuccessfully) to build conceptual understanding. The discussion marked a frame shift from teacher characteristics to student characteristics, with some comments related to student thinking. One example of this shift was the idea of students’ prior knowledge. Rather than focusing their analysis on the teacher, principals considered how students’ backgrounds influenced their opportunities to learn. When discussing student learning in the case, Nancy responded, “What prior activities had [students] done with this? Manipulatives might be something new to them.” Steven elaborated,

[W]e commented on the lack of prior experience with manipulatives. It’s also probably a lack of prior experience with a problem-solving approach . . . And I’m not sure that I’m familiar, as a student in this case, with the patterns of thinking that have to go together to create that type of solution.

Principals moved from considering student characteristics to a student thinking frame. Specifically, they discussed students’ struggles and frustration as influencing opportunities to learn. One phase of the discussion centered on students’ frustration in not knowing a procedure (FOIL³) for the task, and indicated that this may have caused them to struggle with how to think through the solution. This comment marked a shift toward a student thinking frame.

Principals’ analysis of the third case situated primarily in the student thinking frame. One prominent idea in the analysis of the Case of Ed Taylor was levels of thinking—a term principals brought from prior experiences whose meaning was not immediately explicated. Steven commented on how students made sense of the quadratic pattern in the case:

The student is getting observations and visualizations that they haven’t received before, but it requires, I think, a more skilled level or depth of thinking . . . It was so much simpler just to learn to jump through the hoops, show me the formula, I do that thing. Tell me when to apply the formula, and we’re good to go.

Level of thinking was used here to describe the different expectations students face when they are asked to do mathematics that goes beyond simply applying a formula. Notable in this discussion was the integration of principals’ pre-existing constructs into a student frame promoted by the PD. As the conversation went on, the notion of levels of thinking became connected to a student thinking construct (the Mathematical Tasks Framework) and a teacher thinking construct (teacher questioning). Principals were connecting their own ideas to PD constructs and shifting from characteristic-based to thinking-based frames.

Principals’ discussion of “understanding” is an example of this shift. Initially, understanding was described vaguely and categorized as a general frame. Later discussions of understanding featured more nuance and integration. For example, Garrett integrated understanding with the PD-introduced practices for orchestrating productive discussions (Smith & Stein, 2011), focusing on monitoring student thinking:

As far as impact that’s student-wide, I think monitoring is always a good way to maybe help students understand or see where other people are coming up with their ideas, or where they’re getting their solutions. As a student, it might be easier for me to learn from what Nancy’s doing than from what the teacher is actually discussing. So any time you monitor, I think it helps the students to kind of understand what they’re trying to get accomplished.

By using both constructs, Garrett connected teacher thinking (the goals behind monitoring) and student thinking frames. Similarly, Mona discussed a practice for orchestrating productive discussions (connecting), which encompasses teacher and student thinking:

But then, [Ed Taylor] does move on to that connecting piece where, he drew the tiles in their new position and he asked why making the square is a good thing to do. And they said you could multiply it easier, and so what he said was that it’s more important for him, yes, to get the right answer, but to be able to process it and explain the answer. So, to me that’s a significant connecting piece.

In this comment, Mona linked expectations for student thinking (justify their answers) to teacher thinking (connecting mathematical ideas) using specific math ideas from the case.

The final case discussion (the Case of Edith Hart) showed continued linking of teacher and student thinking frames. Specifically, principals linked teacher thinking (questioning) to student learning. For example, several principals noted that Edith Hart elicited multiple mathematical approaches from students. Cassandra argued that Edith Hart’s question—“Does anyone else have any observations or ideas?”— communicated to students that there were multiple solutions, which played out in the work students produced. Chad also noted that “[Edith Hart] did not want to give away the big secret, so she asked them what would happen if they bought a card and never used it.” Chad noted that this was the “right question at the right time” because it supported connecting the task’s context and the graphical representation.

We see two significant shifts in principals’ use of frames across the four case discussions. The first shift was from general or characteristic-based frames to thinking-based frames. The second shift linked teacher and student thinking frames, frequently through the PD-introduced constructs. During this shift, principals linked their own constructs explicitly to the PD constructs. Both shifts represent significant change in LCK at the school level, as they mark changes in the features of classroom practice that principals noticed.

Two features of the PD catalyzed these opportunities to consider new frames and deepen LCK at the school level. First, the constructs introduced for analyzing the cases supported this shift. All PD-introduced constructs focused on thinking frames and connected teacher and student thinking. The constructs provided opportunities to shift to thinking frames and connect them, but by no means guaranteed the shift. Second, the questions posed in the case discussions (see Table 1) evolved from general (similarities and differences between teachers) to particular (identifying pedagogical moves and their implications on student learning). As constructs were introduced, prompts encouraged principals to use new constructs but did not guarantee their use.

Tracing Shifts in LCK: The Case of Steven

In this section, we provide a longitudinal view of one principal’s learning by tracing shifts in Steven’s LCK over time. Steven had substantial teaching and leadership experience (see Table 2); bachelor’s degrees in social studies, education, accounting, and business administration; and a number of master’s credits. We focused on Steven because he was initially reluctant to embrace the PD, but underwent significant shifts in his thinking about algebra teaching and learning. Steven intended to retire the next year and as such, could have easily only engaged superficially. He did not have a robust mathematics background, which might have suggested challenges in doing the mathematical tasks in the PD. At the start, Steven represents a principal whose views might be challenging to change. In this brief case study, we draw on data from the written assessments, case discussions, and an individual interview at the close of the Phase I PD.

In his post-PD interview, Steven talked at length about how he changed his perspective on algebra teaching and learning as the PD unfolded, showing a shift in Steven’s LCK at the school level related to what it means to teach and learn algebra:

I think you actually waited for us to begin to change our perception ourselves. You didn’t start out by saying, look, you’re going to look at teaching algebra differently than you have ever looked at it before. You didn’t say that this is the right way to do it. You just took us through the exercises until we began to see value in what it is that we’re doing.

Steven’s pre-assessment showed fragmented knowledge of algebra content. He could not provide a correct definition of function and did not provide examples or non-examples. On the pattern task, he described the connections between his equation and the aspects of the diagram. After the assessment, Steven remarked, “this is a short exercise, but if you don’t remember the concept of a function, it’s really going to be a struggle.” Steven’s initial analysis of the narrative case attended largely to teacher actions, such as assigning homework and team learning, but did not describe how these actions influenced student learning. Similarly, in the first PD case discussion, Steven focused on teacher characteristics, specifically related to teacher experience. He was “concerned about new teachers” dealing with policy pressure, argued that new teachers “don’t have the same confidence in what they are doing,” and noted the role of mentoring.

In later sessions, Steven made explicit connections between his experiences in doing mathematics and analyzing the cases. In thinking about changing practices in his district, Steven repeatedly expressed concern for the increased time needed to do high cognitive demand tasks. An activity in which principals mapped state standards to a mathematical task they had solved (identifying more than 20 standards) marked a turning point for Steven:

I gave Mike [first author] some resistance, to start with. Our teachers, they would say, “there’s no way in the world I could spend an hour and a half on one problem.” All right. Which is why I figured he brought the part of making all the connections with the [standards]. I figured that was just for me, because I kept complaining, not complaining but kept commenting so much about it. I’m disappointed now, it wasn’t just for me, it was planned all the time.

These excerpts suggest Steven initially approached the PD with skepticism. Although Steven found work on the mathematical tasks interesting and brought his own analytical frames to the cases, he wondered openly about the trade-offs inherent in the PD-promoted approaches to teaching algebra. Through engagement in the PD activities, Steven came to know, understand, and value a new approach to algebra teaching and learning. The frames Steven used to analyze cases changed later in the PD, attending how students’ prior mathematical experiences their work. He considered students’ mathematical thinking, observing that students were being exposed to “observations and visualizations that they haven’t received before” that furthered their thinking. Steven’s post-assessment showed evidence of these shifts in LCK at the classroom and school levels. Steven correctly provided a definition of function, examples, and non-examples. His case analysis connected teacher questions and student learning goals, attended to how the teacher supported mathematical confidence, and identified a specific mathematical goal for the lesson.

Steven’s case shows change in LCK related to algebra at the classroom and school levels. At the classroom level, his ability to define function, provide examples and non-examples, and connect mathematical representations improved. At the school level, Steven’s analysis of algebra teaching and learning showed greater depth and a focus on specific issues of teacher and student thinking. The use of specific mathematical ideas in his case analysis and the links between the mathematics and teacher and student thinking reinforce the notion that the postholing approach on which the PD was built supported the multifaceted development of principals’ LCK.

Discussion

In this study, we made practical use of the LCK theory by designing and implementing a PD experience for principals related to algebra content and teaching. The focus on algebra at the classroom and school levels was important for two reasons. First, principals directly mobilized their knowledge to design district algebra improvement initiatives with similarly prepared teacher-leaders. Second, the experience immersed principals in the work of teaching and learning mathematics, shifting thinking toward substantive, observable features of effective mathematics teaching and learning. Stein and Nelson (2003) posited the LCK construct as critical to effective instructional leadership, illustrating examples of LCK and calling for research relating LCK to instructional leadership and systemic change. This research represents an effort to design PD using the postholing construct, with the goal of developing LCK at multiple levels with a focus on algebra. Following Stein and Nelson, development of multiple levels of LCK would set the stage for meaningful systemic change in how principals support the teaching and learning of algebra.

The results suggest that aspects of LCK can be systematically developed in an intervention with principals; specifically, in this case, knowledge of algebra at the classroom and school levels. Through the PD, principals were better able to define function and to move flexibly between representations. This knowledge helps principals understand function’s importance in secondary mathematics and the role that representations play in mathematics more generally. This knowledge of algebra at the classroom level was foundational to the development of principals’ knowledge at the school level. The shifts in frames principals used to analyze cases of teaching show change in LCK at the school level. Principals integrated their own constructs with the constructs introduced by the PD, first shifting from characteristic- to thinking-based frames and then connecting the frames of teacher and student thinking. These moves mark a shift from attending to surface-level characteristics of teaching to considering what good teacher and student thinking looks like in algebra. Although the notion of LCK and postholing has been widely cited in leadership literature, few researchers have used these constructs as design principles for PD. This study, focused on the first two levels of LCK, shows that a postholing approach can lead to meaningful learning.

The postholing approach provided principals with a sustained look at a slice of mathematics content—in this case, algebra as the study of patterns and functions—at the classroom (content), school (instructional), and district (systemic) levels. This work supported principals in generalizing lessons learned in the PD to the teaching and learning of mathematics beyond the focal content. As such, we anticipate the shifts seen in principals as a result of this project will engender three important changes in administrators’ practice as instructional leaders. First, principals are likely to use their new shared knowledge and language to engage with their mathematics teachers and provide support focused on research-based effective teaching and meaningful student outcomes (Robinson, 2006). Second, principals have new lenses for observing teaching practice tailored more closely to a conceptual approach to teaching mathematics. Rather than looking only for clearly written objectives and student engagement, principals might now ask about the mathematical tasks teachers are using or their patterns of questioning as measures of instructional quality. Third, principals developed new ideas about mathematics teaching that generalize beyond algebra that will inform their work as instructional leaders at the systemic level. The frames of teacher and student thinking, along with constructs such as questioning, anticipating student thinking, and the use of multiple representations, are observable beyond algebra. Although the PD goals centered on algebra, these constructs and frames can support instructional change in mathematics broadly.

These results contribute to the ongoing conversation about LCK, and about the connections between research outcomes related to effective content instruction and educational leadership. They indicate that postholing experiences can lead to important changes in LCK, even in a challenging content area such as algebra. We do acknowledge important limitations to the study. First, principals were voluntary participants and may not represent a broad cross-section of principals. Second, this intervention was a design-based experiment, with the research team engaged in development both before and during the PD. Third, we have little direct data about the ways in which this new LCK influenced principals’ day-to-day practice, although there are promising indicators in this regard. Although the analysis of the impact on algebra teaching in the districts is beyond the scope of this article, we close with a brief reflection on the ways in which administrator engagement provides leading indicators of deep and durable change.

Looking Forward: How Deep and Durable Is the Change?

To conjecture about the broader influences of the PD on districts, we consider two perspectives: the durability of principals’ learning about algebra, and the influence of this learning on leadership practice. As principals’ leadership content knowledge changed, so did their positions as mathematical authorities. In the first few sessions, principals praised the contributions of others while diminishing their own contributions. For example, in Session 4, Chad explained another participant’s work by saying, “It was actually mathematical genius” and then subsequently, “Don’t ask for an explanation, because I can’t give you one.” In Session 5, in revisiting the question, “What is a function?” Steven’s response was, “From a non-math person?” In both cases, principals mediated participation by identifying as a person not skilled at mathematics.

In later sessions, principals asserted more mathematical authority while doing mathematics. In a conversation in the final session, Wade noted, “The beauty of it is, we were always able to solve the problem; we weren’t ever left hanging.” These changes paralleled shifts toward student thinking frames, evident in Steven’s comment about connecting representations:

A key word there is “relationships,” because all too often in our math classes, I think . . . relationships are seen as hoops that you jump through. And given this equation, jump through the hoop, and make that graph. Now, from this graph or set of data, jump back through the hoop, then create the equation. And I’m not sure that the students see the relevance in that.

These comments indicate that principals changed their relationships toward mathematics, which in turn are likely to shift how they observe and evaluate mathematics teachers.

Principals also raised issues about issues of staffing, teacher evaluation, and policy, constructs that relate to Stein and Nelson’s (2003) LCK at the district level. This phase of the project did not target this aspect of LCK; these comments, however, suggest shifts in principals’ thinking about district-level leadership. One theme was curricular coverage and the notion that high cognitive demand tasks allow more content coverage despite taking more time. In Douglas’s reflective interview, he imagines a conversation with teachers around this idea:

[Y]ou might have 115 content expectations but if you look at these tasks, one task might cover 40 of those . . . and that’s the thing we have to understand. If you walk into our teachers’ rooms [and ask] what are you covering today, they’ll tell you one thing because that’s what they think they’re supposed to be doing. I’m covering this content expectation. Look, I’ve even got it written on the board. I’m doing my teacher thing . . . I’ve got to cover these 95 things, I’ve got 180 days to do it, I’m going to cover these this day, these this day, these this day. [But with rich tasks,] I can cover 30 of them, it might take us a week or two with this task, but I’m going to get [more] covered.

Another policy focal point was teacher evaluation. Principals noted that they had been observing mathematics teachers for short durations using general criteria, as Steven observes: “we tended to do it with fairly broad brushstrokes . . . I realized attention to detail makes a difference.” Principals described how new perspectives on teacher and student thinking would change their evaluation practices. More than half of the principals noted changes in their fluency and confidence in talking with mathematics teachers about practice, as Wade noted in his reflection:

Well, as I said, to be a learner first, I think, solidifies that now I can be the evaluator and the teacher of sorts. And now I can speak with some knowledge behind it. [Before, I could tell teachers that] your questions [have] real simple answers. Now I can elaborate and say, these are the problem solvers we want our student to be. Therefore, these are the kinds of questions we need to have. And, “Hey, I listed all the questions you asked in the observation. You tell me, how many higher-order thinking questions do you think you have? How many rote-answer questions do you think you have?” And that’s very telling. And I can do it in a very non-threatening way. Because now I can just say, I wrote down the questions I heard. That’s it. I’m just a big dumb right hand who wrote them down. You tell me where these questions fall in the spectrum.

Wade’s comments exemplify the ways in which principals’ postholing experiences influenced thinking about their roles as instructional leaders. Wade explicitly attributes his change to having engaged in the content from both learner and teacher perspectives through doing mathematical tasks and analyzing classroom episodes. These comments suggest that the LCK developed in the PD is likely to influence principals’ instructional leadership practice.

The goal of this study was to explore the ways in which a postholing approach can influence principals’ LCK related to algebra. The analysis demonstrates that the postholing design approach can produce meaningful shifts in how principals make sense of mathematics teaching and learning, which sets the stage for meaningful change in instructional leadership practice. Future research can move this work forward in meaningful ways by further investigating how changes in principals’ understanding of effective teaching and learning in a content area influence their practice in teacher evaluation and instructional leadership.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This article was supported in part by a grant from the Michigan State Department of Education Title II Teacher Quality Grant Program (grant number 2314).

Notes

Author Biographies

Michael D. Steele is an associate professor in the Department of Curriculum and Instruction in the School of Education at the University of Wisconsin–Milwaukee. His work focuses on supporting secondary math teachers in developing mathematical knowledge for teaching, and integrating content and pedagogy through teacher preparation and professional development.

Kate R. Johnson is an assistant professor in the Department of Mathematics Education at Brigham Young University. Her interests include illuminating and understanding the influence of teacher identity on the practice of teaching mathematics for social justice and learning to teach. Current research projects focus on developing understanding about how teachers learn to teach mathematics for social justice, how people discuss the role of race and gender in mathematics education, and how religious identity influences understanding about issues of social justice.

Samuel Otten is an assistant professor in the Department of Learning, Teaching, and Curriculum at the University of Missouri. His research interests include standards for mathematical practice, student participation in secondary mathematics classrooms, and discourse.

Beth A. Herbel-Eisenmann is an associate professor of teacher education in the Department of Teacher Education at Michigan State University. Her research interests include bringing a discourse perspective to the study of written, enacted, and hidden curriculum in mathematics classrooms. She is not only interested in interrogating the norms that are embedded in and carried by teacher and textbook discourse patterns but also dedicated to understanding how these patterns may affect diverse students in mathematics classrooms, especially in terms of their mathematical understanding, dispositions, and epistemology.

Cynthia L. Carver is an associate professor in the Department of Educational Leadership at Oakland University. Her research and teaching interests are focused on the preparation and practice of school leaders—both principals and teachers—for democratic schools. Her expertise includes leadership for instructional reform, teacher leadership, teacher/leader learning and development, and qualitative research methods.

References

Ball

D. L.

Thames

M. H.

Phelps

(2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59, 389-407.

Boaler

Humphreys

(2005). Connecting mathematical ideas. Portsmouth, NH: Heinemann.

Boyd

Grossman

Ing

Lankford

Loeb

Wyckoff

(2011). The influence of school administrators on teacher retention decisions. American Educational Research Journal, 48, 303-333.

Chazan

(2000). Beyond formulas in mathematics and teaching: Dynamics of the high school algebra classroom. New York, NY: Teachers College Press.

Common Core State Standards Initiative. (2010). Common core standards for mathematics. Available from http://www.corestandards.org/

Cooney

T. J.

Beckmann

Lloyd

G. M.

(2011). Developing essential understandings of functions for teaching mathematics in grades 9-12. Reston, VA: National Council of Teachers of Mathematics.

Copland

M. A.

(2000). Problem-based learning and prospective principals’ problem-framing ability. Educational Administration Quarterly, 36, 585-607.

Coxford

A. F.

Fey

J. T.

Hirsch

C. R.

Schoen

H. L.

Burrill

Hart

E. W.

Watkins

A. E.

(with Messenger M. J., & Ritsema B.). (1997). Contemporary mathematics in context: A unified approach (Course 1). Columbus, OH: Glencoe/McGraw-Hill.

Even

(1993). Subject-matter knowledge and pedagogical content knowledge: Prospective secondary teachers and the function concept. Journal for Research in Mathematics Education, 24, 94-116.

10.

Hiebert

J. A.

Morris

A. K.

Berk

Jansen

(2007). Preparing teachers to learn from teaching. Journal of Teacher Education, 58, 47-61.

11.

Hoekstra

Korthagen

(2011). Teacher learning in the context of educational change: Informal learning versus systematically supported learning. Journal of Teacher Education, 62, 76-92.

12.

Kaufman

J. H.

Stein

M. K.

(2010). Teacher learning opportunities in a shifting policy environment for instruction. Educational Policy, 24, 583-601.

13.

Kazemi

Elliott

Mumme

Carroll

Lesseig

Kelley-Petersen

(2011). Noticing leaders’ thinking about videocases of teachers engaged in mathematics tasks in professional development. In Sherin

M. G.

Jacobs

V. R.

Philipp

R. A.

(Eds.), Mathematics teacher noticing: Seeing through teachers’ eyes (pp. 188-203). New York, NY: Routledge.

14.

Loveless

(2008). The misplaced math student: Lost in eighth-grade algebra. Washington, DC: Brown Center on Educational Policy.

15.

Markovits

Smith

M. S.

(2008). Cases as tools in mathematics teacher education. In Tirosh

Wood

(Eds.), International handbook of mathematics teacher education: Tools and processes in mathematics teacher education (Vol. 2, pp. 39-64). Rotterdam, The Netherlands: Sense Publishers.

16.

November

Alexander

van Wyk

M. M.

(2010). Do principal-educators have the ability to transform schools? A South African perspective. Teaching and Teacher Education, 26, 786-795.

17.

Remillard

J. T.

Herbel-Eisenmann

B. A.

Lloyd

G. M.

(Eds.). (2009). Mathematics teachers at work: Connecting curriculum materials and classroom instruction. New York, NY: Routledge.

18.

Robinson

V. M. J.

(2006). Putting education back into educational leadership. Leading & Managing, 12, 62-75.

19.

Robinson

V. M. J.

Lloyd

C. A.

Rowe

K. J.

(2008). The impact of leadership on student outcomes: An analysis of the differential effects of leadership types. Educational Administration Quarterly, 44, 635-674.

20.

Smith

M. S.

(2001). Practice-based professional development for teachers of mathematics. Reston, VA: National Council of Teachers of Mathematics.

21.

Smith

M. S.

Silver

E. A.

Stein

M. K.

(2005). Improving instruction in algebra: Using cases to transform mathematics teaching and learning. New York, NY: Teachers College Press.

22.

Smith

M. S.

Stein

M. K.

(2011). Five practices for orchestrating productive discussions. Reston, VA: National Council of Teachers of Mathematics.

23.

Spillane

J. P.

(2000). Cognition and policy implementation: District policymakers and the reform of mathematics education. Cognition and Instruction, 18, 141-179.

24.

Steele

M. D.

Hillen

A. F.

(2012). Content-focused methods courses: Integrating pedagogy and mathematical content that is central to the 7-12 curriculum. Mathematics Teacher Educator, 1, 52-69.

25.

Steele

M. D.

Hillen

A. F.

Smith

M. S.

(2013). Developing mathematical knowledge for teaching in a methods course: The case of defining function. Journal of Mathematics Teacher Education, 16, 451-482.

26.

Stein

M. K.

Baxter

J. A.

Leinhardt

(1990). Subject-matter knowledge and elementary instruction: A case from functions and graphing. American Educational Research Journal, 27, 639-663.

27.

Stein

M. K.

Nelson

B. S.

(2003). Leadership content knowledge. Educational Evaluation and Policy Analysis, 25, 423-448.

28.

Stein

M. K.

Silver

E. A.

Smith

M. S.

(1998). Mathematics reform and teacher development: A community of practice perspective. In Greeno

J. G.

Goldman

(Eds.), Thinking practices in mathematics and science learning (pp. 17-52). Hillsdale, NJ: Lawrence Erlbaum.

29.

Stein

M. K.

Smith

M. S.

Henningsen

M. A.

Silver

E. A.

(2009). Implementing standards-based mathematics instruction: A casebook for professional development (2nd ed.). New York, NY: Teachers College Press.

30.

Stein

M. K.

Spillane

J. P.

(2005). What can researchers on educational leadership learn from research on teaching? Building a bridge. In Firestone

W. A.

Riehl

(Eds.), A new agenda for research on educational leadership (pp. 28-45). New York, NY: Teachers College Press.

31.

Stevens

R. J.

(2004). Why do educational innovations come and go? What do we know? What can we do? Teaching and Teacher Education, 20, 389-396.

32.

St. John

Fuller

K. A.

Houghton

Tambe

Evans

(2005). Challenging the gridlock: A study of high schools using research-based curricula to improve mathematics. Inverness, CA: Inverness Research.

33.

Strauss

Corbin

(1990). Basics of qualitative research: Grounded theory procedures and techniques. Newbury Park, CA: Sage.

34.

Thompson

C. L.

Zeuli

J. S.

(1999). The frame and the tapestry: Standards-based reform and professional development. In Darling-Hammond

Sykes

(Eds.), Teaching as the learning profession: Handbook of policy and practice (pp. 341-375). San Francisco, CA: Jossey-Bass.

35.

Tickle

B. R.

Chang

Kim

(2011). Administrative support and its mediating effect on US public school teachers. Teaching and Teacher Education, 27, 342-349.