With mathematics teachers attempting to implement Common Core State Standards in Mathematics and guided by the National Council of Teachers in Mathematics’ vision for comprehensive mathematics reform, this is an especially interesting time to understand how teachers think about their own practice in mathematics education is more generally, it is necessary to understand how teachers learn and think about their practice over time. Examination of teachers’ visions allow educational researchers insight into how teachers envision their ideal practice. Researchers explore how teachers’ visions align or do not align with the visions of mathematics education reformers, policymakers, district leaders, and teacher educators. Vision serves as a way to look back at how mathematics instruction has been and forward toward what it could be. Vision is helpful, because it frees teachers from the constraints of their classroom.
Introduction
Three facets of mathematics education guide my inquiry: vision, practice, and reform. These dimensions come together to guide my study: How might instructional vision allow teachers to enact change in their practice in pursuit of mathematics education reform? Through an iterative process of thinking about the aforementioned dimensions, consulting the literature, and reworking my conceptualization of vision, practice, and reform, I arrived at the following questions.
1. What is instructional vision?
2. What does the research suggest about teachers’ visions of mathematics instruction?
3. What does the research on instructional visions suggest about teacher learning?
4. What are the connections of vision and practice?
5. What does this suggest for mathematics education reform?
I examined core texts to conceptualize vision, its nature, and its role in education reform before conceptualizing visions of mathematics instruction. In selecting articles, I prioritized those that addressed visions of mathematical instruction, teaching practice, and mathematics education reform. The majority of articles selected address visions of mathematical instruction directly. Few articles serve to conceptualize vision broadly or offer helpful insight to the connections of teacher learning and teaching practice. I afforded special attention to studies that examined visions of ambitious instruction (in mathematics or otherwise), enactment of vision, and preservice and novice teachers’ experiences.
The studies that follow center a variety of stakeholders’ instructional visions including teacher education programs, mathematics education reformers, policies, teacher colleagues, school districts, and teachers. Each of the articles consider the teachers’ instructional vision and how it influences and is influenced by other stakeholders, context, and practice.
After a brief conceptualization of vision, three parts organize the paper: visions of instruction, teacher growth, and influencing vision and practice. A unifying vision emerges: The literature on visions of mathematics instruction is in pursuit of a (particular) vision of mathematics instruction for conceptual understanding. This theme is addressed throughout and explicitly in the conclusion.
Conceptualizing Vision
Vision is powerful, concrete, and practical. Hammerness (2001) refers to vision as “a set of images of ideal classroom practice for which teachers strive” (p. 143). Vision looks both backward and forward: “Educators use vision as not only a guide for the future and a motivating image of the possible, but also a means of looking back and reflecting upon past work and purposes” (Hammerness, 2010, p. 1041). Through interview-based analyses, Hammerness (2001) examined variance among three dimensions of vision and formed constellations. These constellations serve as a bases for understanding teachers’ instructional vision.
Focus, range, and distance operate as dimensions to understand the development of teachers’ work and careers. Focus refers to the center of interest of the vision and the clarity of the interest. This dimension captures the aspects of practice that receive the most attention and the clarity of the focus. Range refers to scope, capturing the breadth of the focus. Distance refers to how close or far a teacher’s vision is from their practice. These three dimensions work within a context of support, and vision cannot be isolated from the circumstances in which the teacher works (Hammerness, 2001). Differences in these dimensions yield four profiles, constellations, of vision.
“Close-Clear Constellation” visions are “clearly and narrowly focused, fairly close to practice and in a supportive or indifferent context.” (Hammerness, 2001, p. 147). In interviews, teachers in this vision could explain how they envisioned themselves as teacher and their students while providing vivid examples. Teachers mentioned their focus frequently. Teachers in this constellation felt optimistic about and relatively close to attaining their visions. Vision played a significant, positive, motivating role.
“Close-Cloudy Constellation” visions have “a fuzzy and narrow focus, are quite close to practice, and are in supportive contexts” (Hammerness, 2001, p. 150). Teachers in this constellation did not clearly articulate their visions and did not provide examples or elaborations to their vision. Visions in this constellation lacked a central attention and played a minimal role with teachers describing vision as working in their “unconscious.” Teachers in this vision reported that their practice was quite close to their vision.
“Distant-Clear Constellation” visions have “a clear and narrow focus are quite distant from practice and are in a context that is at best indifferent, or at worst, inimical” (Hammerness, 2001, p. 153). Teachers in this group felt the gap between their practice and vision was vast. They were discouraged by their distant vision. These teachers said vision played a minimal role but talked extensively about their vision. These teachers talked about the unsupportive nature of their school contexts.
“Far-Clear Constellation” visions have “a clear and broad focus, are far from practice and are in a supportive context” (Hammerness, 2001, p. 156). In this constellation, vision acted as a significant measure and guide, but teachers felt vision was quite far from practice. Teachers in this constellation were able to test and refine their practice against their vision and plan future paths toward their vision. All the teachers in this constellation selected their school context so they may better attain their visions. “Vision invited them to reflect and analyze past curriculum, shaped plans for the future, and encouraged them to continue to refine and revise their work” (p. 158).
Visions of Instruction
Parsons, Vaughn, Malloy, & Piercynski (2017) build on Duffy’s (2002) conceptualization of vision as a personal stance that evolves, is shaped by experience and context, and informs decision making. In examining teachers' visions for teaching, they attended to the dimensions of vision (Hammerness, 2001) and identified five foci: successful learners, motivated learners, lifelong learners, classroom environment, and nonacademic skills. Teachers visions are multifaceted, attending to multiple goals for students.
Teachers hold two types of visions, characterized by range (Hammerness, 2001): an immediate vision and a long-term vision. Immediate visions included a focus on a student outcome for the end of an academic year, such as reading with fluency and writing sentences. Long term visions for students focused on a vision of students as adults, such as having a job and improving their circumstances (Parsons et al., 2017).
Teachers had "core visions" that were stable across the seven-year study, but their visions were not singular. That is, teachers expressed visions of both lifelong learners and successful learners, but over time, the teachers' visions maintained a consistent core. Visions were complex, addressing short and long-range goals for students, and multidimensional, attending to a variety of goals for students (Parsons et al., 2017).
Jansen, Gallivan, and Miller (2018) examined the impact of a mathematics teacher education program by studying the visions of teachers two to three years after graduation. Unlike Hammerness (2001), visions aligned with (or did not align with) a particular vision promoted by the teacher education program.
Participants self-reported vision, whether and how the vision had changed, and their awareness of the vision promoted by the teacher education program. Seven elements of mathematics instruction were mentioned by 25% or more participants (Jansen et al., 2018).
1. Promote conceptual understanding
2. Productive struggle
3. Multiple strategies and representations
4. Small group work
5. Differentiated instruction
6. Teacher as direct instructor
7. Teacher as facilitator
Five of the seven elements aligned with faculty members’ intention to promote conceptual understanding, opportunity for productive struggle, multiple strategies and representations, group work, and teacher as facilitator. The other two elements, differentiated instruction and direct instruction, did not align with the faculty’s reported intended vision (Jansen et al., 2018).
Much like Hammerness (2001), Jansen et al. (2018) identified profiles of vision of mathematical instruction. Unlike Hammerness, Jansen et al’s profiles are built from elements of mathematics instruction and compared with elements of ambitious mathematics instruction promoted by the teacher education program:
Profile 1: “Teach concepts through multiple representations or strategies and real-world problem contexts” (Jansen et al., 2018, p. 13).
Profile 2: “Teach concepts through promoting productive struggle and facilitating students’ mathematical thinking through discourse” (Jansen et al., 2018, p. 13).
Profile 3: “Differentiate through direct instruction and collaborative work” (Jansen et al., 2018, p. 13).
Most graduates reported visions that aligned with the experiences in the teacher education program. Two of the three profiles aligned with the vision of the teacher education program but illustrated different installations of alignment. Different visions of mathematical instruction align with teacher education; there is more than one way to develop a vision of mathematical instruction that aligns with teaching for conceptual understanding (Jansen et al., 2018).
Jansen et al. (2018) and Parsons et al. (2017) both demonstrate an evolution of vision. Vision changes over time to become what it is now and is influenced by teacher education programs and practice (Jansen et al., 2018). Visions are complex, with multiple facets that come into and out of focus. Over time, a central core remains consistent (Parsons et al., 2017).
Vision also acts a means of resistance. Hara and Sherbine (2018) explored how student teachers make sense of neoliberal discourses, experience the process of teacher visioning in neoliberal discourses, and the effect of visioning on the response to the discourses. Visioning is a generative process of articulating and reflecting on beliefs. Unlike Hammerness (2001) and the authors that follow her, Hara and Sherbine do not distinguish visioning from beliefs, with the former acting as the verb for the latter (i.e. engage in visioning about beliefs).
Visioning serves to engage and reflect on articulated beliefs but does not allow teachers to actively disrupt neoliberal pressures. Teacher visioning can serve as an impetus for resistance to neoliberal pressures; teachers draw on the process of visioning to anticipate and respond to neoliberal pressures. Teacher visions evolve. Student teachers did not initially articulate resistance in initial visions, but over the course of the semester, their beliefs became more thoroughly developed. Visioning allows teachers to conceptualize their role in reproducing and resisting hegemonic discourses (Hara & Sherbine, 2018).
Hammerness (2001) introduces the idea of enacting vision into practice and the use of vision as a measurement for how close a teacher is to his or her ideal practices. Teachers attempt to enact their visions through their teaching and planning practices, creating positive classroom environments, and with classroom management (Hara & Sherbine, 2018). Teachers experience affordances, such as a supportive context, and report obstacles including institutional policy and diverse learners. Despite these obstacles, teachers stayed true to their visions, negotiating challenges and adapting instruction (Hara & Sherbine, 2018). Teachers (sometimes) actively pushed back against policy and more frequently quietly subverted policy to enact their vision (Parsons et al., 2017).
Hara and Sherbine (2018) report on the constraints that contribute to a gap between beliefs and practice. Obligation to established practice and the expectations of curriculum accounted, in part, for a gap in beliefs and practice. Connected of student teaching, Participants perceived surveillance of their practice as an obstacle.
[Hammerness on the dark side of vision.]
Teacher Growth
The Jansen et al. (2018) project took place within a larger study investigating the effects of teacher education program with respect to outcomes including mathematical and pedagogical content knowledge and teaching practice. The teacher education program faculty sought to enact a shared instructional vision of teaching mathematics for conceptual understanding. Coursework in the program emphasized the shared vision and addressed how teachers may provide learning opportunities aligned with the vision.
Munter and Correnti (2017) and Horn (2010) examined vision as related to teacher growth and teacher learning, respectively. Munter and Correnti show teacher vision predicted teacher growth in practice. Horn specifies how conversation can support teacher learning through re-visioning. Like Jansen et al. (2018) and unlike Hammerness (2001), Munter and Correnti were particularly interested in understanding the development of a particular mathematics instructional practice. They studied teachers within districts that were implementing comprehensive mathematics instruction reform.
The opportunity to make sense of and solve mathematical problems while engaging in discussion was central to the districts’ reform efforts. Districts supported teachers with both conceptual and practical tools to promote mathematics instruction for conceptual understanding. These tools included principles for supporting students’ learning, tools for reflecting on planning, vision statements, observation protocols, curriculum materials, diagnostic assessments, and Accountable Talk prompts to help students in facilitating discussion (C Munter & Correnti, 2017).
Munter and Correnti (2017) collected data on quality of instruction, mathematical knowledge for teaching, and instructional vision. They examined the relation of instructional vision and mathematical knowledge for teaching to the quality and development of mathematics instruction. Teachers’ mathematical knowledge for teaching at the beginning of the study was predictive of instructional practice in the first two years of the study but not beyond. There was no association between initial mathematical knowledge of teaching and growth. Teachers’ initial visions of high quality mathematics instruction were predictive of their growth in instructional quality.
Horn (2010) examined interactions among colleagues. Like Munter and Correnti, this study examined vision within teacher development and teacher learning. Unlike Munter and Correnti, Horn examined informal teacher learning through collegial conversations. Two forms of discourse were important for representing and learning about practice: replays and rehearsals.
Teaching replays provide “blow-by-blow” accounts of classroom events. Teachers narrate or act out their part as teacher. Teaching rehearsals allow for teachers to narrate or act out interaction in an imagined or anticipatory manner. Replays and rehearsals allowed for a re-visioning routine in which teachers reconsidered or revised their understanding of teaching practice (Horn, 2010).
As part of revisioning, replays and rehearsals “organize interaction, create alignments, and position teachers in relationship to each other and the students they teach” (Horn, 2010, p. 231). Accounts of practice changed over the course of interaction. First, a teacher would render an event, usually via replay. Eventually, a colleague prompted the teacher to elaborate. Sometimes the process of elaboration led to a re-vision of the initial account.
Replays and rehearsals serve four purposes. (a) They provide support. (b) They provide evidence-based consultation. Teachers provided specific images of practice, bringing the classroom into the teacher community. (c) They provided a means of developing knowledge for teaching, linking separate classroom events with a unifying principle. (d) They served to create alliance among teachers. Horn’s (2010) study specifies how particular aspects of collaboration account for teacher learning and suggests that conversations that employ the outlined components support adaptive expertise in teaching.
Like Munter and Correnti (2017), Allen, Web, and Matthews (2016) recognize vision as one component of teacher practice. Where Munter and Correnti consider mathematical knowledge for teaching and vision, Allen et al. consider Science, Technology, Engineering, and Mathematics (STEM) pedagogical content knowledge, a constructivist paradigm for teaching and learning, and instructional vision. Following pedagogical content knowledge (Shulman, 1986), mathematical knowledge for teaching argues mathematics teachers need specialized knowledge of content (Hill et al., 2004).
Adaptive teaching in STEM is understood as “a process that teachers initiate when the recognize and gauge their students’ STEM-related conceptual development, inquiry processes, and real-world connections and then maneuver their instruction to further develop these features of students’ learning” (Allen et al., 2016, p. 217). Teachers with well-developed STEM pedagogical content knowledge, a constructivist paradigm, and the ability to draw on a vision are better positioned to engage in adaptive teaching. STEM pedagogical content knowledge includes a teacher’s knowledge of how students think about STEM topics, strategies for engaging students in inquiry, and real-world STEM connections. Teacher who frame teaching and learning within a constructivist paradigm believe learning to be the result of meaning making from experience, personalization means connecting with students’ experiences, and the processes of pursuing knowledge and understanding is more important than knowing facts (Allen et al., 2016).
While teachers are taught methods of STEM instruction, the models and paradigms waver as novice teachers transition from theory to practice. Envisioning the teacher one hopes to become and reflecting on and in practice assists teachers in becoming adaptive. A vision of inquiry-based instruction allows for and requires adaptation in its pursuit. Drawing on a vision of practice allows the teacher to navigate challenges that arise in practice (Allen et al., 2016).
Influencing Vision and Practice
Taken together, Jansen et al (2018), Munter and Correnti (2017), and Horn (2010) illustrate an influence on vision and practice. Munter and Correnti demonstrate how vision and instructional practice change within districts implementing comprehensive reform efforts. In examination of vision statements to develop rubrics for assessing visions of high quality mathematics instruction (Charles Munter, 2014), teachers’ visions of high quality mathematics instruction became more sophisticated over the four year study. Not only does this demonstrate a change in vison, but it also demonstrates increased sophistication over time within districts implementing comprehensive reform efforts. Both projects show change toward a particular vision.
Horn (2010) illustrates how informal learning occurs through collegial conversations. Teachers engage in replays and rehearsals. Through explanation of a classroom moment, lesson, or experience, colleagues make conversational moves that demonstrate learning. A colleague requesting an elaboration or clarification, prompts the teacher to reinterpret the event he or she initially recounted. Sometimes, this results in a reinterpretation or re-visioning process of a past or future event. In this case, colleagues influence and prompt a re-visioning of classroom events.
van Es, Cashen, Barnhar, and Auger (2017) investigate the development of preservice teachers’ noticing of ambitious mathematics pedagogy. Teacher noticing, the ability to attend to and reason about teaching and learning, captures teachers’ moment-by-moment decision making, is essential to adaptive teaching. Adaptive teaching, and thus noticing, is central to student centered, responsive approaches to learning.
Like Horn (2010), vision serves as one piece of teacher learning. Noticing includes three practices. First, attending to features of instruction captures the preservice teacher’s ability to decide what to focus on. Second, elaborating on observations captures the candidates’ descriptions of observations. Third, interrogating observations to reason abut instruction captures the connections between phenomena and instruction. Within the third practice, candidates draw on a range of frameworks to analyze instruction. This is called “blending visions.” Some candidates are guided by a vision of mathematics focused on student ideas. van Es et al. (2017) refer to this as “using a vision of ambitious teaching to systematically analyzing instruction” (p. 176).
Like Jansen et al. (2018), van Es et al. (2017) investigate the development of a particular vision within a teacher education program. van Es et al. investigate teaches’ noticing of ambitious mathematics instruction and whether teachers develop ways of noticing features of ambitious mathematics pedagogy. Over the course of a semester, candidates noticing developed to attend to the features of instruction promoted in the course, provide more detail in elaboration, and utilizing the vision promoted in the course.
Jansen et al. (2018) studied a teacher education program’s effectiveness of promoting a shared instructional vision. The study acted as an assessment to content and pedagogy coursework. Most graduates reported a vision of mathematical instruction for understanding, the intended vision of the teacher education program. Visions for enacting mathematics instruction for understanding differed. Instructional vision has residue from teacher education to or three years past graduation.
Perrin (2006) does not examine influence on vision or practice directly. Instead, Perrin examined the extent to which mathematics teachers were aware of and agreed with NCTM’s vision of mathematics instruction. NCTM (2009) calls for mathematics education reform and acknowledges that to achieve their vision teachers and students must make changes so that teachers become guides of experience. Secondary certified mathematics teachers had greater awareness and stronger beliefs in NCTM’s vision than elementary certified mathematics instructors (Perrin, 2006).
Visions of Mathematics Instruction for Conceptual Understanding
Instructional vision provides an understanding to teachers’ development and practice. Visions are complex and multifaceted, evolving over time. They influence and are influenced by teacher education and experiences in the classroom. Schools, districts, and colleagues influence instructional vision through promoting instructional practices, re-visioning classroom experiences, and with reform efforts. Conversely, vision influences instructional practice, context, and learning. The literature on instructional vision and teachers’ visions of mathematics instruction are summarized in Table 1.
Visions of mathematical instruction focus on teacher learning, enactment of vision, and vision within reform efforts. The literature on teacher learning, practice, and reform are summarized in Table 2. Instructional vision is one facet of teacher practice that is independent and interwoven with teachers’ pedagogical content knowledge and learning paradigm.
Hammerness (2001) examines instructional vision independent from a particular vision. Throughout the visions of mathematics instruction, a unifying theme emerges: the pursuit of a (particular) vision of mathematics instruction for conceptual understanding (Allen et al., 2016; Jansen et al., 2018; C Munter & Correnti, 2017; Perrin, 2006; van Es et al., 2017). Furthermore, mathematics instruction for understanding requires adaptive teaching (Allen et al., 2016; Parsons et al., 2017). Vision, along with pedagogical content knowledge (Allen et al., 2016; Jansen et al., 2018; C Munter & Correnti, 2017) and a constructivist paradigm (Allen et al., 2016) is a necessary component to position teachers to be adaptive.
Conclusion
Vision, practice, and reform guided my inquiry: How might instructional vision allow teachers to enact change in pursuit of mathematics education reform? Specifically, I explored the following questions. 1) What is instructional vision? 2) What does the research suggest about teachers’ visions of mathematics instruction? 3) What does the research on instructional visions suggest about teacher learning? 4) What are the connections of vision and practice? 5) What does this suggest for mathematics education reform.
The research presented represent a variety of stakeholders’ instructional visions including faculty members of teacher education programs, mathematics education reformers, policies, teacher colleagues, school districts, and teachers. Each study gave some consideration to the teachers’ instructional vision, including how it influenced and was influenced by context and practice. After a conceptualization of vision, three parts revealed a unifying vision. The literature on visions of mathematics instruction is in pursuit of a (particular) vision of mathematics instruction for conceptual understanding.