Abstract
We designed a video-based course to develop preservice teachers’ vision of ambitious instruction by decomposing instruction to learn to attend to student thinking and to examine how particular teaching moves influence student learning. In this study, we examine the influence that learning to systematically analyze ambitious pedagogy in the course has on preservice teachers’ classroom practice. Analysis of preservice teachers’ videos from the Performance Assessment for California Teachers Teaching Event reveals that they engaged in more student-centered practices compared with a cohort of candidates who did not participate in the course—creating opportunities to see student thinking, noticing student thinking during instruction, and pursuing student ideas to learn more about their thinking. We also found that their probing of student thinking focused primarily on correct answers and procedural fluency. These findings have implications for defining a pedagogy of teacher preparation to develop beginning teacher competency.
Keywords
The vision of instruction proposed by policy makers and the education research community advocates for teachers to adopt a student-centered approach to instruction that deviates from the cultural norms of U.S. schooling (Ball & Cohen, 1999; Common Core Standards Initiative [CCSI], 2011; Levin, Hammer, & Coffey, 2009; National Council of Teachers of Mathematics [NCTM], 2000; National Research Council [NRC], 2001). This vision of teaching, referred to recently as ambitious pedagogy (Jackson & Cobb, 2010; Lampert, Beasley, Ghousseini, Kazemi, & Franke, 2010; Thompson, Windschitl, & Braaten, 2013), calls for teachers to design high-quality, cognitively demanding tasks and to facilitate meaningful discussions where teachers elicit student ideas, attend to and reason about student thinking as it unfolds during a lesson, and then use what they learn to make informed instructional decisions (Ball & Cohen, 1999; CCSI, 2011; Lampert et al., 2010; Levin & Richards, 2011; NCTM, 2000; NRC, 2001; Stein, Engle, Smith, & Hughes, 2008). When teachers adopt such responsive teaching practices, both teacher and student learning result (Franke, Carpenter, Levi, & Fennema, 2001; Jacobs, Franke, Carpenter, Levi, & Battey, 2007). The question becomes how to prepare preservice teachers to learn to enact these practices and to make them part of one’s beginning instructional repertoire.
Recent research advocates for teacher education to place an increased focus on preparing beginning teachers to learn to enact high-leverage practices (Grossman, Hammerness, & McDonald, 2009; Lampert et al., 2010). The idea is that teaching involves a set of routine practices that occur with high frequency in a lesson and that beginning teachers can develop competency in enacting them through approximations of practice (Grossman & McDonald, 2008). Another approach focuses on preparing beginning teachers to learn in and from practice through systematic analysis of teaching (Hiebert & Morris, 2012; Hiebert, Morris, Berk, & Jansen, 2007; Santagata & Guarino, 2011). This approach involves preparing beginning teachers to design and enact instruction that provides them opportunities to gain insight into student thinking and to use what they notice as evidence to determine the extent to which the lesson’s learning goals have been met.
We locate our work at the intersection of these two approaches. In particular, we view learning to analyze teaching in systematic ways as a core practice of teaching that involves learning to decompose instructional practice, to attend to particular events and interactions that are consequential for student learning, and to interpret the meaning behind those events to make informed teaching decisions. Just as preservice teachers need to learn how to launch a task and orchestrate a discussion, for example, they also need to learn how to look at and make sense of teaching—what has recently been referred to as teacher noticing (Sherin, Jacobs, & Philipp, 2011). At the same time, we view a central component of ambitious pedagogy as noticing in practice. By noticing in practice, we mean creating opportunities for student thinking to emerge during classroom activity so that teachers can attend to and reason about student ideas and use what they learn to inform their teaching, similar to what some researchers call responsive teaching (Levin, Hammer, Elby, & Coffey, 2013). We propose that learning to attend to and make sense of student thinking as captured in classroom artifacts, and particularly in video, as well as learning to decompose the work of teaching in systematic ways can help preservice teachers learn practices for eliciting and responding to student thinking during instruction.
A good deal of research suggests that when preservice teachers are provided opportunities to learn to systematically analyze teaching in the context of a course, they can improve their ability to attend to noteworthy features of instruction and analyze the relationship between teaching and student learning (Santagata & Guarino, 2011; Star & Strickland, 2008; van Es & Sherin, 2002). However, less of this research explores the relationship between learning to observe and analyze teaching and preservice teachers’ own practice. In prior research, we found that preservice teachers developed more sophisticated ways of analyzing instruction in the context of a video-based course, as well as several months after the conclusion of the course (Barnhart & van Es, 2015; van Es, Cashen, & Auger, 2011). In this study, we explore whether learning to analyze teaching in the context of this course influenced secondary preservice teachers’ classroom instruction. More specifically, we seek to answer two questions: (a) Do secondary preservice teachers who enrolled in a video-based course focused on learning to analyze teaching enact practices that are in the spirit of ambitious pedagogy? (b) If they do enact such practices, what about student thinking and learning do they notice and take up as objects of discussion? By observing preservice teachers’ practice, along with drawing inferences about their noticing through the discourse that unfolds during instruction, we can gain insight into ways that preservice teachers seek to enact practices that are responsive to student ideas.
To situate this work, we begin with a brief overview of the vision of ambitious instruction that frames our study. We then review recent research related to developing pedagogies of teacher preparation to prepare teachers for enacting ambitious practices and the use of video for supporting preservice teachers in moving toward this goal. We then describe core features of the Learning to Learn from Teaching (LLfT) course, our data analysis procedures, followed by the results.
Developing Preservice Teacher Competency for Ambitious Instruction
The backdrop of our work is a vision of instruction that places student thinking and learning at the center of classroom activity. In mathematics education, for example, research and policy documents advocate for teachers to shift their instruction from a primary focus on computational accuracy to mathematical reasoning to develop students’ procedural fluency, conceptual understanding, and productive dispositions as mathematics knowers and doers (Kazemi & Stipek, 2001; Kilpatrick, Martin, & Schifter, 2003; NCTM, 2000; NRC, 2001). In such classrooms, students represent their thinking, develop solutions to complex problems, debate the merits of different solution strategies, and construct arguments and critique one another’s reasoning. Similarly, proposals to improve science education emphasize teaching students to collect, interpret, and evaluate evidence to construct scientific explanations (NRC, 2007, 2012; Windschitl, Thompson, & Braaten, 2008). Students need to learn to assess the quality of data, draw inferences from data, and question, reason about, generate, and test scientific explanations and models (American Association for the Advancement of Science, 2009, NRC, 2012). Across all disciplines, there is a focus on students learning to analyze sources of information, draw inferences from that information, and construct evidence-based arguments that are rooted in core ideas of the discipline (CCSI, 2011; Reisman & Wineburg, 2012). Classrooms that seek to make progress toward this vision become discourse-rich spaces, and teachers need to attend closely to student ideas—noting similarities and differences in student thinking, monitoring more and less complex reasoning, and assessing students’ disciplinary knowledge as it progresses in a lesson (Ball, Lubienski, & Mewborn, 2001; Levin et al., 2009).
This vision of instruction is not particularly new, yet it is well documented that it is quite challenging for preservice teachers to adapt their instruction based on student ideas. Studies of expert and novice teachers find that novice teachers are less responsive to student input during instruction, even when students raise interesting points for discussion (Berliner, 2001; Westerman, 1991). Moreover, beginning teachers typically focus on themselves and the extent to which they complete the lesson as planned so they are not attuned to students and their experience in the lesson (Kagan & Tippins, 1991). In addition, preservice teachers have few opportunities to observe models of teachers adapting their instruction based on student ideas. Such an approach is not typical of U.S. teachers so they have limited opportunities to see these practices in action early in their career (Stigler & Hiebert, 1999).
Over a decade ago, Feiman-Nemser (2001) proposed that three central components of teacher preparation include developing an understanding of learners and learning, a beginning repertoire of teaching, and tools to study practice. Recent research on improving teacher education have made progress on these dimensions—one focused on identifying a core set of high-leverage teaching practices that can equip novice teachers for meeting the demands of ambitious instruction and another directed at preparing teachers to systematically analyze teaching to learn in and from one’s practice. We review these two approaches and consider how learning to decompose the work of teaching—to notice features of instruction and student thinking in particular—can complement these efforts and support preservice teachers in developing a disposition to use their instructional contexts as a site to learn about students and their thinking and to use what they learn to inform instructional decisions.
Recent efforts to improve teacher education focus on preparing teachers to enact high-leverage practices that will provide them insight into student thinking, as well as equipping them with strategies for learning from practice (Grossman & McDonald, 2008; Lampert et al., 2010; McDonald, Kazemi, & Kavanagh, 2013). Through structured activities that approximate the work of teaching, preservice teachers can learn to enact a core set of practices that will place them on a trajectory for achieving the vision of ambitious pedagogy (Ball, Sleep, Boerst, & Bass, 2009; Kazemi, Franke, & Lampert, 2009; Lampert et al., 2010; Thompson et al., 2013). Grossman et al. (2009) argue that much of teacher preparation focuses on observing and reflecting on teaching, with less emphasis on preparing teachers to do the work of teaching. They propose that pedagogies of enactment be added to current teacher preparation designs to help novices develop skilled practice early in their careers.
Other researchers propose an alternative approach—one centered on making evidence-based claims about the causal relationship between teaching and learning (Hiebert & Morris, 2012; Santagata, Zannoni, & Stigler, 2007). From this perspective, the work of teacher preparation involves learning to identify clear learning goals, to collect evidence of students making progress toward the goal, and to examine how particular teaching moves influence student learning and use this analysis to inform next steps. The idea is that through careful, structured analysis of practice connected to the core work of teaching, preservice teachers will develop a framework for learning from practice—collecting data from instruction to evaluate the extent to which students have made progress toward the learning goals on an ongoing basis.
Central to both approaches is learning to look at and make sense of instructional interactions, what is referred to as teacher noticing (Sherin et al., 2011). Teacher noticing captures what teachers attend to, how they reason about what they see, and how they use what they learn to inform their instructional decisions (Jacobs, Lamb, & Philipp, 2010; Star & Strickland, 2008; van Es & Sherin, 2008). To prepare preservice teachers to enact high-leverage practices involves learning to both see and enact these practices during instruction, as well as learning to attend to students’ thinking about content in the moment of teaching and how to navigate their ideas during instruction. Similarly, to engage in systematic analysis of teaching involves developing ways of capturing and interpreting student thinking to gauge one’s instructional effectiveness. In this way, learning to notice instructional interactions guided by a framework of teaching becomes a central goal for teacher preparation.
Attending to the Substance of Student Ideas
Classrooms are complex settings and teachers typically notice a wide range of features of classroom interactions, such as the classroom environment, content of the lesson, student thinking and engagement, and classroom management and discourse (Frederiksen, Sipusic, Sherin, & Wolfe, 1998; Luna, 2014; Sherin et al., 2011; Star & Strickland, 2008; van Es & Sherin, 2008). Research on teachers’ noticing of student thinking finds that teachers typically attend to correct and incorrect answers, the accuracy of students’ ideas, and errors and misconceptions (Santagata, 2004; Schleppenbach, Flevares, Sims, & Perry, 2007; Schleppenbach, Perry, Miller, Sims, & Fang, 2007). Moreover, research finds that it can be challenging to hear students’ ideas because teachers interpret student ideas from their own experiences, understanding, and curricular goals (Brodie, 2010; Chamberlin, 2005). Recent research advocates for teachers to explicitly focus on the disciplinary substance of student thinking (Coffey, Hammer, Levin, & Grant, 2011; Levin & Richards, 2011; Walkoe, 2014). The idea is that a more careful focus on the particulars of student thinking as they are tied to the content of instruction can help preservice teachers develop a student-centered frame to inform their instructional decision-making (Levin et al., 2009). We propose that learning to attend to the substance of student thinking can afford preservice teachers opportunities to develop practices for inquiring into student ideas—practices for seeing noteworthy ideas in student work or student talk, for probing student thinking, and for considering alternative explanations of student ideas.
Affordances of Video for Learning to Notice Classroom Interactions
We draw on research that shows the power of video to help preservice teachers learn to look at and develop a vision of ambitious instruction (Borko, Jacobs, Eiteljorg, & Pittman, 2008; T. Hatch & Grossman, 2009; Santagata & Guarino, 2011; van Es & Sherin, 2002). Video can capture the complexity of teaching and can be paused and reviewed multiple times, allowing beginning teachers to slow down the work of teaching and study classroom interactions as they unfold during instruction (Brophy, 2004). It can also reflect the vision of teaching that reform documents advocate, providing representations of “classroom discourse” and “student reasoning” as they come alive during instruction. In addition, video can capture classroom interactions at different grain sizes—brief interactions that occur in a single classroom lesson allowing the viewer to hone in on student thinking of content or particular pedagogical practices, as well as whole lessons that take place over the course of several days, weeks, or months to represent the work of teaching as it unfolds over time (T. Hatch & Grossman, 2009; Lampert & Ball, 1998).
An extensive body of literature suggests that viewing video records of practice, along with structured protocols, can lead preservice teachers to more productively analyze and reflect on teaching and learning (Santagata & Guarino, 2011; Star, Lynch, & Perova, 2011; Star & Strickland, 2008; Stockero, 2008; van Es & Sherin, 2002)—focusing on student learning of content, attending to a range of student ideas and contemplating a variety of interpretations of their understanding, and making connections between particular teaching moves and student learning. What is less understood, however, is the relationship between developing such observation and analysis skills in the context of a video-based course and preservice teachers’ instructional practice. That is, while observing and analyzing teaching with video can help preservice teachers develop a more precise framework for observing teaching and become more reflective about the relationship between their teaching, student learning, and content, less is known about the influence that developing these capacities through video analysis has on the kinds of learning opportunities preservice teachers’ create in their own classrooms.
There is some evidence that suggests that using video to study teaching and learning can influence teachers’ classroom practice. For instance, van Es and Sherin (2010) found that elementary mathematics teachers who participated in a video club shifted their instruction over time—making space for student thinking to emerge, probing student thinking, and learning about students’ ideas during instruction. Similarly, Kiemer, Gröschner, Pehmer, and Seidel (2014) found that secondary mathematics and science teachers who participated in a year-long video-based professional development program facilitated more productive classroom discourse by engaging students more prominently in classroom discourse and scaffolding student learning in these contexts with more concrete, learning-oriented feedback. This study seeks to contribute to this limited knowledge base and examines the influence that learning to systematically analyze classroom interactions may have on preservice teachers’ instructional practice, with a particular focus on whether and how they enact student-centered practices and if so, what about student thinking they focus on in these interactions.
We designed a video-based course, LLfT, to attune secondary teachers to features of ambitious instruction. In this study, we focus our inquiry on a cohort of mathematics preservice teachers’ instruction for several reasons. Research shows that students’ mathematics achievement and motivation declines in the secondary school years (Köller, Baumert, & Schnabel, 2001). Designing instruction to cultivate mathematics teachers adopting a more relational, responsive approach to instruction may play an important role in helping secondary students have more positive experiences in mathematics in upper grades. In addition, greater numbers of students are enrolling in Algebra because it is a gateway for advanced mathematics learning, but fewer of them experience success (Stein, Kaufman, Sherman, & Hillen, 2011). Thus, learning to see teaching as an interactional activity that is largely informed by student thinking may be a crucial piece for improving student learning and persistence at this critical juncture. Finally, policy documents for improving mathematics and science education promote classroom environments focused on student thinking (CCSI, 2011; NCTM, 2000; NRC, 2001); yet, secondary mathematics teachers do not typically design instructional interactions focused on student ideas (see Chamberlin, 2005; Stockero, 2008; Van Zoest, 1995). For these reasons, we are specifically interested in understanding how secondary mathematics preservice teachers learn to enact a more student-centered, responsive approach to instruction.
Research Design
Study Context
This study took place in a 9-month single-subject teaching credential program at a large western university. During the 2008-2009 academic year, faculty created the LLfT project to develop elementary and secondary preservice teachers’ ability to learn to systematically analyze teaching practice through video analysis (Santagata & Guarino, 2011; Santagata & van Es, 2010). The course for secondary preservice teachers was heavily influenced by research on teacher noticing, lesson analysis, and teacher reflection (Hiebert et al., 2007; Rodgers, 2002; Santagata et al., 2007; van Es & Sherin, 2002). The LLfT course took place during the first quarter of a three quarter credential program. All students were also assigned field placements in which they observed a teacher for 1 to 2 hr each week.
The LLfT course used video cases of teaching (e.g., Boaler & Humphreys, 2005; Seago, Mumme, & Branca, 2004) along with established frameworks to support preservice teachers in learning to notice features of ambitious pedagogy, with a particular focus on describing and interpreting student thinking and the influence of reform-oriented teaching moves on student learning. The course used the Framework for Analyzing Teaching (Hiebert et al., 2007) and the Reflective Cycle (Rodgers, 2002) as tools for structuring conversations for video analysis. To aid preservice teachers in the process of attending to student learning, the reflective cycle engages them in the process of learning to see, describe, and contemplate noteworthy events and interactions that occurred during teaching (Rodgers, 2002). Through this reflective cycle, preservice teachers focus on describing in detail noteworthy student thinking and constructing a range of interpretations of what they observed. The course also provided preservice teachers with frameworks for learning to attend to particular aspects of ambitious pedagogy in practice, including, the cognitive demand of the tasks (Stein, Smith, Henningsen, & Silver, 2009), the nature and substance of teacher questions (Boaler & Humphreys, 2005), the nature and quality of classroom discourse (Hufferd-Ackles, Fuson, & Sherin, 2004), the use of tools for creating and representing thinking, and the development of classroom norms that promote understanding (Carpenter & Lehrer, 1999).
Data Sources
Data come from two cohorts of secondary mathematics preservice teachers, one cohort of 26 candidates who enrolled in the LLfT course 1 and another cohort of 12 candidates from a previous year who did not take the course because it was not offered as part of the program of study at that time. Aside from the addition of the LLfT course, there were no other changes to the program between the two cohorts.
Data for the study consist of videos of teaching that preservice candidates submit as part of the Performance Assessment for California Teachers (PACT) Teaching Event (Pecheone & Chung, 2006). The PACT Teaching Event is a portfolio assessment that measures preservice teachers’ abilities to plan, enact, assess, and reflect on a lesson sequence focused on domain-specific goals; for mathematics, these include developing students’ procedural fluency, conceptual understanding, and mathematical reasoning. As part of Task 3 of the PACT Teaching Event, preservice teachers videotape a lesson and self-select either a continuous 15- to 20-min clip or two lesson segments edited together lasting no longer than 20 min that demonstrate their ability to teach mathematics targeted toward the three aspects of mathematics learning. Preservice teachers complete the PACT assessment about 3 months after completion of the LLfT course, during their student teaching assignment.
Data Analysis
Data analysis occurred in two phases related to the two research questions. The first phase included analysis of teaching practices in the video lessons. To begin, we selected a random sample of preservice teachers’ videos from the cohort of participants who enrolled in the LLfT course and the cohort who did not enroll in the course and analyzed them for evidence of reform-oriented mathematics practices, such as eliciting student thinking, exploring and probing student ideas, and comparing and contrasting student-generated strategies (Carpenter & Lehrer, 1999; CCSI, 2011; Franke et al., 2009; NCTM, 2000; NRC, 2001; Sherin, 2000). In our first pass of viewing the videos, we observed differences between the participants’ videos from each cohort. We then drew on prior research (van Es & Sherin, 2010) to systematically code these differences in 2-min time intervals (see Borko et al., 2008; Sherin & van Es, 2009; van Es & Sherin, 2010), identifying evidence of preservice teachers seeking to make student thinking visible, probing student thinking, and learning in the context of teaching. While we started with these codes, we wrote analytic memos to note alternative practices that emerged from our analysis (Miles, Huberman, & Saldaña, 2014). We used these memos to identify other teaching practices that preservice teachers appeared to enact, which were reflective of those advocated by mathematics education research. From the written analytic memos and preliminary coding results, the framework was then modified to consist of three categories: Making Space for Student Thinking, Attending to and Taking up Student Ideas, and Pursuing Student Thinking (see Table 1).
Teaching Practices for Making Thinking Visible.
We then returned to the entire data set and coded all 38 videos using the modified coding framework. To check the reliability of the framework, a sample of 20% of the videos was randomly selected and coded by a trained research assistant. Percent agreement for Making Space for Student Thinking, Attending to and Taking Up Student Ideas, and Pursuing Student Thinking were 83%, 91%, and 83%, respectively. The two coders discussed the disagreements in each category until consensus was reached. To compare the two cohorts quantitatively, frequency counts were tabulated for each practice. In addition, the occurrence and average rate of occurrence per 2-min segment for each practice was calculated. Because a number of segments did not contain instances of these practices, the average rate could be less than one. We conducted a t test to compare differences between the means on these three dimensions.
Our second question sought to understand what about student thinking preservice teachers attended to during instruction. While creating discourse-rich interactions focused on student thinking is certainly a worthwhile goal, this did not provide insight into the types of student thinking that preservice teachers elicited and pursued in their teaching. Thus, we explored the nature and substance of what they attended to related to student thinking and how they took these ideas up for discussion. We focused specifically on the videos of the preservice teachers who participated in the LLfT course because our findings revealed that they pursued student thinking to a greater extent. We focused our analysis on instances where they pursued student thinking because it is in these interactions that we were able to gain insight into the nature of student mathematical thinking to which preservice teachers attended. Of the 26 preservice candidates, 20 of them pursued student thinking in their lessons. Across this group, there were a total of 66 episodes of candidates pursuing student thinking. In our analysis, an episode consisted of an interaction where a preservice teacher pressed students for explanations regarding their answers, explanations, or responses. Using discourse analysis (J. A. Hatch, 2002), we used the nature of the teacher questioning and subsequent responses to infer what preservice teachers noticed in the student work or student thinking (Boaler & Humphreys, 2005; Hiebert & Wearne, 1993). For instance, if a preservice teacher asked, “How did you get this answer?” we inferred that the focus of discourse was on the student’s procedure or strategy for arriving at the answer. If the preservice candidate asked, “Can you explain to me why you drew the line that way?” we inferred that the focus was on the student’s mathematical reasoning. This approach is consistent with Franke and colleagues’ (2009) analysis of teacher questions to elicit and extend student mathematical thinking.
In our initial coding of the 66 episodes, we used an inductive coding approach to classify the purpose and content of the questions preservice teachers asked. We then wrote analytic memos as we examined the episodes related to what the preservice teachers noticed based on their reaction to the student(s) response to their questions. Because we sought to understand the kind of learning opportunity that preservice teachers created through these interactions, we used methods for modeling the complex practice of teaching (Leinhardt & Greeno, 1986; Leinhardt & Steele, 2005) to develop our own model for the process of attending and responding in these episodes. In particular, we created a working model that illustrates the process preservice teachers used to notice and pursue student thinking during instruction to draw inferences about the learning opportunities they created for students, particularly as they related to those being assessed in the PACT Teaching Event.
Results
We found that preservice teachers in the LLfT cohort enacted a variety of responsive teaching practices and that they did so at a greater frequency than the cohort of candidates who did not enroll in the course. Here, we describe the practices we observed among the preservice teachers in the LLfT course and illustrate them with examples from the data. We then present the differences between the two cohorts, followed by a more in-depth analysis of the substance of student thinking that preservice teachers noticed.
Preservice Teachers’ Enactment of Responsive Teaching Practices
Our analysis revealed three ways that the preservice teachers sought to enact responsive teaching practices—Making Space for Student Thinking, Attending to and Taking Up Student Ideas, and Pursuing Student Thinking (see Figure 1).
Average rate of occurrence in teaching practices between two cohorts.
Making space for student thinking
Preservice teachers who enrolled in the video-based course made space for student thinking to emerge during instruction in two different ways. One approach involved eliciting a range of student ideas and responses by inviting students to share novel ideas, solutions, and explanations. In the videos, we observed preservice teachers asking questions like, “Who thought about this a different way?” or “Does anyone have another explanation?” The second approach involved providing time for students to think. While wait time has proven to be an important practice for advancing student understanding (Cazden, 2001), we found other ways that the LLfT candidates provided time for thinking to emerge. One way included directly stating to the class that they wanted students to take time to think about a problem or an idea, making statements like, “Just think about it for a moment.” We also observed instances when a teacher asked a student a question and the student did not have a response immediately, and another student wanted to interject with an answer. The preservice teacher would stop the conversation and say, “Let’s give your classmate time to think.” This type of move signals to the class that it is okay to take time to think about an idea and reflects a preservice teacher’s effort to slow down instruction to promote thinking. There were also instances when preservice teachers structured time for students to work together, and they stated explicitly while framing the task that the students were to use the time to think about a situation, solution, or explanation. Finally, preservice teachers recognized and publicly associated ideas with particular students. For example, in one lesson, a preservice teacher saw that a student had her hand raised, and she said, “Julia has another idea.” Public recognition of the ideas gives the student ownership over the idea while also signaling to the class that they will take time to consider it further. These approaches all reflect a stance focused on eliciting thinking and explicitly providing time and space for students to think about the mathematics.
Attending to and taking up student ideas
Another practice we observed was noticing student thinking by attending to and taking up student ideas during instruction. This is different from the first category because this category involves highlighting ideas that are worthy for exploration and publicly interpreting them, either in an exchange with the student or for the whole class. In some videos, candidates were leading a whole class discussion and a student offered an unsolicited idea. Rather than proceeding with the lesson, the candidate paused and considered the idea, question, or response with the student. During one lesson, for instance, a preservice teacher concluded the warm-up portion of the lesson and as he started to shift to the next portion, a student asked a question about the problem they had just discussed. The teacher stopped what he was doing, turned to the student to listen to the question, and then asked several questions to make sense of the student’s question. Another way preservice teachers attended to and took up student ideas involved re-voicing or rephrasing student ideas. For example, in one lesson, a preservice teacher re-voiced a student’s response as a way to highlight the idea for the class:
Madison just told us for this particular number line: 3, 4, 5, 5, 6, 7, 10 there’s only one mode. But she said if there were like two 6s or two 10s, then there would also be another mode. Does that make sense?
One preservice teacher re-voiced a student’s response to verify whether or not her own interpretation was correct, “So, you’re saying it will go out like that? Is that what you’re saying or would it need to be crossed like that?” This approach invited students to clarify their ideas and allowed the preservice teacher to publicly interpret the student’s thinking.
Pursuing student thinking
The final category refers to moves preservice teachers made to further examine student ideas. We identified four different ways that preservice teachers pursued student thinking. One approach included asking students to explain their mathematical reasoning. This approach is focused on having students explain why their answer or their procedure made sense. For example, questions such as “So, why do you think this?” or “Why do you think it should be an open circle?” demonstrate the candidate actively pursuing student thinking related to procedural and mathematical reasoning. The second approach focused on preservice teachers asking students how they got their answer. This approach focused less on why a response made sense (reasoning) and more on what procedures the student followed to get his or her answer. For example, questions such as “How did you get 2?” or “So how did you guys do this? Can you tell me how to do it?” are ways that preservice teachers attempted to gain better insight into the student thinking that occurred. Third, candidates pressed for additional explanations, asking students to elaborate more on their previous explanation. This practice may occur because a preservice teacher does not understand a student’s response, wants to gain deeper insight student thinking, or wants to check for student understanding. The following dialogue represents an example of pressing for additional student explanations:
’Cause it says negative 6.
Okay, but I could point to negative 6 on the x-axis and I could have a vertical line there.
Oh ’cause it’s y, ’cause y is that (points to line on graph)
But still, how do you tell the difference?
Finally, we observed some instances when preservice teachers posed alternate examples for students to consider to clarify student understanding. Posing alternate examples occurred when the candidate inferred student confusion or determined that the class did not fully understand a concept. In one lesson, for instance, the class was exploring formulas to compute simple and compound interest. The preservice teacher noticed a student misconception regarding the importance of why the interest rate matters between calculating simple and compound interest. In response, she posed an alternate example to have students consider different interest rates to address this misconception.
Overall, we observed the preservice teachers who participated in the video-based course enacting responsive teaching practices—making space for student thinking to emerge, attending to and taking up novel student ideas, and pursuing student ideas to get insight into students’ thinking over the course of a lesson. Because the literature shows that it is not typical for novice teachers to enact such practices, we compared the cohort of preservice teachers from the LLfT course to a prior cohort who did not have the course as part of their program of study. We found that preservice teachers who enrolled in the video-based course enacted these practices with greater frequency across all three categories (see Figure 1). The average rate of occurrence per 2-min segment for Making Space for Student Thinking was 0.28 compared with 0.14, Attending to and Taking up Student Ideas was 0.14 and 0.11, and Pursuing Student Thinking was 0.38 and 0.13. All three categories showed more frequent enactment of these practices for the LLfT cohort. While there is an overall trend that the LLfT participants more frequently enacted these practices, the difference was statistically significant at the .05 level for Making Space for Student Thinking, t(33) = 2.23, and at the .01 level for Pursuing Student Thinking, t(35) = 2.92.
We find these results encouraging. It is not typical for preservice teachers to elicit a range of student ideas, to pause a lesson to take up novel ideas or questions, or to probe students for explanations and reasoning. We conjecture that what preservice teachers attend to when they elicit and probe student ideas provides greater insight into the learning opportunities they create for students. Thus, we examined how, through their pursuit of student ideas, candidates promote interactions where students have opportunities to develop robust mathematical understanding.
Examining the Substance of Preservice Teacher Noticing During Instruction
Our second research question concerned what in particular about student thinking the candidates attended to and how they took up these ideas during instruction. Analysis reveals that in the episodes where candidates pursued student thinking, they focused primarily on the correctness or incorrectness of student answers, as well as mathematical thinking related to procedural understanding, such as recall of steps, strategies, or factual mathematical terms. Second, across the candidates, we observed a similar pattern in the progression of the interactions for pursuing student thinking. In particular, the ways candidates noticed student answers and their evaluation of the correctness or incorrectness of the answer determined their path for pursuing student thinking. Figure 2 illustrates our working model of how candidates noticed and pursued student thinking during instruction. We use this model to highlight the relationship between what is noticed about student thinking and the opportunities for mathematical learning that candidates then created based on their noticing. We identified three types of interactions that unfolded across the candidates’ videos.
Relationship between noticing and pursuing student thinking during instruction.
One path we observed the candidates follow included noticing the correct answer and attempting to pursue student thinking by asking questions directed at having students justify and explain why their answer was correct. Students’ explanations and justifications to support their correct answers were typically aligned with the recall of procedural steps, strategies, or mathematics information/terms (i.e., formula, definition, or mathematics properties). Moreover, because candidates tended to know that students have the correct answer prior to asking for justification, the students’ ability to recall procedural knowledge further supported the candidates’ evaluation that the students understand and therefore they moved on to another student or another problem.
The following episode illustrates a situation where a preservice teacher focused on the correct answer and the student’s strategy for arriving at the correct answer.
90 degrees and 90 degrees.
90 degrees and 90 degrees, how did you get those?
Um, the sum of the exterior angles is 360. So there are four angles and 360 divided by 4 is 90. And then a square has four right angles.
okay it has 4 right angles.
and the sum of the interior angles of a square is 360.
ok, yes.
In this episode, the candidate attended to the student’s correct answer first and asked the student to articulate how she came to that answer. In line 6, we infer that the candidate’s utterance of “ok, yes” highlights what she is noticing related to the student’s thinking—she is noticing the correct answer and the student’s recall of the strategy and procedure to solve the problem. The candidate’s lack of pressing for more elaboration after her initial questioning suggests that the candidate was interested in the recall of strategies and procedures in getting the correct answer. The candidate perceived the student’s ability to recall the strategy and steps she used to arrive at her answer to be sufficient evidence of student understanding.
Candidates also noticed whether an answer was incorrect and then attended to the reason for the incorrect answer, whether it was a mistake in calculation or a procedural error. A candidate’s attention to an incorrect answer due to a calculation mistake or procedural error was reflected in the leading questions that he or she subsequently asked the student, questions that were typically aimed at guiding students to arrive at the correct answer. The following example demonstrates this type of interaction.
How did you get 2?
Negative two times zero.
Negative two times zero is . . . zero right? So let’s rewrite this. Negative two times . . . what is our x?
Zero.
Zero. Do we know what y is yet?
No.
No. Let’s just put plus y then. We still have a variable. What does our equation equal to?
Zero.
So what’s negative two times zero?
Zero.
Zero. Okay let’s rewrite it then. Zero plus what?
Plus two.
Plus two . . . Is equal to?
Two.
Then what does that mean y is equal to?
y is equal to two.
Very nice. That’s how you got it, right? When x is one, you put the one here, and then you solve for y, right? Because we’re trying to find the ordered paired, any solutions that make this statement true, right? So what happens when x is two? What do you do?
Times . . .
Okay, go ahead and write it down. So what would you get? Let me see what you’re going to write down. Very nice. Okay so how can you simplify this now?
Throughout this episode, the candidate guided the student in solving the problem correctly and was thus focused on accuracy and correctness in the procedure. The very first line of question was aimed at bringing the student’s attention to the mistake in his calculation. As the student corrected his calculation error in line 3, the candidate continued with her line of questioning to help the student complete each step accurately, ultimately arriving at the correct answer. From this interaction, we infer that the candidate was attending to the student’s correct use of procedures and calculations.
An alternative path candidates followed revealed their attention to student confusion or misunderstanding of mathematical ideas when students had an incorrect answer. The following excerpt illustrates a candidate attending to a student’s misunderstanding of how many points are needed to graph a line. In this exchange, a student is using a projector to graph points on the line and students are working together to determine how many points are needed.
You need one more.
One more.
Do you want to do one more, James?
(several students tell the student where to place the point on the line)
All the way at the top.
Eight.
The eight.
Well, here, I have a question for you guys. How many points do you need to make a line?
Four.
Two.
Two.
Shh. Lisa, why two?
Because, you connect point A to point B.
Point A to point B. If you have two points, can you connect those two points and make a line? Okay so right here we have four points already, right? So we can definitely make a line. But even with just two points we can still make a line, right? Okay, so James, go ahead and make your line.
At this point, one student, Anna, calls out if it matters how many points they graph and the teacher took up this idea.
Who asked that? Anna? So, does it matter? If you have four points, can you make a line?
Yeah.
If you have two points can you make a line?
Yeah.
Three points?
Yeah, even one point.
Yes. Can you make a line with one point?
Maybe?
’Cause you know what direction it’s going in?
Yes.
How? But how do you know it goes this way or this way with just one point?
In the beginning of this exchange, the candidate noticed that her students had a misunderstanding about the minimum amount of points that are needed to graph a line. Her question in line 7 illustrates her noticing this idea. Instead of telling students how many points they needed and moving on, she took the opportunity to pursue this question. As students continued to share the number of points they thought were needed, the candidate took notice of a correct answer and decided to pursue the thinking a bit further. This is seen in line 11 when she called on Lisa to justify her response. What is interesting about this excerpt is that in the midst of continuing with the lesson, the candidate also noticed another student, Anna, had a question regarding the point of discussion. In line 14, the candidate took up Anna’s confusion and made it an object of inquiry. Again, in line 21, another student voiced confusion and the candidate pressed the student to explain his thinking regarding the necessary components that are needed to graph a line. Throughout the rest of the episode, the candidate’s intent was to show that there were different ways to graph a line—knowing two points or knowing a point and a slope. Although attending to student confusion and pursuing student thinking in this way was less common, we saw evidence of candidates listening to students in the midst of teaching and taking time to work through the confusion with the students and the class. In this case, instead of glossing over Anna’s comment of “why does it matter,” the candidate took time to address this confusion.
Overall, the preservice teachers’ attention to and pursuit of student thinking was primarily focused on students’ answers and procedural accuracy, with less attention to their reasoning or conceptual understanding. Although the candidates engaged in practices to elicit and pursue student thinking, because the substantive focus was predominately on the accuracy and correctness of procedures, the learning opportunities created were limited to developing students’ procedural understanding. We now turn to discuss these results.
Discussion and Conclusion
To adopt a more responsive approach to instruction, prospective teachers need opportunities to learn to enact practices for eliciting and responding to student ideas. Lampert and colleagues (2010) argue that “such deliberately responsive and discipline-connected instruction” (p. 130) is a challenging but important skill for new teachers to develop. Our analysis shows that learning to systematically analyze teaching with video can help preservice teachers learn to enact practices that afford opportunities to access and examine student thinking. Several studies have found that preservice teachers attend more closely to student thinking in their reflections on teaching after learning to analyze lessons in systematic ways (Santagata et al., 2007; Star & Strickland, 2008; Stockero, 2008). The findings from this study extend this work by showing that preservice teachers who systematically analyze teaching can also begin to enact practices to enable them to focus more closely on student thinking during instruction. In particular, they created space during instruction for student thinking to become visible and available for the class to consider, they attended to and took up noteworthy student ideas, and they pursued student ideas.
When we compare the two cohorts, we observed statistically significant differences on two of the three dimensions of our analysis, making space for student thinking and pursuing student thinking. Consistent with other research, we found that analyzing artifacts of teaching can lead preservice teachers to develop expert-like teaching practices (Windschitl, Thompson, & Braaten, 2011). While the differences on the second dimension, attending to and taking up student ideas, were not statistically significant, we are not discouraged by this result. The expert-novice literature shows how difficult it is for teachers to veer from a planned lesson to notice and take up student comments, particularly unsolicited student ideas (Berliner, 2001; Jacobs, Lamb, & Philipp, 2010; Kagan & Tippins, 1991). Moreover, research suggests that this sort of impromptu response requires deep and flexible knowledge of the discipline and close attention to students’ ideas as they relate to the disciplinary goals of the lesson (Ball & Cohen, 1999; Hill et al., 2008; Kersting, Givvin, Thompson, Santagata, & Stigler, 2012). The fact that the preservice teachers from the video-based course paused to consider unsolicited ideas and made them part of the classroom discourse, as well as re-voiced student ideas for the class to consider more frequently than the other cohort, suggests that they were beginning to notice student thinking in practice (Levin et al., 2009). Thus, not only did they develop a vision of ambitious instruction to inform their noticing of ambitious pedagogy when they reflected on practice (Barnhart & van Es, 2015; van Es et al., 2011), but they also enacted practices to notice during instruction. Our analysis suggests that the preservice teachers shaped classroom interactions in such a way so that they could notice student thinking. At the same time, they did not simply dismiss a surprising idea but they took them up and used them to inform instruction. This is particularly noteworthy because research finds that teachers’ classrooms can become valuable sites for both teacher and student learning (Franke et al., 2001; Lampert, 2010; Sherin, 2002).
We also found, however, that the candidates’ attention in these interactions was often on student answers, the process and strategies for solving problems to arrive at an answer, and moving students toward solving problems correctly. This is consistent with prior research that shows that teachers often ask follow-up questions to probe both correct and incorrect responses (Franke et al., 2009). Research shows that engaging students in extended discourse after providing a correct answer can result in developing deeper understanding (Schleppenbach, Flevares, et al., 2007). Moreover, it is well documented that both students and teachers benefit from student talk, leading to increased content knowledge and understanding (see Franke et al., 2009). While the focus of the preservice teachers’ inquiry was on procedural accuracy and correct and incorrect answers, it may be that learning to navigate classroom discourse around procedures can become a gateway for eliciting, attending to and probing student mathematical reasoning and conceptual understanding. That is, teacher educators can more deliberately design learning opportunities to build on preservice teachers’ attention to correct and incorrect answers to help them learn to notice student reasoning of core disciplinary ideas and develop practices for extending mathematical discourse to cultivate more complete disciplinary understanding.
An important consideration is how analyzing video may have helped preservice teachers learn to enact such practices. Video affords close observation of instructional episodes. The ability to stop, pause, and rewind a video segment to study a classroom interaction multiple times allows candidates opportunities to see important events they may not notice upon first viewing, to interpret what is observed from multiple perspectives and to explore those perspectives with others. We conjecture that by viewing and decomposing ambitious pedagogy as represented in video records of practice, preservice teachers learned to see practices for making thinking visible during instruction, such as strategies for structuring opportunities for students to think during a lesson or for eliciting a range of student ideas (T. Hatch & Grossman, 2009). In addition, by engaging in multiple cycles of structured video analysis, they appeared to learn practices for noticing noteworthy ideas and ways of questioning those ideas, the same sort of practices for attending to and pursuing student ideas during instruction. Practice-based approaches to teacher preparation have focused on designing approximations of teaching for preservice teachers to learn to enact ambitious pedagogy (Grossman et al., 2009; McDonald et al., 2013). Typically, video is used as a tool to aid in reflection of instruction (Santagata & Guarino, 2011; Stockero, 2008). However, we see here that structured video analysis can also serve as way to approximate the work of teaching by fostering preservice teachers’ practices for shaping instruction to get insight into student thinking, identifying noteworthy ideas in classroom interactions, and asking a range of questions about those ideas to gain insight into student thinking and understanding. In this way, analyzing video not only supports developing a vision of ambitious teaching (van Es et al., 2011) but also functions to develop preservice teachers’ beginning repertoire of teaching.
Although these results are encouraging, we recognize that there are several limitations to this study. First, the nature of the tasks that preservice teachers posed play an important role in the kinds of learning experiences that they were able to construct for their students. Analysis of the relationship between the task quality, the opportunities for noticing of student thinking these tasks afforded, and instructional practices is an important area for future inquiry. Second, the data of teaching practice come from only one source, the PACT Teaching Event. Candidates videotape themselves early on in the student teaching experience, so it is a limited representation of their practice. In addition, the candidates are responsible for videotaping themselves and self-selecting the clips, which raises questions about identifying clips that represent the practices being assessed in the PACT. Third, we used classroom discourse as captured in these videos to gain insight into the candidates’ noticing during instruction. More recent advances in video technology for research (see Luna, 2014) may provide more reliable data for analyzing teacher noticing in practice. Finally, although the video cases the candidates analyzed in the course were chosen to highlight practices advocated by reform, as well as those that are assessed in the PACT Teaching Event, the candidates may have interpreted the practices in relation to promoting procedural fluency because this is a typical framing of mathematics teaching for novice teachers (Bartell, Webel, Bowen, & Dyson, 2013; Star & Strickland, 2008). An important area for future research concerns how teacher educators help teacher candidates develop a more robust framing of instruction. Research on facilitation of video-based professional development provides insight into strategies for orchestrating productive discussions with video (Coles, 2013; van Es, Tunney, Goldsmith, & Seago, 2014; Zhang, Lundeberg, & Eberhardt, 2011). Understanding how teacher educators help candidates learn to see and make sense of important events in video will contribute to research on improving teacher education.
Teacher education is challenged to prepare candidates to learn to enact responsive teaching practices. This study provides preliminary evidence that developing a vision of ambitious instruction through video analysis of teaching can lead preservice teachers to learn to elicit, attend to, and pursue student ideas during instruction. We see an important next step as identifying principles for designing a set of teacher preparation experiences to achieve this goal. Such research will move the field forward in building a knowledge base for teaching and teacher preparation (Hiebert, Gallimore, & Stigler, 2002; McDonald et al., 2013) and the construction of shared pedagogical practices to improve the work of teacher education.
Footnotes
Authors’ Note
The opinions expressed are those of the authors and do not necessarily reflect the opinions of the supporting agency.
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 research was supported by the Knowles Science Teaching Foundation.