One primary concern of education research is to study the nature and the quality of teaching and learning in school settings. Over the past 30 years, education researchers have produced a wealth of representations of classroom teaching, in both text and video forms (Sherin, 2003). These representations assist researchers in their scholarly analysis of classroom interactions between teachers and students. Although in the educational research literature, researchers discuss important epistemological and methodological issues associated with the creation and analysis of textual and video representations (e.g., Brophy, 2003; Parks & Schmeichel, 2014), these issues are not always disclosed in research reports or accounted for in policy and practice implications of such studies.

More importantly, video representations of teaching have become common tools in the curriculum and pedagogy of teacher education, and videos of classroom teaching are increasingly being created and used in teacher education contexts to facilitate conversations and analysis of the quality of interactions between the teacher and students (see Brophy, 2003). Furthermore, video representations of teaching have become more easily and increasingly accessible online through video libraries and are used to engage teachers and researchers in online professional communities that view, comment, and discuss selected video clips of classroom instruction (e.g., Teaching Channel). Alongside the proliferation and use of videos of teaching for research and teacher education purposes, the tendency to use evaluative language and frames to analyze teaching episodes continues to persist.

Some solutions have been offered to address the persistent problem of evaluative and deficit language when discussing or analyzing video representations of teaching in teacher education contexts. One approach has been to shift the frame of the analysis from the teacher to the teaching (e.g., Stigler & Hiebert, 1999), while another has been to shift the tone of the analysis of teaching episodes by promoting inquiry-oriented language (e.g., Smith, 2009). Yet another has been to shift the frame of the video camera. Sherin, Linsenmeier, and van Es (2009), for example, suggest that video clips that are focused on the students’ talk and participation as opposed to clips that are focused on the teachers’ teaching moves seem to be particularly generative in the context of teachers viewing and discussing together video episodes of mathematics teaching.

Tobin and colleagues (1991, 2009) approached this problem by adding a cross-cultural lens to the analysis of video representations of preschool classrooms in order to help make visible familiar and locally entrenched classroom practices. Herbst and Chazan (2003), on the other hand, bypassed video representations altogether and resorted to animated classroom episodes in order to elicit different analytical lenses and conversations about an observed episode of mathematics teaching. Others (e.g., Oslund & Crespo, 2014) have turned to using classroom photographs rather than videos of mathematics teaching to support prospective teachers’ ways of seeing and connecting students’ mathematics inside and outside of the classroom.

The embracing-of-contraries approach that I propose here is not simply one more solution to the aforementioned problem. Differently from the solutions listed above, an embracing of contraries does not try to bypass the temptation to evaluate teaching practice but rather requires facing that temptation head-on. Furthermore, unlike most popular video representations of teaching that accentuate either the positives or negatives, the embracing of contraries does both. One example of what this looks like can be found in Dateline’s (1998) Equal Possibilities, a documentary focused on a high school teacher (Mr. Workman) teaching the same math course to two sections of students—one co-ed and the other girls only—using two very different teaching methods. The differences between the two classrooms were striking. The co-ed classroom was eerily quiet, with the teacher walking the rows as a drill sergeant giving occasional short comments to individual students: “right” and “try harder.” In the girls-only classroom, the teacher acted very differently. The students were expected to work with their group members and help each other figure out problems and only ask for help once they were all stuck.

In the co-ed classroom the teacher used teaching techniques discussed in the literature as rapid-fire questioning and step-by-step instructions (i.e., Stevenson & Stigler, 1992; Stigler & Hiebert, 1999), with an initiation-response-evaluation (IRE) style of classroom talk where the teacher is always in control of the content, direction, and pace of classroom talk (Cazden, 2001). In the girls-only classroom the teacher used more student-centered teaching methods whereby the students were expected to work with their group members and help each other figure out problems and only ask group questions. This latter type of learning environment has come to be associated with reform mathematics classrooms, where teachers promote student-student interactions that are focused on getting students to explain, reexplain, refine, and revise their thinking over a number of interactions with classmates and their teacher (Jacobs & Morita, 2002; Stigler & Hiebert, 1999).

The power of this video documentary is that it challenges the viewer to resist the temptation to categorize Mr. Workman’s teaching of mathematics as solely traditional or solely reform. Instead this teacher shows viewers that teaching practice is complex, flexible, and necessarily responsive, responsible, and dependent on the context, the students, and the mandated curricular goals. More importantly, this teacher was portrayed as deliberate and purposeful, and as having well-articulated pedagogical reasons to guide his teaching of mathematics using two completely different and some might say opposite methods of instruction. The embracing-of-contraries approach that I propose here resembles the Equals Possibilities documentary, and I offer it here as one way to make visible our tendency toward documenting and describing the extremes of teaching practice and by doing so disrupt this familiar way of viewing and discussing videos of teaching practice.

The tensions I draw attention to in this article about representing the complex nature of mathematics teaching are not new. I join other educators who have also documented opposing or conflicting practice from the same teacher as a natural occurrence in the work of teaching. However, these authors have done so at a much larger grain size than the one I report on here. In Parks’s (2009) case, she documented a wide repertoire of both student- and teacher-centered teaching practices throughout a year of observations in one elementary school teacher’s classroom. In a different project, researchers documented reform and traditional teaching that coexisted within the same high school mathematics teacher, as each day she taught two distinct versions of the same algebra course in response to parents, students, and her district’s expectations (Herbel-Eisenmann, Theuli-Lubienski, & Id-Deen, 2006).

In my case I not only draw attention to these issues, I propose and illustrate an alternative strategy for analyzing videotaped teaching episodes using an 8-minute video of Ms. Kelly’s teaching of mathematics in a kindergarten classroom. I edited it so as to illustrate contraries in her mathematics teaching by creating two versions—one that represented Ms. Kelly’s teaching as “Reform” and another that represented Ms. Kelly’s teaching as “Traditional”—that coexisted within a short, 8-minute video of her teaching. This approach to making video representations is inspired by Cardinal Richelieu’s quote alluded to in the epigraph of this article and draws on Elbow’s (1983) discussion about complex processes, such as teaching, that require and demand the embracing of contraries.

Elbow (1983) describes the process of teaching as managing opposite mind-sets and skills. He suggests that successful teaching demands the ability to be extremely supportive and extremely critical, to play the role of ally and adversary, without letting one mentality prosper at the expense of the other or being halfhearted in both (Elbow, 1983). More specifically, he stated that conflicting obligations—to students and the subject matter (a tension widely described by many educators)—explain much of what to outsiders appears as paradoxical incoherence in teachers’ daily teaching practice. He further suggests that striving for a harmonious middle in teaching practice is futile and leads to ineffective teaching. Instead he suggests that alternating between opposite mind-sets is how most teachers cope with conflicting intentions in their teaching practice.

Researchers have documented these kinds of dilemmas and tensions in the work of teaching. Lampert (2001) writes about pedagogical dilemmas and tensions she faced and managed in her own teaching of mathematics to fifth graders. Kennedy (2004) has spent her career studying the tensions in teachers’ professional vision and their enacted practice. She writes about the living contradictions in the work of teaching, and she traces them to parallel contraries in societal hopes and expectations for teachers and schools, such as expecting teachers to create positive learning environments in their classrooms but also be efficient and goal oriented, to socialize students to accommodate prevailing cultural norms while also teaching them to be critical thinkers and question authority. The list of tensions that pull teachers in polar opposite directions is large and growing. Learning to recognize such tensions and manage them in productive ways inside and outside the classroom is important, especially for new teachers who are the most at risk of teacher burnout and leaving the profession within the first three years of teaching.

In this section I illustrate the translation of Elbow’s (1983) theory of embracing contraries in the work of teaching to embracing contraries when analyzing teaching practice, in particular when working with video representations of teaching. But first I introduce Ms. Kelly and the context of my research project to explain why and how I decided to try Elbow’s (1983) embracing of contraries when composing and creating video representations of teaching.

Ms. Kelly, a beginning teacher participating in my research project (Crespo, 2006) was working in a charter urban school located in a refurbished old warehouse building. She and her students wear school uniforms. Most of her students are Latinos who are new immigrants to the United States learning English as a second language. She is friendly, mild mannered, and confident. Three researchers collected data in Ms. Kelly’s classroom, but the entire research team viewed videos of her teaching and engaged in the process of constructing a video collection of about 10 clips of video spanning 2 years of her teaching. These video clips were drawn from a data set of six complete mathematics lessons recorded over the span of 2 school years. In addition, each lesson was recorded both with a stationary camera mounted on a tripod and with a movable flip camera that could get closer to the teacher and the students’ actions.

Ms. Kelly’s teaching practice challenged our research team’s visions of kindergarten classrooms as spaces where students learn through play and explorations. Hence Ms. Kelly’s case challenged the research team’s commitment to taking an agnostic stance toward the project participants’ teaching of mathematics. It did not take very long for the research team to, in Cardinal Richelieu’s words, “hang” Ms. Kelly as a prototypical authoritarian and controlling mathematics teacher, especially when compared to other beginning teachers in the project.

Even before the first visit, the research team had opinions about the overly prescribed and scripted mathematics textbook (Saxon, 2010) Ms. Kelly was required to use. Other school practices and the requirement that students and teachers wear uniforms further suggested imposed conformity and uniformity as the school norm and expectation. So even before seeing her first math lesson, Ms. Kelly’s teaching practice was deemed suspect and expected to be the antithesis to any idealized version of reform mathematics teaching. Unsurprisingly, the team’s analyses of Ms. Kelly’s mathematics teaching practices matched those initial impressions.

It would have been a lot simpler to let the research team’s impressions of Ms. Kelly’s teaching of mathematics go unchallenged and move on to other pressing research tasks. However, I questioned whether the research team had a fair impression of Ms. Kelly’s mathematics teaching or whether we had closed off our minds too soon. I decided to do a mini-experiment inspired by Richelieu’s quote at the start of this article to see if by rearranging Ms. Kelly’s six lines (in this case, video footage of her teaching) I could find some evidence of reform mathematics teaching hidden within the traditional mathematics teaching methods the research team kept noticing and discussing during project meetings.

Ms. Kelly’s video records offered a hard case for testing the embracing-contraries approach that Elbow (1983) theorized and that I propose here to making and using video records of teaching practice. I reasoned that if this approach were to allow my research team to view Ms. Kelly’s math teaching differently, then perhaps it could work with other video records that prompted such singular interpretations. Using video footage of Ms. Kelly’s mathematics teaching, I illustrate how an embracing-contraries heuristic can be a more generative process for working with classroom videos than the typical single chunking and interpretation of a selected segment of a videotaped mathematics lesson. I illustrate and discuss next what this approach entails.

Composing Two Different Versions of Ms. Kelly’s Mathematics Teaching

For this experiment I chose a video-recorded lesson with a different structure from the “I do, we do, you do” sequence typical of the Saxon mathematics textbook lessons Ms. Kelly was required to follow. Unlike typical Saxon math lessons, this particular one had the students work more independently than in most other lessons in Ms. Kelly’s video collection of lessons. It focused on copying a geometric design from a geocard onto a physical geoboard (Saxon 2010, Lesson 86, p. 4). Each kindergartner was given a geocard with a geometric design that was completely different from the others. No two designs were exactly alike. A sample of geocards and a brief lesson description are included in Figure 1.

Figure 1. Mathematics textbook lesson Ms. Kelly taught in her kindergarten classroom.

This lesson was observed and recorded during Ms. Kelly’s second year of teaching. The entire mathematics lesson lasted about 45 minutes, the standard math period for Ms. Kelly’s kindergarten classroom. The embracing-of-contraries mini-experiment used 8 minutes of continuous video footage at the start of that lesson. These 8 minutes of raw footage were broken up into smaller chunks, a total of 8 chunks ranging from 9 seconds to 2 minutes. These 8 chunks were then put back together to construct two versions of Ms. Kelly totaling about 4 minutes each. These two versions are illustrated in Table 1, which also includes a brief description of each video clip to highlight the contrast between each of the opposing parts and also the whole for these two video representations of Ms. Kelly’s teaching of mathematics.

Table 1. Selected Video Clips from Ms. Kelly’s Mathematics Teaching

Clip	Ms. Kelly Traditional	Ms. Kelly Reform
1	2:05 min: Teacher reads the task to the students and specifies acceptable behaviors.	0:34 min: Teacher presents the task to the students and sends them to work on the task.
2	0:09 min: Teacher monitors closely and publicly praise students who follow procedure.	1:02 min: Students explore shapes in their geoboard monitored but unassisted by the teacher.
3	0:16 min: Two students are praised for finishing their work and sitting quietly.	0:55 min: Teacher monitors at a distance, “Look at your picture and your design, are they the same?”
4	1:40 min: Teacher assists a student with wrong work by taking over the materials and the task. “No, no, no, it is this way!” Completes the design for the student.	1:33 min: Teacher assists a student with wrong work—“How can we check that? What do we need to do?” Teacher watches the student correct their work. “You’re so smart!”

Ms. Kelly’s Teaching Remixed as Traditional

In this remixed video representation, Ms. Kelly introduces the math task to her students as something the textbook wants them to do. Students sit around an oval carpet, and the teacher sits at one end of the carpet waiting for students to settle and quiet down to start the lesson. She launches a monologue about what not to do with the geoboard materials: “You should not have your bag open yet your hand should not be inside of it. You should look like Magaly, Magaly has her geoboard in front of her, she has her geobands in her bag on top of her geoboard and her design is in front and she’s sitting crisscross applesauce.” During the exploration phase of the lesson the students work independently, and Ms. Kelly monitors. She is frequently heard making statements that are focused more prominently on students’ behaviors rather than on their thinking. Ms. Kelly praises students’ compliant behaviors to those stipulated at the start of the lesson: “I love how Armando and Maria have finished their shapes and they have their hands in their lap.” Ms. Kelly is quick to jump in when she sees students struggling with the task. She does so by taking over the task and showing them how to correct their geoboard designs. She uses gestures and discursive moves that have been widely documented in the mathematics education research literature as “funneling” students toward task completion and correctness of their answers (i.e., Herbel-Eisenmann & Breyfogle, 2005). This is further illustrated in the following transcription of video clip 4, the last segment of the Ms. Kelly Traditional video representation. Four video frames associated with this video clip and transcript are included in Figure 2.

Figure 2. Ms. Kelly Traditional takes over and shows how to correct the geoboard design.

Video clip 4 (Traditional)—correcting students’ mistakes

A student waves his geoboard and the design card at the teacher and says, “I made a copy.” The student’s design matches the shapes in the geocard but not their size and orientation. Figure 2 shows four frames of Ms. Kelly’s interaction with the student. She is crouched down on the carpet and has lined up the student’s geoboard and geocard so they both have the same orientation. Note that in all four images it is the teacher rather than the student who manipulates the geoboard and the rubber band. Their dialogue is as follows (T = Teacher; S = Student):

T:

Okay let’s have a look.

T:

Is your triangle the same as this triangle?

S:

[Looks at the design card and the shape he made in the geoboard and then looks back to the teacher—says nothing.]

T:

Well, let’s try. Alright, so where does your triangle need to start [removing the triangle-shaped rubber band]?

S:

Right here [pointing in the card to the midpoint where the line segment meets the triangle’s leg].

T:

Okay so that’s where that is, where does it go to? Right where? Here? [Pointing inside the geoboard] Right here? Okay. So leave it, hold on one minute. Start there. Okay, move it over to that one [pointing in the geoboard to a peg where vertex of the triangle should be].

S:

[Manipulates the rubber band and creates a horizontal rather than diagonal line segment.]

T:

No, no, no [undoes the rubber band and points to the peg that needs to be used]. … No, let’s look! Straight down! So where does it go? It goes there but then where does it go? This way? Keep going! One more peg! There you go!

S:

[Student again stretches the rubber band from the correct peg to the one farther down from where it needs to be]

T:

No no no! You were good! [pulls the rubber band back to previous peg]. Now look at them, do they look the same?

S:

They look the same!

T:

Nods and moves back to the center of the carpet.

Using a discourse analysis frame (e.g., Cazden, 2001) on the interaction between Ms. Kelly and the young student in the video, we hear the prevalent IRE pattern of teacher-student classroom talk. We notice that the word “no” is uttered by the teacher multiple times (seven times). When the teacher asks questions, she quickly answers them herself (e.g., “Where does this go? Here?”), and she uses imperative verbs and tone (e.g., “keep going,” “leave it,” “hold on,” “start here”).

Using the perspective of gesture and body research (e.g., Goldin-Meadow, 2000) on this episode, we can notice hand gestures, body motions, and eye glances. The young student is seen closely watching the teacher cues and appears tentative and unsure when manipulating the materials. Moreover, in the video we can see the teacher completely “erasing” the student’s work by removing the green triangle rubber band from the geoboard and restarting the green triangle from scratch without connecting it to the work or thinking the student had produced earlier. The teacher takes over the geoboard manipulatives and completes the work for the student, effectively positioning him as a spectator rather than an active math doer.

This interaction is clearly teacher-directed, and the goal seems to be more about task completion than mathematical reasoning, as the student seems to be guessing and following the teacher’s directives. These are the kinds of classroom interactions that mathematics education reformers would advocate changing so as to make those interactions more student-centered and focused on students’ mathematical thinking. Using the NCTM’s (1991) “from-to” bulleted list in the “Embracing Contraries with Video Representations of Teaching” section, we can say that this remixed episode clearly represents the five features of traditional mathematics instruction listed in the NCTM Professional teaching standards document:

• •

  Classroom as a collection of individuals not working together.

• •

  Teacher as the sole authority for right answers.

• •

  Student is not engaged in doing mathematical reasoning.

• •

  Emphasis on the mechanistic finding of answers.

• •

  Treats mathematics as a body of isolated concepts and procedures.

Without any evidence to the contrary it would be very tempting for the author and perhaps the readers of this article to conclude that Ms. Kelly is enacting the prototypical traditional mathematics teaching that has been shunned in the mathematics education literature. However, in the next section, Ms. Kelly is represented as enacting very different teaching moves, not in a different day’s lesson or with a different group of students but within the same 8 minutes of video footage where Ms. Kelly Traditional is also found.

Ms. Kelly’s Teaching Remixed as Reform

In contrast to Ms. Kelly Traditional, Ms. Kelly Reform presents the task to the students using her own words rather than by reading the textbook script; she sounds excited about the lesson as she shows the materials to be used during it. The task is presented as something they will be exploring and learning—not just something to be quickly done. Students are on the carpet sitting all the way around the oval. The teacher sits on a chair holding materials that she is showing as she presents the task to the students. Materials are passed down until all 30 students have one of each in front of them.

The classroom is buzzing with excitement, and students are working independently while the teacher is standing at the center of the carpet overseeing and monitoring their work. The camera focuses on a student working on his own and having difficulties reproducing the given design. The student has placed a red rubber band at the top of the geoboard rather than across the middle of it and is working with the blue rubber band to copy a blue rhomboid but makes a trapezoidal shape instead. The student compares his geoboard to the geocard and removes all rubber bands. The camera then moves to the neighboring student who is making an entirely different design with two different size squares and a triangle in between the two. The student has the larger square copied onto the geoboard but in a wrong position. The teacher is still in the middle of the carpet overseeing the students’ work, not yet assisting anyone, and is overheard saying: “Alright, I want you to look at your paper and look at your geoboard, make sure your shapes are going the right way. Check yours first then each other’s.”

Video clip 4 (Reform)—questioning students’ mistakes

As the students check their own work and then check with their neighbor, Ms. Kelly walks over to a student whose geoboard does not exactly match the design in the geocard. She approaches this by asking questions rather than by telling or showing the student how to correct his work. Four frames associated with this dialogue are included in Figure 3. These four frames show the hands of the teacher acting as holder and pointer of the materials, and the hands of the student acting on the materials. The contrast between these frames and the frames included in Figure 2 further suggest the distinction in the two types of video representations of Ms. Kelly. Their interaction sounds as follows:

T:

Let’s look at yours [crouching on the floor].

Figure 3. Ms. Kelly Reform encourages and waits for the student to correct the geoboard shape.

S:

[Places the paper and geoboard side by side.]

T:

Alright, is the green going the same way?

S:

[Nods yes with his head.]

T:

Okay, is the red one going the same way?

S:

[Nods yes with his head.]

T:

Okay! Is the blue one going the same way?

S:

[Nods yes with his head.]

T:

Let’s look. This one has an open peg over here … does this have an open peg over here? [Points to the pegs but does not touch the students’ incorrect shape.]

S:

[Indicates no by shaking his head side to side.]

T:

Noooo, so how do we fix that? What do we need to do?

S:

[Moves the rubber band to the correct peg.]

T:

Good, now let’s look at this one. It has an open peg down here and a rubber band up here. So what do we need to do?

S:

[Moves rubber band to the correct peg.]

T:

Okay! You are so smart! Good job! [Walks back to the center of the carpet and calls everyone’s attention.]

In this short video segment we can again use a discourse analysis lens (Cazden, 2001) to notice the same IRE pattern that is so prevalent between teacher and students. But when we zoom in more closely to the questions and patterns of questioning the teacher is using as she works with this student, we can notice that she asks questions quite differently from Ms. Kelly Traditional. In this case she asks about all of the shapes in the design, including those that correctly match the geocard design. When the student nods yes instead of no with regard to the orientation of an incorrectly oriented blue parallelogram, the teacher again moves to correction mode but takes a different approach. She uses questions such as “How do we fix that?” and “What do we need to do?” rather than directives and commands. The teacher thus invites the student to manipulate the materials, and the student does so without much prompting.

A nonverbal body and gesture framework (Goldin-Meadows, 2000) in turn helps us see a teacher that holds back on jumping in at the first sign of student struggle, and instead allows her students to experience productive struggle when doing mathematics (Hiebert & Grouws, 2007). Students in this clip can be seen as confident learners as they grab the materials with no hesitation or look for the teacher’s approval or confirmation. In the interaction with a student, the teacher points to pegs and holds the geoboard to keep it in place but not once touches the rubber bands that are used to make the shapes. She positions herself as the spectator of the students’ mathematical work. The student is guided but not told what to do or think.

The nature of this videotaped interaction can be described as more student-centered than teacher-centered. The teacher uses more probing types of teacher questions, and her patterns of questioning are more supportive than stifling students’ independent mathematical thinking. Because the student is the one who manipulates the rubber band from one peg to the next, working from one shape to make another one, he has the opportunity to develop intuitive and flexible understanding of shapes and their transformations, potentially recognizing the relation between the initially incorrect shape and then the corrected one.

When considering this interaction in relation to the NCTM (1991) professional standards, we can agree that this version of Ms. Kelly’s mathematics teaching is more closely aligned with the more desirable reform-like practices. Using the NCTM’s (1991) “from-to” bulleted list, we can see that this remixed episode includes all five principles of reform mathematics instruction listed in the NCTM Professional teaching standards document:

• •

  Students working together and checking each other’s work.

• •

  Emphasis on student’s intellectual independence and sense-making.

• •

  Student is invited to reason mathematically.

• •

  Emphasis on problem solving rather than mechanistic answer finding.

• •

  More opportunities for students to seek and make mathematical connections.

In this article I have argued that there is a need for creating more complex video examples that push the limits of reform versus traditional categorizations of teaching and that make it possible for viewers to see more of the complex nature of mathematics teaching. I illustrated what this might look like using an embracing-of-contraries heuristic modeled after Elbow’s (1983) theoretical discussions of the work of teaching as embodying competing mind-sets. This entailed editing a video episode using contrasting categorizations of teaching practice (e.g., student- or teacher-centered) and remixing those video chunks to create two versions of the same teaching episode that highlighted teaching practices that are recognizable as contradictory.

For my research team, considering the contraries within Ms. Kelly’s teaching of mathematics at such grain size was a turning point in the ways in which we talked about and analyzed her teaching of mathematics and that of other research participants. We could no longer view or generalize Ms. Kelly’s mathematics teaching as the antithesis to reform mathematics teaching and instead began to see her teaching as a complex system that embodied both teacher and student-centered pedagogies. These insights have implications beyond this one particular mini-experiment and Ms. Kelly’s teaching of mathematics.

An embracing-of-contraries approach to generating and analyzing video representations of teaching builds on and extends the conversation about the use of video records of practice as research and educational tools in teacher education (i.e., Brophy, 2003; Chazan & Herbst, 2011; Sherin et al., 2009). This line of research has focused on developing video representations and tools that elicit and/or develop teachers’ frames for noticing and learning in and from teaching practice. It also brings this line of research into relation with broader conversations about the persistence of overly evaluative and deficit discourses about students and teachers, and the role of educational research and researchers in reproducing and disrupting such frames and discourses (i.e., Comber & Kamler, 2004; Davis, 2004; Shields, Bishop, & Mazawi, 2005).

The approach I described here creates a new type of video representation of teaching that makes visible the analytical frames that were used to create it. It purposefully brings into focus what typically is hidden in video representations of teaching, namely the frames that assisted in the selection and production of these representations. It purposefully focuses attention on contradictory teaching practices and by doing so elevates them as worthy of noticing and discussing rather than simply dismissing. The contribution of this alternative becomes apparent when we consider that the most common criteria for selecting and using video representations for teacher learning purposes are the degree to which they represent authentic practice and that they illustrate either exemplars or dilemmas of practice (see Sherin et al., 2009). This narrow focus in the selection, use, and production of video representations of teaching limits what is possible for researchers and teachers to learn from video representations of teaching.

A simultaneous focus on contraries within a given episode of teaching provides a different lens for analyzing video records than what others have documented in the research literature (Chazan & Herbst, 2011; Tobin & Hsueh, 2006). Most typical video representations are made and used to encourage viewers to identify, name, and classify or evaluate observed instructional interactions (Sherin et al., 2009). Some teacher educators design and use video observation protocols to focus and press the noticing and language of teachers, especially novices, when analyzing those instructional interactions (e.g., McDuffie et al., 2014). In contrast, video representations that embrace contraries highlight, in very practical terms, the evaluative frames and discourses that are persistently used in the viewing and discussing of teaching practice. Rather than suppress, silence, or prescribe noticing frames, this approach embeds and plays back the evaluative frames and discourses about teaching practice that teacher educators seek to problematize. For researchers and teacher educators, this approach offers a productive way to work with as well as subvert the evaluative frames and language that are often brought to bear in the viewing and discussing of video episodes of teaching.

In other words, an embracing-of-contraries video representation forcefully challenges the hyperrealism and familiarity that tends to limit how and what we can learn from video observations (MacLure, Holmes, MacRae, & Jones, 2010). It provides a heuristic for decomposing teaching practice in ways that are accessible to novices but without overly simplifying it (Grossman & McDonald, 2008). By doing so, this approach invites researchers and practitioners to interrogate and play with old, new, and different kinds of frames for viewing, selecting, generating, and discussing video representations of teaching.

Said another way, video representations of mathematics teaching are important tools in the study of teaching practice. They enable the study of classroom interactions at various levels of detail, and they support the important work of developing common vocabulary and terminology to describe and analyze classroom interactions (Chazan & Herbst, 2011). They also allow teacher educators to make teaching practice accessible to novices who are less expert at parsing out what is challenging or crucial to address when teaching mathematics (Brophy, 2003). However, too often video representations of teaching become exemplars and illustrative examples of desirable teaching practice (Sherin et al., 2009), thereby reinforcing and reproducing the very same overly critical and evaluative frames and discourse about teaching that researchers and teacher educators are seeking to challenge and redress (Davis, 2004).

Reform rhetoric compounds this problem with its tendency to idealize and oversimplify the divide between traditional and reform mathematics teaching (Kennedy, 2004). In mathematics education, descriptions and characterizations of procedural, calculational, and chalk-and-talk mathematics lessons are often represented in opposition, not in combination to the more desirable forms of reform-minded teaching. As Parks (2009) suggests, such dichotomies construct two mutually exclusive images of mathematics teaching, rendering everything that is not reform practice as unacceptable. This closes off opportunities for inquiry into the depth and range of nonreform practices that could perhaps be used to shed light on how such practices may function in different classroom contexts, or when teaching different mathematics topics, or when supporting students with different orientations to learning mathematics. An embracing-of-contraries heuristic offers another approach to analyze video episodes of mathematics teaching that is potentially more generative.

For researchers focusing on understanding classroom practice by closely observing video records of teaching, an embracing-of-contraries heuristic offers a tool for making visible teachers’ successes—not only their failures—with respect to enacting ambitious forms of teaching. By representing pockets of mathematics teaching that resemble reform teaching within any and every teacher’s video representation of mathematics teaching, researchers can begin to counter the overabundance of examples of questionable mathematics teaching in the research literature and the media. These counterimages and counternarratives are important, as they can challenge the very prevalent deficit framing and discourse about teachers’ competence and ability to enact ambitious forms of mathematics teaching.

For teacher educators and for teachers working with video observations of mathematics teaching, an embracing-of-contraries heuristic can be very useful when viewing and selecting videos of mathematics teaching to be used in video clubs or shown in teacher preparation classes. These kinds of video representations would promote and support different kinds of conversations than those focused either on students’ mathematical work or on the teacher teaching moves. Imagine the possible kinds of conversations that one could have with Ms. Kelly about her teaching of mathematics by offering two different images about her teaching. An illustration of ambitious mathematics teaching captured within her own teaching (even if aided by some creative editing) offers a mirror and a window on her teaching potential in ways that a video of an experienced veteran teacher will not. Rather than offering feedback and critique on her teaching of mathematics that could undermine her sense of competence, offering two different images of the same teaching episode allows for a rethinking of practice with (rather than without) a sense of teacher agency and empowerment. This is especially important for the newest members to the teaching profession if they are to become the new generation of teachers who will challenge deficit frames and discourses about teachers and their teaching practice.