2 Method
2.1 Source of data
The data were drawn from video recordings of six middle-school mathematics lessons. Five of the lessons took place in public schools in a mid-size Midwestern city; the remaining one took place in a parochial school in the same community. Four of the teachers were female, and two were male. Lessons ranged from 40 to 61 min in length. Lesson topics and grade levels are listed in Table 1. Participating teachers were aware of the research team’s interest in verbal and non-verbal instructional communication. Before the lesson, each teacher completed a brief written survey that focused on the planned lesson content the teacher’s expectations for students during the lesson, and the difficulties the teacher expected students to have during the lesson. After the lesson, each teacher participated in a brief oral interview with a member of the research team. The interview included questions about the teacher’s view of how the lesson went, whether anything about the lesson or students’ reactions was surprising, and which portions of the lesson were new material and which were review.
2.2 Video analysis
The video recordings were transcribed and coded in several passes. In the first pass, we transcribed teachers’ and students’ speech, and identified students’ and teachers’ turns at talk. In a second pass, we identified trouble spots. A trouble spot is a part of a conversational exchange that contains a mistake or that is not sufficiently clear, so that a repair is subsequently initiated (Golab et al. 2009). Given our focus on student learning, we defined trouble spots as points in the discourse where students made errors or otherwise displayed lack of understanding or uncertainty about the lesson material—that is, points where students’ under-standing of the lesson was compromised or disrupted. We identified three types of trouble spots: (1) student-initiated questions regarding the instruction (see, e.g., Sidnell 2010); (2) incorrect responses by student(s) to teachers’ questions or statements (called ‘‘errors’’ by Schleppenbach et al., 2007); and (3) dysfluent utterances on the part of students, defined as utterances in which students produced incoherent statements or directly expressed lack of certainty (see, e.g., Schegloff et al. 1977). These categories emerged from our analysis of the data, and were also informed by the literature. All of the trouble spots that we identified fell into one of these three broad categories. All three categories indicate a lack of under-standing on the part of the student(s), and thus, at face value suggest that common ground had not been established or maintained. Note that all of the trouble spots we identified involved student-initiated utterances reflecting lack of understanding; we did not code instances in which teachers expressed uncertainty or lost their train of thought. After each trouble spot was identified, we identified the teacher turns preceding and following the trouble spot. Turns were typically defined by change of speaker. How-ever, in some cases, students made back-channel responses (e.g., ‘‘Oh’’ or ‘‘I get it’’) or spoke during the teacher’s turn. If the teacher did not cede the floor, these student responses were not coded as a change of turn. In cases where teachers held the floor for an extended series of utterances preceding or following the trouble spot, we defined the turn for our analysis purposes as the teacher’s talk on the same idea unit or topic (Brintoon and Fujiki 1989). In the turn preceding a trouble spot, teachers often initiated a new topic explicitly in speech (Hurtig 1977). For example, one teacher introduced a new topic by saying, ‘‘Ok, let’s take a look at this next one.’’ In the turn following a trouble spot, teachers often finalized the topic. For example, in response to a question about the difference between brackets and braces, one teacher said, ‘‘A bracket looks like this,’’ and drew a bracket, answering the question and finalizing the topic. In our analysis, we compared teachers’ use of gestures in turns that preceded and followed trouble spots. This insured that there was some degree of similarity both in the discourse context and in the content of the utterances being compared. To code gestures, the stream of manual activity was first segmented into individual gestures. Gestures were segmented from one another based on changes in hand shape, motion or placement of the hands. Each gesture was then classified into one of the following categories, based on the system developed by McNeill (1992): (1) pointing gestures, which indicate objects or locations, typically with an extended finger or hand (e.g., pointing to a number on the board to indicate that number); (2) representational gestures, which depict aspects of semantic content via hand-shape or motion trajectory, either literally (e.g., representing a line by tracing it in the air) or metaphorically (e.g., representing an ‘‘idea’’ as an object held in a cupped hand); (3) beat gestures, which are simple, up-and-down rhythmic movements that do not depict semantic content, but instead align with the prosody or discourse structure of speech.1 We also identified writing gestures, defined as writing or drawing actions that had an indexical or pointing function (e.g., underlining or circling something) and that were integrated with speech in the way that hand gestures typically are. Writing gestures could be easily distinguished from writing as a functional act (e.g., writing equations on the board). We suggest that all of these types of gestures may contribute to students’ comprehension of instructional material. Pointing and writing gestures serve to highlight particular elements of objects or inscriptions within the complex visual field (Goodwin 2007). For example, if a teacher points to or circles the 3 in x3 while saying the word ‘‘exponent’’, the gesture may facilitate comprehension for students who are uncertain as to what the term ‘‘exponent’’ refers. Representational gestures may help listeners to understand unfamiliar terms. For example, if a teacher produces a representational gesture that depicts a line with negative slope while talking about the idea of negative slope, students may be more likely to grasp the meaning of the term. Finally, beat gestures may draw listeners’ attention to the speaker. If a teacher uses beat gestures, students may be more likely to attend to the teacher, rather than to daydream or attend to something else. Thus, teachers’ gestures might foster shared understanding in a variety of ways.
2.3 Reliability of coding
To establish reliability in identifying trouble spots, a second coder reviewed one of the six lessons and identified trouble spots. Agreement between coders was 86 %. To establish reliability in coding gestures, a second coder recoded teachers’ gestures in the turns preceding and following eight trouble spots (13 % of the corpus). Agreement was 86 % for identifying individual gestures from the stream of manual behavior, and 79 % (N = 33) for classifying gestures as point, representational, beat, or writing gestures.