CGI Teachers' Patterns of Interacting with Male and Female Students during Their First and Second Years of Practice

This article reports on an exploratory study of the shifts of teachers' interactions with boys from their first to their second year of using cognitively guided instruction (CGI) in their math instruction. Interaction analysis was used to analyze 22 videos, two per each of 11 teachers, who applied CGI instruction in their classrooms (a) after participating in a CGI professional development program (year 1) and (b) after practicing CGI for one year (year 2). In the eleven year 1 videos, some teachers differentiated their attention to boys and girls based on two kinds of interactions: (a) during one-to-one interactions, teachers tended to distribute their attention more to boys as compared to girls; often, that attention was unevenly distributed among boys with some boys receiving more attention than others; and (b) during whole group interactions, teachers tended to ask more boys than girls to share their strategies. In the eleven year 2 videos, some noticeable shifts occurred: (a) during one-to-one interactions, teachers tended to distribute their attention to boys and girls and among students in more balanced patterns than during the previous year; and (b) during whole group interactions, teachers tended to balance how often they called on boys and girls to share. Microanalysis of selected episodes suggests that both shifts coincided with teachers' adapting their teaching to be more aligned with CGI principles of instruction, such as attending to students' mathematical thinking processes.

Keywords: Cognitively guided instruction; Mathematics education; Teacher-student interaction; Gender difference

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Mathematics teachers tend to differentiate their attention and interactions according to gender, a trend that holds from elementary to college grades (Bailey, [3]; Duffy et al., [15]; Jones & Dindia, [34]; Sadker & Sadker, [46]). Even when teachers and students perceive their interactions as gender-equal/fair, observations, and interviews show that there are differences in how teachers treat boys and girls (Lehman, [38]). The relevant literature mostly focuses on traditional teacher-centered environments, in which the teacher asks questions and calls students for a question-response interaction. Less studies focus on gender differences in the patterns of teacher-student interactions in student-centered environments, in which teacher undertakes the role of facilitator, such as Cognitively Guided Instruction (CGI). On the other hand, gender-based studies, focused on the effectiveness of CGI instructions, report mainly about the different strategies that boys and girls might employ, rather than on the teacher-student interactions (Fennema & Carpenter, [19]; Fennema et al., [23], [24]). Hyde and Jaffee ([33]) suggested that teachers hold gender stereotypes about strategies and that, somehow, those biases may be activated and transmitted during instruction, but few studies use video recordings of teacher-student interactions to explore teachers' patterns of attending to boys and girls in CGI instruction. This study aims to fill this gap. Because this is a relatively underexplored area, the current study is a small-scale exploratory study.

In this research, we explore the ways in which teachers who received two years of CGI professional development (PD) changed the pattern of their interactions with boys and girls between their first and second year.

The traditional model of teaching has been often criticized as being focused on the transmission of knowledge—what Freire has called the "banking model" of education (Freire & Macedo, [26]). In such teaching, the teacher adopts the role of the central authority of the classroom and "delivers" the concept to be learned to the students (Ramis-Conde & Hope, [43]).

Commonly encountered teacher-student interactions in traditional math classrooms is the classroom question-response interactions—in which, during instruction, the teacher is asking questions to the whole class and calls students to answer these questions. Teachers use the question-response interactions as a means of attending to students. In the context of teacher-centered classrooms (NCEE, [39]), attending to students means that the teacher checks if the "delivery" of the concept to be learned has been successful. In the literature mentioned in the following section, the term "attending" refers most often to the classroom question-response interactions initiated by the teacher.

Teachers paying more attention to boys than to girls is an old-age problem, as reported since at least as early as the 1970s (e.g. Serbin et al., [50]). The pattern of "over-attending" to boys continued for the next decades, and this pattern was similar for all grades and accentuated in specific subjects, such as math and science (Bailey, [3]; Duffy et al., [15]; Jones & Dindia, [34]; Sadker & Sadker, [46]). The overattention to boys is manifested through various ways. For example, several studies found that teachers tend to call more boys than girls during the question-response interactions (Altermatt et al., [2]). The tendency to attend more to boys than to girls is manifested also by implicit ways; for example, Fennema et al. ([21]) found that teachers had a more accurate knowledge about which boys rather than girls were successful in their classrooms.

Several scholars noticed, furthermore, that teachers' attention was not evenly distributed to all boys in the classroom, but that a few boys—sometimes one or two—received a considerable percentage of attention compared to other students (Bailey, [3]; Eccles, [16]). Some scholars attributed teachers' tendency to attend to boys as a reaction to boys' demand for attention—a demand that students expressed sometimes with disruptive behavior and sometimes with calling out answers (Sadker & Sadker, [46]). In any case, teachers tended to respond to boys' calling out answers positively, while at the same time, they reprimanded girls from calling out answers (Bailey, [3]). Other scholars attributed teachers' tendency to attend to boys solely to teachers' bias that boys are better mathematicians. For example, Altermatt et al. ([2]) and Duffy et al. ([15]) found that teachers' tendency to pay more attention to boys was independent of boys' volunteering rates to participate in the discussion. Fennema et al. ([21]) found that teachers tended to attribute boys' success to their ability and girls' success to their effort. Teachers' biases and tendency to attend more to boys may be evident from classroom observations even if surveys and/or interviews reveal that teachers and/or students are not conscious of the gender differences in their interactions (Bailey, [3]; Lehman, [38]).

Similar findings raise the concern that teachers hold conscious or unconscious biases and beliefs that might affect the ways they interact with their students (Copur-Gencturk et al., [14]; Staats, [54]). It is suggested that teachers' beliefs are mediated through instruction and hence play a role for gender differences in students' math performance (Felton-Koestler, [17]; Fennema & Peterson, [20]), and especially girls in elementary grades are susceptible to the beliefs and attitudes of their female teachers because they identify with them (Boaler, [6]). Teacher's gender-based behaviors send implicit messages to the students that may affect (a) students' beliefs about their mathematical identity, (b) students' participation and consequent performance in the classroom (Bailey, [3]; Lehman, [38]; Peterson et al., [42]; Staats, [54]), and (c) students' choices about STEM-related careers (Boaler, [5]).

Constructivism brought a different perspective about how learning happens, leading to models of teaching that are student-centered and dialogic in nature. In these learning environments the "banking model" is rejected, and teachers start from what learners know, in alignment with Freire's (1997) suggestions about teaching. When teachers start from what learners know, they abandon the idea that the teacher is the central authority in the classroom and undertake the role of a facilitator who pays attention to students' thinking.

CGI is a PD program that aims to provide teachers with knowledge about how students learn mathematics. CGI, in alignment with constructivism, claims that when students enter school, they already possess knowledge and strategies that can help them solve key problems without the need of explicit instruction from the teacher. Teachers in CGI programs learn research-based evidence about the development of students' mathematical thinking such as the development of understanding basic numerical concepts (Carpenter et al., [10]; Schoen et al., [47]). The goal and challenge are for teachers to start developing practices that will help them understand students' mathematical thinking as it happens in the context of the classroom interactions.

In the CGI PD program, teachers encounter (a) examples of the strategies and tools that elementary students may use to solve different types of mathematical problems, (b) examples of what teacher-students' interactions may look like in a CGI classroom, and (c) models of how students' strategies may evolve in time. But teachers' actual learning of how to develop and sustain the practices that help them understand and extent students' mathematical thinking happens during their practice, in the context of the classroom interactions. In other words, teachers learn about children's thinking while they attend to (listen and struggle to understand) their students' mathematical thinking (Carpenter & Franke, [8]). Therefore, in CGI classrooms, two kinds of learning happen: students learn how to solve problems; teachers learn about students' mathematical thinking and how to support this thinking in the context of their practice (Carpenter & Franke, [8]).

The learning that takes place within CGI classrooms can also be viewed through theories of situated learning, in which participants learn as they engage in activity (Lave & Wenger, [37]). As learning emerges in the activity, participants shift their roles: students start to think of themselves as "mathematicians," and teachers start to acknowledge that their students are competent problem-solvers and that they themselves (the teachers) are facilitators (Steinberg et al., [56]). For researchers who study such learning environments, it is important to look at how participants engage in the activity, how their roles shift, the role that their interactions play both for their learning and for their identities' shifts, what practices assist these processes, and what is the role of tools and artifacts (Carpenter & Franke, [8]). CGI challenges the preconceived beliefs that teachers hold about how learning happens, what the role of the student and of the teacher might be in the classroom, and what the nature of the teacher-student interactions might become (Carpenter & Franke, [8]; Steele, [55]). Some of these preconceived beliefs may be strong; therefore, the change is difficult and may need some time to happen (Carpenter et al., [11]; Steinberg et al., [56]; Swars Auslander et al., [59]; Swars et al., [58]).

Sharrock and Rubenstein ([51]) identify CGI teaching practices used in elementary classrooms; these practices can be applied in three distinct phases: initiation, mathematical inquiries, and whole group sharing.

Teacher introduces to the students a mathematical problem and makes sure that students understand the problem (e.g. teacher reads the problem, asks students to repeat the problem, asks students to explain the problem to a peer, asks general clarifying questions about the problem).

Students work independently (or sometimes in small groups) to solve the problem. Teacher makes sure that students understand (a) that they can solve the problem using whatever resources/tools they want (e.g. cubes, drawings, sentential and/or numerical representations); (b) that they are not expected to use an instructed strategy, but they are free to devise their own strategies; (c) that they will be expected to share their solution and strategy; therefore, they should keep track of the process they adopt (e.g. by writing down a description, using numerical representations, keeping their tools).

At this phase, the teacher attends to students' mathematical thinking by (a) observing their working process and (b) engaging in one-to-one interactions in which teacher asks the student how she solved the problem. By asking follow-up or more detailed questions, the teacher aims to genuinely understand the strategy that the student used so that to inform her (the teacher's) instructional decisions (Carpenter et al., [9]). In this phase it is important that the teacher does not impose preferred strategies on students.

When students are provided with time to devise their own strategies that are being valued, listened to, and respected by the teacher, they start seeing themselves as capable problem solvers (Carpenter et al., [9]). In the next phase, students will have the opportunity to share their strategies with their peers.

During the last phase of CGI, the teacher purposefully chooses some students to share their strategies and mathematical ideas with the rest of the classroom. The teacher selects from a variety of strategies and leads a discussion that will extend students' thinking by guiding them to compare different strategies and to make connections that will deepen students' understanding of mathematical ideas (Sharrock & Rubenstein, [51]). During the whole group interactions, students learn from each other's ideas and strengthening; as a result, their perceptions of themselves as capable math solvers.

CGI changes the ways that teachers interact with their students. The above description of the CGI phases highlights two teacher-student interactions whose nature is different from those of traditional teacher-centered math instruction. In the traditional question-response interaction, teacher selects a student to answer a question related to teacher's or textbook's strategy to solve a problem. Therefore, teacher's attention to student is related to predetermined and prescribed answers. In the CGI interactions, attention to student has different weight and value; namely, (a) in the one-to-one teacher-student interaction, teacher attends to student's process of mathematical thinking, a process that is not predetermined nor prescribed, and (b) in the whole group sharing interaction, the teacher selects students' strategies to guide comparisons and connections that will extend students' mathematical thinking.

As noted above, the CGI PD program is based on evidence about the development of students' mathematical thinking. In general, cognitive science researchers consider the patterns of children's evolution of mathematical thinking as universal, and they are rarely interested in studying gender differences in these patterns (Fennema, [18]). For example among divergent groups of students, the strategies for addition and subtraction evolve from counting strategies to more abstract ones, and this pattern is consistent in both one-digit and multi-digit problems (Carpenter, [7]; Nunes, [41]). Although gender differences are rarely studied, some CGI scholars have reported differences in how girls and boys solve problems. For example, longitudinal studies showed that girls in grades 1–3 prefer concrete (and observable) strategies, such as modeling and counting, while boys prefer more advanced strategies, such as mental counting, derived facts, and invented algorithms (Carr & Jessup, [13]; Fennema et al., [23], [24]).

Research on CGI teachers' perceptions about their students provides some interesting findings. Weisbeck ([63]) found that although teachers described their male and female students using similar characteristics, they reported that they thought more about boys during the instruction. This indicates a possible tendency to attend more to boys than to girls, in similar patterns found in studies in traditional teacher-centered math classrooms.

In summary, although there is some evidence about the differentiation of attention that teachers show to boys and girls, the relevant literature focuses on interactions in teacher-centered math classrooms. Less is known about the differentiation of attention (if any) in student-centered classrooms, in which the nature of teacher-student interactions is different. The claim that the nature of interactions differs is based on the following rationale: in teacher-centered math classrooms, the attention of the teacher is constrained to the end-result as "delivered" by the instructor; in student-centered math classrooms, the attention of the teacher is focused on the process of students' mathematical thinking.

On the other hand, gender-based research in CGI classrooms focuses mainly on the strategies that boys and girls use and rarely on how teachers differentiate their attention and interactions according to gender. Knowledge about these differentiations is important to break a perennial cycle of sending to students (girls) "messages" that affect how they see their role as "mathematicians." Therefore, there is a need for studies that explicitly explore if and how teachers differentiate their interaction patterns according to gender in CGI classrooms. Furthermore, because teachers need some time to learn how to use CGI practices, it is important to study the shifts in their patterns as they get more experienced in the CGI instruction. Therefore, the initial, exploratory, open-ended research question that guided this study was are there differences in the patterns in which teachers interact with male and female students after practicing CGI for 1 year?

The teacher sample comes from a 2-year study (2013–2015) that aimed at improving early elementary teachers' mathematics instruction through CGI professional development (PD; Schoen et al., [47]). The larger study used a multi-site cluster randomized control trial (RCT) design to select 11 elementary schools from two Florida public school districts. The PD workshops were spread across 2 years. For the purposes of this study, we used data from 11 teachers who participated in both year 1 and year 2 CGI PD. Teachers did not teach the same group of students in year 2 nor the same lesson. All teachers are White females. Eight taught first grade and three taught second grade. One teacher had less than five years of teaching experience; ten teachers had more than 5 years of teaching experience.

Teachers participated in 14 days of PD (8 days in 2013-–2014; 6 days in 2014–2015). The PD workshops were conducted by CGI leaders from the Teachers Development Group (TDG) under the leadership of xxx (line hidden for blind review purposes).

During the first year, the CGI PD focused on providing teachers with a conceptual model of student's thinking (Carpenter et al, [10]). Teachers also watched and discussed students' interview videos and problem-solving activities, observed the teaching of TDG facilitators, and conducted on-site cognitive interviews with students. The basic PD activities and structure of the second-year program were similar to the first year's while the content was more advanced. The PD materials mainly concentrated on addition, subtraction, multiplication, division, base ten concepts, and solving equations (see Schoen et al., [49] for more details on the PD materials).

The data consisted of 22 classroom videos of 11 teachers that participated in both year 1 and year 2 of the CGI PD program. Two video cameras were used: one camera followed the teacher of the class, and another was observing the whole class.

The current study uses interaction analysis for analyzing video data. Interaction analysis is a method of video analysis that includes ways for exploring the details in interactions (Jordan & Henderson, [35]). The methodology is appropriate for analyzing videos from a CGI classroom, because it is useful for studying interactions that display a moment-to-moment contingency—in which a person's action (verbal or nonverbal) depends on the preceding actions (verbal or non-verbal) of the other interlocutor(s). As previously mentioned, in the CGI interactions, the teacher's moment-to-moment actions depend on students' preceded actions, because teacher's purpose is to understand the student's thinking process and not to impose her own thinking as often happens in traditional lecture-type classrooms. The purpose in interaction analysis is to find recurrent patterns as they emerge in collective behaviors (Jordan & Henderson, [35]). These patterns are occurring in ways that participants are usually not conscious of, because people often do not realize to what cues they are responding to and what cues they are sending to others during interactions. Video recordings in interaction analysis allow researchers to observe how participants interact and communicate meaning through multiple channels such as speech, prosody, gesture, gaze, and body position, rewatch videos paying attention to different channels each time, consider their different interpretations, and rely on evidence from the videos rather than participants' explanations of their behavior. After watching the videos, the first author created content logs, including headings, notes, and/or summaries that listed the events as they occurred in the video recordings (Jordan & Henderson, [35]). Then, she watched the videos for each teacher and wrote memos and graphic visuals to compare teachers' interactions between year 1 and year 2. The memos included information on the questions that teachers asked in one-to-one interactions with girls and boys: how they responded to students' solutions and strategies; whether they asked them to use another strategy; if they asked questions to extend students' thinking; in what ways they used praise both verbally and nonverbally; if they showed respect to students' thinking or tried to impose their own thinking and/or preferred strategies (both verbally and nonverbally); whether they asked students to use more advanced strategies; and how they scaffolded students' thinking. The identification of boys and girls was based on students' appearance and names.

The visual graphics included a visual representation of the timeline of teacher-student interactions in year 1 and year 2, facilitating a quick comparison between years for each teacher (see Fig. 1 for an example). Approximately one third (n = 4 out of 11) of the visual graphics were validated by the first and second authors to support the trustworthiness of the visual graphics. The process of writing memos and visual graphics was an iterative process from which two themes emerged: (a) shifts in callings for sharing students' work and (b) shifts in one-to-one interactions.

Graph: Fig. 1Visual graphic representing the timeline of teacher's interactions with students

For the micro-analysis of specific episodes of interest, the two first authors watched selected videos multiple times and discussed their different perspectives and interpretations about the differentiations in interactions to reach consensus (Sidnell, [52]). The episodes for the micro-analysis were chosen because they were considered representative episodes for each theme. To enrich our perspectives, we watched the episodes in different ways (without sound, slower, or quicker). For the analysis, we attended not only to verbal, but also to nonverbal (e.g. gaze, gesture) ways of communication (Goodwin & Goodwin, [29]; Streeck, [57]).

Analysis of the visual graphics of the 22 videos showed: (a) teachers (n = 5) called more boys to share during year 1 but balanced how they called on boys and girls during year 2; (b) teachers (n = 4) balanced their callings for boys and girls in both years; (c) teachers (n = 2) did not shift their patterns and continued calling more boys to share in both years.

Closer analysis of the videos showed that the gender-balanced and imbalanced patterns related to whether teachers seemed purposeful or not about their selection of which students would share (Table 1). Cases in which teachers asked most boys (or only boys) to share their strategies with the rest of the class coincided with seemingly arbitrary choices, meaning that teachers showed no indication about why they chose one student to share and not another. On the other hand, cases in which teachers asked both boys and girls to share in balanced ways tended to coincide with seemingly purposeful choices, meaning that teachers seemed to choose a student to share because of the specific strategy the student used (indicated, e.g. by showing appreciation when the student was describing the strategy to the teacher in their previous one-to-one interaction, by explaining to the group why teachers selected a student to share, by linking the strategies of different students that were asked to share suggesting that teacher held a specific plan when she asked students to share, or by verbal statements during the sharing showing that the teacher remembered the strategy of the student she chose to share).

Table 1 The distribution of the cases according to teachers' patterns for callings and type of their choices

The following representative cases taken from the video corpus illustrate examples of seemingly purposeful and seemingly arbitrary choices. Micro-analysis of the selected episodes suggests that seemingly purposeful choices coincided with teacher adopting CGI principles (such as attending to student's mathematical thinking and considering student as a competent mathematics problem solver). The reverse stands for seemingly arbitrary choices, in which teacher seemed to prefer to impose her own strategies rather than attending to student's mathematical thinking.

In both years of the study, Anne asked students how they reached a solution and what tools they used. However, during year 1, it was not clear why she chose some students (boys) to share and not others (girls) who had used the same strategy. She also spent more time with boys in the one-to-one interactions. During year 2, when Anne encountered a student who used a novel strategy, she immediately recognized that and asked the student to share later, saying things like "I like your strategy, will you do me a favor to share?" or "can you go to the carpet?" (The carpet was the place students who would share went). This shift between the 2 years—Anne explicitly stating the reasons for choosing who to share in year 2—coincided with a shift in the percentages of boys and girls who would share.

Specifically, in year 1, Anne assigned two tasks to the students. For the first task, she asked three students to share (all boys). For the second task, she asked three students to share (two boys and one girl—one of the boys shared during both tasks). Considering the population of girls/boys in the classroom, the percentage of girls who shared was 11%, while the percentage of the boys who shared was 50%, and considering that one boy shared twice, we should note that 83% of the sharing interactions involved boys. In year 2, Anne assigned one task and she asked 6 students to share (three boys and three girls). Considering the population of girls/boys in the classroom, the percentage of girls who shared was 30%, while the percentage of boys who shared was 50%. This time, the 50% of the sharing interactions was with girls, and 50% of the sharing interactions was with boys. The following episode is representative of Anne's one-to-one interactions with students in year 2, in which Anne attended to a student's thinking and seemed to purposefully select who would share. Students were working on the following problem: Mrs. Eping has 136 ornaments. If she put 108 ornaments on her tree, how many more ornaments does Mrs. Epling have left to put on her tree?

• Teacher: What did you do Sonia? (Teacher leans forward to the student).
• Sonia: Ok, I took 8 from the 10 and 30 and left me 2, and I took the 20 left of the 22, and I counted the six ones and I got 28.
• Teacher: Oh my goodness. (Teacher smiles and sits down and closer to Sonia, with her head at the same level to Sonia's). Al right, say that again? Ok, I am gonna write down. (Teacher takes a pencil and positions her hand with the pencil in front of Sonia's paper). Tell me. Go ahead.

The first extract shows that Sonia's strategy captured the teacher's attention, which was conveyed both by verbal and nonverbal ways. Anne asked Sonia what she did and leaned closer to her, and when she heard student's response, she expressed her excitement by smiling and saying, "Oh my goodness." In the end of the first extract, Anne asked Sonia to tell her what she did, and she took a pencil to write down Sonia's step-by-step answer. This action might be a way for Anne to model to Sonia the use of numerical expressions. However, some of Anne's actions here and in the following extract show that Anne did not first understand what Sonia's strategy was, although she could see that it was right. Writing down in a step-by-step process seemed to be indicative of how Anne attended to Sonia's thinking process. In the next 85 s that separate the first and the second extract, Anne was writing down in numerical expressions what Sonia was saying, asking questions such as "How did you do that?" and "And what do that equal?".

36 Sonia: And then I add 2 to the 20, that equals 22.37 (Teacher writes "20 + 2," adds an arrow and writes "22"). Aha!38 Sonia: So, 22 I added 6 to 22.39 Teacher: Where did you get 6?40 Student: These ones (Student opens her palm and shows to the teacher the yellow cubes-that represent ones-that she was holding).41 Teacher: Oh, the ones that were on the 36. (Teacher writes down). Ok, plus six. Now I totally understand what you did Sonia. Aha! (Anne smiles).

Anne's open-ended questions are indicative of her adopting the CGI principle that teachers should attend to student's mathematical thinking. Her last statement "Now I totally understand what you did Sonia" confirms that this whole time Anne was trying to understand Sonia's thinking process. Writing down the process seemed to help Anne in understanding it.

In the next minutes, Anne kept writing down what Sonia was explaining to her, and, in the end, teacher said: "And that's totally right? Awesome. Will you bring your paper and the tools that you hold on your hand, cause I like the way you still have those six on your hands, you do not forget about them." By asking Sonia to bring her paper and tools to the carpet, Anne conveys her intention to ask Sonia to share her strategy during the whole group sharing. The moment-to-moment interaction with Sonia shows that this decision seemed to be purposeful and based on the strategy that Sonia used—a strategy that Anne showed interest and excitement about, indicated both by verbal and nonverbal actions (such as smiling and looking at Sonia in excitement). Anne's suggestion to Sonia to bring also her tools suggests that Anne intended to ask Sonia to share with the class how she used these tools to solve the problem (something confirmed later during the group sharing).

In year 1, Martha assigned 3 tasks to the students and asked 7 out of 7 boys to share (100%) and 2 out of 8 girls (25%) to share. Considering that one boy shared twice, 80% of the sharing interactions in the group was with boys, and 20% was with girls. In year 2, Martha assigned 4 tasks to the students and asked 3 out of 7 boys to share (42.85%) and 4 out of 7 girls to share (57.14%). In year 2, 42.85% of the sharing interactions was with boys, and 57.14% of the sharing interactions was with girls.

The following episode from year 1 was selected as an example of seemingly arbitrary choice and is representative of teacher's tendency to impose her own strategy to students rather than attending to their own thinking process, suggesting that teaching arbitrary choices for sharing coincided with a failure to adapt to the CGI principles, attending to student's thinking process, and/or seeing students as competent mathematics problem solvers. In this episode, Martha asked Andrian to get up to share, but there was no indication why she chose this student and not another, since she ended up guiding him how to solve the problem in a step-by-step process. Students have been working on the following problem: Andy has 82 books. If he donates 47 books to the library, how many books does Andy have?

32 Teacher: What are you going to build first? Wherever we are starting?33 Andrian: Eighty—34 Teacher: Eighty what baby?35 Andrian: 82.36 Teacher: Ok, build 82, that's what he is doing. I suggest all of you do this at the same time with your cubes. Everyone do this at the same time please.37 (The teacher goes to another student and looks at him). Teacher: Honey, I am asking you to do the same thing, I am asking you to build 82.

Notice that Martha did not ask Andrian to share his strategy of solving the problem, although this is the purpose of the whole group sharing in CGI. Already from turn 32, the teacher directed Andrian's mathematical thinking process by asking: "What are you going to build first?" The word build was already indicative of what the teacher expected Andrian to do. Martha asked all students to follow the same procedure. In turn 36 she said to students: "I suggest all of you do this at the same time with your cubes. Everyone do this at the same time please." Following this instruction, in turn 38, she repeated to a student that she was supposed to do the same thing with Andrian. Martha's expectation that all students should follow the same procedure continued during the whole sharing interaction.

41 (Andrian puts cubes ("ones") on the paper that is projected to the class. He puts one "one" on the left of the "tens" and another "one" on the right of the "tens").42 Teacher: Honey, put your ones together, ok?43 (Andrian moves one of the "ones" at the same side with the second one).44 Teacher: Ok. This is 82. And what do you have to take away from 82? (0.2) Everybody? It tells you!45 Students' voices heard together: 47.46 Teacher: 47, can we take away forty s—47 (Andrian takes one "ten" away from the paper).48 Teacher: Hold on honey hold on hold on.49 (Andrian puts back on the paper the "ten" he took away).50 Teacher: Can we take away 47 from this?51 Students in slow voice: No.52 Teacher: NO, so we don't have seven one's to take away, do we?

Martha not only asked students to follow her own thinking process as in typical teacher-centered classrooms, but, at times, she seemed to want students to follow exactly what she wanted. For example, in turn 42, she asked Andrian "Honey, put your ones together," following the suggestion with a quick, "ok?" Five turns later, when Andrian took away one "ten" (Turn 47), she said "Hold on honey hold on hold on" (Turn 48). The urge hold on repeated three times was indicative of her tendency to also set the pace. Indeed, Martha's tone and pace revealed a rush to get the answers. This was indicative in various interruptions during the whole episode. For example, in turn 34, she interrupted Andrian's answer (eighty—) and asked "Eighty what baby?").

Martha showed lack of attention to Andrian's and the other students' mathematical thinking, in contrast to CGI principles. This lack of attention was demonstrated in various ways: (a) the teacher did not give much time to student(s) to respond to her questions, and sometimes she interrupted them, (b) she asked all students to follow the same procedure and to use the same tools, (c) when students answered wrong, she quickly corrected them without asking questions to understand their thinking, (d) teacher directed students in a step by step solving process, without permeating them any autonomy in their thinking, even in simple actions, such as how to place the tools on the paper.

Hence Martha's seemingly arbitrary choices coincided with her inattention to students' thinking process, while purposeful choices in the case of Anne coincided with attending to students' thinking processes. Martha in year 1 seemed to prefer (and impose) her own mathematical thinking and did not leave to the students much space for practicing their own problem-solving methods, in contrast with Anne who showed an appreciation and excitement for students' mathematical thinking and problem-solving methods.

In terms of choosing with whom to interact in the one-to-one interactions, the analysis of the visual maps did not reveal clear patterns. For the purposes of this study, the most-interesting cases (n = 4) include teachers who shifted their distribution of attention among students across the two years. These teachers went only to some students—with a preference to specific boys and one of these teachers went only to boys in year 1. But in year 2, the same teachers went to more students, and this resulted in including equal/balanced numbers of boys and girls in their one-to-one interactions. Micro-analysis of the four cases showed that this shift coincided with teachers' tendency to attend more to students' mathematical thinking in year 2, than those in year 1 (as explained in the following example). On the other hand, some clear cases (n = 3) of teachers who went to most of the students in the classroom in both years (one teacher went to all students and two to all but two), coincided with teachers' tendency to attend highly to students' mathematical thinking in both years.

In the previous section, we demonstrated how a whole group interaction unfolded in Martha's class during year 1. The interaction revealed Martha's tendency to impose her own thinking instead of attending to students' thinking. Her tendency—not to attend to students' mathematical process—was also observed in one-to-one interactions in year 1. In year 2, Martha seemed to attend to students' mathematical thinking more than she did in year 1. In contrast to her actions in year 1, she did not impose a preferred strategy; instead, she seemed to respect students' pace of work; she did not correct them when they made mistakes, but rather she asked them to check their work or to check what the problem was asking. Martha's increased attendance to students' thinking coincided with a higher distribution of her attention to the class in year 2. During year 1, Martha interacted with all the boys (100% of the boys' population); however, she did not interact with four out of 8 girls (50% of the girls' population). Not only was the distribution of her attention unequal between boys and girls, but furthermore, it was unequally distributed among boys. (With two of the boys she interacted four times and with two other boys she interacted three times). During year 2, Martha interacted with 6 out of 7 girls and with 8 out of 10 boys, and she did not interact with any student more than twice.

The following episode is representative of the one-to-one interactions Martha had during year 2, and it illustrates Martha's tendency to attend to students' thinking. Students were working on the following problem: Charlie has 5 bags with 8 legos in each bag. How many legos does he have altogether?

50 Teacher: What are you doing body?51 (Teacher leans to Nick's desk).52 (Nick is pointing to the teacher the five boxes he has drawn on the paper).53 Teacher: How many boxes?54 Nick: 555 Teacher: Ok... And then what did you do?56 Nick: I tried to figure out how many legos to put.57 (Nick points with his pencil at the dots inside the boxes—he has already drawn 8 dots inside the first three boxes).58 Teacher: Does it tell you how many legos?59 Nick: Eight. So, I am doing the 8.60 Nick continues putting dots in the other boxes.61 Teacher: Ok, so you put 8 in each box, ok.

Martha started by asking Nick what he did, and she leaned towards him. Although it was evident by his work on the paper that he had drawn 5 boxes, Martha explicitly asked him "How many boxes?" inviting him to say more about what he did. Getting a one-word response, Martha continued asking "And then what did you do?" (turn 55). To Nick's answer "And then I tried to figure out how many legos to put," Martha continued with a question "Does it tell you how many legos?" By "it," Martha was referring to the problem. With her open-ended question, Martha avoided something that she did during year 1 (when she explicitly was telling students what they should do). In the next eight turns, Martha waited for Nick to count his dots on the paper, showing a patience that she did not demonstrate in year 1, when she did not give enough time to students to answer her questions and she was rather imposing her own (quick) pace to students' work. After counting, Nick wrote a wrong answer to the paper (i.e. 38).

79 Teacher: Ok, how did you get that?80 Nick: I counted.81 Teacher: Can I hear you? Can I hear you do it?82 Nick: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18 (Nick points as he counts with the back of his pencil at the dots of his drawing).83 Nick: 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39.84 (Nick stops counting aloud and continues pointing at the dots as if counting).85 (Nick erases the number 38 and writes 48 on his paper).86 Teacher: See how you checked your work? And when you checked your work what did you get?

Martha showed again a different behavior than the previous year, when she did not correct Nick, but asked "Ok, how did you get that?" To Nick response I counted, Martha continued demonstrating a willingness to follow his thought, by not correcting him again, but instead asking him "Can I hear you? Can I hear you do it?" Martha let Nick count again without interrupting him, and only when Nick wrote the correct answer, she told him "See how you checked your work? And when you checked your work what did you get?".

The purpose of this exploratory study was to explore the ways in which teachers who had experienced CGI PD changed their interactions with boys and girls from their first to their second year of using CGI in their math instruction. Findings indicate noticeable shifts in two levels of teacher-student interactions: (a) one-to-one and (b) whole group interactions.

In both kinds of interactions, some teachers in year 1 over-attended to boys while others balanced their attention to boys and girls—no noticeable over-attention to girls was observed. During year 2, fewer teachers over-attended to boys. Before discussing the reason for this shift, let's first look into how over-attention to boys was manifested through the different interactions. In one-to-one interactions, teachers chose to interact with more boys than girls; in whole group interactions, teachers asked more boys than girls to share their work. These results align with previous (older) studies in teacher-centered classrooms, in which teachers showed over-attention to boys in comparison to girls (Bailey, [3]; Duffy et al., [15]; Jones & Dindia, [34]; Sadker & Sadker, [46]). The current study also confirmed previous findings that some teachers distribute their attention to a few students (boys; Bailey, [3]; Eccles, [16]). The finding that some teachers over-attend to boys is interesting, considering how long ago the previous studies took place. Furthermore, previous studies were conducted in teacher-centered classrooms, in which the nature of attending to students is different than the nature of attending to students in student-centered classrooms, as in the current study.

It has been previously suggested that teachers' beliefs, previous experiences, and/or biases influence their tendency to over-attend to boys (Altermatt et al., [2]; Duffy et al., [15]). The current study indicates that these beliefs, previous experiences and/or biases affect teacher-student interactions regardless the nature of these interactions. Yet, some questions emerge: Why did some teachers shift their patterns in attending to boys and girls in year 2, and/or why did some teachers, as early as year 1, seem to balance their interactions between boys and girls? Were the shifts coincidental, or were there other factors and forces at play? Our analysis of the video corpus suggests that other factors might be at play, namely, the knowledge that teachers had obtained using CGI and adhering to its instructional principles.

In the examined cases (n = 11, 22 videos), teachers' purposeful choices about who would share tended to coincide with balanced callings between boys and girls. On the other hand, seemingly arbitrary choices tended to coincide with teachers asking more boys than girls to share. It is possible that arbitrary callings might perpetuate (a) biases (e.g. boys are better in math than girls; Copur-Gencturk et al., [14]; Hand et al., [32]; Vander Heyden et al., [60]) or (b) management techniques (e.g. boys should share so that they do not misbehave; Glock & Kleen, [28]). Regardless the reasons for this imbalance, this study argues that when teachers adopted CGI principles, such as attending to students' thinking, they reflected on the reasons why they ask specific students to share. This reduces the arbitrary choices and automatically leads to more equitable choices, at least in terms of gender. This is not surprising, if we consider that individual differences are natural in any group of students; therefore, when students are encouraged to solve problems using their own strategies, they will manifest observable variations in strategies that are (probably) distributed equally within boys and girls. Additionally, in the examined cases, teachers tended to distribute their attention (almost) equally between boys and girls in the one-to-one interactions when they showed evidence that they were actively attending to students' mathematical thinking. Teachers not only asked "tell me what you did," but they followed up with questions, respected students' pace and thought processes, and refrained from imposing their preferred strategies. Therefore, a hypothesis that emerges from this study is that the adoption of the CGI principle that teachers should attend to students' thinking shifts teachers' distribution of attention to more gender-equitable ways.

In CGI, teachers change their belief that they should adopt an authoritative role in the classroom. The research literature suggests that this is a challenging shift (Kinser-Traut & Turner, [36]; Ryder, [45]). Furthermore, in CGI, teachers shift their previously held beliefs that students should follow an instructed procedure for solving problems and start to acknowledge students as competent mathematicians who use their own strategies and processes for problem solving (Carpenter et al., [11]; Schoen et al., [47]). Fennema et al. ([22]) conducted a 3-year longitudinal study and found that different teachers shift these beliefs at different rates and reach the desirable level of adopting CGI principles at different points in time. Other case studies that followed, confirmed that adopting CGI principles requires some time for some teachers (Carpenter et al., [12]; Franke et al., [25]; Steinberg et al., [56]). Indeed, although CGI PD programs inform teachers about the practices that will help them understand students' mathematical thinking, teachers' actual learning happens during their practice, in the context of their classroom interactions (Carpenter & Franke, [8]). Shifts in teachers' practices, such as asking generic questions to promote students' autonomous thinking, require time and undergo stages (Gal, [27]). According to theories of situated learning, when participants shift their roles (e.g. students start to think of themselves as "mathematicians," and teachers start to acknowledge their students as competent problem-solvers), their new role emerges gradually as they engage in the activities (Lave & Wenger, [37]; Weinberg et al., [62]). Thus, in the current study, some teachers needed more time than others to adopt the CGI principles, and this shift coincided with equitable shifts in both types of teacher-student interactions. These results align with literature suggesting that constructive practices of teaching are consistent with inclusive practices (Al-Shammari et al., [1]; Grier-Reed & Williams-Wengerd, [30]).

It is important to note some limitations of the current study. The study is small in nature; therefore, the findings have limited generalizability, although thematic generalizability is possible. The sample of the teachers/participants is small. Additionally, there was a restricted number of video observations per teacher. Specifically, the findings of the study were based on one classroom observation per year for 11 teachers that participated. Regarding the first limitation, it is common for exploratory studies to use a relatively small sample, so that bigger samples can be used by future studies after the development of more refined research designs, which are based on the findings from the exploratory studies. Regarding the second limitation, it should be noted that interaction analysis assumes that the interactions that people initiate are following consistent patterns (Hall & Stevens, [31]). Moreover, research suggests that teachers initiate interactions with students in consistent ways as to how teachers distribute their attention to students (e.g. Read, [44], reviewed studies in Beaman et al., [4]).

Despite its limitations, the study's findings serve as a starting point for designing future studies to test the tentative hypothesis that adoption of CGI principles could lead to more equitable ways of teaching in terms of gender. Future studies should further investigate the relation between teachers' level of adoption of CGI principles with the equitable shifts in their instruction. Future mixed-methods research designs can investigate the reasons for teachers' differentiations and shifts in their interactions with students and the reasons for their differentiations in the level of adoption of CGI principles (e.g. teachers' teaching experience, beliefs, biases, personal commitment, attitudes). Interviews triangulated with observations in future qualitative descriptive research designs can shed more light to teachers' perceptions about their interactions with students and to the classroom events that might trigger teachers' adoption of CGI principles during practice. It also remains to be explored whether CGI PD might lead to more equitable instruction also regarding race and ethnicity.

We would like to acknowledge the contribution of Robert Schoen (PI and Director), Amanda Tazazz, and Kristopher Childs for videotaping the corpus videos used in this study and to thank them for their overall support. Uma Gadge, Naomi Iuhasz, and Qiuqing Zhang have coded the videos; some of these codes helped as anchors for the construction of the visual graphics described in this study. Also, many thanks to Changzhao Wang and Lei Sun for their feedback on the study while the data analyses were being carried out. We are grateful to the three reviewers who dedicated their time and effort towards valuable suggestions, helping us to improve the quality of this manuscript. We'd like to dedicate this study to Elizabeth Fennema upon whose pioneering work on gender differences in mathematics and as a co-creator of CGI this study rests.

The research reported here was supported by the Institute for Education Sciences through Award Numbers R305A120781 and R305A180429 and by the United States Department of Education through Award Number U423A180115, all to Florida State University.

All data and materials support our published claims and comply with the field's standards.

Not applicable.

This work was conducted under the oversight of the University of Miami IRB through protocol numbers 20120677, 20181099 and the Florida State University IRB through protocol numbers 2018.23852 and 2019.27476. Informed consent was obtained from all teacher participants included in the study.

The authors declare no competing interests.