Changes in teachers’ video analyses in the video-club context
Analysis of the teachers’ comments from the early to late meetings reveal that the teachers shifted to notice and interpret student mathematical thinking (see Table 3). They increased in the percentage of comments they made about the students and mathematical thinking. They also came to more frequently interpret the events viewed, and their comments became more specific and more grounded in the events they viewed in the video segments.
Table 3 Teachers’ overall analytic focus in second and final video-club meetings
Using the z-test to examine differences in the percentages of comments made by dependent samples reveals that in all five areas in which we hypothesized there would be an increase, there was a statistically significant difference at the .05 level. The two-tailed z-test statistic for the difference between the teachers’ comments in the early and late meetings on Student was 3.17; on Mathematical Thinking, the two-tailed z-test statistic was 3.0; on Interpret, 6.0; on Specificity, 2.3; and on Video-focus, 5.6. Thus, in the video-club context, the teachers shifted to analyzing students’ mathematical thinking in detailed ways based on the events in the video clips they viewed.
The following examples illustrate this shift. In one of the early meetings, the group viewed a segment from Elena’s class in which students shared solutions at the board for problems related to addition and subtraction of decimals. Upon viewing the segment, the facilitator asked what they found interesting in the clip:
- Yvette::
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I like that they were using dry erase [boards]. I think that makes it more interesting… It was kind of fun maybe [for kids] to do while the teacher was doing it, maybe more motivational.
- Frances::
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I noticed they all went from right to left. I know this particular [curriculum] series doesn’t teach them to go from right to left. Do you teach them to go from right to left?
- Elena::
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Some of them add the hundreds, then the tens, then the ones, and then we have some that just automatically go from right to left.
- Frances::
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Well it’s the conventional method. I’ve had parents come to me and say, “I don’t want them to learn that way because I can’t help them at all, because I don’t know what they’re doing.” We were over at that meeting at [the middle school] and the [teachers] told us the same thing. They don’t want them learning these other methods.
- Yvette::
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I think a child has to be strong in at least one of [the methods]. So, I think an introduction is good, but if you’re over frustrating them, I think one strategy is important.
- Daniel::
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My students are like, “Why are we doing this step if we have to eventually go back and do it the other way?”
- Frances::
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That was my next question, how do you feel about teaching both ways? Do you teach both then, or do you stick to just the conventional way?
This exchange represents the kind of discussions the teachers had early in the series of meetings. Yvette’s initial response focused on a classroom climate issue, how the use of dry erase boards motivates students. Frances then inquired about what methods the teachers use to teach addition and subtraction, whether they use what the teachers refer to as the conventional method or partial sums. After Elena responded that students use both approaches, Frances, Yvette, and Daniel shared their experiences teaching partial sums, concluding with Frances asking the group again what method they teach. Thus, we see here that they discussed a range of topics, with a primary focus on pedagogy, particularly as it relates to the reform curriculum, and classroom climate. They commented on both the students and themselves as teachers, and their comments were mostly descriptive and evaluative and both general and specific, drawing on some events they viewed in the clip, but with a greater focus on what occurs in their own classrooms.
Later in the series of meetings, though, the teachers initiated and sustained a detailed focus on examining and interpreting students’ mathematical thinking based on the events in the clip. For example, the teachers viewed a clip from Daniel’s classroom in which a student, Maria, solved the problem 0.2 × 8.0 at the board. Her method involved calculating each of the partial products. She sometimes treated 8.0 as 80 and other times as 8.0, yet she consistently recorded each answer with two digits to the right of the decimal point.
While viewing the clip, Daniel asked the facilitator to stop the videotape and then turned to a colleague and said, “You’re nodding, like yes.” The teacher, Wanda, along with several other members of the group, proceeded to interpret the method the student used to solve the problem.
- Wanda::
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She’s doing it a little early. She’s [recording the decimal point] at each one down, but when you multiply decimals you count the numbers to the right of the decimal and then you count over that amount in your answer. She’s just doing it each time.
- Frances::
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She doesn’t have to do that until the very end. She’s just doing it each time…
- Daniel::
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That’s the way [the curriculum designers] are trying to keep them away from counting over for decimals, but I guess, yeah, okay…
- Elena::
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But it kind of shows that she’s understanding the whole part about decimals and moving over.
- Daniel::
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I was really trying to figure out, ‘cause I’m like, she’s got the right answer, she knows something here.
- Wanda::
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It’s partial products.
- Facilitator::
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…but if she’s doing 8 times 0.2, there’s only one number behind the decimal point, so why is she saying there’s two?
- Frances::
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…She’s not multiplying 8 times 2.
- Daniel::
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Eighty…
- Facilitator::
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She’s doing 80 times 0.2? She can’t say that her answer is both 16 and that she has to have two numbers to the right of the decimal.
This excerpt is rather different from the discussion early in the series of meetings. Here, the teachers seek to understand and explain Maria’s method, using details from the video clip to interpret what they think she understands.
While we were encouraged to observe that the teachers came to interpret particular student mathematical ideas in detailed ways, we were also interested in investigating to what extent this may have influenced them outside of the video-club context.
Teachers’ self-reports of changes in thinking and practice
Analysis of the exit interviews reveals three ways the teachers perceived they were influenced by participating in the video club. Specifically, they claimed to have learned the importance of attending to student thinking, to attend to student thinking during instruction, and about their school’s mathematics curriculum.
Learning about the importance of attending to student thinking
All seven teachers reported learning about the importance of listening to student thinking as a result of participating in the video club. Throughout the exit interviews, teachers made numerous comments indicating that the video club helped them recognize that students can have quite interesting mathematical ideas. For example, Daniel remarked that the video club “made me conscious that there are so many times when [the students]…may be thinking in a way that I’ve never thought of.” Similarly, Linda commented that in the video club she found herself looking closely at students’ ideas and realizing “they really do work through a lot of things.” Along the same lines, Yvette stated that “It was interesting to hear [students’] thoughts…You know,…what was she thinking about, what was he thinking about, where are they coming from…It was thought provoking and interesting.” These comments are reminiscent of what Cohen (2004, p. 46) refers to as teachers coming to understand that students are “havers” of mathematical ideas.
The teachers also indicated that the video club helped them to recognize that it was valuable for them as teachers to pay attention to student ideas. Elena commented, “Listening to the video is kind of like ‘Okay!’ If you listen a couple of times you just hear more…[The student’s idea] makes a lot more sense.” Another teacher, Frances, stated the following:
[In class] a child gives an answer and you’ve got to go onto the next [problem], … you watch the clock… [But in the video club] it’s really focused on the kids [and] if you understood what she was saying. You could really understand if she got this or not …[even if] she didn’t spit it out the way you wanted her to [or] didn’t use the terminology…. But, if you really stop and think, ‘Yeah, she did get it…she was just saying it a different way or she got part of it.’
Frances emphasized the need to think carefully about students’ ideas and the ways in which reflecting on video can support this process. Similarly, Drew explained, “Well again, it goes back to…listening to the children give their remarks… That really hit home on the videos. I know, for myself…I want to do more of that.” Research suggests that there is often a disconnect between teachers’ practices and their beliefs (e.g., Forgasz and Leder 2008). In this case, however, we found that both the teachers’ self-reports and our own analysis of the video-club discussions indicate participants’ increased awareness of student thinking.
Attending to student thinking while teaching
Six teachers commented that they believed that as a result of participating in the video club they paid closer attention to students’ ideas while teaching. For instance, Drew, a first-year teacher, said, “I notice different things that they don’t understand and I have them talk through it.” Drew further explained that he found himself adapting his teaching based on the information students provided about their mathematical understandings and any difficulties that he observed. Elena also remarked that she shifted her instruction. “[I] look to have kids really explain more of how they’re thinking…” Elena also believed that she would often consider how to support learning in her classroom based on what she came to know about students’ understanding from their explanations. Another teacher, Frances, explained, “I think [I’ve come to] slow down and really listen to kids. …[I try to] focus and really listen to kids and find out ‘What exactly do you mean by this?’” Similarly, Linda asserted that she had become “more patient” during class and “probably step[s] back a little bit more” to take in the ideas students are sharing about the mathematics being discussed in class. She also described holding herself back from giving the answer in order to let students work through problems on their own.
Thus, not only did the teachers claim that they learned about the importance of student thinking in the video-club meetings, but these six teachers believed that they also increasingly paid attention to student ideas while teaching. This result is particularly noteworthy given that the video-club discussions were not focused explicitly on helping teachers learn to attend to student thinking during instruction. For the teachers, analyzing student thinking via video seems to have had a strong connection to analyzing student thinking in their classrooms, similar to Schön’s (1983) claims about the potential of reflection-on-action to influence reflection-in-action.
Learning about curriculum
All seven teachers reported that they learned about their school’s mathematics curriculum as they participated in the video club. They explained that having opportunities to view lessons from other teachers’ classrooms and from other grade levels gave them valuable insights into the details of lessons and the broad goals of the curriculum. Recall that the school as a whole was in the third year of using a reform-based mathematics curriculum, though this was a new endeavor for several participants. They expressed concerns about not having been familiar with the overall design and goals of the curriculum, and a few mentioned that they had felt insufficiently prepared to use the materials with their students.
For example, when asked what he thought was the most valuable part of the video club, Daniel, responded:
Just seeing… oh, [the fifth graders are] doing fractions there too! Not that I didn’t know that before, but just to be able to see that and how it’s going on and hear the teachers talking about what their expectations are, what [the students] need to know and how they go about teaching their math. And [seeing] oh, they use this key word, or this phrase, or this vocabulary… We just don’t have enough time in the day to talk about these things. I should know at least all the chapter titles and all the concepts they do in fifth grade, but nobody gives you any time to do that… So, that was valuable.
Being provided the time and space to view their colleagues’ curriculum implementation was a common sentiment across all teachers. Another teacher, Frances, emphasized this point when she said, “Especially, for me, having taught third [grade] and gone to fifth, it was interesting to see how what I taught in third grade really related to fourth and what they’re teaching at fourth, how that has helped me in fifth.” Again, the video club was not designed to support learning about curriculum, but the teachers reported that viewing video of one another’s teaching helped them accomplish this goal.
Changes in teachers’ instruction
The analysis of teachers’ classroom practice identified three changes in teachers’ instruction over time. Specifically, teachers increasingly made space for students’ thinking to become public in the classroom, they more frequently probed students’ thinking, and they took on the stance of learner in the context of teaching.
Making space for student thinking
The first shift we observed is the teachers more often made space for student thinking to emerge in the classroom. They did so in three ways: publicly recognizing students have ideas to contribute, providing extended opportunities for student thinking, and eliciting multiple methods or solutions (see Table 4).
Table 4 Characteristics of making space for student thinking
Publicly recognizing unsolicited student ideas refers to the teacher indicating that a student has a question or idea to contribute to the lesson. For example, late in the year, Yvette led a discussion on converting percents to decimals. She wrote on the overhead several percents and asked the students to represent them in decimal form. She then asked the students to share their answers to each problem. After students provided their answers, Yvette saw that a student had his hand raised, and she turned to him and said, “What’s your question, Steven?” inviting him to participate in the discussion. In another observation late in the year, Yvette noticed a student’s hand raised and she turned to the class and said, “Max has a thought.” The important point here is that she recognized that these students wanted to contribute and that she made space for unsolicited questions to become part of the classroom discourse.
This practice was not a common feature of the teachers’ early classroom observations, occurring in only 24% of the 125 2-min segments coded. Later in the year, the teachers recognized unsolicited ideas in 68% of them 109 2-min segments that were coded. This shift is significant at the .05 level (two-tailed z-test statistic = 4.7).
Teachers also provided extended opportunities for students to work through the mathematics. One way they did this was to encourage students to take more time to work through their ideas, making comments like, “Let’s give some people some time to think” and “I’m not going to tell you. You have to figure it out.” A second approach was to allow students multiple opportunities to express their thinking. For example, when students made an error solving a problem, the teachers allowed them time to rework the problem and then share their revised thinking. This practice indicates that the teachers were creating an environment where students could generate and rework their ideas in order to develop their thinking, both important goals of mathematics education reform (Lampert 2003; National Council of Teachers of Mathematics [NCTM] 2000). In the early observations, we observed this practice in 31% of the total 2-min segments, as compared to 71% of the total coded segments in the late observations. This shift is significant at the .05 level (two-tailed z-test statistic = 5.2).
Finally, the teachers elicited multiple methods or solutions from the students. We observed the teachers posing tasks in ways that prompted students to generate their own solution strategies and then inviting students to the board to share how they solved a problem and explain their solutions. In an observation of Linda, she placed a grid with 100 cubes on the overhead projector and asked the students to come up with a way to divide it into 20 equal parts. She then invited three students to draw on the grid to illustrate their approach. Similarly, when we observed Drew conducting a chapter review, he posed several problems and then invited two or three students to the board to share and explain how they solved each. Across the teachers, we saw evidence of this practice in 20% of the total 2-min segments in the early observations compared to 59% of the total segments in observations late in the year. Again, this shift is significant at the .05 level (two-tailed z-test statistic = 6.3).
Probing student thinking
The second area that we examined concerns teachers’ probing of student thinking. We observed that the teachers probed student thinking in two ways, asking for explanations and probing students’ explanations. The first approach refers to teachers asking follow-up questions when students provide an answer without an explanation. The second concerns teachers probing students’ explanations to gain deeper insight into their thinking.
For example, in Elena’s classroom, she posed the following problem: there are 8 boxes of cupcakes. There are 6 cupcakes per box. How many cupcakes are there in all? One student, Angelina, responded that the answer is 14 and Elena responded, “14? Ok, how did you come up with that number, Angelina?” After she explained her approach, Elena asked Peter for the answer. Following his response, “48”, the teacher said, “Okay, 48. How did you get that?” These two exchanges show Elena asking for an explanation for the students’ answers, with a particular focus on how they arrived at their answers. The teachers also asked for explanations related to students’ reasoning, posing questions like, “Why do you think that?” or “You said something about 20, what do you mean?” In the early observations, the teachers asked students to provide explanations for an answer in 36% of the total number of 2-min segments, compared to 82% of the total number of segments late in the year. This shift is significant at the .05 level (two-tailed z-test statistic = 4.7).
We also observed teachers probing students’ explanations to gain deeper insight into their thinking. Consider the example from Daniel’s classroom when he taught a lesson on multiplying decimals. Maria explained how she solved the problem 8.0 × 2.0. She essentially approached the problem using the partial products method, but her explanation was unclear. Daniel prompted her to explain her solution, and as she talked through her approach the second time, Daniel interrupted her and asked her: “What do you mean, since there’s two things behind the line?”; “You said, ‘Two over’. What do you mean ‘two over’?”; “Where is that coming from? What is our cue to do that?”; “So, are you doing partial products as if there’s no decimal point in there?” These questions further probed the students’ explanation to understand her mathematical reasoning. We observed a shift in teachers’ practice in this dimension as well. Early on, teachers probed student explanations in 5% of the total two-minute segments; whereas, late in the year, they probed explanations in 48% of total number of segments. This shift is significant at the .05 level (two-tailed z-test statistic = 7.2).
Learning while teaching
Finally, the third shift is characterized as learning while teaching. We defined this practice as teachers viewing themselves as mathematics learners in the context of the classroom. We observed teachers adopting a contemplative stance, expressing uncertainty in the act of teaching, pausing in the midst of instruction to consider a student idea, or working through the mathematics during instruction. We also observed the teachers verbalize their confusion during instruction, making comments like, “Hmmm… I’m a little confused. Let me think about this” or “I hadn’t thought of that before.” The teachers also paused during instruction to make sense of an idea about which they appeared confused. In the case of Daniel trying to understand Maria’s strategy, he made statements like: “I’d like to know how that came out but I’m not sure I followed it” and “Now, I’m interested. I’ve never heard of this. Is this just a coincidence that it works?” Furthermore, we observed teachers pausing in the midst of instruction to solve a problem on their own. In one observation of Wanda, for example, the students were working in groups to solve probability problems using factor trees. As they worked together, Wanda solved a problem on the board, checked her answer against her curriculum guide, and then re-solved the problem before pulling the groups back to a whole-class discussion. These teacher moves suggest that the teachers were not always certain about an idea a student raised or about the mathematics and that they were positioning themselves as learners in the classroom. In the early classroom observations, we observed this practice in 6% of the total segments, contrasting that with 41% in the total number of late classroom observations. This shift is significant at the .05 level (two-tailed z-test statistic = 6.2).