classroom. Beth also maximized engagement time by having students work with each other and by asking students to take on roles as helpers and to “make their way around the room” (Beth, observation period). This intervention reflected support from the administration. When I asked Beth for factors that support her practice, she said that having behavioral counselors and a
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dedicated vice-principal allowed her to make plans for students that were not prepared to learn on a given day. I had seen this arrangement in practice as early as the first day when I conducted an acclimation visit. Beth was managing classroom discussions by having students pass a red ball. If a student received the ball, the student was expected to give input and/or to “phone a friend” if the student needed help. A student refused to engage in class with a confrontational tone. Beth said to her, “this is your choice, if you are not ready to be with us and learn, it is best for you to go”. The student picked up her belongings. As she walked out the door, Beth asked her to write the homework that she needed to bring back the next day. I was puzzled by the interaction because I was not used to see a student’s oppositional behavior matched with a calm exit and a reminder to do homework. During the check-in, Beth explained that she had made arrangements for the student to do work at the counselor’s office so that her student would not just miss class, but have an alternative place to continue learning. This was also the case with students placed on in-school suspension. Beth and her colleagues had a dedicated service period for in-school suspension. When they headed out to the room, they would tell each other what their students needed to work on and would then follow up with their students in the suspension room. I was able to do in school suspension duty with Beth one day. She expected maximum engagement of all students in the suspension room too.
Forms of knowledge informing ‘maximizes engagement time’. Figure 4.2 above, depicts
the forms of knowledge that reflected association with this intervention. This intervention was informed by mathematical forms of knowledge such as: foundational gaps, perseverance,
strengths and errors. Through maximized learning engagement, Beth helped students
overcome gaps, errors and persevere. She also had students that exhibited particular strengths help classmates and facilitate discussions with them. I asked Beth in a check-in why she had
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students work together. My original understanding was that Beth valued collective components in teaching. Her answer confirmed a different intent. She said, “because there is only one of me and 25 of them” (check-in). Having students help each other was a way for Beth to maximize everyone’s learning engagement.
This intervention was also informed by other nonmathematical forms of knowledge. As seen in the examples above, Beth maximized learning engagement consistently, paying attention to students under particular conditions such as those in suspensions or needing behavioral counseling (personal conditions). Additionally there was an association to forms of knowledge on dissonance with school culture and on conflicting priorities. As described on intervention 1 (expands the boundaries of the learning space), Beth experienced challenges with having students consistently do their homework. She also stated on multiple occasions that “their 46 minutes” were the only times within her control (check-ins). Maximized learning engagement was found to respond to challenges on completion of homework and ongoing preparation that were expected to take place outside the classroom.
Findings: Central phenomena in Beth’s practice. I now revisit Beth’s interventions to
take a closer look at central phenomena in her practice. These phenomena include her
perspective on learning, the interplay between her interventions and her teaching goals and her understanding of her students’ mathematical learning experiences.
Beth’s Interventions and her perspective on learning. Beth made a strong use of psychology, as well as reflective tasks and prompts for her students. Her use of psychology was evident for both, mathematical learning needs and non-mathematical learning needs.
Mathematically, Beth prompted and expected her students to pay attention to why and how they interpreted and applied the mathematical expressions that they worked with. In non-
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mathematical aspects, Beth would also communicate to students through questions or by posing statements that prompted students to consider how their behaviors affected others (or
themselves) and their consequences. For example, on various occasions I heard Beth tell her students the phrase, “perception is reality”. She used this phrase to help students re-think situations and to re-think how these situations had an impact on others (check-in and post- observation interview). At the post-observation interview, I shared with Beth that I had noticed this pattern in her behaviors. Beth’s responses best reflected her perspective on learning.
I asked Beth if there was anything in her background or experiences that led her to this choice. Although my original question did not point to either mathematical or non-mathematical aspects of her students, Beth’s response incorporated both. She explained that when she was in her senior year in high school she took Pre-Calculus, but she wasn’t “that AP Calc AP student” (post-observation interview). She said she did like numbers, but that she had “her struggles”. Beth stated:
So I think that helped me, because I wasn’t the genious kid, I had to focus on how am I thinking about this. And then in college, I worked in the math lab a lot, so even working with classmates, I had to figure out “why are they getting so stuck?” (post-observation interview)
Mathematically, Beth recognized that learning needed a sense of self-awareness and focus on how thinking was taking place. This was consistent with the ways that Beth asked her students to approach their work in the classroom. Beth created a learning environment that facilitated a focus on self-awareness in how thinking was taking place. Beth’s psychology-type of strategies transcended into non-mathematical learning areas as she helped students find ways to develop behaviors that would support their learning. During the interview Beth continued to explain that where she taught had an impact on her need to look for ways to help her students develop
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and where she had learned as a student. She described her own learning experience as “good morning, you will now work on problems one through fifty even, and keep it quiet”. She said that would work for half of her students, “and the other have would go insane” (post-observation interview).
Interplay between Beth’s interventions. Beth expanded the boundaries of the learning space to make students’ learning more accessible. She also demonstrated maintaining high expectations in learning that supported her need to look for ways to have her students continue learning even after assessments had been taken and graded. Beth maximized their learning engagement in the classroom, but in also expanding the boundaries of their learning space, she looked for ways to develop the understanding that learning needed to continue to take place outside the classroom. Beth learned from her students that they responded to her when she demonstrated genuine care for them, and she used this to re-frame their roles as students, as something that was good and as something that they needed to care about as much as she did. The interventions I have described up to this point made learning more accessible, but in a way that positioned students to reciprocate to Beth’s efforts to support their learning. “Meaningful learning”, using Beth’s own words, was further supported in two different ways. Beth used psychology and reflective techniques to help students focus on their thinking and this focus transcended to both mathematical and nonmathematical aspects of their learning. She facilitated meaningful mathematical conversations to foster classroom norms where students focused on how they were thinking, making particular use of the structure and language of math.
During the pre-observation interview, Beth described herself as consistent and flexible. While there was evidence in the data to support both descriptors, my interpretation is that Beth held unwavering high standards on her own practice and this, in turn, resulted in her consistency.
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Beth demonstrated flexibility in that she found different ways to help students meet her learning expectations. My interpretation, however, was that Beth demonstrated being highly resourceful and creative. She looked for alternative ways to make learning more accessible. She made use of as many resources as possible (e.g. family, guidance counselors and students themselves) to advance her students’ learning.
Model of the learning phenomena in Beth’s case. Up to this point I have presented and
described the forms of knowledge of the student used by Beth as well as how these have informed her interventions in the classroom. I now present a model for students’ learning experiences in Beth’s classroom. I used the model in Figure 4.3 as an analytical tool to further understand and explain the learning phenomena in Beth’s practice. I describe the components in this model. I end this section with examples from students’ learning experiences depicted through the model.
Interventions. Beth used different forms of knowledge of her students to inform the
teaching interventions she implemented in her practice. These forms of knowledge and their association to each intervention were depicted in Figure 4.2. Beth’s interventions revealed a form of dual preparation. While students learned math, they also learned how to support their own learning through different forms (e.g. asking questions, coming for help, writing their thinking).
Buy-In. In the process of implementing these interventions, I noticed a high incidence of
attempts from Beth to help students learn how to support their own learning. Through our discussions of her particular students’ experiences at the post-observation interview, I confirmed a pattern I had observed from classroom observations and check-ins. Beth’s students were at different stages in a gradual development of personal skills to support their own learning. The
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first step was buy-in or trust in the learning process, which Beth had particularly supported through her demonstartion of genuine care. She had also supported it throug her high
expectations. Student’s did not necessarily develop full buy-in or trust at the beginning. This was more of an initial step into a cyclical process of additional work between Beth and her students.
Figure 4.3. Model of Learning Phenomena in Beth’s Case
Skills to persevere mathematically and non-mathematically. Beth’s continuous work
through her interventions tended to students’ needs mathematically and non-mathematically. Students were expected to support this learning through their preparation, their questions and through their communication of their thinking inside and outside the classroom. In Beth’s case these perseverance behaviors were closely related to her conception of college and career
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readiness skills. Beth valued them highly because they did not just serve as supports for learning, they often times preceded learning (see Steve’s case below). Thus, while there was mathematical learning in the classroom, meaningful learning needed attention to the other non- mathematical components. But, the data shows that these skills were not developed overnight. Beth described her work as one that needed consistency and time. This meant that the process required for students to also continue persevering in learning both mathematically and non- mathematically.
Experienced success. Students’ work with Beth led to an initial improvement that helped
develop additional buy-in and trust.
Ownership over their learning. The model reflects a cyclical phenomenon where the
consistent and continuous engagement in: buy-in – perseverance – success led to an outcome on students’ ownership over their learning. This ownership however, entailed different types of behaviors depending on the student. In some cases, ownership required that a student learned to manage a personal challenge (Selena) in order to continue supporting her own learning. In other cases, ownership required developing behaviors that a student may not have been comfortable with before (Steve). I used double arrows for this outcome in the model because Beth described the development of these behaviors as taking place over time.
I end this section by revisiting some of Beth’s students to describe their learning experiences through this model.
Selena. Selena used to work on her own. A few times I saw Hannah ask Selena a
question where she sat, but for the most part, Slena was one of the few students that did not join a group. I had noticed during the observational period that she asked Beth questions and that Beth would sometimes provide a quick answer. Othe times Beth would take her answer to the
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board and work out the problem with her. While Selena waited to get help from Beth, she would move her arms and body as if she was swaying to a song in her head. She often moved her arms as if she played the drums. Up until the post-observation interview, my field notes on Selena recorded the questions from Selena to Beth and their interactional dialogue to work though a problem. During the post-observation interview, I asked Beth to desrcibe her thoughts on how her students did on the test and to pick any particular student examples that in her opinion reflected learning improvement. Beth picked Selena.
I did not recognize her by the name, because Selena did not look on appearance Hispanic, but her name and her parent’s name (from the informed consent form) reflected otherwise. Beth proceded to explain to me how she had made arrangements with the Vice-Principal’s office because of Selena’s behaviors. Beth explained that she would get angry as she worked on
problems and would start swearing. Although expanding the boundaries of the learning space was a form of support for Selena, based on the observational data, Beth worked with her through
meaningful mathematical conversations. Beth also said that in her case, she learned to “let her
ride it out”. Beth had her work on consequences with the office and then start fresh the next day. Beth also noted that seeing Selena persevere through the test without outbursts was an
accomplishment. Selena reflected trust in Beth in the way that she looked to work with her. According to my data, Selena had also reflected perseverance in the observation period. There were no outbursts that disrupted class. In our discussions about Selena, Beth pointed out that she not only had a good grade (80), she had also reflected confidence in her skillset through her responsese to Beth’s reflective questions at the end of the test. Beth had done all her
homework and had completed the review problems. She had demonstarted ownership over her
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Steve. Steve was another student that worked independently in the classroom. He did not
seat near Beth though. I had noted that he often times put on head phones during classwork. He did not wear the headphones during class discussions, but he also did not contribute answers to these discussions. Beth identified him as another student that had reflected improvement. He used to not ask questions in class. Beth said that she continued to encourage him to come for help and to ask questions, until he started to use the texting App. A short excerpt from Steve’s texting trail is included in Appendix F. The full trail evidences Steve’s questions throughout the year. His first question was: “ion [sic] understand. all these notes, still don’t [sic] know how to find range smh. i think yuh [sic] shou [sic] post more of specific notes that actually tells me how to do the steps than class videos” (Steve, app text trail). Steve’s first question in the app
reflected a lack of buy-in into Beth’s flipped lessons. Steve’s eventual continued use of the App evidences how the intervention of “expands the boundaries of the learning space” helped Steve. Beth’ responses also demonstarted genuine care. This was the beginning of Beth’s response to Steve’s first post:
Steve 2:22am is admarable!! Thank you for trying. Class videos and what I gave for the notes is exactly what's needed to do this. What's missing is you asking questions in class! Ask!!! Waaaaay back when we started this topic, as soon as you weren't seeing where something came from, I need you to ask. (Beth, app text trail).
This response also demonstrates Beth’s efforts to help Steve develop the necessary behaviors to support his learning. In explaining to Steve when he needed to ask questions, Beth was trying to help Steve develop a set of skills that would help support his learning. This was a form of skill that empowered Steve to persevere through learning. Beth explained that with time, Steve used the app more regularly. Beth explained his learning as a two step process. She said, “well, it started with an improvement on effort, accuracy came later” (post-observation interview). In this case, Steve’s learning cycle demonstrated the need to develop nonmathematical skills that
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were associated to perseverance and that resulted in his ultimate learning. In developing a
regular cycle of communication, doing his work outside of class and checking his understanding through the app and pictures of his work, Steve demonstrated ownership over his learning.
Hannah. Specifics about Hannah’s case were presented in table 4.5 (Language). I revisit
her case here to showcase a different outcome from the cycle of consistent work with Beth. In Hannah’s case, according to Beth, she benefited from a flipped classroom because it allowed Beth to give students such as Hannah more one-on-one attention and time to engage in
individual conversations. In Hannah’s case, she needed to develop the skillset to communicate
her thinking. This was a skill that she needed in order to persevere in her learning. Beth also added that as Hannah spoke more about her work, she was able to also write more to support her understanding. Beth said that Hannah was reflecting improved performance in her tests. In class, there were multiple instances where Hannah approached classmates for help and also worked with classmates in groups. My interpretation is that Hannah made improvements through her learning cycle. She was making progress in the process of developing the skillset necessary to continue persevering and evidencing full ownership over her learning.
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