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Each of the following cases reiterates some of the information in Table 1 and supplements it with additional context as well as extracts from transcripts. The substantial use of transcript extracts was deliberate in showing the reader close ties to the framing of the findings and the claims made in the following chapter. In other words, this chapter serves to provide the conditions, or context, to support the variation map of Chapter 5. Case 1: Mr. SA-A Mr. SA-A taught at a public high school in South Africa and has taught mathematics for 28 years. He only implemented the Student Response Form in one of his 10th-grade classes, and this class had 30 students. This class was his homeroom class, and the class’ average grade in mathematics was significantly above the average for the entire grade. Mr. SA-A reported that all of his students had access to the Internet either through their own mobile devices or a mobile device of a friend or parent. Students would have paid plans or access free Wi-Fi should they reach their data cap. While this use of technology was completely new to him, it was not outside of his technological range. In total, he received three student responses that were unprompted. He emphasized that forgetfulness due to a pressured routine caused him to check the spreadsheet less often than he would have liked. He reported that he received, “without exception, responses from students who were very quiet in class and serious about their studies.” He saw these responses as an “aansluiting,” or segue, which he elaborated as “a little something, like even if it is a little story or something interesting, etc. that enters the experiential field of the child that which matters to the student is made a reality.” Mr. SA-A noted that “there [were] definitely students that previously were quiet and introverted… [who] later popped out as students who wanted to participate, which [I] was 130 very proud of. Whether this is due to the Google Form is not something [I could] say for sure – how can you prove that? However, [I do] believe that perhaps it is possible”. The interview took place two days after Mr. SA-A had accepted another position at a different school and he clearly stated the Student Response Form could not be implemented before issues of discipline and norms were first addressed. He was very sorry to part ways with his students and believed there was a kind of bonding, “…something special that I have not really experienced from another class before” although he maintained that he cannot comment on the potential influence of the Student Response Form on the unique classroom atmosphere. Case 2: Mrs. SA-B Mrs. SA-B also taught at a public high school in South Africa and has taught mathematics for 20 years. She implemented the Student Response Form in one 8th -grader class of 23 students. Two students in her class submitted all six responses that she received. All students had access to mobile devices but also had problems concerning limited data plans as mentioned before. She mentioned that maintaining the Student Response Form requires some commitment in remembering to check the spreadsheet often for any new responses, as she did not access the Internet every day. She mentioned a sense of urgency in this regard because it felt as though “[I’m] actually kind of dropping [the student who submitted a response] by not going to have a look… every night.” The students who submitted responses were also the students who “constantly had their hands up in class” rather than the students who were not performing well in class, which “was a pity”. Mrs. SA-B said that the Student Response Form made her more attentive to the actual effectiveness of communicating some idea, which a teacher might otherwise assume obvious. She gave an example of how a student response made her realize that her example of b−1 = 1 𝑏 131 does not reveal properties of exponents as well as 𝑏−2 = 1 𝑏2 as some student thought that 𝑏−2 = 2 𝑏 when using the prior example rather than the latter. She mentioned that one of the two students who used the Student Response Form was an interesting case to her because he entered the class with a failing grade but a recent test showed he was passing and doing significantly better. She talked to him and “he said he just wasn’t interested in the subject in elementary school (the previous grade). [And now] he was one of those who would often ask questions… so with him I can specifically see that he asks questions frequently in class… and suddenly became more interested in the subject. I don’t say it’s just the spreadsheet but overall his attitude changed towards the subject. I do think something like this might be able to change a student’s attitude because it is something interesting. It is something different, another subject does not offer him that and I kind of want to use the word ‘thrill’ but that’s probably not the right word…” Case 3: Ms. SA-C Ms. SA-C taught at a high school in South Africa and has taught mathematics and for four years. She introduced the Student Response Form to all six of her classes, 161 students in all. She received a total of ten responses and mostly within the first week. As she made use of the Student Response Form after teaching these classes for a couple of months, she was also able to comment on perceived indirect influences of the form. She mentioned that due to the low response rate it was easy to become forgetful and not check the spreadsheet on some days. Ms. SA-C perceived one of the main capabilities of using the Student Response Form was that it allowed her to “approach students in a slightly different way if [I knew] what the need is for the students instead of just speaking in general with 40 different personalities every day.” For example, one of the responses was “Ma’am, how does value-added tax work?” This surprised Ms. SA-C (“I did not expect kids [to] know this little about the topic”) because she 132 assumed the concept of value-added tax to be a prominent topic in past work. This response later led the teacher to revisit the topic during class. She noticed that some students were multiplying by 14 rather than 14% and “…they were getting these crazy big answers, and I was like ‘Where’s your percentage button?!’ and the student said ‘Must I press the percent button?’, so definitely in the future I will explain the percent [button]…” She added that once the student started to ask questions “…it started helping and we could make some progress [instead of] most students… sitting back thinking… ‘I don’t know how to get it but it’s okay because she said it is so, so it is so.’” In addition to reporting that she got a better idea of her student needs, she added that, after the Student Response Form was implemented, the “shy kids… felt a little bit more ‘vrywillig’ [candid/frank/free-willing] to ask questions once they knew, above and beyond the form itself, once they knew that they can ask me questions.” Specifically, she was able to name one student who “… sits right in front of the class and if I stand close to him then he becomes all jittery and stuff. But once he knew that he could use the form, even though he hasn’t sent a response yet, he just felt that he can be a little different, he felt a little bit more at ease…” Case 4: Mr. US-A Mr. US-A has three years of experience and taught at a public high school in Georgia, USA. He made use of the Student Response Form in three Advanced Algebra courses with students aged 15 to 18. The three class sizes ranged from “low 20s to low 30s,” and students had submitted 220 responses by the time of the interview. Mr. US-A’s school was transitioning to providing each student with a personal computer. Because this school was a “Google for Education” school, students and teachers were familiar with Google services like Gmail, Google Docs, and Google Drive. He reported that not all of his students had mobile phones so he was 133 concerned that there might be some “socioeconomic bullying,” but fortunately “there was nothing like that.” This teacher also had a co-instructor, and they would often switch roles during class allowing one to teach upfront and the other to walk around and monitor student work. The Student Response Form was mostly used for “a closing question towards the end of class. A few times, we use that like a little halfway point in class to introduce an error like a common error… I think it’s ‘Answer a question posed by the teacher’ and then I just give them free response.” He might also provide an example of student work and have students submit a response to comment on the correctness of the work. Mr. US-A also reported that he and his co-instructor would actively and continuously try to involve as many students as possible in classroom discussions. One example of how the Student Response Form was used in introducing the concept of extraneous solutions. Mr. US-A said “We thought a lot of students wouldn’t know what the word extraneous meant… but the answers that we got showed a lot of kids have an idea of what the word extraneous meant… so somebody said ‘Put a lot of effort in’, or ‘irrelevant’… well, that’s pretty close… that’s good… that’ll probably fit when you define certain things as extraneous but we have to… stretch your definition of extraneous a little.” Mr. US-A customized the functionality of a separate Student Response Form to aid him in monitoring academic performance of the students for whom he was the baseball coach. The students were required to self-report their academic progress at regular intervals with the aim of Mr. US-A being able to provide timely support to students should they require it. He reported this practice to be highly successful and recommended it as being able to make a “significant difference” in an athlete’s academic performance. 134 Case 5: Dr. US-B This teacher had five years of experience and taught at a private school in Florida. The school emphasizes project-based learning and values the use of technology as an integral part of the students’ lives. For instance, every student is required to have a computer and the school does not have any policies against the use of mobile phones in class. Students also have a lower “seat time” when compared to students in a public school, as more time is allocated to project-based learning and individual catering to students. His routine consisted of teaching geometry and algebra to four classes. Two of these classes were mixed seventh and eighth graders. They meet four times a week, spending one or two days on an activity or some problem-solving tasks, and two days using a flipped classroom model. This teacher received roughly 200 responses from all four classes during the time of the study. The most significant advantage afforded by the Student Response Form for Dr. US-B lay in helping him “to focus on providing closure to lessons… and this was a kind of structured way for me to… [get] some indication of what [they] know.” While it did somewhat inform the direction which a lesson might take. The teacher noted “[I do] not think that [I] did a very good job of incorporating… the responses into [my] practice. That is something that [I] would’ve liked to do more.” He gave an example of the use of the Student Response Form where the teacher prompted the students to submit a response where they explained to another student how to get the equation of a line. He mentioned that responses stood out that had “proper Mathematical vocabulary” and ones that did not simply state something like “I don’t get this” but rather offered an “honest assessment” of what they understand. For an example of an honest assessment, he shared a response that stated: “Do the change in Y over the change in X so you get the slope, but then you plug that into… but then I really don’t know how to get the B…” The teacher also 135 noted that he had an email-based practice of students working until they got stuck and then sending him an email with a screenshot and a brief description. In addition, Dr. US-B expressed the need for being able to use mathematical symbols in the responses.

Case 6: Mrs. US-C This teacher had 17 years of experience and taught at a public high school in Georgia, United States. She used the Student Response Form with two grade 10 classes for analytic geometry, of which one was a support class. She received roughly 500 responses from 45 students during the time of the study and the only instance where a teacher’s use of the Student Response Form increased over time. She reported that the Student Response Form was complementary to existing practices that she had “because we just kind of slid it right in there… it doesn’t take long… that’s the beauty of it.” The first existing practice required students to report the clearest point and the muddiest point of a lesson as a ticket out the door. Mrs. US-C initially made use of notecards that were passed out to students on which they would write the responses whereas she is now using the Student Response Form. The second existing practice was called My favorite no where she noted that she “[likes] to have the students tell [me] something right about what we did and maybe an error in understanding… they have to mention something good and then something that was a mistake.” This practice was initially carried out through discussion or notecards during class time whereas she is now using the Student Response Form. The Student Response Form was used a couple of occasions a week. One example of how Mrs. US-C made use of the Student Response Form was a ticket- out-the-door where she prompted the students as follows: “Tell me what is clear to you from today’s lesson. Tell me what is still muddy.” One student response was “Help! I don’t get any of this!” and another response was “I get the SOHCAHTOA, but I’m still shaky on how to decide 136 which one to use when solving a right triangle.” The teacher also reported that “I was very pleased at how much the students wrote especially from the students who don’t like to speak up during class… I think [that] was my biggest pleasure, that ‘Okay!’ they’re really taking the time to answer the questions even more so than they did on notecards…” Case 7: Dr. US-D This instructor had five years of experience and used the Student Response Form for a content course for pre-service teachers with a focus on trigonometry and more specifically the covariation of variables in a function; this class also had a strong focus on mathematical thinking and students are expected to voice their struggles or difficulties with tasks. The class had ten students of whom most were 3rd -year students. Common practices of the class included regular group work where students work on large whiteboards with dry erase markers or using dynamic geometry software; the instructor and teaching assistants would walk around the class facilitating discussion either within the groups or with the class as a whole. The Student Response Form was used from the onset of the semester and received a total of around 170 responses. The Student Response Form for his students “as part of their homework assignment to say… [which] questions were the most confusing, what was the most difficult, what questions they have about the homework… that sort of stuff… that I would want them to come to me within office hours.” The Student Response Form was customized according to problem sets and the respective questions of a problems set. Dr. US-D said that one of the significant struggles for him was getting “students to come to office hours and ask questions outside of class.” He added that: “That’s a very… I don’t know if it’s a power structure that exists with faculty and students… I know as a student I was always afraid to approach faculty so I can see [why] students are uncomfortable doing that… So I think 137 in terms of changing the practice it created an environment for them to feel… safe, comfortable, just sharing…” One of the examples that he offered was a response by a student who stated: “I found question three the hardest to explain because I never thought of how to explain arc length to someone who does not understand the concept.” The instructor went on to say that he “never thought about how, to them, that might be such a foreign concept — to think about how somebody else is thinking… so that was pretty cool…” The instructor noted that student responses tended to change over time from problems with questions to more content-focused problems. He gave an example of the student response that was more focused on content: “I think number seven was the most confusing. I am not sure we ever defined what is a function or determined what is not a function, but I figured that most graphs were functions because they showed a relationship between two variables.” Case 8: The Researcher This instructor had seven years of experience and taught a college level methods course for 24 preservice teachers who were in their second or third years. The course was focused on teaching methods to teach K – 5 Mathematics, and it had a field component where students worked with first graders at a nearby elementary school and reflected on their planning and teaching throughout the course. The Student Response Form was a central part of the course as students were required to submit between one and five responses before each class. The required submissions ranged from responding to an assigned reading, commenting on an assigned short video, or simply whether they would like to share out the responsibility of bringing food to class and how much they would be willing to spend. Several assignments were broken down into 10 to 20 responses on a given topic and combined into a single submission. For example, students had to review all their submitted responses after watching short videos of teachers working on 138 problems with elementary school students, and the pre-service teachers had to try and notice changes in their own responses over time. Submissions were graded by the teaching assistant with zero points for no submission, one point for an incomplete submission, and two points for a complete submission. By the end of the semester the Student Response Form had nearly 2000 responses. The Student Response Form was customized by response topic and number (e.g., CML video, task 7). One example of a student response submitted after watching a video is: “Some of the children understood relationships between numbers and thinking strategies. These children were able to see the 3+3=6 and find that 3+4=7 since 4 is after 3 and 7 is after 6. This demonstrated their understanding of number relationships.” Another response required the student to discuss a strand of Mathematical Proficiency (Kilpatrick, Swafford, Findell, & Committee, 2001) called Productive Disposition: “I think that the last student also showed productive disposition. Productive disposition was evident because he shows that Mathematics is useful by using known Mathematics facts to solve a more challenging Mathematics problem”. These student responses were also incorporated into discussions by printing off all the student responses for some task and selecting a few students to talk about their responses in more detail. They were handed the printout to familiarize themselves with their response if they were not able to recall what they submitted. A class list was maintained to keep track of making sure that each student was called on at least a couple of times a semester, although this practice only persisted for the first two months of the course.