Modern day mathematics teaching focuses heavily on inquiry. This sort of mathematics teaching, often labeled as reform-oriented or inquiry-oriented mathematics teaching, emphasizes conceptual understanding and procedural fluency as opposed to speed and recall (National Council of Teachers of Mathematics (2000). Teachers\u2019 instruction, therefore, revolves around understanding how students think, specifically the strategies that students create when trying to solve problems for the first time.<\/p>\n
For instance, suppose a teacher is interested in how students solve a proportional reasoning problem, such as follows: \u201cTeddy read 30 pages of a book in 45 minutes. How many pages should he be able to read in 120 minutes?\u201d A general teaching flow for this type of problem might involve a proportion with an unknown variable, 30\/45 = x<\/em>\/120, and then teaching cross-multiplication to form the equation 45x<\/em> = (30)(120).<\/p>\n This approach, while efficient, limits students\u2019 understanding of why the teacher converted the proportion to an equation or how the procedure connects to the number of pages Teddy has read. Inquiry-oriented teaching, on the other hand, requires students to explore this problem on their own, to attempt to understand what is being asked, and to formulate a strategy. Then, the teacher can connect each student\u2019s strategies to other strategies in the class and, perhaps, to a general strategy.<\/p>\n Understanding exactly how a student solved a problem, unraveling the layers of steps and missteps a student took, however, requires a patience and attention aimed at individual students. This understanding does not happen when a student writes and explains a quick explanation on the board. Nor does it happen when a teacher circulates around the room, hovering over students as they work. Rather, the most effective way to understand and listen to the way a student thinks mathematically is through a one-on-one investigative interview between teacher and student, a technique honed by Piaget and referred to as a clinical interview<\/em> (Ginsburg, 1997) or diagnostic interview<\/em> (Huff & Goodman, 2007).<\/p>\n Teachers rarely engage in these one-on-one interviews for a number of reasons (Zazkis & Hazzan, 1998). First, teachers seldom have time in a busy school day to sit with a student for a one-on-one interview (Hunting, 1997). Second, learning how to question, listen, and respond to a student are highly refined teaching skills that do not simply manifest without organized support (Jacobs, Lamb, & Philipp, 2010). Yet, few teachers have access to support that helps them focus on noticing mathematical thinking. Third, whenever teachers work with students, certain student attributes affect their disposition toward that student (Dunn, 2004). That is, teachers cannot help but notice certain student characteristics, such as gender, ethnicity, familiarity with mathematical vocabulary, or even the clothes a student is wearing. These factors consciously and subconsciously affect how a teacher hears what a student is saying, inevitably creating prejudices that reify a teacher\u2019s perception of a student and obstruct an opportunity to focus on active listening of a student\u2019s mathematical thinking. Additionally, providing spaces for teachers to practice listening to children\u2019s mathematical thinking, particularly children they might not know or work with regularly, focuses teachers\u2019 attention completely on the child\u2019s thinking rather than\u00a0subconsciously evaluating a student\u2019s physical attributes.<\/p>\n We attempt to address these problems by introducing an idea formulated in a current technology tool that brings the teacher-to-student interview into the modern era and helps to develop a teacher\u2019s mathematical noticing skills. We have built technology that allows teachers and students to interact without having to be physically next to each other, helping to mitigate pre-conceived biases so teachers can focus on building their skill in noticing student thinking.<\/p>\n In our study, we asked the research question: When using smartphone technology for a one-on-one teacher\/student mathematics interviews, what is revealed about how mathematics teachers notice through the way they question, listen to, and respond to student mathematical thinking?<\/p>\n Literature Review<\/span><\/p>\n Recent research on teaching mathematics (Jacobs et al., 2010; Smith & Stein, 2011) as well as the Common Core State Standards for Mathematics Teaching Practices<\/em> (National Governors Association Center for Best Practices and Council of Chief State School Officers, 2010) have outlined the importance of a particular set of teacher skills: the ability to question, listen to, and respond to a students\u2019 mathematical thinking. We refer to the term noticing<\/em>, particularly, mathematics teacher noticing<\/em>, to encompass these constructs. While the act of noticing often refers to the ways a teacher attends to, interprets, and responds to students\u2019 thinking within a classroom environment (Jacobs et al., 2010), we apply the construct of noticing to a one-on-one interview environment.<\/p>\n Questioning<\/p>\n For teachers, learning how to question is the first step in attempting to understand a student\u2019s mathematical thinking. Questioning not only evaluates a student\u2019s mathematical knowledge, but also helps teachers understand how a student thinks (Aizikovitsh-Udi & Star, 2011). Furthermore, good teacher questions help students in their own thinking by guiding their attention, loosening up their thinking, or forcing them to articulate their ideas (Smith & Stein, 2011).<\/p>\n Good questioning involves focusing on concepts rather than calculations and allows for wait time after<\/em> a student\u2019s response (Herbel-Eisenmann & Cirillo, 2009). Therefore, in order for a platform to allow for effective mathematics teacher questioning, it must (a) allow for multiple types of questions that focus on student thinking and concepts and (b) integrate wait time that occurs after a student\u2019s response.<\/p>\n Listening<\/p>\n Teaching mathematics for understanding requires responsive listening\u2013not only to what a student says or shows, but also to what a student is thinking. Listening is the heart of effective mathematics teaching practice\u2013the bridge between a teacher\u2019s questioning and subsequent response (Duckworth, 2001; Empson & Jacobs, 2008). Sherin and van Es\u2019s (2009) work on teacher video clubs found that when teachers stepped outside the classroom and learned to interpret events as opposed to judging whether they were good or bad, they were able to stop critiquing the teaching and focus on understanding what students knew. These teachers honed their listening skills through continually rewatching and discussing video of their own student\/teacher interactions.<\/p>\n Furthermore, Empson and Jacobs (2008) found that learning to listen required repeated viewings of supported student\/teacher one-on-one interactions. Therefore, any platform that allows for teacher listening\/noticing must (a) move outside the whole-group classroom environment, (b) help teachers learn to interpret their own interactions with students, as opposed to evaluate them, and (c) allow teachers multiple viewings of their interactions with students.<\/p>\n Responding<\/p>\n Of the three noticing skills, learning to respond to children\u2019s thinking is the hardest skill for teachers to develop, yet the most effective in extending student learning (Jacobs & Ambrose, 2008). While teachers develop questioning and listening skills over time, the ability to respond does not automatically develop with experience. Smith and Stein (2011) found that teachers needed extensive sustained practice in learning how to respond to student thinking in order to develop a repertoire of appropriate responses. Therefore, any platform that helps teachers respond to student mathematical thinking must provide ample opportunity and support for teachers to practice how to respond.<\/p>\n Technology Framework<\/span><\/p>\n Our work extends prior scholarship on mathematics teacher noticing, particular in its subconstructs of how teachers question, listen, and respond to students within a one-on-one interview setting. Little work exists exploring the nuances of the one-on-one mathematics interview, which may be because previous technologies did not allow for precision in capturing data within the interview itself (Hunting, 1997).<\/p>\n The technological features that seem necessary in order to explore mathematics teacher noticing within a problem solving interview include (a) the capture of every utterance or artifact, (b) ability to re-watch anything previously written or recorded, (c) virtual interactions so both parties do not have to be physically present, and (d) immediate access of a fellow math educator for teacher support.<\/p>\n By using existing technology found in smartphones, we have developed a system that captures every artifact and utterance made between teacher and student. These small words, phrases, and diagrams are crucial in understanding how teachers and students talk to each other. Previously, capturing every audio utterance or facial expression required massive amounts of video and audio data collection and storage. Through utilizing smart phones as our communication conduit, all utterances are documented as part of the back-and-forth nature of mobile chats,\u00a0revealing a digital transcript of text messages, images, and video that teachers and students use to discuss and share their mathematical thinking.<\/p>\n Second, smartphone textual, photo, and video-based communications create a visual record that both teacher and student can recall and refer to in the midst of their interview. This ability to watch a student\u2019s explanation repeatedly or to scroll back in time adds precision to the interview.<\/p>\n Third, our technology situates the interview virtually, so the teacher and student never see each other. By physically separating teacher and student, we attempt to mitigate preconceived notions that both teacher and student might make about each other in a teacher training exercise to develop their\u00a0noticing skills.<\/p>\n Finally, a fellow mathematics educator is available for immediate access to the teacher for support, either physically or through the technology itself. The teacher can ask for guidance as to how to question, listen to, or respond to student thinking, or to articulate next steps.<\/p>\n We find the use of smartphone-based technology beneficial when working with current pre- and in-service mathematics teachers for three reasons.<\/p>\n Research Methods The purpose of this exploratory study was to investigate how teachers notice students\u2019 mathematical thinking within a one-on-one interview when given the opportunity to use technology. We collected qualitative data to gain a better understand of how teachers question, listen to, and respond to student thinking, as well as their feedback to the entire virtual interaction.<\/p>\n Participants<\/p>\n Our study involved three mathematics teachers in the northeastern region of the United States engaged in a one-on-one interview with a student using our technology. Teachers were recruited through existing networks of mathematics teachers that the authors were members of, being mathematics teacher educators and professional developers. Teachers were selected based on interest in piloting technology that helps teachers learn to listen to student mathematical thinking.Mildred had taught seventh and eighth-grade for less than 5 years at a public science, technology, engineering, and mathematics (STEM) focused magnet middle school. Justin had taught ninth-grade algebra and 12th-grade calculus and precalculus at a public high school for less than 5 years. Skyler had taught 2nd and 1st-grade in a private elementary school for 5 years.<\/p>\n Data Collection<\/p>\n Teachers each used their smartphones to interact with what they believed to be a student who shared a video of how he solved a prechosen mathematical problem. These initial student videos were under 2 minutes each and consisted of point-of-view shots from a student\u2019s perspective of how he attempted to solve the mathematical task.<\/p>\n The student for this study was the first author pretending to be a student. This deception was necessary in order to create an idealized case to explore how teachers interacted with this technology. To do this, we met as a research team to detail specifically how the student should respond to provide systematic strategy, thinking, and responses. By controlling exactly what the student did and said, we were able to focus on the teacher and examine what might be possible with this technology. In the videos,\u00a0one\u00a0of the authors had his 5-year old daughter play the part of the\u00a0elementary student. For ease of reading, we refer to the researcher deceptively interacting with the teacher as \u201cthe student.\u201d<\/p>\n Before the student interview began, the teachers were given the opportunity to review and attempt to solve the task and ask questions about the process. Then, each teacher sat with a member of the research team for a 30-60 minute student interview session, interacting with the student through a smartphone. During the student interview session, we videotaped the teachers in order to capture their reactions and responses to the student. We refer to these data as the \u201cteacher video,\u201d whereas all other references to video refer to those sent between the teacher and student about the mathematics problem.<\/p>\n The student interview session began when the student sent the teacher a video of how he or she solved or attempted to solve a mathematical task. The videos involved the student narrating his or her strategy for the problem, pointing to written work on paper. Each video was shot from the student\u2019s perspective, with the student\u2019s finger pointing out specific parts of his or her strategy during the course of the explanation.<\/p>\n For Mildred and Justin, who work with secondary students, the student worked with a ratio and proportional reasoning problem: \u201cTeddy read 30 pages of a book in 45 minutes. How many pages should he be able to read in 120 minutes?\u201d (See Video 1.)<\/p>\n\n
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