Building powerful mathematical voices [electronic resource] : co-constructing public sensemaking

What does it mean to be "good at math?" If you ask many American adults, they will reference speed and accuracy. This is no accident as traditionally, mathematics instruction in the United States has embraced notions that learning mathematics is about replicating procedures quickly, and avoiding mistakes. Learning in mathematics classrooms has long been a solitary and silent endeavor, with teachers presenting material, and students silently absorbing. This version of mathematics instruction positions students as "passive recipients, " and can be thought of as transmission-based (Boaler, 2015; Ruef, 2013). Research tells a different story, that in fact, productive discourse is the greatest driver of learning in classrooms (Cohen & Lotan; 2014). This maps to the third common core state standard for mathematical practice: Construct viable arguments and critique the reasoning of others. There is a new vision for what it means to be "good at math, " and it includes the development of a powerful mathematical voice. This voice may be used to leverage change in the world (e.g. Gutstein, 2007). This form of instruction and learning can be thought of as sensemaking. Developing and learning a mathematical voice means shifting from being a passive recipient of mathematics to an active agent of mathematical sensemaking. It means taking risks in sharing your thinking, because in transmission-based instruction and learning, mistakes are bad, and being wrong publicly may mark you as less "good at math." Becoming a risk-taking, sensemaking, collaborative learner means overcoming that fear, and re-casting mistakes as necessary, and even good. This study examines the ways in which students re-wrote the story of "good at math, " and embraced new identities as mathematical learners along the way. This transformation requires a supportive sensemaking classroom environment, and this study also chronicles the co-construction of how students and their teacher proposed, negotiated, and came to terms on the norms and practices that support what I refer to as public sensemaking. The term co-construction is important in this process. As shown by Kris Gutierrèz and colleagues, classroom cultures are always negotiated spaces (1999). Even in the most authoritarian of classes, students must comply with a teacher's demands and expectations in order to actualize them. Social interaction is a composite space, made up of the actions and inactions of all members of the community. In a public sensemaking classroom, norms and practices such as working to understand the perspective of another person; taking risks in sharing one's thinking; presenting, critiquing, revising, and refining arguments; respecting alternate opinions; and embracing mistakes and productive struggle as part of the learning process. In seeking to document the ways in which a group of students and their teacher worked together to construct a productive public sensemaking class, this study sought to answer these questions: • How do students and teachers co-construct mathematical public sensemaking cultures? • How do students' identities as "mathematical sensemakers" shift as they move from a transmission model math classroom culture to a sensemaking model classroom? The study took place at City School , a Bay Area middle school that serves a diverse student body drawn from across a large urban setting. 94 percent of City School's students are eligible for free or reduced lunch. According to the school's website, more than 91 percent of students qualify for free lunch, and in early 2009, the student body was comprised of 74 percent Latinx/ Hispanic, 11% African-American, 11% Asian, 2% Filipinx, 1% Native American and 1% White students. City School is a public magnet middle and high school, with a mission of preparing students for careers in life sciences. The participants in the study include Isabella Mayen, a second-year teacher trained and experienced in facilitating public sensemaking, and her 62 students, all sixth graders. Isabella's students identified as 76% Latinx, 15% African American, 8% Asian, and 1% Filipinx. The study was primarily ethnographic in nature, with data comprised of video records capture from two cameras, field notes, audio and video records of interviews with students and teachers, pre- and post-surveys of student beliefs and attitudes about learning mathematics, and classroom artifacts such as exit tickets (short answer responses to questions handed in as a "ticket" to exit class). Analysis was both emergent and framed in terms of existing pedagogical and identity theories. The emergent theory developed from the iterative and reflective practices described by Charmaz (1995). Research and theories on teaching and learning that informed this work include complex instruction (Cohen & Lotan, 2014); productive discourse (e.g. Michaels & O'Connor, 2012; Stein & Smith, 2011); and open mathematics (Boaler, 2015). I have framed mathematical identity as both collective and individual. The collective vision of identity borrows from Cobb, Gresalfi, and Hodge's "normative identity, " the set of expectations that become personal obligations if one is to consider oneself, and be considered by others, as "good at mathematics" in a classroom microculture (2009). Normative identity is the framework by which I considered how students collectively constructed, and understood, what it meant to be "good at math." Personal identity follows from Sfard & Prusak's work on the ways in which stories define who we are (2005). In their framing, stories do not inform how we see ourselves, but rather become our identity. These stories come from what people say about us to us, and to others about us. I extend their framing to include implied stories, stories told through body language, and stories we tell ourselves about ourselves (metacognition). Sfard and Prusak describe the process of becoming and envisioned future self as one of gap-closing, and learning as the work of closing that gap. Analysis was dependent on the type of data. I used analytic memos to inform and revise my data collection, and to construct early theory that evolved across the course of the study (Emerson, Fretz, & Shaw, 1995). I also coded all student artifacts, revised the codes, checked for inter-rater reliability, and sought emergent themes from the data, using the Dedoose software package. The pre- and post-surveys included 26 Likert-scale items and three short-answer questions. The Likert-scale items were analyzed using Statistical Package for the Social Science (SPSS) V.21. The themes that emerged from all data sources were triangulated, and I sought disconfirming evidence to check the robustness of the emerging theory (Miles & Huberman, 1994). The findings of this study are organized in four chapters. Chapter 4, Coming to Know City School, offers a composite description of City School and its community members, and offers a counterpoint description to illustrate the important role of the researcher in collecting and analyzing data. Chapter 5, Early days—Co-constructing an Effective Public Sensemaking Culture, describes the negotiations Isabella and her students engaged to build their productive classroom learning environment. Chapter 6, The Power of Being Wrong: Inviting Students Into Mathematical Apprenticeships, offers a close examination of how one of the classes worked across three days to learn the art and importance of agency in risk-taking in sharing one's thinking, critiquing mathematical arguments, and that authority for determining the validity of a mathematical answer lies in consensus rather than the teacher's say-so. Chapter 7, Constructing Identity, reports the normative identities of "good at math" that the students collectively constructed at the beginning of the year in August 2015, in February of 2016, and in March of 2016. This snapshots allowed me to see that the students were shifting the story of "good at math" to include the importance of valuing mistakes and understanding each others thinking in addition to being good at explaining one's own thinking, and being helpful. Chapter 7 also offers five portraits of individual students as mathematical learners, framed in terms of the stories they told about themselves, and others told about them. The final chapter offers a discussion and conclusion that this study documents the important work of building effective public sensemaking classes from the voices of the students and their teacher. This is and existence proof that students who are often written off as "less able" based on the color of their skin and their socioeconomic status are, in fact, highly capable learners of powerful mathematics (e.g. Ladson-Billings, 2006). This study will inform the literature on practices of teaching and learning, and is a strong indication that policy should consider the important voices and knowledge of professional teachers, who hold the most immediate and considered power to positively impact learning opportunities for their students. Finally, this study makes the case that learning a mathematical voice is important in preparing students to be agents of change in their lived worlds. In the words of Montserrat, "Math is about living.".

Keywords: Education.
Advisor: Boaler Jo
Goldman Shelley V. advisor. Thesis advisor
Langer-Osuna Jennifer advisor. Thesis advisor
Lotan Rachel A. advisor. Thesis advisor