# Your search: "author:"Schoenfeld, Alan H.""

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## Scholarly Works (11 results)

What happens to pedagogy when a teacher’s personal goals of supporting students’ productive dispositions toward learning collide with her professional identity as a successful teacher whose students perform well on standardized tests? This dissertation is a mixed-methods case study that shows how context shapes one teacher’s identity and decision-making, such that she seems to be two drastically different teachers in two different instructional contexts – a summer course in which she had complete flexibility over the curriculum, goals, and achievement measures and an academic year course in which she felt bounded by the state standards test. The dissertation examines the very real consequences these pedagogical decisions have for students.

Using qualitative classroom observations and quantitative survey and assessment data, this dissertation examines why, despite the teacher’s strong commitment to growth mindset instruction and equity in both contexts, the teacher implemented pedagogical moves that contributed to distinctly different opportunities for students to engage with rich mathematics in each, and what those shifts meant for students’ mathematical identities and learning.

The different cultural contexts in the summer and academic years offered the teacher identity resources about what was valued as good teaching, which led to distinct pedagogical decisions that aligned with the salient aspects of her professional identity in each context. Despite her commitment to growth mindset instruction in both contexts, this teacher implemented pedagogical moves that contributed to distinctly different opportunities for students to engage with rich mathematics and develop productive mathematical self-concepts.

This dissertation examines the ways the institutional context shifted and practices changed subtly as a result, and uses these comparisons to unpack which elements of the whole system of teaching for a growth mindset are necessary to contribute to productive changes in student mindsets or dispositions toward mathematics, engagement, and persistence with learning. Using Ms. M as a case study, this dissertation sheds light on the ways in which school contexts - in concert with a teachers’ multifaceted identity - contribute to decision-making while setting instructional goals.

There is a strong push from within mathematics education reform to incorporate representations in math classrooms (Behr, Harel, Post, & Lesh, 1993; Kieren, 1993; NCTM, 2000). However, questions regarding what representations should be used (for a given topic) and how representations should be used (such that students gain a deep understanding of that topic and a deep understanding of the representations) remain largely unanswered. Hence, we need a well-specified and general theoretical treatment of how students co-develop domain and representational competence.

In this dissertation study, I use design-based research (DBR) to investigate and support growth and change in students' knowledge of rational number operations. "Among all the topics in K-12 curriculum, rational numbers arguably hold the distinction of being the most protracted in terms of development, the most difficult to teach, the most mathematically complex, the most cognitively challenging, and the most essential to success in higher mathematics and science" (Lamon, 2007). In order to shed some light on the domain of rational number operations, I designed a learning environment centered on the Area Model for Fraction Multiplication (AM-FM) representation, a computer-based tool intended to help students develop a deep understanding of fraction multiplication.

Data for the dissertation were collected from an urban school with a racially and socio-economically diverse student population. I met with ten students once a week for four weeks. During the first and last session students were asked to think-aloud through a pretest and posttest. The second and third sessions consisted of semi-structured clinical interviews during which students were asked to solve fraction multiplication problems using the AM-FM representation. All sessions were videotaped and transcribed. Two students were chosen to serve as cases of knowledge growth and change.

Findings indicate that both students followed a particular learning trajectory for making sense of fraction multiplication when using the AM-FM representation and their emergent knowledge was context sensitive. Futhermore, DBR is predicated on (a) design refinement and (b) local theory development (diSessa & Cobb, 2004; Schoenfeld, 2006). With respect to design, the AM-FM representation and the clinical interview protocol was refined based on analysis of the data. With respect to local theory, I offered a decomposition of competence with fraction multiplication (i.e., domain competence) and the AM-FM representation (i.e., representational competence). Local theory was also refined based on an analysis of the data.

Discussions of race in educational research have focused primarily on performance gaps and differential access to advanced coursework. Thus, very little is known about how race mediates the learning process, particularly with respect to classroom participation and student identity formation. This dissertation examines mathematics learning as a context for illuminating the racial dynamics of learning in everyday classroom activity.

Although mathematics and race may seem strange bedfellows, a poststructural analysis reveals specific linkages between them that suggest that their discourses are actually well aligned. To conceptualize this alignment, this dissertation introduces the theoretical frame of racial-mathematical discourse, which establishes the groundwork for the empirical investigation reported here. Observations took place in four mathematics classrooms at a racially diverse high school over the course of a school year. Interviews (n=35) were conducted with students from the focal classrooms. Data were analyzed to explore how students make sense of racial-mathematical discourse, and to gauge the discourse's impact on learning.

Findings indicate that racial-mathematical narratives were central to students' sense making. All students reported awareness of the "Asians are good at math" narrative, as part of a web of racial ideology. Importantly, students linked it to narratives about other groups' mathematical inferiority (e.g., "Blacks are bad at math"). They also connected racial-mathematical narratives to broader racialized discourses outside mathematics (e.g., perceptions of intelligence). Students observed the presence of these narratives in locations outside the school setting, such as media imagery and international comparisons.

Data further suggest that racial-mathematical discourse is not a static belief system. Rather, it emerges and is reified as students engage in typical classroom practices, such as noting which classmates get asked for help. This is consequential for learning, in that the deployment of racial narratives in social interaction frames students' opportunities to build identities as capable learners. This dissertation develops a framework leveraging insights from sociocultural and poststructural theory to trace the impact of racialized classroom episodes on students' identity formation. It highlights critical issues that need to be taken into account in the design of equitable learning environments, especially for students of color from persistently marginalized backgrounds.

What does it mean to be “good at math”? Traditionally, schools have valued getting the right answer quickly—a perspective that excludes important aspects of mathematics, as well as many students. This multi-site case study investigates how teachers work together to redefine mathematics and mathematical competence. The study involved more than a year of ethnographic observations and interviews at two diverse urban high schools on the West Coast of the United States, where teachers expressed strong commitments to serving all students, especially students from non-dominant backgrounds.

The dissertation tells a complex story of teacher learning, as viewed through the lenses of classroom instruction (Chapter 2), collegial conversation (Chapter 3), and the organization of teachers’ professional support networks (Chapter 4). Drawing on scholarship that takes learning as a negotiation of meaning through engagement in social practices (Vygotsky, 1986; Wenger, 1998; Saxe, 2012), the dissertation examines the relationships between extra-local systems of meaning and moment-to-moment interactions. Extending beyond prior work, the dissertation elucidates the negotiation of intensely conflicting meanings—namely, culturally dominant definitions of mathematics as a discipline, students as learners, and teachers as professionals, and non-dominant definitions that attempt to expand both teachers’ and students’ opportunities to engage with rich, challenging, and rewarding learning experiences.

In each of the contexts studied, navigating tensions between dominant, restrictive meanings and non-dominant, expansive meanings was a challenge for all of the teachers. Dominant discourses frame mathematical activity as consisting primarily of computation and memorization; mathematical ability as innate, fixed, and distributed along a bell-shaped curve; and the work of teaching as private, autonomous, and grounded in personal style and preference. In contrast, equity- and reform-oriented discourses frame mathematical activity as inclusive of a wide variety of skills and practices; position all students as capable learners; and position teachers as learners who benefit from ongoing collaboration and support. Dominant discourses are restrictive: they limit students’ opportunities to learn rich mathematics and teachers’ opportunities to negotiate equity- and reform-oriented shifts in their practice. But as teachers engage with non-dominant meanings that potentially expand learning opportunities, commonsense meanings do not simply disappear. Rather, they interact with non-dominant meanings in messy and complex ways that require careful study in order to understand how and what teachers learn.

The theme of negotiating meaning is laid out in Chapter 1, with a discussion of the dissertation’s underlying theoretical perspective. Research sites are introduced; the dissertation’s structure is presented; and major findings and contributions are highlighted.

Chapter 2, “(Re)Framing Mathematical Competence in Everyday Instruction: Struggles and Successes of Equity-Oriented Teachers,” examines tensions and contradictions in teachers’ classroom practice. It shows that despite the best intentions of the teachers in this study, many of their efforts to support all students position some students as capable of engaging with challenging mathematics and others as just the opposite. Conversely, teacher moves that are counterintuitive within dominant frames of teaching, employed by two of the teachers in the study, are shown to expand students’ opportunities to develop positive mathematical identities. The chapter thus contributes to conversations about what equitable mathematics instruction looks like, while illuminating obstacles—cultural as well as technical—that teachers face as they attempt to enact classroom practices that support all students.

Chapter 3, “Tensions in Equity- and Reform-Oriented Learning in Teachers’ Collaborative Conversations,” examines how collaborative conversations open up and close down opportunities for teachers to navigate the tensions between restrictive and expansive discourses of mathematical competence, through close analysis of a 9½-minute segment of a routine meeting of mathematics teachers. Although the group appeared to be an ideal professional learning community in many ways, and the focal interaction and others like it were generative in a number of respects, teacher talk enacted both restrictive and expansive discourses. The existence of tensions between these discourses presented opportunities for the teachers to negotiate non-dominant meanings for themselves, i.e., to learn; but the ways that teachers framed their own collaborative work interfered with these opportunities. By highlighting conversational norms that impede collaborative learning, the chapter contributes to the field’s understanding of the challenges of equity- and reform-oriented learning in teachers’ professional communities.

Ways of supporting teachers to negotiate expansive meanings are examined in Chapter 4, “Supporting Teachers’ Equity-Oriented Learning and Identities: A Resource-Centered Perspective.” The chapter investigates two cases of ongoing teacher engagement with non-dominant practice and two cases of relative disengagement, illustrating how various resources come together to support teachers’ learning and identity development (or not). Four types of resources are found to be critical, and learning and identity processes are shown to intertwine in mutually informing ways as teachers interact with these different resources.

In elucidating both challenges and supports associated with making sense of non-dominant meanings, this dissertation contributes to the field’s understanding of equity- and reform-oriented teacher learning and why it is so difficult. It also points to ways that the contexts in which teachers work might be constructed to support their engagement with non-dominant, expansive meanings, so that they can support all of their students to engage with rich, challenging mathematics and to develop identities as powerful learners and doers of mathematics.

This dissertation explores the question of how strategic and conceptual knowledge co-develop over the course of several episodes of mathematical problem solving. The core analytic work involves an in-depth microgenetic case study of a single pre-algebra student, Liam, who over six hours of videotaped interaction with a tutor/researcher constructs a deterministic and essentially algebraic algorithm for solving algebra word problems that have an underlying linear structure. Over six hours of videotaped interaction with a tutor/researcher, Liam's later and conceptually more sophisticated strategy is seen to emerge as a gradual refinement of his initial strategy. This focal case study is used to develop a theoretical model of how strategic and conceptual knowledge co-evolve. A novel aspect of the present analysis is that both strategies and the knowledge needed to implement them in problem solving are modeled as complex knowledge systems. The analytic methodology employed in developing the theoretical model is a coordination of Knowledge Analysis (diSessa, 1993; Sherin, 2001) and Microgenetic Learning Analysis (Parnafes & diSessa, submitted; Schoenfeld, Smith, & Arcavi, 1993). The model of co-development of strategic and conceptual knowledge that is developed through the analysis is one of mutual bootstrapping: (1) Within a given strategic frame, a solver activates a particular projection of conceptual knowledge and (2) As the solver creates new conceptual schemes in the context of working within a given particular strategic frame, novel refinements to existing strategies can emerge.

This dissertation is an exploratory study that investigates what and how teachers learn from use of instructional materials. Over the course of one school year, eight secondary mathematics teachers in two urban schools interspersed their usual instruction with “formative assessment lessons” (FALs), based on lessons and lesson guides designed to allow teachers to use student thinking to inform instructional decision-making. The focus of the study was on understanding how the teachers interpreted the goals and uses of these materials for their own and their students’ learning and how the materials supported teachers’ learning about formative assessment.

Teachers were expected to enact five FALs, which they selected from the Mathematics Assessment Project web site (map.mathshell.org) to place in curriculum units in accordance with their instructional goals. The lessons were planned and enacted without formal professional development. However, five collaboration meetings allowed teachers time to plan with colleagues and reflect on their experiences. Teacher surveys, classroom observations, and classroom artifacts were analyzed in order to examine the placement and enactment of the lessons as well as the influence of enacting the lessons on teachers’ classroom practices and use of formative assessment. The teachers’ choices in selecting, placing, planning, and implementing the lessons significantly limited their potential to support teacher learning. However, analyses of student explanations and the teachers’ interactions with small groups of students provide evidence of changes in classroom practices that allowed instruction to be better informed by student thinking.

Two case studies detail how teachers’ knowledge, goals, and beliefs, their perceptions of the lesson designers’ intentions, and the school environment influenced their use of the FALs and other materials, e.g., a monitoring sheet. Although the materials provided affordances that could support a range of teacher learning, the teachers’ choices played a fundamental role in determining what they learned from teaching the FALs and using the materials. In one case, the teacher developed and refined new classroom routines that reflected his goals for his students about communication and accountability. In the other case, the teacher began to change her teaching practices when using the FALs and monitoring sheet, but later adapted her use of the FALs and altered the monitoring sheet to be more consistent with her routine instruction, limiting her opportunities for exploration and learning. The implications of these findings for research on teacher learning, for revision of the FALs, and for professional development are discussed.

This study uses self-generated representations (SGR) - images produced in the act of explaining - as a means of uncovering what university calculus students understand about infinite series convergence. It makes use of student teaching episodes, in which students were asked to explain to a peer what that student might have missed had they been absent from class on the day(s) when infinite series were introduced and discussed. These student teaching episodes typically resulted in the spontaneous generation of several SGR, which provided physical referents with which both the student and an interviewer were able to interact. Students' explanations, via their SGR, included many more aspects of what they found important about that content than did the standard research technique of asking students to answer specific mathematics tasks.

This study was specifically designed to address how students construct an understanding of infinite series. It also speaks to the broader goal of examining how students use SGR as a tool for explaining concepts, rather than simply as tools for solving specific problems. The main analysis indicates that both students and their professors/textbook, when introducing the topic of infinite series, make use of the following five different image types: plots of terms, plots of partial sums, areas under curves, geometric shapes, and number lines. However, the aspects of the mathematical concepts that the students and the professors/textbooks highlight in their explanations and modes of use for those image types are different, and at times conflicting. In particular, differences emerged along three dimensions of competence - limiting processes (Tall, 1980), language, and connections.

While students using SGR generated many of the images that had been used by their professors, the limiting processes that they discussed via those images contrasted sharply. The professors and textbook chapter prioritized the limiting processes represented in particular image types to support mathematically sound conclusions. In contrast, many student explanations focused on limiting processes that did not lead to valid arguments about series convergence. There were also differences in use of language, in that students often assumed much more meaning than was intended in their professors' language choices, leading to problems with their explanations. Finally, while the experts connected their representations in meaningful ways, using other images to clarify or exemplify those that were used to define, students connected their understanding in different ways that were not always supportive of the convergence arguments that they were trying to make.

This study expands the literature on students' understanding of infinite series topics, pointing to gaps in student understanding and ways in which students mis-applied what teachers had presented. In doing so, it suggests many avenues for improving infinite series instruction. In addition, the methods employed in this study are general, and open up ways of looking at student thinking that can be applied to many problematic areas in the curriculum. Typical studies ask students to address tasks and issues framed by a researcher. This study instead asked students to explain the content, thereby providing a much larger window into what counts, from the student perspective.

Students with mathematical learning disabilities (MLDs) experience persistent challenges learning even the most elementary mathematics. While prior research on MLDs has classified students on the basis of test scores and documented performance differences between groups, this dissertation focuses on the qualitative differences in individual students with MLDs as each student attempted to learn. This study extends the understanding of MLDs by (1)focusing on the mathematical domain of fractions, (2)selecting for study students whose difficulties with mathematics are clearly due to an MLD and not other factors, (3)conducting detailed diagnostic analyses of video-taped tutoring sessions, and (4)contrasting the difficulties that the students with MLDs experience to data collected with five typically achieving fifth grade control students. The two case study students, "Lisa" (19-year-old community college student) and "Emily" (18-year-old high school student) each participated in a pretest, four weekly tutoring sessions, and a posttest focused on fraction concepts. Achievement test scores, interviews, and videotaped tutoring session data were used to establish that Lisa and Emily met classic MLD qualifications, and more stringent response-to-intervention criteria. Both students demonstrated unexplained persistent low math achievement and neither student benefited from a tutoring protocol that had been effective for typically achieving fifth grade students. The tutoring sessions were analyzed using microgenetic methods. Analysis indicated that each student relied upon a unique collection of atypical understandings, which reoccurred across the sessions, were resistant to standard instructional approaches, and proved to be highly consequential for the student's ability to understand more complex fraction concepts. A cross case analysis revealed surprising similarities in the atypical understandings displayed by both case study students. These atypical understandings stemmed from and contributed to the student's inability to conceptualize and manipulate representations of fractional quantity. These atypical understandings were not similarly problematic for the fifth grade control students, but did appear in one additional student with an MLD. This suggests that there are qualitative differences in the difficulties experienced by students with MLDs and that it may be possible to design screening measures to identify these indicators of atypicality. In addition, remediation approaches should take into account and specifically target these atypicalities.

Professional learning and development in and from practices have focused either on how individuals learn from formally organized contexts or on how a group of teachers collectively develop within contexts. Thus there is little research on individual teacher professional learning through practices. This dissertation investigates the ways in which two experienced mathematics teachers develop the means of making sense of content and student thinking, and change their practices through their interactions with innovative curriculum materials. This dissertation also conceptualizes a new framework for teaching practices termed student thinking responsive teaching practices, which is grounded in the literature and this dissertation’s empirical investigations.

In my dissertation, I conducted case studies with two teachers, observing their classrooms over the course of a year to capture their everyday teaching practices during regular lessons, as well as their teaching practices during lessons involving innovative curriculum materials. I also conducted teacher interviews and observed professional development workshop sessions to inform my analysis of how the teachers changed their teaching practices in the classroom. I analyzed the collected data using a mixed methods approach. First, I conducted a qualitative analysis with an existing coding scheme from the literature and a new coding scheme I specifically developed. Second, I quantified my qualitative analysis findings to capture how and whether the two teachers changed toward becoming more responsive to student mathematical thinking in their everyday practices.

My dissertation findings clarify some of the mechanisms for change in teacher practices. First, when teachers are supported in, and make use of, instructional moves that focus on student mathematical thinking, not only is student learning advanced, but teacher learning is also advanced. Second, when teachers’ interactions with curriculum materials involve adapting and developing new pedagogical strategies for student thinking responsive teaching practices, they create opportunities to making sense of content and student thinking. In contrast, when they adapt curriculum materials to align with their existing practices, those opportunities are lost. Lastly, in addition to teachers’ beliefs and knowledge, their professional identity is critical for understanding the ways in which teachers interact with and implement curriculum materials. It is important for teachers to see and make sense of themselves as learners and as professionals who are continuously reflecting on and through their practices. Such an identity may become a catalyst for changing their practices and need to be supported not only by innovative curriculum materials, but also at the school and district level. These results have implications for the design of professional development systems that support the positive aspects of curriculum-based learning.