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Constructing Goals for Student Learning through Conversation

Jessica Hunt and Mary Kay Stein

and procedures that they will ultimately be accountable for knowing. Meeting this challenge demands well‐thought‐out learning goals, not only to plan lessons but also to select cognitively demanding tasks that support the goal and to plan questioning

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Mathematical Micro-Identities: Moment-to-Moment Positioning and Learning in a Fourth-Grade Classroom

Marcy B. Wood

Identity is an important tool for understanding students' participation in mathematics lessons. Researchers usually examine identity at a macro-scale: across typical classroom activity and in students' self-reports. However, learning occurs on a micro-scale: in moments during a lesson. To capture identity in these moments, I used positioning theory to develop a framework of micro-identity and then to examine the identities and learning of 1 fourth-grade student during 1 mathematics lesson. This study demonstrates how mathematical identities can shift in dramatic ways in response to minor changes in context so that a student might be, in one moment, engaged in an identity that undermines learning and then later engaged in an academically productive identity. These shifting micro-identities have important implications for mathematical learning, classroom contexts, and macro-identities.

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Understanding Learning Systems: Mathematics Education and Complexity Science

Brent Davis and Elaine Simmt

Complexity science may be described as the science of learning systems, where learning is understood in terms of the adaptive behaviors of phenomena that arise in the interactions of multiple agents. Through two examples of complex learning systems, we explore some of the possible contributions of complexity science to discussions of the teaching of mathematics. We focus on two matters in particular: the use of the vocabulary of complexity in the redescription of mathematical communities and the application of principles of complexity to the teaching of mathematics. Through the course of this writing, we attempt to highlight compatible and complementary discussions that are already represented in the mathematics education literature.

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Learning to Estimate in the Mathematics Classroom: A Conversation-Analytic Approach

Michael A. Forrester and Christopher D. Pike

In contrast to contemporary estimation researchers who have focused primarily on children's computational estimation abilities, we examined the ideas surrounding the teaching and learning of measurement estimation in the classroom. Employing ethnomethodologically informed conversation analysis, we focused on 2 teachers' instructions during estimation lessons and on pupils' (aged 9-11 years) talk during small-group follow-up activities. The results indicated that estimation is understood as discursively interdependent with measurement and is associated both with teacher-formulated accountability and with vagueness, ambiguity, and guessing. Furthermore, the meaning of what it is to estimate is embedded in practical action. In concluding comments we consider the advantages of using conversational analysis as a method for highlighting the relationships between language and mathematics in the classroom.

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ZPC and ZPD: Zones of Teaching and Learning

Anderson Norton and Beatriz S. D'Ambrosio

The goal of this article is to examine students' mathematical development that occurs as a teacher works within each of 2 zones of learning: students' zones of proximal development (ZPD) and students' zones of potential construction (ZPC). ZPD, proposed by Vygotsky, is grounded in a social constructivist perspective on learning, whereas ZPC, proposed by Steffe, is grounded in a radical constructivist perspective on learning. In this article, we consider potential pragmatic differences between ZPD and ZPC as instantiated during a semester-long teaching experiment with 2 Grade 6 students. In particular, we examine the constructions that a teacher fostered by working with these students in each zone of learning. The data suggest that operating in their ZPD but outside of their ZPC impacts the learning opportunities and resulting constructions of the students. Finally, we characterize aspects of ZPD and teacher assistance that foster the development of mathematical concepts.

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Abstract or Concrete Examples in Learning Mathematics? A Replication and Elaboration of Kaminski, Sloutsky, and Heckler's Study

Dirk De Bock, Johan Deprez, Wim Van Dooren, Michel Roelens, and Lieven Verschaffel

Kaminski, Sloutsky, and Heckler (2008a) published in Science a study on “The advantage of abstract examples in learning math,” in which they claim that students may benefit more from learning mathematics through a single abstract, symbolic representation than from multiple concrete examples. This publication elicited both enthusiastic and critical comments by mathematicians, mathematics educators, and policymakers worldwide. The current empirical study involves a partial replication–but also an important validation and extension–of this widely noticed study. The study's results confirm Kaminski et al.'s findings, but the accompanying qualitative data raise serious questions about their interpretation of what students actually learned from the abstract concept exemplification. Moreover, whereas Kaminski et al. showed that abstract learners transferred what they had learned to a similar abstract context, this study shows also that students who learned from concrete examples transferred their knowledge into a similar concrete context.

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Learning From Teaching: Exploring the Relationship Between Reform Curriculum and Equity

Jo Boaler

Some researchers have expressed doubts about the potential of reform-oriented curricula to promote equity. This article considers this important issue and argues that investigations into equitable teaching must pay attention to the particular practices of teaching and learning that are enacted in classrooms. Data are presented from two studies in which middle school and high school teachers using reform-oriented mathematics curricula achieved a reduction in linguistic, ethnic, and class inequalities in their schools. The teaching and learning practices that these teachers employed were central to the attainment of equality, suggesting that it is critical that relational analyses of equity go beyond the curriculum to include the teacher and their teaching.

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Teaching as Assisting Individual Constructive Paths Within an Interdependent Class Learning Zone: Japanese First Graders Learning to Add Using 10

Aki Murata and Karen Fuson

The framework of Tharp and Gallimore (1988) was adapted to form a ZPD (Zone of Proximal Development) Model of Mathematical Proficiency that identifies two interacting kinds of learning activities: instructional conversations that assist understanding and practice that develops fluency. A Class Learning Path was conceptualized as a classroom path that includes a small number of different learning paths followed by students, and it permits a teacher to provide assistance to students at their own levels. A case study illustrates this model by describing how one teacher in a Japanese Grade 1 classroom assisted student learning of addition with teen totals by valuing students' informal knowledge and individual approaches, bridging the distance between their existing knowledge and the new culturally valued method, and giving carefully structured practice. The teacher decreased assistance over time but increased it for transitions to new problem types and for students who needed it. Students interacted, influenced/supported one another, and moved forward along their own learning paths within the Class Learning Path.

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Untangling Teachers' Diverse Aspirations for Student Learning: A Crossdisciplinary Strategy for Relating Psychological Theory to Pedagogical Practice

David Kirshner

The Learning Principle propounded in Principles and Standards for School Mathematics (NCTM, 2000) rehearses the familiar distinction between facts/procedures and understanding as a central guiding principle of teaching reform. This rhetorical stance has polarized mathematics educators in the “math wars,” (Becker & Jacob, 1998), while creating the discursive space for mathematics teaching reform to be reified into a unitary “reform vision” (Lindquist, Ferrini-Mundy, & Kilpatrick, 1997)—a vision that teachers can all too easily come to see themselves as implementing rather than authoring. Crossdisciplinarity is a strategy for highlighting the discrete notions of learning that psychology thus far has succeeded in coherently articulating. This strategy positions teachers to consult their own values, interests, and strengths in defining their own teaching priorities, at the same time marshaling accessible, theory-based guidance toward realization of its diverse possibilities.

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Teaching Algebra to Students with Learning Disabilities

Marcee M. Steele and John W. Steele

In many states, students with learning disabilities are required to take algebra in high school. These students are usually served in inclusive settings (the general education classroom), where they often struggle with algebra because the content is so abstract. Although limited research and literature are available on algebra instruction for students with special needs, recognized strategies that promote students' learning can help make the algebra experience more enjoyable and successful for them and for the other students in the class, as well. This article highlights characteristics of students with learning disabilities, reviews current literature on algebra and students with learning disabilities, summarizes some of the recommendations, and describes how they can be put into practice.