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Mathematics in the Junior High School: Textbook series and learning theory

G. W. Brown

Edited by Lucien B. Kinney and Dan T. Dawson

Do textbook series in arithmetic develop the mathematical concepts and skills in accordance with current learning theory? This is an important question for the junior high school mathematics teacher. The answer determines the extent to which the teacher may rely on the textbook for guidance in the learning sequence. Since, for many students, mathematics instruction ceases at ninth grade, mathematical strands considered by teachers and textbook writers to be important must be brought to a culminating point by that time.

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Research Commentary: Amidst Multiple Theories of Learning in Mathematics Education

Martin A. Simon

Currently, there are more theories of learning in use in mathematics education research than ever before (Lerman & Tsatsaroni, 2004). Although this is a positive sign for the field, it also has brought with it a set of challenges. In this article, I identify some of these challenges and consider how mathematics education researchers might think about, and work with, the multiple theories available. I present alternatives to views of the competition or supersession of theories and indicate the kinds of discussions that will support effective theory use in mathematics education research. I describe the potential for mathematics education researchers to make informed, justified choices of a theory or theories to address particular research agendas.

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A Case of Interpretations of Social: A Response to Steffe and Thompson

Stephen Lerman

In their response to my (1996) article, Steffe and Thompson argued that I have taken an early position of Vygotsky's and that his later work is subsumed in and developed by von Glasersfeld. I argue that the two theories, Vygotsky's and radical constructivism, are, on the contrary, quite distinct and that this distinction, when seen as a dichotomy, is productive. I suggest that radical constructivists draw on a weak image of the role of social life. I argue that a thick notion of social leads to a complexity of sociocultural theories concerning the teaching and learning of mathematics, a perspective that is firmly located in the debates surrounding cultural theory of the last 2 decades.

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Nonstandard Student Conceptions About Infinitesimals

Robert Ely

This is a case study of an undergraduate calculus student's nonstandard conceptions of the real number line. Interviews with the student reveal robust conceptions of the real number line that include infinitesimal and infinite quantities and distances. Similarities between these conceptions and those of G. W. Leibniz are discussed and illuminated by the formalization of infinitesimals in A. Robinson's nonstandard analysis. These similarities suggest that these student conceptions are not mere misconceptions, but are nonstandard conceptions, pieces of knowledge that could be built into a system of real numbers proven to be as mathematically consistent and powerful as the standard system. This provides a new perspective on students' “struggles” with the real numbers, and adds to the discussion about the relationship between student conceptions and historical conceptions by focusing on mechanisms for maintaining cognitive and mathematical consistency.

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Conjectures and Refutations in Grade 5 Mathematics

David A. Reid

This article makes a contribution toward clarifying what mathematical reasoning is and what it looks like in school contexts. It describes one pattern of reasoning observed in the mathematical activity of students in a Grade 5 class and discusses ways in which this pattern is or is not mathematical in order to clarify the features of a pattern of reasoning that are important for making such a judgment. The pattern involves conjecturing a general rule, testing that rule, and then either using it for further exploration, rejecting it, or modifying it. Each element of reasoning in this pattern is described in terms of the ways of reasoning used and the degree of formulation of the reasoning. A distinction is made between mathematical reasoning and scientific reasoning in mathematics, on the basis of the criteria used to accept or reject reasoning in each domain.

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“We Want a Statement That Is Always True”: Criteria for Good Algebraic Representations and the Development of Modeling Knowledge

Andrew Izsák

This article presents a case study in which two eighth-grade students developed knowledge for modeling a physical device called a winch. In particular, the students learned (a) to distinguish equations that are true for any value of the independent variable from equations that constrain the independent variable to a unique value and (b) to solve the latter type of equation to determine when specific physical events occur. The analysis of how these understandings emerged led to two results. First, the analysis demonstrated that students have and can use criteria for evaluating algebraic representations. Second, the analysis led to a theoretical frame that explains how students can develop modeling knowledge by coordinating such criteria with knowledge for generating and using algebraic representations. The frame extends research on students' algebraic modeling, cognitive processes and structures for using mathematical representations, and the development of mathematical knowledge.

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The Use of Symbols, Words, and Diagrams as Indicators of Mathematical Cognition: A Causal Model

Curtis L. Pyke

This article reports on the results of a study that investigated the strategic representation skills of eighth-grade students while they were engaged in a set of tasks that involved applying geometric knowledge and using algebraic equations. The strategies studied were derived from Dual Coding Theory (DCT) (Paivio, 1971, 1990), and they were elicited with task-specific prompts embedded in an assessment developed for the study. The purpose of the study was to test a model that highlights strategic representation as a mediator of the effects of reading ability, spatial ability, and task presentation on problem solving. The proposed model was tested using the linear structural equations modeling approach to causal analysis and the data did not reject the model. The results showed that students' use of symbols, words, and diagrams to communicate about their ideas each contribute in different ways to solving tasks and reflect different kinds of cognitive processes invested in problem solving.

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Learning Mathematics in a Classroom Community of Inquiry

Merrilyn Goos

This article considers the question of what specific actions a teacher might take to create a culture of inquiry in a secondary school mathematics classroom. Sociocultural theories of learning provide the framework for examining teaching and learning practices in a single classroom over a two-year period. The notion of the zone of proximal development (ZPD) is invoked as a fundamental framework for explaining learning as increasing participation in a community of practice characterized by mathematical inquiry. The analysis draws on classroom observation and interviews with students and the teacher to show how the teacher established norms and practices that emphasized mathematical sense-making and justification of ideas and arguments and to illustrate the learning practices that students developed in response to these expectations.

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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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Playing Mathematical Instruments: Emerging Perceptuomotor Integration With an Interactive Mathematics Exhibit

Ricardo Nemirovsky, Molly L. Kelton, and Bohdan Rhodehamel

Research in experimental and developmental psychology, cognitive science, and neuroscience suggests that tool fluency depends on the merging of perceptual and motor aspects of its use, an achievement we call perceptuomotor integration. We investigate the development of perceptuomotor integration and its role in mathematical thinking and learning. Just as expertise in playing a piano relies on the interanimation of finger movements and perceived sounds, we argue that mathematical expertise involves the systematic interpenetration of perceptual and motor aspects of playing mathematical instruments. Through 2 microethnographic case studies of visitors who engaged with an interactive mathematics exhibit in a science museum, we explore the real-time emergence of perceptuomotor integration and the ways in which it supports mathematical imagination.