In describing the role research should play in the new National Council of Teachers of Mathematics (NCTM) Principles and Standards for School Mathematics: Discussion Draft (1998), the authors noted, “With an emphasis on understanding mathematics that is fundamental, educators can be supported as they move beyond some of the superficial interpretations that have been made of ideas from the previous Standards documents. Currently, there are instructional programs that have emphasized solely some of the pedagogical intentions of standards--such as discourse, worthwhile mathematical tasks, or learning from problems--without sufficient attention to students' learning of mathematics content. (p. 17)” However, a major issue in the field of mathematics education is what type or types of research should play this critical role (Kelly & Lesh, 1999). In particular, a major focus in the current debates is whether controlled experimental or quasi-experimental research--conducted in real classroom settings--should be the predominant determinant of policy and practice

### Douglas Carnine and Russell Gersten

### Jennifer K. Jacobs and Eiji Morita

This article describes a novel assessment method used to examine Japanese and American teachers' ideas about what constitutes effective mathematics pedagogy. Forty American and 40 Japanese teachers independently evaluated either an American or Japanese mathematics lesson captured on videotape. Their comments were classified into over 1600 *idea units*, which were then sorted into a hierarchy of categories derived from the data. Next, the authors hypothesized underlying *ideal* instructional scripts that could explain the patterns of responses. Whereas the U.S. teachers were supportive of both traditional and nontraditional elementary school mathematics instruction and had different scripts for the two lessons, the Japanese teachers had only one ideal lesson script that was closely tied to typical Japanese mathematics instruction. The findings suggest that U.S. teachers may have more culturally sanctioned options for teaching mathematics; however, Japanese teachers may have a more detailed and widely shared theory about how to teach effectively.

### Eva Thanheiser, Amy Ellis and Beth Herbel-Eisenmann

In this Research Commentary, 3 *JRME* authors describe the process of publishing their research in *JRME*. All 3 authors published parts of their dissertation in *JRME* and are sharing their stories to help (new) researchers in mathematics education better understand the process and to offer (experienced) researchers in mathematics education a tool that can be used to mentor their less experienced colleagues and students. The authors address preparing, conceptualizing, and writing a manuscript as well as responding to reviewers.

### Margaret Walshaw

In an era when familiar categories of identity are breaking down, an argument is made for using post-structuralist vocabulary to talk about ethical practical action in mathematics education. Using aspects of Foucault's post-structuralism, an explanation is offered of how mathematical identifications are tied to the social organization of power. An analysis of 2 everyday instances is provided to capture the oppressive conditions in which ordinary people involved in mathematics are engaged. Describing how systemic constraints become lived as individual dilemmas offers a way of understanding what we might do to effect change, and what we might do to produce tangible results.

### Raffaella Borasi, Marjorie Siegal, Judith Fonzi and Constance F. Smith

In this study we explore the potential for mathematics instruction of four reading strategies grounded in transactional reading theory. On the basis of the descriptive study of 18 instructional episodes developed in 4 secondary mathematics classes as a result of collaborative action research, we show that encouraging mathematics students to talk, write, draw, and enact texts can provide them with concrete ways to construct and negotiate interpretations of what they read. In addition to helping students better understand the text read, acting on and acting out a text allow students to use that text as a springboard for sense-making and discussion of important mathematical ideas and issues about the nature of mathematics, especially when these reading experiences are supported by compatible classroom norms and values.

### David W. Stinson

This article shows how equity research in mathematics education can be decentered by reporting the “voices” of mathematically successful African American male students as they recount their experiences with school mathematics, illustrating, in essence, how they negotiated the White male math myth. Using post-structural theory, the concepts discourse, person/identity, and power/agency are reinscribed or redefined. The article also shows that using a post-structural reinscription of these concepts, a more complex analysis of the multiplicitous and fragmented robust mathematics identities of African American male students is possible—an analysis that refutes simple explanations of effort. The article concludes, not with “answers,” but with questions to facilitate dialogue among those who are interested in the mathematics achievement and persistence of African American male students—and equity and justice in the mathematics classroom for all students.

### Jane M. Watson and Jonathan B. Moritz

A key element in developing ideas associated with statistical inference involves developing concepts of sampling. The objective of this research was to understand the characteristics of students' constructions of the concept of *sample*. Sixty-two students in Grades 3, 6, and 9 were interviewed using open-ended questions related to sampling; written responses to a questionnaire were also analyzed. Responses were characterized in relation to the content, structure, and objectives of statistical literacy. Six categories of construction were identified and described in relation to the sophistication of developing concepts of sampling. These categories illustrate helpful and unhelpful foundations for an appropriate understanding of representativeness and hence will help curriculum developers and teachers plan interventions.

### 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.

### Jane M. Watson and Jonathan B. Mortiz

One hundred eight students in Grades 3, 5, 6, 7, and 9 were asked about their beliefs concerning fairness of dice before being presented with a few dice (at least one of which was “loaded') and asked to determine whether each die was fair. Four levels of beliefs about fairness and four levels of strategies for determining fairness were identified. Although there were structural similarities in the levels of response, the association between beliefs and strategies was not strong. Three or four years later, we interviewed 44 of these students again using the same protocol. Changes and consistencies in levels of response were noted for beliefs and strategies. The association of beliefs and strategies was similar after three or four years. We discuss future research and educational implications in terms of assumptions that are often made about students' understanding of fairness of dice, both prior to and after experimentation.

### Andreas L. Stylianides

Many researchers and curriculum frameworks recommend that the concept of *proof* and the corresponding activity of *proving* become part of students' mathematical experiences throughout the grades. Yet it is still unclear what “proof” means in school mathematics, especially in the elementary grades, and what role teachers have in cultivating proof and proving among their students. In this article, I propose a conceptualization of the meaning of proof in school mathematics and use classroom episodes from third grade to elaborate elements of this conceptualization and to illustrate its applicability even in the early elementary grades. Furthermore, I use the conceptualization to develop a tool to analyze the classroom episodes and to examine aspects of the teachers' role in managing their students' proving activity. This analysis supports the development of a framework about instructional practices for cultivating proof and proving in school mathematics.