Kay McClain

Participating in Classroom Mathematical Practices

In this article, we describe a methodology for analyzing the collective learning of the classroom community in terms of the evolution of classroom mathematical practices. To develop the rationale for this approach, we first ground the discussion in our work as mathematics educators who conduct classroom-based design research. We then present a sample analysis taken from a 1st-grade classroom teaching experiment that focused on linear measurement to illustrate how we coordinate a social perspective on communal practices with a psychological perspective on individual students’ diverse ways of reasoning as they participate in those practices. In the concluding sections of the article, we frame the sample analysis as a paradigm case in which to clarify aspects of the methodology and consider its usefulness for design research.

Situating Teachers’ Instructional Practices in the Institutional Setting of the School and District:

In this article, we describe an analytic approach for situating teachers’ instructional practices within the institutional settings of the schools and school districts in which they work. In doing so, we draw on an ongoing collaboration with a group of teachers in an urban school district to illustrate both the approach and its usefulness in guiding the development of analyses that feedback to inform such collaborations. The approach involves delineating communities of practice within a school or district and analyzing three types of interconnections between them that are based on boundary encounters, brokers, and boundary objects.

An Approach for Supporting Teachers’ Learning in Social Context

Our purpose in this chapter is to outline a general approach to collaborating with teachers in order to support the establishment of a professional teaching community. As will become apparent, our goal is to help teachers develop instructional practices in which they induct their students into the ways of reasoning of the discipline by building systematically on their current mathematical activity. We develop the rationale for the approach we propose by describing how our thinking about in-service teacher development has evolved over the last thirteen years or so. To this end, we first revisit work conducted in collaboration with Erna Yackel and Terry Wood between 1986 and 1992 in which we supported the development of American second- and third-grade teachers. In doing so, we tease out aspects of the approach we took that still appear viable and discuss two major lessons that we learned. In the next section of the chapter, we draw on a series of teaching experiments we have conducted over the past seven years in American elementary and middle-school classrooms both to critique our prior work and to develop three further aspects of the approach we propose. We conclude by highlighting broad features of the approach and by locating them in institutional context.

Principles of Instructional Design for Supporting the Development of Students’ Statistical Reasoning

This chapter proposes design principles for developing statistical reasoning in elementary school. In doing so, we will draw on a classroom design experiment that we conducted several years ago in the United States with 12-year-old students that focused on the analysis of univariate data. Experiments of this type involve tightly integrated cycles of instructional design and the analysis of students’ learning that feeds back to inform the revision of the design. However, before giving an overview of the experiment and discussing specific principles for supporting students’ development of statistical reasoning, we need to clarify that we take a relatively broad view of statistics. The approach that we followed in the classroom design experiment is consistent with G. Cobb and Moore’s (1997) argument that data analysis comprises three main aspects: data generation, exploratory data analysis (EDA), and statistical inference. Although Cobb and Moore are primarily concerned with the teaching and learning of statistics at the college level, we contend that the major aspects of their argument also apply to the middle and high school levels. EDA involves the investigation of the specific data at hand (Shaughnessey, Garfield, & Greer, 1996). Cobb and Moore (1997) argue that EDA should be the initial focus of statistics instruction since it is concerned with trends and patterns in data sets and does not involve an explicit consideration of sample-population relations. In such an approach, students therefore do not initially need to support their conclusions with probabilistic statements of confidence. Instead, conclusions are informal and are based on meaningful patterns identified in specific data sets. Cobb and Moore’s (1997) proposal reflects their contention that EDA is a necessary precursor to statistical inference. Statistical inference is probabilistic in that the intent is to assess the likelihood that patterns identified in a sample are not

Guest Editor's Introduction: Analyzing Tools: Perspectives on the Role of Designed Artifacts in Mathematics Learning

Concentrating the attention on one aspect makes it leap into the foreground and occupy the square, just as, with certain drawings, you have only to close your eyes and when you open them the perspective has changed. (Calvino, 1983, p. 7)

The Collective Mediation of a High-Stakes Accountability Program: Communities and Networks of Practice

This article describes an analytic approach for situating teachers’ instructional practices within the institutional settings of the schools and school districts in which they work. The approach treats instructional leadership and teaching as distributed activities and involves first delineating the communities of practice within a school or district whose enterprises are concerned with teaching and learning and then analyzing three types of interconnections between them: boundary encounters, brokers, and boundary objects. We illustrate the analytic approach by focusing on one urban school district in which we have conducted an ongoing collaboration with a group of middle school teachers. In doing so, we clarify the critical role that school and district-level leaders can play in mediating state and federal high-stakes accountability policies. We conclude by discussing the implications of the analysis for the process of upscaling and the diffusion of instructional innovations.

An analysis of students’ initial statistical understandings: developing a conjectured learning trajectory ☆

Abstract This paper reports the analysis of performance assessment tasks administered in a seventh-grade classroom. The purpose of the assessments was to obtain data on students’ current statistical understandings that would then inform future instructional design decisions in a classroom teaching experiment that focused on statistical data analysis. The tasks were designed to provide information about students’ current understandings of creating data, organizing data, and assessing the center and “spreadoutness” of data. In considering the analysis, we found that the students typically viewed the mean as a procedure that was to be used to summarize a group of numbers regardless of the task situation. Data analysis for these students meant “doing something with the numbers.” Based on this analysis, a goal that emerged as significant for the classroom teaching experiment was to support a shift in students’ reasoning towards data analysis as inquiry rather than procedure. The influence of the students’ prior experiences of doing mathematics in school was also apparent when they developed graphs. They were primarily concerned with school-taught graphical conventions rather than with what the graphs signified. In the course of the analysis we distinguished between additive and multiplicative reasoning about data. This distinction is significant given that the transition from additive to multiplicative reasoning constitutes the overriding goal of statistics instruction at the middle-school level.

Guest Editor's Introduction: Appendix: The Object and the Context: What Our Data Are and Where They Come From

The data discussed in the articles in this issue are taken from a classroom teaching experiment conducted in the fall semester of 1997 with a group of 29 American seventh-grade students (age 12). During the 12 weeks of the teaching experiment, the research team1 assumed total responsibility for the class sessions, including teaching.2 The primary goal for the teaching experiment was to investigate ways to proactively support middle-school students’ ability to reason about data while developing statistical understandings related to exploratory data analysis. An integral aspect of that understanding entailed students coming to view data sets as distributions. In that process, they would structure and organize data sets multiplicatively as they created data-based arguments that were grounded in their analysis. The research team’s interest was motivated by current debates about the role of statistics in school curricula (cf. Burrill, 1996; Burrill & Romberg, in press; Cobb, 1997; Gal, 2000; Lajoie, in press; Lajoie & Romberg, in press; Lipson & Jones, 1996; National Council of Teachers of Mathematics, 1989, 1991, 2000; Shaughnessy, 1992, 1996). The guiding image that emerged as the research team read and synthesized THE JOURNAL OF THE LEARNING SCIENCES, 11(2&3), 163–185 Copyright