Journal for Research in Mathematics Education Monograph

Mathematical knowledge, primarily the practice of arithmetic, which is developed outside of educational institutions by people engaged in their daily working and domestic activities, has been documented. The idea that different forms of mathematics may develop as a result of different modes of thinking and different environmental demands across cultural groups, has been suggested by research in ethnomathematics. However, there exist no clear guidelines for recognising and describing such culture-specific mathematics, which may differ significantly from the conventional view of mathematics. This article describes a study of the mathematical ideas of a group of carpenters, focusing mainly on their geometric ideas, and provides an account of the methodological strategy which was used. The researcher conducted an ethnographic study over the period of six months, working as an apprentice with a group of carpenters in Cape Town, South Africa. One descriptive example of an interaction with two carpenters involved in problem solving is analyzed in this article. The results show that the carpenters engaged in a wide variety of mathematizing activities in order to solve the problem.

Recent emphasis on discourse in mathematics classrooms has spurred a line of inquiry about different forms of talk in these settings. If mathematical thinking is understood to be a set of practices that includes mathematical discourse, argumentation, which has an especially important role in mathematics, requires analytic attention. In particular, the following questions arise: How can classroom discourse be organized to support mathematical disagreements that (a) are intellectually productive, and (b) minimize social discomfort? This paper investigates the interactional organization of public disagreements in Deborah Ball's third grade classroom by describing a participant structure called accountable argumentation. The norms, expectations, interactional roles, and use of history employed during accountable argumentation are explicated and then applied to the analysis of two public peer disagreement episodes that take place during the same whole-class discussion. These two episodes illustrate the ways in which accountable argumentation supports mathematical learning through disagreement, while mitigating the potentially uncomfortable feelings typically expected in such interactions. (Contains 35 references.) (Author/ASK)

This monograph represents 5 years of collaborative discussions among individuals interested in improving the teaching and learning of mathematics beginning in 1985 with the planning of a conference and culminating in the production of this collection of 13 papers to be presented at the conference. Chapters include: (1) an introduction "Constructivist Views on the Teaching and Learning of Mathematics"; (2) "Constructivism in Mathematics Education" (Nel Noddings); (3) "An Exposition of Constructivism: Why Some Like it Radical" (Ernst von Glasersfeld); (4) "Epistemology, Constructivism, and Discovery Learning Mathematics" (Gerald A. Goldin); (5) "Children's Learning: A Cognitive View" (Arthur J. Baroody & Herbert P. Ginsburg); (6) "The Nature of Mathematics: What Do We Do When We'Do' Mathematics?" (Robert B. Davis & Carolyn A. Maher); (7) "Teacher's Learning: Building Representations of Children's Meanings" (Carolyn A. Maher & Robert B. David); (8) "Discovery Learning and Constructivism" (Robert B. Davis); (9) "What Constructivism Implies for Teaching," (Jere Confrey); (10) "Classrooms as Learning Environments for Teachers and Researchers" (Paul Cobb, Terry Wood, & Erma Yackel); (11) "Teacher Development in Mathematics in a Constructivist Framework" (Carolyn A. Maher & Alice Alston); (12) "On the Knowledge of Mathematics Teachers" (Leslie P. Steffe); and (13) a conclusion, "Suggestions for the Improvement of Mathematics Education" (Robert B. Davis, Carolyn A. Maher & Nel Noddings). (CW)

This monograph chronicles a field-based investigation of children's learning in a second grade mathematics classroom over the course of a school year. The interrelated aspects of the processes by which children learn mathematics with meaning are illustrated here. The investigation was expanded to include a sociological analysis in an attempt to understand the relation between individual mathematical constructions and classroom social interactions. Data include clinical interviews of individual children's mathematical constructions and video recordings of classroom lessons. Microanalyses of specific episodes from the classroom are included in order to illustrate the theoretical constructs that evolved over the course of the investigation. The material is organized in three parts that relate to multiple perspectives of learning mathematics; culture, community, and the meaning of mathematics; and commentary on issues and reform in mathematics education. Contains 164 references. (DDR)

"[Note: This abstract applies to all articles within this issue."] The chapters in this monograph describe qualitative research methods used to investigate students' and teachers' interactions with school mathematics. Each contributing author uses data from his or her own research to illustrate a particular technique or aspect of research design. The different chapters present a wide range of methods, representing a variety of goals and perspectives. Rather than a comprehensive reference manual, this monograph illustrates the diversity of methods available for qualitative research in mathematics education. The monograph begins with a discussion of key elements that contribute to the dynamic and evolving domain of mathematics education research. Background information is then provided that relates to the philosophical and epistemological assumptions underlying all qualitative research. In the chapters that follow, actual studies present the contexts for discussions of research design and techniques. Issues of research design include the importance of making explicit the underlying theoretical assumptions; the selection of an appropriate methodology; the interpretative, intersubjective nature of analysis; and the establishment of reliability and validity. Specific data collection techniques include clinical interviews, stimulated recall interviews, open-ended survey questions, and field notes and video or audio taping to record classroom events. Methods of analysis include participant validation, the categorization of data through constant comparison and software indexing and retrieval, phenomemographic analysis, and the identification of empirical examples of theoretical constructs. The monograph ends with a discussion of general issues, including the role of theory and the establishment of criteria for judging the goodness of qualitative research.

This chapter examines three concepts to consider in creating a bridge between everyday mathematical practices and mathematics in school: everydayness, mathematization, and context familiarity. I discuss each concept in detail using examples from my work with in-service teachers, my experiences in curriculum development, and the research literature.

This chapter examines two recommendations for classroom practices from curriculum and instruction Standards (National Council of Teachers of Mathematics [NCTM], 1989; 2000) and research in mathematics education. One recommendation, informed in part by research uncovering the mathematical practices in everyday activities (Carraher, Carraher, & Schliemann, 1985; D'Ambrosio, 1985, 1991; Lave, 1988; Saxe, 1991), is to close the gap between learning mathematics in and out of school by engaging students in real-world mathematics rather than mathematics in isolation from its applications. The second recommendation is to make mathematics classrooms reflect the practices of mathematicians (Cobb, Wood, & Yackel, 1993; Lampert, 1986, 1990; Schoenfeld, 1992). These two proposals have different implications for changing classroom practices: one emphasizes the need for classroom activities to parallel everyday mathematical practice; the other emphasizes classroom activities that parallel those of academic mathematical practices. This chapter explores the tensions between these two proposals by juxtaposing everyday and academic mathematical practices.

This chapter examines the mathematical activities in a middle-school mathematics classroom during an architectural design project. By looking at three instances of assessment activities during this project--design reviews, final presentations, and classroom conversations--I examine student mathematical activities during assessments and compare mathematical activities in different classroom settings.

This monograph summarizes the findings from five related studies carried out by the authors in Sandy Bay, Tasmania, Australia, in 1979-80. The overall purpose of the studies was to examine whether children in grades 1-3 who differed in cognitive capacity learned to add and subtract in different ways. The first study was a cross-sectional survey designed to determine the memory capacity of a population of children. The second study was designed to portray performance differences on a variety of mathematically related developmental tasks for the same population of children. Data from these two studies were used to form groups of children who differed in cognitive capacity. Six groups were formed via cluster analysis, with memory capacity being the primary distinguishing characteristic. The third, fourth, and fifth studies each used a sample of students from the six cluster groups across grades. The third study examined both the performance and the strategies these children used to solve a structured set of addition and subtraction word problems. The fourth study involved repeated assessment of the children's performance on items measuring objectives related to addition and subtraction. In the last study these children and their teachers were observed during classroom instruction in mathematics to see how addition and subtraction were taught and whether or not instruction was related to the children's cognitive capacity. The results show that children's differences in capacity were reflected in their performance on both verbal and standard problems and in the strategies they used to solve problems. However, instruction did not vary for these children within classrooms. The picture that emerges is one of children struggling to learn a variety of important concepts and skills. Some children were limited by their capacity to process information. Most were able to solve a variety of problems by using invented strategies, those that had not been taught. They dismissed or failed to see the value of the taught procedures in solving these problems. Finally, the capacity of children to process information, the procedures students invented to solve a variety of problems, and the way in which instruction was carried out in schools did not seem related to each other.

The difficulties adolescents have learning algebra have often been used to justify keeping arithmetic and algebra education separate and in a particular order in Grades K-12 curricula. We believe this justification is unwarranted and based on faulty assumptions about the nature of mathematics and the nature of children's capabilities. The operations of arithmetic can be understood as procedures for acting on particular numbers. When one construes them as operations on variables, however, the operations of arithmetic assume the character of functions. The work we have been undertaking in a public school in greater Boston over the past 3 years has convinced us that children as young as 8 and 9 years old are capable of thinking of operations in these broad terms-as procedures that can be carried out on a set of input values to produce a set of output values. In this paper, we present evidence that these children can meaningfully use algebraic notation through their use of letters to stand for variables and not simply unknowns. We are struck by the contrast between what young learners are expected to learn and what they are capable of learning when exploring the algebraic character of arithmetic. If our students are at all representative of children their age, as we believe they are, then educators have compelling reasons for redesigning curricula and reconceptualizing the goals and expectations in early mathematics education.

In this study, we focus on the first steps of qualitative modeling of temporal phenomena. In particular, we analyze subtleties involved in the analysis of images of "hand-made motion." We start with the description of the rationale of the environment on which this study is based. We then describe two pairs of seventh graders who are involved in modeling a given physical phenomenon. Episode 1 (approximately one minute) involves the efforts of one pair of students to understand the correlation between graphs and the path produced by the motion of a hand. We attempt to identify various issues students encounter in explaining how the trajectory of the hand is represented in graphs of "x over time" and "y over time." Episode 2 (approximately two minutes) concerns a dialogue between two students about the role and representation of time in the graphs. Their discussion brings up issues regarding time as an independent variable and as a causal influence on the graph. We close by discussing connections among different fields of modeling as reflected in the design of the learning environment engaged in this study.

This chapter explores the tensions and compromises resulting from what seem to be different conceptions of what mathematics is and of what mathematics children should learn in school. Our work in a fifth-grade class has allowed us to combine elements of mathematicians' mathematics with the students' everyday mathematics. On the one hand, we work towards having children doing mathematics like mathematicians by working on open-ended and investigative situations, sharing ideas and strategies, and jointly negotiating meanings (Cobb, 1991; Lampert, 1986; Schoenfeld, 1991). On the other hand, we also want to develop school activities that build on the students' experiences with everyday mathematics in an effort to bridge the gap between outside and inside school experiences (Bishop, 1994; Lave, 1988; Nunes, 1992; Saxe, 1991). This work has made us reflect on the different beliefs, values, and practices in mathematics that inform our actions in the classroom. But questions still remain. We do not want students' everyday mathematics to serve simply as a source of motivation. Yet, how far can we push everyday mathematics? By mathematizing everyday situations, we may be losing what made them appealing in the first place, but we hope to advance the students' learning of generalization and abstraction in mathematics. But how do the patterns of classroom participation change as we mathematize these situations?

"[Note: This abstract applies to all articles within this issue."] The chapters in this monograph describe qualitative research methods used to investigate students' and teachers' interactions with school mathematics. Each contributing author uses data from his or her own research to illustrate a particular technique or aspect of research design. The different chapters present a wide range of methods, representing a variety of goals and perspectives. Rather than a comprehensive reference manual, this monograph illustrates the diversity of methods available for qualitative research in mathematics education. The monograph begins with a discussion of key elements that contribute to the dynamic and evolving domain of mathematics education research. Background information is then provided that relates to the philosophical and epistemological assumptions underlying all qualitative research. In the chapters that follow, actual studies present the contexts for discussions of research design and techniques. Issues of research design include the importance of making explicit the underlying theoretical assumptions; the selection of an appropriate methodology; the interpretative, intersubjective nature of analysis; and the establishment of reliability and validity. Specific data collection techniques include clinical interviews, stimulated recall interviews, open-ended survey questions, and field notes and video or audio taping to record classroom events. Methods of analysis include participant validation, the categorization of data through constant comparison and software indexing and retrieval, phenomemographic analysis, and the identification of empirical examples of theoretical constructs. The monograph ends with a discussion of general issues, including the role of theory and the establishment of criteria for judging the goodness of qualitative research.

By examining middle school students' perceptions of how they use mathematics outside the classroom, this study attempts to learn more about and build on students' everyday mathematics practice and to close the gap between students' use of mathematics in school and their use of it outside school. Twenty middle school students were interviewed before and after a week of keeping a log in which they recorded their everyday use of mathematics. The interviews and log sheets revealed that the mathematics that the middle school students perceived that they used outside the classroom were the six fundamental mathematical activities identified by Bishop (1988). The study also found that students' perceptions of their out-of-school mathematics practice were strongly influenced by their view of mathematics.