Experiencing School Mathematics

Traditional and Reform Approaches To Teaching and Their Impact on Student Learning, Revised and Expanded Edition

Jo Boaler

• Publisher: Routledge
• Year: 2002
• ISBN:
  9781135642402
• Edition: 1
• Topic: Education
• Subtopic: Education General

NORTH AMERICAN RIGHTS ONLY: This is a revised edition of Experiencing School Mathematics first published in 1997 by Open University Press, © Jo Boaler. This revised edition is for sale in North America only.
The first book to provide direct evidence for the effectiveness of traditional and reform-oriented teaching methods, Experiencing School Mathematics reports on careful and extensive case studies of two schools that taught mathematics in totally different ways. Three hundred students were followed over three years, providing an unusual and important range of data, including observations, interviews, questionnaires, and assessments, to show the ways students' beliefs and understandings were shaped by the different approaches to mathematics teaching. The interviews that are reproduced in the book give compelling insights into what it meant to be a student in the classrooms of the two schools. Questions are raised about and new evidence is provided for:
* the ways in which "traditional" and "reform oriented" mathematics teaching approaches can impact student attitude, beliefs, and achievement;
*the effectiveness of different teaching methods in preparing students for the demands of the "real world" and the 21st century;
*the impact of tracking and heterogeneous ability grouping; and
*gender and teaching styles--the potential of different teaching approaches for the attainment of equity. The book draws some radical new conclusions about the ways that traditional teaching methods lead to limited forms of knowledge that are ineffective in non-school settings. This edition has been revised for the North American market to show the relevance of the study results in light of the U.S. reform movement, the "math wars" and debates about teachers, assessment, and tracking. The details of the study have been rewritten for an American audience and the results are compared with research conducted in the U.S. This is an important volume for mathematics teachers and researchers, education policymakers, and for students in mathematics education courses. NOTE: This is a revised edition of Experiencing School Mathematics first published in 1997 by Open University Press, © Jo Boaler. This revised edition is for sale in North America only.

1

Introduction

The question of which approach we should use to teach mathematics in schools is one that has perplexed parents, teachers, mathematicians, and others for decades (Benezet, 1935a, 1935b, 1936). In some ways, it is incredible that opinions are so divided around this question (Becker & Jacob, 2000), but in others it is not. Both because teaching is a highly complex event, but also because we have lacked careful research on the impact of different approaches to mathematics teaching and learning. Part of the aim of this book is to go some way toward changing this by communicating the experiences of two groups of students – approximately 300 in all – who experienced vastly different approaches to mathematics teaching and learning. The two schools that are the focus of this book, and the teachers who willingly offered their classrooms as sites for analysis, provide an unusual opportunity for us to learn about the ways that students’ experiences in classrooms impact the knowledge, beliefs, and understanding they develop. The stories told by the students in the pages that follow are important: They give detailed insights into the ways that mathematics teaching affects mathematics learning. Their ideas, combined with various data on their work, stand as rare testimony to the potential of different forms of mathematics teaching and learning.

Within mathematics education, there is an established concern that many people are unable to use the mathematics they learn in school in situations outside the classroom. In various research projects, individuals have been observed using mathematics in real-world situations such as street markets, factories, and shops. In these settings, school-learned mathematical methods and procedures are rarely used (Lave, Murtaugh, & de la Rocha, 1984; Lave, 1988; Masingila, 1993; Nunes, Schliemann, & Carraher, 1993). Lave (1988) has used these research findings to challenge the traditional conception of mathematics as an abstract and powerful tool that is easily transferred from one situation to another. She proposed that knowledge, rather than being a free-standing, transferable entity, is shaped or constituted by the situation or context in which it is developed and used. Such ideas have received strong support in the fields of anthropology, psychology, and education and are now pivotal to emerging theories about human cognition. Indeed, Lave and others in the field of situated learning (Brown, Collins, & Duguid, 1989; Greeno & MMAP, 1998; Young, 1993) have been instrumental in raising awareness of the importance of the situation, the context, or the community of practice (Lave & Wenger, 1991) in which ideas are encountered for the capabilities that learners subsequently develop. One of the aims of this research study was to explore the notion of situated learning and investigate the experiences of students when they needed to transfer mathematics from one situation to another. I was interested to discover whether different teaching approaches would influence the nature of the knowledge that students developed and the ways that students approached new and different situations. To do this, I monitored the impact of the students’ contrasting mathematical environments on the beliefs and understandings that students developed and the effectiveness of these in different situations, including the national school leaving examination as well as more applied and realistic tasks.

In designing my research, I was aware that previous studies had evaluated the experiences of students taught using contrasting approaches to mathematics, most of these demonstrating advantages of open or activity-based approaches for students’ performance on tests (Athappilly, Smidchens, & Kofel, 1983; Keedy & Drmacich, 1994; Maher, 1991; Resnick, 1990; Sigurdson & Olson, 1992). However, there appeared to be little research that examined the nature and form of the teaching that contributed toward differential test achievement (Hiebert et al., 1997). My aim was not only to monitor the effectiveness of two schools’ approaches, but to examine the intricate and complex ways in which the different approaches influenced students, including the influence of the curriculum used, the teaching decisions made, and the teacher-student interactions (Cohen, Raudenbush, & Ball, 2000; Stein, Smith, Henningsen, & Silver, 2000). To achieve this, I conducted in-depth, longitudinal studies of students taught using different curriculum approaches over 3 years, combining analyses of their achievement with data on their teaching and learning experiences.

Over recent years, there has been recognition that teaching occurs in particulars (Ball & Cohen, 1999), and that the effectiveness of any teaching and learning situation will depend on the actual students involved, the actual curriculum materials used, and the myriad of decisions that teachers make as they support student learning. Teachers have traditionally been offered general principles and strategies about education, abstracted from the particulars of specific teaching and learning events. Such abstract knowledge can, in certain circumstances, be extremely powerful, but it leaves to teachers the task of translating it into practical action in their own classroom (Black & Wiliam, 1998; Schwab, 1969). An exciting development of recent years is the awareness that actual records of teaching and learning – videos of classrooms, teacher reflections, student work, and other materials – are extremely powerful sites for learning (Ball & Cohen, 1999). Teachers and others are finding that analyses grounded in actual practice allow a kind of awareness and learning that has not previously been possible. I hope this book provides such an example of practice and that the details that are portrayed will serve as sites for learning. In offering a detailed case study of different teaching approaches, I am not providing evidence of hundreds of examples of the same approach, but details that will allow teachers and researchers to make their own decisions about the aspects from which they may generalize and from which they may learn.

The story that will be told in this book concerns the mathematics teaching and learning in two schools. But this is not just an account of different mathematics approaches; it is also about different educational systems, popularly characterized as traditional and reform; about tracked and mixed-ability teaching; about gender and learning styles; and about the ways these factors play out in the day-to-day experiences of students in classrooms. The messages that emerged from the two schools were varied and, at times, unexpected. My ability to tell the story and communicate the systematic differences between the teaching methods at the two schools derives from the clear and open ways in which students reported their experience of the learning process. The students portrayed in this book will take the reader some way toward the worlds of school mathematics as the students experienced them. Furthermore, the students’ actions, reflections, and descriptions provide important insights into the influence of the teaching approaches they encountered on the understanding they developed. The fact that this research is focused on two schools raises questions about its generalizability, but I am happy for these questions to be raised and for the answers to be sought within the pages of this book.

In Chapters 1 and 2, I introduce the study, students, and research methods. Chapter 3 introduces readers to the two schools and mathematics departments, and it gives an overview of the mathematics teaching approaches. Through descriptions of lessons and extracts from interviews, the reader will learn about the main features of the two teaching approaches. Chapters 4 and 5 introduce readers to Amber Hill and Phoenix Park schools in more depth, with descriptions of the characteristics of the two approaches that emerged as central, and students’ responses to them. In these chapters the reader will begin to enter the worlds of school mathematics as students’ experienced them, reading about the students’ experiences and beliefs through extracts of lessons and interviews. Chapters 1 to 4 give the reader a sense of school mathematics at Amber Hill and Phoenix Park, the events that transpired in the classrooms of the two schools, the important teaching moments, and students’ responses to them. Chapters 6 to 8 offer an exploration and analysis of the students’ understanding of mathematics, their beliefs about mathematics, and the ways these were influenced by the different teaching approaches. Chapters 9 and 10 deal particularly with issues of equity. Chapter 9 considers the ways in which the two approaches impacted girls’ opportunities to learn and the implications that the new insights from this study have for analyses of gender. Chapter 10 deals specifically with tracking and the ways in which it shaped students’ learning and precluded the attainment of equity. Chapter 11 summarizes and reflects on the main research results considering what the Amber Hill and Phoenix Park students’ experiences, understandings, and mathematical beliefs tell us about the effectiveness of different curriculum programs, teaching practices, and classroom environments.

2

The Schools, Students, and Research Methods

In the chapters that follow, the schools and students are described in some depth. The purpose of this chapter is to give a brief introduction to the two research settings, the students within them, and the methods used to monitor and understand their experiences.

Research Methods

To contrast two different mathematical approaches, I conducted 3-year ethnographic case studies (Eisenhart, 1988) of the mathematical environments in two schools. As part of these case studies, I performed a longitudinal cohort analysis of a year group of students in each school as they moved from ages 13 to 16. In England, this would be called Year 9 to Year 11, but in this book I refer to the grade levels as Years 8, 9, and 10 to make them equivalent to U.S. grade levels. In England, students stay together in age cohorts all through school, and they take mathematics in every year from ages 5 to 16. Beyond age 16, mathematics becomes optional. In Grades 1 through 10, students do not choose mathematics courses, and mathematics is not divided into algebra and geometry as it is in the United States. The students are taught mathematics as a whole, each year, and the content as decreed by the National Curriculum¹ is made up of four content strands: number, algebra, shape and space, and data handling. The curriculum also includes a strand called Using and Applying Mathematics, which includes mathematical processes such as generalizing, justifying, communicating, proving, and reasoning, which are intended to infuse the other four content strands. Because students stay in the same age cohorts, and usually the same mathematics classes, throughout secondary school, I was able to monitor the incoming Year 8 students over 3 years of school as they went from ages 13 to 16. At age 16, students take the same national examinations in every subject –called the General Certificate of Secondary Education (GCSE). The mathematics GCSE served as one helpful indicator of the mathematics students learned in the different approaches I followed. This examination is made up of a large collection of short questions. Unlike the large-scale assessments used in the United States, these are all open response –there are no multiple-choice questions. These examinations are graded nationally by examination officials (generally experienced teachers).

The case studies I conducted of the teaching and learning in the two schools included a variety of qualitative and quantitative methods. An overview of the research methods used is given in Appendix 1.1 chose to combine these different research strategies partly because of a belief that qualitative and quantitative techniques are not only compatible, but complementary. I also used a number of different techniques in an attempt to represent what Ball (1995) terms the “mobile, complex, ad hoc, messy and fleeting qualities of lived experience” (p. 6). Ball (1995) and Miles (1982) both warn of the danger of reducing the complexity of experience and striving toward a theory that it “all makes sense” (Miles, 1982, p. 126). In analyzing the practices of two schools, I did not wish to provide a definitive explanation of events, but a way of thinking that raised issues and questions about various features of school life. To this end, my research design was governed by the need to view events from a number of different perspectives and conceptualize factors such as enjoyment and understanding in different ways.

To understand the students’ experiences of mathematics, I observed approximately 100 one-hour lessons in each school, usually taking the role of a participant observer (Eisenhart, 1988; Kluckhohn, 1940). I interviewed 32 students in Year 9 and 44 students in Year 10. I analyzed comments elicited from students and teachers about classroom events (Beynon, 1985). I gave questionnaires to all of the students in my case study year groups each year. I interviewed teachers at the start and end of the research, and I collected an assortment of background documentation. These methods, particularly the lesson observations and student interviews, enabled me to develop an understanding of the students’ experiences and begin to view the worlds of school mathematics from the students’ perspectives (Hammersley, 1992). To locate the students’ perspectives within a broad understanding of the two schools, I also spent time “hanging out” (Delamont, 1984) in the faculty rooms and corridors of the schools; I socialized with teachers, and I tried to develop a sense of the two schools in as many ways as possible.

In addition to these methods, I gave the students various assessments during the 3-year period. Most of these I designed myself, but I was also given permission to conduct a detailed analysis of the students’ GCSE examination responses. The various assessment activities and questions I used during the 3 years involved individual and group work and written and practical work. All of the research methods employed within the study were used to inform each other in a continual process of interaction and reanalysis (Huberman & Crandall, 1982). Observation data were collected and analyzed using a grounded theory approach (Glaser & Strauss, 1967), and field notes and interviews were analyzed through a process of open coding (Strauss & Corbin, 1990). Table 2.1 gives an overview of the different methods employed. For more details on the methodology and methods, see Boaler (1996a).

Table 2.1
Research Methods Used in the Study

The Students Involved

The overall aim of my research study was to monitor the experiences of a year group of students as they moved from Years 8 to 10, but time constraints meant that some of my research methods needed to be focused on particular groups of students within the two year groups. For example, my lesson observations, interviews, and applied assessments could not be conducted with all of the mathematics groups in each year because of the time required by these methods. At Amber Hill, the year group was divided into eight ability groups (Sets 1–8) who were all taught mathematics at the same time. In England, most mathematics departments teach students in ability groups, known as sets, with Set 1 students perceived to be the most able. The different sets are taught similar content, but the higher sets are generally taught at a faster pace and cover more difficult material. Trigonometry, for example, is usually only introduced to students in higher groups. The national mathematics examination that students took at age 16 was divided into three levels: foundation, intermediate, and higher. These have some common questions, but the higher paper has some m...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright
• Dedication
• Contents
• Foreword
• Preface
• 1. Introduction
• 2. The Schools, Students, and Research Methods
• 3. An Introduction to Amber Hill and Phoenix Park Schools
• 4. Amber Hill Mathematics: Experiences and Reflections
• 5. Phoenix Park Mathematics: Experiences and Reflections
• 6. Finding Out What They Could Do
• 7. Exploring the Differences
• 8. Knowledge, Beliefs, and Mathematical Identities
• 9. Girls, Boys, and Learning Styles
• 10. Ability Grouping, Equity, and Survival of the Quickest
• 11. Looking to the Future
• References
• Author Index
• Subject Index