Understanding Mathematics and Science Matters

Thomas A. Romberg, Thomas P. Carpenter, Fae Dremock, Thomas A. Romberg, Thomas P. Carpenter, Fae Dremock

• Publisher: Routledge
• Year: 2005
• eBook ISBN: 9781135620219
• Topic: Education
• Subtopic: Education General

The research reported in this book provides reliable evidence on and knowledge about mathematics and science instruction that emphasizes student understanding--instruction consistent with the needs of students who will be citizens in an increasingly demanding technological world.

The National Center for Improving Student Learning in Mathematics and Science--established in 1996 as a research center and funded by the U.S. Department of Education--was instrumental in developing instructional practices supportive of high student achievement in and understanding of mathematics and science concepts. NCISLA researchers worked with teachers, students, and administrators to construct learning environments that exemplify current research and theory about effective learning of mathematics and science. The careful programs of research conducted examined how instructional content and design, assessment, professional development, and organizational support can be designed, implemented, and orchestrated to support the learning of all students. This book presents a summary of the concepts, findings, and conclusions of the Center's research from 1996-2001.


In the Introduction, the chapters in Understanding Mathematics and Science Matters are situated in terms of the reform movement in school mathematics and school science. Three thematically structured sections focus on, respectively, research directed toward what is involved when students learn mathematics and science with understanding; research on the role of teachers and the problems they face when attempting to teach their students mathematics and science with understanding; and a collaboration among some of the contributors to this volume to gather information about classroom assessment practices and organizational support for reform.


The goal of this book is to help educational practitioners, policymakers, and the general public to see the validity of the reform recommendations, understand the recommended guidelines, and to use these to transform teaching and learning of mathematics and science in U.S. classrooms.

Part I

INTRODUCTION

Chapter 1

Standards-Based Reform and Teaching for Understanding

Thomas A. Romberg

Thomas P. Carpenter

Joan Kwako

University of Wisconsin—Madison,

The U.S. “standards-based” reform movement in school mathematics and science has been based on the belief that, in most classrooms at all school levels, mathematics and science instruction is neither suitable nor sufficient to adequately equip our children with the concepts and skills needed for the 21st century. Furthermore, unless something is done to alter current schooling trends, conditions are likely to get worse in the coming decades. In 1983, the need for reform in classroom practices was brought vividly to the attention of the U.S. public with the publication of A Nation at Risk (National Commission on Excellence in Education, 1983) and Educating Americans for the 21st Century (National Science Board Commission on Precollege Education in Mathematics, Science, and Technology, 1983). The authors of those documents claimed that competing in a global environment depended on a workforce knowledgeable about the mathematical, scientific, and technological aspects of the emerging information age and that our schools were failing to prepare students for their future. This concern was reechoed in Before It’s Too Late: A Report to the Nation (National Commission on Mathematics and Science Teaching for the 21st Century, 2000) and in the No Child Left Behind Act of 2001.

The failure to educate our students was seen as a product of the traditional, if simplistic, view of learning and teaching commonly practiced in U.S. schools. Both school mathematics and science were seen as a long list of established concepts and procedures. Learning was seen as acquiring these concepts and procedures through memorization and repeated practice. The job of teaching involved planning, presenting, and keeping order; a teacher’s responsibility basically ended when he or she told students what they had to remember. The school mathematics and science curricula were characterized as superficial, underachieving, and diffuse in content (McKnight & Schmidt, 1998; Peak, 1996). As a consequence, school mathematics and science in most classrooms was a tedious, uninteresting path to follow, with numerous hurdles to clear. It bore little resemblance to what a mathematician, a scientist, or user of either discipline does.

The goal of the research at the National Center for Improving Student Learning and Achievement in Mathematics and Science (NCISLA) was to study and document the effects of alternative notions of the ways school mathematics and science can be organized and taught so that students learn, with understanding, the concepts, skills, and practices of science and mathematics.

MATHEMATICAL AND SCIENTIFIC LITERACY

Our vision of alternative instruction in mathematics and science is based on the expectation that students should become mathematically and scientifically literate. Language literacy, or the ability to read, write, listen, and speak a language is the primary tool through which human social activity is mediated. Each human language and each human use of language has both an intricate design and a variety of functions, which are linked in complex ways (Gee, 1998). For a person to be literate in a language implies that the person knows many of the design resources of the language and is able to use those resources for several different social functions. Analogously, considering mathematics or science as a language implies that students must learn not only the concepts and procedures of specific mathematics or science domains (its design features), but also how to use such ideas to solve nonroutine problems and to model or mathematize in a variety of situations (its social functions). For example, when learning a foreign language one must learn the nouns, verbs, and unique sentence structure (its design features), and one must learn how these are used to communicate in a variety of different social situations (e.g., carry on a discussion, ask questions, read a newspaper). Similarly, when learning algebra one must learn how to write and graph linear equations (design features) and to use such equations to represent a variety of contextual problems involving the joint variation of two variables. Developing the languages of mathematics and science and learning to use them in a variety of situations are essential ways of understanding the worlds we experience . Several chapters in this book address issues associated with student appropriation of the languages of mathematics and science. For example, in chapter 3 Rosebery, Warren, Ballenger, and Ogonowski examine the relationship between “everyday” and “scientific” knowledge and learning, and in chapter 8 Nemirovsky, Barros, Noble, Schnepp, and Solomon argue the idea that learning mathematics entails becoming familiar with “symbolic places.”

The National Council of Teachers of Mathematics (NCTM) used this analogy in 1986 when it charged the Commission on Standards for School Mathematics to do the following:

• Create a coherent vision of what it means to be mathematically literate both in a world that relies on calculators and computers to carry out mathematical procedures and in a world where mathematics is rapidly growing and is extensively being applied in diverse fields; and

• Create a set of standards to guide the revision of the school mathematics curriculum and its associated evaluation toward this vision. (NCTM, 1989, p. 1)

This commission (chaired by Thomas Romberg and contributed to by several NCISLA researchers) produced NCTM’s (1989, 1991, 1995, 2000) four standards documents.

Similarly in science, both the American Association for the Advancement of Science (1993) and the National Research Council (1996; report produced by a group chaired by former NCISLA Associate Director Angelo Collins) stressed science literacy as the primary goal for reform in science teaching. The Organization for Economic Cooperation and Development (OECD) also used this analogy in its development of instruments to monitor reading literacy, mathematical literacy, and scientific literacy for its Program for International Student Assessment (PISA; OECD, 1999, 2001, 2002). (NCISLA PI Jan de Lange chaired the Mathematics Functional Expert Group, of which Thomas Romberg was a member, that produced the mathematics framework and test items for this study.)

To become literate in mathematics and science requires a shift in epistemology about the learning of mathematics and science and, as a consequence, a shift in pedagogy. The epistemological shift involves moving from judging student learning in terms of mastery of concepts and procedures to judging student understanding of the concepts and procedures and student ability to scientifically model or to mathematize problem situations. For example, in many of the chapters in this book, instruction involves posing open contextual problems for students (a social function) in order to generate a need for ways to communicate their notions (develop design features).

FEATURES OF MATHEMATICAL AND SCIENTIFIC LITERACY

In this section, we explore the idea of scientific and mathematical literacy by considering how instruction might address the content and practices of these disciplines.

Domain-Based Knowledge

To teach for mathematics and science literacy, the first issue to be addressed involves deciding what content students should understand. Our strategy has been to define the range of content in a domain by using a phenomenological approach to describing the mathematical and scientific concepts, structures, or ideas in relation to domain phenomena and the kinds of problems commonly used. Note that although both mathematics and science have long been described in terms of common general topics (e.g., arithmetic, algebra, geometry; biology, chemistry, physics), the domain approach focuses on problem areas (big ideas) that give rise to those topics. This conception is based on the fact that mathematics and science are composed of diverse domains of knowledge in response to particular problem areas. Much of the Center’s work focused on the important features of specific content domains in mathematics and science that we expect students to learn with understanding (e.g., univariate and bivariate exploratory data analysis in chap. 6, arithmetic equivalence in chaps. 4 and 5, genetics and EMS astronomy in chap. 7).

We expect students to gradually acquire the concepts and skills in a domain as a consequence of solving problems, be able to relate those ideas to each other (and other ideas in other domains), and to use those ideas in new problem situations. This phenomenological organization for mathematical content is not new. Two well-known publications, On the Shoulders of Giants: New Approaches to Numeracy (Steen, 1990) and Mathematics: The Science of Patterns (Devlin, 1994), described mathematics in this manner As Hans Freudenthal (1983) observed, “Our mathematical concepts, structures, ideas have been invented as tools to organize the phenomena of the physical, social, and mental world” (p. ix). Kitcher (1993) described science similarly in The Advancement of Science: Science Without Legend, Objectivity Without Illusions. In keeping with this work, we identified in each domain the key features and resources important for students to discover, use, or even invent for themselves.

This domain view of mathematics and science differs from current perspectives in at least three important ways. Initially, the emphasis is not with the parts of which things are made, but with the whole of which they are part (i.e., how concepts and skills in a domain are related) and in turn how those parts are related to other parts, other domains, and ideas in other disciplines. Secondly, this conception rests on the signs, symbols, terms, and rules for use, that is, the language that humans have invented to communicate with each other about the ideas in the domain. Students should experience the need for the elements of the languages of mathematics and science; as a consequence, teachers must introduce and negotiate with the students the meanings and use of those elements. In chapter 3, Rosebery et al. illustrates the embodied imagining students use in attempting to understand and explain motion down a ramp. In this example, as well as many others, the choice of scientific and mathematical methods and representations is often dependent on the situations in which the problems are presented.

Pedagogically, the domain view of mathematics and science requires a shift from the “assembly-line” metaphor based on assumptions prevalent in the industrial age of the past century to a “domain-based” metaphor. Greeno (1991) argued that students develop understanding when

a domain is thought of as an environment, with resources at various places in the domain. In this metaphor, knowing is knowing your way around in the environment and knowing how to use its resources. This includes knowing what resources are available in the environment as well as being able to find and use those resources for understanding and reasoning. Knowing includes interactions with the environment in its own terms—exploring the territory, appreciating its scenery, and understanding how its various components interact. Knowing the domain also includes knowing what resources are in the environment that can be used to support your individual and social activities and the ability to recognize, find, and use those resources productively. Learning the domain, in this view, is analogous to learning to live in an environment: learning your way around, learning what resources are available, and learning how to use those resources in conducting your activities productively and enjoyably. (p. 175)

In particular, in chapter 8, Nemirovsky et al. use the metaphor of becoming familiar with a new place (e.g., a town, a train station) and the process of learning a new mathematical concept.

Learning Corridors

In planning instruction, curriculum developers often refer to scope and sequence, suggesting that both what is taught and the order in which it is taught are important. Center researchers have addressed the issue of sequence of instruction that leads to student understanding of science and mathematics. Our conception is based on the notion that an instructional sequence for students and teachers should follow a variety of paths along a “broad corridor” (Brown 1992; Brown & Campion, 1996). The notion of “learning corridor” has also been called “hypothetical learning trajectory” (Cobb, 2001; Simon, 1995), both to emphasize the speculative nature of the possible paths and to focus on the trajectory because the “learning for understanding” perspective sees students progressing from informal ideas in a domain to more formal ideas over time and, in so doing, reconfiguring their knowledge by developing strategies and models that help them discern when and how facts and skills are important. For example, in chapter 6, McClain, Cobb, and Gravemeijer describe the task sequence that evolved in their study of students’ learning of statistical data analysis.

To engineer classroom instruction based on the domain-based metaphor, a collection of problem situations is needed that engage students in exploring the environment of each domain in a structured manner. Students then have the opportunity to construct their mathematical or scientific knowledge from these purposeful activities. To accomplish this, two distinct dimensions need to be considered. The first is that the domain needs to be well mapped. This is not an easy task, as Webb and Romberg (1992) argued:

[K]nowledge of a domain is viewed as forming a network of multiple possible paths and not partitioned into discrete segments.... Over time, the maturation of a student’s functioning within a conceptual field should be observed by noting the formation of new linkages, the variation in the situations the person is able to work with, the degree of abstraction that is applied, and the level of reasoning applied. (p. 47)

The second is that activities that encourage students to explore the domain need to be identified and organized in a structured manner that allows for such learning. Doing both is not easy. For example, although there is no doubt that many interesting activities exist or can be created, whether they lead anywhere is a serious question. Keitel (1987) argued that an activity approach in mathematics can lead to “no mathematics at all,” and Romberg (1992) noted that

too often a problem is judged to be relevant thro...

Table of contents

• Cover
• Half Title
• Studies in Mathematical
• Full Title
• Copyright
• Contents
• Preface
• PART I: INTRODUCTION
• PART II: LEARNING WITH UNDERSTANDING
• PART III: TEACHING FOR UNDERSTANDING
• PART IV: CROSS-CUTTING STUDIES
• Postscrip
• Author Index
• Subject Index