Mathematics Classrooms That Promote Understanding

Elizabeth Fennema, Thomas A. Romberg, Elizabeth Fennema, Thomas A. Romberg

• Publisher: Routledge
• Year: 1999
• ISBN:
  9781135676490
• Edition: 1
• Topic: Pedagogía
• Subtopic: Educación general

Mathematics Classrooms That Promote Understanding synthesizes the implications of research done by the National Center for Research in Mathematical Sciences on integrating two somewhat diverse bodies of scholarly inquiry: the study of teaching and the study of learning mathematics. This research was organized around content domains and/or continuing issues of education, such as equity and assessment of learning, and was guided by two common goals--defining the mathematics content of the K-12 curriculum in light of the changing mathematical needs of citizens for the 21st century, and identifying common components of classrooms that enable students to learn the redefined mathematics with understanding. To accomplish these goals, classrooms in which instruction facilitated the growth of understanding were established and/or studied. This volume reports and discusses the findings which grew out of this research, and subsequent papers and discussions among the scholars engaged in the endeavor. Section I, "Setting the Stage, " focuses on three major threads: What mathematics should be taught; how we should define and increase students' understanding of that mathematics; and how learning with understanding can be facilitated for all students. Section II, "Classrooms That Promote Understanding, " includes vignettes from diverse classrooms that illustrate classroom discourse, student work, and student engagement in the mathematics described in Chapter 1 as well as the mental activities described in Chapter 2. These chapters also illustrate how teachers deal with the equity concerns described in Chapter 3. Section III addresses "Developing Classrooms That Promote Understanding." The knowledge of the teaching/learning process gained from the research reported in this volume is a necessary prerequisite for implementing the revisions called for in the current reform movement. The classrooms described show that innovative reform in teaching and learning mathematics is possible. Unlike many volumes reporting research, this book is written at a level appropriate for master's degree students. Very few references are included in the chapters themselves; instead, each chapter includes a short annotated list of articles for expanded reading which provides the scholarly basis and research substantiation for this volume.

Setting the Stage

Chapter 1
Mathematics Worth Teaching, Mathematics Worth Understanding

Thomas A. Romberg
University of Wisconsin-Madison

James J. Kaput
University of Massachusetts-Dartmouth

When authors in this book describe classrooms that promote mathematical understanding, they each attempt to answer the question: What mathematics is it that students are expected to understand? The answers these authors give, however, may surprise some readers because although the titles of the mathematical domains described—number, geometry, quantities, statistics, and algebra—are familiar, the approach to discussing them is not. Rather than providing a list of specific concepts and skills to be mastered as has often been done in the past, each author focuses on a few key ideas in a domain, the interconnections between them, and examples of how students can grasp an understanding of the ideas as a consequence of exploring problem situations. The descriptions of mathematical content in this book are based on the epistemological shift in expectations about school mathematics that has emerged during the past decade; the chapters reflect attempts to redefine content in light of that shift.

The purpose of this chapter is to examine the scope of the mathematical content we expect students to understand after they have participated in the mathematical curricula described. This chapter has been organized under four headings: (a) traditional school mathematics, to clarify what the shift is away from; (b) mathematics as a human activity, to portray the direction the shift is toward; (c) mathematics worth teaching, to provide an overview to the answers provided in the following chapters to the initial question (What is it that students are expected to understand?); and, finally, (d) speculations about school mathematics in the future.

Traditional School Mathematics

Mathematics is perceived by most people as a fixed, static body of knowledge. Its subject matter includes the mechanistic manipulation of a variety of numbers and algebraic symbols and the proving of geometric deductions. This perception has determined the scope of the content to be covered and the pedagogy of the school mathematics curriculum. Specific objectives, which students are to master, have been stated; the teacher’s role has been to demonstrate how a manipulation is to be carried out or to explain how a concept is defined; and students have been expected to memorize facts and to practice procedures until they have been mastered.

The arithmetic of whole numbers, fractions, decimals, and percents represents the majority of the first 7 of 8 years of the school mathematics curriculum for all students. For those students who successfully complete the hurdles along the arithmetic path, yearlong courses in algebra and geometry follow. Then, for many college-bound students, what follows is a second-year course of algebra and perhaps a year course of precalculus mathematics. Finally, for a few students, a course in calculus is often available. In each course, the emphasis is on mastering a collection of fixed concepts and skills in a certain order. This layer-cake structure that we have inherited is deeply embedded not only in our curricular structure but in our larger society’s expectations regarding schools. This approach to mathematics education reinforces the tendency to design each course primarily to meet the prerequisites of the next course. This layer-cake filtering system is responsible for the unacceptably high attrition from mathematics that plagues our schools.

The traditional three-segment lesson, which has been observed in many classes, involves an initial segment where the previous day’s work is corrected. Next, the teacher presents new material, often working one or two new problems followed by a few students working similar problems at the chalkboard. The final segment involves students working on an assignment for the following day.

This mechanistic approach to instruction of basic skills and concepts isolates mathematics from its uses and from other disciplines. The traditional process of symbol manipulation involves only the deployment of a set routine with no room for ingenuity or flair, no place for guesswork or surprise, no chance for discovery; in fact, no need for the human being. Furthermore, the pedagogy of traditional instruction includes a basal text (which is a repository of problem lists), a mass of paper-and-pencil worksheets, and a set of performance tests. There is very little that is interesting to read. Workbook mathematics thus gives students little reason to connect ideas in today’s lesson with those of past lessons or with the real world. The tests currently used ask for answers that are judged right or wrong, but the strategies and reasoning used to derive answers are not evaluated. This portrayal of school mathematics—a tedious, uninteresting path to follow, with lots of hurdles to clear—bears little resemblance to what a mathematician or user of mathematics does. What is clear is that students do not do mathematics in traditional school lessons. Instead, they learn a collection of techniques that are useful for certain purposes. Because mastery of techniques is defined as knowledge, the acquisition of those techniques becomes an end in itself, and the student spends his or her time absorbing what other people have done.

Traditional school mathematics has failed to provide students with any sense of the importance of the discipline’s historical or cultural importance, nor any sense of its usefulness. Is it any wonder that many students dislike mathematics and fail to learn it? The premise of this book is that traditional teaching and learning of mathematics has not enabled students to learn mathematics with understanding and that our first step must be to redefine mathematics.

Mathematics as a Human Activity

School mathematics should be viewed as a human activity that reflects the work of mathematicians—finding out why given techniques work, inventing new techniques, justifying assertions, and so forth. It should also reflect how users of mathematics investigate a problem situation, decide on variables, decide on ways to quantify and relate the variables, carry out calculations, make predictions, and verify the utility of the predictions. Underlying this perspective of human activity is our use of Thurston’s (1990) metaphor of a tree to describe the discipline of mathematics:

Mathematics isn’t a palm tree, with a single long straight trunk covered with scratchy formulas. It’s a banyan tree, with many interconnected trunks and branches—a banyan tree that has grown to the size of a forest, inviting us to climb and explore, (p. 7)

For school mathematics, this vision emphasizes human actions, “climbing and exploring.” Thurston’s image of mathematics as the everspreading banyan tree reflects the notion that mathematics is a plural noun in that there are several intertwined trunks or branches (strands or domains). Furthermore, this image extends to the tree’s ever-widening root system, drawing in the wide range of activities, interests, linguistic capability, kinesthetic sense, and informal knowledge that students bring to us. It is this broadly inclusive metaphor of mathematics, and the exploration of problems as a way of learning mathematics, that underlies all the subject matter examples of this book.

The discipline of mathematics involves a vast assemblage of ideas in several related content domains and is defined by the community of mathematicians, mathematics educators, and users of mathematics (Hersh, 1997). The content domains have been derived via a continually evolving, culturally shared science of patterns and languages, which is extended and applied through systematic forms of reasoning and argument. Mathematics is both an object of understanding and a means of understanding. These patterns and languages are an essential way of understanding the worlds we experience—physical, social, and even mathematical. Mathematics is ever alive—as alive as any branch of science today—and is neither static nor fixed in time, but changing. Because mathematics evolves, so also must the school curriculum—the educational organization and sequencing of that content.

The emerging redefinition of school mathematics is based on an epistemological shift in perspective regarding what is important to students to know and understand and is reflected in curriculum documents from several countries (e.g., Australia, Japan, the United Kingdom), as well as from the United States. The problem being faced is that there has been a conflict between school mathematics as it has been organized and taught and the emerging perspective about mathematics. To provide even a sketch of a content outline or sequence for K through 12 mathematics that reflects this epistemological shift is well beyond the scope of this chapter and this book. Instead, our aim is to provide a background to the choices about what mathematics to teach (for understanding) from the many choices that are or will become available. Note that in contrast to the traditional classroom, doing mathematics from this perspective cannot be viewed as a mechanical performance or an activity that individuals engage in solely by following predetermined rules.

Curriculum activities that reflect this perspective are those that involve students in problem solving and that encourage mathematization. Such tasks include situations that are subject to measure and quantification, that embody quantifiable change and variation, that involve specifiable uncertainty, that involve our place in space and the spatial features of the world we inhabit and construct, and that involve symbolic algorithms and more-abstract structures. In addition, they encourage the use of mathematical languages for expressing, communicating, reasoning, computing, abstracting, generalizing, and formalizing. These systems of signs and symbols extend the limited powers of the human mind in many directions, and they make possible a long-term (cross-generational) cultural growth of the subject matter. Finally, such situations embody systematic forms of reasoning and argument to help establish the certainty, generality, consistency, and reliability of the individual’s mathematical assertions.

If mathematics is to serve students’ needs to make sense of experience arising outside of mathematics instruction and mathematics itself, including making sense in the various sciences, it must be firmly rooted in and connected to that experience. And its systems of signs and symbols must be learned and experienced as genuine, functioning languages—for expressing, communicating, reasoning, computing, abstracting, generalizing, and formalizing—that the student experiences as serving his or her real needs. Similarly, the systematic logical forms of reasoning and argument must be learned through their satisfying personally and socially experienced needs for certainty and reliability—for establishing, for the student, what is true and what is not true.

In summary, there are several ways of viewing the discipline and choices to be made about which aspects of mathematics are to be included in the school curriculum, but the patterns, signs, symbols, and rules from certain branches of contemporary mathematics should be learned by all students and ideas from other branches known by some students. Furthermore, students should not only know the concepts and procedures for some parts of mathematics but also understand how mathematics is created and used.

This view of mathematics is, above all, integrative: It sees everything as part of a larger whole, with each part sharing reciprocal relationships with other parts. It stresses the acquisition of understanding by all (including the traditionally underprivileged), to the highest extent of their capability, rather than the selection and promotion of an elite. It is a philosophy that simultaneously stresses erudition and common sense, integration through application, and innovation through creativity. Most important, it stresses the creation of knowledge. Against this broad and ambitious view of mathematics, traditional school mathematics appears thin, lifeless, and isolated.

If students follow our approach to mathematical content, we believe that they will learn to formulate problems and develop and apply strategies to find solutions in a range of contexts. By exploring problems, they will learn to verify and interpret results and generalize solutions. In so doing, they will learn to apply mathematical modeling and become confident in their ability to address real-world problem situations. If they reason through their problem situations, students will develop the habit of making and evaluating conjectures and of constructing, following, and judging valid arguments. In the process, they will deduce and induce; apply spatial, proportional, algebraic, and graphical reasoning; construct proofs; and formulate counterexamples. In this sense, problem solving should be the basis of the mathematics curriculum.

Mathematics Worth Teaching

Because students are more likely to learn mathematics in the context of problem situations, the work of the National Center for Research in Mathematic Sciences Education (NCRMSE) Principal Investigators focused on curricular ideas organized in a vertical strand structure of mathematics. There are two aspects to this work. The first involved specifying the scope of the mathematics in each strand that students are expected to know and understand, and the second involved outlining the characteristics and sequence of instructional activities that would engage students in learning mathematics with understanding. Our initial step in specifying the content was to identify the content domains (or strands) that society expects all students to learn with understanding. The domains we examined were number, quantities, algebra, geometry, and statistics. The domains were selected because of their generality and ability to subsume more specialized components of the discipline.

The second step involved identifying the big ideas in each domain that are worthy of extended time and effort. These big ideas are important for students to find, discover, use, or even invent for themselves. For example, the initial related concepts of addition and subtraction of whole numbers (a substrand of the whole-number strand) involve:

• symbolic statements that characterize the domain (e.g., a + b = c and a - b = c, where a, b, and c are natural numbers);
• the implied task (or tasks) to be carried out (for addition and subtraction, this may involve describing the situations where two of the three numbers [a, b, and c] in the statements are known and the third is unknown);
• rules that can be followed to represent, transform, and carry out procedures to complete the task (e.g., “Find the unknown number using one or more of such procedures as counting; basic facts; symbolic transformations, such as a+ [ ] = c ↔ c - a = [ ]; and place value [e.g., 345 = 3 (100) + 4 (10) + 5 (1)]);
• computational algorithms for larger numbers (e.g., adding 34 and 28 involves finding the sums of 30 + 20, 4 + 8; regrouping 12 into 10 + 2, and finding the sum of 30 + 20 + 10 + 2);
• the set of situations that have been used to make the concepts, the relationships between concepts, and the rules meaningful (e.g., join-separate, part-part-whole, compare, equalize).

Note that the set of situations that give meaning to the concepts and rules are considered equally as important as learning to follow the procedural rules. In fact, it is assumed that they are the means by which the procedural rules are understood. The structure of each strand also acknowledges that the big and powerful ideas and connections between them require years to develop and are understood in a variety of forms. Thus, the big ideas are stressed for all students throughout the school years rather than at the end for an elite minority.

From this example, it should be clear each strand comprises several closely related substrands. For example, the whole-number strand includes assigning numbers to sets or objects by counting or measuring, understanding place value, adding and subtracting, multiplying and dividing, and exploring the properties of such numbers (e.g., evenness and primeness). The big ideas in each substrand must be identified as was outlined earlier for addition and subtraction. Such detailed specifications describe the scope of the mathematical topics and ideas students should be expected to know. Understanding these ideas, however, involves relating a key idea to other ideas, explaining and justifying relationships, and so forth. Furthermore, the key ideas in different strands are closely interrelated. For example, measuring lengths, areas, volumes, angles, distances, and so on are important aspects of geometry.

Thus, to sequence instruction based on the tenets of mathematics as a human activity implies that tasks cannot be organized in a particular sequence to cover each detail, as is done in traditional curricula. Instead, instructional units or tasks that focus on investigation of problem situations nee...