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THE MATH WARS By Paul R. Trafton, Barbara J. Reys, and Deanna G. Wasman The authors describe six central characteristics of "standards-based" mathematics curriculum materials and discuss how these characteristics are related and how each one sets the stage for and supports the others. |
OVER THE past decade, many efforts have been made to reform school mathematics to reflect recommendations made by the National Council of Teachers of Mathematics (NCTM) in 1989 in the Curriculum and Evaluation Standards for School Mathematics. The release of updated standards, Principles and Standards for School Mathematics (hereafter referred to as the Standards), will doubtless prompt additional efforts to ensure that every student experiences a high-quality mathematics program.1 One consistent and very tangible focus of reform efforts is the adoption of new forms of curriculum materials. By curriculum "materials," we mean those resources that serve as daily guides for students and teachers in directing activity related to math instruction and learning.
During the past decade, the phrase "standards-based" has been loosely used to describe a variety of materials ranging from individual units to collections of classroom activities, to pieces "added on" to traditional materials. In this article, we discuss comprehensive sets of curriculum materials that embody the ideas expressed in the Standards. These materials differ in substantive ways from traditional textbooks used in the U.S., which tend to focus on acquisition of skills, to cover many topics superficially, and to be highly repetitive.2
Prime examples of "standards-based" curriculum materials are programs developed with support from the National Science Foundation explicitly to reflect the Standards.3 (For a list of these NSF-funded programs see "NSF-Funded Standards-Based Curriculum Projects," page 257.) These are the materials that are at the center of recent controversy, drawing criticism from some research mathematicians and others. The controversy raises such questions as, What is meant by standards? What are standards-based materials? How are they different from traditional textbooks? In this article we describe six central characteristics of "standards-based" mathematics curriculum materials and discuss how these characteristics are related and how each one sets the stage for and supports the others.
Standards-Based Materials Are Comprehensive
A primary concern in all curriculum reform is the inclusion of knowledge, understandings, processes, and skills that constitute competency in a field. Most long-time observers of U.S. mathematics programs have been struck by their narrow focus on skills and procedures. In On the Shoulders of Giants: New Approaches to Numeracy, Lynn Steen notes that "school tradition has it that arithmetic, measurement, algebra, and a smattering of geometry represent the fundamentals of mathematics" -- a limited framework that has produced programs dominated by a heavy emphasis on skills. "But there is much more," Steen continues, "to the root system of mathematics -- deep ideas that nourish the growing branches of mathematics."4
Thus the first characteristic of standards-based materials is a focus on core mathematics for all students. This mathematical core must address the general literacy goals of school mathematics, the mathematics that is considered central to preparing America's youth for the work force, as well as the mathematics necessary to provide a foundation for continued, advanced study of the subject. Within this framework, mathematics includes such familiar strands as arithmetic, algebra, geometry, statistics, and probability. It also addresses broad themes, such as dimension, quantity, uncertainty, shape, and change.
The Standards lists five content standards for preschool through 12th-grade mathematics: 1) number and operations, 2) algebra, 3) geometry, 4) measurement, and 5) data analysis and probability. Under each standard are three or four statements describing what all students should know and be able to do across all grade levels. These essential capacities include "understand the meanings of operations and how they relate to one another," "represent and analyze mathematical situations and structures using algebraic symbols," "use visualization, spatial reasoning, and geometric modeling," and "develop and evaluate inferences and predictions that are based on data." Thus standards-based curricula incorporate a range of important mathematics. They also link this mathematics content with mathematical processes that students should learn and use: problem solving, reasoning and proof, communication, connections, and representation.
Along with concepts and mathematical processes, mathematically comprehensive materials also pay attention to the development of useful skills. In contrast to traditional texts, these skills are developed in the context of understanding, rather than in isolation, with attention to those skills that remain important in a technological era. As Steen notes, "The key issue . . . is not whether to teach fundamentals but which fundamentals to teach and how to teach them."5
In standards-based materials, the importance and interconnectedness of understanding and skills are recognized. A balanced approach to knowledge and skills can help students acquire the mathematics they need for solving problems and also establish a foundation for later study of more sophisticated mathematics. Although career paths may differ in terms of mathematical expertise required, all students need a core of mathematical knowledge (content, processes, understandings, and skills) if they are to keep career options open and find satisfying and productive jobs or occupations.
Standards-Based Materials Are Coherent
Coherence refers to the presentation of mathematics so that the core ideas of the subject are highlighted and cause students to see it as an integrated whole. If students are to think mathematically and use mathematics as a tool for solving problems, coherence is crucial, and establishing connections among the big ideas of mathematics fosters coherence. When carefully established, these big ideas provide anchoring for the important pieces of mathematics -- terminology, definitions, notation, and skills -- that emerge from them. Standards-based materials promote coherence through an initial focus on big ideas, with an emphasis on connections and links to related mathematical ideas and applications.
In Mathematician's Delight, Walter Sawyer observes that every subject has a shadow or imitation. He writes: "One can learn imitation history -- kings and dates, but not have the slightest idea of the motives behind it all; imitation literature -- stacks of notes of Shakespeare's phrases, and a complete destruction of the power to enjoy Shakespeare."6 Unfortunately, mathematics programs have traditionally focused on the details of the discipline without paying much attention to its unifying themes, and so most students have experienced imitation mathematics. They comprehend the subject as a collection of "bits and pieces" consisting largely of symbols, definitions, and procedures. Many traditional mathematics textbooks break knowledge into disconnected segments that consist of small units of knowledge, called "lessons," with little effort to connect them. Focusing on individual pieces and rules does not promote coherence -- in mathematics or in learning.
The study of measurement in many school texts illustrates this disconnected orientation. Traditional texts focus on the end products of knowledge, such as the names of units of measure, relationships among them, and the use of rulers and formulas. Little attention is paid to such important ideas as the notion of a unit, what it means to "measure" an object, and the uses of measurement in various contexts.
In contrast, the study of area in standards-based curricula begins with exploring what it means for one surface to be "bigger" than another. Students might explore ways of finding the area of irregular shapes, engage in discussions of an appropriate unit of area, and create strategies for finding the areas of complex figures. They use their knowledge to solve applied problems, such as finding the "best buy" among various floor-covering materials or planning such large-scale projects as building a home. They also explore the connections between perimeter and area and move toward procedures for finding the area of standard figures, such as squares, rectangles, and triangles. These important aspects of measuring area occur in a broad, meaningful context that helps students learn and retain what they learn.
Even as students must have a comprehensive mathematics program, they must learn in ways that help them see mathematics as a coherent whole and result in learning that is coherent. Learning that is connected and "hangs together" results in higher achievement, greater applicability, and less susceptibility to forgetting. As Sawyer notes, "It is far easier to learn the real subject properly, than to learn the imitation badly."7
Standards-Based Materials Develop Ideas in Depth
The Third International Mathematics and Science Study (TIMSS) describes textbooks used in the U.S. in recent years as being "a mile wide and an inch deep."8 This comment refers to the large number of topics covered at each grade and to the shallow development of each. It is not unusual to see a superficial presentation of important topics, such as symmetry and similarity, repeated at nearly the same shallow level from grade to grade with the primary focus on rules and techniques rather than on mathematics. By contrast, standards-based instructional materials recognize that there are important ideas in mathematics that must be developed at varying levels of depth as students mature mathematically.
In standards-based materials, important ideas are frequently introduced early in a student's school career and revisited continually throughout the grades, with the focus on developing deeper layers of sophistication. For example, in grades K-2 students gather, organize, and represent data that interest them. In grades 3-5, students continue to organize and represent data and also focus on ways to characterize a set of data as a whole and to compare one set of data to another. In grades 6-8, students make arguments and predictions about trends in data. In grades 9-12, students continue to analyze data by curve-fitting and by exploring measures of correlation and variability. At each level, students probe more deeply than before, and the expectations grow for student understanding and use of the knowledge.
The increased sophistication in the way mathematical ideas are treated, together with the coherent development of mathematical ideas, helps students toward deeper understanding. In-depth learning is more likely to occur when the curriculum concentrates on a few big ideas and their interconnections and when teachers design instruction to engage students deeply with these ideas. This approach is characteristic of many industrialized countries that outperformed the U.S. on TIMSS. The additional time gained by in-depth attention to fewer ideas permits students to delve more deeply into mathematical relationships and properties and allows them to develop increasing levels of sophistication. The result is greater student learning and less review and repetition from one year to the next.
There is a natural connection between coherence and developing mathematics in an in-depth way. When the key ideas of the discipline are presented coherently and instruction promotes coherence in learning, then there is a greater inclination for teachers and students to pursue ideas in greater depth and for discussions to focus on fundamental notions.
Standards-Based Materials Promote Sense-Making
One repeatedly hears adults remark, "Math never made sense to me." These individuals then identify some point in their schooling when they no longer understood the mathematics they were learning or why they were learning it. Unfortunately, they never made it past the symbols, definitions, rules, and formulas of mathematics. When given the opportunity to make sense of a mathematical idea, they are surprised -- and delighted -- by the experience. For example, most adults use a "borrowing" procedure to subtract multidigit numbers, but many do not know why the procedure works. When they have opportunities to think about and model subtraction problems, they see what the "borrowing marks" they've made for years actually represent. And they are generally quite pleased with this discovery and, at the same time, frustrated that no one ever helped them understand this process.
One of the most consistent findings of research over the past 50 years is that understanding increases the ability to learn, remember, and use mathematics.9 As James Hiebert and his colleagues note, "Understanding is crucial because things learned with understanding can be used flexibly, adapted to new situations, and used to learn new things. Things learned with understanding are the most useful things to know in a changing and unpredictable world."10 They also argue that developing a deep, connected understanding of mathematics (i.e., making sense of mathematics) promotes the learning of skills. What's more, learning with understanding helps students develop confidence in their mathematical abilities.
But what do we mean by "understanding"? Is it simply reproducing a definition or remembering the steps of a procedure? In recent years, cognitive science has provided new insights into the nature of understanding. Rather than view learning as collecting and stacking pieces of knowledge in our heads, it is more productive to view understanding as a web of interconnected ideas. Hiebert and Thomas Carpenter capture this notion: "A mathematical idea or procedure or fact is understood if it is part of an internal network. . . . A mathematical idea or procedure is understood thoroughly if it is linked to existing networks with stronger or more numerous connections."11
Standards-based materials help students make sense of mathematics in several ways. Sense-making is promoted by spending substantial time on the fundamental ideas of a mathematical domain, such as rational numbers. Having time to "think things through" helps students connect their everyday, informal knowledge with their new knowledge. Two other powerful ways of promoting sense-making are letting students create and use their own ways of thinking to solve problems and having them share their thinking with other students and the teacher. These "seminars," in which students discuss various solution strategies, allow them to reflect on and analyze their own thinking and that of others under a teacher's guidance.
For example, in classrooms in the primary grades, children typically use several methods to attack the following problem: "Maria has 3 boxes of cupcakes. Each box holds 4 cupcakes. How many are left after 5 cupcakes have been eaten?"
Some children draw a picture of the three boxes of four cupcakes and cross off one box and one cupcake. Others model the problem with cubes. Mental methods include skip counting by fours up to 12 and then counting back five or stating that four and four make eight, counting up four more from eight, and then using the fact that 12 -4 = 8 to find that 12 -5 = 7. When computing with larger numbers -- say, adding 28 and 28 -- young children may change the computation to 30 and 26, or first add 25 and 25 and then count six more, or add 20 and 20, eight and eight, and then add 40 and 16. It is not just the actual sharing of these approaches that is important; teachers must use this time to cause all the children to reflect on the various approaches and discuss why they work.
Students' ability to make sense of mathematics in the classroom depends on the expertise of the teacher to select good tasks, engage students in thoughtful reflection, and create a classroom environment that supports reflection and communication. Standards-based curricula have an important role in providing guidance to the teacher and posing problems and questions in the text materials that encourage reflection.
Standards-Based Materials Engage Students
Standards-based materials engage students physically and intellectually through problems and tasks. By engagement we mean more than hands-on tasks, and the purpose goes far beyond "making math fun." The tasks are carefully selected to draw students into the study of mathematics by directing and focusing their thinking on important mathematics. The emphasis is on intellectual engagement. Problems and tasks that raise students' curiosity are posed. When a task is intriguing and poses a challenge, students are more likely to pursue a solution and explanation.
Seymour Papert describes young children's reaction to intellectually engaging tasks as "hard fun."12 Children find such tasks to be "fun" because of their complexity. The tasks also engage them because the work is interesting and within their reach. Tackling a tough problem is common practice, and by these experiences students develop strategies and a disposition to pursue the solutions to problems. One seventh-grader described his feelings this way: "Yes, some of it I did find hard . . . but you would get into it, and you wouldn't want to stop."13
Physical engagement is also an important aspect of the development of many mathematical ideas. Thus many tasks incorporate manipulatives as tools to help students engage in and explore mathematics. These manipulatives provide concrete representations of ideas or models of various sorts (e.g., three-dimensional models, drawings, tables, simulations, graphs) that help students understand the mathematical features of a situation or problem. Mathematical tasks often emerge from a real-life context, which enhances student interest and helps students interpret the mathematics embodied in the problem. For example, an everyday setting helps students understand and identify with the following task: "Jane just finished eating half a pizza. Joe also ate half a pizza. Jane insists that she ate more pizza than Joe. Is this possible?"14
As students reflect on and perhaps model the problem, it becomes clear that one half is not always the same size as another half. Rather, it depends on the size of the original whole. The real-life context helps bring meaning to the question of whether a half is always equal to a half. Although important and useful, context is neither necessary nor sufficient to engage students. The task must provide appropriate challenge, capture student curiosity, and incorporate important mathematics.
A standards-based curriculum encourages student engagement
by the use of interesting and inviting tasks. It depends on a
classroom environment that encourages and nurtures exploration,
risk taking, and mathematical justification. Students sometimes
work in pairs or in small groups; at other times, they work independently.
Emphasizing both physical and mental engagement through interesting
and challenging mathematical tasks is quite different from the
way many curriculum materials are organized. There is often the
tendency to offer loosely related tasks that lack an internal
structure. In standards-based curricula, the use of contexts,
problems, projects, and other tasks to engage students and to
connect mathematical ideas provides a platform for learning and
allows for the development of a belief that math is not only important
but also interesting. Student engagement is further supported
by a coherent, in-depth treatment of mathematics in lessons that
emphasize making sense.
Standards-Based Materials Motivate Learning
The sixth characteristic of standards-based instructional materials addresses a long-standing issue in school mathematics: linking mathematics with its applications. While it has been common for traditional materials to include a few "application problems" in exercise sets or even an entire lesson on a particular application, instructional materials have not substantively incorporated applications as part of their core. And most students fail to see the practical relevance or usefulness of the mathematics they learn. Every teacher has repeatedly heard the question, When are we ever going to have to use this?
Today, technology has spurred growth in the study of applied mathematics and has increased the need for and uses of mathematics in more and more disciplines. Thus establishing a strong and substantive relationship between mathematics and its widespread uses is more urgent than ever. As a result, applications are being woven into instructional materials in powerful new ways. No longer relegated primarily to exercises at the end of chapters, entire units are now often set in an applied context. Applications become part of understanding mathematics and provide ways to interpret mathematics.
Linear relationships provide a good example of how standards-based instructional materials link mathematics and applications. The study of variables and the relationships between variables are major ideas in algebra, and the idea of change can be a powerful setting for introducing students to these concepts. Students begin by representing change between a pair of variables, such as the speed at which one walks and the distance covered, using tables and coordinate graphs. In both representations students learn to describe the pattern in the data. Linear equations are then introduced as a way of describing the relationship symbolically, and they become a third way of representing the data.
A study of the relationship between the rate of walking and the distance covered shows that the faster one walks, the greater the distance one covers. This relationship is reflected in the "steepness" of the graph, in the constant rate of change in the values in the table, and in the value of m (the slope) in the general equation for a linear relation, y = mx + b. Slope, an important idea in linear equations, also emerges from its applications in the building trades. The "steepness" of a set of stairs can be expressed in terms of the ratio of the vertical change ("rise") to the horizontal change ("run").
In this example, new mathematics is set in the context of "real" applications and emerges from it. Further, the ideas are shaped by the setting (investigation of rate of change) and by such multiple representations as tables, coordinates, and equations. Finally, links between the physical application and the mathematical representation are woven into the instructional development.
The connections between mathematics and its applications are reflected in other ways. Rather than a few unrelated "applied" exercises in the text, applications often occur as the subject of short units of work and frequently incorporate a variety of mathematical concepts and skills. Thus the student experiences a curriculum in which mathematics emerges from multiple contexts. The intent is to help students learn more, understand that mathematics is useful, and realize that knowledge is not just an end in itself but a tool for solving problems.
The use of applications to contextualize mathematical study is an important characteristic of standards-based materials. Identifying good applications that reveal the underlying mathematics without overshadowing it is a challenge. However, when designed well, such materials promote coherence and sense-making. They also stimulate student interest and engagement and the development of a healthy, accurate view of mathematics as a useful discipline.
Implications for Teaching
The development of curricular materials based on the six characteristics presented here is crucial to improving opportunities for students to learn important mathematics. Training new and current teachers has also been identified as a particularly important factor.15 Throughout this article, we have portrayed a view of the work of teachers that is very different from the prevailing one in the United States.16 The NCTM's Professional Standards for Teaching Mathematics acknowledges this difference:
The kind of teaching envisioned in these standards is significantly different from what many teachers themselves have experienced as students in mathematics classes. Because teachers need time to learn and develop this kind of teaching practice, appropriate and ongoing professional development is crucial. . . . For teachers to be able to change their role and the nature of their classroom environment, administrators, supervisors, and parents must expect, encourage, support, and reward the kind of teaching described in this set of standards.17
Our experience suggests that the changes that teachers must adopt are neither trivial nor quickly attained and will require ongoing support over an extended period. Teachers need the opportunity to work through new instructional materials, to confront issues associated with new teaching strategies, and to increase their own knowledge of mathematical content. In some cases, school districts have created and facilitated professional development programs that coincide with curriculum adoption and implementation. In other cases, universities have created and facilitated programs that make use of standards-based curriculum materials. These programs typically blend a focus on the mathematics with ways of helping students learn it.
Summary
Reaching the goal of developing mathematical power for all students requires the creation of a curriculum and an environment that are both very different from much of current practice.18 This means that significant change must occur at many levels if the important and necessary goals of mathematics education reform are to be met.
Student learning has to go beyond the learning of specific concepts and skills. It needs to include attention to the development of a mathematical disposition that causes students to be confident in using mathematics, flexible in exploring mathematical ideas, perseverant in working on mathematical tasks, interested and inventive in doing mathematics, inclined to reflect on their thinking and to value mathematics and its uses and to appreciate its role in our culture.19
Instructional materials have a particularly important role in making these changes happen, for they affect the mathematics students encounter and how they encounter it, the processes students use, the way teachers teach, and what is assessed. They are also important because of their central place in American education. As Deborah Ball and David Cohen have noted, "Unlike frameworks, objectives, assessments, and other mechanisms that seek to guide curriculum, instructional materials are concrete and daily. They are the stuff of lessons and units, of what teachers and students do."20
We have discussed six central characteristics of instructional programs that represent efforts to capture the vision and essential elements of the NCTM Standards documents, and we have shown ways in which they collectively support greater student learning in mathematics. We believe that unless teachers and students have access to such materials, we will fail to achieve our common goal of high levels of mathematical literacy for all students.
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Last updated 8 November 2001
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