Rudiments Of Mathematics Class 11 Pdf

Suggested Citation:"4 THE STRANDS OF MATHEMATICAL PROFICIENCY." National Research Council. 2001. Adding It Up: Helping Children Learn Mathematics. Washington, DC: The National Academies Press. doi: 10.17226/9822.

4
THE STRANDS OF MATHEMATICAL PROFICIENCY

During the twentieth century, the meaning of successful mathematics learning underwent several shifts in response to changes in both society and schooling. For roughly the first half of the century, success in learning the mathematics of pre-kindergarten to eighth grade usually meant facility in using the computational procedures of arithmetic, with many educators emphasizing the need for skilled performance and others emphasizing the need for students to learn procedures with understanding.¹ In the 1950s and 1960s, the new math movement defined successful mathematics learning primarily in terms of understanding the structure of mathematics together with its unifying ideas, and not just as computational skill. This emphasis was followed by a "back to basics" movement that proposed returning to the view that success in mathematics meant being able to compute accurately and quickly. The reform movement of the 1980s and 1990s pushed the emphasis toward what was called the development of "mathematical power," which involved reasoning, solving problems, connecting mathematical ideas, and communicating mathematics to others. Reactions to reform proposals stressed such features of mathematics learning as the importance of memorization, of facility in computation, and of being able to prove mathematical assertions. These various emphases have reflected different goals for school mathematics held by different groups of people at different times.

Our analyses of the mathematics to be learned, our reading of the research in cognitive psychology and mathematics education, our experience as learners and teachers of mathematics, and our judgment as to the mathematical knowledge, understanding, and skill people need today have led us to adopt a

composite, comprehensive view of successful mathematics learning. This view, admittedly, represents no more than a single committee's consensus. Yet our various backgrounds have led us to formulate, in a way that we hope others can and will accept, the goals toward which mathematics learning should be aimed. In this chapter, we describe the kinds of cognitive changes that we want to promote in children so that they can be successful in learning mathematics.

Recognizing that no term captures completely all aspects of expertise, competence, knowledge, and facility in mathematics, we have chosen mathematical proficiency to capture what we believe is necessary for anyone to learn mathematics successfully. Mathematical proficiency, as we see it, has five components, or strands:

conceptual understanding—comprehension of mathematical concepts, operations, and relations
procedural fluency—skill in carrying out procedures flexibly, accurately, efficiently, and appropriately
strategic competence—ability to formulate, represent, and solve mathematical problems
adaptive reasoning—capacity for logical thought, reflection, explanation, and justification
productive disposition—habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one's own efficacy.

These strands are not independent; they represent different aspects of a complex whole. Each is discussed in more detail below. The most important observation we make here, one stressed throughout this report, is that the five strands are interwoven and interdependent in the development of profi ciency in mathematics (see Box 4–1). Mathematical proficiency is not a one-dimensional trait, and it cannot be achieved by focusing on just one or two of these strands. In later chapters, we argue that helping children acquire mathematical proficiency calls for instructional programs that address all its strands. As they go from pre-kindergarten to eighth grade, all students should become increasingly proficient in mathematics. That proficiency should enable them to cope with the mathematical challenges of daily life and enable them to continue their study of mathematics in high school and beyond.

The five strands are interwoven and interdependent in the development of proficiency in mathematics.

The five strands provide a framework for discussing the knowledge, skills, abilities, and beliefs that constitute mathematical proficiency. This frame-

Box 4–1 Intertwined Strands of Proficiency

work has some similarities with the one used in recent mathematics assessments by the National Assessment of Educational Progress (NAEP), which features three mathematical abilities (conceptual understanding, procedural knowledge, and problem solving) and includes additional specifications for reasoning, connections, and communication.² The strands also echo components of mathematics learning that have been identified in materials for teachers. At the same time, research and theory in cognitive science provide general support for the ideas contributing to these five strands. Fundamental in that work has been the central role of mental representations. How learners represent and connect pieces of knowledge is a key factor in whether they will understand it deeply and can use it in problem solving. Cognitive

scientists have concluded that competence in an area of inquiry depends upon knowledge that is not merely stored but represented mentally and organized (connected and structured) in ways that facilitate appropriate retrieval and application. Thus, learning with understanding is more powerful than simply memorizing because the organization improves retention, promotes fluency, and facilitates learning related material. The central notion that strands of competence must be interwoven to be useful reflects the finding that having a deep understanding requires that learners connect pieces of knowledge, and that connection in turn is a key factor in whether they can use what they know productively in solving problems. Furthermore, cognitive science studies of problem solving have documented the importance of adaptive expertise and of what is called metacognition: knowledge about one's own thinking and ability to monitor one's own understanding and problem-solving activity. These ideas contribute to what we call strategic competence and adaptive reasoning. Finally, learning is also influenced by motivation, a component of productive disposition.³

Although there is not a perfect fit between the strands of mathematical proficiency and the kinds of knowledge and processes identified by cognitive scientists, mathematics educators, and others investigating learning, we see the strands as reflecting a firm, sizable body of scholarly literature both in and outside mathematics education.

Conceptual Understanding

Conceptual understanding refers to an integrated and functional grasp of mathematical ideas. Students with conceptual understanding know more than isolated facts and methods. They understand why a mathematical idea is important and the kinds of contexts in which is it useful. They have organized their knowledge into a coherent whole, which enables them to learn new ideas by connecting those ideas to what they already know.⁴ Conceptual understanding also supports retention. Because facts and methods learned with understanding are connected, they are easier to remember and use, and they can be reconstructed when forgotten.⁵ If students understand a method, they are unlikely to remember it incorrectly. They monitor what they remember and try to figure out whether it makes sense. They may attempt to explain the method to themselves and correct it if necessary. Although teachers often look for evidence of conceptual understanding in students' ability to verbalize connections among concepts and representations, conceptual understanding need not be explicit. Students often understand before they can verbalize that understanding.⁶

Conceptual understanding refers to an integrated and functional grasp of mathematical ideas.

A significant indicator of conceptual understanding is being able to represent mathematical situations in different ways and knowing how different representations can be useful for different purposes. To find one's way around the mathematical terrain, it is important to see how the various representations connect with each other, how they are similar, and how they are different. The degree of students' conceptual understanding is related to the richness and extent of the connections they have made.

For example, suppose students are adding fractional quantities of different sizes, say They might draw a picture or use concrete materials of various kinds to show the addition. They might also represent the number sentence as a story. They might turn to the number line, representing each fraction by a segment and adding the fractions by joining the segments. By renaming the fractions so that they have the same denominator, the students might arrive at a common measure for the fractions, determine the sum, and see its magnitude on the number line. By operating on these different representations, students are likely to use different solution methods. This variation allows students to discuss the similarities and differences of the representations, the advantages of each, and how they must be connected if they are to yield the same answer.

Connections are most useful when they link related concepts and methods in appropriate ways. Mnemonic techniques learned by rote may provide connections among ideas that make it easier to perform mathematical operations, but they also may not lead to understanding.⁷ These are not the kinds of connections that best promote the acquisition of mathematical proficiency.

Knowledge that has been learned with understanding provides the basis for generating new knowledge and for solving new and unfamiliar problems.⁸ When students have acquired conceptual understanding in an area of mathematics, they see the connections among concepts and procedures and can give arguments to explain why some facts are consequences of others. They gain confidence, which then provides a base from which they can move to another level of understanding.

With respect to the learning of number, when students thoroughly understand concepts and procedures such as place value and operations with single-digit numbers, they can extend these concepts and procedures to new areas. For example, students who understand place value and other multidigit number concepts are more likely than students without such understanding to invent their own procedures for multicolumn addition and to adopt correct procedures for multicolumn subtraction that others have presented to them.⁹

Thus, learning how to add and subtract multidigit numbers does not have to involve entirely new and unrelated ideas. The same observation can be made for multiplication and division.

Conceptual understanding helps students avoid many critical errors in solving problems, particularly errors of magnitude. For example, if they are multiplying 9.83 and 7.65 and get 7519.95 for the answer, they can immediately decide that it cannot be right. They know that 10×8 is only 80, so multiplying two numbers less than 10 and 8 must give a product less than 80. They might then suspect that the decimal point is incorrectly placed and check that possibility.

Conceptual understanding frequently results in students having less to learn because they can see the deeper similarities between superficially unrelated situations. Their understanding has been encapsulated into compact clusters of interrelated facts and principles. The contents of a given cluster may be summarized by a short sentence or phrase like "properties of multiplication," which is sufficient for use in many situations. If necessary, however, the cluster can be unpacked if the student needs to explain a principle, wants to reflect on a concept, or is learning new ideas. Often, the structure of students' understanding is hierarchical, with simpler clusters of ideas packed into larger, more complex ones. A good example of a knowledge cluster for mathematically proficient older students is the number line. In one easily visualized picture, the student can grasp relations between all the number systems described in chapter 3, along with geometric interpretations for the operations of arithmetic. It connects arithmetic to geometry and later in schooling serves as a link to more advanced mathematics.

As an example of how a knowledge cluster can make learning easier, consider the cluster students might develop for adding whole numbers. If students understand that addition is commutative (e.g., 3+5=5+3), their learning of basic addition combinations is reduced by almost half. By exploiting their knowledge of other relationships such as that between the doubles (e.g., 5+5 and 6+6) and other sums, they can reduce still further the number of addition combinations they need to learn. Because young children tend to learn the doubles fairly early, they can use them to produce closely related sums.¹⁰ For example, they may see that 6+7 is just one more than 6+6. These relations make it easier for students to learn the new addition combinations because they are generating new knowledge rather than relying on rote memorization. Conceptual understanding, therefore, is a wise investment that pays off for students in many ways.

Procedural Fluency

Procedural fluency refers to knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently. In the domain of number, procedural fluency is especially needed to support conceptual understanding of place value and the meanings of rational numbers. It also supports the analysis of similarities and differences between methods of calculating. These methods include, in addition to written procedures, mental methods for finding certain sums, differences, products, or quotients, as well as methods that use calculators, computers, or manipulative materials such as blocks, counters, or beads.

Procedural fluency refers to knowledge of procedures, knowledge of when and how to use them appropriately, and skill in performing them flexibly, accurately, and efficiently.

Students need to be efficient and accurate in performing basic computations with whole numbers (6+7, 17–9, 8×4, and so on) without always having to refer to tables or other aids. They also need to know reasonably efficient and accurate ways to add, subtract, multiply, and divide multidigit numbers, both mentally and with pencil and paper. A good conceptual understanding of place value in the base-10 system supports the development of fluency in multidigit computation.¹¹ Such understanding also supports simplified but accurate mental arithmetic and more flexible ways of dealing with numbers than many students ultimately achieve.

Connected with procedural fluency is knowledge of ways to estimate the result of a procedure. It is not as critical as it once was, for example, that students develop speed or efficiency in calculating with large numbers by hand, and there appears to be little value in drilling students to achieve such a goal. But many tasks involving mathematics in everyday life require facility with algorithms for performing computations either mentally or in writing.

In addition to providing tools for computing, some algorithms are important as concepts in their own right, which again illustrates the link between conceptual understanding and procedural fluency. Students need to see that procedures can be developed that will solve entire classes of problems, not just individual problems. By studying algorithms as "general procedures," students can gain insight into the fact that mathematics is well structured (highly organized, filled with patterns, predictable) and that a carefully developed procedure can be a powerful tool for completing routine tasks.

It is important for computational procedures to be efficient, to be used accurately, and to result in correct answers. Both accuracy and efficiency can be improved with practice, which can also help students maintain fluency. Students also need to be able to apply procedures flexibly. Not all computational situations are alike. For example, applying a standard pencil-and-paper algorithm to find the result of every multiplication problem is neither neces-

sary nor efficient. Students should be able to use a variety of mental strategies to multiply by 10, 20, or 300 (or any power of 10 or multiple of 10). Also, students should be able to perform such operations as finding the sum of 199 and 67 or the product of 4 and 26 by using quick mental strategies rather than relying on paper and pencil. Further, situations vary in their need for exact answers. Sometimes an estimate is good enough, as in calculating a tip on a bill at a restaurant. Sometimes using a calculator or computer is more appropriate than using paper and pencil, as in completing a complicated tax form. Hence, students need facility with a variety of computational tools, and they need to know how to select the appropriate tool for a given situation.

Procedural fluency and conceptual understanding are often seen as competing for attention in school mathematics. But pitting skill against understanding creates a false dichotomy.¹² As we noted earlier, the two are interwoven. Understanding makes learning skills easier, less susceptible to common errors, and less prone to forgetting. By the same token, a certain level of skill is required to learn many mathematical concepts with understanding, and using procedures can help strengthen and develop that understanding. For example, it is difficult for students to understand multidigit calculations if they have not attained some reasonable level of skill in single-digit calculations. On the other hand, once students have learned procedures without understanding, it can be difficult to get them to engage in activities to help them understand the reasons underlying the procedure.¹³ In an experimental study, fifth-grade students who first received instruction on procedures for calculating area and perimeter followed by instruction on understanding those procedures did not perform as well as students who received instruction focused only on understanding.¹⁴

Without sufficient procedural fluency, students have trouble deepening their understanding of mathematical ideas or solving mathematics problems. The attention they devote to working out results they should recall or compute easily prevents them from seeing important relationships. Students need well-timed practice of the skills they are learning so that they are not handicapped in developing the other strands of proficiency.

When students practice procedures they do not understand, there is a danger they will practice incorrect procedures, thereby making it more difficult to learn correct ones. For example, on one standardized test, the grade 2 national norms for two-digit subtraction problems requiring borrowing, such as 62–48=?, are 38% correct. Many children subtract the smaller from the larger digit in each column to get 26 as the difference between 62 and 48 (see Box 4–2). If students learn to subtract with understanding, they rarely make

Box 4–2 A common error in multidigit subtraction

this error.¹⁵ Further, when students learn a procedure without understanding, they need extensive practice so as not to forget the steps. If students do understand, they are less likely to forget critical steps and are more likely to be able to reconstruct them when they do. Shifting the emphasis to learning with understanding, therefore, can in the long run lead to higher levels of skill than can be attained by practice alone.

If students have been using incorrect procedures for several years, then instruction emphasizing understanding may be less effective.¹⁶ When children learn a new, correct procedure, they do not always drop the old one. Rather, they use either the old procedure or the new one depending on the situation. Only with time and practice do they stop using incorrect or inefficient methods.¹⁷ Hence initial learning with understanding can make learning more efficient.

When skills are learned without understanding, they are learned as isolated bits of knowledge.¹⁸ Learning new topics then becomes harder since there is no network of previously learned concepts and skills to link a new topic to. This practice leads to a compartmentalization of procedures that can become quite extreme, so that students believe that even slightly different problems require different procedures. That belief can arise among children in the early grades when, for example, they learn one procedure for subtraction problems without regrouping and another for subtraction problems with regrouping. Another consequence when children learn without understanding is that they separate what happens in school from what happens outside.¹⁹ They have one set of procedures for solving problems outside of school and another they learned and use in school—without seeing the relation between the two. This separation limits children's ability to apply what they learn in school to solve real problems.

Also, students who learn procedures without understanding can typically do no more than apply the learned procedures, whereas students who learn

with understanding can modify or adapt procedures to make them easier to use. For example, students with limited understanding of addition would ordinarily need paper and pencil to add 598 and 647. Students with more understanding would recognize that 598 is only 2 less than 600, so they might add 600 and 647 and then subtract 2 from that sum.²⁰

Strategic Competence

Strategic competence refers to the ability to formulate mathematical problems, represent them, and solve them. This strand is similar to what has been called problem solving and problem formulation in the literature of mathematics education and cognitive science, and mathematical problem solving, in particular, has been studied extensively.²¹

Strategic competence refers to the ability to formulate mathematical problems, represent them, and solve them.

Although in school, students are often presented with clearly specified problems to solve, outside of school they encounter situations in which part of the difficulty is to figure out exactly what the problem is. Then they need to formulate the problem so that they can use mathematics to solve it. Consequently, they are likely to need experience and practice in problem formulating as well as in problem solving. They should know a variety of solution strategies as well as which strategies might be useful for solving a specific problem. For example, sixth graders might be asked to pose a problem on the topic of the school cafeteria.²² Some might ask whether the lunches are too expensive or what the most and least favorite lunches are. Others might ask how many trays are used or how many cartons of milk are sold. Still others might ask how the layout of the cafeteria might be improved.

With a formulated problem in hand, the student's first step in solving it is to represent it mathematically in some fashion, whether numerically, symbolically, verbally, or graphically. Fifth graders solving problems about getting from home to school might describe verbally the route they take or draw a scale map of the neighborhood. Representing a problem situation requires, first, that the student build a mental image of its essential components. Becoming strategically competent involves an avoidance of "number grabbing" methods (in which the student selects numbers and prepares to perform arithmetic operations on them)²³ in favor of methods that generate problem models (in which the student constructs a mental model of the variables and relations described in the problem). To represent a problem accurately, students must first understand the situation, including its key features. They then need to generate a mathematical representation of the problem that captures the core mathematical elements and ignores the irrelevant features. This

step may be facilitated by making a drawing, writing an equation, or creating some other tangible representation. Consider the following two-step problem:

In a common superficial method for representing this problem, students focus on the numbers in the problem and use so-called keywords to cue appropriate arithmetic operations.²⁴ For example, the quantities $1.83 and 5 cents are followed by the keyword less, suggesting that the student should subtract 5 cents from $1.13 to get $1.08. Then the keywords how much and 5 gallons suggest that 5 should be multiplied by the result, yielding $5.40.

In contrast, a more proficient approach is to construct a problem model— that is, a mental model of the situation described in the problem. A problem model is not a visual picture per se; rather, it is any form of mental representation that maintains the structural relations among the variables in the problem. One way to understand the first two sentences, for example, might be for a student to envision a number line and locate each cost per gallon on it to solve the problem.

In building a problem model, students need to be alert to the quantities in the problem. It is particularly important that students represent the quantities mentally, distinguishing what is known from what is to be found. Analyses of students' eye fixations reveal that successful solvers of the two-step problem above are likely to focus on terms such as ARCO, Chevron, and this, the principal known and unknown quantities in the problem. Less successful problem solvers tend to focus on specific numbers and keywords such as $1.13, 5 cents, less, and 5 gallons rather than the relationships among the quantities.²⁵

Not only do students need to be able to build representations of individual situations, but they also need to see that some representations share common mathematical structures. Novice problem solvers are inclined to notice similarities in surface features of problems, such as the characters or scenarios described in the problem. More expert problem solvers focus more on the structural relationships within problems, relationships that provide the clues for how problems might be solved.²⁶ For example, one problem might ask students to determine how many different stacks of five blocks can be made using red and green blocks, and another might ask how many different ways hamburgers can be ordered with or without each of the following:

catsup, onions, pickles, lettuce, and tomato. Novices would see these problems as unrelated; experts would see both as involving five choices between two things: red and green, or with and without.²⁷

In becoming proficient problem solvers, students learn how to form mental representations of problems, detect mathematical relationships, and devise novel solution methods when needed. A fundamental characteristic needed throughout the problem-solving process is flexibility. Flexibility develops through the broadening of knowledge required for solving nonroutine problems rather than just routine problems.

Routine problems are problems that the learner knows how to solve based on past experience.²⁸ When confronted with a routine problem, the learner knows a correct solution method and is able to apply it. Routine problems require reproductive thinking; the learner needs only to reproduce and apply a known solution procedure. For example, finding the product of 567 and 46 is a routine problem for most adults because they know what to do and how to do it.

In contrast, nonroutine problems are problems for which the learner does not immediately know a usable solution method. Nonroutine problems require productive thinking because the learner needs to invent a way to understand and solve the problem. For example, for most adults a nonroutine problem of the sort often found in newspaper or magazine puzzle columns is the following:

One solution approach is to reason that all 36 have at least two wheels for a total of 36×2=72 wheels. Since there are 80 wheels in all, the eight additional wheels (80–72) must belong to 8 tricycles. So there are 36–8=28 bikes.

A less sophisticated approach would be to "guess and check": If there were 20 bikes and 16 tricycles, that would give (20×2)+(16×3)=88 wheels, which is too many. Reducing the number of tricycles, a guess of 24 bikes and 12 tricycles gives (24×2)+(12×3)=84 wheels—still too many. Another reduction of the number of tricycles by 4 gives 28 bikes, 8 tricycles, and the 80 wheels needed.

A more sophisticated, algebraic approach would be to let b be the number of bikes and t the number of tricycles. Then b+t=36 and 2b+3t=80. The solution to this system of equations also yields 28 bikes and 8 tricycles.

A student with strategic competence could not only come up with several approaches to a nonroutine problem such as this one but could also choose flexibly among reasoning, guess-and-check, algebraic, or other methods to suit the demands presented by the problem and the situation in which it was posed.

Flexibility of approach is the major cognitive requirement for solving nonroutine problems. It can be seen when a method is created or adjusted to fit the requirements of a novel situation, such as being able to use general principles about proportions to determine the best buy. For example, when the choice is between a 4-ounce can of peanuts for 45 cents and a 10-ounce can for 90 cents, most people use a ratio strategy: the larger can costs twice as much as the smaller can but contains more than twice as many ounces, so it is a better buy. When the choice is between a 14-ounce jar of sauce for 79 cents and an 18-ounce jar for 81 cents, most people use a difference strategy: the larger jar costs just 2 cents more but gets you 4 more ounces, so it is the better buy. When the choice is between a 3-ounce bag of sunflower seeds for 30 cents and a 4-ounce bag for 44 cents, the most common strategy is unit-cost: The smaller bag costs 10 cents per ounce, whereas the larger costs 11 cents per ounce, so the smaller one is the better buy.

There are mutually supportive relations between strategic competence and both conceptual understanding and procedural fluency, as the various approaches to the cycle shop problem illustrate. The development of strategies for solving nonroutine problems depends on understanding the quantities involved in the problems and their relationships as well as on fluency in solving routine problems. Similarly, developing competence in solving nonroutine problems provides a context and motivation for learning to solve routine problems and for understanding concepts such as given, unknown, condition, and solution.

There are mutually supportive relations between strategic competence and both conceptual understanding and procedural fluency,

Strategic competence comes into play at every step in developing procedural fluency in computation. As students learn how to carry out an operation such as two-digit subtraction (for example, 86–59), they typically progress from conceptually transparent and effortful procedures to compact and more efficient ones (as discussed in detail in chapter 6). For example, an initial procedure for 86–59 might be to use bundles of sticks (see Box 4–3). A compact procedure involves applying a written numerical algorithm that carries out the same steps without the bundles of sticks. Part of developing strategic competence involves learning to replace by more concise and efficient procedures those cumbersome procedures that might at first have been helpful in understanding the operation.

Box 4–3 Subtraction Using Sticks: Modeling 86–53=?

Begin with 8 bundles of 10 sticks along with 6 individual sticks. Because you cannot take away 9 individual sticks, open one bundle, creating 7 bundles of 10 sticks and 16 individual sticks. Take away 5 of the bundles (corresponding to subtracting 50), and take away 9 individual sticks (corresponding to subtracting 9). The number of remaining sticks—2 bundles and 7 individual sticks, or 27—is the answer.

Students develop procedural fluency as they use their strategic competence to choose among effective procedures. They also learn that solving challenging mathematics problems depends on the ability to carry out procedures readily and, conversely, that problem-solving experience helps them acquire new concepts and skills. Interestingly, very young children use a variety of strategies to solve problems and will tend to select strategies that are well suited to particular problems.²⁹ They thereby show the rudiments of adaptive reasoning, the next strand to be discussed.

Adaptive Reasoning

Adaptive reasoning refers to the capacity to think logically about the relationships among concepts and situations. Such reasoning is correct and valid, stems from careful consideration of alternatives, and includes knowledge of how to justify the conclusions. In mathematics, adaptive reasoning is the glue that holds everything together, the lodestar that guides learning. One uses it to navigate through the many facts, procedures, concepts, and solution methods and to see that they all fit together in some way, that they make sense. In mathematics, deductive reasoning is used to settle disputes and disagreements. Answers are right because they follow from some agreed-upon assumptions through series of logical steps. Students who disagree about a mathematical answer need not rely on checking with the teacher, collecting opinions from their classmates, or gathering data from outside the classroom. In principle, they need only check that their reasoning is valid.

Adaptive reasoning refers to the capacity to think logically about the relationships among concepts and situations.

Many conceptions of mathematical reasoning have been confined to formal proof and other forms of deductive reasoning. Our notion of adaptive reasoning is much broader, including not only informal explanation and justification but also intuitive and inductive reasoning based on pattern, analogy, and metaphor. As one researcher put it, "The human ability to find analogical correspondences is a powerful reasoning mechanism."³⁰ Analogical reasoning, metaphors, and mental and physical representations are "tools to think with," often serving as sources of hypotheses, sources of problem-solving operations and techniques, and aids to learning and transfer.³¹

Some researchers have concluded that children's reasoning ability is quite limited until they are about 12 years old.³² Yet when asked to talk about how they arrived at their solutions to problems, children as young as 4 and 5 display evidence of encoding and inference and are resistant to counter suggestion.³³ With the help of representation-building experiences, children can demonstrate sophisticated reasoning abilities. After working in pairs and

reflecting on their activity, for example, kindergartners can "prove" theorems about sums of even and odd numbers.³⁴ Through a carefully constructed sequence of activities about adding and removing marbles from a bag containing many marbles,³⁵ second graders can reason that 5+(–6)=–1. In the context of cutting short bows from a 12-meter package of ribbon and using physical models to calculate that 12 divided by is 36, fifth graders can reason that 12 divided by cannot be 72 because that would mean getting more bows from a package when the individual bow is larger, which does not make sense.³⁶ Research suggests that students are able to display reasoning ability when three conditions are met: They have a sufficient knowledge base, the task is understandable and motivating, and the context is familiar and comfortable.³⁷

One manifestation of adaptive reasoning is the ability to justify one's work. We use justify in the sense of "provide sufficient reason for." Proof is a form of justification, but not all justifications are proofs. Proofs (both formal and informal) must be logically complete, but a justification may be more telegraphic, merely suggesting the source of the reasoning. Justification and proof are a hallmark of formal mathematics, often seen as the province of older students. However, as pointed out above, students can start learning to justify their mathematical ideas in the earliest grades in elementary school.³⁸ Kindergarten and first-grade students can be given regular opportunities to talk about the concepts and procedures they are using and to provide good reasons for what they are doing. Classroom norms can be established in which students are expected to justify their mathematical claims and make them clear to others. Students need to be able to justify and explain ideas in order to make their reasoning clear, hone their reasoning skills, and improve their conceptual understanding.³⁹

It is not sufficient to justify a procedure just once. As we discuss below, the development of proficiency occurs over an extended period of time. Students need to use new concepts and procedures for some time and to explain and justify them by relating them to concepts and procedures that they already understand. For example, it is not sufficient for students to do only practice problems on adding fractions after the procedure has been developed. If students are to understand the algorithm, they also need experience in explaining and justifying it themselves with many different problems.

Adaptive reasoning interacts with the other strands of proficiency, particularly during problem solving. Learners draw on their strategic competence to formulate and represent a problem, using heuristic approaches that may provide a solution strategy, but adaptive reasoning must take over when

they are determining the legitimacy of a proposed strategy. Conceptual understanding provides metaphors and representations that can serve as a source of adaptive reasoning, which, taking into account the limitations of the representations, learners use to determine whether a solution is justifiable and then to justify it. Often a solution strategy will require fluent use of procedures for calculation, measurement, or display, but adaptive reasoning should be used to determine whether the procedure is appropriate. And while carrying out a solution plan, learners use their strategic competence to monitor their progress toward a solution and to generate alternative plans if the current plan seems ineffective. This approach both depends upon productive disposition and supports it.

Productive Disposition

Productive disposition refers to the tendency to see sense in mathematics, to perceive it as both useful and worthwhile, to believe that steady effort in learning mathematics pays off, and to see oneself as an effective learner and doer of mathematics.⁴⁰ If students are to develop conceptual understanding, procedural fluency, strategic competence, and adaptive reasoning abilities, they must believe that mathematics is understandable, not arbitrary; that, with diligent effort, it can be learned and used; and that they are capable of figuring it out. Developing a productive disposition requires frequent opportunities to make sense of mathematics, to recognize the benefits of perseverance, and to experience the rewards of sense making in mathematics.

A productive disposition develops when the other strands do and helps each of them develop. For example, as students build strategic competence in solving nonroutine problems, their attitudes and beliefs about themselves as mathematics learners become more positive. The more mathematical concepts they understand, the more sensible mathematics becomes. In contrast, when students are seldom given challenging mathematical problems to solve, they come to expect that memorizing rather than sense making paves the road to learning mathematics,⁴¹ and they begin to lose confidence in themselves as learners. Similarly, when students see themselves as capable of learning mathematics and using it to solve problems, they become able to develop further their procedural fluency or their adaptive reasoning abilities. Students' disposition toward mathematics is a major factor in determining their educational success. Students who view their mathematical ability as fixed and test questions as measuring their ability rather than providing opportunities to learn are likely to avoid challenging problems and be easily dis-

couraged by failure.⁴² Students who view ability as expandable in response to experience and training are more likely to seek out challenging situations and learn from them. Cross-cultural research studies have found that U.S. children are more likely to attribute success in school to ability rather than effort when compared with students in East Asian countries.⁴³

Most U.S. children enter school eager to learn and with positive attitudes toward mathematics. It is critical that they encounter good mathematics teaching in the early grades. Otherwise, those positive attitudes may turn sour as they come to see themselves as poor learners and mathematics as nonsensical, arbitrary, and impossible to learn except by rote memorization.⁴⁴ Such views, once adopted, can be extremely difficult to change.⁴⁵

The teacher of mathematics plays a critical role in encouraging students to maintain positive attitudes toward mathematics. How a teacher views mathematics and its learning affects that teacher's teaching practice,⁴⁶ which ultimately affects not only what the students learn but how they view themselves as mathematics learners. Teachers and students inevitably negotiate among themselves the norms of conduct in the class, and when those norms allow students to be comfortable in doing mathematics and sharing their ideas with others, they see themselves as capable of understanding.⁴⁷ In chapter 9 we discuss some of the ways in which teachers' expectations and the teaching strategies they use can help students maintain a positive attitude toward mathematics, and in chapter 10 we discuss some programs of teacher development that may help teachers in that endeavor.

An earlier report from the National Research Council identified the cause of much poor performance in school mathematics in the United States:

Some of the most important consequences of students' failure to develop a productive disposition toward mathematics occur in high school, when they have the opportunity to avoid challenging mathematics courses. Avoiding such courses may eliminate the need to face up to peer pressure and other sources of discouragement, but it does so at the expense of precluding careers in science, technology, medicine, and other fields that require a high level of mathematical proficiency.

Research with older students and adults suggests that a phenomenon termed stereotype threat might account for much of the observed differences in mathematics performance between ethnic groups and between male and female students.⁴⁹ In this phenomenon, good students who care about their performance in mathematics and who belong to groups stereotyped as being poor at mathematics perform poorly on difficult mathematics problems under conditions in which they feel pressure to conform to the stereotype. So-called wise educational environments⁵⁰ can reduce the harmful effects of stereotype threat. These environments emphasize optimistic teacher-student relationships, give challenging work to all students, and stress the expandability of ability, among other factors.

Students who have developed a productive disposition are confident in their knowledge and ability. They see that mathematics is both reasonable and intelligible and believe that, with appropriate effort and experience, they can learn. It is counterproductive for students to believe that there is some mysterious "math gene" that determines their success in mathematics.

Hence, our view of mathematical proficiency goes beyond being able to understand, compute, solve, and reason. It includes a disposition toward mathematics that is personal. Mathematically proficient people believe that mathematics should make sense, that they can figure it out, that they can solve mathematical problems by working hard on them, and that becoming mathematically proficient is worth the effort.

Properties of Mathematical Proficiency

Now that we have looked at each strand separately, let us consider mathematical proficiency as a whole. As we indicated earlier and as the preceding discussion illustrates, the five strands are interconnected and must work together if students are to learn successfully. Learning is not an all-or-none phenomenon, and as it proceeds, each strand of mathematical proficiency should be developed in synchrony with the others. That development takes time. One of the most challenging tasks faced by teachers in pre-kindergarten to grade 8 is to see that children are making progress along every strand and not just one or two.

Learning is not an all-or-none phenomenon, and as it proceeds, each strand of mathematical proficiency should be developed in synchrony with the others.

The Strands of Proficiency Are interwoven

How the strands of mathematical proficiency interweave and support one another can be seen in the case of conceptual understanding and procedural fluency. Current research indicates that these two strands of proficiency con-

tinually interact.⁵¹ As a child gains conceptual understanding, computational procedures are remembered better and used more flexibly to solve new problems. In turn, as a procedure becomes more automatic, the child is enabled to think about other aspects of a problem and to tackle new kinds of problems, which leads to new understanding. When using a procedure, a child may reflect on why the procedure works, which may in turn strengthen existing conceptual understanding.⁵² Indeed, it is not always necessary, useful, or even possible to distinguish concepts from procedures because understanding and doing are interconnected in such complex ways.

Consider, for instance, the multiplication of multidigit whole numbers. Many algorithms for computing 47×268 use one basic meaning of multiplication as 47 groups of 268, together with place-value knowledge of 47 as 40+7, to break the problem into two simpler ones: 40×268 and 7×268. For example, a common algorithm for computing 47×268 is written the following way, with the two so-called partial products, 10720 and 1876, coming from the two simpler problems:

Familiarity with this algorithm may make it hard for adults to see how much knowledge is needed for it. It requires knowing that 40×268 is 4×10×268; knowing that in the product of 268 and 10, each digit of 268 is one place to the left; having enough fluency with basic multiplication combinations to find 7×8, 7×60, 7×200, and 4×8, 4×60, 4×200; and having enough fluency with multidigit addition to add the partial products. As students learn to execute a multidigit multiplication procedure such as this one, they should develop a deeper understanding of multiplication and its properties. On the other hand, as they deepen their conceptual understanding, they should become more fluent in computation. A beginner who happens to forget the algorithm but who understands the role of the distributive law can reconstruct the process by writing 268×47=268×(40+7)=(268×40)+(268×7) and working from there. A beginner who has simply memorized the algorithm without understanding much about how it works can be lost later when memory fails.

Proficiency is Not All or Nothing

Mathematical proficiency cannot be characterized as simply present or absent. Every important mathematical idea can be understood at many levels and in many ways. For example, even seemingly simple concepts such as even and odd require an integration of several ways of thinking: choosing alternate points on the number line, grouping items by twos, grouping items into two groups, and looking at only the last digit of the number. When children are first learning about even and odd, they may know one or two of these interpretations.⁵³ But at an older age, a deep understanding of even and odd means all four interpretations are connected and can be justified one based on the others.

The research cited in chapter 5 shows that schoolchildren are never complete mathematical novices. They bring important mathematical concepts and skills with them to school as well as misconceptions that must be taken into account in planning instruction. Obviously, a first grader's understanding of addition is not the same as that of a mathematician or even a lay adult. It is still reasonable, however, to talk about a first grader as being proficient with single-digit addition, as long as the student's thinking in that realm incorporates all five strands of proficiency. Students should not be thought of as having proficiency when one or more strands are undeveloped.

Proficiency Develops Over Time

Proficiency in mathematics is acquired over time. Each year they are in school, students ought to become increasingly proficient. For example, third graders should be more proficient with the addition of whole numbers than they were in the first grade.

Acquiring proficiency takes time in another sense. Students need enough time to engage in activities around a specific mathematical topic if they are to become proficient with it. When they are provided with only one or two examples to illustrate why a procedure works or what a concept means and then move on to practice in carrying out the procedure or identifying the concept, they may easily fail to learn. To become proficient, they need to spend sustained periods of time doing mathematics—solving problems, reasoning, developing understanding, practicing skills—and building connections between their previous knowledge and new knowledge.

How Mathematically Proficient Are U.S. Students Today?

One question that warrants an immediate answer is whether students in U.S. elementary and middle schools today are becoming mathematically proficient. The answer is important because it influences what might be recommended for the future. If students are failing to develop proficiency, the question of how to improve school mathematics takes on a different cast than if students are already developing high levels of proficiency.

The best source of information about student performance in the United States is, as we noted in chapter 2, the National Assessment of Educational Progress (NAEP), a regular assessment of students' knowledge and skills in the school subjects. NAEP includes a large and representative sample of U.S. students at about ages 9, 13, and 17, so the results provide a good picture of students' mathematical performance. We sketched some of that performance in chapter 2, but now we look at it through the frame of mathematical proficiency.

Although the items in the NAEP assessments were not constructed to measure directly the five strands of mathematical proficiency, they provide some useful information about these strands. As in chapter 2, the data reported here are from the 1996 main NAEP assessment except when we refer explicitly to the long-term trend assessment. In general, the performance of 13-year-olds over the past 25 years tells the following story: Given traditional curricula and methods of instruction, students develop proficiency among the five strands in a very uneven way. They are most proficient in aspects of procedural fluency and less proficient in conceptual understanding, strategic competence, adaptive reasoning, and productive disposition. Many students show few connections among these strands. Examples from each strand illustrate the current situation.⁵⁴