# Adding It Up: Helping Children Learn Mathematics(2001)

## Chapter: 7 DEVELOPING PROFICIENCY WITH OTHER NUMBERS

« Previous: 6 DEVELOPING PROFICIENCY WITH WHOLE NUMBERS
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## 7DEVELOPING PROFICIENCYWITH OTHER NUMBERS

In this chapter, we look beyond the whole numbers at other numbers that are included in school mathematics in grades pre-K to 8, particularly in the upper grades. We first look at the rational numbers, which constitute what is undoubtedly the most challenging number system of elementary and middle school mathematics. Then we consider proportional reasoning, which builds on the ratio use of rational numbers. Finally, we examine the integers, a stepping stone to algebra.

### Rational Numbers

Learning about rational numbers is more complicated and difficult than learning about whole numbers. Rational numbers are more complex than whole numbers, in part because they are represented in several ways (e.g., common fractions and decimal fractions) and used in many ways (e.g., as parts of regions and sets, as ratios, as quotients). There are numerous properties for students to learn, including the significant fact that the two numbers that compose a common fraction (numerator and denominator) are related through multiplication and division, not addition.1 This feature often causes misunderstanding when students first encounter rational numbers. Further, students are likely to have less out-of-school experience with rational numbers than with whole numbers. The result is a number system that presents great challenges to students and teachers.

Moreover, how students become proficient with rational numbers is not as well understood as with whole numbers. Significant work has been done, however, on the teaching and learning of rational numbers, and several points

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can be made about developing proficiency with them. First, students do have informal notions of sharing, partitioning sets, and measuring on which instruction can build. Second, in conventional instructional programs, the proficiency with rational numbers that many students develop is uneven across the five strands, and the strands are often disconnected from each other. Third, developing proficiency with rational numbers depends on well-designed classroom instruction that allows extended periods of time for students to construct and sustain close connections among the strands. We discuss each of these points below. Then we examine how students learn to represent and operate with rational numbers.

#### Using Informal Knowledge

Students’ informal notions of partitioning, sharing, and measuring provide a starting point for developing the concept of rational number.2 Young children appreciate the idea of “fair shares,” and they can use that understanding to partition quantities into equal parts. Their experience in sharing equal amounts can provide an entrance into the study of rational numbers. In some ways, sharing can play the role for rational numbers that counting does for whole numbers.

In some ways, sharing can play the role for rational numbers that counting does for whole numbers.

In view of the preschooler’s attention to counting and number that we noted in chapter 5, it is not surprising that initially many children are concerned more that each person gets an equal number of things than with the size of each thing.3 As they move through the early grades of school, they become more sensitive to the size of the parts as well.4 Soon after entering school, many students can partition quantities into equal shares corresponding to halves, fourths, and eighths. These fractions can be generated by successively partitioning by half, which is an especially fruitful procedure since one half can play a useful role in learning about other fractions.5 Accompanying their actions of partitioning in half, many students develop the language of “one half” to describe the actions. Not long after, many can partition quantities into thirds or fifths in order to share quantities fairly among three or five people.

An informal understanding of rational number, which is built mostly on the notion of sharing, is a good starting point for instruction. The notion of sharing quantities and comparing sizes of shares can provide an entry point that takes students into the world of rational numbers.6 Equal shares, for example, opens the concept of equivalent fractions (e.g., If there are 6 chil-

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dren sharing 4 pizzas, how many pizzas would be needed for 12 children to receive the same amount?).

It is likely, however, that an informal understanding of rational numbers is less robust and widespread than the corresponding informal understanding of whole numbers. For whole numbers, many young children enter school with sufficient proficiency to invent their own procedures for adding, subtracting, multiplying, and dividing. For rational numbers, in contrast, teachers need to play a more active and direct role in providing relevant experiences to enhance students’ informal understanding and in helping them elaborate their informal understanding into a more formal network of concepts and procedures. The evidence suggests that carefully designed instructional programs can serve both of these functions quite well, laying the foundation for further progress.7

#### Discontinuities in Proficiency

Proficiency with rational numbers, as with all mathematical topics, is signaled most clearly by the close intertwining of the five strands. Large-scale surveys of U.S. students’ knowledge of rational number indicate that many students are developing some proficiency within individual strands.8 Often, however, these strands are not connected. Furthermore, the knowledge students acquire within strands is also disconnected. A considerable body of research describes this separation of knowledge.9

As we said at the beginning of the chapter, rational numbers can be expressed in various forms (e.g., common fractions, decimal fractions, percents), and each form has many common uses in daily life (e.g., a part of a region, a part of a set, a quotient, a rate, a ratio).10 One way of describing this complexity is to observe that, from the student’s point of view, a rational number is not a single entity but has multiple personalities. The scheme that has guided research on rational number over the past two decades11 identifies the following interpretations for any rational number, say : (a) a part-whole relation (3 out of 4 equal-sized shares); (b) a quotient (3 divided by 4); (c) a measure ( of the way from the beginning of the unit to the end); (d) a ratio (3 red cars for every 4 green cars); and (e) an operation that enlarges or reduces the size of something ( of 12). The task for students is to recognize these distinctions and, at the same time, to construct relations among them that generate a coherent concept of rational number.12 Clearly, this process is lengthy and multifaceted.

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Instructional practices that tend toward premature abstraction and extensive symbolic manipulation lead students to have severe difficulty in representing rational numbers with standard written symbols and using the symbols appropriately.13 This outcome is not surprising, because a single rational number can be represented with many different written symbols (e.g., 0.6, 0.60, 60%). Instructional programs have often treated this complexity as simply a “syntactic” translation problem: One written symbol had to be translated into another according to a sequence of rules. Different rules have often been taught for each translation situation. For example, “To change a common fraction to a decimal fraction, divide the numerator by the denominator.”

But the symbolic representation of rational numbers poses a “semantic” problem—a problem of meaning—as well. Each symbol representation means something. Current instruction often gives insufficient attention to developing the meanings of different rational number representations and the connections among them. The evidence for this neglect is that a majority of U.S. students have learned rules for translating between forms but understand very little about what quantities the symbols represent and consequently make frequent and nonsensical errors.14 This is a clear example of the lack of proficiency that results from pushing ahead within one strand but failing to connect what is being learned with other strands. Rules for manipulating symbols are being memorized, but students are not connecting those rules to their conceptual understanding, nor are they reasoning about the rules.

Another example of disconnection among the strands of proficiency is students’ tendency to compute with written symbols in a mechanical way without considering what the symbols mean. Two simple examples illustrate the point. First, recall (from chapter 4) the result from the National Assessment of Educational Progress (NAEP)15 showing that more than half of U.S. eighth graders chose 19 or 21 as the best estimate of These choices do not make sense if students understand what the symbols mean and are reasoning about the quantities represented by the symbols. Another survey of students’ performance showed that the most common error for the addition problem 4+.3=? is .7, which is given by 68% of sixth graders and 51% of fifth and seventh graders.16 Again, the errors show that many students have learned rules for manipulating symbols without understanding what those symbols mean or why the rules work. Many students are unable to reason appropriately about symbols for rational numbers and do not have the strategic competence that would allow them to catch their mistakes.

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#### Supporting Connections

Of all the ways in which rational numbers can be interpreted and used, the most basic is the simplest—rational numbers are numbers. That fact is so fundamental that it is easily overlooked. A rational number like is a single entity just as the number 5 is a single entity. Each rational number holds a unique place (or is a unique length) on the number line (see chapter 3). As a result, the entire set of rational numbers can be ordered by size, just as the whole numbers can. This ordering is possible even though between any two rational numbers there are infinitely many rational numbers, in drastic contrast to the whole numbers.

It may be surprising that, for most students, to think of a rational number as a number—as an individual entity or a single point on a number line—is a novel idea.17 Students are more familiar with rational numbers in contexts like parts of a pizza or ratios of hits to at-bats in baseball. These everyday interpretations, although helpful for building knowledge of some aspects of rational number, are an inadequate foundation for building proficiency. The difficulty is not just due to children’s limited experience. Even the interpretations ordinarily given by adults to various forms of rational numbers, such as percent, do not lead easily to the conclusion that rational numbers are numbers.18 Further, the way common fractions are written (e.g., ) does not help students see a rational number as a distinct number. After all, looks just like one whole number over another, and many students initially think of it as two different numbers, a 3 and a 4.

Research has verified what many teachers have observed, that students continue to use properties they learned from operating with whole numbers even though many whole number properties do not apply to rational numbers. With common fractions,19 for example, students may reason that is larger than because 8 is larger than 7. Or they may believe that equals because in both fractions the difference between numerator and denominator is 1. With decimal fractions,20 students may say .25 is larger than .7 because 25 is larger than 7. Such inappropriate extensions of whole number relationships, many based on addition, can be a continuing source of trouble when students are learning to work with fractions and their multiplicative relationships.21

The task for instruction is to use, rather than to ignore, the informal knowledge of rational numbers that students bring with them and to provide them with appropriate experiences and sufficient time to develop meaning for these new numbers and meaningful ways of operating with them. Systematic errors can best be regarded as useful diagnostic tools for instruction since they more

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often represent incomplete rather than incorrect knowledge.22 From the current research base, we can make several observations about the kinds of learning opportunities that instruction must provide students if they are to develop proficiency with rational numbers. These observations address both representing rational numbers and computing with them.

#### Representing Rational Numbers

As with whole numbers, the written notations and spoken words used for decimal and common fractions contribute to—or at least do not help correct— the many kinds of errors students make with them. Both decimals and common fractions use whole numbers in their notations. Nothing in the notation or the words used conveys their meaning as fractured parts. The English words used for fractions are the same words used to tell order in a line: fifth in line and three fifths (for ). In contrast, in Chinese, is read “out of 5 parts (take) 3.” Providing students with many experiences in partitioning quantities into equal parts using concrete models, pictures, and meaningful contexts can help them create meaning for fraction notations. Introducing the standard notation for common fractions and decimals must be done with care, ensuring that students are able to connect the meanings already developed for the numbers with the symbols that represent them.

Research does not prescribe a one best set of learning activities or one best instructional method for rational numbers. But some sequences of activities do seem to be more effective than others for helping students develop a conceptual understanding of symbolic representations and connect it with the other strands of proficiency.23 The sequences that have been shown to promote mathematical proficiency differ from each other in a number of ways, but they share some similarities. All of them spend time at the outset helping students develop meaning for the different forms of representation. Typically, students work with multiple physical models for rational numbers as well as with other supports such as pictures, realistic contexts, and verbal descriptions. Time is spent helping students connect these supports with the written symbols for rational numbers.

In one such instructional sequence, fourth graders received 20 lessons introducing them to rational numbers.24 Almost all the lessons focused on helping the students connect the various representations of rational number with concepts of rational number that they were developing. Unique to this program was the sequence in which the forms were introduced: percents, then decimal fractions, and then common fractions. Because many children

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in the fourth grade have considerable informal knowledge of percents, percents were used as the starting point. Students were asked to judge, for example, the relative fullness of a beaker (e.g., 75%), and the relative height of a tube of liquid (e.g., 30%). After a variety of similar activities, the percent representations were used to introduce the decimal fractions and, later, the common fractions. Compared with students in a conventional program, who spent less time developing meaning for the representations and more time practicing computation, students in the experimental program demonstrated higher levels of adaptive reasoning, conceptual understanding, and strategic competence, with no loss of computational skill. This finding illustrates one of our major themes: Progress can be made along all strands if they remain connected.

Another common feature of learning activities that help students understand and use the standard written symbols is the careful attention the activities devote to the concept of unit.25 Many conventional curricula introduce rational numbers as common fractions that stand for part of a whole, but little attention is given to the whole from which the rational number extracts its meaning. For example, many students first see a fraction as, say, of a pizza. In this interpretation the amount of pizza is determined by the fractional part and by the size of the pizza. Hence, three fourths of a medium pizza is not the same amount of pizza as three fourths of a large pizza, although it may be the same number of pieces. Lack of attention to the nature of the unit or whole may explain many of the misconceptions that students exhibit.

A sequence of learning activities that focus directly on the whole unit in representing rational numbers comes from an experimental curriculum in Russia.26 In this sequence, rational numbers are introduced in the early grades as ratios of quantities to the unit of measure. For example, a piece of string is measured by a small piece of tape and found to be equivalent to five copies of the tape. Children express the result as “string/tape=5.” Rational numbers appear quite naturally when the quantity is not measured by the unit an exact number of times. The leftover part is then represented, first informally and then as a fraction of the unit. With this approach, the size of the unit always is in the foreground. The evidence suggests that students who engage in these experiences develop coherent meanings for common fractions, meanings that allow them to reason sensibly about fractions.27

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##### Computing with Rational Numbers

As with representing rational numbers, many students need instructional support to operate appropriately with rational numbers. Adding, subtracting, multiplying, and dividing rational numbers require that they be seen as numbers because in elementary school these operations are defined only for numbers. That is, the principles on which computation is based make sense only if common fractions and decimal fractions are understood as representing numbers. Students may think of a fraction as part of a pizza or as a batting average, but such interpretations are not enough for them to understand what is happening when computations are carried out. The trouble is that many students have not developed a meaning for the symbols before they are asked to compute with rational numbers.

Proficiency in computing with rational numbers requires operating with at least two different representations: common fractions and finite decimal fractions. There are important conceptual similarities between the rules for computing with both of these forms (e.g., combine those terms measured with the same unit when adding and subtracting). However, students must learn how those conceptual similarities play out in each of the written symbol systems. Procedural fluency for arithmetic with rational numbers thus requires that students understand the meaning of the written symbols for both common fractions and finite decimal fractions.

What can be learned from students’ errors? Research reveals the kinds of errors that students are likely to make as they begin computing with common fractions and finite decimals. Whether the errors are the consequence of impoverished learning of whole numbers or insufficiently developed meaning for rational numbers, effective instruction with rational numbers needs to take these common errors into account.

Some of the errors occur when students apply to fractions poorly understood rules for calculating with whole numbers. For example, they learn to “line up the numbers on the right” when they are adding and subtracting whole numbers. Later, they may try to apply this rule to decimal fractions, probably because they did not understand why the rule worked in the first place and because decimal fractions look a lot like whole numbers. This confusion leads many students to get .61 when adding 1.5 and .46, for example.28

It is worth pursuing the above example a bit further. Notice that the rule “line up the numbers on the right” and the new rule for decimal fractions “line up the decimal points” are, on the surface, very different rules. They

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prescribe movements of digits in different-sounding ways. At a deeper level, however, they are exactly the same. Both versions of the rule result in aligning digits measured with the same unit—digits with the same place value (tens, ones, tenths, etc.). This deeper level of interpretation is, of course, the one that is more useful. When students know a rule only at a superficial level, they are working with symbols, rules, and procedures in a routine way, disconnected from strands such as adaptive reasoning and conceptual understanding. But when students see the deeper level of meaning for a procedure, they have connected the strands together. In fact, seeing the second level is a consequence of connecting the strands. This example illustrates once more why connecting the strands is the key to developing proficiency.

A second example of a common error and one that also can be traced to previous experience with whole numbers is that “multiplying makes larger” and “dividing makes smaller.”29 These generalizations are not true for the full set of rational numbers. Multiplying by a rational number less than 1 means taking only a part of the quantity being multiplied, so the result is less than the original quantity (e.g., which is less than 12). Likewise, dividing by a rational number less than 1 produces a quantity larger than either quantity in the original problem (e.g.,).

As with the addition and subtraction of rational numbers, there are important conceptual similarities between whole numbers and rational numbers when students learn to multiply and divide. These similarities are often revealed by probing the deeper meaning of the operations. In the division example above, notice that to find the answer to 6÷2=? and the same question can be asked: How many [2s or ] are in 6? The similarities are not apparent in the algorithms for manipulating the symbols. Therefore, if students are to connect what they are learning about rational numbers with what they already understand about whole numbers, they will need to do so through other kinds of activities.

One helpful approach is to embed the calculation in a realistic problem. Students can then use the context to connect their previous work with whole numbers to the new situations with rational numbers. An example is the following problem:

I have six cups of sugar. A recipe calls for of a cup of sugar. How many batches of the recipe can I make?

Since the size of the parts is less than one whole, the number of batches will necessarily be larger than the six (there are nine in 6). Useful activities

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might include drawing pictures of the division calculation, describing solution methods, and explaining why the answer makes sense. Simply teaching the rule “invert and multiply” leads to the same sort of mechanical manipulation of symbols that results from just telling students to “line up the decimal points.”

What can be learned from conventional and experimental instruction? Conventional instruction on rational number computation tends to be rule based.30 Classroom activities emphasize helping students become quick and accurate in executing written procedures by following rules. The activities often begin by stating a rule or algorithm (e.g., “to multiply two fractions, multiply the numerators and multiply the denominators”), showing how it works on several examples (sometimes just one), and asking students to practice it on many similar problems. Researchers express concern that this kind of learning can be “highly dependent on memory and subject to deterioration.”31 This “deterioration” results when symbol manipulation is emphasized to the relative exclusion of conceptual understanding and adaptive reasoning. Students learn that it is not important to understand why the procedure works but only to follow the prescribed steps to reach the correct answer. This approach breaks the incipient connections between the strands of proficiency, and, as the breaks increase, proficiency is thwarted.

Conventional instruction on rational number computation tends to be rule based.

A number of studies have documented the results of conventional instruction.32 One study, for example, found that only 45% of a random sample of 20 sixth graders interviewed could add fractions correctly.33 Equally disturbing was that fewer than 10% of them could explain how one adds fractions even though all had heard the rules for addition, had practiced the rules on many problems, and sometimes could execute the rules correctly. These results, according to the researchers, were representative of hundreds of interviews conducted with sixth, seventh, and ninth graders. The results point to the need for instructional materials that support teachers and students so that they can explain why a procedure works rather than treating it as a sequence of steps to be memorized.

Many researchers who have studied what students know about operations with fractions or decimals recommend that instruction emphasize conceptual understanding from the beginning.34 More specifically, say these researchers, instruction should build on students’ intuitive understanding of fractions and use objects or contexts that help students make sense of the operations. The rationale for that approach is that students need to under

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stand the key ideas in order to have something to connect with procedural rules. For example, students need to understand why the sum of two fractions can be expressed as a single number only when the parts are of the same size. That understanding can lead them to see the need for constructing common denominators.

One of the most challenging tasks confronting those who design learning environments for students (e.g., curriculum developers, teachers) is to help students learn efficient written algorithms for computing with fractions and decimals. The most efficient algorithms often do not parallel students’ informal knowledge or the meaning they create by drawing diagrams, manipulating objects, and so on. Several instructional programs have been devised that use problem situations and build on algorithms invented by students.35 Students in these programs were able to develop meaningful and reasonably efficient algorithms for operating with fractions, even when the formal algorithms were not presented.36 It is not yet clear, however, what sequence of activities can support students’ meaningful learning of the less transparent but more efficient formal algorithms, such as “invert and multiply” for dividing fractions.

Although there is only limited research on instructional programs for developing proficiency with computations involving rational numbers, it seems clear that instruction focused solely on symbolic manipulation without understanding is ineffective for most students. It is necessary to correct that imbalance by paying more attention to conceptual understanding as well as the other strands of proficiency and by helping students connect them.

### Proportional Reasoning

Proportions are statements that two ratios are equal. These statements play an important role in mathematics and are formally introduced in middle school. Understanding the underlying relationships in a proportional situation and working with these relationships has come to be called proportional reasoning.37 Considerable research has been conducted on the features of proportional reasoning and how students develop it.38

Proportional reasoning is based, first, on an understanding of ratio. A ratio expresses a mathematical relationship that involves multiplication, as in \$2 for 3 balloons or of a dollar for one balloon. A proportion, then, is a relationship between relationships. For example, a proportion expresses the fact that \$2 for 3 balloons is in the same relationship as \$6 for 9 balloons Ratios are often changed to unit ratios by dividing. For example, the unit ratio dollars per balloon is obtained by “dividing” \$2 by 3 balloons. The

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ratio or rate, \$ per balloon, is called the unit rate because it is the cost of one balloon. The unit rate may be useful to students when they think about real situations.39 In this case it describes the precise manner by which any number of dollars can be compared with any number of balloons at the same price.

Proportional reasoning has been described as the capstone of elementary school arithmetic and the gateway to higher mathematics, including algebra, geometry, probability, statistics, and certain aspects of discrete mathematics.40 Nevertheless, U.S. seventh and eighth graders have not performed well on even simple proportion problems such as finding the cost of 6 pieces of candy if 2 pieces cost 8 cents and if the price of the candy is the same no matter how many are sold.41 On the 1996 NAEP, only 12% of eighth-grade students could solve a problem involving the comparison of two rates, 8 miles every 10 minutes and 20 miles every 25 minutes.42

Research tracing the development of proportional reasoning shows that children have some informal knowledge of proportions. Studies with second graders have suggested that their intuitive understanding is insufficient for solving certain proportion problems.43 Proficiency grows as students connect different aspects of proportional reasoning.44 Three aspects are especially important. First, students’ reasoning is facilitated as they learn to make comparisons based on multiplication rather than just addition. For example, consider two marigolds that were 8 inches and 12 inches tall two weeks ago and 11 inches and 15 tall inches now. Which plant grew more? There are two different correct responses to this question. An additive or absolute comparison focuses on the difference and concludes that each plant grew the same, 3 inches. A multiplicative or relative comparison looks at the change relative to the original height; the shorter plant grew of its original height, while the larger plant grew less, just of its original height. Either answer is correct depending on whether “grew more” is interpreted in absolute or relative terms. The ability to reason about comparisons in relative terms is closely tied to reasoning proportionally.45

A second aspect is that students’ reasoning is facilitated as they distinguish between those features of a proportion situation that can change and those that must stay the same.46 In a proportion the quantities composing a ratio can change together in such a way that the relationship between them (the quotient) remains the same. Some students are inclined to take a more simplistic view, believing that if something changes, everything changes. In a proportion the numbers in the ratios can change but the multiplicative relationship must stay the same (e.g., \$2 for 3 balloons expresses the same relationship as \$4 for 6 balloons). The physical situation is not the same because the

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second ratio refers to twice as many dollars and balloons as the first. What is the same is the multiplicative relationship between the dollars and the balloons or, said another way, the cost of a single balloon (the unit rate). Written symbolically, without labels, the statement becomes But notice how the important contextual framework is lost with this abstract notation.

Proportional reasoning is further enhanced as the first two aspects are connected with a third: Students’ reasoning is facilitated as they learn to build composite units, or units of units. The rate “\$2 for 3 balloons” or “2-for-3” is a composite unit.47 The ability to use composite units is one of the most obvious differences between students who reason well with proportions and those who do not.48 Students who reason correctly about proportional situations often choose one ratio as a composite unit and use it as a comparative base. For example, they might use “2-for-3” to examine whether another ratio, such as 12-for-24, has the same relationship. By building up the 2-for-3 units (2-for-3, 4-for-6, 6-for-9, 8-for-12, 10-for-15, 12-for-18), the students realize 2-for-3 is not proportional to 12-for-24, because 12-for-24 cannot be generated with the 2-for-3 composite unit. There is a danger, of course, in using this essentially additive building-up process to generate equivalent ratios because students may not understand that the relationship is multiplicative. They need to see that 2-for-3 and 6-for-9, for example, express the same relationship or unit rate because 9 is the same multiple of 3 as 6 is of 2. But building from composite units does provide many students with a useful tool for working with proportional situations.

The conceptual aspects of proportional reasoning usually play out in three types of proportion problems. Missing value problems present three values and ask students to find the fourth or missing value (e.g., If 3 balloons cost \$2, then how much do 24 balloons cost?). Numerical comparison problems ask students to determine which of two given ratios represents more or less (e.g., Which is the better value: 3 balloons for \$2 or 24 balloons for \$12?). Qualitative comparison problems ask students to evaluate the effect on a ratio of a qualitative change in one or both of the quantities involved (e.g., What happens to the price of a balloon if you get more balloons for the same amount of money?). Traditionally, instruction has focused on missing-value problems, with some attention to numerical comparisons. For both kinds of problems, traditional textbooks tend to emphasize formal strategies from the beginning49—setting up a correct equation (3:2=24:x), using a variable for the missing value, and using a “cross-multiplication” algorithm (3x=48 or x=16).

It should be clear from the previous analysis that moving directly to the cross-multiplication algorithm, without attending to the conceptual aspects

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of proportional reasoning, can create difficulties for students. The aspects of proportional reasoning that must be developed can be supported through exploring proportional (and nonproportional) situations in a variety of problem contexts using concrete materials or situations in which students collect data, build tables, and determine the relationships between the number pairs (ratios) in the tables.50 When 187 seventh-grade students with different curricular experiences were presented with a sequence of realistic rate problems, the students in the reform curricula considerably outperformed a comparison group of students 53% versus 28% in providing correct answers with correct support work.51 These students were part of the field trials for a new middle school curriculum in which they were encouraged to develop their own procedures through collaborative problem-solving activities. The comparison students had more traditional, teacher-directed instructional experiences.

Proportional reasoning is complex and clearly needs to be developed over several years.52 One simple implication from the research suggests that presenting the cross-multiplication algorithm before students understand proportions and can reason about them leads to the same kind of separation between the strands of proficiency that we described earlier for other topics. But more research is needed to identify the sequences of activities that are most helpful for moving from well-understood but less efficient procedures to those that are more efficient.

Ratios and proportions, like fractions, decimals, and percents, are aspects of what have been called multiplicative structures.53 These are closely related ideas that no doubt develop together, although they are often treated as separate topics in the typical school curriculum. Reasoning about these ideas likely interacts, but it is not well understood how this interaction develops. Much more work needs to be done on helping students integrate their knowledge of these components of multiplicative structures.

### Integers

The set of integers comprises the positive and negative whole numbers and zero or, expressed another way, the whole numbers and their inverses, often called their opposites (see Chapter 3). The set of integers, like the set of whole numbers, is a subset of the rational numbers. Compared with the research on whole numbers and even on noninteger rational numbers, there has been relatively little research on how students acquire an understanding of negative numbers and develop proficiency in operating with them.

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A half-century ago students did not encounter negative numbers until they took high school algebra. Since then, integers have been introduced in the middle grades and even in the elementary grades. Some educators have argued that integers are easier for students than fractions and decimals and therefore should be introduced first. This approach has been tried, but there is very little research on the long-term effects of this alternative sequencing of topics.

#### Concept of Negative Numbers

Even young children have intuitive or informal knowledge of nonpositive quantities prior to formal instruction.54 These notions often involve action-based concepts like those associated with temperature, game moves, or other spatial and quantitative situations. For example, in some games there are moves that result in points being lost, which can lead to scores below zero or “in the hole.”

Various metaphors have been suggested as approaches for introducing negative numbers, including elevators, thermometers, debts and assets, losses and gains, hot air balloons, postman stories, pebbles in a bag, and directed arrows on a number line.55 Many of the physical metaphors for introducing integers have been criticized because they do not easily support students’ understanding of the operations on integers (other than addition).56 But some studies have demonstrated the value of using these metaphors, especially for introducing negative numbers.57

Students do appear to be capable of understanding negative numbers far earlier than was once thought. Although more research is needed on the metaphors and models that best support students’ conceptual understanding of negative numbers, there already is enough information to suggest that a variety of metaphors and models can be used effectively.

#### Operations with integers

Research on learning to add, subtract, multiply, and divide integers is limited. In the past, students often learned the “rules of signs” (e.g., the product of a positive and negative number is negative) without much understanding. In part, perhaps, because instruction has not found ways to make the learning meaningful, some secondary and college students still have difficulty working with negative numbers.58

Alternative approaches, using the models mentioned earlier, have been tried with various degrees of success.59 A complete set of appropriate learn-

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ing activities with integers has not been identified, but there are some promising elements that should be explored further. Students generally perform better on problems posed in the context of a story (debts and assets, scores and forfeits) or through movements on a number line than on the same problems presented solely as formal equations.60 This result suggests, as for other number domains, that stories and other conceptual structures such as a number line can be used effectively as the context in which students begin their work and develop meaning for the operations. Furthermore, there are some approaches that seem to minimize commonly reported errors.61 In general, approaches that use an appropriate model of integers and operations on integers, and that spend time developing these and linking them to the symbols, offer the most promise.

### Beyond Whole Numbers

Although the research provides a less complete picture of students’ developing proficiency with rational numbers and integers than with whole numbers, several important points can be made. First, developing proficiency is a gradual and prolonged process. Many students acquire useful informal knowledge of fractions, decimals, ratios, percents, and integers through activities and experiences outside of school, but that knowledge needs to be made more explicit and extended through carefully designed instruction. Given current learning patterns, effective instruction must prepare for interferences arising from students’ superficial knowledge of whole numbers. The unevenness many students show in developing proficiency that we noted with whole numbers seems especially pronounced with rational numbers, where progress is made on different fronts at different rates. The challenge is to engage students throughout the middle grades in learning activities that support the integration of the strands of proficiency.

A second observation is that doing just that—integrating the strands of proficiency—is an even greater challenge for rational numbers than for whole numbers. Currently, many students learn different aspects of rational numbers as separate and isolated pieces of knowledge. For example, they fail to see the relationships between decimals, fractions, and percents, on the one hand, and whole numbers, on the other, or between integers and whole numbers. Also, connections among the strands of proficiency are often not made. Numerous studies show that with common fractions and decimals, especially, conceptual understanding and computational procedures are not appropriately linked. Further, students can use their informal knowledge of propor-

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tionality or rational numbers strategically to solve problems but are unable to represent and solve the same problem formally. These discontinuities are of great concern because the research we have reviewed indicates that real progress along each strand and within any single topic is exceedingly difficult without building connections between them.

A third issue concerns the level of procedural fluency that should be required for arithmetic with decimals and common fractions. Decimal fractions are crucial in science, in metric measurement, and in more advanced mathematics, so it is important for students to be computationally fluent—to understand how and why computational procedures work, including being able to judge the order-of-magnitude accuracy of calculator-produced answers. Some educators have argued that common fractions are no longer essential in school mathematics because digital electronics have transformed almost all numerical transactions into decimal fractions. Technological developments certainly have increased the importance of decimals, but common fractions are still important in daily life and in their own right as mathematical objects, and they play a central role in the development of more advanced mathematical ideas. For example, computing with common fractions sets the stage for computing with rational expressions in algebra. It is important, therefore, for students to develop sound meanings for common fractions and to be fluent with ordering fractions, finding equivalent fractions, and using unit rates. Students should also develop procedural fluency for computations with “manageable” fractions. However, the rapid execution of paper-and-pencil computation algorithms for less frequently used fractions (e.g., ) is unnecessary today.

Finally, we cannot emphasize too strongly the simple fact that students need to be fully proficient with rational numbers and integers. This proficiency forms the basis for much of advanced mathematical thinking, as well as the understanding and interpretation of daily events. The level at which many U.S. students function with rational numbers and integers is unacceptable. The disconnections that many students exhibit among their conceptual understanding, procedural fluency, strategic competence, and adaptive reasoning pose serious barriers to their progress in learning and using mathematics. Evidence from experimental programs in the United States and from the performance of students in other countries suggests that U.S. middle school students are capable of learning more about rational numbers and integers, with deeper levels of understanding.

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### Notes

 1 See Harel and Confrey, 1994. Rational numbers, ratios, and proportions, which on the surface are about division, are called multiplicative concepts because any division problem can be rephrased as multiplication. See Chapter 3. 2 Behr, Lesh, Post, and Silver, 1983; Confrey, 1994, 1995; Empson, 1999; Kieren, 1992; Mack, 1990, 1995; Pothier and Sawada, 1983; Streefland, 1991, 1993. 3 Hiebert and Tonnessen, 1978; Pothier and Sawada, 1983. 4 Empson, 1999; Pothier and Sawada, 1983. 5 Confrey, 1994; Pothier and Sawada 1989. 6 Confrey, 1994; Streefland, 1991, 1993. 7 Cramer, Behr, Post, and Lesh, 1997; Empson, 1999; Mack, 1995; Morris, in press; Moss and Case, 1999; Streefland, 1991, 1993. 8 Kouba, Zawojewski, and Strutchens, 1997; Wearne and Kouba, 2000. 9 Behr, Lesh, Post, and Silver, 1983; Behr, Wachsmuth, Post, and Lesh, 1984; Bezuk and Bieck, 1993; Hiebert and Wearne, 1985; Mack, 1990, 1995; Post, Wachsmuth, Lesh, and Behr, 1985; Streefland, 1991, 1993. 10 Kieren, 1976. 11 Kieren, 1976, 1980, 1988. 12 Students not only should “construct relations among them” but should also eventually have some grasp of what is entailed in these relations—for example, that Interpretation D is a contextual instance of E—namely, you multiply the number of green cars by to get the number of red cars, while thinking of as three times (Interpretation A), and thinking of it as 3 divided by 4, is the equation which is basically the associative law for multiplication. 13 Behr, Wachsmuth, Post, and Lesh, 1984; Hiebert and Wearne, 1986. 14 Hiebert and Wearne, 1986; Resnick, Nesher, Leonard, Magone, Omanson, and Peled, 1989. 15 Carpenter, Corbitt, Kepner, Lindquist, and Reys, 1981. 16 Hiebert and Wearne, 1986. 17 Behr, Lesh, Post, and Silver, 1983. 18 Davis, 1988. 19 Behr, Wachsmuth, Post, and Lesh, 1984. 20 Resnick, Nesher, Leonard, Magone, Omanson, and Peled, 1989. 21 Behr, Wachsmuth, Post, and Lesh, 1984. 22 Resnick, Nesher, Leonard, Magone, Omanson, and Peled, 1989. 23 Cramer, Post, Henry, and Jeffers-Ruff, in press; Hiebert and Wearne, 1988; Hunting, 1983; Mack, 1990, 1995; Morris, in press; Moss and Case, 1999; Hiebert, Wearne, and Taber, 1991. 24 Moss and Case, 1999. 25 Behr, Harel, Post, and Lesh, 1992. 26 Davydov and Tsvetkovich, 1991; Morris, in press; Schmittau, 1993. 27 Morris, in press.
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 28. Hiebert and Wearne, 1986. 29. Bell, Fischbein, and Greer, 1984; Fischbein, Deri, Nello, and Marino, 1985. 30. Hiebert and Wearne, 1985. 31. Kieren, 1988, p. 178. 32. Mack, 1990; Peck and Jencks, 1981; Wearne and Kouba, 2000. 33. Peck and Jencks, 1981. 34. Behr, Lest, Post, and Silver, 1983; Bezuk and Bieck, 1993; Bezuk and Cramer, 1989; Hiebert and Wearne, 1986; Kieren, 1988; Mack, 1990; Peck and Jencks, 1981; Streefland, 1991, 1993. 35. Cramer, Behr, Post, and Lesh, 1997; Huinker, 1998; Lappan, Fey, Fitzgerald, Friel, and Phillips, 1996; Streefland, 1991. 36. Huinker, 1998; Lappan and Bouck, 1998. 37. Lesh, Post, and Behr, 1988. 38. Tourniaire and Pulos, 1985. 39. Behr, Harel, Post, and Lesh, 1992; Cramer, Behr, and Bezuk, 1989. 40. Post, Behr, and Lesh, 1988. 41. Lesh, Post, and Behr, 1988. 42. Wearne and Kouba, 2000. 43. Ahl, Moore, and Dixon, 1992; Dixon and Moore, 1996. 44. Lamon, 1993, 1995. 45. Lamon, 1993. 46. Lamon, 1995. 47. The term composite unit refers to thinking of 3 balloons (and hence \$2) as a single entity. The related term compound unit is used in science to refer to units such as “miles/hour,” or in this case “dollars per balloon.” 48. Lamon, 1993, 1994. 49. Heller, Ahlgren, Post, Behr, and Lesh, 1989; Langrall and Swafford, 2000. 50. Cramer, Post, and Currier, 1993; Kaput and West, 1994. 51. Ben-Chaim, Fey, Fitzgerald, Benedetto, and Miller, 1998; Heller, Ahlgren, Post, Behr, and Lesh, 1989. 52. Behr, Harel, Post, and Lesh, 1992; Karplus, Pulas, and Stage, 1983. 53. Vergnaud, 1983. 54. Hativa and Cohen, 1995. 55. English, 1997. See also Crowley and Dunn, 1985. 56. Fischbein, 1987, ch. 8. 57. Duncan and Sanders, 1980; Moreno and Mayer, 1999; Thompson, 1988. 58. Bruno, Espinel, Martinon, 1997; Kuchemann, 1980. 59. Arcavi and Bruckheimer, 1981; Carson and Day, 1995; Davis, 1990; Liebeck, 1990; Human and Murray, 1987. 60. Moreno and Mayer, 1999; Mukhopadhyay, Resnick, and Schauble, 1990. 61. Duncan and Saunders, 1980; Thompson, 1988; Thompson and Dreyfus, 1988.
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Adding It Up: Helping Children Learn Mathematics Get This Book
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Adding It Up explores how students in pre-K through 8th grade learn mathematics and recommends how teaching, curricula, and teacher education should change to improve mathematics learning during these critical years.

The committee identifies five interdependent components of mathematical proficiency and describes how students develop this proficiency. With examples and illustrations, the book presents a portrait of mathematics learning:

• Research findings on what children know about numbers by the time they arrive in pre-K and the implications for mathematics instruction.
• Details on the processes by which students acquire mathematical proficiency with whole numbers, rational numbers, and integers, as well as beginning algebra, geometry, measurement, and probability and statistics.

The committee discusses what is known from research about teaching for mathematics proficiency, focusing on the interactions between teachers and students around educational materials and how teachers develop proficiency in teaching mathematics.

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