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309 7 Pipes, Tubes, and Beakers: New Approaches to Teaching the Rational-Number System Joan Moss PEANUTS reprinted by permission of United Feature Syndicate, Inc. Poor Sally. Her anger and frustration with fractions are palpable. And they no doubt reflect the feelings and experiences of many students. As mathematics education researchers and teachers can attest, students are of- ten vocal in their expression of dislike of fractions and other representations of rational numbers (percents and decimals). In fact, the rational-number system poses problems not only for youngsters, but for many adults as well.1 In a recent study, masters students enrolled in an elementary teacher-train- ing program were interviewed to determine their knowledge and under- standing of basic rational-number concepts. While some students were con- fident and produced correct answers and explanations, the majority had difficulty with the topic. On attempting to perform an operation involving fractions, one student, whose sentiments were echoed by many, remarked, “Oh fractions! I know there are lots of rules but I can’t remember any of them and I never understood them to start with.”2
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310 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM We know from extensive research that many people—adults, students, even teachers—find the rational-number system to be very difficult.3 Intro- duced in early elementary school, this number system requires that students reformulate their concept of number in a major way. They must go beyond whole-number ideas, in which a number expresses a fixed quantity, to un- derstand numbers that are expressed in relationship to other numbers. These new proportional relationships are grounded in multiplicative reasoning that is quite different from the additive reasoning that characterizes whole num- bers (see Box 7-1).4 While some students make the transition smoothly, the majority, like Sally, become frustrated and disenchanted with mathematics.5 Why is this transition so problematic? A cursory look at some typical student misunderstandings illuminates the kinds of problems students have with rational numbers. The culprit ap- pears to be the continued use of whole-number reasoning in situations where it does not apply. When asked which number is larger, 0.059 or 0.2, a major- ity of middle school students assert that 0.059 is bigger, arguing that the number 59 is bigger than the number 2.6 Similarly, faulty whole-number reasoning causes students to maintain, for example, that the fraction 1/8 is larger than 1/6 because, as they say, “8 is a bigger number than 6.”7 Not surprisingly, students struggle with calculations as well. When asked to find the sum of 1/2 and 1/3, the majority of fourth and sixth graders give the answer 2/5. Even after a number of years working with fractions, some eighth graders make the same error, illustrating that they still mistakenly count the numerator and denominator as separate numbers to find a sum.8 Clearly whole-number reasoning is very resilient. Decimal operations are also challenging.9 In a recent survey, research- ers found that 68 percent of sixth graders and 51 percent of fifth and seventh graders asserted that the answer to the addition problem 4 + .3 was .7.10 This example also illustrates that students often treat decimal numbers as whole numbers and, as in this case, do not recognize that the sum they propose as a solution to the problem is smaller than one of the addends. The introduction of rational numbers constitutes a major stumbling block in children’s mathematical development.11 It marks the time when many students face the new and disheartening realization that they no longer un- derstand what is going on in their mathematics classes.12 This failure is a cause for concern. Rational-number concepts underpin many topics in ad- vanced mathematics and carry significant academic consequences.13 Stu- dents cannot succeed in algebra if they do not understand rational numbers. But rational numbers also pervade our daily lives.14 We need to be able to understand them to follow recipes, calculate discounts and miles per gallon, exchange money, assess the most economical size of products, read maps, interpret scale drawings, prepare budgets, invest our savings, read financial statements, and examine campaign promises. Thus we need to be able to
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311 NEW APPROACHES TO TEACHIING THE RATIONAL-NUMBER SYSTEM Additive and Multiplicative Reasoning BOX 7-1 Lamon,15 whose work on proportional reasoning and rational number has made a great contribution to our understanding of students’ learning, elucidates the dis- tinction between relative and absolute reasoning. She asks the learner to con- sider the growth of two fictitious snakes: String Bean, who is 4 feet long when the story begins, and Slim, who is 5 feet long. She tells us that after 5 years, both snakes have grown. String Bean has grown from 4 to 7 feet, and Slim has grown from 5 to 8 feet (see the figure below). She asks us to compare the growth of these two snakes and to answer the question, “Who grew more?” Lamon suggests that there are two answers. First, if we consider absolute growth, both snakes grew 3 feet, so both grew the same amount. The second answer deals with relative growth; from this perspective, String Bean grew the most because he grew 3/4 of his length, while Slim grew only 3/5 of his length. If we compare the two fractions, 3/4 is greater than 3/5, and so we conclude that String Bean has grown proportionally more than Slim. Lamon asks us to note that while the first answer, about the absolute differ- ence, involves addition, the second answer, about the relative difference, is solved through multiplication. In this way she shows that absolute thinking is additive, while relative thinking is multiplicative.
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312 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM understand rational numbers not only for academic success, but also in our lives as family members, workers, and citizens. Do the principles of learning highlighted in this book help illuminate the widespread problems observed as students grapple with rational num- ber? Can they point to more effective approaches to teaching rational num- ber? We believe the answer to both these questions is “yes.” In the first section below we consider each of the three principles of How Students Learn, beginning with principle 2—the organization of a knowledge net- work that emphasizes core concepts, procedural knowledge, and their con- nections. We then turn to principle 1—engaging student preconceptions and building on existing understandings. Finally we consider metacognitive in- struction as emphasized in principle 3. The second section focuses on instruction in rational number. It begins with a description of frequently used instructional approaches and the ways in which they diverge from the above three principles. We then describe an experimental approach to teaching rational number that has proven to be successful in helping students in fourth, fifth, and sixth grades understand the interconnections of the number system and become adept at moving among and operating with the various representations of rational number. Through a description of lessons in which the students engaged and proto- cols taken from the research classrooms, we set out the salient features of the instructional approach that played a role in shaping a learning-centered classroom environment. We illustrate how in this environment, a focus on the interconnections among decimals, fractions, and percents fosters stu- dents’ ability to make informed decisions on how to operate effectively with rational numbers. We also provide emerging evidence of the effectiveness of the instructional approach. The intent is not to promote our particular cur- riculum, but rather to illustrate the ways in which it incorporates the prin- ciples of How People Learn, and the observed changes in student under- standing and competence with rational numbers that result. RATIONAL-NUMBER LEARNING AND THE PRINCIPLES OF HOW PEOPLE LEARN The Knowledge Network: New Concepts of Numbers and New Applications (Principle 2) What are the core ideas that define the domain of rational numbers? What are the new understandings that students will have to construct? How does a beginning student come to understand rational numbers? Let us look through the eyes of a young student who is just beginning to learn about rational number. Until this point, all of her formal instruction in
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313 NEW APPROACHES TO TEACHIING THE RATIONAL-NUMBER SYSTEM arithmetic has centered on learning the whole-number system. If her learn- ing has gone well, she can solve arithmetic problems competently and easily makes connections between the mathematics she is learning and experi- ences of her daily life. But in this next phase of her learning, the introduc- tion of rational number, there will be many new and intertwined concepts, new facts, new symbols that she will have to learn and understand—a new knowledge network, if you will. Because much of this new learning is based on multiplicative instead of whole-number relations, acquiring an under- standing of this new knowledge network may be challenging, despite her success thus far in mathematics. As with whole-number arithmetic, this do- main connects to everyday life. But unlike whole numbers, in which the operations for the most part appear straightforward, the operations involved in the learning of rational numbers may appear to be less intuitive, at odds with earlier understandings (e.g., that multiplication always makes things bigger), and hence more difficult to learn. New Symbols, New Meanings, New Representations One of the first challenges facing our young student is that a particular rational number can take many forms. Until now her experience with sym- bols and their referents has been much simpler. A number—for example, four—is represented exclusively by one numeral, 4. Now the student will need to learn that a rational number can be expressed in different ways—as a decimal, fraction, and percent. To further complicate matters, she will have to learn that a rational-number quantity can be represented by an infinite number of equivalent common and decimal fractions. Thus a rational num- ber such as one-fourth can be written as 1/4, 2/8, 3/12, 4/16, 0.25, 0.250, and so on. Not only does the learning of rational number entail the mastery of these forms and of the new symbol systems that are implied, but the learner is also required to move among these various forms flexibly and efficiently.16 Unfortunately, this flow between representations does not come easily.17 In fact, even mature students are often challenged when they try to understand the relations among the representations.18 To illustrate how difficult translat- ing between fractions and decimals can be, I offer two examples taken from our research. In a recent series of studies, we interviewed fourth, sixth, and eighth graders on a number of items that probed for rational-number understand- ing. One of the questions we asked was how the students would express the quantity 1/8 as a decimal. This question proved to be very challenging for many, and although the students’ ability increased with age and experience, more than half of the sixth and eighth graders we surveyed asserted that as a decimal, 1/8 would be 0.8 (rather than the correct answer, 0.125).
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314 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM In the next example, an excerpt taken from an interview conducted as part of a pretest, Wyatt, a traditionally trained fifth-grade student, discussed ordering a series of rational numbers presented to him in mixed representa- tions. Interviewer Here are 3 numbers: 2/3, 0.5, and 3/4. Could you please put these numbers in order from smallest to largest? Wyatt Well, to start with, I think that the decimal 0.5 is bigger than the fractions because it’s a decimal, so it’s just bigger, because fractions are really small things. The response that 1/8 would equal 0.8 should be familiar to many who have taught decimals and fractions. As research points out, students have a difficult time understanding the quantities involved in rational number and thus do not appear to realize the unreasonableness of their assertion.19 As for Wyatt’s assertion in the excerpt above that decimals and fractions cannot be compared, this answer is representative of the reasoning of the majority of the students in this class before instruction. Moreover, it reflects more general research findings.20 Since most traditional instruction in rational num- ber presents decimals, fractions, and percents separately and often as dis- tinct topics, it is not surprising that students find this task confusing. Indeed, the notion that a single quantity can have many representations is a major departure from students’ previous experience with whole numbers; it is a difficult set of understandings for them to acquire and problem-laden for many.21 But this is not the only divergence from the familiar one-to-one corre- spondence of symbol to referent that our new learner will encounter. An- other new and difficult idea that challenges the relatively simple referent-to- symbol relation is that in the domain of rational number, a single rational number can have several conceptually distinct meanings, referred to as “subconstructs.” Now our young student may well become completely confused. The Subconstructs or the Many Personalities of Rational Number What is meant by conceptually distinct meanings? As an illustration, consider the simple fraction 3/4. One meaning of this fraction is as a part– whole relation in which 3/4 describes 3 of 4 equal-size shares. A second interpretation of the fraction 3/4 is one that is referred to as the quotient interpretation. Here the fraction implies division, as in 4 children sharing 3
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315 NEW APPROACHES TO TEACHIING THE RATIONAL-NUMBER SYSTEM pies. As a ratio, 3/4 might mean there are, for example, 3 red cars for every 4 green cars (this is not to be confused with the part–whole interpretation that 3/7 of the cars are red). Rational numbers can also indicate a measure. Here rational number is a fixed quantity, most frequently accompanied by a number line, that identifies a situation in which the fraction 1/4 is used repeatedly to determine a distance (e.g., 3/4 of an inch = 1/4, 1/4, 1/4). Finally, there is the interpretation of rational number as a multiplicative operator, behaving as an operation that reduces or enlarges the size of an- other quantity (e.g., the page has been reduced to 3/4 its original size). The necessity of coordinating these different interpretations requires a deep understanding of the concepts and interrelationships among them. On the one hand, a student must think of rational numbers as a division of two whole numbers (quotient interpretation); on the other, she must also come to know these two numbers as an entity, a single quantity (measure), often to be used in another operation. These different interpretations, generally referred to as the “subconstructs” of rational number, have been analyzed extensively22 and are a very important part of the knowledge network that the learner will construct for rational number. Reconceptualizing the Unit and Operations While acquiring a knowledge network for rational-number understand- ing means that new forms of representation must be learned (e.g., decimals, fractions) and different interpretations coordinated, the learner will encoun- ter many other new ideas—ideas that also depart from whole numbers. She will have to come to understand that rational numbers are “dense”—mean- ing that between any two rationals we can find an infinity of other numbers. In the whole-number domain, number is discrete rather than continuous, and the main operation is counting. This is a very big change indeed.23 Another difficult new set of understandings concerns the fundamental change that students will encounter in the nature of the unit. In whole num- bers, the unit is always explicit (6 refers to 6 units). In rational numbers, on the other hand, the unit is often implied. But it is the unstated unit that gives meaning to the represented quantities, operations, and the solutions. Con- sider the student trying to interpret what is meant by the task of multiplying, for example, 1/2 times 1/8. If the student recognizes that the “1/8” in the problem refers to 1/8 of one whole, she may reason correctly that half of the quantity 1/8 is 1/16. However since the 1 is not stated but implied, our young student may err and, thinking the unit is 8, consider the answer to be 1/4 (since 4 is one-half of 8)—a response given by 75 percent of traditionally instructed fourth and sixth graders students in our research projects.
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316 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM New Conceptualizations: Understanding Numbers As Multiplicative Relations Clearly the transition to learning rational numbers is challenging. Funda- mentally, students must construct new meanings for numbers and opera- tions. Development of the network of understandings for rational numbers requires a core conceptual shift: numbers must be understood in multiplica- tive relationship. As a final illustration, I offer one more example of this basic shift. Again, consider the quantity 3/4 from our new learner’s perspective. All of our student’s prior learning will lead her to conclude that the 3 and 4 in 3/4 are two separate numbers that define separate quantities. Her knowledge of whole numbers will provide an additive understanding. Thus she will know that 3 and 4 are contiguous on the number line and have a difference of 1. But to interpret 3/4 as a rational number instead of considering these two numbers to be independent, as many students mistakenly continue to do,24 our student must come to understand this fraction as a new kind of quantity that is defined multiplicatively by the relative amount conveyed by the sym- bols. Suddenly numbers are no longer simple. When placed in the context of a fraction, 3 and 4 become a quantity between 0 and 1. Obvious to adults, this numerical metamorphosis can be confusing to children. How can children learn to make the transition to the complex world of rational numbers in which the numbers 3 and 4 exist in a relationship and are less than 1? Clearly, instruction will need to support a major conceptual change. Looking at students’ prior conceptions and relevant understandings can provide footholds to support that conceptual change.25 Students’ Errors and Misconceptions Based on Previous Learning (Principle 1) As the above examples suggest, students come to the classroom with conceptions of numbers grounded in their whole-number learning that lead them astray in the world of rational numbers. If instruction is to change those conceptions, it is important to understand thoroughly how students reason as they puzzle through rational-number problems. Below I present verbatim interviews that are representative of faulty understandings held by many students. In the following excerpt, we return to our fifth grader, Wyatt. His task was to order a series of rational numbers in mixed representations. Recall his earlier comments that these representations could not be compared. Now as the interview continues, he is trying to compare the fractions 2/3 and 3/4. The interview proceeds:
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317 NEW APPROACHES TO TEACHIING THE RATIONAL-NUMBER SYSTEM Interviewer What about 2/3 and 3/4? Which of those is bigger? Wyatt Well, I guess that they are both the same size because they both have one piece missing. Interviewer I am not sure I understand what you mean when you say that there is one piece missing. Wyatt I’ll show you. [Wyatt draws two uneven circles, roughly partitions the first in four parts, and then proceeds to shade three parts. Next he divides the second circle into three parts and shades two of them (see Figure 7-1). O.K., here is 3/4 and 2/3. You see they both have one part missing. [He points to the unshaded sections in both circular regions.] You see one part is left out, so they are both the same. FIGURE 7-1 Wyatt’s response is typical in asserting that 2/3 and 3/4 must be the same size. Clearly he has not grasped the multiplicative relations involved in rational numbers, but makes his comparisons based on operations from his whole-number knowledge. When he asserts that 2/3 and 3/4 are the same size because there is “one piece missing,” Wyatt is considering the differ- ence of 1 in additive terms rather than considering the multiplicative rela- tions that underlie these numbers. Additive reasoning is also at the basis of students’ incorrect answers on many other kinds of rational-number tasks. Mark, a sixth grader, is working on a scaling problem in which he is attempting to figure out how the length and width of an enlarged rectangle are related to the measurements of a smaller, original rectangle. His challenge is to come up with a proportional relation and, in effect, solve a “missing-term problem” with the following relations: 8 is to 6 as 12 is to what number? Interviewer I have two pictures of rectangles here (see Figure 7-2). They are exactly the same shape, but one of them is bigger than the other. I
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318 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM made this second one bigger by taking a picture of the first one and then enlarging it just a bit. As you can see, the length of the first rectangle is 8 cm and the width is 6 cm. Unfortunately, we know only the length of the second one. That is 12 cm. Can you please tell me what you think the width is? Mark Well, if the first one (rectangle) is 8 cm and 6 cm, then the next one is 12 cm and 10 cm. Because in the 8 and 6 one (rectangle) you subtract 2 from the 8 (to get the difference of the width and the length). So in the bigger rectangle you have to subtract 2 from the 12. So that’s 10. So the width of the big rectangle is 10. 6 cm ? cm 8 cm 12 cm FIGURE 7-2 Mark’s error in choosing 10 instead of the correct answer of 9 is cer- tainly representative of students in his age group—in fact, many adults use the same kind of faulty reasoning.26 Mark clearly attempts to assess the relations, but he uses an additive strategy to come up with a difference of 2. To answer this problem correctly, Mark must consider the multiplicative relations involved (the rectangle was enlarged so that the proportional rela- tionship between the dimensions remains constant)—a challenge that eludes many. It is this multiplicative perspective that is difficult for students to adopt in working with rational numbers. The misconception that Mark, the sixth grader, displays in asserting that the height of the newly sized rectangle is 10 cm instead of the correct answer of 9 cm shows this failure clearly. Wyatt
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319 NEW APPROACHES TO TEACHIING THE RATIONAL-NUMBER SYSTEM certainly was not able to look at the relative amount in trying to distinguish between the quantities 2/3 and 3/4. Rather, he reasoned in absolute terms about the circles, that “. . . both have one piece missing.” Metacognition and Rational Number (Principle 3) A metacognitive approach to instruction helps students monitor their understanding and take control of their own learning.27 The complexity of rational number—the different meanings and representations, the challenges of comparing quantities across the very different representations, the un- stated unit—all mean that students must be actively engaged in sense mak- ing to solve problems competently.28 We know, however, that most middle school children do not create appropriate meanings for fractions, decimals, and percents; rather, they rely on memorized rules for symbol manipulation. The student errors cited at the beginning of this chapter indicate not only the students’ lack of understanding of rational number, but also their failure to monitor their operations and judge the reasonableness of their responses.29 If classroom teaching does not support students in developing metacognitive skills—for example, by encouraging them to explain their reasoning to their classmates and to compare interpretations, strategies, and solutions—the consequences can be serious. Student can stop expecting math to make sense. Indeed for many students, rational number marks the point at which they draw this conclusion. INSTRUCTION IN RATIONAL NUMBER Why does instruction so often fail to change students’ whole-number conceptions? Analyses of commonly used textbooks suggest that the prin- ciples of How People Learn are routinely violated. First, it has been noted that—in contrast to units on whole-number learning—topics in rational num- ber are typically covered quickly and superficially. Yet the major conceptual shift required will take time for students to master thoroughly. Within the allotted time, too little is devoted to teaching the conceptual meaning of rational number, while procedures for manipulating rational numbers re- ceive greater emphasis.30 While procedural competence is certainly impor- tant, it must be anchored by conceptual understanding. For a great many students, it is not. Other aspects of the knowledge network are shortchanged as well, in- cluding the presentation and teaching of the notation system for decimals, fractions, and percents. Textbooks typically treat the notation system as some- thing that is obvious and transparent and can simply be given by definition at a lesson’s outset. Further, operations tend to be taught in isolation and
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340 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM Then I did 10 percent of 160, which is 16. Then I did 5 percent, which was 8. I added them [16 + 8] to get 24, and added that to 80 to get 104. For anyone who has seen a colleague pause when asked to compute a percentage, as one must, say, to calculate a tip, the ease with which these students worked through these problems is striking. Knowledge Network These are only a few examples from the posttest interviews that illus- trate the kinds of new understandings and interconnections students had been able to develop through their participation in the curriculum. Overall, our analyses of the children’s thinking revealed that students had gained (1) an overall understanding of the number system, as illustrated by their ability to use the representations of decimals, fractions, and percents interchange- ably; (2) an appreciation of the magnitude of rational numbers, as seen in their ability to compare and order numbers within this system; (3) an under- standing of the proportional- and ratio-based constructs of this domain, which underpins their facility with equivalencies; (4) an understanding of percent as an operator, as is evident in their ability to invent a variety of solution strategies for calculating with these numbers; and (5) general confidence and fluency in their ability to think about the domain using the benchmark values they had learned, which is a hallmark of number sense. Our research is still in an early stage. We will continue to pursue many questions, including the potential limitations of successive halving as a way of operating with rational numbers, downplaying of the important under- standings associated with the quotient subconstruct, as well as a limited view of fractions. Furthermore, we need to learn more about how students who have been introduced to rational numbers in this way will proceed with their ongoing learning of mathematics. While we acknowledge that these questions have not yet been answered, we believe certain elements of our program contributed to the students’ learning, elements that may have implications for other rational-number curricula. First, our program began with percents, thus permitting children to take advantage of their qualitative understanding of proportions and com- bine that understanding with their knowledge of the numbers from 1 to 100, while avoiding (or at least postponing) the problems presented by fractions. Second, we used linear measurement as a way of promoting the multiplica- tive ideas of relative quantities and fullness. Finally, our program empha- sized benchmark values—of halves, quarters, eighths, etc.—for moving among equivalencies of percents, decimals, and fractions, which allowed students to be flexible and develop confidence in relying on their own procedures for problem solving.
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341 NEW APPROACHES TO TEACHIING THE RATIONAL-NUMBER SYSTEM CONCLUSION: HOW STUDENTS LEARN RATIONAL NUMBER Principle #1: Prior Understandings For years mathematics researchers have focused their attention on un- derstanding the complexities of this number system and how to facilitate students’ learning of the system. One well-established insight is that rational- number teaching focused on pie charts and part–whole understandings rein- forces the primary problem students confront in learning rational number: the dominance of whole-number reasoning. One response is to place the multiplicative ideas of relative quantity, ratio, and proportion at the center of instruction. However, our curriculum also builds on our theory and research find- ings pointing to the knowledge students typically bring to the study of ratio- nal number that can serve as a foundation for conceptual change. Two separate kinds of understandings that 10-year-olds typically possess have a multiplicative orientation. One of these is visual proportional estimation; for children, this understanding usually functions independently of numbers, at least initially. The second important kind of understanding is the numerical procedure for repeated halving. By strengthening and merging these two understandings, students can build a solid foundation for working flexibly with rational numbers. Our initial instructional activities are designed to elicit these informal understandings and to provide instructional contexts that bring them to- gether. We believe this coordination produces a new interlinked structure that serves both as foundation for the initial learning of rational number and subsequently as the basis on which to build a networked understanding of this domain. Principle #2: Network of Concepts At the beginning of this chapter, I outlined the complex set of core concepts, representations, and operations students need to acquire to gain an initial grounding in the rational-number system. As indicated above, the central conceptual challenge for students is to master proportion, a concept grounded in multiplicative reasoning. Our instructional strategy was to de- sign a learning sequence that allowed students to first work with percents and proportion in linear measurement and next work with decimals and fractions. Extensive practice is incorporated to assure that students become fluent in translating between different forms of rational number. Our inten- tion was to create a percent measurement structure that would become a central network to which all subsequent mathematical learning could be
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342 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM linked. This design is significantly different from traditional instruction in rational number, in which topics are taught separately. Principle #3: Metacognition In this chapter, I have not made detailed reference to students’ develop- ing metacognition. Yet the fostering of metacognition is in fact central to our curriculum. First, as the reader may have noted, we regularly engaged the students in whole-group discussions in which they were asked to explain their reasoning and share invented procedures with their classmates. We also designed the lessons so that students worked in small groups to col- laborate in solving problems and constructing materials; we thereby pro- vided students with a forum to express and refine their developing under- standings. There were also many opportunities for students to consider how they would teach rational number to others, either younger students or their own classmates, by designing their own games and producing teaching plans for how these new concepts could be taught. In all these ways, we allowed students to reflect on their own learning and to consider what it meant for them and others to develop an understanding of rational number. Finally, we fostered metacognition in our program through the overall design and goals of the experimental curriculum, with its focus on interconnections and multiple representations. This focus, I believe, provided students with an overview of the number system as a whole and thus allowed them to make informed decisions on how best to operate with rational numbers. Final Words I conclude this chapter with an interchange, recorded verbatim, be- tween a fourth-grade student and a researcher. Zach, the fourth grader, was being interviewed by the researcher as part of a posttest assessment. The conversation began when Zach had completed two pages of the six-page posttest and remarked to the interviewer, “I have just done 1/3 of the test;...that is 33.3 percent.” When he finished the third page, he noted, “Now I have finished 1/2 or 50 percent of the test.” On completing the fourth page he remarked, “Okay, so I have now done 2/3 of the test, which is the same as 66 percent.” When he had completed the penultimate page, he wondered out loud what the equivalent percentage was for 5/6: “Okay, let’s see; it has got to be over 66.6 percent and it is also more than 75 percent. I’d say that it is about 80 percent....No, wait; it can’t be 80 percent because that is 4/5 and this [5/6] is more than 4/5. It is 1/2 plus 1/3…so it is 50 percent plus 33.3 percent, 83.3 percent. So I am 83.3 percent finished.” This exchange illustrates the kind of metacognitive capability that our curriculum is intended to develop. First, Zach posed his own questions,
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343 NEW APPROACHES TO TEACHIING THE RATIONAL-NUMBER SYSTEM unprompted. Further, he did not expect that the question had to be an- swered by the teacher. Rather, he was confident that he had the tools, ideas, and concepts that would help him navigate his way to the answer. We also see that Zach rigorously assessed the reasonableness of his answers and that he used his knowledge of translating among the various representations to help him solve the problem. I conclude with this charming vignette as an illustration of the potential support our curriculum appears to offer to stu- dents beginning their learning of rational number. Students then go on to learn algorithms that allow them to calculate a number like 83.3 percent from 5/6 efficiently. But the foundation in math- ematical reasoning that students like Zach possess allow them to use those algorithms with understanding to solve problems when an algorithm has been forgotten and to double check their answers using multiple methods. The confidence created when a student’s mathematical reasoning is secure bodes well for future mathematics learning. NOTES 1. Armstrong and Bezuk, 1995; Ball, 1990; Post et al., 1991. 2. Moss, 2000. 3. Carpenter et al., 1980. 4. Ball, 1993; Hiebert and Behr, 1988; Kieren, 1993. 5. Lamon, 1999. 6. Hiebert and Wearne, 1986; Wearne and Hiebert, 1988. 7. Hiebert and Behr, 1988. 8. Kerslake, 1986. 9. Heibert, 1992. 10. National Research Council, 2001. 11. Carpenter et al., 1993. 12. Lamon, 1999. 13. Lesh et al., 1988. 14. Baroody, 1999. 15. Lamon, 1999. 16. National Council of Teachers of Mathematics, 1989, 2000. 17. Markovits and Sowder, 1991, 1994; Sowder, 1995. 18. Cramer et al., 1989. 19. Sowder, 1995. 20. Sowder, 1992. 21. Hiebert and Behr, 1988. 22. Behr et al., 1983, 1984, 1992, 1993; Kieren, 1994, 1995; Ohlsson, 1988. 23. Hiebert and Behr, 1988. 24. Kerslake, 1986. 25. Behr et al., 1984; Case, 1998; Hiebert and Behr, 1988; Lamon, 1995; Mack, 1990, 1993, 1995; Resnick and Singer, 1993.
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344 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM 26. Hart, 1988; Karplus and Peterson, 1970; Karplus et al., 1981, 1983; Cramer et al., 1993; Noelting, 1980a, 1980b. 27. National Council of Teachers of Mathematics, 1989, 2000; National Research Council, 2001. 28. Ball, 1993. 29. Sowder, 1988. 30. Baroody, 1999; Heibert, 1992; Hiebert and Wearne, 1986; Moss and Case, 1999; Post et al., 1993. 31. Armstrong and Bezuk, 1995; Ball, 1993; Hiebert and Wearne, 1986; Mack, 1990, 1993; Markovits and Sowder, 1991, 1994; Sowder, 1995. 32. Confrey, 1994, 1995; Kieren, 1994, 1995; Post et al., 1993; Streefland, 1991, 1993. 33. Kieren, 1994, 1995; Mack, 1993, 1995; Sowder, 1995; Streefland, 1993. 34. Kieren, 1994, p. 389. 35. Kieren, 1992, 1995. 36. Confrey, 1995. 37. Lachance and Confrey, 1995. 38. Streefland, 1991, 1993. 39. Mack, 1990, 1993. 40. Lamon, 1993, 1994, 1999. 41. As of this writing, this curriculum is being implemented with students of low socioeconomic status in a grade 7 and 8 class. Preliminary analyses have shown that it is highly effective in helping struggling students relearn this number system and gain a stronger conceptual understanding. 42. Kalchman et al., 2000; Moss, 1997, 2000, 2001, 2003; Moss and Case, 1999. 43. National Research Council, 2001. 44. Parker and Leinhardt, 1995. 45. Case, 1985; Noelting, 1980a; Nunes and Bryant, 1996; Spinillo and Bryant, 1991. 46. Confrey, 1994; Kieren, 1994. 47. Resnick and Singer, 1993. 48. Case, 1985. 49. Confrey, 1994; Kieren, 1993. 50. Case and Okomoto, 1996. 51. Case, 1998; Kalchman et al., 2000. 52. Parker and Leinhardt, 1995. 53. Lembke and Reys, 1994. 54. While the activities and lessons we designed are organized in three phases, the actual order of the lessons and the pacing of the teaching, as well as the particular content of the activities described below, varied in different class- rooms depending on the needs, capabilities, and interests of the participating students. 55. These materials are available at any building supply store. 56. Parker and Leinhardt, 1995. 57. Hiebert et al., 1991. 58. Resnick et al., 1989.
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345 NEW APPROACHES TO TEACHIING THE RATIONAL-NUMBER SYSTEM 59. Kalchman et al., 2000; Moss, 1997, 2000, 2001; Moss and Case, 1999. 60. From pre- to posttest, achieving effect sizes between and 1 and 2 standard deviations. REFERENCES Armstrong, B.E., and Bezuk, N. (1995). Multiplication and division of fractions: The search for meaning. In J.T. Sowder and B.P. Schappelle (Eds.), Providing a foun- dation for teaching mathematics in the middle grades (pp. 85-120). Albany, NY: State University of New York Press. Ball, D.L. (1990). The mathematics understanding that prospective teachers bring to teacher education elementary school. Journal, 90, 449-466. Ball, D.L. (1993). Halves, pieces and twoths: Constructing and using representational contexts in teaching fractions. In T.P. Carpenter, E. Fennema, and T.A. Romberg (Eds.), Rational numbers: An integration of research (pp. 157-196). Mahwah, NJ: Lawrence Erlbaum Associates. Baroody, A.J. (1999). Fostering children’s mathematical power: An investigative ap- proach to K-8 mathematics instruction. Mahwah, NJ: Lawrence Erlbaum Associ- ates. Behr, M.J., Lesh, R., Post, T.R., and Silver, E.A. (1983). Rational-number concepts. In R. Lesh and M. Landau (Eds.), Acquisition of mathematics concepts and pro- cesses (pp. 91-126). New York: Academic Press. Behr, M.J., Wachsmuth, I., Post, T.R., and Lesh, T. (1984). Order and equivalence of rational numbers: A clinical teaching experiment. Journal for Research in Math- ematics Education, 15(4), 323-341. Behr, M.J., Harel, G., Post, T.R, and Lesh, R. (1992). Rational number, ratio, and proportion. In D.A. Grouws (Ed.), Handbook of research on mathematics teach- ing and learning (pp. 296-333). New York: Macmillan. Behr, M.J., Harel, G., Post, T.R, and Lesh, R. (1993). Rational numbers: Towards a semantic analysis: Emphasis on the operator construct. In T.P. Carpenter, E. Fennema, and T.A. Romberg (Eds.), Rational numbers: An integration of re- search (pp. 13-48). Mahwah, NJ: Lawrence Erlbaum Associates. Carpenter, T.P., Kepner, H., Corbitt, M.K., Lindquist, M.M., and Reys, R.E. (1980). Results of the NAEP mathematics assessment: Elementary school. Arithmetic Teacher, 27, 10-12, 44-47. Carpenter, T.P., Fennema, E., and Romberg, T.A. (1993). Toward a unified discipline of scientific inquiry. In T. Carpenter, E. Fennema, and T.A. Romberg (Eds.), Rational numbers: An integration of research (pp. 1-12). Mahwah, NJ: Lawrence Erlbaum Associates. Case, R. (1985). Intellectual development: Birth to adulthood. New York: Academic Press. Case, R. (1998, April). A psychological model of number sense and its development. Paper presented at the annual meeting of the American Educational Research Association, San Diego, CA.
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347 NEW APPROACHES TO TEACHIING THE RATIONAL-NUMBER SYSTEM Kieren, T.E. (1992). Rational and fractional numbers as mathematical and personal knowledge. In G. Leinhardt, R. Putnam, and R.A. Hattrup (Eds.), Analysis of arithmetic for mathematics teaching (pp. 323-372). Mahwah, NJ: Lawrence Erlbaum Associates. Kieren, T.E. (1993). Rational and fractional numbers: From quotient fields to recur- sive understanding. In T. Carpenter, E. Fennema, and T.A. Romberg (Eds.), Ra- tional numbers: An integration of research (pp. 49-84). Mahwah, NJ: Lawrence Erlbaum Associates. Kieren, T.E. (1994). Multiple views of multiplicative structure. In G. Harel and J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 387-397). Albany, NY: State University of New York Press. Kieren, T.E. (1995). Creating spaces for learning fractions. In J.T. Sowder and B.P. Schappelle (Eds.), Providing a foundation for teaching mathematics in the middle grades. Albany, NY: State University of New York Press. Lachance, A., and Confrey, J. (1995). Introducing fifth graders to decimal notation through ratio and proportion. In D.T. Owens, M.K. Reed, and G.M. Millsaps (Eds.), Proceedings of the seventeenth annual meeting of the North American chapter of the International Group for the Psychology of Mathematics Education (vol. 1, pp. 395-400). Columbus, OH: ERIC Clearinghouse for Science, Math- ematics and Environmental Education. Lamon, S.J. (1993). Ratio and proportion: Children’s cognitive and metacognitive processes. In T.P. Carpenter, E. Fennema, and T.A. Romberg, (Eds.), Rational numbers: An integration of research (pp. 131-156). Mahwah, NJ: Lawrence Erlbaum Associates. Lamon S.J. (1994). Ratio and proportion: Cognitive foundations in unitizing and norming. In G. Harel and J. Confrey (Eds.), The development of multiplicative reasoning in the learning of mathematics (pp. 89-122). Albany, NY: State Uni- versity of New York Press. Lamon, S.J. (1995). Ratio and proportion: Elementary didactical phenomenology. In J.T. Sowder and B.P. Schappelle (Eds.), Providing a foundation for teaching mathematics in the middle grades. Albany, NY: State University of New York Press. Lamon, S.J. (1999). Teaching fractions and ratios for understanding: Essential con- tent knowledge and instructional strategies for teachers. Mahwah, NJ: Lawrence Erlbaum Associates. Lembke, L.O., and Reys, B.J. (1994). The development of, and interaction between, intuitive and school-taught ideas about percent. Journal for Research in Math- ematics Education, 25(3), 237-259. Lesh, R., Post, T., and Behr, M. (1988). Proportional reasoning. In M. Behr and J. Hiebert (Eds.), Number concepts and operations in the middle grades (pp. 93- 118). Mahwah, NJ: Lawrence Erlbaum Associates. Mack, N.K. (1990). Learning fractions with understanding: Building on informal knowl- edge. Journal for Research in Mathematics Education, 21, 16-32. Mack, N.K. (1993). Learning rational numbers with understanding: The case of infor- mal knowledge. In T.P. Carpenter, E. Fennema, and T.A. Romberg (Eds.), Ratio- nal numbers: An integration of research (pp. 85-105). Mahwah, NJ: Lawrence Erlbaum Associates.
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348 HOW STUDENTS LEARN: MATHEMATICS IN THE CLASSROOM Mack, N.K. (1995). Confounding whole-number and fraction concepts when build- ing on informal knowledge. Journal for Research in Mathematics Education, 26(5), 422-441. Markovits, Z., and Sowder, J.T. (1991). Students’ understanding of the relationship between fractions and decimals. Focus on Learning Problems in Mathematics, 13(1), 3-11. Markovits, Z., and Sowder, J.T. (1994). Developing number sense: An intervention study in grade 7. Journal for Research in Mathematics Education, 25(1), 4-29, 113. Moss, J. (1997). Developing children’s rational number sense: A new approach and an experimental program. Unpublished master’s thesis, University of Toronto, Toronto, Ontario, Canada. Moss, J. (2000). Deepening children’s understanding of rational numbers. Disserta- tion Abstracts. Moss, J. (2001). Percents and proportion at the center: Altering the teaching se- quence for rational number. In B. Littweiller (Ed.), Making sense of fractions, ratios, and proportions. The NCTM 2002 Yearbook (pp. 109-120). Reston, VA: National Council of Teachers of Mathematics. Moss, J. (2003). On the way to computational fluency: Beginning with percents as a way of developing understanding of the operations in rational numbers. In Teach- ing children mathematics (pp. 334-339). Reston, VA: National Council of Teach- ers of Mathematics. Moss, J., and Case, R. (1999). Developing children’s understanding of rational num- bers: A new model and experimental curriculum. Journal for Research in Math- ematics Education, 30(2), 119, 122-147. National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics. National Council of Teachers of Mathematics. (2000). Curriculum and evaluation standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics. National Research Council. (2001). Adding it up: Helping children learn mathemat- ics. Mathematics Learning Study Committee, J. Kilpatrick, J., Swafford, J., and B. Findell (Eds.). Center for Education. Division of Behavioral and Social Sciences and Education. Washington, DC: National Academy Press. Noelting, G. (1980a). The development of proportional reasoning and the ratio con- cept. Part I: Differentiation of stages. Educational Studies in Mathematics, 11, 217-253. Noelting, G. (1980b). The development of proportional reasoning and the ratio con- cept. Part II: Problem-structure at successive stages: Problem-solving strategies and the mechanism of adaptive restructuring. Educational Studies in Mathemat- ics, 11, 331-363. Nunes, T., and Bryant, P. (1996). Children doing mathematics. Cambridge, MA: Blackwell. Ohlsson, S. (1988). Mathematical meaning and applicational meaning in the seman- tics of fractions and related concepts. In J. Hiebert and M. Behr (Eds.), Number concepts and operations in the middle grades (vol. 2, pp. 53-92). Mahwah, NJ: Lawrence Erlbaum Associates.
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Representative terms from entire chapter: