How Students Learn Mathematics in the Classroom (2005) / Chapter Skim
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7 Pipes, Tubes, and Beakers: New Approaches to Teaching the Rational-Number System
7 Pipes, Tubes, and Beakers: New Approaches to Teaching the Rational-Number System
Pages 121-162
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From page 121... ...
_4 W: :0~51 'at e 1977 United Features Syndicare inc. PEANUTS repan ad by permission of Uni ad Feature Syndicate, Inc Poor Sally.
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The introduction of rational numbers constitutes a major stumbling block in children s mathematical development." It marks the time when many students face the new and disheartening realization that they no longer understand what is going on in their mathematics classes.' This failure is a cause for concern Rational-number concepts underpin many topics in advanced mathematics and car y significant academic consequences.'3 8tudents cannot succeed in algebra if they do not understand rational numbers. But rational numbers also pervade our daily lives.'4 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.
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~ S %00 Stan 15 [eel} String Bean (7 feet) Slim (R feetl Lao on suggests that there are two answers.
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The second section focuses on instruction in rational number It begins with a description of frequendy used instructional approaches and The ways in which they diverge Tom The above dlree pnnciples. We then describe an experimental approach to teaching rational number That has proven to be successful in helping students in fourth, ffdh, and srxdh 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 protocols 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.
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But in This next phase of her learning, The introduction 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 tills new learning is based on multiplicative instead of whole-number relations, acquiring an understanding of dais new knowledge network may be challenging, despite her success Thus far in mathematics.
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From page 126... ...
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.:' But this is not the only divergence from the familiar one-to-one correspondence of symbol to referent that our new learner will encounter Another new and difficult idea that challenges the relatively simple referent-tosymbol 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?
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From page 127... ...
, often to be used in another operation. These different interpretations, generally referred to as The "subconstructs" of rational number, have been analyzed extensively and are a very important part of The knowledge network dhat the learner will construct for rational number, Reconceptualizing the Unit and Operations While acquiring a knowledge network for rational-number understanding means dhat new forms of representation must be learned (e.g., decimals, fractions)
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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.
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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.
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To answer this problem correctly, Mark must consider the multiplicative relations involved (the rectangle was enlarged so that the proportional relationship between the dimensions remains constant challenge that eludes many. It is this multiplicative perspective that is difficult for students to adopt in working with rational numbers.
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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 unstated unit—all mean that students must be actively engaged in sense making to solve problems competendy.r We know, however, that most middle school children do not create appropriate meanings for fractions, decimals, and percents; radher, they rely on memorized rules for symbol manipulation, The student errors cited at the beginning of tills 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 senous. Student can stop expecting madh to make sense.
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From page 132... ...
The central problem with most textbook instruction, many researchers agree ~ is the failure of textbooks to provide a grounding for the major conceptual shift to multiplicative reasoning that is essential to mastering rational number To support this claim, let us look at how rational number is typically introduced in traditional practice. Pie Charts and a Part-Whole Interpretation of Rational Numbers Most of us learned fractions with the model of a pie chart, and for many people, fractions remain inextricably linked to a picture of a partly shaded shape.
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Kieren33 has developed a program for teaching fractions that is based on the multiplicative operations of splitting. As part of his approach he used paper folding rather than pie charts as its primary problem situation.
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Pipes, Tubes, and Beakers: A New Approach to Rational-Nurnber Learning Percents as a Starting Point In our curriculum, rather than teaching fractions and decimals first, we introduce percents which we believe to be a "pnvileged" proportion in that it only involves fractions of the base 100.44 We do this through students' everyday understandings. We situate the initial learning of percent in linear measurement contexts, in which students are challenged to consider the relative lengths of different quantities.
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" These ideas of percents and proportion serve as an anchoring concept for the subsequent learning of decimals and fractions, and then for an overall understanding of the number system as a whole. Starting Point Visual Proportional Estimation and Halving and Doubling Our starting point in developing our curnculum was to consider students' informal knowledge and the intuitions they have developed that could serve as a foundation.
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The context we chose was to have students work with percents and linear measurement. As will be elaborated below, students were engaged from the start of the instructional sequence in estimating proportional relations based on length and in using their knowledge of halving to compute simple percent quantities.
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, but They also had a strong qualitative understanding of what different numerical values "mean." For example, students commented that 100 percent means "everything," 99 percent means "almost everything," 50 percent means "exactly half," and 1 percent means "almost nothing. As one student remarked, "You know if you are on a diet you should drink 1 percent milk instead of 4 percent milk." Pipes and Tubes A Repr-eserrtatiorr for- Fullness To further explore students' intuitions and informal understandings, we presented them with a set of props specie cally designed for The lessons.
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Again The students were full of ideas, many of which are central to The knowledge network for rational number First students demonstrated Their understanding of The unit whole, as mentioned earlier, a concept That is often elusive in traditional instruction: "Each of these pipes is 100 percent." They also demonstrated understanding of The part-whole construct: "If you raise The tube up here [pointing to dhreequarters of The length of The pipe] , Then The part That is covered is 75 percent, and The part That is left over is 25 percent." Students also naturally displayed Their sense of rational number as operator: "This is 50 percent of The tube, and if we cut it in had again it is 25 percent." In addition, students demonstrated insights for proportions: "50 percent on tills bigger pipe is bigger than 50 percent on tills little pipe, but They're bodh still 50 percent." The idea of rational number as a measure was also embedded in the students' reasoning!
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328 HOW STUDENTS LEARN: MATHEMAT CS N THE CEASSFOOM FIGURE 75a F~YT1'1 1 1~ I_ r ~ - .~ ~ ..,' ~ '::.~::~_ ~;.~r' ~—it| W1i 1
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, String Challenges Guessing Mystery Objects String measurement activities also proved to be an excellent way of considering percent quantities and calculating percentages using benchmarks. A string challenge that became a regular feature of classroom life was what we called "The Mystery Ob ect Challenge." In this activity, which often started the lessons, the teacher held up a piece of string that was cut to the percent of the length of a certain object in the room.
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As students often remarked, "Our string lengths are different even though all of our percents are the same." Sutntnary of Lessons Part I The first phase of the lessons began with estimations and then calculations of percent quantities. These initial activities were all presented in the context of linear measurement of our specially designed pipes and tubes, beakers of water, stung, and number lines.
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The activities consistency helped students integrate their sense of visual proportion widh Their ability to do repeated halving. Our goal in all of These initial activities was to create situations in which These two kinds of informal understandings could become linked and serve as a foundation for The students' further learning of dais number system.
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, we devised many activities to help the students work with ordering decimals. The first of these activities was the "Stop-Start Challenge." In this exercise, students attempted to start and stop the watch as quickly as possible, several times in succession.
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Students worked on many activities that helped them first understand how decimals and percents are related and then learn how to represent decimals symbolically. As the decimal lessons proceeded, we moved on to activities designed to help students to consider and reflect on magnitude.
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This allowed Them to increase Their understanding of The possibility of fluid movement between representations. Card Games In one set of lessons, I gave the students a set of specially designed cards depicting various representations of fractions, decimals, and percents (e.g., there was a 3/8 card, a card widh .375, and a card That read 37 1/2 percent)
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A second card game employing the same deck of cards, invented by a pair of students, had as its goal not only The comparison of decimals, fractions, and percents in mixed representations, but also the addition and subtraction of The differences between These numbers. This game again used The LCt)
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From page 149... ...
From the very first lessons, students demonstrated and used their everyday knowledge of percents and worked successfully with percents in situations that called on their understanding of proportion. Our particular format also allowed students to express their informal knowledge of other concepts and meanings that are central to rational number understanding Recall that when working with the pipes and tubes and the beakers of water, students successfully incorporated ideas of the rational-number subconstructs of measure, operator, and ratio.
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338 HOW STUDENTS LEARN: MATHEMAT CS N THE CEASSFOOM to students from classrooms in which textbook instruction had been pro vided.59 Briefly, we found that students in the experimental group had improved significantly 60 Further, the scores that they obtained after instruction were often higher than those of children who had received instruction in conventional classrooms and who were many years older Not only were students in the experimental classrooms able to answer more questions than did the "textbook" students, but the quality of their answers was better Specifically, the experimental group made more frequent reference to proportional concepts in justifying their answers than did the students in the nonexperimental group. What follows are some examples of changes in students' reasoning following participation in the experimental program, consisting of selections from interviews that were conducted following the conclusion of the expenmental classes.
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From page 151... ...
Aldhough this type of computation was performed regularly in our classrooms, 65 percent of 160 was a significantly more difficult calculation Than those The students had typically encountered in their lessons. Furthermore, dais item required That students work with 10 percent as well as with the familiar benchmarks (25 percent, 50 percent, 75 percent, and 12 I/2 percent)
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From page 152... ...
Second, we used linear measurement as a way of promoting the multiplicative ideas of relative quantities and fullness. Finally, our program emphasized 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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Our initial instructional activities are designed to elicit These informal understandings and to provide instructional contexts that bring Them together We believe this coordination produces a new interlinked structure that serves bodh as foundation for The initial learning of rational number and subsequently as The basis on which to build a networked understanding of dais domain. Principle #2: Network of Concepts At He beginning of tills 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.
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There were also many opportunities for students to consider how they would teach rational number to odhers, eidher 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 redect on their own learning and to consider what it meant for Them and odhers to develop an understanding of rational number Finally, we fostered metacognition in our program Through the overall design and goals of The experimental curnculum, with its focus on interconnections and multiple representations.
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I conclude with this charming vignette as an illustration of The potential support our curriculum appears to offer to students beginning their learning of rational number Students Then go on to learn algoridlms Tat allow Them to calculate a number like 83.3 percent from 5/6 efficiently. But The foundation in madhematical reasoning That students like Bach possess allow Them to use Those algondhms with understanding to solve problems when an algoridlm has been forgotten and to double check their answers using multiple methods.
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41. As of this writing, this curriculum is being implemented with students of low socioeconomic status in a grade 7 and 8 c ass P¢lim nary analyses have shown that it is highly effective in helping struggling students relearn this number system and gain a st onger conceptual understanding.
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Schappelle (Eds.) , Providing afoundatSon for teach Sng mathematics in the middle grades (p p.
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. Mahwah, NJ: Lawrence Erlbaum Associates.
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In J.T Sowder and B.P Schappelle (Eds.) , Providing afoundationfor teach in g mathematics in the middle grades A bany, NY: State University of New York Press.
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. The development of proportional reasoning and the ratio concept Part 11 Problem-stnuctu¢ at successive stages Problem-solving strategies and the mechanism of adaptive rest ucturing.
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In J.T Sowder and B.P Schappelle (Eds.) , Providing afoundationfor teaching mathematics in the middle grades.
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