How Students Learn History, Mathematics, and Science 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 309-350
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From page 309... ...
As mathematics education researchers and teachers can attest, students are often vocal in their expression of dislike of fractions and other representations of rational numbers (percents and decimals)
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From page 310... ...
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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From page 311... ...
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.
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From page 312... ...
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.
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From page 313... ...
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 this new learning is based on multiplicative instead of whole-number relations, acquiring an understanding of this new knowledge network may be challenging, despite her success thus far in mathematics.
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From page 314... ...
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.
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From page 315... ...
, 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.
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From page 316... ...
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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From page 317... ...
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 difference of 1 in additive terms rather than considering the multiplicative relations that underlie these numbers.
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From page 318... ...
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.
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From page 319... ...
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 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.
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From page 320... ...
The central problem with most textbook instruction, many researchers agree,32 is the failure of textbooks to provide a grounding for the major conceptual shift to multipli cative 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.
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From page 321... ...
Kieren35 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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From page 322... ...
. Our instructional strategy is to help students to further develop these informal understandings and then integrate them into a developmentally sequenced set of activities designed to help them develop a network of concepts and relations for rational num bers (Principle 2)
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From page 323... ...
. BOX 7-2 Students' Informal Knowledge Proportional Understandings While we know that formal proportional reasoning is slow to develop47 it has none theless been shown that children from a very early age have a strong propensity for making proportional evaluations that are nonnumerical and based on perceptual cues.
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From page 324... ...
The context we chose was to have students work with percents and linear measurement. As will be elaborated below, students were en gaged from the start of the instructional sequence in estimating propor tional relations based on length and in using their knowledge of halving to compute simple percent quantities.
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From page 325... ...
, 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 Representation for Fullness To further explore students' intuitions and informal understandings, we presented them with a set of props specifically designed for the lessons. The set included a series of black drainage pipes (of varying heights)
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From page 326... ...
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 partwhole construct: "If you raise the tube up here [pointing to three quarters of the length of the pipe]
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From page 329... ...
. String Challenges: Guessing Mystery Objects String measurement activities also proved to be an excellent way of considering percent quantities and calculating percentages using benchmarks.
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From page 331... ...
Students were not given formal instruction in specific calculating procedures; rather, they naturally employed procedures of their own that involved percent benchmarks and repeated halving. While percent was the only form of rational number that we officially introduced at this point, students often referred to fractions when working on these initial activities.
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From page 332... ...
The activities consistently helped students integrate their sense of visual proportion with 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 foun dation for the students' further learning of this number system.
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From page 333... ...
Magnitude and Order in Decimal Numbers To illuminate the difficult concepts of magnitude and order (recall Wyatt's assertion that 2/3 = 3/4 and others' comments that 0.2 is smaller than 0.059) , we devised many activities to help the students work with ordering decimals.
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From page 334... ...
As the decimal les sons proceeded, we moved on to activities designed to help students to consider and reflect on magnitude. Thus the final activities included situa tions in which students engaged in comparing and ordering decimals.
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From page 335... ...
In a favorite game called "Crack the Code," students moved between representations of rational numbers as they were challenged to stop the watch at the decimal equivalent of different fractions and percents. For example, given a relatively simple secret code, e.g., 2/5, students stopped the watch at close to 40 centiseconds or 0.40 seconds as possible.
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From page 336... ...
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, frac tions, and percents in mixed representations, but also the addition and sub traction of the differences between these numbers. This game again used the LCD stopwatches introduced earlier in the lessons.
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From page 337... ...
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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From page 338... ...
Specifically, the experimental group made more frequent reference to proportional con cepts 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 experi mental classes.
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From page 339... ...
Furthermore, this item required that students work with 10 percent as well as with the familiar benchmarks (25 percent, 50 percent, 75 percent, and 12 1/2 percent) that served as a basis for most of their classroom work.
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From page 340... ...
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
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From page 341... ...
Our instructional strategy was to design 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.
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From page 342... ...
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.
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From page 343... ...
But the foundation in math 5 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.
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From page 344... ...
Preliminary analyses have shown that it is highly effective in helping struggling students relearn this number system and gain a stronger conceptual understanding.
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From page 345... ...
. Order and equivalence of rational numbers: A clinical teaching experiment.
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From page 346... ...
. Intellectual development beyond elementary school II: Ratio, a survey.
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From page 347... ...
. Teaching fractions and ratios for understanding: Essential con tent knowledge and instructional strategies for teachers.
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From page 348... ...
. Developing children's understanding of rational num bers: A new model and experimental curriculum.
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From page 349... ...
Schappelle (Eds.) , Providing a foundation for teaching mathematics in the middle grades.
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