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7 Developing Proficiency with Other Numbers
Pages 231-254

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From page 231...
... Then we consider proportional reasoning, which builds on the ratio use of rational numbers. Finally, we examine the integers, a stepping stone to algebra.
From page 232...
... 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.
From page 233...
... 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.
From page 234...
... 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.
From page 235...
... 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. 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.
From page 236...
... 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.
From page 237...
... 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 unity 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, 3 of a pizza.
From page 238...
... 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. \/Vhat 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.
From page 239...
... 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.
From page 240...
... Researchers express concern that this kind of learning can be "highly dependent on memory and subject to deterioration."3~ 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.
From page 241...
... 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.
From page 242...
... 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.4~ 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.
From page 243...
... 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.
From page 244...
... 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.5~ 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.
From page 245...
... 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.
From page 246...
... 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.
From page 247...
... 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.
From page 248...
... 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 3 to get the number of red cars, while thinking of 3 as three times 4 (Interpretation A)
From page 249...
... 47. The term composite unit refers to thinking of 3 balloons (and hence $2)
From page 250...
... (1998~. Proportional reasoning among 7th grade students with different curricular experiences.
From page 251...
... (1989~. Proportional reasoning: The effect of two concept variables, rate type and problem setting.
From page 252...
... (1994~. Missing-value proportional reasoning problems: Factors affecting informal reasoning patterns.
From page 253...
... (2000~. Three balloons for two dollars: Developing proportional reasoning.
From page 254...
... . Fractions in realistic mathematics education: A paradigm of developmental research.


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