Order and Equivalence of Rational Numbers: A Clinical Teaching Experiment

Behr, Merlyn J.; Wachsmuth, Ipke; Post, Thomas R.; Lesh, Richard

doi:10.2307/748423

Cited by 184 publications

(116 citation statements)

References 9 publications

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“…Pitkethly & Hunting, 1996). Some researchers have analyzed different subconstructs of fractions, such as fraction as ratio and fraction as operator, that might constitute fractional and rational number knowledge (e.g., Behr, Harel, Post, & Lesh, 1993;Behr, Wachsmuth, Post, & Lesh, 1984;Davis, Hunting, & Pearn, 1993;Kieren, 1988Kieren, , 1995. Other researchers have emphasized the importance of real-world contexts and the informal knowledge that students bring to their activity with fractions in helping students develop fractional knowledge (e.g., Mack, 1990Mack, , 1995Mack, , 2001Streefland, 1991).…”

Section: Introductionmentioning

confidence: 99%

Units coordination and the construction of improper fractions: A revision of the splitting hypothesis

Hackenberg

2007

The Journal of Mathematical Behavior

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Section: Introductionmentioning

confidence: 99%

Units coordination and the construction of improper fractions: A revision of the splitting hypothesis

Hackenberg

2007

The Journal of Mathematical Behavior

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“…Research on ratio and proportion word problem solving was particularly prominent in the 1980s and early 1990s (e.g., Behr, Wachsmuth, Post, & Lesh, 1984;Carpenter, Fennema, & Romberg, 1993), with scholars in mathematics education continuing to explore ways to improve students' learning of this important, yet challenging topic. Although there is evidence to support the efficacy of interventions from the mathematics education community for enhancing the mathematics performance of high and average achieving students (Cohen & Hill, 2001;Fuson, Carroll, & Drueck, 2000;Schoenfeld, 2002), research on how to address low achieving students' difficulties with word problem solving is somewhat conflicting.…”

mentioning

confidence: 99%

Improving seventh grade students’ learning of ratio and proportion: The role of schema-based instruction

Jitendra

Star

Starosta

et al. 2009

Contemporary Educational Psychology

141

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The present study evaluated the effectiveness of an instructional intervention (schema-based instruction, SBI) that was designed to meet the diverse needs of middle school students by addressing the research literatures from both special education and mathematics education.Specifically, SBI emphasizes the role of the mathematical structure of problems and also provides students with a heuristic to aid and self-monitor problem solving. Further, SBI addresses well-articulated problem solving strategies and supports flexible use of the strategies based on the problem situation. One hundred forty eight seventh-grade students and their teachers participated in a 10-day intervention on learning to solve ratio and proportion word problems, with classrooms randomly assigned to SBI or a control condition. Results suggested that students in SBI treatment classes outperformed students in control classes on a problem solving measure, both at posttest and on a delayed posttest administered four months later.However, the two groups' performance was comparable on a state standardized mathematics achievement test. KEYWORDS: word problem solving, ratio and proportion, middle school students, schema- One of the major difficulties noted in this domain is that students tend to use additive rather than multiplicative solution methods (e.g., "to solve 6:14 = ?:35, they find the difference between 6 and 14 and subtract it from 35 to find 27:35 rather than seek multiplicative relationships," Fuson & Abrahamson, 2002, p. 213). Research in this domain has examined how children reason in various proportionality tasks and the extent to which developmental or SCHEMA-BASED INSTRUCTION 4 instructional factors influence proportional reasoning (e.g., Bright, Joyner, & Wallis, 2003;Lamon, 1993Lamon, , 2002Lo & Watanabe, 1997;Post et al., 1988). Proportional reasoning develops over time and "understanding at one level forms a foundation for higher levels of understanding" (Lamon, 2007, p. 637). Recommendations outlined in the research for developing children's proportional thought processes have included providing ratio and proportion tasks in a wide range of contexts (e.g., measurements, prices, rates) and ensuring that students have experienced conceptual methods before presenting symbolic methods such as cross-product algorithm for solving proportional problems (Lamon, 1999;Van de Walle, 2007).The topics of ratio and proportion are frequently encountered in elementary and middle schools in the form of word problems. Traditionally, word problems are used to teach "mathematical modeling and applied problem solving" (Van Dooren, De Bock, Hessels, Janssens, & Verschaffel, 2005, p. 58). That is, the word problem context provides a description of a problem-solving situation that requires responding to a question by executing "one or more operations (+, -, x, ÷) on the quantities in the problem" (Van Dooren et al., 2005, p. 58).However, word problem solving has proved to be a significant challenge for students, in part because it requires s...

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“…The emergence of this practice involved an important shift in the way in which most of the students construed the relation between the size of a unit of measure and the number of iterations it required to cover a certain length. At the beginning of the instructional intervention, most of the pupils seemed to rely on what Behr et al (1984) called the whole-number-dominance strategy. They regarded numbers that would come later in the counting sequence as always accounting for larger sizes.…”

Section: Actual Learning Trajectorymentioning

confidence: 99%

“…This idea is integral to understanding the reciprocal of a basic multiplicative relation, in which the scalar operator is a whole number, and is thereby central to proportional reasoning (Thompson and Saldanha 2003). At the same time, it has proven difficult for students to grasp (Behr et al 1984;Streefland 1991;Tzur 2007) and is one of those ideas where employing thoughtful means of support can make, we believe, much difference to students' learning.…”

Section: Introductionmentioning

confidence: 99%

Unit fractions in the context of proportionality: supporting students' reasoning about the inverse order relationship

2014

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We analyze a classroom design experiment, conducted in a fourth grade classroom, that served to explore an instructional path in which the introduction of unit fractions and supporting proportional reasoning coincide. Central to this path is the use of means of support in which the objects that unit fractions quantify are not characterized as equal-sized parts of a whole, but as entities that are always separate from a reference unit. We argue that such a path is crucial for helping students develop deep quantitative understandings of fractions, where fraction quantities are, from the very start, linked to the reciprocal and multiplicative relations that their use implies. We focus on the first part of the design experiment in which we helped the students make sense of a concept that is important for initial fraction learning and proportional reasoning, the inverse order relationship among unit fractions.

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Order and Equivalence of Rational Numbers: A Clinical Teaching Experiment

Cited by 184 publications

References 9 publications

Units coordination and the construction of improper fractions: A revision of the splitting hypothesis

Units coordination and the construction of improper fractions: A revision of the splitting hypothesis

Improving seventh grade students’ learning of ratio and proportion: The role of schema-based instruction

Unit fractions in the context of proportionality: supporting students' reasoning about the inverse order relationship

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