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## Giant Pencils: Developing Proportional Reasoning

Teachers can help students develop proportional reasoning and explore measurement with an open-ended activity.

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## Proportional Reasoning in Nursing Practice

We investigate how expert nurses undertake the calculation of drug dosages on the ward. This calculation is error-critical in nursing practice and maps onto the concepts of ratio and proportion. Using episodes of actual drug administration gleaned from ethnographic study, we provide evidence that experienced nurses use a range of correct proportional-reasoning strategies based on the invariant of drug concentration to calculate dosage on the ward instead of the single taught method they describe outside of the practice. These strategies are tied to individual drugs, specific quantities and volumes of drugs, the way drugs are packaged, and the organization of clinical work.

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## Using Artwork to Explore Proportional Reasoning

By examining ratios in paintings and using a free educational app, students can size up artists' use of proportional reasoning in their creations.

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## Take Time for Action: Proportional Reasoning

Developing facility with proportional reasoning should be “one of the hallmarks of the middle grades mathematics program” (NCTM 1998, 213). Such reasoning has long been a problem for students, however, because of the complexity of thinking that it requires. Several standards-based curriculumreform projects have explored new approaches to developing students' proportional reasoning concepts and skills. Instead of offering direct instruction on standard algorithms for checking equivalence of ratios or solving proportion equations, these new approaches encourage students to build understanding and strategies for proportional reasoning through guided collaborative work on authentic problems.

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## Connecting Research to Teaching: Proportional Reasoning

The attainment of proportional reasoning is considered a milestone in students' cognitive development. According to the NCTM's Curriculum and Evaluation Standards (1989), this ability is “of such great importance that it metits whatever time and effort must be expended to assure its careful development” (p. 82). As teachers and researchers know, students' understanding of proportionality develops slowly over a number of years. This article reports research findings regarding the learning and teaching of proportional reasoning that have potential for making contributions to classroom practice.

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## Using Proportional Reasoning to Solve Geometric Problems

The proportional nature of geometric work, and the ways in which proportional reasoning can be employed as a generalized problem–solving tool. A geometric construction is presented for students to solve. The author gives reasoning via proof and proportions and provides extension activities.

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## Contextual Factors Affecting Errors in Proportional Reasoning

Although instruction in proportions generally begins in the middle school years. proportional reasoning remains problematic for many college students (Adi & Pulos, 1980; Niaz, 1989; Renner & Paske, 1977; Thornton & Fuller, 1981). A review of the proportional reasoning literature by Tourniaire and Pulos (1985) suggests that a number of factors relating to the context of the problem are responsible for variability in performance. Such fac tors include the type of problem (e.g., rate, mixture, or physical problems), presence of discrete versus continuous quantities, and familiarity of problem content. It is apparent that most students' understanding of proportion concepts is relatively fragile and easily influenced by structural variations in the problem. Recent research in mathematics education has emphasized the importance of the interaction between intuitive knowledge (i.e., knowledge about a subject matter gai ned through natural expe rience) and the acq uisition of formal mathematics concepts (see Behr, Hare!, Post, & Lesh, 1992). It would be useful to determine exactly which factors in proportion problems elicit an intuitive understanding of the underlying concept; instruction in proportional reasoning could then use that knowledge as the basis for developing a more generalizable system of symbols and algorithms.

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## Improving Middle School Teachers' Reasoning about Proportional Reasoning

Article describes a professional development project to increase teachers' understanding of proportional reasoning, the thinking patterns associated with proportional reasoning, and the applications of proportional reasoning across the middle-grades curriculum.

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## Crossing the Bridge to Formal Proportional Reasoning

Mathematics educators agree that ratio and proportion are important middle school mathematics topics. In fact, researchers have stated that proportional reasoning involves “watershed concepts” that are at the “cornerstone of higher mathematics” (Lamon 1994; Lesh, Post, and Behr 1988). Yet assisting students in developing robust understanding of the many concepts and procedures that are related to using ratios, rates, and proportions is not straightforward. For example, being able to reason proportionally and being able to represent that reasoning symbolically do not always go hand-inhand. As with many complex topics, students' understanding grows with time and experience.

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## Exploring Proportional Reasoning through Movies and Literature

Have you ever watched a toddler climb onto a chair and compared the height of the chair with the height of the toddler? Have you wondered how big a chair would have to be for you, as an adult, to have the same experience of struggling to get into it? Have you looked at a Barbie doll's neck or high heels and wondered how long each would be if Barbie were an adult female of average height? When watching a movie, such as Honey, I Shrunk the Kids, have you ever wondered how large objects like silverware and blades of grass must have been to give the appearance that a child had been shrunk? Satisfying our curiosity about these questions requires proportional reasoning.