RATIOS AND PROPORTIONALITY
The concept of proportionality is essential to understanding much in science, mathematics, and technology. Many familiar variables, such as speed, density, and map scales are ratios (or rates) themselves. Students' understanding of proportionality progresses along three strands of benchmarks that address the relationship between parts and wholes, experience with and understanding of numerical descriptions and comparisons, and basic computation skills relevant to ratios and proportions. The next edition of Atlas will include several topics—number sense, measurement, shape, and scale—that relate to ratios and proportions.
The 6-8 benchmark "The expression a/b can mean different things..." provides a center point for this map, drawing from and contributing to every strand of development. Though it may seem no more than a technical detail about mathematical notation, this benchmark reflects the basic equivalence between fractions, division, and ratios—that they all can be expressed as quotients.
When something is bigger than something else, we can characterize the relationship by how much bigger it is or how many times bigger. (The trick, of course, is to decide when the additive or multiplicative relationship is appropriate.) The fact that a/b implies a special kind of comparison of a to b is critical to this map.
Two benchmarks on this map do not appear in Benchmarks. The K-2 benchmark "An important kind of relationship..." clarifies the importance of thinking about parts and wholes. The 6-8 benchmark "Some interesting relationships..." draws attention to a constant ratio between variables as an alternative to a constant difference (on which students often get stuck), and it also appears on the Describing Change map.
The benchmarks in the computation strand identify skills that contribute to working with ratios, rates, and proportions. Of course, general computation skills would include many more skills than are given here. Benchmarks for other, equally important skills (and the general topic of mathematical operations) can be found in Benchmarks Chapter 12: HABITS OF MIND and will be included in the next edition of Atlas.
Research in Benchmarks
Upper elementary- and middle-school students may exhibit limited understanding of the meaning of fractional number (Kieren, 1992). For example, many 7th-graders do not recognize that 5 1/4 is the same as 5 + 1/4 (Kouba et al., 1988). In addition, elementary-school students may have difficulties perceiving a fraction as a single quantity (Sowder, 1988), but rather see it as a pair of whole numbers. An intuitive basis for developing the concept of fractional number is provided by partitioning (Kieren, 1992) and by seeing fractions as multiples of basic units—for example, 3/4 is 1/4 and 1/4 and 1/4 rather than 3 of 4 parts (Behr et al., 1983).
Elementary- and middle-school students make several errors when they operate on decimals and fractions (Benander & Clement, 1985; Kouba et al., 1988; Peck & Jencks, 1981; Wearne & Hiebert, 1988). These errors are due in part to the fact that students lack essential concepts about decimals and fractions and have memorized procedures that they apply incorrectly. Interventions to improve concept knowledge can lead to increased ability by 5th-graders to add and subtract decimals correctly (Wearne & Hiebert, 1988).
Students of all ages misunderstand multiplication and division (Bell et al., 1984; Graeber & Tirosh, 1988; Greer, 1992). Commonly held misconceptions include "multiplication always makes larger," "division always makes smaller," "the divisor must always be smaller than the dividend." Students may correctly select multiplication as the operation needed to calculate the cost of gasoline when the amount and unit cost are integers, then select division for the same problem when the amount and unit cost are decimal numbers (Bell et al., 1981). Numerous suggestions have been made to improve student concepts of multiplication (Greer, 1992), but further research is needed to determine how effective these suggestions will be in the classroom.
Lower middle-school students may have difficulties understanding the relationship between fractions and decimal numbers (Markovits & Sowder, 1991). They may think that fractions and decimals can occur together in a single expression, like 0.5 + 1/2, or they might believe that they must not change from one representation to the other (from 1/2 to 0.5 and back) within a given problem. Instruction that focuses on the meaning of fractions and decimals forms a basis on which to build a good understanding of the relationship between fractions and decimals. Instruction that merely shows how to translate between the two forms does not provide a conceptual base for understanding the relationship (Markowits & Sowder, 1991).