Grade K-12 Instruction
Beyond understanding whole numbers, young children can be encouraged to understand and represent commonly used fractions in context, such as 1/2 of a cookie or 1/8 of a pizza, and to see fractions as part of a unit whole or of a collection. Teachers should help students develop an understanding of fractions as division of numbers, and in the middle grades, in part as a basis for their work with proportionality, students need to solidify their understanding of fractions as numbers.
Representing numbers with various physical materials should be a major part of mathematics instruction in the elementary school grades. By the middle grades, students should understand that numbers can be represented in various ways, so that they see that 1/4, 25%, and 0.25 are all different names for the same number. Students’ understanding and ability to reason will grow as they represent fractions and decimals with physical materials and on number lines and as they learn to generate equivalent representations of fractions and decimals.
As students gain understanding of numbers and how to represent them, they have a foundation for understanding relationships among numbers. In grades 3 through 5, students can learn to compare fractions to familiar benchmarks such as 1/2. And, as their number sense develops, students should be able to reason about numbers by, for instance, explaining that 1/2 + 3/8 must be less than 1 because each addend is less than or equal to 1/2. In grades 6-8, it is important for students to be able to move flexibly among equivalent fractions, decimals, and percents and to order and compare rational numbers using a range of strategies.
In grades 6-8, operations with rational numbers should be emphasized. Students’ intuitions about operations should be adapted as they work with an expanded system of numbers (Graeber and Campbell, 1993). For example, multiplying a whole number by a fraction between 0 and 1 (e.g. 8 x 1/2) produces a result less than the whole number. This is counter to students’ prior experience (with whole numbers) that multiplication always results in a bigger number.
The development of rational number concepts is a major goal for grades 3-5, which should lead to informal methods for calculating with fractions. For example, a problem such as 1/4 + 1/2 should be solved mentally with ease because students can picture 1/2 and 1/4 or can use decomposition strategies, such as 1/4 + 1/2 = 1/4 + (1/4 + 1/4). . . . When asked to estimate 12/13 + 7/8, only 24 percent of thirteen-year-old students in a national assessment said the answer was close to 2 (Carpenter et al. 1981). Most said it was close to 1, 19, or 21, all of which reflect common computational errors in adding fractions and suggest a lack of understanding of the operation being carried out. If students understand addition of fractions and have developed number sense, these errors should not occur.