Fraction concepts are among the most complex and important mathematical ideas that children encounter before their secondary school years (Behr et al. 1984). Perhaps because of their complexity, fractions are also among the concepts least understood by students. On the 1990 National Assessment of Educational Progress, for example, only 50 percent of eighth graders could express in decimal form a fraction with a denominator of 10. Only 49 percent could tell the weight on the Moon of an object that weighs thirty pounds on Earth, given that a weight on the Moon is 1/6 the weight on Earth (National Assessment Governing Board 1991). American students are not alone in having trouble (Strang 1990; Kerslake 1986). Why do students have so much difficulty with this area of mathematics?
Part of the difficulty lies in the nature of fractions. The word fraction may have any one of several meanings. The fraction-bar symbol itself may express several distinctly different mathematical ideas, and the concept of rational number has several different subconstructs (Ohlsson 1988; Behr et al. 1992). Fairly sharing a pizza among four friends is conceptually different from predicting the likelihood of rain, given the past record of rain on an average of one day in four. Using four eggs from a carton of a dozen is quite different from cutting a third from a stick of butter. The "simple" fraction that we try to teach our students turns out to be not so simple on closer examination.
Of How Much Help Is the Textbook?
Because most teachers rely on the mathematics textbook in planning their instruction, looking at what is available from textbook publishers seems appropriate. What meaningful learning experiences with fractions do textbooks offer? A recent examination of three 1992 textbook series for grades 1 through 5, randomly selected from those published by major companies and available for adoption in the state of Virginia, gives some indication of what today's textbooks offer.
This investigation of textbooks' treatment of fractions was guided by several ideas from recent research. For example, Lesh's model (adapted from Bruner and discussed in Behr et al. ) suggests five modes of representation (see question 2 that follows): it was noted whether these modes were reflected in the textbooks. Also noted was the presence of a reasonable balance in the inclusion of different subconstructs of rational number, for example, part-whole, ratio, and decimal.
Questions Considered in the Textbook Examination
Question 1: How much of a textbook is devoted to instruction in fractions?
Rationale. One indication of the importance the authors place on fractions is the amount of space they allot to them.
Results. As expected, the percent of pages devoted to fractions increases along with the grade levels, from an average of 2 percent of the pages in the first-grade textbook to an average of 34 percent in the fifth-grade textbook. The numbers were fairly consistent across the three series, and, in general, the number of fraction-related pages increased sharply from the fourth to the fifth grade.
Question 2: Did the textbook address the various modes of representing fractions?
Rationale. Lesh has described five different modes of representing rational number: spoken symbols, written symbols, manipulative materials, pictures, and real-world situations. The processes of transforming among these five modes of representation and translating within each mode make the mathematical ideas meaningful to the child (Behr et al. 1992).
TABLE 1 Percent of Pages Devoted to Fraction Instruction Series 1 Series 2 Series 3 Percent Percent Percent No. of of Total No. of of Total No. of of Total Pages Pages Pages Pages Pages Pages Grade 1 8 2% 6 2% 10 2% Grade 2 16 4% 16 4% 23 5% Grade 3 58 12% 37 7% 61 12% Grade 4 68 14% 86 19% 93 18% Grade 5 146 29% 154 34% 197 38%