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DESPITE THE GREAT AMOUNT OF TIME THAT MIDDLE-GRADES TEACHers devote to teaching fractions and decimals, converting between these two representations continues to be a difficult task for students. According to the results of the sixth National Assessment of Educational Progress (NAEP) conducted in 1992, although 90 percent of eighth-grade students correctly paired a simple fraction with its pictorial representation, only 63 percent of students successfully shaded a fractional portion of a given rectangular region using equivalent fractions. Likewise, 92 percent correctly identified 14.9 seconds as being the decimal representation closest to 15 seconds, but when comparing common fractions with decimal notation, only 51 percent of eighth-grade students chose 1/2 as being the fraction closest to 0.52. Twenty-nine percent of eighth graders chose the fraction 1/50 as being closest in value to 0.52 (Kouba, Zawojewski, and Struchens 1997).
Why is this task so difficult for students? As students progress through school, teachers begin to use symbols to represent mathematical ideas, and the symbols begin to take on a life of their own. Students no longer expect mathematics to make sense. Instead, they find themselves immersed in learning that focuses heavily on the rules for working with fractions and decimals. Students look for meaning in the patterns of the symbols and the syntax rather than try to understand what they are doing. Hiebert and Wearne (1985) found that students often rely solely on syntactic rules. Confusion arises when the symbolic configuration of a problem is similar to problems learned earlier, and students end up using inappropriate rules. It has been noted that most errors are not a result of the incorrect use of a rule but rather the use of the wrong rule for a particular situation (Hiebert and Wearne 1985).
Another common error that students make is trying to modify a rule to get it to produce the answer that they think looks right. These errors occur so frequently that educators can predict the type of mistakes that are likely to be made, which explains why students tend to make the same errors throughout their schooling. Those students who do improve often do so by increasing their ability to discern differences in syntax, as ok posed to developing a better conceptual understanding of the mathematics involved.