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Results from the Sixth Mathematics Assessment

of the National Assessment of Educational Progress

National Council of Teachers of Mathematics, 1997. Edited by Patricia Ann Kenney and Edward A. Silver.

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This assessment included 18 items on the representation and comparison of rational numbers. Half of the items involved fractions only and three items addressed fraction-decimal relationships. The following tables and comments were condensed from Chapter 5, "What Do Students Know About Numbers and Operations", by Vicki L. Kouba, Judith S. Zawojewski, and Marilyn E. Strutchens.)

The set of items described in table 5.13 shows patterns of performance on items in which rational numbers were represented as parts of geometric regions or number lines. Of special interest are those items that were administered at more than one grade, which afforded the opportunity to examine student performance across grade levels. For item 1 in the table, the majority of students at all three grade levels chose the correct picture of a region for a simple fraction. Performance increased sharply from grade 4 to grade 8 – 69 percent to 90 percent, respectively – and then remained essentially the same from grade 8 to grade 12. The growth between grades 4 and 8 most likely reflects the curricular emphasis on fractions in the middle grades. Notable was the less successful performance on the examples that dealt with equivalent fractions, as illustrated by items 2, 3, and 4. This is consistent with Behr et al.’s (1983) reported difficulty that students have when a geometric region needs to be interpreted visually in more than one way. For example, a region divided into eighths can be reinterpreted as fourths if certain partitions can be visually ignored. The very poor performance on item 3 by 4th-grade students is very likely due to lack of curricular experience with equivalent fractions, coupled with the fact that this item required a constructed response. While performance was lower for 8th-grade students on items 3 and 4, the difference compared to item 2 was not as drastic as it was for the fourth grade, which perhaps reflects the middle grades’ emphasis on rational numbers.

Items 5, 6, and 7 were each based on a picture of a number line. Performance on these items was lower than that of the first set, which involved geometric regions. Items involving number line representations for rational numbers often have a number of features that may add to difficulty in interpretation.

Item 7 from table 5.13, which is shown in detail in table 5.14, was answered correctly by only half of the 12th-grade students. The item had at least three features that may have made it especially difficult: it used a mix of decimal and fraction in the prompt; it involved a number line that was two units long; and it had three segments per unit, while asking the student to locate a decimal with an understood denominator of 100.

Results for the Pizza Comparison based on the five performance levels for extended construction-response questions are in table 5.15 above, and descriptions of the performance levels and sample responses for each level are shown below in figure 5.3. The descriptions and sample responses are as they appeared in an NAEP publication (Dossey, Mullis, and Jones 1993, pp. 93-95). About one-fourth of the 4th-grade students produced either extended or satisfactory responses that communicated the relationship between fractional part and relative size. Another 18 percent of the students declared that Ella was right, and their reasoning involved the misconception that "1/2 always equals 1/2". There is some indication that this question was difficult for 4th-grade students: 7 percent did not attempt to answer it and another 49 percent did not use fraction concepts in communicating their responses.

Figure 5.3