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 Results from the Seventh Mathematics Assessment of the National Assessment of Educational Progress National Council of Teachers of Mathematics, 2000. Edited by Edward A. Silver and Patricia Ann Kenney. . This assessment included 24 items on fractions with varying numbers of items on the 4th-, 8th-, and 12-grade assessments. The data provided evidence that fractions continue to be difficult for students, particularly for 4th-grade students. (The following tables and comments were condensed from Chapter 7, "Rational Numbers", by Diana Wearne and Vicki L. Kouba.)  The 1996 NAEP contained items in which the students were asked either to write or to identify the fraction associated with a given pictorial representation or to represent the fraction on a region. Table 7.1 contains both a description of, and performance results for, those items. Fourth grade students found representing a fraction or writing or identifying the associated fraction to be easier if the unit was a region (item 1, 73 percent correct) than if it was a set (item 2, 54 percent correct). This is consistent with earlier NAEP results (Post 1981; Carpenter et al. 1978). This suggests that students may have a fragile understanding of "unit" in their work with fractions. The meaning that some students have constructed for the unit when fractions are represented as regions is more robust than the meaning attached to the unit when fractions are represented as sets, as collections of objects. Essential to the notation of the fraction n/d is that the unit has been divided into d equal parts. It is difficult for students to understand equivalent fractions, and operations with fractions, without recognizing this. As shown in table 7.2, only 50 percent of students in grade 4 were able to state how many fourths were in a whole. Because this item assesses fundamental knowledge of fractions, a topic introduced in the first grade, the results are disturbing. Perhaps equally disturbing is the fact that one-sixth of the students omitted this item, the initial item in that item block. This may show, in another way, the difficulty students have in fully understanding the meaning of the unit in fraction tasks. An understanding of equivalent fractions is important in developing a sense of the relative size of fractions and helping students connect their intuitive understandings and strategies to more general, formal methods. As shown in the table, 70 percent of the 8th-grade students identified the diagram illustrating the equivalence of two fractions (item 1). However, fewer than 50 percent of the 4th-grade students were successful on this item. Similar 4th-grade results, with fewer than half of the students responding correctly, occurred on item 2 in which the students were presented with three equivalent fractions and asked to write two more; the three given fractions were all equivalent to a unit fraction that was not included in the set (for example, if the unit fraction had been 1/3, fractions included in the item could have been 2/6 and 3/9). Table 7.3 also includes two items related to the ordering of fractions, items 5 and 6. Item 5 asks the 4th-grade students to tell which of two unit fractions was the larger and to explain their reasoning. This question was posed within a context to facilitate students being able to explain, in words or by a diagram, why one unit fraction was larger than another. Only about one-sixth of the 4th-grade students responded correctly. These results are similar to those noted in other studies (Post et al. 1985). The other ordering item (item 6) asked the 8th-grade students to identify which sets of fractions were ordered from least to greatest. Despite the fact that all the fractions were less than 1 and in reduced form, only 35 percent of the 8th-grade students chose the correct ordering. Results on items 5 and 6 are troubling because it is difficult to imagine how students can work meaningfully with fractions if they do not have a sense of their relative size. Table 7.4 contains the performance results for a released item in which the 8th-grade students were asked to identify a fraction equivalent to 4/12, the fraction associated with the shaded portion of the rectangle. The percents of 8th-grade students selecting 1/3 and 1/4 have remained constant over the last three NAEP assessments, with 65 percent selecting 1/3 and 25 percent selecting 1/4. Table 7.5 contains additional items assessing knowledge of fractions, including items that can be categorized as applications. The first three items asked students to identify the size of the whole set given the size of a specific fractional part of the set or, given the size of the set, to find the size of the subset left after a specified part of the original set had been removed. As indicated in the table, finding the size of the whole given an indicated fractional part (item 1) was the easier of the two types of tasks for 4th-grade students; this multiple-choice task proved not to be difficult for the students in grades 8 and 12. Items 2 and 3 contain descriptions of tasks that asked students to find the size of a subset after a specified fractional part of the original set had been removed. The remaining items, items 4-8 in table 7.5, were administered only at the upper two grades. This set included writing a story problem to go with a given division by a fraction number sentence (item 4) and solving four multistep problems (items 5-8) in which students had to apply basic fraction ideas. As reported in the table, at most one-third of the students in grade 12 responded correctly to these items. However, significantly more students in grade 12 responded correctly to item 8 on the 1996 NAEP (25 percent) than had on the 1992 NAEP (20 percent). Given the weak understanding younger students showed for some of these ideas, perhaps it is not surprising that students have trouble applying these ideas as they get older.