Middle school students' understanding of number sense related to percent

School Science and Mathematics, Jan 1997 by Gay, A Susan, Aichele, Douglas B

Allinger and Payne (1986) suggest that the proportion method and the factor-factor-product method are the two common teaching approaches to percent, and textbooks often present both methods (e.g., Orfan et al., 1987; Quast, Cole, Sparks, Haubner, & Allen, 1987). Allinger and Payne (1986) claim that the way percent is taught encourages students to rely exclusively on rules and procedures to solve percent problems. This would imply that little attention is being given in classrooms to helping students develop a number sense about percent.

The present study examined middle school students' number sense related to percent. To provide some insight into students' understanding of percent, three research questions were posed:

1. Can students interpret a quantity expressed as a percent given a pictorial discrete set or continuous region with part or all of the area shaded?

2. Do students understand the meaning of a quantity expressed as a percent of a number?

3. What strategies do students use to make comparisons about percents in both pictorial and abstract settings?

Subjects

The subjects were 106 seventh-grade and 93 eighth-grade students enrolled in average mathematics classes in a large urban district within the metropolitan Oklahoma City area during the 199091 school year. Approximately half of each group were girls and half were boys. The students had not received instruction on percent concepts during this school year.

Instrumentation Researchers consider the understanding of a percent as a fractional part of a whole to be fundamental before working percent problems (Allinger & Payne, 1986; Schminke et al., 1973). In work with rational numbers, some researchers have claimed that children have more difficulty conceptualizing a discrete set as a whole than viewing a continuous region as a whole (Behr & Post, 1988; Payne, 1984). Both of these findings were considered along with the results of the 1986 NAEP as the written test was developed for this study.

The written test was used to measure students' performance on a sample of questions focusing on percent. The three parts of the test used the same percents (50%, 25%, 100%, 60%, 110%, 33 1/3%, 87%) in the order of most familiar to least familiar, as determined by students participating in a pilot study. The seven multiple-choice questions in Part 1 of the test were designed to determine students' ability to compare a given percent with a quantity represented as a shaded part of a pictorial discrete set of circles. A question from this part of the test is presented in Figure 1. The seven multiple-choice questions in Part 2 asked students to compare a shaded part of a continuous rectangular region with a given percent. Questions on this part of the test used the same format as questions in Part 1, except that one continuous rectangular region was used. Seven multiple-choice questions in Part 3 were based on a released exercise (see Figure 2) by the National Assessment of Educational Progress during its fourth assessment (Dossey, Mullis, Lindquist, & Chambers, 1988) and asked students to determine if a quantity expressed as a percent of a number was less than, greater than, or equal to the given number. The question in Figure 2 was the seventh question in Part 3 of the test. An open-ended test item in Part 3 asked each student to explain how he or she decided on a response to the preceding multiple-choice test question comparing 87% of 10 to 10. In the pilot study, the KuderRichardson Formula 21 (Isaac & Michael, 1981) was used to determine a reliability coefficient of .855 for the written instrument.