Achievement at the Upper Quarter Benchmark
Exhibit
2.6 describes performance at the Upper Quarter Benchmark. Eighth-grade
students performing at this level applied their mathematical knowledge
and understandings in a wide variety of relatively complex problem
situations. For example, they demonstrated facility with fractions
in various formats, as illustrated by Example Item 5 shown in Exhibit
2.7. This item required students to shade squares in a rectangular
grid to represent a given fraction. Since the grid is divided into
squares that are a multiple of the fraction’s denominator, more
than one step is required to solve the problem. Internationally, about
half the students (49 percent on average) were able to shade in nine
of the 24 squares to represent 3/8 of the region. Eighty percent or
more of the students in Singapore, Hong Kong, Belgium (Flemish), Korea,
and Chinese Taipei answered the question correctly. No Benchmarking
entities performed that well, but students in the First in World Consortium,
Naperville, the Michigan Invitational Group, and Massachusetts performed
significantly above the international average.
Example Item 6 is a proportional reasoning word problem that students
at the Upper Quarter Benchmark typically answered correctly (see Exhibit
2.8). Given the number of magazines sold by each of two boys and
the total amount of money made from the sales, students were to calculate
how much money one of the boys made by selling his 80 magazines. On
average, 44 percent of students internationally answered this question
correctly. In Singapore and Chinese Taipei at least three-quarters
of the students answered correctly. No Benchmarking participant performed
significantly above the international average, and students in Maryland,
the Michigan Invitational Group, the Chicago Public Schools, the Rochester
City School District, and the Miami-Dade County Public Schools performed
significantly below the international average.
Students reaching the Upper Quarter Benchmark generally were able
to apply knowledge of geometric properties. In Example Item 7 in Exhibit
2.9, students needed to use their knowledge of the properties
of parallelograms and rectangles to solve for the area of the rectangle
(dimensions not labeled) that was part of a different figure with
given dimensions. Three-quarters or more of the students in Singapore,
Japan, Hong Kong, Korea, and Chinese Taipei answered the item correctly.
Internationally, however, less than half the eighth-grade students
(43 percent on average) did so. The United States performed significantly
below the international average, as did eight of the Benchmarking
entities: North Carolina, South Carolina, Missouri, the Delaware Science
Coalition, and the public school systems in Jersey City, Chicago,
Miami-Dade, and Rochester.
Example Item 8 shown in Exhibit
2.10 asks students for the number of triangles of a given dimension
needed to cover a rectangle of given dimensions. The international
average on this item was 46 percent correct. Many students (approximately
29 percent internationally) incorrectly chose Option A, which is half
the number of required triangles needed to fill the rectangle but
just enough to cover the perimeter. Japanese students had the highest
performance on this item, with 80 percent answering correctly. About
two-thirds or more of the students in Korea, Hong Kong, Singapore,
Belgium (Flemish), and the Netherlands answered the item correctly.
Performance among the Benchmarking participants ranged from 62 percent
correct responses in Naperville to 30 percent in Miami-Dade. The United
States as a whole performed at about the international average, and
most of the Benchmarking jurisdictions performed similarly.
Unlike students at lower benchmarks, those reaching the Upper Quarter
Benchmark typically could solve simple linear equations. As illustrated
by Example Item 9 in Exhibit
2.11, for example, students successfully solved for the value
of x in a linear equation involving the variable on both sides of
the equation. Eighty percent or more of the students in Japan, Hong
Kong, and Korea answered this item correctly. Even though the United
States did relatively well in algebra (see Chapter 3), this problem
posed difficulties for students in the Benchmarking entities. Naperville
(72 percent) and First in the World (61 percent) were the only Benchmarking
participants that performed significantly above the international
average of 44 percent correct responses. The United States performed
below average (34 percent) on this question, as did students in 11
of the Benchmarking entities.
