Achievement at the Top 10% Benchmark
Exhibit
2.1 describes performance at the Top 10% Benchmark. Students reaching
this benchmark demonstrated the ability to organize information in
problem-solving situations and to apply their understanding of mathematical
relationships. They typically demonstrated success on the knowledge
and skills represented by this benchmark, as well as those demonstrated
at the three lower benchmarks.
Example Item 1 in Exhibit
2.2 illustrates the type of measurement item a student performing
at the Top 10% Benchmark generally answered correctly. As can be seen,
students had to apply their knowledge of the area of rectangles and
inscribed shapes to solve a two-step problem about the area of a garden
path. The international average for this item was 42 percent correct,
indicating that this was a relatively difficult item for eighth graders
around the world. Nevertheless, more than two-thirds of the students
answered the item correctly in Hong Kong, Singapore, Japan, Chinese
Taipei, and Korea. Among the Benchmarking participants, eighth graders
in the Naperville School District did as well as their counterparts
in the high-performing Asian countries, with 69 percent answering
correctly. Generally, however, students in the United States –
in the country as a whole and in the Benchmarking entities –
performed relatively less well than students internationally on measurement
questions involving relationships between shapes. No other Benchmarking
entity performed significantly above the international average on
this test question, and students in six Benchmarking entities and
in the United States overall performed significantly below the international
average. On average internationally, more than 20 percent of students
chose Option A, solving for the area of the larger rectangle rather
than that of the path. Option C was an equally popular distracter,
selected by more than 20 percent of students internationally.
Unlike students performing at lower benchmarks, students reaching
the Top 10% Benchmark typically could correctly answer multistep word
problems. Example Item 2 in Exhibit
2.3 requires students to select relevant information from two
advertisements to solve a complex multistep word problem involving
decimals. Given the price for each issue of a magazine and a certain
number of free issues, students were asked to calculate which of the
two magazine subscriptions was the less expensive for 24 issues. Students
received full credit if they showed correct calculations for at least
one of the subscriptions, identified the less expensive magazine,
and calculated the difference between the two subscriptions. With
an international average of 24 percent correct (for full credit),
this item was among the most difficult in 魅影直播 1999. Singapore, Korea,
and Chinese Taipei were the only countries where the majority of the
students answered correctly. The best performance by a Benchmarking
entity was in Naperville, where 41 percent of the eighth graders answered
correctly. Students in the First of World Consortium (36 percent)
and Montgomery County (35 percent) also performed significantly above
the international average.
Students reaching the Top 10% Benchmark exhibited an understanding
of the properties of similar triangles, as shown by Example Item 3
(see Exhibit
2.4). Given two angle measurements, the length of a side of a
triangle, and the dimensions of a second similar triangle, students
needed to find the length of an unlabeled side of the first triangle.
Internationally, most eighth-grade students had not mastered the concept
of proportionality of corresponding sides or could not solve the resulting
equation; only 37 percent, on average, answered the question correctly.
In comparison, top-performing Korea had 70 percent correct responses.
Among the 魅影直播 1999 countries, only in Korea, Japan, Singapore, Hong
Kong, Chinese Taipei, and Belgium (Flemish) did at least half the
students answer correctly. In the Benchmarking jurisdictions, correct
responses were provided by more than half the eighth graders in Naperville
(56 percent) and the First in the World Consortium (52 percent).
The eighth-grade students reaching the Top 10% Benchmark typically
were able to apply a generalization to solve a sequence problem like
the one shown in Example Item 4 in Exhibit
2.5. In this algebra problem, given the initial terms in a sequence
and the 50th term of that sequence, students generalized to find the
51st term. Even though results are presented only for Part C, this
problem was presented in three parts, A, B, and C. To provide some
scaffolding, parts A and B asked students to indicate how many circles
would be in the 5th and 7th figures, respectively, if the pattern
were extended. On average internationally, 65 percent of the students
answered Part A correctly and 54 percent successfully extended the
sequence to the 7th figure in Part B.
To receive full credit for Part C, students had to show or explain
how they arrived at their answer by providing a general expression
or an equation and by calculating the correct number of circles for
the 51st figure. Internationally on average, 30 percent of the students
received full credit for their responses. In comparison, about two-thirds
of the students in Korea, Chinese Taipei, Japan, and Singapore received
full credit. Although eighth graders in six Benchmarking entities
– First in the World, Naperville, the Michigan Invitational Group,
Montgomery County, the Academy School District, and Oregon –
performed significantly above the international average, their performance
was below that of the top performers, ranging from 54 to 39 percent
correct. Most students added the sequence number to the number of
circles in the preceding figure: 1275 + 51 = 1326. Very few calculated
the answer by a general expression: n(n+1)/2 or 51(52)/2 (although
13 percent of the Dutch students did so).
