Have Students Been Taught the Topics Tested by 魅影直播?
In interpreting the achievement results, it is important to consider
how extensively the topics tested are taught in the participating
entities. As shown in Exhibits
5.20 through 5.24,
the five major mathematics content areas assessed in 魅影直播 1999 were
represented by 34 topic areas. For each area, teachers indicated whether
their students had been taught the topics before this year (i.e.,
the eighth grade), one to five periods this year, more than five periods
this year; whether the topics had not yet been taught; or whether
the teacher did not know. Exhibits
5.20 through 5.24
show the percentages of students in each entity reported to have been
taught each topic before or during the year of testing.
According to their teachers, nearly all students in all the comparison
countries had been taught the topics in fractions and number sense,
as shown in Exhibit
5.20. The international average for each topic exceeded 90 percent
of students, with the exception of “square roots (of perfect
squares less than 144), small integer exponents” and “concepts
of ratio and proportions; ratio and proportion problems,” with
averages of 83 and 87 percent, respectively. Teachers in the United
States overall as well as in the Benchmarking jurisdictions reported
similar percentages, with 90 percent or more of the students in each
jurisdiction being taught each topic with the exception of the two
topics relating to square roots and ratio/proportion.
However, Exhibit
R2.7 in the reference section indicates that internationally many
students had instruction in these topics before the eighth grade,
while students in several Benchmarking jurisdictions were taught them
during that grade. For example, high-performing Chinese Taipei reported
that 90 percent of its students were taught more than 80 percent of
the fractions and number sense topics before the eighth grade and
not again during the eighth grade. Only eight percent of U.S. students
were taught more than 80 percent of these topics before the eighth
grade only. Similarly, all but one of the Benchmarking jurisdictions
had less than one-fifth of their students taught more than 80 percent
of fraction and number sense topics before the eighth grade only.
In the U.S. overall and across the Benchmarking jurisdictions, a larger
proportion of students were taught, or were continued to be taught,
fractions and number sense topics at the eighth grade than were students
internationally. This echoes the findings of the
魅影直播 1995 curricula analysis that showed that states in the U.S.
intended to cover far more than the average number of mathematics
topics commonly covered internationally, and that topics in the U.S.
were often added as students progressed through school at the same
rate as in other countries but without dropping other topics that
had been taught previously.(9)
Instructional coverage was high for the measurement topics presented
in Exhibit
5.21. At least 87 percent of students, on average internationally,
were taught six of the seven topics. The topic with the lowest coverage
was “scales applied to maps and models,” with an international
average of 77 percent. Two topics, “units of measurement; standards
metric units” and “perimeter and area of simple shapes –
triangles, rectangles, and circles,” were taught to 96 percent
of students on average internationally. The United States as a whole
and most of the Benchmarking jurisdictions reported percentages above
the international average for a majority of the topics. While teachers
in Jersey City reported that all students were taught all measurement
topics, teachers in the Fremont/Lincoln/Westside Public Schools reported
percentages of students below the international averages for six of
the seven measurement topics.
As indicated by Exhibit
R2.8 in the reference section, measurement topics received less
emphasis in the eighth grade than did fractions and number sense topics
(see Exhibit
R2.7). As with fractions and number sense, substantial percentages
of students internationally had studied the measurement topics before
the eighth grade, whereas among the Benchmarking jurisdictions, greater
percentages began or continued to study them during the eighth grade.
Montgomery County was the only jurisdiction reporting a greater percentage
of students than internationally (22 percent, on average) who were
taught more than 80 percent of the measurement topics before the eighth
grade and not again during the eighth grade.
Corresponding to the reports for the intended curricula, teachers
reported lower average percentages internationally across the data
representation, analysis, and probability topics, shown in Exhibit
5.22. Teachers were asked about three topics in this content area,
including “representation and interpretation of data in graphs,
charts, and tables” and “arithmetic mean.” While the
international average for students who were taught these two topics
was 75 and 70 percent, respectively, all Benchmarking jurisdictions
and the United States overall reported that at least 88 percent of
their students were taught each of these topics. The international
average percentage of students taught the other topic in this content
area, “simple probabilities – understanding and calculations,”
was 43 percent. Coverage of this topic varied widely, from just three
or four percent in Japan and Chinese Taipei to 99 percent in Korea.
The Benchmarking jurisdictions generally resembled the United States
overall, where 79 percent were taught this topic.
For students in most countries, the data representation, analysis,
and probability topics received moderate attention in the eighth grade,
with few students having been taught them only in earlier grades,
and one-third having not yet been taught half or more of the topics
by the end of the eighth grade (see Exhibit
R2.9). In comparison, however, relatively greater percentages
of students in the United States and in the Benchmarking entities
were reported to have been taught these topics during the eighth grade.
In the U.S. overall, 79 percent of students were taught more than
half the topics during the eighth grade. All Benchmarking jurisdictions
had a much greater percentage of students than internationally (39
percent, on average) who were taught more than half the topics during
the eighth grade, ranging from 60 percent in Rochester to 99 percent
in Chicago.
Teachers reported a range of instructional coverage across topics
in geometry, presented in Exhibit
5.23. “Simple two dimensional geometry – angles on a
straight line, parallel lines, triangles and quadrilaterals”
was reported to have been taught internationally on average to 95
percent of the students, and “visualization of three-dimensional
shapes” to only 57 percent. The topics showing the greatest variation
across countries were “symmetry and transformations” and
“visualization of three-dimensional shapes.” For example,
the percentage of students taught “symmetry and transformations”
ranged from less than 30 percent in Chinese Taipei to 98 percent in
Japan. The other four geometry topics were taught to more than 90
percent of the students in high-performing Japan, Korea, and Singapore.
The United States was similar to the international averages in coverage
of the geometry topics, as were most of the Benchmarking participants,
although they did show variation, particularly the districts and consortia.
For example, in Jersey City, Montgomery County, and Naperville, 90
percent of more of the students were taught each of the geometry topics.
However, in the Academy School District, Miami-Dade, and Rochester,
less than 50 percent of the students were taught “symmetry and
transformations” and “visualization of three-dimensional
shapes,” the two topics that had the lowest coverage both internationally
and in the U.S.
As shown in Exhibit
R2.10 in the reference section, only small percentages of students
had completed instruction in the geometry topics before the eighth
grade, and relatively large percentages had not yet been introduced
to many geometry topics by the end of the eighth grade. According
to the teachers in the United States, 25 percent of the students had
not been taught half or more of the geometry topics by the end of
eighth grade, close to the international average of 22 percent. This
was exceeded only by Chinese Taipei (33 percent) among the comparison
countries. In the Czech Republic, Italy, Japan, Korea, and Singapore,
less than only 10 percent of the students had not yet been taught
half or more of these topics. One-quarter or more of the students
in six Benchmarking states and four districts and consortia had not
been taught half or more of the geometry topics by the end of the
eighth grade, with the greatest percentage in the Academy School District
(49 percent).
Teachers across countries reported that most students had been taught
the algebra topics, as shown in Exhibit 5.24.
More than 85 percent of students internationally, in the U.S. overall,
and in all the Benchmarking entities were taught each of these topics,
with the exception of “solving simple inequalities,” which
had an international average of 66 percent. The percentages of students
taught the algebra topics in the United States and in the Benchmarking
entities generally exceeded the international averages. In North Carolina,
the Academy School District, Jersey City, Montgomery County, and Naperville,
90 percent or more of the students were taught each of the algebra
topics.
For many jurisdictions, teachers reported presenting algebra topics
during the eighth grade for substantial percentages of students (see
Exhibit R2.11). Teachers in all Benchmarking jurisdictions except
Rochester reported that at least half the students were taught more
than half the topics for more than five periods during the eighth
grade. Similarly, teachers in all Benchmarking jurisdictions reported
that less than 10 percent of the students had been taught half or
more of the topics before the eighth grade only. In contrast, 85 percent
of the students in Chinese Taipei and 35 percent in Japan were taught
the topics before the eighth grade.
