What Activities Do Students Do in Their Mathematics Lessons?
Because it can affect pedagogical strategies, class size is shown
in Exhibit
6.7. Teachers’ reports on the size of their eighth-grade
mathematics class reveal that across countries the average was 31
students, but there was considerable variation even among the higher-performing
countries – from 42 students in Korea to 19 in Belgium (Flemish).
Average class size was relatively uniform across all of the Benchmarking
entities, ranging from 22 to 30 students. The relationship between
class size and achievement is difficult to disentangle, given the
variety of policies and practices and the fact that smaller classes
can be used for both advanced and remedial learning. It makes sense,
however, that teachers may have an easier time managing and conducting
more student-centered instructional activities with smaller classes.
Extensive research about class size in relation
to achievement indicates that the existence of such a relationship
is dependent on the situation.(3) Dramatic reductions
in class size can be related to gains in achievement, but the chief
effects of smaller classes often are in relation to teacher attitudes
and instructional behaviors. Also, the research is more consistent
in suggesting that reductions in class size have the potential to
help students in the primary grades. The 魅影直播 1999 data support the
complexity of this issue. The five highest-performing countries –
Singapore, Korea, Chinese Taipei, Hong Kong, and Japan – were
among those with the largest mathematics classes. Within countries,
several show little or no relationship between achievement and class
size, often because students are mostly all in classes of similar
size. Within other countries, there appears to be a curvilinear relationship,
or those students with higher achievement appear to be in larger classes.
In some countries, larger classes may represent the more usual situation
for mathematics teaching, with smaller classes used primarily for
students needing remediation or for those students in the less-advanced
tracks.
Exhibit
6.8 presents a profile of the activities most commonly encountered
in mathematics classes around the world, as reported by mathematics
teachers. As can be seen from the international averages, the two
predominant activities, accounting for nearly half of class time on
average, were teacher lecture (23 percent of class time) and teacher-guided
student practice (22 percent). In general for the United States overall
and the Benchmarking entities, teachers’ reports on the frequency
of these activities matched the international profile. According to
U.S. mathematics teachers, class time is spent as follows: 15 percent
on homework review; 20 percent on lecture style teacher presentation;
35 percent on teacher-guided or independent student practice; 12 percent
on re-teaching and clarification; 11 percent on tests and quizzes,
six percent on administrative tasks; and four percent on other activities.
One noteworthy exception is 26 percent of class time in Naperville
spent on homework review, compared with 15 percent for the United
States.
As shown in Exhibit
6.9, most students internationally (86 percent on average) agreed
with teachers’ reports about the prevalence of teacher-guided
activities, saying that their teachers frequently showed them how
to do mathematics problems. Just as found in the 1995 videotapes,
it appears that in the U.S. the teacher states the problem, demonstrates
the solution, and then asks the students to practice. Ninety-four
percent of U.S. eighth graders reported that their teachers showed
them how to do mathematics problems almost always or pretty often
during mathematics lessons. More than 90 percent of the students in
each of the Benchmarking entities reported this also.
Compared with their counterparts internationally (59 percent), more
U.S. students reported that working independently on worksheets or
textbooks occurred almost always or pretty often (86 percent). Working
on their own on worksheets or textbooks was also quite pervasive throughout
the Benchmarking entities, where more than 80 percent of the students
in each jurisdiction reported doing this activity that frequently.
As for working on mathematics projects, the Benchmarking states typically
were below the international average (36 percent), ranging from 22
to 33 percent. There was considerable variation across the districts
and consortia. Less than one-fifth of the students reported frequent
project work in the Academy School District, the First in the World
Consortium, and Naperville. At the other end of the continuum, 63
percent so reported in Jersey City, followed by 34 to 38 percent in
Chicago, the Fremont/Lincoln/Westside Public Schools, Miami-Dade,
and Rochester.
Compared with students internationally, eighth graders in each of
the Benchmarking jurisdictions and in the United States overall reported
an unusually large amount of classroom time devoted to working on
homework. Internationally, 55 percent of the students reported frequently
discussing their completed homework. The figure for the United States
was 79 percent, and it ranged from 70 to 91 percent for the Benchmarking
jurisdictions. An even greater difference was evident for frequently
beginning homework in class – 42 percent internationally compared
with 74 percent for the United States. In the Benchmarking jurisdictions,
from 43 to 90 percent of the students reported beginning their homework
in class almost always or pretty often.
As might be anticipated, students reported that use of the board
was an extremely common presentational mode in mathematics class (see
Exhibit
6.10). On average internationally, 92 percent of students reported
that teachers used the board at least pretty often, and 60 percent
reported that students did so. Using the board seems to be less common
in the United States, especially for students (37 percent). In the
United States, use of an overhead projector is a popular presentational
mode, especially for teachers – 59 percent compared with 19 percent
internationally. This mode was used frequently for more than 80 percent
of the students in Maryland, North Carolina, Oregon, the Academy School
District, the Fremont/Lincoln/Westside Public Schools, Guilford County,
Montgomery County, and Naperville.
Educators, parents, employers, and most of the public support the
goal of improving students’ capacity for mathematics problem-solving.
To examine the emphasis placed on that goal, 魅影直播 created an index
of teachers’ emphasis on mathematics reasoning and problem-solving
(emrps). As shown in Exhibit
6.11, the index is based on teachers’ responses about how
often they asked students to explain the reasoning behind an idea,
represent and analyze relationships using tables, charts, or graphs,
work on problems for which there was no immediate solution, and write
equations to represent relationships. Students were placed in the
high category if, on average, they were asked to do these activities
in most of their lessons. The medium level represents students asked
to do these activities in some to most lessons, and students in the
low category did them only in some lessons or rarely.
Nearly half the Japanese students were at the high index level, compared
with the international average of 15 percent. Across countries, most
students (61 percent on average) were in the medium category. An emphasis
on problem-solving was related to performance, with students at the
high and medium levels having higher average achievement than those
at the low level, both internationally and for most entities. There
was tremendous variation among the Benchmarking participants on this
index. From 41 to 46 percent of the students were in the high category
in Jersey City, First in the World, and the Michigan Invitational
Group, compared with eight to nine percent in Chicago and Oregon.
Exhibit
R3.7 in the reference section shows the percentages of students
asked in most or every lesson to engage in each of the activities
included in the problem-solving index. For comparison purposes, the
exhibit also shows the percentages of students asked to practice computational
skills in most or every lesson. According to their teachers, internationally
on average nearly three-fourths of the students (73 percent) were
asked to practice their computational skills in most or every mathematics
lesson. Nearly as many (70 percent) were asked to explain the reasoning
behind an idea this frequently. The other three problem-solving activities
occurred much less often. Forty-three percent of students, on average
across countries, wrote equations representing relationships in most
or every lesson, but only about one-fourth (26 percent) represented
and analyzed relationships using tables or graphs, and about one-fifth
(21 percent) worked on problems for which there was no immediately
obvious method of solution. While the Benchmarking entities did not
vary greatly from the international profile, there were differences.
For example, twice as many students as internationally reported spending
time in most or every lesson working on problems for which there was
no immediately obvious method of solution in the First in the World
Consortium, the Jersey Public Schools, and the Michigan Invitational
Group (44 to 51 percent). More than 90 percent of the students in
Jersey City and the Michigan Invitational Group were frequently asked
to explain the reasoning behind an idea, and 90 percent of the Naperville
students were frequently asked to write equations to represent relationships.
Teachers were not asked about the emphasis placed on using things
from everyday life in solving mathematics problems, but students were
(see Exhibit
R3.8). In most of the countries, students reported a moderate
emphasis on doing this type of problem in mathematics class. Nearly
two-thirds (65 percent), on average internationally, said these activities
occur once in a while or pretty often, and an additional 15 percent
said they occur almost always. The figures were somewhat higher for
the United States and most Benchmarking jurisdictions. More than 60
percent of the students in Maryland, North Carolina, the Academy School
District, the Fremont/Lincoln/Westside Public Schools, Jersey City,
and the Michigan Invitational Group reported that they use things
from everyday life in solving mathematics problems almost always or
pretty often.
