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 TIMSS 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.
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, TIMSS 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.
TIMSS 1999 is a project of the International
Boston College, Lynch School of Education