When student assessments reveal differences in performance across nations or states or within population groups of the magnitude that they have displayed in the assessments analyzed here, there is a compelling policy need to explore the sources of these disparities. A better understanding of why some groups of students perform well in mathematics and science while others do not can help educators and policymakers in deciding which facets of the education system require more or less attention.

Many recent analyses have focused on differences in the educational experiences of students. The Third International Mathematics and Science Study provides more comprehensive information on the educational experiences of students than any
international (and many national) studies conducted to date. Within this large-scale study, a curriculum analysis provides country profiles of the mathematics and science that students are expected to learn at each grade.**[9]** Student and teacher surveys provide information on the subject matter content and activities that make up a lesson; and a video study (for the United States, Germany, and Japan) provides observational information on what
actually takes place in a sample of eighth grade mathematics classrooms.

In most countries, curricula focus on a limited number of topics at each grade. Each topic is introduced in the grade sequence and continues until a point when it is discontinued in favor of a new topic. In contrast, U.S. curricula follow a spiral approach: a topic is introduced in its simplest terms in early grades and continues in more advanced forms into later grades. Topics thus "spiral" throughout the curriculum-in theory, providing greater depth, elaboration, and complexity at each appearance. Three central ideas underlie the U.S. approach. First, content is more easily mastered when broken into "bite-sized" pieces. Second, the pieces are best learned when presented in order of difficulty and complexity. Third, students must master each piece before moving on to the next.

However, this approach when put into actual practice has important consequences for learning and instruction that are not always consistent with the theory. The U.S. curricula include a great deal of repetition over grades, and despite the intent to present new aspects of a topic at each appearance, much of the information seems to get rehashed from previous levels. On average, topics remain in the mathematics curriculum as a whole two years longer than is the norm internationally. And the curriculum includes a large number of topics since few are dropped as others are added. On average, the U.S. mathematics curriculum covers more topics than are covered in 75 percent of countries that participated in the 1995 international study.

Analyses of topics covered at various grade levels in mathematics textbooks across the world illustrate this point. At fourth grade, the five most emphasized mathematics topics accounted for 60 percent of page space in U.S. textbooks but over 85
percent internationally. In eighth grade mathematics, the five most emphasized topics in U.S. (nonalgebra) texts accounted for less than 50 percent of total coverage, compared with 75 percent internationally.**[10]** U.S. eighth grade textbooks for regular, nonalgebraic mathematics cover approximately 36 different topics, compared with an average of 8 topics in Japanese and 4.5 topics in German texts.**[11]** Findings are similar for the 4th and 12th grades. (See figure 1-14.)

A review of the topics emphasized at each grade level reveals that U.S. mathematics texts are also often out of step with the international norm. For example, at eighth grade-where U.S. students perform relatively poorly in mathematics compared with
other nations-the international norm is to focus on algebra and geometry. In the United States, eighth grade texts place greater emphasis on whole numbers, decimals, and fractions-topics that most other countries have already completed. Videotaped
lessons confirm this finding. Lessons in German and Japanese classrooms were focused on algebra and geometry, while, in about 40 percent of the cases, U.S. lessons focused on arithmetic (NCES 1996c).**[12]**

Overall, the U.S. science curriculum has more in common with the curricula of other countries than is the case for U.S. mathematics. Still, U.S. science curricula reflect some of the patterns observed in mathematics. In the United States, new topics are introduced at regular intervals in the first five grades. Much of the content seems repetitive until about 10th grade, when general science is replaced by courses devoted to specific areas of science such as biology, chemistry, or physics.

However, in the elementary and middle grades, U.S. students take general science courses that cover more topics than are covered in most of the participating countries. General science textbooks in the United States tend toward inclusiveness, covering more distinct topics than are covered in texts in 75 percent of the other countries. The typical U.S. science textbook covers between 53 and 67 topics, depending on grade level. In Japan, the range is 8 to 17 topics. In Germany, where data were available only for eighth grade, the average is nine topics. (See figure 1-15.)

This tendency toward inclusive coverage means that most general science textbooks in the United States touch on topics rather than concentrating on them. As an example, the five most emphasized topics in U.S. fourth grade science texts accounted for 25 percent of the total textbook space, compared with an international average of 70 to 75 percent. In eighth grade, the five most emphasized topics in U.S. general science texts accounted for 50 percent of textbook space, compared with 60 percent internationally.

Textbooks and curriculum guides are not the only critical factors in curriculum and instruction. Equally critical from the perspective of educational reformers are instructional considerations such as the amount of time students spend engaged with subject matter, the kinds of tasks used to facilitate their problem-solving and thinking capacities, and the technological tools available to support active student learning.

Differences in student performance outcomes are determined, at least to some degree, by differences in instructional practice and instructional quality. Science instruction in the United States may be roughly comparable to science instruction in other countries. But, as revealed in the recent international comparison, eighth grade mathematics classes in the United States are pitched at a lower level than in higher achieving countries. While U.S. eighth graders are still working on "high-end arithmetic," their peers in other countries are learning algebra and geometry.

The international comparison also revealed differences in goals, activities, and overall lesson quality in the United States, Germany, and Japan. The goal of mathematics lessons in the United States and Germany was most often to have students learn a particular skill, while the goal in Japanese classrooms was more often to help students develop deep understandings of mathematics (see NCES 1997c). These differences in goals translated into differences in other aspects of instruction. For example, 71 percent of Japanese teachers provide learning activities that require high-level thinking and reasoning. In comparison, only 29 percent of German teachers and 24 percent of U.S. teachers engaged students in this kind of learning (NCES 1997c).

On the basis of a videotaped sample of eighth grade mathematics classrooms in the three countries, judges rated most lessons from U.S. classrooms to be of low quality (87 percent), compared with 40 percent of lessons from German classrooms and just 13 percent of Japanese lessons. These judgments were made independently of detailed summaries that documented the exact sequence of mathematical statements and equations presented and the learning activities used. Any words that provided clues to the identity of the country were disguised.

None of the lessons from U.S. mathematics classrooms were rated high on quality, compared with 30 percent of lessons from Japanese classrooms and 23 percent from German classrooms. Moreover, most of the expert judges viewed lessons in Japanese classrooms as more consistent with U.S. mathematics standards than lessons in U.S. classrooms. However, 75 percent of the U.S. teachers of those same lessons judged their own instruction to be in "some accord" with the standards.

Aside from the issue of instructional quality, there has been some empirical evidence to support the common-sense notion that the more time students spend engaged in learning, the more they will learn. This is the primary reason why time is considered an important instructional variable. It is considered so crucial, in fact, that many educators believe systemic change cannot be successful in schools unless ways are found to provide students with more learning time (National Education Commission on Time and Learning 1994). Still, questions remain about just how much influence instructional time has on achievement.

Through the recent international comparative study, it has become clear that, at the very least, the relationship is not as simple as has been assumed. In fact, no consistent relationship was observed between class time and achievement in either
subject at either fourth or eighth grade.**[13]** This finding suggests that how teachers and students spend their instructional time is more important than the amount of time available for mathematics and
science instruction during the school day. For example, eighth grade students in Belgium, the Czech Republic, and the Slovak Republic-all high-performing nations-reported spending more time than the average on mathematics. But so too did students in
Kuwait, who were among the lowest scorers. South Korean and Japanese eighth graders reported spending the international average amount of class time on mathematics but were among the highest achievers.

U.S. students spend at least as much class time on mathematics and science as students in most countries. At eighth grade, over half of U.S. students spend 31/2 to 5 classroom hours on mathematics each week compared with an international norm of 2 to
31/2 hours (Beaton, Mullis et al. 1996; and Beaton, Martin et al. 1996).**[14]** Almost half of fourth grade U.S. students spend five or more
hours of instructional time each week on mathematics and three hours or more on science. In most other countries, fourth graders spend about three to four hours on mathematics and two hours on science (see Martin et al.
1997 and Mullis et al. 1997).**[15]**

Although learning time can be extended through homework and study before or after the school day, no consistent relationship has been found between international achievement and the amount of time students reported spending on homework. In some
high-achieving countries such as Hungary, Singapore, and Slovenia, students spend considerably more time than the norm on homework. However, students in low-achieving countries such as Iran and Kuwait also reported considerable time on homework. In
Denmark, Scotland, and the Netherlands-which are middle- to high-achieving countries-one-quarter to one-half of the students reported spending no time at all on homework on a normal day.**[16]**

Students in most countries reported spending an hour of nonschool time on mathematics on a normal day and a half-hour to an hour on science. U.S. students averaged
48 minutes to one hour on mathematics homework and between 36 and 48 minutes on science, depending on grade level (Beaton, Mullis et al. 1996; Beaton, Martin et al. 1996; Martin et al. 1996; and Mullis et al. 1996).**[17]** (See appendix table 1-17.)

Homework competes with extracurricular activities for students' attention, and television often turns out to be the prime competitor. In most countries, eighth grade students spend two to three hours a day watching television. (See figure 1-16) The habit of U.S. students are consistent with these patterns: eighth graders reported spending 2.6 hours watching television, compared with 2.3 hours doing their school homework or studying. Not only was this within the international norm, but it was virtually identical to patterns exhibited by Japan and Hong Kong, two of the top-scoring economies. Students in other high-scoring countries such as Singapore and Belgium spent somewhat more time studying than watching television; however, students in the Czech Republic spent more time watching television than studying.

The relationship of achievement to time spent viewing television is more consistent than the relationship between achievement and time spent on homework-but it turns out to be a curvilinear relationship. Students who watched one to two hours of television were the highest achievers in most countries. Students who watched more than two hours of television or less than one hour had lower mathematics and science achievement on average. More significant perhaps was the finding that eighth grade students who watched television for five or more hours each day, and fourth grade students who watched TV for four or more hours, were the lowest achievers in all participating countries. The United States had a fair number of students who spent this much time watching television-17 percent of fourth grade students and 13 percent of eighth grade students (Beaton, Mullis et al. 1996; Beaton, Martin et al. 1996; Martin et al. 1997; and Mullis et al. 1997).

Educational standards in both mathematics and science acknowledge the potential benefits of technology and recommend that students have regular access to computers and other tools such as calculators. Although there are studies of individual schools or districts where the use of computers and access to the Internet have yielded learning gains, there are no national data that affirm that the presence of technology in itself is spurring achievement gains in mathematics and science nationwide. It is probably often the case that information technologies, when available, are not being used effectively in the classroom; nor does it seem from empirical analysis that educators have yet understood how to integrate technology into programs of reform on a wide scale.

By 1994, more than half of U.S. middle and high school students reported access to computers in school for mathematics instruction; of that number, about 62 to 70 percent actually used the computers to solve mathematics problems. This represents a large increase from 1978 when only 56 percent of 13-year-olds and 46 percent of 17-year-olds used computers for problem solving during instruction. (See text table 1-2.)

Teacher responses from recent international comparisons paint a slightly more limited picture of computer use for mathematics instruction. When asked about the use of computers in mathematics instruction, three-quarters of U.S. teachers at the eighth
grade level reported that students never or hardly ever solve mathematics problems using a computer. Sixty percent of U.S. fourth grade teachers reported that students never or hardly ever use the computers in solving mathematics problems.**[18]** However, mathematics teachers reported frequent instructional use of calculators. More than half of eighth grade mathematics teachers in the United States reported that students in their classes use
calculators for basic tasks such as checking answers and performing routine computations. More than half also reported having their students use calculators to solve complex problems and more than one-third to explore number concepts (Williams et al. 1997). (See appendix table 1-23.)

Across the world, computers are used quite rarely for mathematics and science instruction. Except in Denmark, England and Wales, and Slovenia, less than one-fifth of eighth grade students used computers for problem solving in science. And except in the United States, Austria, Denmark, England and Wales, and Sweden, less than one-third of fourth grade students used computers at least some of the time according to teachers' reports. (See appendix table 1-16.)

Limited availability of computers at school can be offset by access to computers at home, even though home computers are often used for other than academic purposes. During the 1994/95 school year, about half of U.S. students had a computer at home. Students in England and Wales, Iceland, Ireland, the Netherlands, and Scotland were most likely to own computers (about 75 percent); students in Colombia, Iran, Latvia, Romania, and Thailand were least likely (less than 20 percent). (See text table 1-3.)

The vision of tomorrow's classroom held by many educational reformers not only includes access to computers by students and teachers but also widespread access to the Internet. Although most U.S. schools are quite far from this vision, Internet
access in schools has increased substantially in the last several years. A recent survey indicated that in fall 1996, 65 percent of public schools reported access to the Internet-a gain of 30 percentage points over 1994 figures. Internet access was
more likely in secondary than in elementary schools (three-quarters versus under two-thirds); in more affluent than less affluent schools (78 percent versus 53 to 58 percent); and in schools with low to moderate minority enrollments, as compared with
schools with high minority enrollments (65 to 72 percent versus 56 percent). (See appendix table 1-25.) As with computers, access to the Internet does not always translate into use by students and teachers,
nor does it ensure effective use. Although close to two-thirds of U.S. schools could connect to the Internet, access was possible from only 14 percent of instructional rooms (e.g., classrooms, computer labs, library media centers) according to recent
surveys (NCES 1997a). (See figure 1-17.)

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