Chapter 1: Elementary and Secondary Education

Curriculum and instructional methods influence what students learn and whether they can apply knowledge and skills to new problems or applications (Schmidt et al. 2001). This section summarizes data regarding methods of teaching mathematics and science in the United States. It presents findings about textbooks, curricular content, and aspects of teachers' instructional practices and provides international comparisons when available.

Approaches to Teaching Mathematics and Science

Proponents of different curricular emphases and teaching methods, particularly in mathematics, have argued in recent years over the effectiveness of various approaches. Some emphasize computational skills and number operations, and others stress mathematical understanding and reasoning skills (Reys 2001). NRC and others have concluded that students need to develop these and other skills so that they reinforce and complement one another (Kilpatrick and Swafford 2002 and NCTM 2000). Mathematics proficiency, according to NRC, consists of five essential components, or strands, that should be integrated to support effective learning. These strands are:

Few national data exist linking curricular reforms to changes in student achievement, although some state and local studies suggest standards-based curricula that integrate a range of skills with knowledge may lead to overall higher achievement and help reduce gaps between minority and white students (Briars 2001, Mullis et al. 2001, Riordan and Noyce 2001, Schneider et al. 2002, and Schoenfeld 2002). Some research also supports the potential effectiveness of inquiry-based instruction in science, in which students learn primarily by conducting experiments to test ideas and answer questions (Amaral, Garrison, and Klentschy 2002; Stoddart et al. 2002; and Stohr-Hunt 1996).

Textbooks

Textbook content can affect teaching and learning. Systematic expert ratings of how well textbooks address nationally recognized content and curriculum standards for mathematics and science have taken place, although the available research does not include rigorous studies that relate textbook content to student achievement.

Starting in 1999, AAAS Project 2061 assigned teams of mathematics and science professors and K–12 teachers to evaluate textbooks, teachers' guides, and related instructional materials in categories based on subject and grade level. Using selected criteria from Benchmarks for Science Literacy (AAAS 1993), reviewers in one Project 2061 evaluation (AAAS 1999b) measured how well middle school mathematics textbooks addressed 6 central mathematics concepts/skills and how well the textbooks incorporated 24 instructional criteria consistent with NCTM standards (NCTM 1989 and 2000). Project 2061 rated 4 of the 12 textbooks it evaluated as excellent but judged the remaining 8 to be inadequate overall and merely satisfactory in teaching number and geometry skills. At the time, those eight were among the most widely used middle school mathematics texts in the United States.

Project 2061 also conducted evaluations of algebra textbooks (AAAS 2000a), middle school science materials (AAAS 1999a), and high school biology textbooks (AAAS 2000b). Overall, reviewers judged most to have deficits in teaching students many thinking skills identified by standards documents; they also lacked some content identified in subject standards. Commonly found weaknesses included emphasizing detail and terminology at the expense of core concepts (a problem more prevalent in science materials), insufficiently developing students' reasoning abilities, and providing inadequate guidance for students and teachers to discover and correct misconceptions. Reviewers also identified several common positive attributes: most materials covered content thoroughly and accurately, provided a range of applications and hands-on activities, and used inviting graphics to illustrate ideas. Project 2061 noted that some newer texts showed improvement over older ones.

The American Institute of Biological Sciences (AIBS) assessed how well 10 high school biology textbooks and related materials (Morse 2001) adhered to standards embodied in NSES. Overall ratings ranged from just below adequate to slightly below excellent. In general, AIBS concluded that the materials conveyed life science content very well but were not as effective in providing guidance for teachers and in handling certain non-life-science content. Most instructional materials received high marks for accuracy, attractive illustrations and design, and inclusion of recent developments in biology research. However, AIBS found that most were crammed with too much information and detail, placing a great burden on teachers to select priorities and make links between content areas. In addition, AIBS concluded that most materials failed to fully capitalize on current understanding about how students learn and did not provide useful assessments for tracking and advancing learning.

Reviewers rated some recently developed curriculum materials as strong in areas that rarely receive positive ratings. For example, AIBS concluded that three recently developed instructional packages incorporated the pedagogical recommendations in NSES quite well. An earlier National Science Foundation evaluation of middle school science instructional materials (NSF 1997) also identified several packages that embodied useful standards-based reforms such as organizing content around conceptual themes, emphasizing important concepts in science, balancing breadth and depth of content coverage, and providing assessments tied to instructional goals.

International data indicate that U.S. textbooks tend to address more topics than those used in other countries and to devote less attention to the five most prominent topics. They fail to build more challenging material on simpler content introduced earlier and to make clear connections among content areas (Schmidt, McKnight, and Raizen 1997). As a result, reviewers have criticized U.S. texts as typically less focused and less coherent than those used in many other countries. The data indicate striking differences in textbook length: fourth grade mathematics textbooks in the United States in 1995 averaged 530 pages, more than three times as long as the international average in TIMSS (Valverde and Schmidt 1997). Similar differences in length were found in science textbooks. This greater length results from covering more topics rather than from covering individual topics more thoroughly.

Curriculum

In addition to testing students' learning, the 1995 TIMSS study collected information at the three age and grade levels about the curriculum intended by policymakers, the curriculum that teachers taught, methods of teaching, instructional materials, students' school experiences, and demographic characteristics. TIMSS also examined eighth grade mathematics class practices in the United States, Germany, and Japan through a classroom videotape study and teacher interviews. In TIMSS-R, conducted 4 years later, the videotape component was expanded to include seven countries and to cover science as well as mathematics.[8] Analyses show differences among countries in two important aspects of the mathematics and science curriculum: breadth of coverage and lesson difficulty.

Breadth of Coverage

Consistent with findings about textbooks, research indicates that mathematics and science curricula in the United States generally cover more content areas (NCES 2000a). In eighth grade science, TIMSS-R data showed U.S. students as more likely than the international average to study four of six main content areas: earth science, biology, physics, and scientific inquiry and the nature of science (NCES 2000a).[9] For example, about 95 percent of U.S. eighth graders had received instruction on scientific inquiry before the TIMSS-R assessment compared with an 80 percent international average. The rates for studying the other five topics ranged from 70 to 81 percent in the United States compared with international averages of 53 to 72 percent. (The proportions of U.S. students who studied chemistry and environmental resource issues were comparable to the international average.)

Similarly, eighth grade mathematics classes covered many topics. Higher percentages of U.S. students received instruction in four of the five mathematics content areas in 1999: fractions and number sense; algebra; data representation, analysis, and probability; and measurement. The vast majority of U.S. students had studied these topics by the end of grade 8 (ranging from 91 to 99 percent). Only in geometry did no significant difference exist: 58 percent of eighth graders in the United States had studied that topic compared with 65 percent in other countries (NCES 2000b).

Curriculum in the United States, as observed from curriculum frameworks for both mathematics and science, repeats content across more grades than does curriculum in other countries.[10] In eighth grade mathematics, for example, U.S. curricula often continue to cover topics that no longer appear in the curricula of other nations such as number operations, fractions, percentages, and estimation (Schmidt et al. 2001; Schmidt, McKnight, and Raizen 1997; and Stevenson 1998). U.S. curriculum frameworks generally failed to build more complex content on simpler but related content covered earlier.

In addition, U.S. teachers in 1995 spent significantly less time than German or Japanese teachers on the most emphasized topics (Schmidt, McKnight, and Raizen 1997). U.S. eighth grade mathematics teachers covered 16 to 18 topics during the year with only a single topic receiving more than 8 percent of available teaching time. In Japan, teachers focused extensively on only four topics, allocating two-thirds of total classroom time to these topics (Wilson and Blank 1999). These patterns found in TIMSS reflect findings from the Second International Mathematics and Science Study in the early 1980s (McKnight et al. 1987) and suggest a structural feature of some durability in U.S. elementary and secondary education.

Lesson Difficulty

For the 1999 TIMSS-R mathematics video study, researchers developed a measure of lesson difficulty, procedural complexity, based on the number of steps needed to solve a problem using common methods. The measure is thus independent of a student's prior knowledge and skill (NCES 2003b). Japan stood apart from other participating nations in lesson complexity. In the United States and the other five countries, only 6 to 12 percent of problems had high complexity compared with 39 percent of problems used in Japanese lessons (figure 1-11

).[11] Only 17 percent of problems in Japanese lessons addressed low-complexity problems compared with 63 to 77 percent in the other six nations. U.S. mathematics lessons did not differ significantly from those in the other five nations in the proportion of problems that had high or low complexity.

Using other measures, the 1995 TIMSS classroom video study also revealed differences in lessons' degree of challenge. Mathematics professors were asked to assign a grade level to videotaped eighth grade mathematics classes: they rated U.S. lessons on average at the seventh grade level, German lessons at the end of eighth grade, and Japanese lessons at the beginning of ninth grade (NCES 1997b). In addition, professors evaluated lesson quality based on the percentage of lessons requiring deductive reasoning by students: 0 percent of lessons in the United States, 21 percent in Germany, and 62 percent in Japan required use of deductive reasoning (Schmidt, McKnight, and Raizen 1997). Deductive reasoning, such as that used to prove a theorem, is a higher order skill that experts recommend students practice and an important component of learning in mathematics, science, and other disciplines.

TIMSS data thus portrayed U.S. eighth grade mathematics classes as rarely emphasizing logic or involving students in logical reasoning. In 1995, in U.S. mathematics lessons, teachers stated the rule students should follow to solve problems for nearly 80 percent of topics rather than explaining the rule or having students work on the reasoning. In contrast, students and teachers developed solutions using logic (for example, proving or deriving the answer step by step) for more than 80 percent of topics covered in Japan and nearly 80 percent of topics covered in Germany (Stevenson 1998). German teachers usually proved rules for the class and Japanese teachers tended to give students the assignment of figuring out the solution's proof (NCES 1997b).

Analyzing the topics teachers prioritize provides another way to examine differences in difficulty. As figure 1-12

shows, U.S. eighth grade mathematics students in 1999 were twice as likely as the international average to be in classes where teachers placed the most emphasis on numbers and arithmetic (28 versus 14 percent), and they were three times as likely to be in classes where algebra received the most emphasis (27 versus 8 percent) (Mullis et al. 2001). In contrast, far higher percentages of other nations' eighth graders experienced a combined emphasis on algebra and geometry or on algebra, geometry, numbers, and other topics.

Instructional Practices

The 1999 TIMSS-R video study of mathematics classes in seven nations showed that in the United States teachers spent about half of total lesson time (53 percent) reviewing previously taught material, with the other half nearly equally divided between introducing and practicing new content (NCES 2003b) (figure 1-13

). In Japan teachers spent 60 percent of class time introducing new material, more than in any of the other six countries. Although most lessons in each nation included both review and new material, U.S. teachers presented proportionally many more lessons devoted entirely to reviewing old content than did teachers in Hong Kong or Japan, two economies with particularly high scores.

In 1999, U.S. eighth graders watched the teacher demonstrate how to solve mathematics problems more often than their international peers (NCES 2000b). Compared with the international average, U.S. students were more likely to work alone on mathematics worksheets or textbook problems and to use data from everyday life, but less likely to do projects in their mathematics classes. TIMSS-R also indicated that U.S. eighth grade mathematics students were more likely than the international average (54 versus 43 percent) to write equations to represent mathematical relationships in most, or every, lesson (figure 1-14

). However, no significant differences existed for several other learning activities: explaining their reasoning for an answer, representing or analyzing relationships using tables and graphs, working on problems with no obvious method of solution, and practicing computation (Mullis et al. 2001). Students in all countries quite often explained their reasoning (70 percent of all teachers reported this activity in most lessons compared with 72 percent in the United States) and practiced computational skills (73 percent overall compared with 66 percent in the United States).

Teachers' goals can influence how they teach material and the activities they emphasize. In 1995, eighth grade mathematics teachers in the United States were more likely than those in Japan or Germany to prioritize the goal of developing correct answers to problems. German and Japanese teachers made students' understanding of mathematical concepts the priority.

Science class practices in 1999 tended to emphasize student-directed investigations. Higher proportions of science students in the United States than in TIMSS-R countries overall said that they "pretty often or almost always" explained the reasoning behind an idea, worked on science projects, conducted experiments or investigations, and worked from worksheets or textbooks. On average, U.S. students watched teachers show them how to work through a science problem less often than did students in other countries (NCES 2000a). The frequency of other specific learning practices, including explaining observations, representing or analyzing relationships with tables and graphs, and working on problems with no obvious method of solution, did not significantly differ between the United States and the international average (NCES 2000a).

Although U.S. mathematics (and science) teachers report that they are familiar with and are implementing recent content and pedagogical reforms, detailed observation and analysis of mathematics classroom practice in 1995 suggest otherwise. TIMSS data indicate that Japanese eighth grade mathematics teachers were more likely than their U.S. counterparts to be practicing many of the reforms recommended by national organizations like NCTM (NCES 1997b). Teachers who report reforming their methods may be referring to aspects of practice that have little demonstrated effect on students' thinking. In one study, more than two-thirds of reform-oriented teachers identified either real-world applications or students working in groups as examples of reform practices, and only 19 percent identified activities involving problem solving or mathematical thinking (Hiebert and Stigler 2000).

Footnotes

[8] TIMSS-R, limited to eighth grade, collected data from teacher and student surveys on many topics mentioned for TIMSS, although many items were new or different.

[9] A topic counted as being taught if teachers reported that they spent more than five class periods on it during the current year or that students had studied it in a previous grade.

[10] Based on a sample of state and local curriculum frameworks because the United States lacks a national curriculum.

[11] Japan did not participate in the mathematics video study in 1999. Data reported here for Japan come from the 1995 video study. TIMSS collected data from the other six nations in 1999.