Conference Material

There is no one at the helm of mathematics and science education in the U.S.; in truth, there is no identifiable helm. No single coherent vision of how to educate today's children dominates U.S. educational practice in either subject, nor is there a single, commonly accepted place to turn to for such visions. Our visions to the extent that they exist at all are multiple.

These splintered visions produce unfocused curricula and textbooks that fail to define clearly what is intended to be taught. They influence teachers to implement diffuse learning goals intheir classrooms. They emphasize familiarity with many topics rather than concentrated attention to a few. And they likely lower the academic performance of students who spend years in such a learning environment. Our curricula, textbooks, and teaching all are "a mile wide and an inch deep."

This preoccupation with breadth rather than depth, with quantity rather than quality, probably affects how well U.S. students perform in relation to their counterparts in other countries. It thus determines who are our international "peers" and raises the question of whether these are the peers that we want to have. In today's technologically oriented global society, where knowledge of mathematics and science is important for workers, citizens, and individuals alike, an important question is: What can be done to bring about a more coherent vision and thereby improve mathematics and science education?

Reforms have already been proposed by political, business, educational and other leaders. Extensive efforts are underway to implement these standards, but the implementation process itself is shaped by the prevailing culture of inclusion. Like the developers of curricula and the publishers of textbooks, teachers add reform ideas to their pedagogical quivers without asking what should be taken away.

The study summarized below represents an effort to describe the nature of the diffuse vision of mathematics and science education in the U.S. and to raise questions relevant to policy making.

Purpose of A Splintered Vision

A Splintered Vision (written by William Schmidt, Curtis McKnight and Senta Raizen of the U.S. National Research Center for the Third International Mathematics and Science Study and published by Kluwer Academic Publishers) discusses data from the analysis of 491 curriculum guides and 628 textbooks from around the world as part of the recently completed Third International Mathematics and Science Study (TIMSS). It also presents detailed accompanying data on teacher practices in the U.S. and two other countries: Germany and Japan.

The TIMSS is a large-scale, cross-national comparative study of the national educational systems and their outputs in about 50 countries. Researchers examined mathematics and the sciences curricula, instructional practices, and school and social factors, as well as conducting achievement testing of students. They collected data from representative documents that laid out official curricular intentions and plans, analyzed entire mathematics and science textbooks, and searched entire K-12 textbook series for selected "in-depth" topics (subareas within the subject matter.) In six countries TIMSS conducted classroom observations, teacher interviewing, and videotaping.

The TIMSS curriculum and teacher data are extensive and cannot be explored in a single report. The results of analyses of these data are being reported in a series of volumes, three of which are now available. 1

The present report intends to document and characterize the state of U.S. mathematics and science curricula, textbooks, and teaching practices and place them in a cross-national context. Unfortunately, this study could only provide a snapshot of the "moving target" that is educational practice in the U.S. These data were collected in 1992-93, when the mathematics standards hadonly existed for three years and the science standards were not finalized. 2 The intervening years have been a time of change for state curriculum standards and textbooks. The TIMSS data on teacher practices discussed here were collected in 1995.

This report is meant to be descriptive and, to a lesser extent, interpretive. It is not a plea for specific reforms. We seek to provide data germane to the ongoing public debate about science and mathematics education policies in the U.S.

Unfocused Curricula

Curricula in both mathematics and science in U.S. schools are unfocused in comparison with those in other countries studied. The lack of curricular focus is more true in mathematics than in science, though physical science guides closely resemble mathematics in their fragmentation. U.S. curricula are unfocused in several respects:

• Topics Covered

  Mathematics curricula in the U.S. consistently cover far more topics than is typical in other countries. The number of mathematics topics in the U.S. composite 3 is higher than the 75th percentile internationally in all grades until ninth, when schools typically teach specific courses such as algebra, geometry, etc. In science, the tendency toward inclusion is similar, though less pronounced. The number of science topics in the U.S. composite exceeds the 50th percentile internationally in all but one grade until the tenth, when schools tend to abandon general science approaches for specific courses, such as chemistry and physics.

• Repetition

  In both mathematics and science, topics remained in our composite U.S. curricula for more grades than all but a few other TIMSS countries. The U.S. approach can be characterized as "come early and stay late." In mathematics, the U.S. practice is to add far more topics than other countries do in grades one and two and then repeat these topics until grade seven. In grades nine and 11 the U.S. composite curriculum drops many more topics than other countries. On average, mathematical topics remain in the U. S. composite curriculum for two years longer than the international median. Only five countries have higher average durations. In science, U.S. practice is to introduce new science topics at intervals, especially grades one and five, with little change in the intervening grades. In grades 10 to 12 the U.S. composite curriculum drops many more topics than other countries. Average intended duration is close to the international median in earth sciences and life sciences, but the U.S. average duration in the physical sciences is two years longer than the median and higher than all but seven countries. In mathematics, the tendency to retain topics over many grades may reflect the traditional approach of distributed mastery the idea that mastering pieces of a subject would lead to mastery of a bigger whole. U.S. curricula lack a strategic concept of focusing on a few key goals, linking content together, and setting higher demands on students. This propensity for inclusion extends even to reform proposals. Many reform recommendations simply add to the existing topics (or are implemented by adding to existing content), thereby exacerbating the existing lack of curricular focus.

• Emphasis

  U.S. curricula in mathematics and science seek to do something of everything and less of any one thing. Given roughly comparable amounts of instructional time, this topic diversity limits the average amount of time allocated to any one topic. In mathematics, this accumulation may be a product of our model of distributed mastery over the grades. The reasons for the better results in science are less clear but seem related to general science approaches that move from topic to topic.

• Variations Among States

  While the core of mathematics topics was broad, it varied little among the states. The number of core science topics was much smaller, and the overlap among state curricula was also small. While students in U.S. states might have studied a number of science topics roughly equal to the international median, the differing curricular intentions of various states are such that students in different states likely studied only a few common topics.

• Defining the "Basics"

  Student achievement in mathematics and science in any country is largely shaped by what educational policy makers in that nation regard as "basic" in these subjects and how well they communicate and support those basics. The U.S. mathematics instructional practices defined de facto eighth grade basics of arithmetic, fractions and a relatively small amount of algebra. In Germany, Japan, and internationally, the basics were defined as algebra and geometry. For science, the picture is more complex since U.S. curricula include single area courses, such as physical sciences, life sciences, or earth sciences. These courses defined a more restricted, focused set of basics, but they applied only to the subset of students receiving those particular courses.

Textbooks play an important role in making the leap from intentions and plans to classroom activities. They make content available, organize it and set out learning tasks in a form designed to be appealing to students. Without restricting what teachers may choose to do, textbooks drastically affect what U.S. teachers are likely to do under the pressure of daily instruction. The question thus arises: Do U.S. mathematics and science textbooks add guidance and focus to help teachers cope with unfocused curricula? Unfortunately, the answer is "no." The splintered character of mathematics and science curricula in the U.S. is mirrored in the textbooks used by teachers and students. American textbooks are unfocused in several ways:

• Topics Included

  The U.S. mathematics and science textbooks include far more topics than was typical internationally at all three grade levels analyzed. In mathematics, U.S. textbooks are far above the 75th percentile of the TIMSS countries in the number of topics covered. For example, U.S. mathematics textbooks designed for fourth and eighth graders cover an average of 30 to 35 topics, while those in Germany and Japan average 20 and 10 respectively for these populations. As a result, typical mathematics textbooks in the U.S. look quite different than those of a nation such as Japan. The typical eighth grade U.S. textbook (non-algebra) is larger and more comprehensive than the average Japanese text, but it contains fewer sequences of extended attention to a particularly important topic. The U.S. textbooks' sequences are also shorter and have more breaks. The lesson by lesson organization of the U.S. book is much less focused than the Japanese book, and there is far more skipping among topics in successive segments. In science, the differences are even greater. At all three population levels, U.S. science textbooks included far more topics than even the 75th percentile internationally. The average U.S. science textbook at the fourth, eighth, and 12th grade has between 50 and 65 topics; by contrast Japan has five to 15 and Germany has just seven topics in its eighth grade science textbooks.

• Emphasis

  The propensity of U.S. curricula to do something of everything but little of any one thing is mirrored in textbooks. The few most emphasized topics account for less content than is the case internationally. Among the fourth grade mathematics textbooks investigated, the five topics receiving the most textbook space accounted on average for about 60 percent of space in the U.S. textbooks but over 85 percent of textbook space internationally. At the eighth grade level, the five most emphasized topics in non-algebra U.S. textbooks accounted for less than 50 percent of textbook space compared to an international average of about 75 percent. An exception are U.S. eighth grade algebra books which were highly focused, the five most emphasized topics accounting for 100 percent of the books. Among the U.S. fourth grade science textbooks investigated, the five topics receiving the most attention accounted for an average of just over 25 percent of total space in U.S. textbooks compared to an average of 70 to 75 percent internationally. Among the U.S. eighth grade science textbooks investigated, the five most emphasized topics in more general science texts accounted for about 50 percent of textbook space compared to an international average of about 60 percent. In contrast, U.S. eighth grade science books oriented to a single area were highly focused, with the five most emphasized topics accounting for more of the textbooks than was true in the international average.

• Difficulty

  U.S. eighth grade science textbooks emphasized understanding and using routine procedures, which represent the less complex, more easily taught expectations for student performance. This emphasis was typical of what was done internationally. It is not, however, typical of the diverse and more demanding performances called for in current U.S. science education reform documents. Most U.S. schools and teachers make selective use of textbook contents and rarely, if ever, cover all of a textbook's content. Publishers can reasonably expect that those who adopt and buy a particular textbook will feel free to use only the contents that suit their purposes. Unfortunately, the result is large textbooks covering many topics but comparatively shallowly. Even in the largest textbooks, space is still limited. It is impossible for textbooks so inclusive to help compensate for unfocused official curricula. Thus, our analysis shows that U.S. textbooks support and extend the lack of focus seen in those official curricula.

Teachers in the U.S. are sent into their classrooms with a mandate to implement inclusive, fragmented curricula and armed with textbooks that embody the same "breadth rather than depth" approach. How do they handle such a situation? Not surprisingly, the instructional decisions made by U.S. teachers mirror the inclusive approach of the tools they are given. U.S. teachers handle the splintered vision they get in several ways:

• Topics Covered

  U.S. mathematics and science teachers typically report teaching more topics than their counterparts in other countries, including Germany and Japan. This is true for science teachers even when using a single area textbook such as physical science, life science, or earth science.

• Emphasis

  Since instructional time for science or math within a school year is limited, the data show that teaching more topics means that teachers spend little time on most topics. U.S. eighth grade mathematics teachers indicated that they taught at least a few class periods on all but one topic area included in the teacher survey's questionnaire. They devoted 20 or more periods of in-depth instruction to only one topic area, fractions and decimals. However, in Germany and Japan many other topic areas received this more extensive coverage. According to the survey, the five topic areas covered most extensively by U.S. eighth grade mathematics teachers accounted for less than half of their year's instructional periods. In contrast, the five most extensively covered Japanese eighth grade topic areas accounted for almost 75 percent of the year's instructional periods. U.S. eighth grade science teachers also indicated that they would devote at least some class time to every topic area surveyed. None was omitted completely and no topic was marked to receive more than 13 class periods of attention by eighth grade physical and general science teachers. Additional topic areas received more extensive coverage in Germany and Japan. On average U.S. eighth grade general science teachers' most extensively covered topics accounted for only about 40 percent of their instructional periods, but this percentage was also lower for science in Germany and Japan (about 50 to 60 percent).

• Number of Activities

  U.S. teachers engage in more teaching activities per lesson than their counterparts in other countries. More than 60 percent of U.S. eighth grade mathematics and science teachers reported using six or more activities in a typical class. In Germany only 25 percent reported using six or more activities, and even fewer reported doing so in Japan.

U.S. mathematics and science teachers work hard and often face demanding workplaces. Our data show that they are scheduled to work about 30 periods each week, which is more than teachers in Germany (just over 20 periods) and Japan (fewer than 20). These teachers rarely have the luxury of being idealists. Unfocused curricula and inclusive textbooks set few boundaries for instructional decisions and appear to require a little bit of everything. It is easier for real teachers making real decisions in the real workplaces of U.S. schools to settle for the first alternative that seems good enough rather than search for the best possible instruction. They try to cover as much as they can rather than teach just a little. In a word, they "satisfice" The data shows that U.S. mathematics and science teachers are aware of and believe in more effective, complex teaching styles than they practice. They often have information that would help them do their work more effectively. Their beliefs suggest that they might choose to organize instruction differently under circumstances less consumed by the need for coverage. Effective teachers should not be unusual, nor should effectiveness require extraordinary efforts and dedication by teachers. The reality, however, is that U.S. teachers are placed in situations in which they cannot do their best. We have yet to unleash the effectiveness of U.S. teachers. It seems likely that fundamental changes are needed in teacher knowledge, working conditions, curricula quality, student expectations, and textbook content.

What Can We Expect from U.S. Students?

In mathematics, we have a highly fragmented curriculum, textbooks that are a "mile wide and an inch deep," and teachers who cover many topics but none extensively. We make low demands on students and have a more limited conception of "the basics" than the international norm. It seems highly likely that U.S. student achievement in mathematics will be below international averages. Our science curriculum is less fragmented. Science achievement seems likely to be closer to international averages, but still not what we desire and certainly below some, if not most, of our economic peers. U.S. students' achievements the yield of our aggregate national education "system" in mathematics and the sciences are likely to be disappointing and many of the reasons are not under students' control. We must make substantial changes if we are to compete and to produce a quantitatively and scientifically literate workforce and citizenry.