A Splintered Vision:
An Investigation of
U.S. Science and Mathematics Education
EXECUTIVE SUMMARY
Report Authors
William H. Schmidt
Curtis C. McKnight
Senta A. Raizen
With the collaboration of
Pamela M. Jakwerth
Gilbert A. Valverde
Richard G. Wolfe
Edward D. Britton
Leonard J. Bianchi
Richard T. Houang
U.S. National Research Center for the Third International Mathematics and Science Study,
Michigan State University
Third International Mathematics and Science Study
THE SPLINTERED VISION: AN OVERVIEW
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.
Unfocused Textbooks
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.
How Teachers Deal with the Splintered Vision
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.
Is This The Best Our Teachers Can Do?
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.
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