Approaches
to Teaching Mathematics and Science
Textbooks
Curriculum
Instructional Practices
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:
 Understanding. Comprehending mathematics concepts, operations,
and relations, including mathematical symbols and diagrams.
 Computing. Carrying out mathematical procedures (such
as adding, subtracting, multiplying, and dividing numbers) flexibly,
accurately, efficiently, and appropriately.
 Applying. Being able to formulate problems mathematically
and devise strategies for solving them using concepts and procedures
appropriately.
 Reasoning. Using logic to explain and justify a solution
to a problem or extend from something known to something not yet
known.
 Engaging. Seeing mathematics as sensible, useful, and
doable when one works at it, and being willing to do the work.
Few national data exist linking curricular reforms to changes in
student achievement, although some state and local studies suggest
standardsbased 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 inquirybased
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 StohrHunt 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 handson 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 nonlifescience 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
standardsbased 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 TIMSSR, conducted 4 years later, the videotape
component was expanded to include seven countries and to cover science
as well as mathematics.
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, TIMSSR 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).
For example, about 95 percent of U.S. eighth graders had received
instruction on scientific inquiry before the TIMSSR 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.
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 twothirds
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 TIMSSR 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
111 ).
Only 17 percent of problems in Japanese lessons addressed lowcomplexity
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
112
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 TIMSSR 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
113 ).
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. TIMSSR 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
114 ).
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 studentdirected
investigations. Higher proportions of science students in the United
States than in TIMSSR 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 twothirds of reformoriented
teachers identified either realworld 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).
