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2
Science and Mathematics Education
American public schools are populated by 40 million students
and 2 million teachers. The magnitude and diversity of American
schooling make it impossible to deliver a complete and exact picture
of how well the schools are doing: simplification and abstraction
are necessary and inevitable. In attempting to abstract from the
complexities of American schooling and to propose new indicators of
the quality of science and mathematics education, the committee has
taken on a task that requires a clear sense of definitions, methods,
and goals. This chapter first defines what we take to be the purposes
of science and mathematics education and then presents a conception
of schooling that has shaped our choice of indicators.
SCIENTIFIC AND MATHEMATICAL LITERACY
The recognition that societies are changing rapidly is wide-
spread witness such terms as information age, postindustrial so-
ciety, and gioloal economy that have come into common usage. These
changes are spurred by and give rise to the development of ever more
powerful technologies. Several writers who have pondered the impli-
cations of these changes have made the point that the accelerating
change that characterizes U.S. and other Western societies may well
require a higher degree of scientific and mathematical literacy than
15
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16
INDICATORS OF SCIENCE AND M24TTIEM:A TICS EDUCATION
ever before (Toffler, 1980; Naisbitt, 1982; Zarinnia and Romberg,
1986~.
In general, the committee agrees with this view: we believe
that all students should have the tools of knowledge and judgment
that science and mathematics provide. These tools will enable them
to prepare for careers and cope with rapid changes in tomorrow's
labor markets, make informed choices in their private and family
lives, and understand issues of broader scope. And all students, as
they become adults, should be free to enjoy the enlargement of the
world that comes with scientific and mathematical literacy. Clearly,
then, not only should students leave school literate in science and
mathematics, but they should also have acquired the mental tools
with which they can renew that literacy throughout their lives.
Any useful attempt to determine to what extent schools are
meeting this broad goal must reflect an informed view of what con-
stitutes literacy in science and mathematics. The committee suggests
that there are several dimensions of scientific and mathematical lit-
eracy, each of which needs to be addressed seriously. The following
descriptions of these dimensions are intended to be normative: they
are the ideals against which science and mathematics curricula and
instruction should be compared. Because the sciences and math-
ematics, despite their many interconnections, are quite differently
constituted, literacy in each of these domains is discussed separately.
Literacy in Science'
The dimensions of scientific literacy that should be integral to
any educational program include the nature of the scientific world
view, the nature of the scientific enterprise, scientific habits of mind,
and the role of science in human affairs. The first three of these
dimensions deal with knowledge of science and intellectual skills; the
fourth deals with the relation between science and society. Each is
characterized in somewhat greater detail below.
The Nature of the Scientific World View
Over the last three
centuries or so, scientific activity in many fields has resulted in a set
of interconnected and testable notions about the nature of the world
*We thank F. James Rutherford, American Association for the Advance-
ment of Science, for suggesting the four dimensions of scientific literacy discussed
in this section.
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SCIENCE AND MATHEMATICS EDUCATION
17
and its parts. While the details continue to change with time, these
notions amount to a rather robust and useful construct of ideas that
deserves the name scientific world view. The ideas that constitute
its elements have various levels of complexity or abstraction.
. At the highest level of abstraction, there are grand concep-
tual schemes that bring together and bring order to large numbers
of observations, concepts, and theories, each of which is of lesser
generality and applies only to some narrower field of science. Some
of these grand schemes can be expressed in words or numbers; oth-
ers can only be understood mathematically. Examples include the
Newtonian universe, organic evolution, and plate tectonics.
~ At a more operational level, the scientific world view finds
expression in theories and mathematical models that organize facts
and laws in ways that help one understand a particular aspect of the
world. Examples include gravitation, solid-state science, statistics,
weather systems, and economic determinism.
~ Particular concepts, mathematical tools, or techniques can
reappear in various scientific specialties, thereby not only helping to
suggest new advances but also providing syntheses among different
parts of the total scientific world view. Examples of such recurring
concepts or tools include scale, cycles, waves, estimation, energy,
antibodies, and probability.
The scientific world view also consists of general beliefs that have
shown their worth over time. These include the following notions:
. The world of phenomena is rationally understandable and not
capricious; causal relations may be found.
. Good scientific theories permit deductions that can be
checked against experience. This testing frequently takes the form of
comparing measurements of the real world to numerical predictions.
. Individual data and observations are subject to some uncer-
tainty, but phenomena are consistent.
~ Throughout the history of scientific growth, and despite ma-
jor advances along the way, scientists have often found a few fun-
damental thematic notions useful and motivating: for example, the
search for unity or unification among diverse phenomena; the use of
mechanical or mathematical models; and such notions as simplicity,
parsimony, symmetry, evolution, causality, order or hierarchy, and
continuity or discontinuity.
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18 INDICATORS OF SCIENCE AND MATHEMATICS EDUCATION
The Nature of the Scientific Enterprise The scientific enterprise
subscribes to a set of value commitments in principle an ethos
of science that can be explicitly formulated. For individuals to
understand the conclusions that scientific research yields concerning
the natural world, they must understand these principles and some
of the characteristics of the way science works:
~ Science is both theoretical and empirical, and these two
aspects reinforce each other. Mathematical models without data
are sterile; observations without measurement of some kind remain
largely impressionistic.
. Science is not only a personal, individual calling but also a
social activity carried out by individuals who collaborate over time
and space. Some of this accessibility is a consequence of the univer-
sality of mathematics. This characteristic makes scientific activity,
next to mathematics, one of the most international and shareable
experiences and opens scientific research and teaching to all talents
everywhere.
.
The substance of science at any given time is found in the
consensus among scientists, as reflected largely in current writings,
data bases, and mathematical formulations. As scientific knowledge
becomes more developed and inclusive, findings reached at any par-
ticular time are seen as tentative. As mathematical models attain
greater generality, they provide novel tests of the current formula-
tions. Thus, science is an enterprise concerned with discovery of the
new and with testing (and correction when needed) of the old.
. Necessary conditions for understanding the processes of sci-
ence include familiarity with a wide range of natural phenomena;
asking questions and forming hypotheses; understanding the need
for tests or controlled comparisons; embracing theories of measure-
ment, evidence, and data; and accepting a method of notation or
formalism, most often mathematical, that allows an unambiguous
and replicable depiction of a set of phenomena.
~ Pure scientific research often points the way to practical ap-
plications through engineering development, and the latter in turn
can help make basic experimental science much more effective.
Scientific Habits of Mind Individuals, scientists or not, need
to be capable of analytical thinking in the context of the various
sciences and mathematics. This capability can be taken to include:
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SCIENCE AND MA THEMA TICS ED UCA TION
.
19
Identifying questions or formulating hypotheses relative to a
problem, recognizing when such questions ought to be quantitative,
and being able to express them mathematically if that is appropriate.
. Identifying and seeking out information (numerical or other-
wise) relevant to a problem or issue.
~ Using that information to test the hypothesis or answer the
question, while appreciating the limits placed by samples or intrinsic
uncertainty.
. Playing with information, in the sense of solving puzzles and
raising hypotheses.
. Offering arguments and counterarguments that can be tested
by reference to data or accepted principles.
. Communicating, collaborating, and building consensus in or-
der to develop a common language and common models.
The historic study of most scientific advances shows, however,
that other, more individual, and even aesthetic, elements enter to
assist the purely logical-critical faculty during the creative phase of
scientific work. There is, in short, no single "scientific method."
Moreover, accepting the scientific world view does not disqualify an
individual from sensitivity to, or the appreciation of, artistic and
humanistic achievements. In fact, one of the values of science is
cultural. Participation in science can be satisfying in much the same
way as participation in music and art.
Science and Human Affairs Science and mathematics are im-
portant to society because of their deep connections with and ef-
fects on human events and ideas. For example, mathematical con-
cepts such as estimation, statistics, sampling, and risk underlie most
public-policy concerns. The application of science and mathematics
to such matters as health, industrial processes, agriculture, and the
environment engage a large number of policy makers and individual
citizens. From this it follows that, for a society to be scientifically
literate, people need to understand some of the relationships between
science, policy, and society.
. Tensions exist between science and society, because science
must sometimes assert the presence of uncertainty and ambiguity
when facts are needed by organizations that must make policy deci-
sions or individuals who must make personal ones.
. Scientists may behave differently when they are involved in
public policy decisions than when they are acting as researchers. The
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INDICATORS OF SCIENCE AND MATHEMATICS EDUCATION
ethics of science in public policy calls for objectivity, but this cannot
always be expected in cases in which the self-interest of science itself
is at stake.
~ Science and technology have helped to better the human con-
dition. But progress in science and technology can also have unan-
ticipated negative effects on society. These effects can be monitored
and in some instances modified by the action of alert citizens, pro-
vided they have the necessary educational background to obtain and
interpret information on the ways in which science and technology
influence personal, local, and national affairs.
A historical sense of the way science grows can be useful. One
cannot always predict what science will be of practical value in the
future, because several seemingly unrelated lines of basic research
may come together over time (sometimes decades) and contribute
to breakthroughs. For example, cell culture was initiated to permit
studies of development and other cellular behaviors; it subsequently
made possible such medical advances as the polio vaccine. Much
mathematics originally developed for its own beauty or interest has
later provided tools for scientific or technological applications that
were inconceivable earlier.
Literacy in Mathematics ~
The types of mathematical literacy practical arithmetic, civic
application, professional use of mathematics, and cultural apprecia-
tion-correspond roughly to the central objectives of the four hier-
archical tiers in the educational system: primary, secondary, under-
graduate, and graduate. Although it is useful to postulate levels of
mathematical literacy corresponding to levels of education, some key
elements are integral to all levels:
.
Understanding the fundamental ideas of mathematics. For
example, the Pythagorean theorem has theoretical and practical
importance in all levels of mathematics learning, as does the notion
of symmetry. These (and other) intrinsic mathematical ideas could
provide benchmarks of literacy that transcend educational levels.
. Understanding the role of mathematics as the language of
science and its role in describing the nature of complex systems.
*We thank Lynn Arthur Steen, St. Olaf College, for formulating the four
levels of mathematical literacy discussed in this section.
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SCIENCE AND MATNEMA TICS EDUCATION
21
Understanding that order can beget disorder (as in turbulence) and
vice versa (as in statistical experiments); that mathematical mod-
els for growth can represent phenomena in biology, economics, and
chemistry; and that mathematics is still being created to meet new
needs are examples of perceptions about the nature of mathematics
that should be part of mathematical literacy at every level.
.
Recognizing that mathematics is a dynamic and changing
field, not, as it is generally taught, a static and bounded disci-
pline reflecting recorded knowledge (Confrey, 1985~. Three current
trends have deep implications for what it means to be literate in
mathematics (Hilton, 1986~: the increasing variety of applications in
many other fields, which need to be recognized and understood at
some level by nonmathematicians; a new unification of mathematics,
which calls for breaking down artificial barriers between topics in a
student's education; and the changes that the computer is bringing
about in mathematics (the relative importance of topics, how some
mathematics is done, and the creation of new topics), which need to
infuse mathematical knowledge and understanding at all levels.
Practical Literacy in Mathematics Practical literacy is knowI-
edge that can be put to immediate use in improving basic living
standards. The ability to compare loans, to figure unit prices, to
manipulate household measurements, and to estimate the erects of
various rates of inflation brings immediate real benefit. This kind of
applied arithmetic is one objective of universal primary education.
Civic Literacy in Mathematics Civic literacy involves more so-
phisticated concepts, which enhance public understanding of leg-
isTative issues. Major public debates on nuclear deterrence and nu-
clearpower, economic policy, public health, and the use of resources
frequently center on scientific issues. Inferences drawn from data,
projections concerning future behavior, and interactions among vari-
ables in complex systems involve issues with essentially mathematical
content. A public afraid or unable to reason with figures is unable to
discriminate between rational and reckless claims in the technological
arena. Ideally, secondary education should provide all students with
the mathematical knowledge and understanding needed by today's
"enlightened citizenry" that Thomas Peterson called the only proper
foundation for democracy.
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22 INDICATORS OF SCIENCE AND MATHEMATICS EDUCATION
Using Mathematics as a Too! Literacy that involves using math-
ematics as a too! encompasses the mathematics necessary to study
and work in science, engineeering, and other fields that employ
mathematical language, ideas, and models. It refers to all uses
of mathematics whether in theoretical physics or business man-
agement. As science and industry come to depend increasingly on
mathematical tools, professionals in ever more diverse fields will need
to learn this universal language. The basis for the mathematics that
constitutes use-related literacy must be laid at the secondary school
level, even though these tools are greatly extended and enhanced in
college mathematics courses.
Cultural Literacy in Mathematics
Cultural literacy in mathe-
matics, the most sophisticated of these levels, pertains to the role
of mathematics as a major intellectual achievement. Because cul-
tural literacy lacks an immediate, practical purpose, its appeal may
be limited. Yet the simpler and historically earlier parts of mathe-
matical invention, like the invention of zero or of negative numbers,
are accessible to many people, including quite young students. At
this level of difficulty, an appreciation of mathematics as an intel-
lectual activity engaged in by one's fellows should be part of any
concept of mathematical literacy. As one progresses through the
more complex developments in mathematics, however, the size of the
interested audience may decrease, to an audience perhaps something
like the readership of Scientific American. Pursuing cultural literacy
in mathematics to the more advanced stages enables one to appreci-
ate the seemingly arcane research of twentieth-century mathematics
not only for its potential and unknown practical application but also,
and more important, as an invaluable and profound contribution to
the heritage of human culture. For the most part, individuals at-
tain this sort of literacy through intensive study in some advanced
subject, not necessarily mathematics itself.
A CONCEPTION OF SCHOOLING
How do schools produce the learning that is entailed in scientific
and mathematical literacy? A central principle that guides this re-
port is that teachers and students are the most important resources in
the educational process and that their behaviors determine schooling
outcomes. A second, related principle is that incentives and con-
straints influence the behavior of students and teachers. A third
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SCIENCE AND MATNEMA TICS EDUCATION
23
principle concerns the question "Excellence for whom?" The com-
mittee is concerned not only with the achievement levels of the most
able students, but also with the distribution of knowledge and skills
among students from different backgrounds. We expand our concep-
tion of these three principles in the sections that follow.
Schooling as the Behavior of Students and Teachers
The committee's formulation of indicators is based on the view
that what students and teachers do determines how much learning
takes place. This principle may seem obvious and not worth empha-
sizing. To appreciate its significance, it is useful to review how it
evolved from earlier work on the determinants of children's academic
achievement. Such work comes from several different disciplinary
approaches psychology, sociology, and economics.
In educational psychology, there is a long history of research
on how students learn and how teachers teach. Research on learn-
ing dates back to the behaviorist theories of Thorndike (1932) and
Skinner (1953, 1968), was followed by theories that emphasized the
interaction of the student with the structure of the subject matter
(Brownell, 1947; Piaget, 1954; Bruner, 1960, 1966; Gagne, 1965;
Ausubel, 1968; Dienes and Golding, 1971), and is currently devel-
oping into theories of how children actively construct knowledge for
themselves through their interaction with the environment, including
the formal and informal teaching to which they are exposed (Resnick,
19873. Each of these theories has implications for the behavior of
teachers as they shape their instruction.
Sociology and anthropology also have contributed insights on the
effects of teachers' (and administrators') behavior as they set the con-
text for learning by the way classroom lessons are presented, children
are grouped within the classroom for instruction, and classrooms and
schools are organized. (For a review, see Committee on Research in
Mathematics, Science, and Technology Education, 1985:26-34.)
The 1960s saw the application of economics to the study of
education, sometimes referred to as the estimation of educational
production function models. The goals of this line of research, as
exemplified by the widely known report by Coleman et al. (1966) on
equality of educational opportunity, is to find schooling inputs that
are systematically related to student learning. Initially, this research
treated in parallel fashion such inputs as physical facilities, teaching
materials, and the attributes of teachers and students.
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24 INDICATORS OF SCIENCE AND MATHEMATICS EDUCATION
These different streams of research in education have provided a
great deal of knowledge about the kinds of variables that are impor-
tant in explaining student achievement, including the finding that
the most important resources in the educational process are human
beings, whose behavior influences what is learned in school. The
aspects of human behavior that influence students' achievement are
wide-ranging: they include the decisions of talented college graduates
about whether to become teachers and how long to stay in teach-
ing (SchIechty and Vance, 1983), the decisions of elementary school
teachers about how much time to allocate to mathematics and sci-
ence tWeiss, 1978), and the decisions of students about whether to
take science and mathematics courses (Welch et al., 1982; Bryk et
al., 1984) and how much homework to do or how much television to
watch (Walberg et al., 1986~.
Although the results of educational research studies regarding
the critical importance of the behavior of students and teachers have
been informative, it has been difficult to make linkages between these
results and policies to improve schooling. One reason is that, as these
very studies indicate, the resources most important in explaining chil-
dren's achievement are the human beings whose behavior influences
what is learned in school. And human behavior is not subject to easy
adjustment by managers and policy makers who wish to improve
learning. Policy makers can change the behavior of teachers and
students only to a limited degree.
Incentives and Constrairz~s
In emphasizing that the behavior of students and teachers is
difficult to alter, we do not mean to imply that it cannot be influ-
enced. In fact, a second principle underlying the recommendations
in this report is that the behavior of teachers and students is indeed
influenced by the incentives and constraints they face. Examples of
such incentives include teachers' salaries relative to those offered in
other professions, which may attract or dicourage talented individu-
als, and the quality of the mathematics and science courses available
in a school, which may increase or decrease student enrollment.
These two principles the importance of the behavior of teachers
and students and the responsiveness of the behavior of teachers and
students to the incentives and constraints they face have influenced
both the design of this report and our recommendations. They have
led us to recommend the collection of information on many aspects
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SCIENCE AND MATHEMATICS EDUCATION
25
of the behavior of teachers and students that influence the quality
of mathematics and science instruction and that ultimately influence
the level of science and mathematics literacy in the population. And
they have lect us to recommend the collection of information on many
incentives and constraints that influence the behavior of teachers and
students.
From this perspective, what is the importance of physical re-
sources devoted to mathematics and science instruction, such as
laboratories, teaching materials, and, most important, curriculum?
Don't they matter? Indeed they do. However, we believe that they
matter primarily through their influences on the behavior of teachers
and students. For example, the lack of adequate laboratory facilities
may make it difficult for a school to attract teachers who really want
to teach science and may force teachers who do teach science in that
school to base instruction on memorizing facts rather than on de-
veloping an understanding of scientific principles through hands-on
experiments. By the same token, the lack of facilities and the conse-
quent dullness of the instruction may lead students to avoid taking
science courses.
Our emphasis on looking at physical facilities and curriculum
from the perspective of examining how they influence the behavior
of teachers and students is not intended to downplay the importance
of these resources. The opposite is in fact the case. Some of the
early production function research concluded that physical facilities
do not matter, because the research was based on a design that im-
plicitly held constant which teachers worked in a school and which
courses students took. This research design eliminated some of the
most important mechanisms through which facilities do matter: by
influencing the quality of teachers who are attracted to the school
and the number of students who take science courses. (For a discus-
sion of research on the effects of instructional resources, see Carey,
1986.) Thus, our emphasis is intended to highlight the potential
importance of facilities in influencing the behavior of the key actors
in the educational process.
Similarly, understanding the effects of curriculum on student
learning is often clouded by the lack of distinction between the cur-
riculum laid out in state and school district manuals, what has been
called the mandated or intended curriculum, and the curriculum that
children actually experience, the de facto or actual curriculum. The
difference between the intended curriculum and the actual curricu-
lum stems from the decisions teachers make about what aspects of the
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26 INDICATORS OF SCIENCE AND MATHEA!A TICS EDUCATION
intended curriculum to emphasize and how to adapt the curriculum
(including the textbook) to accommodate their own skills and inter-
ests and their perceptions of their students' skills and interests. As a
result of these decisions by individual teachers about how to use the
intended curriculum, children in different classrooms and in different
schools experience different actual curricula and consequently learn
different things, even when they all attend schools using the same in-
tended curriculum. For this reason, the recommendations presented
in Chapter 7 on indicators of curriculum quality are sensitive to the
distinction between intended curricula and actual curricula.
Another implication of our perspective is that it is important to
pay attention not only to the quality of the physical resources and
curricula in schools, but also to the role teachers play in shaping
curricula and in deciding what supplies and materials are purchased.
For example, teachers are much more likely to use new curricula and
new teaching materials if they have had a hand in the planning and
decision processes (Berman and McLaughlin, 1974-1975~. Therefore,
some of our recommendations include ideas for learning more about
what influences teachers' responses to changes in resources and the
intended curriculum.
The Distribution of Excellence
A third principle underlying the recommendations in this report
is that, in addition to describing the extent to which schools are mak-
ing progress in promoting excellent mathematics and science educa-
tion, indicators should address the question: Excellence for whom?
This is central to promoting scientific and mathematical literacy for
all students and to ensuring that talent will be nurtured wherever
it is found. An example of the comm~ttee's concern regards teacher
qualifications: one needs to know not only about changes in the qual-
ifications of the nation's science teachers as a whole, but also about
the qualifications of science teachers who teach identifiable groups of
children, such as minority group children, urban children, rural chil-
dren, and children not in advanced-placement science courses. This
principle underlies many of our specific recommendations for how
data should be collected and reported, especially data on teacher
qualifications and on student behavior.
Representative terms from entire chapter:
mathematics education
