Today's students will live and work in the twenty-first century, in an
era dominated by computers, by world-wide communication, and by a global
economy. Jobs that contribute to this economy will require workers who
are prepared to absorb new ideas, to perceive patterns, and to solve
Mathematics is the key to opportunity for these jobs. Through
mathematics, we learn to make sense of things around us. As technology
has mathematicized the workplace, and as statistics has permeated the
arena of public policy debate, the mathematical sciences have moved from
being a requirement only for future scientists to being an essential
ingredient in the education of all Americans.
Yet an endless string of reports (e.g., Kirsch 1986, McKnight 1987,
Dossey 1988, Paulos 1988, Lapointe 1989) cite serious deficiencies in
the mathematical performance of U.S. students. Compared to other
nations, we rank very low; compared to our own expectations, we hardly
do any better. Although basic computational skills are reasonable
secure, only one in twenty high school graduates can deal competently
with problems requiring several successive steps (Dossey 1988).
U.S. students drop out of mathematics at alarming rates, averaging about
50% each year after mathematics becomes an elective subject. Blacks,
Hispanics, and other minorities drop out at even greater rates (Oaxaca
1988). Because mathematical preparation is a key to leadership in our
technological society, uneven preparation in mathematics contributes to
uneven opportunity for economic power.
Both economic necessity and concerns of equity demand revitalization of
mathematics education. Job forecasts project a shortfall of well over
half a million scientists and engineers by the year 2000; increased
retirements of teachers matched with rising enrollments in schools
suggest as well the likelihood of a severe shortage of mathematics and
science teachers . It is, therefore, vitally important both to the
nation and to each individual that all students receive a quality
education in mathematics.
Goals for Students
For most of the history of education--beginning with Plato's Academy and
the Roman quadrivium--students have been required to study mathematics
in order to learn to think clearly. As the embodiment of pure reason,
mathematics (especially Euclid) provided an ideal vehicle for teaching
rigorous habits of mind.
About five hundred years ago, as expanding commerce required widespread
use of complex systems of accounting, arithmetic and rudimentary algebra
became part of the educational system. Two hundred years ago "vulgar
arithmetic" was a common entrance requirement for the new American
universities, while " `rithmetic" became the third "R" in the common
expectations for primary education.
Geometry and arithmetic--thinking and calculating--are not only
paradigms of school mathematics but also caricatures of mathematics in
the minds of parents. Today, for quite different reasons, neither goal
is especially relevant. Although most children learn to calculate well
enough, calculators have made this hard-learned skill virtually
obsolete. And although high school students still study proofs in
geometry, little learned there--and little is all it is--transfers to
clarity of thought in other important areas of life.
To help today's students prepare for tomorrow's world, the goals of
school mathematics must be appropriate for the demands of a global
economy in an age of information. The new NCTM Standards for School
Mathematics (NCTM 1989) identify five broad goals required to meet
students' mathematical needs of the twenty-first century:
- To value mathematics. Students must recognize the varied
roles played by mathematics in society, from accounting and finance to
scientific research, from public policy debates to market research and
political polls. Their experiences in school must bring students to
believe that mathematics has value for them--so that they will have the
incentive to continue studying mathematics as long as they are in
- To reason mathematically. Mathematics is, above all else, a
habit of mind that helps clarify complex situations. Students must
learn to gather evidence, to make conjectures, to formulate models, to
invent counter-examples, and to build sound arguments. In so doing,
they will develop the informed skepticism and sharp insight for which
the mathematical perspective is so valued by society.
- To communicate mathematics. Learning to read, to write, and
to speak about mathematical topics is essential not only as an objective
in itself--in order that knowledge learned can be effectively used--but
also as a strategy for understanding. There is no better way to learn
mathematics than by working in groups, by teaching mathematics to each
other, by arguing about strategies, and by expressing arguments in
careful written form.
- To solve problems. Industry expects school graduates to be
able to use a wide variety of mathematical methods to solve problems
wherever they arise. Students must, therefore, experience variety in
problems--variety in context, in length, in difficulty, and in methods.
They must learn to formulate vague problems in a form amenable to
analysis; to select appropriate strategies for solving problems; to
recognize and formulate several solutions whenever that is appropriate;
and to work with others in reaching consensus on solutions that are
effective as well as logical.
- To develop confidence. The ability of individuals to cope
with the mathematical demands of everyday life--as employees, as
parents, and as citizens--depends on the attitudes toward mathematics
conveyed by school experiences. Among the greatest paradoxes of our age
is the spectacle of parents who recognize the importance of mathematics
yet boast of their own mathematical incompetence. Mathematics can
neither be learned nor used unless supported by self-confidence built on
Even when measured against older standards, the prevailing mathematics
pattern in U. S. schools is an "underachieving" curriculum (McKnight
1987). "We have inherited a mathematics curriculum conforming to the
past, blind to the future, and bound by a tradition of minimum
expectations" (NRC 1989).
When compared to contemporary goals--to value mathematics, to reason
mathematically, to communicate mathematics, to solve problems, and to
develop confidence--today's curriculum is totally inadequate. The new
curriculum standards of the National Council of Teachers of Mathematics
(NCTM 1989) make clear that the whole environment of learning must
change: not only what is taught, but also how it is taught and how it is
Recent studies of mathematics education (e.g., McKnight 1987, NRC 1989,
AAAS 1989, NCTM 1989) reveal many similar principles required for
effective learning. While different reports stress certain aspects
(such as international comparisons or the impact of computers) more than
others, there is widespread--and perhaps surprising--agreement on
certain necessary actions:
- Raise expectations. Evidence from other countries as well as
from some districts in the United States shows that if more is expected
in mathematics education, more will be achieved. Despite common belief
of U.S. parents that special talent is required to succeed in
mathematics, in reality all that is required is hard work and
self-confidence. Children can succeed in mathematics, and they
will succeed if we expect them to.
- Increase breadth. The traditional mathematics curriculum
focuses too narrowly on a few topics of limited appeal and utility--on
arithmetic, which leads to algebra, which in turn leads to calculus.
Most students would benefit from a curriculum with a broader vision, one
that reflects the expanding power and richness of the mathematical
sciences. Estimation, chance, measurement, symmetry, data, algorithms,
and visual representation are as much part of mathematics--and for many
students, a more interesting part--than computation and manipulation.
- Use calculators. Nothing better symbolizes the backwards
nature of our mathematics curriculum than the reluctance of teachers and
test-makers to make full and appropriate use of calculators. Research
shows overwhelmingly that appropriate use of calculators enhances both
children's understanding of arithmetic and their mastery of basic
skills. It is more important for children to develop good numer sense
than merely to memorize methods of calculation. As well as offering one
among many important methods of calculation (others being mental
arithmetic, estimation, paper-and pencil, and computers), calculators
provide a powerful tool for developing number sense.
- Engage students. Research in learning has demonstrated
repeatedly, in a variety of ways, that students do not simply learn what
is taught. Rather, their experiences modify prior beliefs, yielding a
mathematical knowledge that is uniquely personal. Clear presentations
alone are insufficient to modify existing misconceptions. To ensure
effective learning, mathematics teachers must involve students in their
own learning by employing classroom strategies that make students active
participants rather than passive receivers of knowledge.
- Encourage teamwork. Employers repeatedly stress the
importance of working with a team on common objectives. Most problems
that people encounter are sufficiently large or complex as to require
the talents of many different people. Students of mathematics must
learn how to work with others to achieve a common goal: to plan, to
discuss, to compromise, to question, and to organize. Teamwork in the
classroom not only teaches these skills, but it is also a very effective
way to learn mathematics--by communication with peers.
- Assess objectives. Assessment is an issue of increasing
importance in American education, but to be effective assessment must be
aligned with the objectives of learning. When assessment is dominated
by standardized multiple choice tests, as it is now, then teachers teach
skills required for those tests regardless of what the official
objectives of instruction may be. Assessment must become an integral
part of the educational process, not just an infrequent objective exam;
it must be designed to reflect what students know and how they think.
Above all else, assessment must align with curricular goals: to value
mathematics, to reason mathematically, to communicate about mathematics,
to solve problems, and to develop self-confidence.
- Require mathematics. Students should study mathematics every
year they are in school. Projections of future jobs as well as patterns
of college course prerequisites show nothing but a relentless increase
in the mathematical demands of employment and careers. There is no
point at which a high school student can correctly conclude that he or
she needs no more mathematics. All students who are college-bound need
four years of mathematics to be prepared for college prerequisites in
courses that they may not, while in high school, anticipate taking.
Students who are not preparing for college must keep up their
mathematics skills in anticipation of vocational or on-the-job training
that they will receive after graduating from high school. Regardless of
career goal, all students should study the same core of broadly useful
mathematics while in high school.
- Demonstrate connections. The power of mathematics derives
both from its internal unity and its external applicability. Everything
is connected. Results in number theory provide clues to problems in
geometry, and are applied in computer science and satellite engineering.
That's the way mathematics works, so students must see this at every
opportunity in their school experience. Connections motivate learning
and reinforce ideas arising in different contexts. We can no longer
afford to let mathematics remain an isolated discipline in the schools,
nor to permit continued fragmentation within the mathematics curriculum
itself into isolated courses, separated topics, and disconnected bits of
- Stimulate creativity. Too often school mathematics is judged
"dull" by students, even by very good students, because teachers,
textbooks, and tests insist that each problem can be solved by one
proper method yielding a single correct answer. Nothing could be
further from the reality of mathematics in practice. Multiplicity of
approaches, invention of new methods, and varieties of solutions are far
more typical than are automated answers. These days computers and
calculators perform most of the routine tasks of mathematics. In a
computer age, students need to use their imagination as much as their
intellect, their judgement as much as their memory.
- Reduce fragmentation. Curriculum planning based on specific
learning objectives has produced an atomized curriculum of particular
techniques practiced on problems specially selected to illustrate
textbook methods. Real problems don't come in compartmentalized form.
In school, the best clue concerning approach to a problem--and approach
is in many cases the most important decision--is which section of the
book it appears in. Fragmenting the mathematics curriculum destroys the
logical unity of mathematics which is the primary source of its unique
power to model the world.
- Require writing. Nothing helps a student learn a subject
better than the discipline of writing about it. Writing in a
mathematics class serves several purposes. It advances the goal of
learning to communicate about mathematics; it helps students clarify
their own understanding as they try to put ideas into coherent written
form; and it provides an opportunity for students who like writing
better than mathematical abstraction to grow in the discipline with a
vehicle more suited to their abilities. Many teachers have reported
positive results from journal writing and other "meta-assignments" in
which students reflect on their experiences in learning mathematics. In
contrast to the common school ritual of mindless mimicry calculations,
writing enhances learning by involving the student in expression of
- Encourage discussion. Most talk in a mathematics class comes
from the teacher, not the students. In a typical class, students take
notes, practice what the teacher has demonstrated, and then work in
isolation to perfect the technique. None of this engages the student's
mind as effectively as does vigorous argument and discussion. Argument
in search of convincing proof is the essence of the mathematical method.
It can be learned only by doing, not by listening.
Curriculum. Any change in curriculum will require significant,
specific change in curricular content. Computers and increased
applications, especially, have made certain parts of mathematics more
important, and others less so. Many areas of mathematics that are
commonly used in both civic and practical contexts are rarely taught in
school, while other topics that have long since out-lived their
usefulness remain in the curriculum simply because they are still on
tests or in texts.
In a revitalized school program, many widely-used areas of mathematics
must receive increased emphasis:
Several other topics, which now consume a significant part of a child's
school mathematics experience, should be reduced significantly:
Geometry and measurement|
Probability and statistics
Patterns and relationships
Observation and conjectures
Three dimensional geometry
Estimation and mental arithmetic
The first four of these items--manual calculations--are now less
important than formerly because calculators and computers are both more
accurate and more reliable. The last item, two-column proofs in
geometry, has never been part of real mathematics: it exists only in
school geometry as an exercise totally isolated from the rich reasoning
so appropriate to geometrical intuition. Geometry can be taught more
effectively without this stereotyped form of proof, and proofs can be
taught more effectively in many contexts besides geometry.
Graphing by hand
These changes in emphases must be implemented in a way that builds more
integrated mathematical experiences from primary school through high
school. Major themes of mathematics such as chance and change, shape
and dimension, quantity and variable should run through the entire
curriculum, being woven together into a single fabric of mathematical
Teaching. Just as content changes, so too must teaching methods.
It makes little difference what is taught unless students are provided
with suitable opportunities to learn. Effective classroom practice will
emphasize such things as:
At the same time, many common current practices must be significantly
reduced, since the evidence shows conclusively that they are not
Testing. Finally, testing must change. No amount of
effort to change curricular content or teaching practice will be
successful unless the instruments of assessment match curricular
objectives. Effective assessment will:
Teaching by telling|
One method, one answer
Curriculum, teaching, and testing must change together to improve
mathematics education. Unless all improve in concert, nothing will
change. The NCTM Standards and other documents provide a clear
blueprint for reconstructing U. S. mathematics education. We know what
needs to be done, and we know how to do it. What's required now is a
commitment to action.
- Be open-ended, not just multiple choice.
- Allow calculators in virtually every context.
- Provide opportunities for students to show what they know and how
they think, not just seek to determine what they do not know.
- Emphasize integration of knowledge and holistic strategies for
approaching problems (e.g., estimation, graphing, models, computers,
- Be integrated with teaching, not separate from it.
- Employ a variety of methods, including observation, oral protocols,
student notebooks, written tests, and group projects.
- American Association for the Advancement of Science. Science
for All Americans. Washington, D.C.: American Association for the
Advancement of Science, 1989.
- Dossey, John A.; Mullis, Ina V. S.; Lindquist, Mary M.; Chambers,
Donald L. The Mathematics Report Card: Are We Measuring Up?
Princeton, NJ.: Educational Testing Service, 1988.
- Kirsch, Irwin S. and Jungeblut, Ann. Literacy Profiles of
America's Young Adults. Princeton, NJ: Educational Testing Service,
- Lapointe, Archie E.; Mead, Nancy A.; Phillips, Gary W. A World
of Differences: An International Assessment of Science and
Mathematics. Princeton, NJ.: Educational Testing Service, 1989.
- McKnight, Curtis C., et. al. The Underachieving Curriculum:
Assessing U.S. School Mathematics from an International Perspective.
Champaign, IL.: Stipes Publishing Co., 1987.
- Mullis, Ina V. S. and Jenkins, Lynn B. The Science Report Card:
Elements of Risk and Recovery. Princeton, NJ.: Educational Testing
- National Council of Teachers of Mathematics. Curriculum and
Evaluation Standards for School Mathematics. Reston, VA.: National
Council of Teachers of Mathematics, 1989.
- National Research Council. Everybody Counts: A Report to the
Nation on the Future of Mathematics Education. Washington, D.C.:
National Academy Press, 1989.
- Oaxaca, Jaime and Reynolds, Ann W. Changing America: The New
Face of Science and Engineering. (Interim Report). Washington, D.C.:
Task Force on Women, Minorities, and the Handicapped in Science and
Technology, September 1988.
- Paulos, John Allen. Innumeracy: Mathematical Illiteracy and
its Consequences. New York: Hill and Wang, 1988.