Teaching Mathematics for Tomorrow's World
Lynn Arthur Steen, St. Olaf College
Educational Leadership, 47:1 (September 1989) 18-22.
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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 unconventional problems.

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 school.

  • 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 success.

Curricular Change

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 assessed.

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:

  1. 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.

  2. 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.

  3. 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.

  4. 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.

  5. 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.

  6. 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.

  7. 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.

  8. 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 knowledge.

  9. 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.

  10. 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.

  11. 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 meaning.

  12. 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.

Changing Emphases

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:

• Geometry and measurement
• Probability and statistics
• Patterns and relationships
• Spatial reasoning
• Collecting data
• Observation and conjectures
         
         
         
         
         
         
         
• Discrete mathematics
• Real problems
• Three dimensional geometry
• Graphical reasoning
• Estimation and mental arithmetic
Several other topics, which now consume a significant part of a child's school mathematics experience, should be reduced significantly:
• Fractions
• Long division
• Graphing by hand
         
         
         
• Paper-and-pencil algorithms
• Two-column proofs
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.

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 method.


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:

• Active learning
• Problem solving
• Concrete materials
• Instructional variety
         
         
         
         
• Oral communication
• Written exercises
• Paragraph answers
• Continual assessment
At the same time, many common current practices must be significantly reduced, since the evidence shows conclusively that they are not particularly effective:

• Teaching by telling
• Rote memorization
• One method, one answer
         
         
         
• Memorizing rules
• Template exercises
• Routine worksheets
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:

  • 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, calculation).
  • Be integrated with teaching, not separate from it.
  • Employ a variety of methods, including observation, oral protocols, student notebooks, written tests, and group projects.
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.

 

References

  1. American Association for the Advancement of Science. Science for All Americans. Washington, D.C.: American Association for the Advancement of Science, 1989.
  2. 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.
  3. Kirsch, Irwin S. and Jungeblut, Ann. Literacy Profiles of America's Young Adults. Princeton, NJ: Educational Testing Service, 1986.
  4. 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.
  5. McKnight, Curtis C., et. al. The Underachieving Curriculum: Assessing U.S. School Mathematics from an International Perspective. Champaign, IL.: Stipes Publishing Co., 1987.
  6. Mullis, Ina V. S. and Jenkins, Lynn B. The Science Report Card: Elements of Risk and Recovery. Princeton, NJ.: Educational Testing Service, 1988.
  7. National Council of Teachers of Mathematics. Curriculum and Evaluation Standards for School Mathematics. Reston, VA.: National Council of Teachers of Mathematics, 1989.
  8. National Research Council. Everybody Counts: A Report to the Nation on the Future of Mathematics Education. Washington, D.C.: National Academy Press, 1989.
  9. 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.
  10. Paulos, John Allen. Innumeracy: Mathematical Illiteracy and its Consequences. New York: Hill and Wang, 1988.

Copyright © 1989. Contact: Lynn A. Steen URL: http://www.stolaf.edu/people/steen/Papers/89edl.html

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