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A Rationale for Change The basic premise of this report is that the United States must restructure the mathematics curriculum- both what is taught and the way it is taught- if our chil- dren are to develop the mathematical knowledge (and the confidence to use that knowledge) that they will need to be personally and professionally compel tent in the twenty-first centu- ry. This restructuring involves more than producing new texts or retraining teachers. Replacing parts is not suffi- cient. What is required is a complete redesign of the content of school mathe- matics and the way it is taught. Changing Conditions We begin our analysis by reflecting on some of the major changes affecting the context of mathematics education: Changes in the need for mathematics, As the economy adapts to information-age needs, workers in every sector from hotel clerks to secretaries, from automobile mechanics to travel agents must learn to interpret intelligently computer-controlled processes, Most jobs now require analytical rather than merely mechanical skills, so most students need more mathemat

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2 Reshaping School Mathematics ical power in school as preparation for routine jobs. Simi- larly, the extensive use of graphical, financial, and statisti- cal data in daily newspapers and in public policy discuss signs compels a higher standard of quantitative literacy for effective participation in a democratic society. Changes in mathematics and how it is used. In the past quarter of a century, significant changes have Qccurred in the nature of mathematics and the way it is usecl. Not only has much new mathematics been discovered, but the types and variety of problems to which mathematics is applied have grown at an unprececlented rate. Most visible, of course, has been the development of computers and the explosive growth of computer applications. Most of these applications of computers have required the cleve~opment of new math- ematics in areas where applications of mathematics were infeasible before the acivent of computers (Howson and K~ahane, 15861. Less visible, but equally important, has been the enormous wealth of ideas generated in several main branches of mathematics linked by unifying concepts of wiclespre:ocl applicability (e.g., Boarcl on Mathematical Sciences (BMS3, 1986~. Students must study the mathematics usecl in such applications in order to grasp the power of mathematics to solve real problems. Changes in the role of technology. Computers ancl calculators have changed profoundly the world of mathematics. They have affected not only what mathematics is important, but also how mathemat- ics is done (Rheinbolcit, 1985~. It is now possible to exe- cute almost ail of the mathematical techniques taught from kindergarten through the first two years of college on hand-held calculators. This fact alone --I the fulfillment in our age of the dream of Pascal-- must have significant effects on the mathematics curriculum (Pea, 1987a). Although most developments at the forefront of a disci- pline cannot generally be expected to have a major effect on the early years of eclucation, the changes in mathematics brought about by computers and calcula- tors are so profound as to require readjustment in the bal- ance and approach to virtually every topic in school mathematics. Changes in American society. As mathematics has changecl, so has American soci- ely. The changing demographics of the country and the changing demancis of the workplace exert extraordinary burdens on mathematics eclucation, burdens that we

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A Philosophy and Framework have not yet successfully borne (National Research Council (NRCl, 19891. In the early years of the next centu- ry, when toclay's school children will enter the work force, most jobs will require greater mathematical skills (John- ston ancl Packer, 1987~. At the same time, white males-the fraclitional base of mathematically trained workers in the United States-will represent a significantly smaller fraction of new workers (Oaxaca and Reynolds, 1988~. Society's need for an approach to mathematics education that ensures achievement across the clemo- graphic spectrum is both compelling and urgent (Office of Technology Assessment, 1988~. Changes in understanding of how stuclents ~earn. Learning is not a process of passively absorbing infor- mation ancl storing it in easily retrievable fragments as a result of repeated practice and reinforcement. Insteacl, students approach each new task with some prior knowl- ecige, assimilate new information, ancl construct their own meanings (Resnick, 1987~. Furthermore, ideas are not isolated in memory but are organized ancl associated with the natural language that one uses ancl the situa- tions one has encountered in the past. This constructive, active view of learning must be reflected in the way mathematics is taught. Changes in international competitiveness. Just as a global economy is emerging as a dominant force in American society, many recent reports have shown that U.S. students do not measure up in their math- ematical accomplishments to students in other countries (e.g., Stevenson et al., 1986; McKnight et al., 1987; Stigler ancl Perry, 1988; Lapointe et al., 19891. Because of wicJeiy different social contexts in which education takes place, comparing the educational systems of different countries is fraught with clangor. Nevertheless, the data are so compelling that they cannot be ignorecl. In particular, most other industrial countries have considerably different expectations about topics taught and level of perfor- mance than is common in American schools. One implication of these reflections is the need for a new practical philosophy of mathematics~education as a basis for curricular reform. Each new generation neecis to step away from current schooling practices, reflect on the mathematical expectations for students ancl society, and restate the assumptions upon which the system for teaching and learning is basecl. Such reflection is especially important in a time of rapid change.

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4 Reshaping School Mathematics Outdated Assumptions The mathematical content of today's school curriculum is about 500 years old. The core of this curriculum arithmetic, geometry, and elementary algebra differs in only superficial ways from the curriculum followed by tutors during the Renais- sance. Advanced topics such as quadratic equations, loga- rithms, and probability are of more recent vintage, but even calculus as taught in today's schools and colleges is three centuries old The tremendous stability of today's curriculum depencis on a guidance system controlled by two unwavering and outbat- ed public assumptions: Mathematics is a fixed and unchanging body of facts and procedures; and To do mathematics is to calculate answers to set prob- lems using a specific catalogue of rehearsed techniques. These principles are the gyroscopes of today's school mathe~ matics. Despite turbulence in schools and revolution in the workplace, mathematics education maintains its course, fold lowing a path little changed through the centuries, To the Romans a curriculum was a rutted course that guicl- ed the path of two-wheeled chariots. Toclay's mathematics curriculum - a course of study-follows a deeply rutted path directed more by events of the past than by the changing neecis of the present, Vast numbers of specific learning objectives, each with associated pedagogical strategies, serve as mileposts along the trail mapped by texts from kindergarten until twelfth grade. Problems are solved not by observing and responding to the natural landscape through which the mathematics curricula lum passes, but by mastering time-testecl routines conveniently placed along the path near every anticipated problem. Stu- dents who progress through this curriculum develop a kind of mathematical myopia in which the goal is to solve artificial word problems rather than realistic world problems.

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5 A Philosophy and Framework Few have the stamina to survive the curriculum of mathe- matics-at least not the way it is now delivered. Of 4 million who begin, only 500,000 are still studying mathematics 12 years later. Most students receive little of lasting value from the final mathematics course they study-typically high school geometry or algebra 11. Many of those who cirop out harbor life-long feelings of guilt or distaste for school mathematics. Some of those who become disenchanted with mathematics become teachers; others help decide educational and research policy for the nation. Very few adults in the United States have had the benefit of successful completion of a mathematics curriculum, Transitions to the Future The price of stability is anachronism, Evidence is mounting from many sources that our present curriculum must change course if it is to serve society well in the twenty-first century, Forces for change, which are growing increasingly powerful, are beginning to redirect the mathematics curriculum in sever- al important ways: The focus of school mathematics is shifting from a clualis- tic mission minimal mathematics for the majority, advanced mathematics for a few to a singular focus on a significant common core of mathematics for all stu- dents. The teaching of mathematics is shifting from an authori- tarian model based on "transmission of knowledge" to a student-centered practice featuring "stimulation of learn- ing." Public attitudes about mathematics are shifting from inclif- ference and hostility to recognition of the important role that mathematics plays in today's society. The teaching of mathematics is shifting from preoccupa- tion with inculcating routine skills to developing broad- based mathematical power. The teaching of mathematics is shifting from emphasis on tools for future courses to greater emphasis on topics that are relevant to stuclents' present and future needs. The teaching of mathematics is shifting from primary emphasis on paper-and-pencil calculations to full use of calculators and computers, These transitions, elaborated in Everybody Counts (NRC, 1989), are bringing about a substantial change in the way

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6 Reshaping School Mathematics mathematics is taught and learnecl. New strategies that promise significant change are emerging in many districts and states (e.g., Denham and O'Mally, 1985; Chambers, 1986; Alli- goocl, 1989~. The ruts of the old curriculum are being eroded by the waves of change sweeping across the landscape of mathematics eclucation. The following chaff from World Almanac gives speeds of animals in km/hr: 98 antelope 48 bear 48 cat 1 12 cheetah 14 chicken 69 coyote 48 deer 40 elephant 72 elk 67 fox 56 jacka/ 5 1 giraffe 62 greyhound 77 horse 45 human 80 lion 18 pig 56 rabbit 24 turkey 48 warthog 64 zebra The human speed is listed as 45 km/hr. What do you think this means? Find the speed of the winner of the most recent Olympic 1500 meter run. How does this compare with the speed listed for the human in the chart? What explanation can you give for this? What is the typical speed for animals? How did you find this speed? How do the speeds differ? Are there any animals whose speeds are similar? Do these animals have anything in common? Are there any animals whose speeds are much different than the rest? Separate the animals into groups according to the kind of food they eat and compare the speeds for each group. What conclusions can you make? Write a paragraph describing the results of your analysis. Data Anatysis