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CHANGE ...mobiliziJ'g for curricular reform Since the publication in 1983 of A Nation at Risk, Amer icans have known that fundamental changes must occur throughout all parts of our educational system in order to: · Raise performance levels significantly in our nation's schools and colleges; · Prepare young people for lifelong learning; · Educate all students well, not only those identified as col- lege bound; · Create learning environments better suited to the needs of disadvantaged groups. The future of our country depends strongly on our ability to bring about these fundamental changes in mathematics education. C~ e ~ - e e · - ~ ~ ~ e ~ ~ ~ e ~ ~ e ontinual change is a natural and essential characteristic of mathematics education Because mathematics is one of the pillars of education, re- form of education must include significant change in the way mathematics is taught and learned. As mathematics and so- ciety change continuously, so must mathematics education. Change is a natural state for education, not just a transition between old and new To ensure continuous responsiveness in the future, mathematics education must adopt structures that will make change permanent; mathematics education must always respond to changes in science, in society, and in mathematics itself. Challenges Mathematics education in the United States is facing ma- jor challenges on nearly every front: · Far too many students, disproportionately minority, leave school without having acquired the mathematical power necessary for productive lives. "If an unfriendly foreign power had attempted to impose on America the mediocre edFucatio1'al performance that exists today, we migh' well have viewed it as an act of war. As it stands, we have al- lowed this to happen to ourselves." - A Nation al Risk 73
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Change "We have drifted into a curriculum by default, a curriculum of mini- mum expectations that resists the changes needed to keep pace with the d~e- ma1'ds of prepari1'gstu- de1tts for contemporary life." -John A. Dossey 74 · The shortage of qualified mathematics teachers in the United States is serious-more serious than in any other area of education- and affects all levels from elementary school to graduate school. · At a time when the percentage of minority students is in- creasing, the shortage of new minority teachers of mathe- matics is particularly acute. · On average, U.S. students do not master mathematical fundamentals at a level sufficient to sustain our present technologically based society. · When compared with other nations, U.S. students lag far behind in level of mathematical accomplishment; the re- sulting educational deficit reduces our ability to compete . . . - in international arenas. Public attitudes, which are reflected and magnified by the entertainment industry, encourage low expectations in mathematics. Only in mathematics is poor school perfor- mance socially acceptable. Curricula and instruction in our schools and colleges are years behind the times. They reflect neither the increased demand for higher-order thinking skills, nor the greatly expanded uses of the mathematical sciences, nor what we know about the best ways for students to learn mathemat- iCS. Calculators and computers have had virtually no impact , . . - · . . . on mathematics Instruction In spite ot their great poten- tial to enrich, enlighten, and expand students' learning of mathematics. · Common methods of evaluation especially standardized, paper-and-pencil, multiple-choice tests of"basic skilIs"- are themselves obstacles to the teaching of higher-order thinking skills as well as to the use of calculators and com- puters. · Undergraduate mathematics is intellectually stagnant, overgrown with stale courses that fail to stimulate the mathematical interests of today's students. The information ~ 1 ~ - ~ morrow's scientist and engineer will need extensive math ematics education, tomorrow's citizen will need a very different type of mathematical education to deal with age Is a mathematical age. Even as to
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...mobilizing for curricular reform mathematics-based tools, equipment, and techniques which will permeate the workplace. Far more than most citizens currently appreciate, mathematics education will play a sub- stantial role in determining which doors are open and which are closed as students leave school and enter the world of work. Counterproductive Beliefs It is mistakenly thought, even by otherwise well-informed adults, that the mathematics they learned in school is ade- quate for their children. Parental and legislative pressures in the past few years, driven largely by frustration over de- clining test scores, have led to many rash actions: · Increased numbers of required courses where there is no agreement on what the added courses should contain or where capable teachers are to be found to teach them; · Increased use of standardized tests where there is very little understanding of what the tests contain or what they . _ ~ O are capable of testing; Increased use of test scores, especially for teacher and school accountability where there is little recognition that the tests reflect only a small part of curricular ob Octaves. The nation is in the grip of a testing mystique that has led to widespread misuse of standardized tests. Public pressures for "back-to-basics" stem from a very limited understanding of the challenges we face. Carried to the extreme, these pres- sures will rob our children of the opportunity to learn what they will need to know of mathematics in their adult lives. Too often, what results from such actions are watered- down curricula, unreliable tests, and diminished morale. The only elective way in which these relatively ill-informed policies can be combated is through a systematic effort to develop in the public a deeper understanding of what works and what does not. It will not be easy to develop better understanding. Of- ten, public discussion about mathematics education masks Myth: Increased requirements yield better prepared students. Reality: Motivation almost al- ways works better than require- ments. Often, increased require- ments have an eject quite the op- posite of what was intended. In Wisconsin, for example, when the university increased from two to three years the number of courses required for admission and also increased the minimum grade point requirement, in some schools the number of students who elected four years of high school mathematics dropped. Once the three-year requirement was met, students skipped se- nior mathematics to protect their grade point averages. In Florida, increased requirements for grad- uation from high school have caused an increase in the num- ber who drop out. 75
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Change Cultural Context Competition and individualism ingrained parts of traditional American culture, are reflected in typical mathematical courses where students work alone to solve set problems. Other cul- tures, including many which are now a growing part of the Amer- ~can scene, stress teamwork and group problem-solving. To the extent that mathematics instruc- tion in the United States contin- ues to stress individualism and competition over cooperation and teamwork, to that extent we con- tinue to introduce unnecessary counterproductive practices for many in our multicultural nation. Adult Attitudes Too many Americans seem to be- lieve that it does not really matter whether or not one learns math- ematics. Only in America do adults openly proclaim their ig- norance of mathematics ("I never was very good at math") as if it were some sort of merit badge. Parents and students in other countries know that mathemat- ics matters. 76 a hidden agenda of values that have traditionally been car- ried forward by the school study of mathematics. Since the demise of Latin as a required school subject, it is to math- ematics that many look as a vehicle to teach such qualities as precision, discipline, neatness, and accuracy. Mathemat- ical truth in popular caricature- is certain, absolute, un- changing, eternal. Mathematics appears to many to be a safe harbor of calm in a turbulent sea of social and educational change. Proposals to change mathematics education appear to threaten time-honored values that are deeply embedded in the public image of mathematics. The need for change in mathematics education is too great to allow stereotypes of mathematics to impede reform. It is important that the public learn not only about the need for change, but also about how the essential qualities of mathematics are conveyed by contemporary as well as traditional views of the field. As an active partner on the rapidly advancing frontier of science, mathematics is con- stantly expanding and changing. Mathematics education, in contrast, has been constrained by societal forces to such a degree that it has hardly changed at all. This contrast in the pace of change virtually ensures that mathematics education is perpetually out of date. N.......... alive nol~c~es rooted in myth impede reform of mathematics education. As a subject with an extensive and substantial history, mathematics more than any other science has been taught as an ancient discipline. A nation that persists in this view of mathematics is destined to fall behind scientifically and economically. Parents who persist in this view deny their children the opportunity to develop and prosper in the in ~ . formation age.
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...mobilizing for curricular reform The American Way The development of more effective strategies for revital- izing mathematics education must be based in part on an understanding of why it is so difficult in the United States to bring about change in education. The truth we shrink from confronting is that most previous reform efforts have failed. A properly skeptical public will rightly ask why any new effort is more likely to succeed. Part of the difficulty we face in mathematics education is a natural reflection of our constitutional dilemma: to reconcile local authority with national need. Most other countries have either national curricula or na- tionwide curricular guidelines. Curricular development is typically a routine function of a ministry of education which taps the best brainpower in the nation to develop complete sets of texts and other resource materials for classroom use. Specific day-to-day syllabi and teacher guides are often pro- vided to schools; in some cases, these syllabi are actually mandated by a ministry of education. In many countries, all children in the same grade study essentially the same material in almost the same way. Such practice, common around the world, reveals a strong tradition of a "top-down" approach in education. in the United States, with our traditional and legal decen- tralization of education, we go about things very differently. Every summer, thousands of teachers work in small teams for periods ranging from one week to two months, charged by their school districts to write new mathematics curric- ula. These teacher teams usually have little training in the complicated process of curricular development, little or no help in coping with changing needs, and little to fall back on except existing textbooks, familiar programs, and tradition. The consequence usually is the unquestioned acceptance of what already exists as the main body of the new curricu- lum, together with a little tinkering around the edges. Many school districts simply adopt series of textbooks as the cur- riculum, making no effort to engage the staff in rethinking curricula; in those places, the status quo certainly reigns. International Expectations Average students in other coun- tries often learn as much mathe- matics as the best students learn in the United States. Data from the Second International Math- ematics Study (1982) show that the performance of the top 5 per- cent of U.S. students is matched by the top 50 percent of stu- dents in Japan. Our~very best students the top 1 percent- scored lowest of the top 1 percent in all participating countries. All U.S. students whether below, at, or above average-can and must learn more mathematics. 77
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Change "Most students seem to think that mathematics courses are chiefly designed to winnow out the weak and grind down the ungifted. We need a change in attitude." Edward E. David, Jr. 78 T· · · . . . . . · . . . . . . · . . . · . . . . . . . . . . . . . raditional U.S. approaches to curricular change make reform impossible. The American process of curricular reform might be de- scribed as a weak form of a grass-roots approach. The record shows that this system does not work. It is not our teachers who are at fault. In fact, teachers should play a dominant role in curricular decision-making. But teachers who work in summer curricular projects are being given an unrealis- tic task in an impossible time frame, with only the familiar status quo to guide them. In static times, in periods of unchanging demands, perhaps our grass-roots efforts would suffice to keep the curriculum current. In today's climate, in which technology and research are causing unprecedented change in the central methods and applications of mathematics, present U.S. practice is totally inadequate. International comparisons of student perfor- mance in mathematics for example, the Second Interna- tional Mathematics Study- show that U.S. students lag far behind their counterparts in other industrialized countries. The top-down systems have beaten us hands down. Modern Mathematics Curricular reforms undertaken in the two decades from 1955 to 1975 under the slogans of "modern mathematics" or "new math" left a mixed legacy to American mathemat- ics education. The movement sprang from many roots and took on many different (and sometimes opposing) forms. Implementation was quite uneven, as were results. Looking back, one can identify several important areas of success and failure: · Certain important seeds sowed during this period (for ex ample, renewed emphasis on geometry, probability, and statistics) have taken root and are now on the verge of blossoming.
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...mobilizing for cllrricuiar reform · Too often, the proposed means to achieve deeper under- standing (for example, sets and commutative law) became ends in themselves, thus opening mathematics education to public ridicule. · Innovative applications of mathematics to nontraditional fields (for example, to biology and business) became ac- cepted as part of the content of school mathematics. · By moving some parts of school curricula into unfamiliar areas, mathematics educators lost the confidence of their most important ally parents. Both educators and parents can learn from the experiences of the modern mathematics era, but the lessons are not so simple as conventional wisdom often suggests. Lessons from the Past The history of the past twenty-five years of curricular re- form gives us only negative examples from which to learn. Few traces remain of the expensive major curricular devel- opment projects so prominent in the 1960's and 1970's. These projects tried to develop, on a national scale, com- plete curricula (including instructional materials) that could be adopted by school districts. But the theorists and planners who developed these curricula were naive about the process of change; big curricular projects failed to take root in Amer- ican schools because they were transplanted fully grown into an environment better suited to locally grown methods. Where teachers were not directly a part of the develop- ment procedure, where their ownership of the product was not ensured, where teachers considered district acceptance of the curriculum as a top-down imposition, the revised pro- grams did not last. Where parents could not (or did not) understand the need for change or the reasons new curric- ular emphases were chosen, resentment and anger resulted and a solid conviction set in that if the "old math" was good enough for parents, it was good enough for their children. As the United States enters a new period of change in mathematics education, we can benefit from several lessons drawn from these previous attempts. First, free-standing, full-service curricular development projects adopted intact 79
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Change 80 by school districts do not work. Second, a superficial, district-by-district approach to curricular overhaul is poten- tially disastrous, given the demanding nature of what educa- tors face. U...... =ffective reform requires strong leadership by teachers, parents, professionals, and . · ~ po Cans. Third, any successful effort to improve mathematics cur- ricula and instruction in the schools will require an extensive public information campaign that reaches all the varied con- stituencies of mathematics education. These diverse publics must be convinced in understandable language that a very different mathematics education is both better and neces- sary for their children and for the country. Effective change requires a great deal from the public: Conviction of the need for change; Consensus on high-quality mathematics education for ev- eryone; · Skepticism of "quick fixes" and simplistic solutions; · Awareness of the general nature of needed changes; · Support for investment of necessary resources; · Recognition of the need for continuing leadership at the national level. General reaction to the many recent calls for school reform has been uneven and fragmented. The pattern of unfocused reaction shows that it is not enough just to get the public's at- tention. Public concern is a necessary, but by no means su~- cient, condition for meaningful educational change. Too of- ten, a partially informed public becomes a poorly informed electorate. The time is ripe for a new approach to curricu- lar reform, one that establishes appropriate national expec- tations supported by broad public support among parents, teachers, and taxpayers.
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...mobilizing for curricular reform T· ~ ransoms In order to meet the challenges of our time, mathematics education is already beginning to negotiate several Biscuit transitions which will dominate the process of change during the remainder of this century. Only gradually, by extensive experience, will teachers find the most effective point along each transition path. Although no one can say in advance where the best balance lies, it is quite clear that present prac- tice is at an ineffective extreme. Transition I: The focus of school mathematics is shiftingirom a dualistic mission- minimal mathematics for the majority, advancer! mathematics for a few to a singular focus on a significant common core of mathematics for ad students. The needs of industry for a quantitatively literate work force compel schools to provide more mathematical educa- tion to more students than ever before. Accomplishing this will pose significant challenges to: · Develop a core of mathematics appropriate for all students throughout each year of school; · Educate well a significantly larger fraction of the popula- tion; . Stimulate able students with the excitement and challenge of mathematics; · Differentiate instruction by approach and speed, not by curricular goals; · Select topics and approaches of broad interest and effec- tiveness. Transition 2: The teaching of mathematics is shifting from an authoritarian mottle! based on "transmission of knowledge" to a student-centered practice featuring "stimulation of ~learn ing' In both schools and colleges, classrooms of passive stu dents who are expected to sit and absorb rules which appear as arbitrary dicta from on high are gradually giving way to learning environments that: · Encourage students to explore; · Help students verbalize their mathematical ideas; Voice of Experience "Math Achievement through Prob- lem Solving is an activity-oriented process that uses small groups to focus on nonroutine problems. It was designed for students with bad work habits who seem to get very little out of traditional high school algebra. We've founds out that a lot of these students know much more than we thought, and many know less. We've been surprised at the high level of thinking that goes on in some of these students." Jon Brace 81
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Charge 82 · Show students that many mathematical questions have more than one right answer; · Provide evidence that mathematics is alive and exciting; · Teach students through experience the importance of care- ful reasoning and disciplined understanding; · Build confidence in ad students that they can learn math- ematics. Transition 3: Public attitudes about mathematics are shifting from ir~di~erer~ce and hostility to recognition of the important role that mathematics Allays in today's society. Although the burden of unfavorable school experiences continues to color public opinion about mathematics, con- temporary events are sending different messages which are gradually being heard: · In other nations where more is expected, more mathemat- ics is learned; · As the role of science and technology expands, so does the importance of mathematics; · To function as an informed citizen, numerary is as impor- tant as literacy. As attitudes about the importance of mathematics improve, so will expectations for mathematics education. Transition 4: The teaching of mathematics is shifting from preoccupation with inculcating routine skills to developing broad-based mathematical power. Mathematical power requires that students be able to dis cern relations, reason logically, and use a broad spectrum of mathematical methods to solve a wide variety of non- routine problems. The repertoire of skills which now un- dergird mathematical power includes not only some tradi- tional paper-and-pencil skills, but also many broader and more powerful capabilities. Today's students must be able to: · Perform mental calculations and estimates with profi- c~ency; · Decide when an exact answer is needed and when an esti- mate is more appropriate;
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...mobilizing for curricular reform · Know which mathematical operations are appropriate in particular contexts; · Use a calculator correctly, confidently, and appropriately; · Estimate orders of magnitude to confirm mental or calcu- lator results; · Use tables, graphs, spreadsheets, and statistical techniques to organize, interpret, and present numerical information; · Judge the validity of quantitative results presented by oth- ers; · Use computer software for mathematical tasks; · Formulate specific questions from vague problems; · Select effective problem-solving strategies. Transition 5: The teaching of mathematics is shifting from emphasis on tools for future courses to greater emphasis on topics that are relevant to students' present and future needs. Most mathematics should be presented in the context of its uses, with appreciation of mathematics as a deductive logical system built up slowly through the rising levels of education. Examples of areas deserving greater emphasis are: Probability, which facilitates reasoning about uncertainty and assessment of risk; · Exploratory data analysis and statistics, which facilitate reasoning about data; Model-building, which facilitates systematic, structured understanding of complex situations; · Operations research, which facilitates planning of complex tasks and achieving performance objectives; · Discrete mathematics, which facilitates understanding of most applications of computers. These new topics imply that observation and experimenta tion will be important in future mathematics programs and that school mathematics will draw closer to other school sub- jects, especially to science. Transition 6: The teaching of mathematics is shifting from primary emphasis on paper-and-pencil calculations to fuR use of calculators and computers. Mathematics teachers at all levels-from elementary school to university are adapting their teaching methods Voice of Experience "I approach each problem as if I didn't already know the conven- tional solution. The students are much more involved and excited!. They become creators. It's as de- scribedt by Felix Klein: The math- ematician himself does not work in a rigorous, deductive manner, but rather uses fantasy." Kenneth Cummins 83
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Change "Over the long term, ba- sic skills only give you the right to compete against the Third Worl~for Third World wages." Marc S. Tucker 84 to include both new approaches to instruction as well as new subject matter appropriate to future-oriented curricula. Calculators and computers make new modes of instruction feasible at the same time that they inject into the learning en- vironment a special sense of wonder which goes with healthy development of mathematical power. Calculators and computers should be used in ways that anticipate continuing rapid change due to technological de- velopments. Technology should be used not because it is seductive, but because it can enhance mathematical learning by extending each student's mathematical power. CaTcula- tors and computers are not substitutes for hard work or pre- cise thinking, but challenging tools to be used for productive ends. Transition 7: The public perception of mathematics is shifting from that of a f xed body of arbitrary rules to a vigorous active science of patterns. Mathematics is a living subject which seeks to understand patterns that permeate both the world around us and the mind within us. Although the language of mathematics is based on rules that must be learned, it is important for mo- tivation that students move beyond rules to be able to express things in the language of mathematics. This transformation suggests changes in both curricular content and instructional style. It involves renewed effort to focus on: · Seeking solutions, not just memorizing procedures; · Exploring patterns, not just learning formulas; · Formulating conjectures, not just doing exercises. As teaching begins to reflect these emphases, students will have opportunities to study mathematics as an exploratory, dynamic, evolving discipline rather than as a rigid, absolute, closed body of laws to be memorized. They will be encour- aged to see mathematics as a science, not as a canon, and to recognize that mathematics is really about patterns and not merely about numbers.
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Representative terms from entire chapter: