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A Framework for Change Curriculum discussions always involve, implicitly if not explicitly, many different curricula of progres- siveiy diminishing scope: · An expected curriculum, that represents future neecis of employers and science. · An ideal curriculum, that establishes actual goals for teaching and learning. · An available curriculum, that can be taught with existing teaching materials and currently trained teachers. · An adopted curriculum, that a school district says should be taught. An implemented curriculum, that teachers actually teach. An assessed curriculum, that is examined by tests or other forms of evaluation. An achieved curriculum, that most students actually master. A major aim of curriculum development should be to close the gap between the expected and the achievecl curricula. To do this, one must make the gaps between each pair of suc- cessive steps in the hierarchy as small as possible. There are, however, no unique curricula that will do this. We aim, there- fore, to provide a framework within which many curricula may bloom.

36 Reshaping School Mathematics Principles Pressure to reshape mathematics education comes from many directions from technology, from society, from research, and from mathematics itself, The broad practical view of mathematics as a science and language of patterns provides a strong foundation for new mathematics curricula. Technological change and research findings suggest direc tions for curricular change, Such change will take many forms, but should be built on certain fundamental principles that fol- low from our view of mathematics and our review of research, Principle 1: Mathematics eclucation must focus on the devel- opment of mathematical power. Mathematical power enables students to understand mathematical concepts and methods and to discern mathemofical relations in a variety of contexts. It helps students to reason logically and to solve a variety of problems, both routine and nonroutine, To be effective, mathematical power requires of students that they be able to read documents using mathematical methods and express quantitative and logi- cal analyses in both oral and written form. Students who achieve sig- nificant mathematical power during their school years will be able to use mathematics in their careers and in everyday lives, They will be intelligent users of mathematical ideas, accepting or rejecting claims that are ostensibly based on mathematical arguments, They will see things mathemat- ically, recognizing when mathematical analyses help to explain events. They will have sufficient mathematical knowledge to pursue a profes- sion or vocation of their choice and to undertake fur- ther study of subjects that require mathematical profi- ciency.

37 A Philosophy and Framework Mathematical power entails the capability to com- municate about mathematics. In acidition to learn ing how to solve problems, stu- clents must also learn to read and understand math- cmatical texts . . . . ; . E::::: := K:::::::::: ~ :::.:.:..: :,:,:,:,:,:::: I::::::::: [2.22.2.. I............ .,:.:,:,:2: :) - ::::::: !............... 2 2''~ . 2 . <a ,i.jj: ~ :::::::: ·:,:,:,:,:,:,:,:'1 ::::::::. :,::::::: a::::::: . ;....... ............. ace:::::. ·:::;:::::: ~ ~ ::::::: :::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::. ::.::__ -:::: ::::::::::::::::::.:.::::::::::::::::::::::::: :. , it. ~:~:~:~:~:~:~:~:~:~:~:~:~:~:~:~:~:~:~:~:~:~:~:~:~:~::::: ::::::::::::::::::::::::::::::::::: :': :':: ::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: Hi,: .,:::::::: :::::j::::: :::::::::: :,., ::::::::: ,:,:,:.:,:2:,:,:2:,.... ·:::::::: :::, 2:':':.:':':':':'-:.: : :::::::::::::::::: and to express to .>j~. others both orally and in writing the results of mathe- matical investiga- tions and problem solving. The mathematics curriculum, therefore, must provide appropriate contexts in which students can learn to read, write, and speak about math- ematics. .. ...... .. ~ . . ~ ~ . . i. i ... ~ . _ . . ~ .............. ~ . . ~ ~ it. i it. ~ ~ ~ ~ ~ ~ ~ ~ ~...... . i ~. ......................... .. . ~_. .... ,,,,,,,,, i. I i ~.~ .~ ~ . ~ ~ · - d>,< ~, ':,:,:,:2:,:,:,:,:,:,:,: :,:,:::.:,:,:.:::,::::: :~: :::::: P. i: :2 . .''' '' ' ' ' ' ' ' ' ' ' . ' ' ' ' ' . ' . . ' . . ' ' . ' ' ' ' ' ' ' ' ' ' . ' 'X ::::::::::::::::::::::::::::::::: i::::::::::::::::::::::::::::::::::::::::::::::::::::.'.~ : :'': '': .:::: :. .:::::::: ~Jo,::::::::: :' ' ' ' 2 ' 2 ' ' ' ' ' ' ' ' ' ' . ' 2 ' ' ' ' ' ' ' ' 2 '' ' ' ' ' ' '' '':: :R : . .: : . ::::::::::.i::::::'::::::~ ~ . :<:::::: i:: ::: . . .:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:.:::::::::::::::::: :~::: :2 .':: id; X':':: :,:,: ~:: :iij,<~:: :,:2:,:,: :,: :,:,:,:,:,:.:: :,:,:,: :,:,:,: :,:,:,: :,:: :2:,:,:,:,: :,:.:,:,: :,: :2:'j'·: i.,i ;2'2. ~Xt< ' 2 ii ,,,, . ''-i''' '''- i'''i-'''' X ~:~:::::::::::::::::::::::::::::::::: D~::::::::::::::::::::: llfflPi:i::::::::::::~:~:~:~:~:~: ~.'.'.:::::::::::::::.::::.:.: ::::.:: ...,: :':':2:::,:2:2:,:2:2:,:,:,:,:':,:~:~:~:~:~:~:~:~:,:,:'' :::::::::::::::::::::::::::::: I ·.'2''.-22'".2' :: ::'2:2:2:': :::::::: ................... ...i22'22'.'...... "j................... i ' . 2 2 ' '' ,X5' i-'' k:':~:~:~:~:~:~:~:~:~:':': :~:~:~:~:~:~:~:::::~ A: :::::::::::: _: ~-.~,:: ::::: ::::::: ::::::::::: ,.. :::: :::,,.,~i: I_::::::::::::::::: ::::::::::::::::::::::::: `:.>j:>>>x~.~j;~,~,,~= Do, ,.-:,:::.: :,:,:.:,:,:,:::,:::: ::::::::::: :,:,:,:,:,:,::::: :,:- ~: ::::::::::: ::~::::: .:::: i::::: ::: id:::::: :::::::::::::::::::::::::::: ,,, do, '::::::::::::~.::,.:::::::.::.::: :::::: :::::.:::::::::::::::,: ~i:::::::: :::::::::::::i::~_~: : :::::::::: :::::::::::::::::::::::::::: _..~. :::::::::,: ~:::::::::.::::,::~::::~:,:,:,::,:,:,: go, :,: :::: i: i:: ::: ,: : :: .::: ,: ::.: :::: ::::: -_.. :::::::::':,:':::': :'::': :': :':::': :2:2: ::: :,::::: :,: ::: :,:::,:::: :,: deer::::::::::: :::::::::::::::: ::::::::::::: :::: Ax: . : :::::.:.:: :.::::::::::::::::::::~:::~:::: ma::::::::::::::: :::::: :::::::::::::::::::::::::: ::::: :': :': :': :2: :':':::':::::'::::: ::: ::: :':2:::'::::: ::: :::::::::: I.:.:.. .:. :...:. .: :.:.:.:.:.:.:.:.:.:.:.:::.:.:.:.:.:.:: ::: :::::: ::::::::: :....: ,,,, :,~,.:,.::.2 ,2, ' :' :': .,~: . ~ ~ it, ~ Principle 2: Calculators and computers should be used throughout the mathematics curriculum. Students will achieve mathematical power only if they see mathematics as a modern, relevant subject. New curricular materials must be designed in the expectation of continuous change resulting from further scientific And technological clevelopments. In mathematics, under- stancling cannot generally be achieved without active participation in the actual process of mathematics-in conjecture and argument, in exploration and reasoning, in formulating and solving, in calculation and verification, Calculators function like "fast pencils," so the mathemati- cal process can be made more useful and efficient than with paper and pencil. Computers, similarly, enable stu- clents to quickly calculate, graph, or simulate processes that are simply impossible to carry out by any other means. Instruction based on calculators and computers has, therefore, the potential to lead to more understand- ing than does traditional instruction, Calculators and computers also appeal to teachers because they introduce excitement and inventiveness to otherwise routine courses. Of course, technology should not be used just because it is appealing. But it must be used when it can enhance the teaching and learning of mathematics. There are very few portions of the curricu- lum where such improvement is not possible.

38 Reshaping School Mathematics Principle 3: Relevant applications should be an integral part of the curriculum. Students need to experience mathematical ideas in the context in which they naturally arise-from simple counting and measurement to applications in business and science. Calculators and computers make it possible now to introduce realistic applications throughout the curriculum. The significant criterion for the suitability of an applica- tion is whether it has the potential to engage students' interests and stimulate their mathematical thinking. Appealing applications should be cirown from the world in which the child lives, from community events, or from other parts of the curriculum-and not just from science, but also from business, geography, art, and other sub- jects. The primary goal of instruction should be for students to learn to use mathematical tools in contexts that mirror their use in actual situations, Mathematical ideas should always be presented and developed in the context of meaningful mathematical activities. Principle 4: Each part of the curriculum should be justified on its own merits. Mothematics offers such a rich array of interesting and useful ideas that choices are necessarily difficult, Howev- er, no concept or skill should remain in the curriculum just because it is there now. Although there is much that is timeless in the present curriculum, we can no longer afford as the chief justification for a topic that it is in the curriculum already. We need, instead, a "zero-based" cur- riculum process in which no idea is immune from careful scrutiny, Revision of curriculum should not be just an exercise in adding more topics. It should be, rather, a discipline of establishing priorities. Some emphases should be dropped, others added, and some retainecl. Even for important priorities that do remain, modern applications or technology may suggest quite different approaches. Often a fresh approach can avoid the rigidity of thought that inhibits desirable change, In many clistricts, the secondary mathematics curricu- lum especially is already full of topics, many treated too quickly, Nevertheless, there is much mathematics that could be made accessible and interesting to students, Unless traditional topics contribute directly to curricular goals, they clutter the curriculum, Neither the number of

39 A Philosophy and Framework topics nor the nature of the topics is as important as a curriculum that instills in students firm command of math- ematical thinking. Principle 5: Curricular choices should be consistent with con- temporary standards for schoo/ mathematics. The new Curriculum and Evaluation Standards for School Mathematics (National Council of Teachers of Mathematics (NCTM), 1989) exemplify the kind of broach curricular standards that should be used as criteria for assessing the merit of topics in school mathematics, Simi- lar stanclarcis are expressed in many recent state docu- ments, for example, California (Denham and O'Malley, 1985) and Wisconsin (Chambers, 1986), Although differing in many cletails, these various clocuments establish signifi- cant new goals for effective school mathematics, Local choices should be made in a manner consistent with these curricular stancdarcis, The pace of change is so great that even current curriculum guides are likely to be inadequate for tomorrow's needs, Curricular change requires sustained effort by people of vision, rooted in the reality of schools, yet with objectives firmly set on the future. Principle 6: Mathematics instruction at all levels should foster active student involvement. The proper use of technology requires new approach- es to teaching mathematics in which students will be much more active learners. Quite asicle from technology, research on how students learn suggests more effective ways to teach mathematics. Mathematics teaching must adapt to both of these clevelopments. It will no longer be appropriate for most mathematics instruction to be in the traditional mode where teachers present material to a class of passive students. No single teaching method nor any single kind of learn- ing experience can develop the varied mathematical abilities implied under the definition of mathematical power (Fey, 1979; Mathematical Sciences Education Board (MSEB], 1987~. What is needed is a variety of activi- ties, including discussion among pupils' practical work, practice of important techniques, problem solving, appli- cation of everyday situations, investigational work, and exposition by the teacher. Teachers should be catalysts who help students learn to think for themselves. They should not act solely as train- ers whose role is to show the "right way" to solve problems. In aciclition, classroom activities should provide ample

40 Reshaping School Mathematics opportunity for students to communicate with each other using the language of mathematics in both written and oral form, A useful metaphor is that of the teacher as an intellec- tual coach. At various times, this will require that the teacher be: · A role model who demonstrates not just the right way, but also the false starts and higher-orcler think- ing skills that lead to the solution of problems; · A consultant who helps inclividuals, small groups, or the whole class to decide if their work is keeping to the subject and making reasonable progress; · A moderator who poses questions to consider, but leaves much of. the decision making to the class; · An interlocutor who supports students during class presentations, encouraging them to reflect on their activities and to explore mathematics on their own; · A questioner who challenges students to make sure that what they are doing is reasonable and purpose- ful, and ensures that students can defend their con- clusions. All these roles serve well the most important aim of education, namely, to wean students from their teachers. Mathematics education must aim to make students self- sufficient, so that they can use mathematics effectively without a teacher present, as they must once they leave the school environment. To do this, schools must foster suf- ficient independence in students that they can function mathematically on their own while still in school, or there will be no possibility of their doing so after leaving school. Goals By themselves, general principles provide insufficient direc- tion to help focus curriculum development. More specific goals, related to the new Curriculum and Evaluation Stan- clarcis for School Mathematics (NCTM, 1989), must be built on the foundation provided by our expression of a practical phi- losophy of mathematics as a language and science of pat- terns, and on the related technological and research per- spectives. These goals offer a more constructive framework for the process of curricular change.

41 A Philosophy and Framework Mathematics education needs to be viewed as an integrat- ed whole, progressing continually from primary school through graduate school. Students learn in different ways and at differ- ent rates, in different directions and at different depths. Such differences cut across grade levels and school-level bound- aries. Many areas of mathematics, not just arithmetic, algebra, and geometry, should be seen as lengthy strands to be woven throughout all of school mathematics. Notwithstanding the continuity of mathematics, goals for different levels of school must reflect different stages in chil- dren's development of mathematical power. As new curricula are developed to meet the challenges of a changing society, they must strive to achieve certain broad goals that form an effective framework for school mathematics: A primary goal of elementary school mathematics is to clevel- op number sense. Student abilities to reason effectively with numerical information requires experience with: · Representation-the ability to use numbers to express quantitative data and relations, · Operations mastery of single-digit arithmetic; ability to determine appropriate arithmetic proce- dures; facility in estimation; experience in select ing appropriate means to carry out complex cal- culations. Interpretatio~the ability to draw inferences from data and check both the data and the infer- ences for accuracy and reasonableness. Elementary school mathematics should use con- crete materials, computer software, and calculators. It should emphasize mental arithmetic, particularly for estimating the results of multidigit computations. At the same time there should be a sharp reduction in time devoted to teaching the traditional written methods of calculation for multidigit numbers, fractions, and deci- mals. An elementary school curriculum that approaches arithmetic from this perspective will be strikingly differ- ent from the arithmetic commonly taught today. The central mathematical task of today's elementary school is to develop manual skill in a wide variety of operations on whole numbers, rational fractions, and decimals. Reducing emphasis on these topics while increasing opportunities for reasoning, for discovering patterns, for identifying correct procedures, and for drawing inferences will require a fundamental change in the conditions of teaching. A school mathematics

42 Reshaping School Mathematics program with this kind of emphasis offers the promise of impressive progress in the level of quantitative reason- ing. Elementary school mathematics should provide an effective foundation for all aspects of mathematics. If students are to be better prepared mathematical- ly for vocations as well as for everyday life, the elemen- tary school mathematics must include substantial sub- ject matter other than arithmetic · Geometry, including properties of two- and three- dimensional objects, symmetry and congruence, constructions of geometric figures, and transfor- mations of geometric figures; · Measurement, including units of measure, telling time, reading temperatures, and computing with money; · Data analysis, including collection, organization, representation, and interpretation of data; con- struction of statistical tables and diagrams; and the use of data for analytic and predictive pur- poses; Probability, introduced with simple experiments and data-gathering; · Discrete mathematics, including basic combina- torial thinking and the use of graphs to model problems. Each of these topics can play a distinctive role in making the elementary school mathematics curriculum more interesting and relevant to stuclents. Geometry provides an obvious window on the physical world, now enhanced through computer graphics, In mathe- matics as in life, a picture is worth a thousand words, Measurement provides meaningful applications even to very young children, as well as reinforcement of number concepts. Data analysis provides a source of interesting and relevant problems, as does probability, which can also be related to familiar games, Concepts from algebra can introduce students to simple aspects of abstraction, while discrete mathematics provides topics to relate mathematics to many areas, particular- ly computers. Moreover, instruction should be integrated so that relations among different areas will be perceived and

43 A Philosophy and Framework reinforcecl. For example, teachers should stress the use of arithmetic in geometry and probability, and the use of geometric concepts in the representation of data. Calculators should be available in all instructional and assess- ment situations. Calculators should be used in school mathematics from kindergarten on as crevices that children use to develop and discover number relationships and to solve problems. The replacement of most paper-and- pencil cirills with calculator-based instruction will not itself be a panacea. Although it is just as possible to assign minciless clri~ls with calculators as with paper and pencil, young children can instead be given activities with calculators that emphasize discovery and explo- ration in ways not possible or practical with paper and pencil. It is as important as ever for children to learn when and how to use addition, subtraction, multiplication, and division. But there is no evidence that drill-and- practice on standard algorithms leads to understand- ing. Mathematics educators must take advantage of ca~culator-based instruction as a tool to help students to achieve this unclerstancling. Students /earning mathematics should use real objects and real data. Observation is as fundamental to mathematics as to science. Young chilciren need to manipulate real objects as they learn to count and to explore arith- metic, To develop sound intuition for length, area, vol- ume, and shape, chilciren studying mathematics must draw, cut, fold, construct, pour, and measure. Children of ail ages must constantly explore the rela- tion between the relatively pristine patterns studied in school mathematics and the messier reoiity of worldly clata. Real data are more convincing than contrived data. The act of gathering data-whether by mea- surement, counting, polls, experiments, or computer simulation-enriches the chilcl's engagement in learn- ing. Moreover, the inevitable dialogue that emerges between the reality of measurement and the reality of calculation-between the experimental and the theo retical-captures the whole science of mathematics.

44 Reshaping School Mathematics Middle school mathematics should emphasize the practical power of mathematics. If instruction is to give students mathematical power, then problem solving needs to be emphasized throughout all grades. Students need to perceive mathematics as more than the subject matter itself-as, in fact, a discipline of reasoning that enables them to attack and solve problems of increasing diffi- culty and complexity, A focus on problems rather than just on exercises is important throughout the curriculum. Broadening the elementary school curriculum has important implications about entry into secondary school mathematics. The middle school grades should not be viewed as a time for consolidation or as a pause for rest, but as an essential part of a child's mathematical development. Its focus should be on mathematics for everyday life, a theme rich in motiva- tion that leads naturally to many important mathemati- cal topics (e.g., data analysis, geometric measure- ment, interest rates, and spreadsheet analysis). Understanding the concepts of elementary school mathematics is essential for the study of secondary school mathematics; however, proficiency in the pro- cedures of hand arithmetic computation should no longer be the critical factor in judging student readi- ness for advanced study. Mathematics in school should reinforce other school subjects, and vice versa. Much of the motivation for the development of mathematics-both historical and personal- is related to science, yet in school there are precious few honest links between mathematics and any of its applications. The applications of mathematics extend far beyond the natural sciences-to business, social science, geography, and various vocational and commercial areas. Young children can learn much mathematics in the context of explorations: experience with data, practice with arithmetic, and exposure to shapes and change. High school students need to experience applications in their mathematics classes as well as to use mathematics extensively in other classes. Since mathematics is both the language of science and a science of patterns, the special links between mathematics and science are far more than just those between theory and applications. The methodology of mathematical inquiry shares with the scientific method a focus on exploration, investigation, conjecture, evi

45 A Philosophy and Framework dence, and reasoning. Firmer school ties between sci- ence and mathematics should especially help strengthen students' grasp of both fields. A major goal of the secondary mathematics curriculum should be to develop symbol sense. The transition from elementary to secondary mathe- matics is characterized by a shift from concrete objects to abstract symbols. Developing fluency with symbols and other abstract entities which can be geometric, algebraic, or algorithmic-must be a cen- tral aim of secondary school mathematics. Student ability to reason effectively with symbols requires expe- rience with: · Representation-the ability to represent mathe- matical problems in symbolic form and to use these symbolic representations in relations, expres- sions, and equations; · Operations--the ability to determine appropriate symbolic procedures and to select appropriate means to solve problems expressed in symbolic form; · Interpretatio~the ability to ciraw inferences by reasoning with symbolic systems to check these results for accuracy and reasonableness. Computers and calcula- tors have, of course, an important role to play in the development of sym- bol sense. Since powerful calculators will have just as profound an effect on how symbolic manipulations are done as they have had on how arithmetic is done, the current empha- sis in secondary school on manipulative skills will need to be replaced by a larger emphasis on unclerstand- ing and problem solving. A valuable impact of tech- nology on the secondary curriculum will surely be the development of sophisticated software that will enable students to discover patterns rather than just to manip- ulate symbols, What type of common household phenomenon is represented by the following graph: E a, 4 - 3500 Be/ / time Mystery Graph

46 Reshaping School Mathematics Secondary school mathematics should introduce the entire spectrum of mathematical sciences. Secondary school mathematics must prepare stu- dents for the workplace, for college, and for citizen- ship, To meet these objectives, the curriculum must include a broad range of topics reflecting the full power of the mathematical sciences: · Algebra, including general algorithms and families of functions (polynomial, trigonometric, exponen tial, logarithmic). · Geometry, including transformational geometry, vector geometry, solid geometry, and analytic geometry. · Data analysis, including measures of uncertainty, probability and sampling distributions, and inferen- tial reasoning. Discrete mathematics, including combinatorics, graph theory, recurrence relations, and recur- sion-all emphasizing algorithmic thinking. Optimization, including mathematical modelling, "what if" analysis, systems thinking, and network flows. Stressing general algorithms in a computer context will make algebra and trigonometry more interesting. Despite its reputation as a subject that is boring and irrelevant, geometry has always been a subject of great potential interest because of its associations with the physical world. Data analysis can easily be related to interesting and significant applications, as can dis- crete mathematics and optimization. In teaching mathematics it is important to illustrate the unity and integrity of the discipline. For example, fractal geometry is quite accessible to high school stu- dents and involves aspects of algebra, geometry, and discrete mathematics, as well as providing fascinating uses of computers, Data analysis Reacts directly to alge- braic and geometric methods, while algebra and geometry themselves are joined in analytic geometry. The ties that bind topics to each other are often as important as the topics themselves.

47 . . . A Philosophy and Framework Students should apprehend that in mathematics, reasoning is the standard of truth. Learning to under- stancl and construct logi- cal, coherent mathe- matical arguments is a major goal of school mathematics. Euclidean geometry, however, is not the only vehicle for teaching students about reasoning. Both algebra and discrete mathemat- ics provide excellent opportunities for argu- ments expressed in oara- graph form; , even flowcharts and spread- sheets can be used to ........... . .... ................... ; ; ;. . .. ......... ~ I ~ ~ ~ ......... . ~ ............................ .. .. ....... . . ~ . ~ ::::::::::::::::::::::::::'.fflt ~ . ~ ,:, ................ ~ ~ ~ .. .............. . . A.< x ............................ ,, ....................................... .. . .... ~ . id. ::::- :; i, i..~....~...~..~. - ·: ::::::::::: ::::::::: ::: ::: ::::: :::: ::::::::: ::.: i:. ~ ...... ... ...... .... w ...... ... ...... ........ ................ ........... .. ................... .................................................................. .. ,,. ,., , ,,.,, . ~ , , .. , ,, , , ~ ........................................................... . . ~. ~ . ~ ............................................................................... .. ... .................... .................. ~ ~ . ............................................. ......... ............... ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: : ::.:.:~:.:~:~:::::::::~:::::~:::::::::::::::::::::::::::::::::::::::::::: : :::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::~:~:::::::::::::::id.:. <::::::::::::::::: - ~ - ~ - ~ ~ - ~ - - ~ :::.::: :: ::: ~ :::::::: ~$ :::::::::::::: :::::::::::::::: :::: :::::::::::::: :::~:::::::::::::::: ::::::::::: :::::::::::::::::::::::::::::: ::: .. ::::::::::::: ~ ~ ~ 1 hi::: :::::::: ~ :::::::::::::::::::::~:~ ~ ::::: :::: ::: :: >: A:::::::::: A:::::::::: :::::::::::: :::::::::::: S:::::::::: S:::::::::: &::::::::: ::::::: ;::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::;:::::::::::::::::::::;:::::::: :::::::::::::::::::::::::: ~:~:::::::::::::::::::::::::::::::::::::::::::::::::: A:::::::::::::::: ; ~... ~ ~ ~ ~ ~ ~.. ~ ~.: .. ~.. .. :.: .: :.:.:,:.:.:,:.:,:~j:.<j~j:~.,~j,~, - :;:;:;,? ~j j~j~j: ::. ::! ::::;::: :::::: \. :::::::: : :::: :::: :::::::::::: : : ::::::: :::::::::::::::::: :::::::::::::::::::::::: : ::::::::::::::::::::::::::: - ~ ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: : :::::::::::::: ::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: illustrate the logical nature of mathematical argument. More important than facility with formal proof is an understanding rooted in a variety of elementary exam- pies that mathematical truth is logical and not purely empirical. Young chilciren can develop a sense of logic from primitive experiences with numbers. Once sym- bols are understoocl, many basic ideas can be provecl, often in a variety of ways. Geometric proofs of alge- braic results (e.g., demonstrations of the Pythagorean formula by rearrangement of squares) are especially convincing to children who are still struggling to uncler- stancl symbolic expressions. All students should study mathematics every year they are in school. Mathematics should play an important role in the education of all students, not only of those preparing for college. The core of secondary school mathematics should be fundamentally the same for all students, although the clepth of presentation may vary. Enrich- ment beyond the core would naturally be clifferentiat- ecl to take into account the differing aspirations and probable further education of students. Students can learn to apply mathematics-incleed, they can often learn new mathematics in other subjects (e.g., science, geography, business) provicled appropriate links are forged between mathematics and these other

48 Reshaping School Mathematics disciplines. Mathematics, like writing, is a subject that should be regularly taught "across the curriculum." Advocating that all students take essentially the same core mathematics may appear unrealistic, given the uneven preparation students bring to high school mathematics. Indeed, the United States separates stu- dents by ability earlier-and into more tracks-than any other industrialized country. Many students, under this system, are tracked out of real secondary school mathematics,-being relegated to cleacl-end courses like "general mathematics." Too little has been expect ecl for too long from too many stuclents. Recognizing this, many give up much too early on studying signifi- cant amounts of mathematics in secondary school. More must be expected of our students than is the case now. Heightened expectation will surely lead to better performance. All high schools should offer four full years of mathe- matics appropriate for all students. It is particularly important for students to take mathematics throughout their last year of high school. All too open, mathemat- ics students who complete all of the mathematics available to them in the eleventh grade find that a year away from mathematics leaves them ill-prepared for college mathematics or for job-related require- ments. Enabling Conditions One cannot separate curriculum and instruction from the brooder context of education, To improve mathematics edu- cation, change must occur simultaneously in curriculum, in teaching, in professional development, in textbooks, and in assessment practices. Although our focus in this Framework is principally on the content of the curriculum, there are impor- tant implications of our recommendations for other parts of the educational context. Professional Development No significant curriculum reform will be possible without an effective program of professional development for mathemat- ics teachers' As teachers implement important, timely, and exciting changes, they will require continuing programs of pro. fessional support. Such programs will require a commitment from local school districts as well as leadership and funding at the state and federal levels.

49 A Philosophy and Framework Although the evidence is more anecdotal than analytical, it is likely that many elementary school teachers would welcome an opportunity to increase their confidence about teaching mathematics. Increased confidence would encourage alter- native styles of teaching that diminish emphasis on formalism and drill-and-practice, Any curriculum developed according to this framework will be more challenging to teach than the present curriculum. Therefore, particular attention must be paid to in-service training of elementary school teachers, as well as to the use of specialist mathematics teachers through- out the elementary school Oracles. Secondary and middle school teachers who are already prepared to some extent with a specialization in mathematics will need extensive continuing education both in new areas of content that are not part of their present repertoire, and in styles of teaching better suited to active student participation. Programs that provide vertical integration of teaching, learn. ing, and research experiences are well suited to this purpose since they enable teachers to experience for themselves the kind of open environment for learning that they should attempt to create for their stuclents. However, neither specialist teachers nor special programs to introduce a new curriculum can by themselves ensure suc- cess, Mathematics teachers, like other professionals, must engage in life-long programs of professional development. As professionals who must keep up with a rapidly changing and technically complex field, mathematics teachers especially need time and opportunity to read, to reflect, to plan, and to exchange ideas with other mathematics teachers. Further- more, for teachers to succeed with a technology-based cur- riculum, they will need properly equipped classrooms and appropriate rewards for the special effort and innovative teaching that will be required. Teachers' working environments must support teachers' professional lives: an improved profes- sional climate for teachers is absolutely critical for improve ment in mathematics education. Instructional Materials In the overwhelming majority of classrooms, the content of the textbook determines what is taught and how it is taught, Teachers may skip topics in textbooks, but they will seldom give significant attention to topics not included in texts, Nei- ther will most teachers approach a topic differently from the way it is treated in the text Therefore, real curriculum change is possible only if it is accompanied by new curricular materials, New textbooks must be designed and written to reflect the important principles of mathematics curricula: genuine prob- lems; calculators and computers; relevant applications; read- ing and writing about mathematics; and active strategies for

50 Reshaping School Mathematics learning. It is not sufficient for publishers to provide ancillary software or supplementary materials to be used with a text- book but that are not discussed in it. Neither will it be accept- able to relegate new material to separate sections, Publishers need to assess the entire structure and philosophy of current textbooks in light of research findings on how chil- ciren construct mathematical images and how they learn to make sense of formal procedures. If textbooks began to inte- arate relevant insights from cognitive research, they would ~ ~ _ , begin to exert a positive influence for change In school matn- ematics, Publishers also need to recognize that mathematics education will undergo significant and continuous change well info the next century. As a consequence, textbooks will become out-of-date more rapidly than in the past. This short- ening of the useful life of a textbook will require considerable adjustment in publishers' plans and in school districts' pur- chase policies. Textbooks must reflect fully a new conception of mathemat- ics education, integrating into the main subject matter of the text all important principles of mathematics curricula and edu- cational research. Unless textbooks help teachers to use actively calculators, computers, and genuine problems, new emphases such as technology and applications will continue to have insignificant impact on the curriculum, Unless text- books contain engaging projects and group activities, few teachers will have time to create them on their own, And unless textbooks include suitable assignments to enhance stu- dents' experience with reading and writing in the language of mathematics, students will remain deficient in their ability to communicate effectively. . . . . . Assessment Textbooks circumscribe what topics may be taught, but tests determine what topics will be taught. Too many of today's standardized tests stray far from both the available and the adopted curriculum; none even gets near the ideal curriculum. Just as new text materials must be developed in parallel with the new curriculum, so also should new strategies of and standarcis for assessment be clevelopecl as the curricu- lum is defined. A curriculum fitting this framework will require methods of assessment quite different from current ones. Guidelines for effective assessment are discussed in the Curriculum and Eval- uation Standarcis for School Mathematics (NCTM, 19891. Assessment must shape and guide instruction and not remain separate from it; it must determine not just what students do not know, but what they do know and how they think. Diag- nostic materials that probe student understanding can pro- vicle a springboard for improved instruction. Assessment must

51 A Philosophy and Framework permit full use of calculators ancl, where appropriate, of com- puters. Instruments can be developed to assess mathematical power rather than merely mathematical skill. But just as careful assessment of writing cannot be accomplished without having students write real essays, neither can mathematical power be assessed unless students have to solve nonroutine problems. Mathematics for the Future We are entering a decade in mathematics education of transition from entrenched precomputer traditions to new structures appropriate to the twenty-first century. These transi- tions will inclucle: Greater breacith of mathematical sciences. · More students who take more mathematics. · Increased use of technology. · More active learning. · Enhancecl professionalism for teachers. · Increased need for higher-order thinking skills. More sophisticated means of assessment. Effective change requires significant movement in each area, coordinated and sustained for an indefinite period. Efforts to orchestrate this change have only just begun, but must be continued. The tapestry of mathematics in the twenty-first century will be woven not just from the ancient threads of algebra and geometry, but also from more contemporary themes such as uncertainty, symmetry, data, algorithm, and computation. As applications expand the variety of roles played by mothemat- ics, and as computers reduce the role of routine calculations, the balance and connections among different parts of math- ematics will change significantly. Important threads in the tapestry extend throughout the entire range of the mathematics curriculum, providing rich opportunity for sustained development of a chilcl's mathemat- ical intuition and power. They lead to deep themes of contem- porary mathematics; they point to ideas that explain and unify the process of mathematical cliscovery; anGI they provide a secure base for mathematics' many applications. The chal- lenge for those who develop new mathematics curricula is to emphasize themes that both advance the power of mathe- matics and at the same time offer developmental opportunity for children's mathematical education.