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Redesign from ~ Technological Perspective _ =_ - _ _ _ ~ . _ . ~ Of the many forces at work that are changing the way mathematics is learned, the impact of technology is both most urgent and most contro- versial. In less than two decades society has moved from primitive electronic cal- culators to desk-top worksta- tions that are as powerful as the largest computers of only a few years ago. The unprece- dented magnitude and speed of change in technol- ogy has created a consider- able degree of professional confusion and public alarm about mathematics education, Mathematicians and parents are divided on the wisdom of early and widespread use of calculators before children have mastered arithmetic by traditional means. Calculators are seen as doing to arithmetic what many believe television has clone to reading. Concern about further deterioration in basic skills fuels a general fear of change that has produced a sus- tained public debate about the wisdom of calculator use, Many who view mathematics as an ideal instrument to filter students into professional and educational tracks believe that calculators make this filtering less effective-by enabling too many students to score well despite weakness in traditional skills. The image of a calculator as an inappropriate intellectu- a~ crutch is so deeply ingrained in many adult's minds-espe- cially among mathematicians and scientists-that most col- lege entrance exams do not permit calculator use. Advocates of early and unrestricted use of calculators- including virtually all mathematics educators argue on the basis of student motivation, classroom realism, and needs of

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18 Reshaping School Mathematics the workplace. Calculators are seen as an effective tool to transform the typical arithmetic lesson from worksheet drucigery into motivated exploration. Appropriate use of cal- culators can enhance opportunity for chilciren to learn higher- orcler thinking skills without first mastering standard computa- tional algorithms. Incleecl, early informal experience with multi- ple approaches to arithmetic problems~including calculators, fingers, ancl other devices-provides a secure base for subse- quent study of stanclarcl techniques. Calculators enable cur- ricula to move beyond emphasis on mechanics to experience with icleas. In contrast to calculators, use of computers in the schools is rather widely supported by the general public. However, among mathematicians and classroom teachers, their use is just as controversial as calculators-ancl for essentially the same reasons. Many mathematicians and teachers fear That time spent learning to use computer programs-whether it be programming languages such as Logo or Basic or packages such as Mathematica-is time subtracted from what they believe to be the central lessons of mathematics: solving prob- lems with paper, pencil, ancl pure thought. Electronic aicis like computers should be used by professionals to implement quickly anci accurately what they have aireacly learnecl, not usecl in education as an alternative to traclitional techniques for developing skill ancl unclerstanding. Despite the controversy, most mathematics educators who have studied the issues ancl the evidence have concluded that the potential benefit to mathematics education is enor- mous, well worth the extra effort ancl increased risk associated with venturing into uncharted territory (Wi~f, 1982; Fey, 1984; Hansen, 1984; Smith et al., 1988~. Increased use of technology in mathematics education is inevitable, but wise use is not automatic. Technology has more to offer education than just high-tech flash carcis. Effective use of calculators ancl com- puters requires objectives for mathematics education that are aligned with the mathematical neecis of the information age. New Opportunities From four-function calculators to clesk-top workstations, computer technology is poised to make an extraordinary impact on the content and presentation of mathematics edu- cation. Computing crevices will: Decrease the value of many manual skills traditionally taught in the school mathematics curriculum;

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19 A Philosophy and Framework Increase the importance of many areas of mathematics that now are rarely taught; Focus attention as much on problem formulation as on problem solving; Make possible tools for teaching and learning of a sophis- tication still largely undreamed of by most mathematics eclucators. More than any other empow- erment of technology, comput- er graphics will, in particular, totally transform the way math- ematics is used. Because the United States still leacis the world in most aspects of computer technology, we have a unique opportunity to grasp the poten- tial of this technology and use it to make dramatic improve- ments in mathematics education. A growing volume of research supports appropriate use of calculators in any grade. It is now clear that an understanding of arithmetic can be developed with a curriculum that uses estimation, mental arithmetic, and calculators, with reduced instruction in manual calculation. ~ncleecl, mental arithmetic may replace written methocis as the basic skill of our computer age. Since few arithmetic calculations are done most efficiently using paper and pencil, the level of arithmetic skill that is the current goal in most elementary school classrooms is far in excess of what is neeclec] for tomorrow's society. IncJeecl, there is some evidence that overemphasis on manual skills hinders the chilcl's learning of when and how to use them. Too often, skill rather than meaning becomes the message. Thus, any reform of school mathematics must entail a major reduction in the time spent on teaching traditional arithmetic skills. Technological developments suggest strongly that even those aspects of the secondary school curriculum that are ori- entecl mainly to development of algebraic skills such as poly- nomial arithmetic no longer serve a compelling purpose. In a computer age, facility in these skills is not an absolute prerequi- site either to the use of mathematics or to further study in mathematically based fields. Use a calculator to find three different numbers whose product is 7429. How many different answers can you find? Write a paragraph explaining what you did, why you did it, and how well it worked. New Priorities Reclucing priority on development of routine skills will allow a variety of clesirab~e consequences. There will be more time to Exploring Numbers

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20 Reshaping School Mathematics develop understanding of processes and reasoning that lie at the heart of mathematical problem solving (Conference Board of the Mathematical Sciences (CBMS), 19831. Indeed, enabling students to solve a variety of problems is one of the main purposes of school mathematics education. By reducing emphasis on manual skills, it will be easier to develop a curriculum that will allow all students some level of mathematical accomplishment while retaining the interest and enthusiasm of the more able students. The current emphasis on manual arithmetic skills, which any observant stu- dent knows are seldom used outside school, contributes to dis- taste for mathematics in many able students. For slower stu- denTs who fail to achieve quick mastery of arithmetic skills, there is no path to future success in mathematics. Less stress on skills will make possible an elementary school mathematics curriculum in which lack of success in one area will not neces- sarily preclude success in another. In such a curriculum, it will be possible to emphasize approx- imation and estimation, topics that play essential roles in many areas of mathematics (Schoen, 19861. Is it more important for a student to be able to multiply 2507 x 4131 precisely or to be able to say that the result is about 10 million? Often, the approximate answer is not only sufficient, but it also provides more insight than the exact answer. Moreover, the approxi- mate answer provides a quick check on the result of any exact proceclure, whether by a hand moving a pencil or by fingers pushing buttons on a calculator, A broader curriculum stressing a variety of mathematical strategies will make it possible to teach material to students in each grade that will be useful to them no matter when they end their mathematics education, At the same time, students preparing for further study of mathematics will be stimulated by early glimpses via the power of the computer~of what lies aheacl. Finally, computers and calculators have changed not only what mathematics is important, but also how mathematics should be taught (Zorn, 1987~: Computers and calculators change what is feasible and what is important. They make the difficult easy and the infeasible possible. For example, computers can display and manipulate mathematical objects such as compli- catecl three-climensional forms that cannot reasonably be studied without computers. As a consequence, stu- clents can solve realistic problems that are relevant to their everyday experiences and that have the potential of stimulating continuing interest in mathematics. Computers free the teacher for those tasks that only a teacher can do. For example, teacher and students can

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21 A Philosophy and Framework together explore conjectures. Computers provide a dynamic and graphic medium that offers many effective ways to present mathematical ideas, Technology makes mathematics realistic. Before the advent of calculators and computers, even the most able students could not perform calculations required for most realistic problems. Nor, for that matter, could teach- ers do such calculations for students without spending far too much time on the computations themselves. Now, computation itself is no longer a barrier. If the problem can be grasped by the stuclents, then it can be solved. Real date from real experiments can be analyzed. Equa- tions that represent significant physical situations can be solved, Many sophisticated concepts can be made more intelligible with computers than through any other means. The Year 2000 The eventual use of technology in the teaching and learn- ing of mathematics can be seen, at best, only dimly today, Few classrooms today are equipped to make the use of com- puters convenient and inviting for teachers. Software, even when of high quality, is open relevant only to narrow curricular objectives. Too often it is neither teacher-friendly nor student- friendly. Rarely is it coordinated with textbooks or curriculum. Despite these current problems, which are legacies of old technology, workstations of the 1990s will offer powerful, flexi- ble environments that will make possible a much improved symbiosis between teacher and technology. Developments in computer technology, both hardware and software, are notoriously hard to predict. Still, enough is known now to be able to make some reasonable predictions about what is desirable and feasible for computers in schools in the year 2000: All students should have available hand-held calculators with a functionality appropriate to their grade level. Cal- culators suitable for secondary school will by then have symbolic and graphics capability sufficient for all high- school level mathematics. All mathematics classrooms should have permanently installed contemporary computers with display units con- venienfly visible by all students. New schools may be equipped with desks that include built-in computers. There should be sufficient computer facilities available for laboratory and out-of-classroom neecis of all students. In

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' Ha, Em. - ,:,:,:,:,:,:,:,:,:,:,: :,:,:::jii:>i.:::::: ::::::::,:,:,:.: Em:::::::::::: : : :: : :::::::::::: :::::::::::::A.:::~:::::: : :::::::::::::::: ~.,x,~:::~ .~ :,: By:? ~,.,..,,,,:,:,:,:,:,:,:,:,:,:,:,:,:,:::::::::::: Sap::::::: hi.:;:: ::i ..~....~::::: ~ ,,,..,::::: ::::::::::a a' ~ ::':::::::::::::::::::'QXR hi,:,:::::::::: it::::::::::::... ::::::: , :,,:,, .. . '' ~ ~ i~:::::::::::::.:.:.:.:.:::..:.:::j$.~::S particular, portable computers should by then be available for students to borrow overnight as they would a book from the library i::::::::: :.:.:.:.:.:.:.:.:.:.:.:.:.:::.:.:.::: ;j,f,.,.. :q ................ ........... ~ ~ . - - >- I. -. . x:: ::: ~ _? ,' ::::: :':':':' ' ' ' ' ' ' ' ' 2 ' qSg38 _:::..: :: :~ _~::::::::::::::::::::::::4W00WO ~_:::::::.::: ::~:.:.:.:.:.:.:2:2:2:.:.:.:.:.:.:i888888 _~::::::::::::::::::2:2:':':':':':':':':':':':':::~ - :::::::::::::::~:.:.:.:.:.:.:.:.:.: : ~ 2 2 2 2 ' ' ' ,., ' ' ' ' ' ' ' ' ' ' ' ' ' jam ;.'.2.2.2.2.,..2 T.... ".'.'.'.'.' .'..2 .2 .2. ~ B2,,, ,,, ,a,, .,,,,,,,,,, 2 :2` t::::::::::::::::::::::::::::.y y:,:,:,:,:,:,:':":":2:, :':':':':':':':':':':':':'' '''2: K::::::::~:::::::::::::::::::::: - - - - - - -.- ~ ......... ,, T. ~'.'.'.' ,~^jj.2.'.'.'.'.2 ' ' , a; . < 222.2 `X:,:,,~,, ,.,.2 ~ . .,,, , 2,,,,,,,,, of,,,,,, ':: ... :::: it: ~ .: . J ~T r.::::: ,8: an. ::::::::::: j'!F.^ a::::::::::: Ad: at: ::::::::::::::::::::::::r~bi>- F: 2~2,5 2 ~,.d ........... ' ' '.~' ' :':. '..i] Facilities like these are beyond the budgets of most school districts toclay. However, by 2000 their relative cost should be much less than it is toclay. Districts need to plan now to ensure calculator and computer facilities ade- quate for all classes and all stuclents. School boards and administrators must plan school budgets to ensure full access to the tools of learning, espe- cially for districts with limited resources. This is an area in which government, business, and industry can effectively enhance education by cooperating on a plan to ensure full technological support in every classroom in Ameri- ca. Curricular blueprints developed today must be based on the technological reality of tomorrow's schools. Research Findings Of all the influences that shape mathematics education, technology stancis out as the one with greatest potential for revolutionary impact. it is also the area of greatest public con- cern, since it is so new, Without a rich base of experience on which to draw, it is very difficult to say just how technology can be most effectively used in mathematics eclucation. For- tunately, sufficient research has been clone by pioneers in this field to suggest general trencis and likely results. The effects of calculators in school mathematics have been studied in over 100 formal investigations during the past 15 years. These studies have tested the impact of a variety of kinds of calculator use-from limited access in carefully select- ed situations to access for all aspects of mathematics instruc- tion and testing. There have been two major summaries of research on calculator usage (Hembree and Dessart, 1986; Suydam, 1986~. In almost every reported stucly, the perfor- mance of groups using calculators equaled or exceeded that of control groups denied calculator use. The recent Hembree and Dessart mesa-analysis of 79 calcu- lator studies sorted out the effects of calculator use on six dimensions of attitude toward mathematics as well as on the

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23 A Philosophy and Framework acquisition, retention, and transfer of computational skill, con- ceptual understanding, and problem-solving ability. The analy- sis led to this conclusion (Hembree and Dessert, 1986~: Students who use calculators in concert with tradi- tional instruction maintain their paper-and-pencil skills without apparent harm. Indeecl, use of calcuia- tors can improve the average student's basic skills with paper and pencil, both in basic operations and in problem solving. Research suggests that access to calculators in a well- plannecl program of instruction is not likely to obstruct achievement of skill in traditional arithmetic proceclures. More optimistically, it appears that when students have access to calculators for learning and achievement testing, they per- form at significantly higher levels on both computation and problem solving. In particular, stuclents using calculators seem better able to focus on correct analysis of problem situations, The earliest educational use of computers was focused on computer-assisted instruction (GAI), often based on pro- grammed learning, most frequently for drill on rote skills. Sever- al reviews of research on fine effectiveness of CAI (e.g., Bangert-Drowns et al., 1985) have concluded that it is general- ly very effective, giving better achievement in shorter time than traditional instruction. Lately, principles of artificial intelligence have been applied to design of sophisticated tutors for algebra, geometry, and calculus. The designers suggest that the use of such tutors can yield ciramatic increases in student achievement. However, no data are yet available about the use of such tutors in realistic classroom settings. There are several kinds of computer-based systems that give students powerful new tools for learning in an exploratory envi- ronment. Best known is Logo; its turtle graphics teach students concepts of geometry, algebra, and higher-order thinking (Papert, 1980~. Although research findings have failed to con- firm the strongest claims that Logo clevelops a high level of general reasoning, a variety of studies have found positive effects on more specific instructional goals (Campbell, 1989~. Moreover, thousands of classroom teachers have been con- vinced by first-hand experience that Logo is a powerful instructional tool. A different sort of computer-based exploratory tool is pro- videcl by the Geometric Supposer (Schwartz and Yerushalmy, 1987) and Geodraw (Bell, 1987~. Each provides students with open but guided environments for exploring the results of geo- metric constructions. Green Globs (Dugdale, 1982) provides a comparable setting for algebraic exploration. Although there is Title formal research describing the effects of these learning

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24 Reshaping School Mathematics ........ . . ,,,.,,. I. ~ ~ . . ~ ~ ~ I. . ....... ......... ... . ~ . ~ ~ ~ ~. ........ . .. . i i;. ;; ~ I,,,, , ,, . ., .,, ~ I, ~ ~ ~ ,, ............ .............. .... ,, .., I, ........................ ... ........ ................. . . ,,, .,,,.., ,.,,,,,...,,,,,,. ~ I...... ... .... ................. ... .... . . . ~ I,. .. . ............................ .. .... ............................ ...... . .... ............................ . . i Hi,,,,, . ........................... . . . ... ..... . . , . .. . .. . . .. . .. ............. ........ ........ .. ....... .. ~i,, I, ...... ,.. ...... .... .. ........ .,.,.a ....... ., . .......... A.., j , ~ . ., ,.. ,. I. ~ . . .' I,,. . ~ I.' ::::.::.:...1 , ......... .... . .. .......... .......... ....... ...... . . . . ... . ,, . I I. ~ ~ . ~ I. ~ ~ . ~ . ... . .. . ~ . . . .. ~ ~ ~ ~ ~ . ~ .. ...................... ........ ................ ... ,, ~ ,,, I,, ,.,,,,,,,,., ~ , . ..... .. . . . ..... .. ....... :::::::::::::::::::::::::::::::::::::::::: .::::::::::~::~:::::::::::::::::::::::::: i::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::: : ::::::::::::::::::::: : : : : : : : : : : : : : : : : : : : : : : : : : : : ::::::::::::::::::::: :::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::: :. :::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::: and teaching tools, there are some suggestions (Yerushalmy et al,, 1987) that students may perform as well or better than control students on traditional criteria while at the same time learning other objectives. Some investigators have studied the effects of computer graphics on student under- standing of mathematical con. cepts like function (Rhoads, 1986; Schoenfelcl, 1988a) or statistics and data analysis (Swift, 1984~. In each case, the computer seems clearly to enhance student interest and understanding of important ideas. Although most studies have focused, one way or another, on finding better ways to reach traditional goals, there have been some daring departures from conventional curriculum priorities. Both Lesh (1987) and Heid and Kunkle (1988) tested the effects of experimental algebra instruction in which stu- dents used symbol manipulation software to perform routine tasks like solving equations. Each found that students who were freed from the traditional symbolic procedural aspects of problem solving became much more adept at problem for- mulation and interpretation. In two similar studies of computer-aided calculus, Heid (1988) and Palmiter (1986) found that students who learned calculus with the aid of computer software developed a much deeper understanding of fundamental concepts than did students in traditional skill-oriented courses. Heid also found that her students picked up needed procedural knowledge in a short time period following careful instruction in conceptual background, and Palmiter found that her students acquired their understanding much more quickly than students in con- ventional courses. Open Questions Most current research addresses fundamental questions of technology applied to the mathematics curriculum: What are the essential interactions among conceptual development, procedural knowledge, and problem solving? This research

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25 A Philosophy and Framework indicates that access to computers and calculators need not hinder attainment of traditional curricular objectives, and that it may substantially advance it. Unfortunately, there is no con- sensus on how to investigate new effects such as the improve- ment of higher-order thinking skills. A series of articles in Ecluca- tional Researcher (Becker, 1987; Papert, 1987; Pea, 1987b; Walker, 1987) illustrate the wide diversity of opinion on this topic, A key concern is the extent to which the development of mathematical power can be inferred from written test per- formance or within the limited time spans of most research studies, It has sometimes been proposed that the availability of computers would, more or less in itself, produce significant improvements in mathematical thinking, From the few attempts that have been made to measure changes in rea- soning power, it is possible to conclude that such advance- ments cannot come from trivial technological fixes, Repeated attempts to document such change has yet to reveal a lasting effect for example, studies of the effect of Logo on planning, of the impact of Pascal on understanding of algebraic syntax, and of the cognitive impact of learning metaprinciples of pro- gramming in Basic, While these results do not necessarily imply that computers will not improve mathematical thinking, they do suggest that simplistic approaches are not likely to pro- duce measurable improvements. Rapid changes in the objectives and strategies of mathe- matics education have outpaced the evidence of effective- ness provided by educational research (NRC, 1985~. Introduc- tion of calculators and computers, especially, opens up many new issues that need careful study: Organization for Learning. Changes in curriculum, in teaching practice, and in the educational role of com- puters and calculators provide both opportunity and compelling need for new research on the effectiveness of different strategies. Computers virtually compel reordering and new combinations of traditional topics. What orders yield optimal learning? Levels of Learning. Technology makes possible earlier introduction of certain topics (e.g., decimals). What is the relation between the stage of introduction and ultimate understanding? Modes of Learning. As instruction recognizes an active role for students in constructing their own knowledge, we need to monitor the long-term impact of this approach on stuclents' abilities to learn and to use mathematical concepts throughout their lives, Manipulative Skills. Powerful calculators compel reexami- nation of traditional priorities for arithmetic and algebraic

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26 Reshaping School Mathematics skills Which skills best support mathematical power, and when must they be taught? Procedural and Conceptual Knowledge. What level of manipulative skill is necessary in order to be able to understand-and thus use-mathematics in a problem- solving context? Transfer of Knowledge. How can school instruction pro- vicle students with a background that will enable them to apply what they have learned in out-of-school contexts? Instructional Uses of Technology. Technological research has just begun to create tools with the power to alter sig- nificantly the traditional process of instruction. What kinds of mathematical comprehension can these new tools fess ter?