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TEA CHINO Hi There is little we do in America that is more important than teaching. Elective teaching of mathematics requires appropriate pedagogical and mathematical foundations, but thrives only in an environment of trust which encourages leadership and innovation. In short, teaching must become more professional. Under current conditions, most school teachers face nearly insurmountable obstacles. Lacking freedom to establish fully either ends or means, teachers rarely have the opportunity to exercise truly professional responsibilities. What emerges, often, are foreshortened ideals and shattered dreams. Evidence from many sources shows that the least e~ec- tive mode for mathematics learning is the one that prevails in most of America's classrooms: lecturing and listening. Despite daily homework, for most students and most teach- ers mathematics continues to be primarily a passive activity: teachers prescribe; students transcribe. Students simply do not retain for long what they learn by imitation from lec- tures, worksheets, or routine homework. Presentation and repetition help students do well on standardized tests and lower-order skills, but they are generally ineffective as teach- ing strategies for long-term learning, for higher-order think- ing, and for versatile problem-solving. Teachers, however, almost always present mathematics as an established doctrine to be learned just as it was taught. This "broadcast" metaphor for learning leads students to expect that mathematics is about right answers rather than about clear creative thinking. In the early grades, arithmetic becomes the stalking horse for this authoritarian mode! of learning, sowing seeds of expectation that dominate student attitudes all the way through college. - , , _ _ . . ~. . . Understanding Mathematics Many students master the formalisms of mathematics without, at first, any real understanding. Some go on to achieve a retrospective understanding after they have reached a more advanced vantage point. Surprisingly, many mathematicians and scientists recall that their own educa ...learning through involvement Myth: Learning mathematics means mastering an immutable set of basic skills. Reality: Skills are to mathe- matics what scales are to mu- sic or spelling is to writing. The objective of learning is to write, to play music, or to solve problems not just to master skills. Practice with skills is just one of many strategies used by good teachers to help students achieve the broader goals of learning. 57
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Teaching Back to School A table of data gives information on stopping distances for several cars in terms of speed, weight, and types of brakes (drum, disc, antilock). The information in- cludes both reaction time dis- tance as well as braking distance. Develop a model for these data in terms of graphs, equations, or computer programs that would enable one to predict how other cars would handle under similar circumstances. Compare what the model predicts with the advice . . . . . given In driver ec ~ucat~on courses. 58 tion fits this model; rarely is anything learned well until it is revisited from a more advanced perspective. The vast majority of students never move beyond for- mal knowledge since they do not persist in subsequent work to reach the point where the veil of confusion is lifted. (Those who do persist are likely to be the ones who sub- sequently go on to careers in science.) Present educational practice in the United States offers students only one path to understanding a long, dimly lit journey through a moun- tain of meaningless manipulations, with the reward of power and understanding available only to those who complete the ourney. P· · · · · . . . . · . . . resent educational practice offers mathematics students only a dim light at the end of a very long tunnel. Most students do not find the light at the end of the tunnel sufficient to illuminate their journey through the mathemat- ics curriculum. Far too many abandon their effort before receiving any benefit from the power of retrospective un- derstanding. To improve mathematics education for all stu dents. we need to exnancl teaching r~rnctin~s that Name ~nr1 . ~ . . ~ . .. . ~ . ~ . . ~ . motivate students as they struggle with their own 1earnlng. In addition to beckoning with the light of future understand- ing at the end of the tunnel, we need even more to increase illumination in the interior of the tunnel. :Learning Mathematics In reality, no one can teach mathematics. Effective teach- ers are those who can stimulate students to learn mathemat- ics. Educational research offers compelling evidence that stu- dents learn mathematics well only when they construct their own mathematical understanding. To understand what they
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...learning through involvement learn, they must enact for themselves verbs that permeate the mathematics curriculum: "examine," "represent," "trans- f 0 r m , " " s 0 ~ v e , " " a p p ~ y , " " p r 0 v e , " " c 0 m m u n i c a t e . " T h i s h a p - pens most readily when students work in groups, engage in discussion, make presentations, and in other ways take charge of their own learning. All students engage in a great deal of invention as they learn mathematics; they impose their own interpretation on what is presented to create a theory that makes sense to them. Students do not learn simply a subset of what they have been shown. Instead, they use new information to modify their prior beliefs. As a consequence, each student's knowledge of mathematics is uniquely personal. _P tudents retain best the mathematics that they learn by processes of internal construction and experience. Evidence that students construct a hierarchy of under- standing through processes of assimilation and accommo- dation with prior belief is not new; hints can be found in the work of Piaget over fifty years ago. Insights from con- temporary cognitive science help confirm these earlier ob- servations by establishing a theoretical framework based on evidence from many fields of study. Engaging Students No teaching can be effective if it does not respond to stu- dents' prior ideas. Teachers need to listen as much as they need to speak. They need to resist the temptation to con- tro! classroom ideas so that students can gain a sense of ownership over what they are learning. Doing this requires genuine give-and-take in the mathematics classroom, both among students and between students and teachers. The Myth: Students learn by remem- bering what they are taught. Reality: Students construct mean- ing as they learn mathematics. They use what they are taught to modify their prior beliefs and be- havior, not simply to record and store what they are told. It is stu- dents' acts of construction and invention that build their math- ematical power and enable them to solve problems they have never seen before. 59
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Teaching Myth: The best way to learn how to solve complex problems is to decompose them into a se- quence of basic skills which can then be mastered one at a time. Reality: Research in learning as well as instructional practice of other countries offer little sup- port for this strategy of teach- ing. There is abundant evidence that mastery of necessary skills is rarely sufficient for solving complex problems. Moreover, many other countries introduce students to complex problems well before they have studied all of the prerequisite skills. Those students often invent effective ap- proaches to the problem, thereby gaining valuable experience in higher-order thinking. Robust Arithmetic How do you add up long lists of numbers? A dozen different people do it in a dozen differ- ent ways-top down, bottom up, grouping by tens, bunching, and various mixtures. There is no sin- gle correct method. 60 best way to develop effective logical thinking is to encourage open discussion and honest criticism of ideas. Clear presentations by themselves are inadequate to re- place existing misconceptions with correct ideas. What stu- dents have constructed for themselves, however inadequate it may be, is often too deeply ingrained to be dislodged with a lecture followed by a few exercises. To change beliefs, students need to have a stake in the outcome. Honest questions by teachers are rare in mathematics classrooms. Most teachers ask rhetorical questions because they are not so much interested in what students really think as in whether they know the right answer. Soon, students are plaguing teachers with their own rhetorical question: "Can't you just tell me the answer?" ·~ rid ~. ~ when students explore mathematics on their own, they construct strategies that bear little resemblance to the canon- ical examples presented in standard textbooks. Just as chil- oren need the opportunity to learn from mistakes, so stu- dents need an environment for learning mathematics that provides generous room for trial and error. In the long run, it is not the memorization of mathematical skills that is particularly important without constant use, skills fade rapidly but the confidence that one knows how to find and use mathematical tools whenever they become necessary. There is no way to build this confidence except through the process of creating, constructing, and discovering mathemat ~cs. · M ............................... athematics teachers must involve students in their own learning. Classes in which students are told how to solve a quadratic equation and then assigned a dozen homework problems to learn the approved method will rarely stimulate much lasting mathematical knowledge. Far better is an approach in which
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...learning through involvemer't students encounter such equations in a natural context; ex- plore several approaches to solutions, including estimation, graphs, computers, and algebra; then compare various ap- proaches and argue about their merits. Of course, classes such as this where active learning is a way of life require more time and energy on the part of both teachers and stu- dents than either is accustomed to giving under present con · ~ c Tons. Teachers' roles should include those of consultant, mod- erator, and interlocutor, not just presenter and authority. Classroom activities must encourage students to express their approaches, both orally and in writing. Students must en- gage mathematics as a human activity; they must learn to work cooperatively in small teams to solve problems as well as to argue convincingly for their approach amid conflicting ideas and strategies. There is a price to pay for less directive strategies of teach- ing. In many cases, greater instructional effort may be re- quired. in those parts of the curriculum where mathematics directly serapes another discipline (for example, engineering), students may not march through the required curriculum at the expected rate. In the long run, however, less teaching will yield more learning. As students begin to take responsi- bility for their own work, they will learn how to learn as well as what to learn. %,, ~ Impact of Computers Calculators and computers compel reexamination of prior- ities for mathematics education. How many adults, whether store clerks or bookkeepers, still do long division (or even long multiplication) by paper and pencil? How many scien .. . . ~ . ~ . ~ tlStS or engineers use paper-and-pencil methods to carry out their scientific calculations? Who would trust a bank that kept its records in ledgerbooks? Those who use mathematics in the workplace-account- ants, engineers, scientists rarely use paper-and-pencil pro- cedures any more, certainly not for significant or complex analyses. Electronic spreadsheets, numerical analysis pack- ages, symbolic computer systems, and sophisticated com Calculators vs. Computers Polls show that the public gener- ally thinks that, in mathematics education, calculators are bad while computers are good. Peo- ple believe that calculators will prevent children from mastering arithmetic, an important burden which their parents remember bearing with courage and pride. Computers, on the other hand, are not perceived as shortcuts to undermine school traditions, but as new tools necessary to soci- ety that children who understand mathematics must learn to use. What the public fails to recog- nize is that both calculators and computers are equally essential to mathematics education and have equal potential for wise use or for abuse. 61
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Teaching Tomorrow's Computers Even as teachers struggle to adapt yesterday's curriculum to today's computers, industrial leaders are designing tomorrow's tech- nology. Multimegabyte memory and gigabyte storage support un- precedented graphics, unleash- ing potential for interactive text- books, remote classrooms, and integrated learning environments. As today's computer visions be- come tomorrow's verities, they will revolutionize the way mathe- matics is practiced and the way it is learned. 62 puter graphics have become the power tools of mathematics in industry. Even research mathematicians now use comput- ers to aid exploration, conjecture, and proof. In spite of the intimate intellectual link between mathe matics and computing, school mathematics has responded hardly at all to curricular changes implied by the computer revolution. Curricula, texts, tests, and teaching habits but not the students are all products of the precomputer age. Little could be worse for mathematics education than an en- vironment in which schools hold students back from learning what they find natural. It is true, as many say, that we are not sure how best to teach mathematics with computers. Nevertheless, despite risks of venturing into unfamiliar territory, society has much to gain from the increasing role of calculators and computers in mathematics education: · School mathematics can become more like the mathemat- ics people actually use, both on the job and in scientific applications. By using machines to expedite calculations, students can experience mathematics as it really is as a tentative exploratory discipline in which risks and failures yield clues to success. · Weakness in algebraic skills need no longer prevent stu- dents from understanding ideas in more advanced math- ematics. Just as computerized spelling checkers permit writers to express ideas without the psychological block of terrible spelling, so will the new calculators enable moti- vated students who are weak in algebra or trigonometry to persevere in calculus or statistics. Calculators in the cIass- room can help make higher mathematics more accessible. · Mathematics learning can become more active and dy- namic, hence more elective. By carrying much of the computational burden of mathematics homework, calcu- lators and computers enable students to explore a wider va- riety of examples; to witness the dynamic nature of math ematical processes; to engage realistic applications using typical not oversimplified data; and to focus on impor- tant concepts rather than routine calculation. · Students can explore mathematics on their own, to ask and answer countless "what if'' questions. Although calculators
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...learning through ir~volrement and computers will not necessarily cause students to think for themselves, they can provide an environment in which student-generated mathematical ideas can thrive. Time invested in mathematics study can build long-lasting intuition and insight, not just short-lived strategies for cal- culation. Innovative instruction based on a new symbio- sis of machine calculation and human thinking can shift the balance of learning toward understanding, insight, and mathematical intuition. . Ten years ago, arithmetic fell to the power of inexpensive hand calculators; five years ago, scientific calculators offered at the touch of a button more sophisticated numerical math- ematics than most students knew anything about. Today's calculators can do a large fraction of all techniques taught in the first two years of college mathematics. Tomorrow's calculators will do what computers do today. Priorities for mathematics education must change to reflect the way computers are used in mathematics. The ready availability of versatile calculators and com- puters establishes new ground rules for mathematics edu- cation. Template exercises and mimicry mathematics- the staple diet of today's texts will diminish under the assault of machines that specialize in mimicry. instructors will be forced to change their approach and their assignments. it will no longer do for teachers to teach as they were taught in the paper-and-pencil era. Education of Teachers Mathematics is taught in every grade throughout the entire thirteen years of school, K-12; consistent growth in skills, Myth: The way to improve stu- dents' mathematical performance is to stress the basics. Reality: Basics from the past, es- pecially manual arithmetic, are of less value today than yesterday- except to score well on tests of basic skills. Today's students need to learn when to use math- ematics as much as they need to learn how to use it. Basic skills for the twenty-first century in- clude more than just manual mathematics. 63
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Teaching Back to School Record the height and weight of everyone in your class and see what relationship you can deter- mine. Do age or sex help make the relationship clear? 64 maturity, and understanding is essential as students move from one level to the next. As a chain that breaks at its weakest link, mathematics instruction is especially vuIner- able to weakness in any grade or course. For this reason, the preparation of mathematics teachers is a crucial factor in revitalizing curricular practice. Too often, elementary teachers take only one course in mathematics, approaching it with trepidation and leaving it with relief. Such experiences leave many elementary teach- ers totally unprepared to inspire children with confidence in their own mathematical abilities. What is worse, experi- enced elementary teachers often move up to middle grades (because of imbalance in enrolIments) without learning any more mathematics. Those who would teach mathematics need to learn con temporary mathematics appropriate to the grades they will teach, in a style consistent with the way in which they will be expected to teach. They also need to learn how students learn what we know from research (not much, but impor- tant), and what we do not know (a great deal). They need to learn science (including technology, business, and social science) so that they can teach mathematics in the contexts where it arises most naturally in measurement, graphs, pre- diction, decisions, and data analysis. And they need to learn the history of mathematics and its impact on society, for it is only through history that teachers will come to know that mathematics changes and to see the differences between contemporary and ancient mathematics. The United States is one of the few countries in the world that continues to pretend-despite substantial evidence to the contrary that elementary school teachers are able to teach all subjects equally well. it is time that we identify a cadre of teachers with special interests in mathematics and science who would be well prepared to teach young children both mathematics and science in an integrated, discovery- based environment.
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...learning through involvement T· - he United States must create a tradition of elementary school specialists to teach mathematics and science. · · - Many models for mathematics specialists are possible, most of which are in place in different school districts today. Implementation can range from paired classes-one teacher for language arts, the other for mathematics and science to certified specialists who lead curricular development and as- sist regular classroom teachers. Many teachers already have the interest, experience, and enthusiasm for such positions; others could qualify through special summer institutes. To encourage more widespread adoption of diverse pat- terns for mathematics specialists, states must alter certifica- tion requirements to encourage these new models. Then uni- versities must implement new courses with open construc- tive instructors so that prospective school teachers can grow in confidence as a result of their university study of mathe- matics. The content of the special mathematics courses for prospective elementary and middle school teachers those who do not undertake a standard mathematics major must be infused by examples of mathematics in the world that the child sees (sports, architecture, house, and home), examples that illustrate change, quantity, shape, chance, and dimen- sion. Teachers themselves need experience in doing mathema- t~cs In exploring, guessing, testing, estimating, arguing, and proving in order to develop confidence that they can re- spond constructively to unexpected conjectures that emerge as students follow their own paths in approaching math- ematical problems. Too often, mathematics teachers are afraid that someone will ask a question that they cannot answer. Insecurity breeds rigidity, the antithesis of mathe- matical power. Since teachers teach much as they were taught, university courses for prospective teachers must exemplify the highest standards for instruction. However, most mathematics that Back to School You have 10 items in your gro- cery cart. Six people are wait- ing in the express lane (10 items or less); one person is waiting in lane 1 and two people are wait- ing in lane 3. The other lanes are closed. What additional infor- mation do you need to know in order to determine which lane to join? 65
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Teaching 66 teachers have studied has been presented only in the author- itarian framework of Moses coming down from Mt. Sinai. Very few teachers have had the experience of constructing for themselves any of the mathematics that they are asked to teach, of listening to students who are developing their own mathematical understandings, or of guiding students to their own discovery of mathematical insights. P· · . . . . . . · . . . . . . . . . . . . . . . . . . . . . . . . . respective teachers should learn mathematics in a manner that encourages active engagement with mathematical ideas. All students, and especially prospective teachers, should learn mathematics as a process of constructing and inter- preting patterns, of discovering strategies for solving prob- lems, and of exploring the beauty and applications of math- ematics. Above all, courses taken by prospective teachers must create in these teachers confidence in their own abil- ities to help students discover richness and excitement in mathematics. Resources Textbooks and their ancillary products (worksheets, home- work exercises, testbanks) dominate mathematics teaching at all levels, from primary school through college. In no other subject do students operate so close to a single prescribed text; neither library work nor laboratory work, neither term papers nor special projects are common parts of mathemat- ics instruction. Classroom mathematics is the study of set texts and set problems that rarely have any parallel either in the world of work or in the many disciplines that depend on mathematics as a tool. Quite apart from the limitations imposed on classroom practice by excessive reliance on textbooks, the very impor
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...learning through involvement lance of the text as the primary medium of instruction poses unique challenges and opportunities: How can major school textbook series or course texts adapt quickly to changes in curricular goals or emerging technol- ogy? · Are we on the verge of a post-Gutenberg revolution in which computer communications can deliver flexible in- teractive texts more readily and efficiently than the print- ing press? How can texts and software act as incentives rather than as brakes for the newly emerging standards for school math- ematics? Even while educators work to reduce the dominance of text-based learning in mathematics classrooms, publishers and teachers need to explore new modes of publication that will enable good innovative ideas to enter expeditiously into typical classroom practice. Texts, software, computer net- works, and databases will blend in coming years into a new hybrid educational and information resource. It is already true that the most common application of school mathemat- ics is to program formulas into computer spreadsheets. As texts evolve and software matures, both must be synchro- nized with forward-Iooking curricular and classroom objec- tives of mathematics education. Assessment Governors and political leaders in all fifty states are advo- cating assessment in order to raise expectations and evaluate programs. Tests serve many important purposes. They allow students to recognize personal success; they enable teach- ers to judge students' progress; they provide administrators means to measure the effectiveness of instruction; and they afford the public accountability for the use of public funds. When designed and used properly, tests and other assess- ment instruments provide feedback that is essential for any system to maintain steady progress toward its objectives. "According to virtually all studies of the matter, textbooks have become the de facto curriculum of the public schools . . It is therefore critical that textbooks stimulate rather than deaden stu- d~ents' curiosity, and that teacher manuals encour- age rather than squelch teachers' initiative and flexibility." Harriet Tyson-Bernstein · · ~ 67
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Teaching Myth: Minimal competency ex . . . . am~nat~ons raise stuc tent accom- plishment. Reality: The typical effect of a . . . minima competency examlna- tion is to reallocate limited in- structional resources from the average students who have no trouble passing such a test to the weakest students who need spe- cial help. Since the effort to bring all students up to the level set by the exam is so great, the progress of the majority is often slowed down while that of a few is im- proved. Many weak students fall further behind; often, the floor is too low for all. On the whole, more students lose than gain from such rigid structures. 68 Unfortunately, tests in mathematics education are rarely used in a manner appropriate to their design. Tests designed for diagnostic purposes are often used for evaluating pro- grams; scores from self-selected populations (for example, takers of Scholastic Aptitude Tests) are used to compare districts and states; and commonly used achievement tests stress simple skills rather than sophisticated tasks, not be- cause such skills are more important, but because they are easier to measure. Tests are dear to the public; they produce winners and losers, as do sports playoffs, primary elections, and lotter- ies. Tests also symbolize commitment to things we value to facts and information that we once learned and that we believe all children should still learn. In America (but not in other countries), objective, multiple-choice tests are the norm; they are efficient, economical, and seemingly fair. l V I .................. 1 ~ Mathematical assessment in America relies excessively on misleading multiple-choice tests. Nonetheless, multiple-choice tests as used in America lead to widespread abuses, which the public rarely recognizes: · Tests become ends in themselves, not means to assess ed- ucational objectives. Knowing this, teachers often teach to the tests, not to the curriculum or to the children. Tests stress lower- rather than higher-order thinking, em- phasizing student responses to test items rather than orig- inal thinking and expression. Test scores are sensitive to special coaching, which aggra- vates existing inequities in opportunities to learn. Tests reinforce in students, teachers, and the public the narrow image of mathematics as a subject with unique correct answers. · Timed tests stressing speed inhibit learning for many stu- dents.
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. · Normed tests ignore the vast differences in rates at which children learn. · Tests provide snapshots of performance under the most stressful environment for students rather than continu- ous information about performance in a supportive atmo- sphere. · Poor test scores lead students to poor self-images, destroy- ing rather than building confidence. Similar problems arise when detailed learner outcomes rather than teacher judgments define the objectives of courses. Like items on objective tests, specific learner out- comes bias teacher effort and constrain student learning. The most important goals for mathematical learning cannot be atomized into tiny morsels of knowledge. Too often, good intentions in testing can lead to very bad results. Minimal competency testing often leads to minimal performance, where the floor becomes a ceiling. In stress- ing the importance of basic skills, such tests fad! to encour- age able students to progress as far as they can. As political pressure for state assessment begins to encompass higher ed- ucation, where assessment is even more complex, it is vitally important that the mathematical community agree on proper standards for assessment. W ............................... hat is tested is what gets taught. Tests must measure what is most important. Assessment should be an integral part of teaching. it is the mechanism whereby teachers can learn how students think about mathematics as well as what students are able to ac- complish. But tests also are used to compare classes and schools, to evaluate teachers, and to place students in future courses or careers. Because assessment is so pervasive and has such powerful impact on the lives of both students and teachers, it is very important that assessment practice align properly both with the purpose of the test and with curricular objectives. ..learning through involvement Myth: Only objective tests yield reliable results. Reality: Experience in evaluat- ing student writing shows that trained readers judging whole es- says produce results more aligned to the goal of high-quality writ- ing than do objective exams of grammar and vocabulary. Sim- ilar experiences show that one can reliably judge scientific un- derstanding by observing student teams in a laboratory. Effective means of assessing operational knowledge of mathematics must be similarly broad, reflecting the full environment in which em- ployees and citizens will need to use their mathematical power. 69
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Teaching Voice of Experience "We weren't happy with per- formance on conceptual and problem-solving parts of national standardized tests. So we threw out the texts ant! are teaching ele- mentary school mathematics with concrete materials and teacher- made worksheets. Now our kids are doing much better on na- tional and state tests and when we hear from parents, it is to fine! out how to get their kids into our program. 70 To assess development of a student's mathematical power, a teacher needs to use a mixture of means: essays, home- work, projects, short answers, quizzes, blackboard work, journals, oral interviews, and group projects. Only broad- based assessment can reflect fairly the important, higher- order objectives of mathematics curricula. As we need standards for curricula, so we need standards for assessment. We must ensure that tests measure what is of value, not just what is easy to test. -Ray Whinnem If we want stu- dents to investigate, explore, and discover, assessment must not measure just mimicry mathematics. By confusing means and ends, by making testing more important than learning, present practice holds today's students hostage to yesterday's mistakes.
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