Teaching Mathematics in The 21st Century

Teaching Mathematics in The 21st Century 15
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KirstyPotts,United States,Professional
Published Date:14-07-2017
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Teaching he first decade of the new millennium is a good time to remember the past, consider the present, Tand plan for the future. The past century brought Mathematics changes that transformed education. Some of the most drastic changes have come in mathematics education. At the turn of the last century, children studied arithmetic in in The the elementary grades. They did sums or long division on slates or, later, in lined paper tablets, and they memorized the times tables. Today, the third- and fourth-generation 21st Century descendents of those schoolchildren log onto the Internet for information about fractals and Fibonacci numbers. In class they work with manipulatives and study economic concepts such as supply and demand; they even person- ally interact with astronauts as they conduct experiments on space shuttles In this chapter we will look at some of the factors that brought about these changes and how the changes are work- ing together to reconstruct or remake mathematics educa- tion for the 21st century. Building a consensus and setting standards for mathematics education have proceeded in the context of national debates over curriculum, evaluation, and professional development-debates sometimes called the “math wars.” From these “wars” have emerged goals and documents such as Principles and Standards for School Math- ematics and Project 2061 as well as standards at the state and local levels. Mathematics educators may hold differing ideas about methods, curriculum, content, and even criteria for excellence. Most, however, share a commitment to increas- ing the scope, accessibility, and excellence of mathematics education in the 21st century. 12 Chapter 1 Being a Teacher in the 21st Century Fifty years ago in a small-town classroom, a teacher with a vision for the future told her students, “By the end of this century you may be living in automatic houses where everything from cooking to cleaning is done for you. You’ll prob- ably wear disposable clothes. You might even vacation on the moon or work on Mars.” What she predicted hasn’t happened yet, although we have taken the first steps toward interplanetary travel; in Canada there are experimental “smart” towns; and our refrigerators may soon be able to talk to us about souring milk or needed items for our grocery lists. The teacher wasn’t totally accurate but she was clairvoyant—a clear seer. What she saw clearly and what she helped her stu- dents see was that the future was filled with wonderful possibilities if only they would “dream big”—set high goals, work to make dreams happen, and believe in themselves. “Dreaming big” will be a prerequisite for teachers in the 21st Century. Never before has so much been expected of us, and never before has so much depended upon us. A hundred years ago a teacher had succeeded if she taught a few things to the many and many things to the few. Those who fell behind or dropped out could always find jobs on the farms and in the factories. Their livelihood didn’t depend upon “school” learning; learning outside the school provided enough to get by in their agrarian, blue-collar world. All of that has changed. Few can live on the wages from semi- or unskilled labor. It’s brains, not brawn, that are needed to survive in the information age, and brains need more than basic training to function at their best; they need knowledge and understanding. Beyond Shop-and-Yard Mathematics The challenge for teachers and their students to “dream big” is perhaps greatest in mathematics education. In the first half of the 20th Century, curriculum de- velopment emphasized shop-and-yard skills. Prompted by the idea of function- alism (education you can use), some educators focused on identifying minimal competencies needed to perform different jobs: dollars-and-cents math for clerk- ing, feet-and-inches math for carpentry, measuring-cups-and-spoons math for cooks and homemakers. The changing needs of a changing world have made this restrictive view not only obsolete but also dangerous. The student who knows no more than shop- and-yard mathematics risks being left behind in a job market that increasingly emphasizes technology and information systems; risks being left out of the na- tional and international discourses about economics, politics, science, and health care; risks, in short, the handicap of mathematical illiteracy. (See Figures 1.1 and 1.2 for examples of the important mathematical topics being tackled by fifth and ever second graders today). In Step with the New Mathematical Literacy The National Council of Teachers of Mathematics (NCTM) has identified five imperatives or needs for all students (NCTM 1998, 45-46). ■ Become mathematical problem solvers. ■ Communicate knowledge.Teaching Mathematics in the 21st Century 3 Figure 1.1 Tackling the information age task of data collection, fifth graders collect data on crater sizes made by dropping different object from different heights. Figure 1.2 Second graders explore the language of probability. ■ Reason mathematically. ■ Learn to value mathematics. ■ Become confident in one’s ability to do mathematics.4 Chapter 1 Teachers Self-Inventory 1. What can I hope to accomplish as a teacher in the 21st century? 2. Am I ready for the challenge of teaching everything to everyone? 3. Am I ready to dream big—to aim for excellence as a teacher of mathematics? 4. Can I instill the ability to dream big in my students—excellence in learning mathematics? 5. Can I go beyond teaching basic skills and model the joy and beauty of mathematics? 6. Do I appreciate mathematics myself? 7. Do I really believe—not just think, but believe— that everyone can learn to reason mathematically? 8. Do I feel confident in my mathematical ability? 9. Have my own mathematical abilities been developed beyond the level of performing basic procedures? 10. Do I understand and can I interpret for my students the mathematical worlds that surround us? Figure 1.3 Teacher’s self-inventory. These are in effect he cornerstones of the new mathe- matical literacy—what’s needed to survive and thrive in the next century. T Topics, Issues, and Explor opics, Issues, and Explorations ations Meeting these imperatives calls for more than hard work and good intentions; it calls for belief—belief in As teachers, we model behaviors and attitudes for our our own abilities to teach and belief in our students’ students—some consciously, some unconsciously. abilities to learn. The Teachers’ Self-Inventory in What kinds of models for teaching mathematics have Figure 1.3 suggests some things to think and talk about you had? Which models do you want to be like? as you set your goals for professional development and Which models do you not want to be like? growth. Changing Views About Who Should Learn Mathematics During much of the 20th Century, opportunities to study mathematics were often unequal. All students studied arithmetic, but only the college-bound elite tackled mathematics. The exclusionary process frequently targeted women and minorities, creating a hierarchy of expectations and opportunities that pushed children in one direction or the other from the earliest grades— often without the children’s or their teachers’ realizing what was happening. Research data show that millions of people have been victims of false as- sumptions about who has the ability to master mathematics.Teaching Mathematics in the 21st Century 5 These assumptions become self-fulfilling expectations, which ultimately undermine the self-concepts of female students, impoverished students and students of color. The single most important change required involves a national con- sciousness raising. Teachers, parents, and the students themselves must recognize that virtually every child has the capacity to master mathemat- ics. . This is true for females as well as for males, for poverty-stricken stu- dents as well as those from more affluent backgrounds, and for persons of every ethnicity (Drew 1996, 2-3). NCTM took a significant step toward “consciousness-raising” by recom- mending the Standards for all students. Instead of tiering objectives—more mathematics for the college bound, less for prospective trade school students, and almost none for at-risk students—the Council asks for more mathematics— more emphasis, more complexity, more challenging goals and objectives—for all students. Tradition and Myths But, you might ask, is this wise? Are we ignoring meaningful differences in apti- tude in the interest of equity and fair play? Won’t expectations be lowered and students who excel in mathematics, shortchanged? Behind these questions lie some of the most damaging of the math education myths: ■ Mathematics is a subject so demanding that few can hope to understand it. ■ Equal treatment to one group somehow subtracts something from another. ■ Mathematics education should be layered—advanced concepts for the few, basic concepts for the many, math facts for the rest. It is a mark of the power of tradition that myths such as these con- tinue to fuel the national debate over reforming mathematics curric- ula. Look outside our own country and the arbitrary nature of some of our curriculum “truths” becomes apparent. In China, where far fewer resources can be devoted to education, almost everyone learns advanced mathematics. “It is assumed,” writes David Drew, “that everyone can master advanced concepts and everyone is expected to do so” (1996, 9). Robert Reich, in The Work of Nations, says, “Japan’s greatest educational success has been to assure than even its slowest learners achieve a relatively high level of proficiency” (1991, 228). In the Trends in International Mathematics and Science Study (TIMSS), the United States has been consistently outperformed by third world countries—countries whose “slowest learners” might have been sus- pected of “holding back” the majority if they had been studying in American classrooms (Gonzales et al. 2004; U.S. Department of Ed- ucation USDE 1998). Equity Reforms In the United States recent reforms probably began to affect perfor- mance in measurable ways in the 1990s (see Figure 1.4 for an ex- Figure 1.4 Mixing English and Spanish in bilingual math classes. ample of one reform, a bilingual math class). However, while test6 Chapter 1 scores for women and minorities have risen signifi- cantly, the performance of students who traditionally do well on achievement tests has neither declined nor T Topics, Issues, and Explor opics, Issues, and Explorations ations fallen behind (National Center for Education Statis- tics 1998, 72–73; Stevens 2003). And study after How do you see knowledge of mathematics—as an study shows the benefits of mainstreaming and inte- abundant or a scarce resource? If you select the former, grating rather than separating students (see also West what are some of the issues involved in the teaching 1991). challenge Drew mentions? If the latter, how would We have two choices as teachers, Drew writes: we you decide who should be given the opportunities to can assume that learn the higher levels of mathematics? 1. virtually everyone can master the material and the challenge is to present it in a manner that allows them to do so, or 2. the material is tough and only a few of the best and For an brightest will be able to learn it activity related to a bilingual (9) lesson in Spanish and English, see Activity 20 on The assumptions we make will not only affect our classroom behaviors and ex- the CD-ROM that pectations but also students’ perceptions about their own abilities and potential accompanies this text. to learn mathematics. Changing Views About How Students Learn Perhaps the most dramatic changes in school mathematics during the 20th cen- tury were in the way children study mathematics. Consider the classroom sce- narios described in the Windows on Learning feature. The children in the first scenario are learning what one writer calls “mus- cle” or “muscular” mathematics (Betz 1948, 203). They exercise their mental muscles with repetitions intended to make responses automatic, without thought. The teacher is the center of the class, in control of learning as well as behavior. The environment of the class is disciplined and quiet. The conse- quences of failure are immediate and devastating—public discussion of errors with a pejorative thrust. The mathematics activity in the second scenario reflects some changes in our perspective, both about learning and about student-teacher roles in the learning process. Instead of drilling and memorizing facts, these children explore ideas like scientists, with a problem to solve, materials to experiment with, and a spirit of inquiry. This is dynamic instead of passive or static learning, and the children rather than a teacher direct and shape the process. Multiple rather than single outcomes are not only possible but also encouraged. The activity is open ended; the learning cooperative. The small group is a learning, team in which the flow of ideas is unstructured and spontaneous and the possibilities, are limitless. Changing Views About What Should be Learned In 1994, NCTM changed the name of its journal for elementary teaching from The Arithmetic Teacher to Teaching Children Mathematics. The change marked a major transition in the way we think and talk about mathematics learning in the elementary grades as well as changes in the content itself. Today mathematics isTeaching Mathematics in the 21st Century 7 Windows on Learning W Windows on Learning indows on Learning The group decided to work on multiplying and “Two Times Two is Four” leave shapes for the next day. The assignment was sim- The year was 1954; the place, Mrs. Taylor’s third-grade ple: find out what happens when you multiply num- classroom at Briscoe Elementary School. bers by themselves. The students sat quietly, their hands folded in “That’s easy,” Angie said. “It’s like adding them up front of them, at desks lined up in five neat rows, with over and over.” six desks to a row. The desks were all filled. The first “Like this,” Hussein agreed and began to arrange group of baby boomers were entering the public blocks on the table in front of them, two sets of two schools, and space and teachers were at a premium. blocks, one on top of the other, for 2 times 2; three sets Mrs. Taylor stood before the class at a chalkboard. of three blocks for 3 times 3. She had just finished correcting the work of six students Shelley sat watching Hussein line up the blocks. who were called to the board to do multiplication prob- She didn’t say anything, but she had a feeling he was lems involving two-digit numbers. The exercise had not missing something by lining the blocks up. gone well, and Mrs. Taylor was frustrated. Meanwhile, Tino was verbalizing what Hussein “Billy, you multiplied 33 times 33 and got 66. You was doing. “You put two sets of two together and get know that can’t be right; 3 plus 3 is 6, so 3 times 3 can’t 4, three sets of three and get 9, four sets of four and get be the same thing. 16, five sets of five and get 25.” “Suzanne, you say 15 times 55 is 770. How can that “Hey, everybody, look at this,” Angie said, looking be if 5 times 5 is 25? The number can’t end with a 0. up from the pad where she had been doodling. “If you “What’s happening, class, is we are forgetting our write all the numbers down, 1’s odd, 4’s even, 9’s odd, times tables.” She picked up a stick and pointed to a 16’s even, 25’s odd.” chart above the chalkboard. Letitia was working with cuisenaire rods, ar- “Everybody stand.” ranging and rearranging them as she looked for The students slipped quickly out of their desks patterns. and stood with their hands at their sides and eyes on Then Shelley reached a tentative hand toward the the chart. Everyone was careful not to look at Billy and blocks in Hussein’s 2 times 2 line. “I think these would Suzanne, who were red faced and embarrassed. look better like this,” she said and quickly rearranged “All right, everyone, together now on the count of the blocks into a square. three.” Hussein saw what she was doing and joined in. Mrs. Powell tapped the chart three times with her “Does it do that every time?” Angie stopped doo- stick, and the children began to chant, “One times 1 is dling to ask. 1, 2 times 2 is 4. . . .” “I don’t know. I think so,” Shelley said and kept on “Two Times Two Is Square” moving blocks. Finally, all of Hussein’s blocks had been arranged into squares. The year was 2004 (half a century later). At Lowell El- “It happens every time. The blocks make a ementary School, in Room 123, fourth graders were ex- square,” Hussein observed. perimenting with multiplying numbers by themselves. “So when you multiply a number by itself, you get Tino, Shelley, Angie, and Letitia were working in a square,” Letitia summarized. a small group around a circular table. In front of them Later when the group discussed the activity with were manipulatives, some colored blocks, various tools one of the class’s team teachers, Ms. Lee, she suggested for measuring, and scratch paper and pencils for they see what the math software the class used had to sketching and trying out ideas. say about squaring. The computer software reinforced “What are we supposed to be doing?” Tino asked. the block arranging Shelley and Hussein had done Letitia consulted a list of objectives in her three- with graphics of squares being multiplied into larger ring journal. and larger squares. It also showed them how to repre- “We’ve done the map activity and the frequency sent the squaring process in math language with a su- count. That leaves multiplying and shapes for this 2 perscript . week. Which one would you rather do?”8 Chapter 1 a foundational discipline. It provides tools and ways of thinking that impact learning across the curriculum. Some factors that influenced the changing mathematics curriculum in the 20th Century included changes in our economic and social worlds, historical events and trends, and new developments in technology and science. Tying the Curriculum to Mental Age and Social Utility Early attempts to design a mathematics curriculum focused on matching content to students’ mental age and, therefore, readiness to learn. For example, in the 1920s, school administrators collected survey data to tie arithmetic topics to chil- dren’s “mental ages.” They used their correlations to sequence the curriculum, “delaying” introduction of many topics such as multiplication and division be- cause of students’ supposed “mental” unreadiness (Washburne 1931, 210, 230–31). Readiness, according to these administrators, could be determined by a combination of intelligence and achievement tests, which would allow teachers to “ability-group” students or individualize instruction. They concluded that arithmetic was too hard for most elementary school students and should be taught in junior high or high school instead (see Brownell 1938, 495-508, for a critique). Just as the Great Depression turned nations inward, the social utilitarians of the mid-20th Century advocated a short-range rather than a long-range view for the mathematics curriculum. Guy Wilson, one of the movement’s leading pro- ponents, believed the schools should teach the skills required to do adult jobs. In 1948 he wrote: The proper basis for functional arithmetic is the social utility theory. This theory posits (1) that the chief purpose of the school is to equip the child for life, life as a child, life as an adult, and (2) that the skills, knowledges, and appreciations should receive attention in school somewhat propor- tional to usefulness in life (321) Wilson (1948) identified basic arithmetic facts needed by the majority of workers and used them to calculate what he called “the drill load of arith- metic”—the facts and skills for a drill mastery program in which “only suc- cess is wanted and only perfect scores” (327, 335). Students, according to Wilson, should memorize 100 primary facts each for addition, subtraciton, and multiplication: (1) Addition—100 primary facts, 300 related decade facts to 39  9, 80 other facts for carrying in multiplication to 9 × 9. . . . Whole num- bers only. . . . (2) Subtraction—100 primary facts, all process difficulties. . .whole num- bers only. . . . (3) Multiplication—100 primary facts, all process difficulties, whole numbers only. . . . (4) Division—emphasis on long division. . . . (5) Common fractions—. . .halves and quarters, thirds, possibly attention to eighths and twelfths separately. (327-28.)Teaching Mathematics in the 21st Century 9 Wilson recommended little or no work with decimals since “decimals rep- resent specialized figuring learned on the job” (1948, 329). Measures, per- centages, geometry, and algebra were relegated for the most part to what he called “appreciation” study—studies undertaken for “fun” and used to “lure” the brightest students forward. Wilson also argued that the metric system should not be taught because English measures were more convenient: “The housewife, even in a metric country, wants a pound of butter” (1948, 327). Light years, parsecs, measurements related to the electronic age—should they be taught? “No, of course not,” wrote Wilson. “The numbers using them are too few” (1948, 337). The social utility argument continues to influence curriculum choices. As recently as the 1980s the National Center for Research in Vocational Education published a series called Math on the Job, with special kinds of numbers for the grain farmer, mechanic, clerk, machinist, cashier, and so forth. Responding to a Bigger World Even as the utilitarians were urging a reduced mathematics curriculum, others were calling for expansion. World War II had shown Americans a bigger world—a world where Swiss students studied calculus in high school, where scientific breakthroughs were needed, not just to win but to survive. The Com- mission on Post War Plans called for more, not less, mathematics in education. In 1947 the President’s Commission on Higher Education proposed increas- ing college enrollments drastically for a minimum of 4 million by 1960—a change that would require a college-track mathematics curriculum for mil- lions. By the time the Soviet Union launched Sputnik in 1957 and galvanized public opinion for the space race, educators were already experimenting with new mathematics curricula. The “new math” as it was popularly called, emphasized mathematics struc- ture. Students studied sets, number systems, different number bases, and num- ber sentences. Teachers guided children to discover concepts rather than lecturing about them. While many ideas of the new math had merit, application may have been flawed. Textbooks were often hard to read and overly formal. Many parents complained that they could not understand their children’s homework. In the meantime, the social revolution of the sixties and seventies flooded colleges with students—many from backgrounds and groups that traditionally had not attended college. To what extent this new college population affected test scores remains unclear, but between 1963 and 1975 SAT scores declined, leading to several major concerns for the mathematics curriculum in the final decades of the century, including how to ■ upgrade the curriculum to match the demands of an increasingly techno- logical society, ■ balance student needs with the needs of society and of mathematics it- self, and ■ teach the expanded curriculum to all of the students. There were no easy answers. A back-to-the-basics movement called for a re- turn to traditional mathematics—teacher lectures, drills, and tests. But many ar- gued that traditional approaches had worked for no more than 5% to 15% of the students; what was needed was a challenging mathematics curriculum that prepared every student to think mathematically—to develop the foundations in10 Chapter 1 mathematical reasoning, concepts, and tools needed for advanced mathematics education as well as enlightened living in the age of technology. The National Council of Supervisors of Mathemat- ics (NCSM) responded with a list of basic skills (1977) and later with “Essential Mathematics for the Twenty- T Topics, Issues, and Explor opics, Issues, and Explorations ations first Century ”(1989). NCTM did the same, producing An effective curriculum is multi-dimensional. It re- an Agenda for Action in 1980, the first version of the sponds to the needs of society, the needs of the indi- Curriculum and Evaluation Standards in 1989, and now vidual, and the needs of the subject. Think about the the Standards 2000 document, Principles and Standards changes in the mathematics curriculum in the 20th for School Mathematics, compiled with the input of thou- Century. Which changes do you think reflected con- sands of mathematics teachers responding over the cerns about which needs? Which changes seem most World Wide Web. Some major points of consensus be- worthwhile or least worthwhile? tween the NCSM and the NCTM recommendations in- clude the following: ■ that all students benefit from a challenging mathematics curriculum; ■ that mathematics reasoning and higher-order thinking skills should be in- tegral to the curriculum; ■ that problem solving should be a priority; ■ that algebraic thinking, geometry, statistics and probability are essential rather than add-on skills; ■ that the emphasis in computation should be on meaning and patterns; ■ that communication of mathematical ideas in a variety of ways (oral, writ- ten, symbolic language, everyday language) is critical to the learning process; ■ that students need opportunities to explore and apply mathematics in hands-on and real-life activities. Building Consensus and Setting Standards Changes in curriculum and pedagogy are not like changes in the seasons, though they may be just as inevitable. Few of the changes described in the previous sections have come smoothly or without controversy. In his 1998 ad- dress, “The State of Mathematics Education: Building a Strong Foundation for the 21st Century,” then–secretary of education Richard W. Riley called for a “ceasefire” in the “math wars” about “how mathematics is taught and what mathematics should be taught.” “We need,” he told the meeting of the Amer- ican Mathematical Society and the Mathematical Association of America, to bring an end to the shortsighted, politicized, and harmful bickering over the teaching and learning of mathematics. I will tell you that if we continue down this road of infighting, we will only negate the gains we have already made—and the real losers will be the students of America. I hope each of you will take the responsibility to bring an end to these battles, to begin to break down stereotypes, and make the importance of mathematics for our nation clear so that all teachers teach better mathe- matics and teach mathematics better. Riley appealed for “civil discourse” and openness to change. The controversy reached mud-slinging levels in the 1990s, with reformers accused of teaching “fuzzyTeaching Mathematics in the 21st Century 11 math” or “placebo math” or “dumbing down to promote classroom equality” (Mathematically Correct 1997; Leo 1997, 14). But reforming the mathematics cur- riculum has always been a stormy process. In 1948 Willian Betz complained, “For nearly six decades we have had unceasing efforts at reform in mathematics,” and, he pointed out, “milestones in this epic struggle” go back to 1892 (197). He wrote, “We have looked at a picture which is no doubt perfectly familiar to every experi- enced teacher of mathematics. It is that of a battle between two sharply contrasting positions regarding the educational role of mathematics (1998, 205). In the Na- tional Society for the Study of Education’s 1970 yearbook, Mathematics Education, Lee Shulman, citing articles published in 1930, 1935, and 1941, says they “can al- most read as a history of controversies, cease-fires, and temporary truces...” (23). Although the tone of the controversies may at times have sunk below the lev- els of civil discourse urged by Secretary Riley, the controversies themselves may not be unproductive. In fact, even the emotionally charged skirmishes may serve a purpose since they tend to involve the public in the dialogue about reform. Nonetheless, if consensus among mathematicians is neither clear nor stable, is it worthwhile to set standards, and can the standards set be worthwhile? If we think of standards as commandments engraved in stone, the answer may be no. However, if we accept setting standards as an ongoing and open-ended process, the answer is yes. According to Webster’s New World Dictionary, the word standard originally meant “a standing place.” The meaning has grown to include flags or banners that symbolize nations, causes, or movements; levels of attainment set as benchmarks; and even foundation supports. Finding out where we stand and establishing goals, benchmarks, and supporting structures for those ideas have all been part of the standard-setting process—or processes since efforts to set standards are ongoing at state and national levels and for a variety of curriculum and development areas. Although driven by a dialogue that has ranged in tone from the rational to the acrimonious, these standards-setting processes have succeeded at several lev- els. First, they have generated research and ideas that have disrupted the status quo, jarring entrenched assumptions about mathematics education and opening the way for new concepts and methods. Second, they have focused attention on critical issues, such as equity and technology in teaching mathematics. And third, they have generated public in- terest and involvement at unprecedented levels. When in our history has mathematics in the schools been dis- cussed and debated with greater intensity and urgency? Making mathematics education a national issue may T Topics, Issues, and Explor opics, Issues, and Explorations ations have been one positive outcome of the math wars. Math- Identify and explain one concept, content area, or ematics, like science, occurs in a social context (see Drew process that you believe should be learned during a 1996, 17). Engaging society in the debate over what is specific grade. Share your ideas in a group or class. taught and how it is taught ensures that reform takes How much agreement or disagreement do you find? place within rather than outside the social context and re- mains responsive to the needs and demands of those most directly affected by the changes. National Standards for Mathematics Education In Goals 2000: Educate America Act, Congress proposed in 1994 “a national frame- work for education reform” and called for “the development and adoption of a voluntary national system of skill standards and certification.” (See Figure 1.5).” The act responded in part to efforts already under way by professional groups12 Chapter 1 The 1994 Goals 2000: Educate America Act challenged schools both to achieve and to compete. (A) By the year 2000, United States students will be first in the world of math- ematics and science achievement. (B) The objectives for this goal are that– (i) mathematics and science education, including the metric system of measurement, will be strengthened throughout the system, espe- cially in the early grades; (ii) the number of teachers with a substantive background in mathe- matics and science, including the metric system of measurement, will increase by 50 percent and (iii)the number of United States undergraduate and graduate students, especially women and minorities who complete degrees in mathe- matics, science, and engineering will increase significantly. (Educate America Act of 1994; see also National Education Goals Panel 1995). Figure 1.5 Setting national goals for mathematics education. such as NCTM and the American Association for the Advancement of Sci- ence (AAAS). Underlying these goals and objectives are several basic assumptions: that having an informed citizenry is essential to national security and productivity; that being informed entails higher levels of achievement in mathematics and sci- ence; that “being first” is a desirable and feasible outcome; that a nation that ex- emplifies diversity can set common standards and achieve common goals in mathematics education. NCTM’s Principles and Standards 2000 Principles and Standards for School Mathematics (2000) integrates areas covered by three earlier Standards publications: Curriculum and Evaluation Standards for School Mathematics (1989), Professional Standards for Teaching Mathematics (1991), and Assessment Standards for School Mathematics (1995). The purpose of the Standards 2000 document is ambitious and broad: “to set forth a compre- hensive and coherent set of goals for mathematics for all students from pre- kindergarten through grade 12 that will orient curricula, teaching, and assessment efforts during the next decades” (NCTM 2000, 6). To this end, the document proposes a vision, principles, and standards to be applied across four grade bands: pre-kindergarten through grade 2, grades 3-5, grades 6-8, and grades 9-12. The vision is both idealistic and far-reaching: NCTM Vision for School Mathematics Imagine a classroom, a school, or a school district where all students have access to high-quality, engaging mathematics instruction. There are am- Reprinted with permission from Principles and Standards for School Mathematics, copyright © 2000 by the National Council of Teachers of Mathematics. All rights reserved. Standards are listed with the permission of the National Council of Teachers of Mathematics (NCTM). NCTM does not endorse the content or validity of these alignment.Teaching Mathematics in the 21st Century 13 bitious expectations for all, with accommodation for those who need it. Knowledgeable teachers have adequate resources to support their work and are continually growing as professionals. The curriculum is mathe- matically rich, offering students opportunities to learn important math- ematical concepts and procedures with understanding. Technology is an essential component of the environment. Students confidently engage in complex mathematical tasks chosen carefully by teachers. They draw on knowledge from a wide variety of mathematical topics, sometimes ap- proaching the same problem from different mathematical perspectives or representing mathematics in different ways until they find methods that enable them to make progress. Teachers help students make, refine, and explore conjectures on the basis of evidence and use a variety of reason- ing and proof techniques to confirm or disprove those conjectures. Stu- dents are flexible and resourceful problem solvers. Alone or in groups and with access to technology, they work productively and reflectively, with the skilled guidance of their teachers. Orally and in writing, students communicate their ideas and results effectively. They value mathematics and engage actively in learning it. (NCTM 2000, 3) NCTM’s Vision for School Mathematics assumes both the importance of knowing mathematics in the 21st Century and the need to continually im- prove mathematics education to meet the challenges of a changing world (see Figure 1.6 for an example of a second grader’s use of modern technology in a counting activity). Understanding and using mathematics is described as an es- sential underpinning of life, a part of our cultural heritage, and a prerequisite for success in the workplace. And providing all students with “the opportunity and the support to learn significant mathematics with depth and understanding” Figure 1.6 Counting with computer graphics.14 Chapter 1 is linked to “the values of a just democratic system” and “its economic needs” (NCTM 2000, 5). NCTM’s Principles for School Mathematics are equally far-reaching: NCTM Principles for School Mathematics ■ Equity: Excellence in mathematics education requires equity— high expectations and strong support for all students. ■ Curriculum: A curriculum is more than a collection of activi- ties: it must be coherent, focused on important mathematics, and well articulated across the grades. ■ Teaching: Effective mathematics teaching requires understand- ing what students know and need to learn and then challenging and supporting them to learn it well. ■ Learning: Students must learn mathematics with understand- ing, actively building new knowledge from experience and prior knowledge. ■ Assessment: Assessment should support the learning of impor- tant mathematics and furnish useful information to both teach- ers and students. ■ Technology: Technology is essential in teaching and learning mathematics; it influences the mathematics that is taught and enhances students’ learning. (NCTM 2000, 11) Together with the Standards, these Principles comprise key components of NCTM’s vision of high-quality mathematics education. The Principles are, in effect, ideals to live by—foundational ideas that influence curriculum and professional development on the larger scale as well as instructional decisions in the classroom on the smaller scale. The Standards are more like building materials. They outline mathematics content and processes for students to learn. Instead of the multiple standards of the 1989 document, Principles and Standards for School Mathematics proposes 10 standards that “specify the un- derstanding, knowledge, and skills students should acquire from kindergarten to grade 12.” The Content Standards—Number and Operations, Algebra, Geometry, Measurement, and Data Analysis and Probability—explicitly describe the content that students should learn. The Process Standards—Problem Solving, Reasoning and Proof, Communication, Connections, and Rep- resentation—highlight ways of acquiring and using content knowledge. (NCTM 2000, 29) Each Standard entails goals that apply across all T Topics, Issues, and Explor opics, Issues, and Explorations ations grades plus differing emphases for the grade bands (see Figure 1.7). For example, number and measurement are NCTM gives detailed “Expections” or specific objec- emphasized in the early grades, while later grades spend tives for each Content Standard by grade group. Study more instructional time on formal algebra and geometry. the Expectations in Principles and Standards for School Arranging the curriculum into 10 standards that span Mathematics for the grade you are teaching or plan to the grades offers a coherent structure for an overall cur- teach. Do any surprise you? Discuss your reactions in riculum. Specific details are left to those who will apply small groups. and implement the ideas.Teaching Mathematics in the 21st Century 15 CONTENT STANDARDS PROCESS STANDARDS Standards Instructional programs from pre-kindergarten Standards Instructional programs from pre-kindergarten through grade 12 should enable all students to— through grade 12 should enable all students to— Number and Operations Problem Solving • Understand numbers, ways of representing numbers, rela- • Build new mathematical knowledge through problem solving tionships among numbers, and number systems • Solve problems that arise in mathematics and in other • Understand meanings of operations and how they relate to contexts one another • Apply and adapt a variety of appropriate strategies to solve • Compute fluently and make reasonable estimates problems • Monitor and reflect on the process of mathematical problem Algebra solving • Understand patterns, relations, and fractions • Represent and analyze mathematical situations and struc- Reasoning and Proof tures using algebraic symbols • Recognize reasoning and proof as fundamental aspects of • Use mathematical models to represent and understand mathematics quantitative relationships • Make and investigate mathematical conjectures • Analyze change in various contexts • Develop and evaluate mathematical arguments and proofs • Select and use various types of reasoning and methods of Geometry proof • Analyze characteristics and properties of two- and three- dimensional geometric shapes and develop mathematical ar- Communication guments about geometric relationships • Organize and consolidate their mathematical • Specify locations and describe spatial relationships using co- thinking through communication ordinate geometry and other representational systems • Communicate their mathematical thinking coherently and • Apply transformations and use symmetry to analyze mathe- clearly to peers, teachers, and others matical situations • Analyze and evaluate the mathematical thinking and strate- • Use visualization, spatial reasoning, and geometric modeling gies of others to solve problems • Use the language of mathematics to express mathematical ideas precisely Measurement • Understand measurable attributes of objects and the units, Connections systems, and processes of measurement • Recognize and use connections among mathematical ideas • Apply appropriate techniques, tools, and formulas to deter- • Understand how mathematical ideas interconnect and build mine measurements on one another to produce a coherent whole • Recognize and apply mathematics in contexts outside of Data Analysis and Probability mathematics • Formulate questions that can be addressed with data and collect, organize, and display data to answer them Representation • Select and use appropriate statistical methods to analyze data • Create and use representations to organize, record, and com- • Develop and evaluate ingerences and predictions that are municate mathematical ideas based on data • Select, apply, and translate among mathematical represen- • Understand and apply basic concepts of probability tations to solve problems • Use representations to model and interpret physical, social, and mathematical phenomena Figure 1.7 NCTM Process and Content Standards (NCTM 2000, 392-403). Reprinted with permission from, copyright© by the National Council of Teachers of Mathematics. All rights reserved. Standards are listed with the permission of the National Council of Teachers of Mathematics (NCTM). NCTM does not endorse the content or validity of these alignments. Project 2061, Science for All Americans A parallel project to NCTM’s Principles and Standards began in 1985, the date of the last visit of Halley’s comet. Sponsored by AAAS, Project 2061 is named for the date when Halley’s comet will return and assumes that children who were be- ginning school in 1985 will see a lifetime of changes in science and technology16 Chapter 1 before the comet’s return in 2061. To prepare them for these changes, Project 2061 proposes educational reforms akin to those promoted by NCTM. Culotta suggests seven major areas of commonality: ■ Less memorization ■ Involvement of teachers in the reform process ■ Integration of disciplines and study ■ Greater emphasis on hands-on activities ■ Greater focus on listening to students’ questions and ideas ■ Connections between discipline and society ■ Emphasis on the scientific process and how problems are solved Project 2061 defines mathematics as “the science of patterns and relation- ships” and describes it as “the chief language of science” (AAAS 1989). In the project’s “Design for Scientific Literacy,” mathematics is included in most of the building blocks for a Project 2061 curriculum: “For purposes of general scien- tific literacy, it is important for students (1) to understand in what sense mathe- matics is the study of patterns and relationships, (2) to become familiar with some of those patterns and relationships, and (3) to learn to use them in daily life” (AAAS 1989). In Project 2061’s “Benchmarks for Scientific Literacy”, as shown in Figure 1.8, specific educational objectives are outlined by grade, with an emphasis upon outcomes or “what students should know” and understand. The Benchmarks emphasize the importance of experiencing mathematics, of establishing con- nections between ideas and areas of inquiry, of “making multiple representa- tions of the same idea and translating from one to another” (AAAS, 2000). Implicit in the various objectives are ties to development; for example, the em- phasis in the early grades is on the specific, concrete, and immediate, with the gradual introduction of abstract ideas and “grand categories” in later grades. “Doing mathematics,” like “doing science,” is encouraged from the earliest grades, and mathematical inquiry leading to the valid development of mathe- matical ideas also starts in the earliest grades when children explore concrete objects to discover what they tell us and what they can be used to show about the world around them. Overall, Project 2061 proposes specific educational objectives within a con- text of scientific values and attitudes, including attitudes about learning: Students in elementary school have a spontaneous interest in nature and numbers. Nevertheless, many students emerge from school fearing math- ematics and disdaining school as too dull and too hard to learn. . . . It is within teachers’ power to foster positive attitudes among their students. If they choose significant, accessible, and exciting topics in sci- ence and mathematics, if they feature teamwork as well as competition among students, if they focus on exploring and un- derstanding more than the rote memorization of terms, and if they make sure all their students know T Topics, Issues, and Explor opics, Issues, and Explorations ations they are expected to explore and learn and have their achievements acknowledged, then nearly all of those Discuss NCTM’s Standards and Project 2061’s Bench- students will indeed learn. And in learning success- marks for learning mathematics. How are they alike? fully students will learn the most important lesson of How are they different? Which Standards and Bench- all—namely that they are able to do so. marks seem most important to you? (AAAS, 1998 chap. 12)Teaching Mathematics in the 21st Century 17 Kindergarten through Grade 2 By the end of the 2nd grade, students should know that: • Circles, squares, triangles, and other shapes can be found in nature and in things that people build. • Patterns can be made by putting different shapes together or taking them apart. • Things move, or can be made to move, along straight, curved, circular, back-and-forth, and jagged paths. • Numbers can be used to count any collection of things. • Numbers and shapes can be used to tell about things. Grades 3 through 5 By the end of the 5th grade, students should know that: • Mathematics is the study of many kinds of patterns, including numbers and shapes and operations on them. Sometimes patterns are studied because they help to explain how the world works or how to solve prac- tical problems, sometimes because they are interesting in themselves. • Mathematical ideas can be represented concretely, graphically, and sym- bolically. • Numbers and shapes—and operations on them—help to describe and predict things about the world around us. • In using mathematics, choices have to be made about what operations will give the best results. Results should always be judged by whether they make sense and are useful. Grades 6 through 8 By the end of the 8th grade, students should know that: • Usually there is no one right way to solve a mathematical problem; dif- ferent methods have different advantages and disadvantages. • Logical connections can be found between different parts of mathematics. • Mathematics is helpful in almost every kind of human endeavor—from laying bricks to prescribing medicine or drawing a face. In particular, mathematics has contributed to progress in science and technology for thousands of years and still continues to do so. • Mathematicians often represent things with abstract ideas, such as num- bers or perfectly straight lines, and then work with those ideas alone. Figure 1.8 Project 2061 Benchmarks in Mathematics for the Elementary Grades through Middle School. Source: AAS (2000). State and Local Standards Efforts to develop national standards have had a signifi- cant impact on mathematics education overall. For ex- ample, in 1996 the framework for the National Assessment of Educational Progress (NAEP) was revised to reflect NCTM curricular emphases and objectives T Topics, Issues, and Explor opics, Issues, and Explorations ations (U.S. Department of Education USDE 1999, 2-3). National standards have also influenced the develop- What standards has your state or district established ment of standards at the state and local levels. Some for mathematics? The information may be available at states have adapted the national standards to fit their your state or county Web site, or you can ask a school own school districts’ needs (see, for example, Colorado librarian for help. How do these standards compare to Model Content Standards 2005.) Others have created NCTM’s Standards 2000 or to Project 2061? their own benchmarks and detail what students should18 Chapter 1 know grade by grade. Georgia’s performance-based standards are actually aligned with Japanese standards as well as the Georgia Criterion-Referenced Compe- tency Tests (www.georgiastandards.org and www.glc.k12.ga.us/). Meeting the Challenges of the 21st Century The 20th Century began the process of reconstructing mathematics education. In 1900, according to a writer in NCTM’s first yearbook, the purpose of teach- ing arithmetic had as much to do with discipline as curriculum. “It was felt that the subject should be hard in order to be valuable, and it sometimes looked as if it did not make so much difference to the school as to what a pupil studied so long as he hated it” (Smith 1926, 18-19). Responding to the period’s rigid and often lifeless teaching methods and materials, the president of the American Mathematical Society, Eliakim Moore, appealed to teachers: Would it not be possible for the children in the grades to be trained in power of observation and experiment and reflection and deduction so that always their mathematics should be directly connected with matters of thoroughly concrete character? . . . The materials and mathematics should be enriched and vitalized. In particular, the grade teachers must make wiser use of the foundations fur- nished by the kindergarten. The drawing and paper folding must lead di- rectly to systematic study of intuitional geometry, including the construction of models . . . with simple exercises in geometrical reason- ing . . . . The children should be taught to represent, according to usual conventions, various familiar and interesting phenomena and study the properties of the phenomena in the pictures to know, for example, what concrete meaning attaches to the fact that a graph curve at a certain point is going down or going up or is horizontal (45-46). Meeting the Challenges as a Nation A hundred years later we can say that many elements of Moore’s vision for learning mathematics are not only possible but also an accomplished fact. Hands-on, dy- namic learning is becoming the norm in elementary classrooms (see Figure 1.9 for an example of a kinder- gartner’s graphing of a hands-on counting activity). Technology has helped us enrich and vitalize the learn- ing process with interactive learning experiences such as the National Center for Education Statistics’ Students’ Classroom (see the Math and Technology Feature, “Ex- plore Your Math Knowledge”). Increasingly, lessons em- phasize understanding and context and deemphasize rote memorization of isolated facts and procedures. The elementary curriculum is no longer limited to arithmetic but includes geometry, algebraic thinking, and mathe- matical reasoning that were once considered too abstract for children. Figure 1.9 Children represent counting jelly beans with a bar graph.Teaching Mathematics in the 21st Century 19 Evidence is mounting that these approaches are working and working well. After decades of declining test scores and public alarm about deficiencies, the M Math and T Math and T ath and Te echnolog echnolog chnology y y trends seem to be reversing as shown in the graph in Figure 1.10. From 1990 to 2005, the National Assess- Explore Your Math Knowledge ment of Educational Progress (NAEP), the nation’s re- The National Center for Education Statistics (NCES) has port card, showed steady gains (2003; Perie, Grigg, developed a Students’ Classroom with activities, games, and Dion 2005). SAT and ACT mathematics scores and learning experiences to encourage mathematics learn- are up. The 2003 Trends in International Mathematics ing. The Web site is http://nces.ed.gov/nceskids/eyk/in- and Science Study (TIMSS) showed both U. S. fourth- dex.asp?flashfalse. and eighth-graders scoring above the international av- The Explore Your Knowledge activity features ques- erage in mathematics and science (Gonzales et al. tions from national tests such as TIMSS. Students re- 2004; NCES 2005; USDE 1997 spond to the questions and then check their answers. The Does this mean that the goals and objectives pro- activity tracks the number of correct answers and posed by the Educate America Act have been reached? prompts students to “Try Again?” In the first decade of the 21st Century, is the U. S. first in the world in mathematics and science achievement? Activity Perhaps yes, perhaps no. If being first is measured by Visit the NCES Students’ Classroom. Explore some of achievements in the world of science and mathematics, the math questions, responding correctly and incorrectly. the U. S. could stand at the top. If we look (as many in Discuss the various resources available at the site, and the national media do) at test scores, our position is brainstorm ways to use the activities in the classroom, in- less clear. cluding the possibilities for using the questions to help Although the results of the 2003 TIMSS placed students prepare for various national tests. U. S. fourth- and eighth graders above the interna- tional averages, U. S. fourth-graders were outper- formed by students in 11 countries and U. S. eighth-graders by students in 9 countries. Students in four Asian countries—Chinese Taipei, Hong Kong SAR, Japan, and Singapore—outperformed both U. S. fourth- and eighth-graders (Plisko 2004). In addi- tion the 2003 Program for International Student As- sessment (PISA) placed U. S. 15-year-olds below the international average for both mathematical and sci- entific literacy (Lemke et al. 2004). Data collected for the NAEP in the 1990s showed no significant im- provement in elementary or middle school teachers’ preparation to teach mathematics (Hawkins, Stancav- age, and Dossey et al. 1998). And the shortage of qualified mathematics teachers continues to grow, and women and minorities continue to be underrepre- sented in mathematics (Seymour 1995a, 1995b; Chaddock 1998). Nonetheless, progress is being made. In the 1991 International Assessment of Educational Progress (IAEP), U. S. elementary school students scored be- low rather than above the international average (USDE 1997). Middle school students’ performance improved significantly since the 1999 TIMSS. Moreover, data from NAEP show positive linear trends or overall increases in mathematics perfor- mance at all age levels tested from 1990 to 2003,20 Chapter 1 279 278 Grade 8 273 272 268 263 270 238 235 Grade 4 226 224 224 220 213 ‘90 ‘92 ‘96 ‘00 ‘03 ‘05 Year Accommodations not permitted Accommodations permitted Figure 1.10 NAEP 1990-2005 trends chart Source: U.S. Department of Education, Institute of Education Science, National Center for Education Statistics. National Assessment of Educational Progress (NAEP), various years, 1990–2005 Mathematics Assessment. Scale score 500 320 307 307 307 308 304 305 310 302 Age 17 300 298 300 306 290 281 276 274 280 Age 13 270 269 269 266 270 264 274 273 260 250 241 Age 9 240 232 231 230 230 222 219 219 219 231 220 230 210 200 0 ‘82 ‘86 ‘90 ‘92 ‘94 ‘96 ‘99 ‘73 ‘78 ‘04 � Nine-year-olds. The average mathematics score of 241 was higher in 2004 than in any pervious assessment year. � Thirteen-year-olds. The average score in 2004 was higher than in any other assessment year. � Seventeen-year-olds. The average score in 2004 did not show a significant change when compared to the score in either 1973 or 1999. Figure 1.11 NAEP 1973-2004 trends chart: National trends in mathematics by average scale scores. Source: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics, National Assessment of Educational Progress (NAEP), selected years, 1973-2004 Long-Term Trend Mathematics Assessments. continuing a positive trend begun in 1973 (see Figure 1.11; Perie et al. 2005; Perie and Moran 2005; USDE 2003, 1). Comparisons of average scores in 1990 and 2005 show that the number of both fourth- and eighth-graders performing at or above the NAEP mathe- matics performance levels increased significantly (Perie et al. 2005, 1). The percentage of fourth graders who can perform basic numerical operations (adding, subtraction, multiplying, and dividing with whole numbers) and solve one-step problems more than doubled from the 1970s to 2004 (20% to

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