The new, a-maze-ing approach to math: a mathematician with a child learns some politics.I am not a mathematics teacher, but I have a degree in mathematics and an intense interest in how the subject is taught. When I retire, I would like to teach math, which is why I started tutoring high school students in my spare time three years ago. My first student was a 9th grader having difficulty with geometry. He stated his problem succinctly suc·cinct
adj. suc·cinct·er, suc·cinct·est
1. Characterized by clear, precise expression in few words; concise and terse: a succinct reply; a succinct style.
2. : "I don't know Don't know (DK, DKed)
"Don't know the trade." A Street expression used whenever one party lacks knowledge of a trade or receives conflicting instructions from the other party. how to do proofs." Confronted with what I thought could be a common problem, I was still unaware that what I was really seeing was a national crisis in mathematics education.
Here's some of what I would soon learn:
-- Only 55 percent of 8th graders taking the National Assessment of Educational Progress The National Assessment of Educational Progress (NAEP), also known as "the Nation's Report Card," is the only nationally representative and continuing assessment of what America's students know and can do in various subject areas. (NAEP NAEP National Assessment of Educational Progress
NAEP National Association of Environmental Professionals
NAEP National Association of Educational Progress
NAEP National Agricultural Extension Policy
NAEP Native American Employment Program ) exam in math correctly answered the question, "How many pieces of string will you have if you divide 3/4 yard of string into pieces each 1/8 yard long?"
-- In an international math test taken by students worldwide in 1995 (the Third International Mathematics and Science Study, or TIMSS TIMSS Trends in International Mathematics and Science Study
TIMSS Third International Math and Science Study ), U.S. student math proficiency for 8th graders fell below the international average (28th out of 41 countries). For 12th grade, U.S. math performance was among the lowest (18th of the 21 countries participating).
-- In 1989 the National Council of Teachers of Mathematics The National Council of Teachers of Mathematics (NCTM) was founded in 1920. It has grown to be the world's largest organization concerned with mathematics education, having close to 100,000 members across the USA and Canada, and internationally. (NCTM NCTM National Council of Teachers of Mathematics
NCTM Nationally Certified Teacher of Music
NCTM North Carolina Transportation Museum
NCTM National Capital Trolley Museum
NCTM Nationally Certified in Therapeutic Massage ) published its Curriculum and Evaluation Standards for School Mathematics--an extensive set of mathematics standards for grades K-12 which de-emphasized memorization mem·o·rize
tr.v. mem·o·rized, mem·o·riz·ing, mem·o·riz·es
1. To commit to memory; learn by heart.
2. Computer Science To store in memory: of number facts, the learning of proofs, and algebraic 1. (language) ALGEBRAIC - An early system on MIT's Whirlwind.
[CACM 2(5):16 (May 1959)].
2. (theory) algebraic - In domain theory, a complete partial order is algebraic if every element is the least upper bound of some chain of compact elements. skills, but encouraged the use of calculators and "discovery learning."
-- The National Science Foundation (NSF NSF - National Science Foundation ) promoted the NCTM standards beginning in 1991 and awarded millions of dollars in grant money for the writing of math texts that embraced them and to state boards state boards Examinations administered by a US state board of medical examiners to license a physician in a particular state; these examinations play an ever-decreasing role in state medical licensure, as these bodies now rely on standardized national examinations of education whose math standards aligned with them (see Figure 1).
Not knowing about these significant events (all of them before last December's release of the Program for International Student Assessment report ranking American 15-year-olds 24th out of 29 in math among industrial countries (see Figure 2)), I had no idea that our children were being deprived of a math education, thanks in no small part to a dubious education theory, watered-down standards, and a well-meaning but intellectually bankrupt federally subsidized sub·si·dize
tr.v. sub·si·dized, sub·si·diz·ing, sub·si·diz·es
1. To assist or support with a subsidy.
2. To secure the assistance of by granting a subsidy. program of math illiteracy illiteracy, inability to meet a certain minimum criterion of reading and writing skill. Definition of Illiteracy
The exact nature of the criterion varies, so that illiteracy must be defined in each case before the term can be used in a meaningful .
What I did know was that my 9th grader didn't know how to do proofs.
I looked through his textbook, one of whose authors was a recent president of NCTM, and I was surprised to find very few proofs of anything. More troubling, most theorems This is a list of theorems, by Wikipedia page. See also
tr.v. mem·o·rized, mem·o·riz·ing, mem·o·riz·es
1. To commit to memory; learn by heart.
2. Computer Science To store in memory: them. The problems at the end of the chapter required students to do only a few simple proofs.
Proofs in geometry class have been a mainstay of mathematics. In fact, proofs were always considered an essential part of high school geometry, not only because of their importance in higher math, but because learning the rules of logical argument and reasoning has applications in science, law, political science, and writing. To see proofs being shortchanged in a geometry textbook was shocking.
Algebra texts were in no better condition, in terms of presentation and content--or, rather, their lack of content. Even if you accept the argument that geometry in general, and proofs in particular, are unnecessary for students to learn, at least algebra should be taught properly, since algebra is the common language of, and gateway to, all of higher math. The absence of clear explanation and logical development left students I later tutored in algebra as lost as my geometry student. Their textbooks (and, probably, their teachers too) encouraged them to use a graphing calculator Graphing Calculator may refer to:
Algebraic equation of particular importance in optimization. A more descriptive name is second-degree polynomial equation. Its standard form is ax2 + bx + c , once the pinnacle of any first-year algebra course. Instead, the quadratic formula quadratic formula
The formula x = [-b is presented for the students to memorize and apply--if it is even mentioned at all.
At the time I started tutoring, my daughter was in 2nd grade. I was concerned that she was not learning her addition and subtraction subtraction, fundamental operation of arithmetic; the inverse of addition. If a and b are real numbers (see number), then the number a−b is that number (called the difference) which when added to b (the subtractor) equals facts. Other parents we knew were saying the same thing. Teachers told them not to worry because kids eventually "get it."
One teacher told me her understanding of the new method. "It used to be that if you missed a concept or method in math, then you were lost for the rest of the year. But the way we do it now, kids have a lot of ways to do things, like adding and subtracting, so that math topics from day to day aren't dependent on kids' mastering a previous lesson."
In a world where it doesn't matter when you learn something, because you'll get it eventually, there seem to be few if any critical junctures, no mastery of procedure, no building on what you've learned--no learning.
The Hell-Hath-No-Fury Postulate postulate: see axiom.
1. Occurring as or resulting from coincidence.
2. Happening or existing at the same time.
co·in , I had the opportunity to find out how my experiences related to the politics of math education. I work for the federal government, which has a program that gives employees a chance to work on Capitol Hill to gain experience and knowledge of legislative and congressional procedures, which is valuable information when working in government. I applied for and received a six-month detail to work in a Democratic senator's office. Senator X (so called, in keeping with mathematical convention to describe a class of variables, because, as I was also to learn, both the good intentions and the shortcomings A shortcoming is a character flaw.
Shortcomings may also be:
1. Raised or grown at home.
2. Originating in or characteristic of a locality: "Rock is homegrown music in the United States, evolved from blues and country and Tin Pan Alley" " breed of scientists and engineers who would then support that state's burgeoning technology industry. Since I thought a likely place to start would be math education, the staffers working the education issue asked me to see what I could come up with.
I compiled a list of questions that I sent by e-mail to various mathematicians Mathematicians by letter: A B C D E F G H I J K L M N O P Q R S T U V W X Y Z See also
The nice thing about working on the Hill is that you almost always get responses to e-mails and phone calls. Fifteen minutes after I sent an e-mail to Harvard mathematics professor Wilfried Schmid, he called. I found out that his initiation into the world of K-12 math education was similar to mine--through his daughter. He explained how she was not being taught her multiplication tables multiplication table
A table, used as an aid in memorization, that lists the products of certain numbers multiplied together, typically the numbers 1 to 12. . He was shocked at the math instruction she was receiving in the 3rd grade. Its substance was shallow, memorization was discouraged, students were kept dependent on mental crutches (her teacher made her work with blocks or count on her fingers), and the intellectual level was well below the capability of most of the kids in his daughter's class.
Schmid's reaction to the problems of math texts and teaching was similar to that of other mathematicians I talked to in the course of my Capitol Hill assignment, particularly those with children. Those dialogues led me to develop an ad hoc For this purpose. Meaning "to this" in Latin, it refers to dealing with special situations as they occur rather than functions that are repeated on a regular basis. See ad hoc query and ad hoc mode. theory that I will postulate (this means I don't have to prove it): Hell hath hath
Third person singular present tense of have. no fury like a mathematician whose child has been scorned by an education system that refuses to know better. In Schmid's case, he talked to parents, school boards, and ultimately with the Massachusetts commissioner of education. Along with others, he succeeded in revamping Massachusetts's math standards, much to the dislike of the education establishment and textbook publishers.
Framing the Debate
The controversy over K-12 math education has come to be known as the "math wars Math wars is the debate over modern mathematics education, textbooks and curricula in the US that was triggered by the publication in 1989 of the Principles and Standards for School Mathematics by the National Council of Teachers of Mathematics (NCTM). ." Like Schmid, mathematicians have been active in this debate, as has the "mathematics community" at large, including not only mathematicians at the university level, but teachers and others involved in the education establishment. They believe that students must master basic skills (the number facts, standard algorithms For computer algorithms, see .
In elementary arithmetic, a standard algorithm or method is an efficient manual method of computation which yields one correct answer, and has been traditionally taught over a long period of time. for adding, subtracting, multiplying, and dividing) in tandem Adv. 1. in tandem - one behind the other; "ride tandem on a bicycle built for two"; "riding horses down the path in tandem"
tandem with larger concepts about mathematics.
On the other side of the debate are the followers followers
see dairy herd. of an education theory that promotes discovery learning, minimization of both teacher instruction and repetitive drills, and a disdain for standard procedures (algorithms). The math being protested--by the mathematics community--is called a variety of things: "reform math," "standards-based math," "new new math new math
Mathematics taught in elementary and secondary schools that constructs mathematical relationships from set theory. Also called new mathematics. ," and, most commonly, "fuzzy math Not to be confused with fuzzy logic.
Fuzzy math (also called "reformed math", "whole math", "constructivist math" or "new-new math") is an educational approach to the teaching of basic mathematics for children. ."
Although the education theories on which much of fuzzy math is based are promoted in many education schools in the United States United States, officially United States of America, republic (2005 est. pop. 295,734,000), 3,539,227 sq mi (9,166,598 sq km), North America. The United States is the world's third largest country in population and the fourth largest country in area. , it is not accurate to say that everyone in the education arena buys into these ideas. Therefore, for purposes of clarity, and to be consistent with a vocabulary used by others describing the math wars, I will use the term "educationist" to refer to those who promote the contested theory of math education known generally as discovery learning.
The math wars revolve around Verb 1. revolve around - center upon; "Her entire attention centered on her children"; "Our day revolved around our work"
center, center on, concentrate on, focus on, revolve about a four-part problem: A disputed theory of education that informs NCTM's standards; state boards of education that base their standards of learning Standards of Learning or (SOL) is a program of the Commonwealth of Virginia. It sets forth learning and achievement expectations for grades K-12 in Virginia's Public Schools. for mathematics on the NCTM standards; textbooks written to incorporate these standards; and teachers and others in the education establishment who are indoctrinated in the disputed education theory and who may not possess enough knowledge of mathematics to overcome the first three factors.
The education theory at the heart of the dispute can be traced to John Dewey, an early proponent One who offers or proposes.
A proponent is a person who comes forward with an a item or an idea. A proponent supports an issue or advocates a cause, such as a proponent of a will.
PROPONENT, eccl. law. of learning through discovery. But for all practical purposes, the story begins on October 4, 1957, when the Soviet Union launched Sputnik Sputnik: see satellite, artificial; space exploration.
Any of a series of Earth-orbiting spacecraft whose launching by the Soviet Union inaugurated the space age. . This event signaled the American malaise in science and mathematics education and the need to overcome it if we weren't to fall further behind the Soviet Union and lose the cold war. Thus the U.S. Congress was motivated to pass appropriations that triggered the development of the "new math," a national effort, spurred by the National Science Foundation, that eventually found its way into most of our schools.
The curriculum, designed primarily by mathematicians, had problems, but it introduced to algebra, geometry, and trigonometry trigonometry [Gr.,=measurement of triangles], a specialized area of geometry concerned with the properties of and relations among the parts of a triangle. Spherical trigonometry is concerned with the study of triangles on the surface of a sphere rather than in the a long-missing formalism Formalism
or Russian Formalism
Russian school of literary criticism that flourished from 1914 to 1928. Making use of the linguistic theories of Ferdinand de Saussure, Formalists were concerned with what technical devices make a literary text literary, apart , logic, and consistency, and it resulted in calculus calculus, branch of mathematics that studies continuously changing quantities. The calculus is characterized by the use of infinite processes, involving passage to a limit—the notion of tending toward, or approaching, an ultimate value. being taught in high school. The problem, however, was that a similar formalism was introduced into K-6 texts and curricula, with the result that students (and elementary school elementary school: see school. teachers, who were caught totally off guard) were exposed to number bases, set theory, and axioms This is a list of axioms as that term is understood in mathematics, by Wikipedia page. In epistemology, the word axiom is understood differently; see axiom and self-evidence. Individual axioms are almost always part of a larger axiomatic system. long before they were ready for them. And soon enough the new math was being blamed for not teaching basic arithmetic: it was often said that new-math students could tell you that 5 + 3 = 3 + 5, but didn't know that it was equal to 8.
Mathematicians have agreed for years that emphasizing sets and number bases in math programs designed for the lower grades was a horrendous mistake. Notwithstanding these errors, however, the difference between the current slew of textbooks and those from the new-math days of the 1960s is definitely worth noting: Accomplished mathematicians wrote many of the texts used in that earlier era, and the math--though misguided and inappropriate for the lower grades and too formal for the high school grades--was at least mathematically correct Mathematically Correct is a website created by educators, parents, citizens and mathematicians / scientists who are concerned about the direction of reform mathematics curricula based on NCTM standards. It is one of the most frequently cited websites in the Math wars. . Some of the high school texts were absolutely first-rate, and new-math--era textbooks like Mary Dolciani's "Structure and Method" series for algebra and geometry continue to be used by math teachers who understand mathematics and how it is to be taught. (They usually use them on the sly, since most teachers are required to use the books that the schools have adopted.)
During the new-math era, which spanned the period between Sputnik to the early 1970s, mathematicians dominated the design of math texts and curricula for the first, and almost last, time. Up to that point mathematicians had been kept out of the math education picture, and K-12 mathematics tended not to include any examination of the logical structure of mathematics itself, with the single exception of Euclidean geometry Euclidean geometry
Study of points, lines, angles, surfaces, and solids based on Euclid's axioms. Its importance lies less in its results than in the systematic method Euclid used to develop and present them. . Students in the first half of the 20th century had instruction in practical matters: consumer buying, insurance, taxation--everything but algebra, geometry, or trigonometry, which, when they were taught, were frequently lacking in depth.
Significantly, the new-math era was one of the only times that mathematicians were given an opportunity to make proper math education available to the masses. (Not until the past few years, working with several state education departments, would they be allowed back into math education decisions.) And some believe that had certain prominent mathematicians who had started working with the development of the new-math programs managed to maintain their influence on those programs, the math education that would have emerged from new math--both lower grades and high school--would have been on par with the best of the math programs overseas.
Eventually, however, the problems with K-6 formalism and the logic and formalism of the program in general doomed new math. The general public, the education community, and even mathematicians themselves judged the new-math programs a failure. Mathematicians were assigned the blame, and the education establishment took back the reins. That establishment received an inadvertent boost in 1983 with the publication of A Nation at Risk, the shockingly pessimistic assessment of the nation's schools by the National Commission on Excellence in Education The National Commission on Excellence in Education produced the 1983 report titled A Nation at Risk. It was chaired by David P. Gardner and included prominent members such as Nobel prize-winning chemist Glenn T. Seaborg. . The report sounded another alarm about student math performance, and the NCTM, increasingly dominated by educationists, took advantage of this new education crisis to write revised math standards. The Curriculum and Evaluation Standards for School Mathematics, published in 1989, purported to put the country back on the math track. But because it was, in part, a reaction to the new math and those believed responsible for it, NCTM did not, as mathematicians point out, promote a lively public debate, as had the creators of the new math, but suppressed it.
Some Secrets about Discovery Learning
The NCTM standards were a brew of progressivism--a nod to the 1920s when math was supposed to be practical--and constructivism constructivism, Russian art movement founded c.1913 by Vladimir Tatlin, related to the movement known as suprematism. After 1916 the brothers Naum Gabo and Antoine Pevsner gave new impetus to Tatlin's art of purely abstract (although politically intended) , which was progressivism that adapted research from cognitive psychology cognitive psychology, school of psychology that examines internal mental processes such as problem solving, memory, and language. It had its foundations in the Gestalt psychology of Max Wertheimer, Wolfgang Köhler, and Kurt Koffka, and in the work of Jean to the task of teaching and called it discovery learning. The standards were based on theories of learning that assumed that children had an innate ability to understand math. The group's math curricula were thus structured to allow children to discover math concepts rather than to be given them, through direct instruction. The standards also expanded their reach to include, in addition to basic arithmetic, algebra, geometry, and trigonometry.
The NCTM's view was that traditional teaching techniques, known as "drill and kill," numbed student minds, turned them off math, and taught them nothing. And so the new standards recommended that students learn "strategies" for learning number facts rather than memorize those facts. It emphasized the use of calculators in all grades. Most important, however, the standards recommended certain areas that should receive "decreased attention" in grades K-4, including "complex paper-and-pencil computations," "long division," "paper-and-pencil fraction computation," "use of rounding to estimate," "rote rote 1
1. A memorizing process using routine or repetition, often without full attention or comprehension: learn by rote.
2. Mechanical routine. practice," "rote memorization of rules," and "teaching by telling." This last item, teaching by telling, is a reference to direct instruction (telling students what they need to know), which NCTM believed should be replaced by "discovery."
Discovery learning has always been a powerful teaching tool. But constructivists take it a step beyond mere tool, believing that only knowledge that one discovers for oneself is truly learned. There is little argument that learning is ultimately a discovery. Traditionalists also believe that information transfer via direct instruction is necessary, so constructivism taken to extremes can result in students' not knowing what they have discovered, not knowing how to apply it, or, in the worst case, discovering--and taking ownership of--the wrong answer. Additionally, by working in groups and talking with other students (which is promoted by the educationists), one student may indeed discover something, while the others come along for the ride.
Texts that are based on NCTM's standards focus on concepts and problem solving problem solving
Process involved in finding a solution to a problem. Many animals routinely solve problems of locomotion, food finding, and shelter through trial and error. , but provide a minimum of exercises to build the skills necessary to understand concepts or solve the problems. Thus students are presented with real-life problems in the belief that they will learn what is needed to solve them. While adherents believe that such an approach teaches "mathematical thinking" rather than dull routine skills, some mathematicians have likened it to teaching someone to play water polo water polo, swimming game encompassing features of soccer, football, basketball, and hockey. The object of the game is to maneuver, by head, feet, or hand, a leather-covered ball 27 to 28 in. without first teaching him to swim.
The Standards were revised in 2000, due in large part to the complaints and criticisms expressed about them. Mathematicians felt that the revised standards, called The Principles and Standards for School Mathematics Principles and Standards for School Mathematics was a document produced by the National Council of Teachers of Mathematics  in 2000 to set forth a national vision for precollege mathematics education in the US and Canada. (PSSM PSSM Principles and Standards for School Mathematics
PSSM Position Specific Scoring Matrix (Protein/DNA sequence comparison)
PSSM Polysaccharide Storage Myopathy
PSSM Pretty Soldier Sailor Moon
PSSM Packet Switch Service Module 2000), were an improvement over the 1989 version, but they had reservations. The revised standards still emphasize learning strategies over mathematical facts, for example, and discovery over drill and kill.
Concept still trumps memorization. Textbooks often make sure students understand what multiplication multiplication, fundamental operation in arithmetic and algebra. Multiplication by a whole number can be interpreted as successive addition. For example, a number N multiplied by 3 is N + N + N. means rather than offering exercises for learning multiplication facts. Some texts ask students to write down the addition that a problem like 4 X 3 represents. Most students do not have a difficult time understanding what multiplication means. But the necessity of memorizing the facts is FACTS I Federal Agencies' Centralized Trial-Balance System still there. Rather than drill the facts, the texts have the students drill the concepts, and the student misses out on the basics of what she must ultimately know in order to do the problems. I've seen 4th and 5th graders, when stumped by a multiplication fact such as 8 X 7, actually sum up 8, 7 times. Constructivists would likely point to a student's going back to first principles as an indication that the student truly understood the concept. Mathematicians tend to see that as a waste of time.
Another case in point was illustrated in an article that appeared last fall in the New York New York, state, United States
New York, Middle Atlantic state of the United States. It is bordered by Vermont, Massachusetts, Connecticut, and the Atlantic Ocean (E), New Jersey and Pennsylvania (S), Lakes Erie and Ontario and the Canadian province of Times. It described a 4th-grade class in Ossining, New York Ossining is the name of two places in New York:
A movement in modern art originating in Moscow in 1920 and characterized by the use of industrial materials such as glass, sheet metal, and plastic to create nonrepresentational, often geometric objects. approach to teaching math and spent one entire class period circling the even numbers on a sheet containing the numbers 1 to 100. When a boy who had transferred from a Catholic school told the teacher that he knew his multiplication tables, she quizzed him by asking him what 23 X 16 equaled. Using the old-fashioned method--one that is held in disdain because it uses rote memorization and is not discovered by the student--the boy delivered the correct answer. He knew how to multiply while the rest of the class was still discovering what multiples of 2 were.
Enter Stage Left: The National Science Foundation
The NCTM standards received a boost of credibility in 1991 when the Education and Human Resources The fancy word for "people." The human resources department within an organization, years ago known as the "personnel department," manages the administrative aspects of the employees. Division of the National Science Foundation funded two grant programs related to math education. The first was for state education departments that aligned their math standards with NCTM's and school districts that adopted constructivist math programs aligned with NCTM's standards; the second, for the development of commercial mathematics texts that also followed the NCTM party line, a spending program that now took NSF into the textbook business.
Eventually, NSF supplemented grants to school districts by funding "distribution centers" that promoted the very programs NSF had helped to create. This "Invasion of the Body Snatchers" technique has worked well in enhancing the NCTM constructivist ethic and the texts that NSF helped develop. Mean-while, NSF continues to fund revisions to some of those texts.
The imprimatur of the National Science Foundation on both NCTM's standards and the various textbooks they funded and NSF's grant program caused states to revise their standards of learning for mathematics. Case in point: In 1992, the California State Board of Education The California State Board of Education is the governing and policy-making body of the California Department of Education. The State Board of Education sets K-12 education policy in the areas of standards, instructional materials, assessment, and accountability. adopted the California Mathematics Framework--a set of standards based prominently on NCTM's standards of 1989.
And Stage Right: the Right
By this time, people were beginning to notice. NSF's embrace of NCTM's philosophy of education, among other problems with the teaching of mathematics, became grist for Lynne Cheney, then an active senior fellow at the American Enterprise Institute The American Enterprise Institute for Public Policy Research (AEI) is a conservative think tank, founded in 1943. According to the institute its mission "to defend the principles and improve the institutions of American freedom and democratic capitalism — limited government, . With well-articulated essays in leading media, Cheney took out after the educationists and won the respect of mathematicians and scientists as she helped raise awareness among a wider audience across the United States. Isolated math revolts began to occur. One of the first was in Silicon Valley, where parents are engineers, scientists, programmers, and mathematicians--and they didn't like the way their children were being taught mathematics. Two local mathematicians, Jim Milgram of Stanford and Hung-Hsi Wu of Berkeley, became key players in the rewriting of California's math standards and the elimination of NSF-funded books from the state's curriculum. Several years later, as I mentioned earlier, Wilfried Schmid from Harvard became active and helped rewrite Massachusetts's standards. Two other states, Minnesota and Michigan (Milgram and Wu went to the Wolverine wolverine or glutton, largest member of the weasel family, Gulo gulo, found in the northern parts of North America and Eurasia, usually in high mountains near the timberline or in tundra. State as well), also just recently revised their math standards. Minnesota's standards were changed due in large part to the efforts of Dr. Larry Gray, chair of the mathematics department at the University of Minnesota (body, education) University of Minnesota - The home of Gopher.
Address: Minneapolis, Minnesota, USA. , and of concerned and outraged parents.
But the education bureaucracy did not roll over, and in the fall of 1999 the U.S. Department of Education released a list of ten recommended math programs, designated as "exemplary" or "promising," all of them aligned with the NCTM standards and based on texts funded by NSF.
The reaction was swift. More than two hundred university professors--including Milgram and Wu, Schmid, and several winners of the Fields Medal, the highest international award in mathematics--wrote an open letter to Secretary of Education Richard Riley Richard Wilson Riley (born January 2, 1933), American politician, was the United States Secretary of Education under President Bill Clinton as well as the Governor of South Carolina, as a member of the Democratic Party. , calling on the Department of Education to withdraw the recommendations. The open letter was also published as a full-page ad in the Washington Post, paid for by the Packard Humanities Institute, long a critic of constructivist education. The Department of Education did not withdraw the recommendations, but instead added two more NCTM-aligned books to the list.
The Education Department's 1999 list, as it represented a more visible--and seemingly overtly political--assertion of political will, was a critical point in the new-math wars. And when Lynne Cheney slammed the Clinton administration Noun 1. Clinton administration - the executive under President Clinton
executive - persons who administer the law as she criticized math textbooks and the NSF in her editorials, she helped ensure that the math wars would become (as they remain) partisan. Republicans tended to be sympathetic to the issues, and some hearings were held. But nothing came of them, and no investigations into NSF and its funding practices were launched. No questions were asked about why the Department of Education didn't rely on mathematicians in the review of proposals for these programs, nor was anyone in the department ever questioned about the NCTM's education philosophy and the millions of tax dollars spent on texts that were the subject of fierce objections from 200 prominent mathematicians and scholars.
An Evidentiary ev·i·den·tia·ry
1. Of evidence; evidential.
2. For the presentation or determination of evidence: an evidentiary hearing.
Adj. 1. Interlude interlude, development in the late 15th cent. of the English medieval morality play. Played between the acts of a long play, the interlude, treating intellectual rather than moral topics, often contained elements of satire or farce.
In the various arguments about how best to teach math, educationists make the point that research shows that their approaches work best. I tend to be suspicious of that research, and apparently others are as well. Ironically, a recent study by the National Academy of Science's Mathematical Sciences Education Board, sponsored by the NSF itself, was also skeptical. "Evaluations of mathematics curricula provide important information for educators, parents, students and curriculum developers," concluded the NAS (1) See network access server.
(2) (Network Attached Storage) A specialized file server that connects to the network. A NAS device contains a slimmed-down operating system and a file system and processes only I/O requests by supporting the popular about 19 specific mathematics curricula, including all 13 NSF-sponsored curricula, that it analyzed. They all "fall short of the scientific standards necessary to gauge overall effectiveness."
While the proponents of these NSF-sponsored math programs may be able to claim that the research shows no evidence that the programs are "ineffective," the mathematics community, and parents who are protesting to the various school boards across the United States, can now claim that the research cannot be used to support claims of superior effectiveness--or any effectiveness at all.
Is there a common ground in this war? It seems that for now both sides agree that subject matter should be as important as pedagogy and that improving the math education of teachers is probably the most important weapon in the battle. What constitutes a proper math education for prospective teachers, however, is subject to debate. A mathematician I know who teaches at a small college told me that a graduate of his education school caused much embarrassment when, during the interview for a job in an elementary school, the job seeker job seeker also job·seek·er
One who seeks employment. was unable to add two fractions when asked to do so. The college's education department subsequently decided to put its math content courses under control of the math department.
Meanwhile, Back on the Hill
Though academic debate about mathematics curricula will no doubt continue, the field of argument is increasingly muddied by politics. It was in this context that I began my investigation into math education in 2002. I recall meeting with Senator X's deputy chief of staff and two other staffers not long after completing my research on math curricula and the battles that had shaped--often, misshaped--them. "So what are your ideas on how math and science education can be enhanced?" they asked. My answer was something like, "You can enhance a car by painting it, but if the car has no engine, it's not going to do much good." This was not what they were expecting to hear. Nor were they expecting to hear that Lynne Cheney had also taken up the cause of antifuzzy math. At that point, the discussion took a decidedly troubling turn. These staffers--Democrats--now worried that they could not support policies that were also advocated by the wife of a powerful Republican.
I told them about the open letter from the two hundred mathematicians and urged them not to confuse the message with the messenger. "This is a real issue," I said. "Kids aren't learning the math they need to learn."
I had discussions and sent e-mails in the hopes that I would at least get a chance to brief Senator X on the issue and, perhaps, persuade him to ask some tough questions of NSF when it came time to fund their programs. But I felt that at any moment everything was going to be whisked away.
And one day it was. The staffers in my office talked with other Democratic staffers on the Hill, who told them that it would be wise to stay away from the "fuzzy math/Lynne Cheney/Bush agenda" issue. Ultimately the staffers I was working with told me they couldn't take a chance on having Senator X "come off like Lynne Cheney."
This development was not surprising to any of the mathematicians with whom I had been working--most of them Democrats, like me. The senator was never briefed, and no investigation into NSF was launched. I was thanked for my hard work. I went back to my regular job and started tutoring middle school students in math at a school in D.C. while continuing to work with high school students in my neighborhood. That year a 9th-grade girl was having problems in geometry and came to me for help. "What seems to be the problem?" I asked. "I don't know how to do proofs," she said.
"I know," I replied. "Don't worry. It isn't you."
All politics is local, I decided.
Textbook Taxes (Figure 1) Since 1990, the National Science Foundation has awarded more than $83 million to programs that developed textbooks promoting approaches that are favored by the National Council of Teachers of Mathematics. Program Grant Organization name amount (in millions of dollars) University of Illinois Math Trailblazer 6.4 at Chicago TERC, Inc. Investigations in Number, 11.4 Data and Space University of Chicago Everyday Mathematics 5.4 Education Development Center MathScape 4.3 Michigan State University Connected Mathematics 4.9 Project University of Wisconsin Mathematics in Context 6.2 at Madison University of Montana MATH Thematics 5.7 Institute for Research Middle School Mathematics 2.8 on Learning through Application Project Connecticut Business and Math Connections 4.1 Industry Association (Secondary Mathematics Education Foundation Core Curriculum Initiative) Western Michigan University Contemporary Mathematics 13.9 in Context (Core Plus) San Francisco State Interactive Mathematics 11.6 University Program Consortium for Mathematics Mathematics: Modeling Our 6.1 and Its Applications, Inc. World (ARISE) Montana Council of Teachers Systemic Initiative for 1.0 of Mathematics Montana Mathematics and Science SOURCE: National Science Foundation The Perils of Fuzzy Math (Figure 2) In math, 15-year-olds in the United States finished 24th out of 29 among Organization for Economic Cooperation and Development (OECD) countries in the triennial Program for International Student Assessment (PISA). Score on 2003 PISA Mathematics Literacy Assessment Finland 544 Korea 542 Netherlands 538 Canada 532 OECD average 500 United States 483 Italy 466 SOURCE: OECD, PISA 2003, 2004 Note: Table made from bar graph.
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The National Science Foundation has funded math texts that meet the standards of the National Council of Teachers of Mathematics. Of these I am most familiar with the Everyday Mathematics (EM) series for grades K-6 because this one was adopted by the Fairfax County school district in Virginia, where I live. EM is now used by some 10 percent of the nation's elementary school students, including in New York City's more than six hundred grade schools. Of particular interest is EM's claim on its web site: "Previous efforts to reform mathematics instruction failed because they did not adequately consider the working lives of teachers."
This may help explain why EM believes that alternative algorithms and "student invented" algorithms make computation easier to understand and master. Students now have a variety of choices to multiply two- and three-digit numbers, including an ancient Egyptian method called "lattice multiplication." And instead of multiplying 45 X 24 in the traditional way, they can multiply 40 X 20, 5 X 20, 4 X 40, and 5 X 4 and add the four products. In fact, EM does not cover the traditional method of multiplying two-digit numbers until the 5th grade; 4th grade is spent mastering the alternatives.
Here is how EM explains its approach to the long-division algorithm, according to according to
1. As stated or indicated by; on the authority of: according to historians.
2. In keeping with: according to instructions.
3. the Teachers Reference Manual:
"The authors of Everyday Math do not believe it is worth the time and effort to develop highly efficient paper-and-pencil algorithms for all possible whole number, fractions and decimal division problems.... It is simply counterproductive coun·ter·pro·duc·tive
Tending to hinder rather than serve one's purpose: "Violation of the court order would be counterproductive" Philip H. Lee. to invest hours of precious class time on such algorithms. The math payoff is not worth the cost, particularly because quotients can be found quickly and accurately with a calculator."
In fact, long division has particular importance, not because of its ability to increase computational fluency, but because what makes it work (the distributive dis·trib·u·tive
a. Of, relating to, or involving distribution.
b. Serving to distribute.
2. property) is an important concept that students will use later in algebra. It also plays an important role in uncovering another significant math concept: why fractions give rise to repeating decimals re·peat·ing decimal
A decimal in which a pattern of one or more digits is repeated indefinitely, for example, 0.353535 ... Also called recurring decimal. . Working out the division of numbers like 1/3 and 1/7 helps students see this; using a calculator does not.
Everyday Mathematics says it bases its claims on "research." For example, this from the Teachers Reference Manual:
"In one study, only 60 percent of U.S. ten-year-olds achieved mastery of the algorithm using the standard regrouping (borrowing) algorithm. A Japanese study found that only 56 percent of 3rd graders and 74 percent of 5th graders achieved mastery of this algorithm."
I e-mailed a set of questions about the study to the contact listed on Everyday Math's web site: What test was used? Were all schools tested? How was "failure" defined? Were any follow-up studies conducted? And similar questions about Japan. I remarked that a generation of adults appears to have mastered the traditional algorithms just fine. A few months later I received a reply from April Hattori, vice president of communications at McGraw-Hill, which publishes EM. She asked if I were a reporter, and how I planned to use the information. I responded that I was thinking of writing an article and my questions were meant to explore EM's claims that students do better when allowed to invent their own methods. I have received no further communications from Ms. Hattori.
Barry Garelick is an analyst with a federal government agency in Washington, D.C.
The opinions expressed in this article are those of the author.