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Mathematical Cognition, Edited by James Royer Copyright by Information Age Publishing, 2003
In January 1998, when U.S. Education Secretary Richard Riley called for an end to the "math wars" in a speech before a joint meeting of the American Mathematical Society and the Mathematical Association of America, he could not have known that within two years, the department he directed would become the focus of the very math wars he sought to quell. In October 1999, the U.S. Department of Education recommended to the nation's 15,000 school districts a list of math books, including several that had been sharply criticized by mathematicians and parents of school children across the country for much of the preceding decade. Within a month of that release, 200 university mathematicians added their names to an open letter to Secretary Riley calling upon his department to withdraw those recommendations. The list of signatories included seven Nobel laureates and winners of the Fields Medal, the highest international award in mathematics, as well as math department chairs of many of the top universities in the country, and several state and national education leaders.1 By the end of the year 1999, the U.S. Secretary of Education had himself become embroiled in the nation's math wars.
Mathematics education policies and programs for U.S. public schools have never been more contentious than they were during the decade of the 1990s. The immediate cause of the math wars of the 90s was the introduction and widespread distribution of new math textbooks with radically diminished content, and a dearth of basic skills. This led to organized parental rebellions and criticisms of the new math curricula by mathematicians and other professionals.
In some respects the education wars of the 1990s have little to distinguish them from earlier periods. There is nothing new about disagreements over the best ways to educate the nation's school children. The periodic waves of education reform from the nation's colleges of education are more similar than they are different. The American education establishment has consistently advocated a progressivist education agenda for the bulk of the 20th century, and the mainstream views of the education community have enjoyed a commanding influence on public schools.2 Recognizing this dominion in the early part of the century, William Bagley in 1926 lamented:
In no other country are the professional students of education so influential. In no other country is school practice so quickly responsive to the suggestions emanating from this group. We may stigmatize our schools as "static, " "reactionary," "slow to change,"-- reluctant to adopt what we, in our wisdom, prescribe. But compared to other countries, ours is the educational expert's paradise.3
Colleges of education exert powerful direct influence on elementary and middle school teachers, and indirect influence on them through other organizations such as state level departments of education and professional teacher organizations. The influence on high school math teachers, while still powerful, has been less direct because of the subject matter specialization of the high school curriculum. The content demands of mathematics itself have limited the direct influence of some pedagogical fashions on high school math teachers. However, because of the hierarchical nature of mathematics and its heavy dependence at any level on prerequisites, high school and even college mathematics courses have at times been strongly affected by progressivist ideas, especially at the end of the 20th century.
The political struggles and policy changes in mathematics education in the 1980s, and especially the 1990s are the major topics of this chapter. However, the events of the final two decades of the 20th century are more easily understood in an historical context. Throughout the 20th century the "professional students of education" have militated for child centered discovery learning, and against systematic practice and teacher directed instruction. In some cases, progressivist math programs of the 1990s were intentionally without student textbooks, since books might interfere with student discovery. The essence of the dictum from educators of the 1990s and late 1980s, that the teacher should be "a guide on the side and not a sage on the stage," was already captured in a statement from the principal of one of John Dewey's "schools of tomorrow" from the 1920s:
The teacher's arbitrary assignment of the next ten pages in history, or nine problems in arithmetic, or certain descriptions in geography, cannot be felt by the pupil as a real problem and a personal problem.4
The next section provides a brief overview of some of the important historical trends and policies leading up to the events of the 1980s and 90s.
Historical Outline: 1920 to 1980
It would be a mistake to think of the major conflicts in education as disagreements over the most effective ways to teach. Broadly speaking, the education wars of the past century are best understood as a protracted struggle between content and pedagogy. At first glance, such a dichotomy seems unthinkable. There should no more be conflict between content and pedagogy than between one's right foot and left foot. They should work in tandem toward the same end, and avoid tripping each other. Content is the answer to the question of what to teach, while pedagogy answers the question of how to teach.
The trouble comes with the first step. Do we lead with the right foot or the left? If content decisions come first, then the choices of pedagogy may be limited. A choice of concentrated content precludes too much student centered, discovery learning, because that particular pedagogy requires more time than stiff content requirements would allow. In the same way, the choice of a pedagogy can naturally limit the amount of content that can be presented to students. Therein lies the source of the conflict.
With roots going back to Jean Jacques Rousseau and with the guidance of John Dewey, progressive education has dominated American schools since the early years of the 20th century. That is not to say that progressive education has gone unchallenged.5 Challenges increased in intensity starting in the 1950s, waxed and waned, and in the 1990s gained unprecedented strength. A consequence of the domination of progressivism during the first half of the 20th century was a predictable and remarkably steady decrease of academic content in public schools.
The prescriptions for the future of mathematics education were articulated early in the 20th century by one of the nation's most influential education leaders, William Heard Kilpatrick. According to E. D. Hirsch, Kilpatrick was "the most influential introducer of progressive ideas into American schools of education."6 Kilpatrick was an education professor at Teachers College at Columbia University, and a protege of John Dewey. According to Dewey, "In the best sense of the words, progressive education and the work of Dr. Kilpatrick are virtually synonymous."7 Kilpatrick majored in mathematics at Mercer College in Macon, Georgia. His mathematical education included some graduate work at Johns Hopkins University, but his interests changed and he eventually attended Teachers College and joined the faculty in 1911. In his 27 years at Teachers College, he taught some 35,000 students and was described by the New York Post as "the million dollar professor" because the fees paid by his students to the college exceeded this amount. In some instances there were more than 650 students in a single one of his auditorium sized classes.8 His book, Foundations of Method, written in 1925 became a standard text for teacher education courses across the country.
Reflecting mainstream views of progressive education, Kilpatrick rejected the notion that the study of mathematics contributed to mental discipline. His view was that subjects should be taught to students based on their direct practical value, or if students independently wanted to learn those subjects. This point of view toward education comported well with the pedagogical methods endorsed by progressive education. Limiting education primarily to utilitarian skills sharply limited academic content, and this helped to justify the slow pace of student centered, discovery learning, the centerpiece of progressivism. Kilpatrick proposed that the study of algebra and geometry in high school be discontinued "except as an intellectual luxury." According to Kilpatrick, mathematics is "harmful rather than helpful to the kind of thinking necessary for ordinary living." In an address before the student body at the University of Florida, Kilpatrick lectured, "We have in the past taught algebra and geometry to too many, not too few."9
Progressivists drew support from the findings of psychologist Edward L. Thorndike. Thorndike conducted a series of experiments beginning in 1901 that cast doubt on the value of mental discipline and the possibility of transfer of training from one activity to another. These findings were used to challenge the justification for teaching mathematics as a form of mental discipline and contributed to the view that any mathematics education should be for purely utilitarian purposes.10 Thorndike stressed the importance of creating many "bonds" through repeated practice and championed a stimulus-response method of learning. This led to the fragmentation of arithmetic and the avoidance of teaching closely related ideas too close in time, for fear of establishing incorrect bonds. According to one writer, "For good or for ill, it was Thorndike who dealt the final blow to the 'science of arithmetic.'"11
Kilpatrick's opinion that the teaching of algebra should be highly restricted was supported by other experts. According to David Snedden, the founder of educational sociology, and a prominent professor at Teachers College at the time, "Algebra...is a nonfunctional and nearly valueless subject for 90 percent of all boys and 99 percent of all girls--and no changes in method or content will change that."12 During part of his career, Snedden was Commissioner of Education for the state of Massachusetts.13
In 1915 Kilpatrick was asked by the National Education Association's Commission on the Reorganization of Secondary Education to chair a committee to study the problem of teaching mathematics in the high schools. The committee included no mathematicians and was composed entirely of educators.14 Kilpatrick directly challenged the use of mathematics to promote mental discipline. He wrote, "No longer should the force of tradition shield any subject from scrutiny...In probably no study did this older doctrine of mental discipline find larger scope than in mathematics, in arithmetic to an appreciable extent, more in algebra, and most of all in geometry."15 Kilpatrick maintained in his report, The Problem of Mathematics in Secondary Education, that nothing in mathematics should be taught unless its probable value could be shown, and recommended the traditional high school mathematics curriculum for only a select few.16
It was not surprising that mathematicians would object to Kilpatrick's report as an attack against the field of mathematics itself. David Eugene Smith, a mathematics professor at Teachers College and renowned historian of mathematics, tried to stop the publication of Kilpatrick's report as a part of the Cardinal Principles of Secondary Education, the full report of the Commission on the Reorganization of Secondary Education, and one of the most influential documents for education in the 20th century. Smith charged that there had been no meeting of the math committee and that Kilpatrick was the sole author of the report. Moreover, Kilpatrick's committee was not representative of teachers of mathematics or of mathematicians.17Nevertheless, Kilpatrick's report was eventually published in 1920 by the U.S. Commissioner of Education, Philander P. Claxton, a friend of Kilpatrick.18
The Kilpatrick committee and leading educational theoreticians had thrown the gauntlet, and the Mathematical Association of America (MAA) responded vigorously. Already in 1916, in anticipation of the Kilpatrick report, E. R. Hedrick, the first president of the MAA, appointed a committee called the National Committee on Mathematical Requirements. It was chaired by J. W. Young of Dartmouth and included mathematicians E. H. Moore, Oswald Veblen, and David E. Smith, in addition to several prominent teachers and administrators from the secondary school system. The reports of this committee were delayed because of World War I, but they were eventually collected into a 625 page volume entitled, The Reorganization of Mathematics for Secondary Education. The report was published in 1923 and is sometimes referred to as the 1923 Report.
Meanwhile in 1920, the National Council of Teachers of Mathematics (NCTM) was founded, largely at the instigation of the MAA. The first NCTM president, C. M. Austin, made it clear that the organization would "keep the values and interests of mathematics before the educational world" and he urged that "curriculum studies and reforms and adjustments come from the teachers of mathematics rather than from the educational reformers." The NCTM was created in part to counter the progressivist educational agenda for mathematics, and it later played an important role in disseminating the 1923 Report.19
The 1923 Reportwas perhaps the most comprehensive ever written on the topic of school mathematics. It included an extensive survey of secondary school curricula, and it documented the training of mathematics teachers in other countries. It discussed issues related to the psychology of learning mathematics, and justified the study of mathematics in terms of its applications as well as its intrinsic value. It even proposed curricula for the schools. In contradiction to the Kilpatrick report, the 1923 Report underscored the importance of algebra to "every educated person."20 The 1923 Report exerted some influence on public education. For example, some of the policies of the College Examination Board were based upon recommendations in the 1923 Report. However, over the next two decades, the views expressed in the Kilpatrick report wielded greater influence than the 1923 Report.21 The NCTM also changed over time. It grew and gradually it "attracted to its membership and to its leadership those in positions much more subject to the influence and pressure of the professional reform movements."22
In the 1930s the education journals, textbooks, and courses for administrators and teachers advocated the major themes of progressivism. The school curriculum would be determined by the needs and interests of children, as determined by professional educators, and not by academic subjects. It became a cliche in the 1930s, just as in the 1990s, for educators to say, "We teach children, not subject matter." The Activity Movement of the 1930s promoted the integration of subjects in elementary school, and argued against separate instruction in mathematics and other subjects. It drew its inspirations from Kilpatrick's writings. The Activity Movement spread rapidly into the nation's elementary schools. High schools were more resistant in part because the teachers were trained in specific subject areas and they were less willing to discard their specialties in favor of an ill defined holism. Some proponents of the Activity Movement did not even acknowledge that reading and learning the multiplication tables were legitimate activities . As in the 1990s, there was public resistance to the education doctrines of this era. Among the critics were Walter Lippman, one of the nation's most widely respected commentators on public affairs, and literary critic, Howard Mumford Jones.23
In the 1940s it became something of a public scandal that army recruits knew so little math that the army itself had to provide training in the arithmetic needed for basic bookkeeping and gunnery.24 Admiral Nimitz complained of mathematical deficiencies of would-be officer candidates and navy volunteers. The basic skills of these military personnel should have been learned in the public schools but were not.25 As always, education doctrines did not sit well with much of the public. Nevertheless, by the mid-1940s, a new educational program called the Life Adjustment Movement emerged from the education community. The basic premise was that secondary schools were "too devoted to an academic curriculum." Education leaders presumed that 60% or more of all public school students lacked the intellectual capability for college work or even for skilled occupations, and those students would need a school program to prepare them for every day living. They would need appropriate high school courses, including math programs, that focused purely on practical problems such as consumer buying, insurance, taxation, and home budgeting, but not on algebra, geometry, or trigonometry. The students in these courses would become unskilled or semiskilled laborers, or their wives, and they would not need an academic education. Instead they would be instructed in "home, shop, store, citizenship, and health."
By 1949 the Life Adjustment Movement had substantial support among educators, and was touted by numerous federal and state education agencies. Some educators even suggested that in order to avoid stigmatizing the students in these programs, non-academic studies should be available to all students. Life Adjustment could meet the needs of all American students.26
However, many schools stubbornly clung to the teaching of academic subjects even when they offered life adjustment curricula as well. Moreover, parents of school children resisted these changes; they wanted their own children educated and not merely adjusted. They were sometimes joined by university professors and journalists who criticized the lack of academic content of the progressivist life adjustment programs. Changes in society at large also worked against the life adjustment agenda. Through the 1940s, the nation had witnessed tremendous scientific and engineering advances. By the end of the decade, the appearance of radar, cryptography, navigation, atomic energy, and other technological wonderments changed the economy and underscored the importance of mathematics in the modern world. This in turn caused a recognition of the importance of mathematics education in the schools. By the end of the 1940s, the public school system was the subject of a blizzard of criticisms, and the life adjustment movement fizzled out. Among the critics was Mortimer Smith. Reminiscent of Bagley's 1926 characterization of "students of education," he wrote in his 1949 book Madly They Teach:
...those who make up the staffs of the schools and colleges of education, and the administrators and teachers whom they train to run the system, have a truly amazing uniformity of opinion regarding the aims, the content, and the methods of education. They constitute a cohesive body of believers with a clearly formulated set of dogmas and doctrines, and they are perpetuating the faith by seeing to it through state laws and the rules of state departments of education, that only those teachers and administrators are certified who have been trained in the correct dogma.27
As would be the case in the final decade of the century, critics of this period complained of a lack of attention to basic skills.28
Progressive education was forced into retreat in the 1950s, and even became the butt of jokes and vitriol.29 During the previous half century, enrollment in advanced high school mathematics courses, and other academic subjects, had steadily decreased, thanks at least in part to progressive education. From 1933 to 1954 not only did the percentage of students taking high school geometry decrease, even the actual numbers of students decreased in spite of soaring enrollments. The following table gives percentages of high school students enrolled in high school math courses.30
Percentages of U.S. High School Students Enrolled in Various Courses
1909 to 1910
1914 to 1915
1921 to 1922
1927 to 1928
1933 to 1934
1948 to 1949
1952 to 1953
1954 to 1955
The "New Math" period came into being in the early 1950s and lasted through the decade of the 1960s. New Math was not a monolithic movement. According to a director of one of the first New Math conferences, "The inception of the New Math was the collision between skills instruction and understanding ...The disagreements between different entities of the New Math Movement were profound. Meetings between mathematicians and psychologists resulted only in determining that the two had nothing to say to each other."31 However, in a 1960 paper delivered to the NCTM, Harvard psychologist Jerome Bruner wrote:
I am struck by the fact that certain ideas in teaching mathematics that take a student away from the banal manipulation of natural numbers have the effect of freshening his eye to the possibility of discovery. I interpret such trends as the use of set theory in the early grades partly in this light--so too the Cuisenaire rods, the use of modular arithmetic, and other comparable devices.32
In spite of disagreements, most projects of that period shared some general features. The New Math groups introduced curricula that emphasized coherent logical explanations for the mathematical procedures taught in the schools. New Math was clearly a move away from the anti-intellectualism of the previous half-century of progressivist doctrine. For the first time, mathematicians were actively involved in contributing to K-12 school mathematics curricula.
The University of Illinois Committee on School Mathematics headed by Max Beberman began in 1951 and was the first major project associated with the New Math era. Beberman's group published a series of high school math textbooks, and drew financial support from the Carnegie Corporation and the U.S. Office of Education. In 1955, the College Entrance Examination Board established a Commission on Mathematics to investigate the "mathematics needs of today's American youth." The Commission, consisting of high school teachers, math educators, and mathematicians, issued a report with recommendations for a curriculum to better prepare students for college, and produced a sample textbook for twelfth grade on probability and statistics.33 The efforts of these and other early groups received little attention until the U.S.S.R launched Sputnik, the first space satellite, in the fall of 1957. The American press treated Sputnikas a major humiliation, and called attention to the low quality of math and science instruction in the public schools. Congress responded by passing the 1958 National Defense Education Act to increase the number of science, math, and foreign language majors, and to contribute to school construction.
That same year, the American Mathematical Society set up the School Mathematics Study Group (SMSG), headed by Edward G. Beg1e, then at Yale University, to develop a new curriculum for high schools. Among the many curriculum groups of the New Math period, SMSG was the most influential. It created junior and senior high school math programs and eventually elementary school curricula as well. The original eight members of SMSG were appointed by the president of the American Mathematical Society, but thereafter the two organizations had no formal connection. SMSG subsequently appointed a 26 member advisory committee and a 45 member writing group which included 21 college and university mathematicians as well as 21 high school math teachers and supervisors.34
The National Council of Teachers of Mathematics set up its own curriculum committee, the Secondary School Curriculum Committee, which came out with its recommendations in 1959. Many other groups emerged during this period including, the Ball State Project, the University of Maryland Mathematics Project, the Minnesota School Science and Mathematics Center, and the Greater Cleveland Mathematics Program. In the late 1950s, individual high school and college teachers started to write their own texts along the lines suggested by the major curriculum groups.35
One of the contributions of the New Math movement was the introduction of calculus courses at the high school level.36 Although, there were important successes in the New Math period, some of the New Math curricula were excessively formal, with little attention to basic skills or to applications of mathematics. Programs that included treatments of number bases other than base ten, as well as relatively heavy emphases on set theory, or more exotic topics, tended to confuse and alienate even the most sympathetic parents of school children. There were instances in which abstractness for its own sake was overemphasized to the point of absurdity.37 Many teachers were not well equipped to deal with the demanding content of the New Math curricula. As a result public criticisms increased.
A substantial number of mathematicians had already expressed serious reservations relatively early in the New Math period. In 1962, a letter entitled On The Mathematics Curriculum Of The High School, signed by 64 prominent mathematicians, was published in the American Mathematical Monthly and The Mathematics Teacher. The letter criticized New Math and offered some general guidelines and principles for future curricula.38
By the early 1970s New Math was dead. The National Science Foundation discontinued funding programs of this type, and there was a call to go "back to the basics" in mathematics as well as in other subjects.39 However, this direction for education did not go unchallenged. Progressive education had recovered from its doldrums of the 1950s, and by the late 1960s and early 1970s, it had regained its momentum. A. S. Niell's book Summerhill, published in 1960, is an account of an ultra progressive school in England. It was one of the most influential books on education of that decade. Founded in 1921 in Suffolk, England as a boarding school for relatively affluent children, Summerhill students determined completely what they would learn, and when. Niell wrote, "Whether a school has or has not a special method for teaching long division is of no significance, for long division is of no importance except to those who want to learn it. And the child who wants to learn long division will learn it no matter how it is taught." By 1970, some 200,000 copies of Summerhill were being sold per year, and it was required reading in 600 university courses.40
Modeled on Summerhill, and supported by the challenges at that time of structures of authority, both within education and the larger society, "free schools" proliferated, and eventually helped give rise to the Open Education Movement. The Open Education Movement was nothing new; it was just a repetition of progressivist programs promoted in the 1920s, but the idea of letting children decide each day what they should learn at activity tables, play corners, or reading centers, was once again promoted as profound and revolutionary.41
The effects of the Open Education Movement were particularly devastating to children with limited resources, due to their lack of access to supplemental education from the home, or tutoring in basic skills outside of school. Lisa Delpit, an African American educator who taught in an inner city school in Philadelphia in the early 1970s wrote about the negative effects of this type of education on African American children. Relating a conversation with another African American teacher, she explained, "White kids learn how to write a decent sentence. Even if they don't teach them in school, their parents make sure they get what they need. But what about our kids? They don't get it at home..." Summarizing the effects of the open classroom movement from her perspective in 1986, Professor Delpit wrote:
I have come to believe that the "open classroom movement," despite its progressive intentions, faded in large part because it was not able to come to terms with the concerns of poor and minority communities.42
Another prominent educator, Nancy Ichinaga, came to similar conclusions about the effects of the Open Education Movement on low income students, based on her experience as principal of Bennett-Kew Elementary school, in Inglewood, California. Ichinaga began a 24 year career as principal of Bennett-Kew in the Fall of 1974, one month before scores from the California's standardized test were released. At that time the school included only grades K-3 and it was called Bennett Elementary school. Bennett's 1974 third grade students ranked at the third percentile in the state, almost the absolute bottom. The school was then in its fourth year of the "Open Structure Program" and the student body throughout her tenure as principal was nearly 100 percent minority and low income. Reacting with shock and dismay at the test scores, Ichinaga confronted the teachers who admitted that their program was not working. The entire student body was illiterate and the student centered mathematics program was in shambles.
With the collaboration of her teachers, Nancy Ichinaga introduced clearly defined and well structured reading and math programs which included practice in basic skills. After a few years, test scores increased to well beyond the 50th percentile, and by the end of the 20th century, her school had earned national acclaim and became a model for others to emulate.43 At an education conference held in May 1999, Principal Ichinaga described the situation in her school in 1974:
My school had been patterned after Summerhill. And that's how bad it was! The kids used to make jello and bake cookies, and I used to tell the teachers, "Do you know what you've accomplished? You just gave them rotten teeth!"44
As in earlier periods of the 20th century, the agenda of progressivist educators was resisted by broad sectors of the public. The majority of states created minimum competency tests in basic skills starting in the mid-1970s, and almost half of them required students to pass these tests as a condition for graduation from high school. Due to public demand, some school districts created "fundamental schools" that emphasized traditional academics and promoted student discipline. While basic skills tests held the Open Education Movement in check, by their nature they could not be used to hold students to very high standards, or to raise existing standards. During the 1970s, standardized test scores steadily decreased and bottomed out in the early 1980s.45