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**Contents:**

According to one writer, "For good or for ill, it was Thorndike who dealt the final blow to the 'science of arithmetic. 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 In 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.

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. 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. Commissioner of Education, Philander P. Claxton, a friend of Kilpatrick. The Kilpatrick committee and leading educational theoreticians had thrown the gauntlet, and the Mathematical Association of America MAA responded vigorously. Already in , in anticipation of the Kilpatrick report, E. It was chaired by J.

Young of Dartmouth and included mathematicians E. 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 page volume entitled, The Reorganization of Mathematics for Secondary Education. The report was published in and is sometimes referred to as the Report. 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 Report was 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.

Standardized Testing: The Good, The Bad, & The Ugly

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 Report underscored the importance of algebra to "every educated person. For example, some of the policies of the College Examination Board were based upon recommendations in the Report. However, over the next two decades, the views expressed in the Kilpatrick report wielded greater influence than the Report.

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.

The second camp holds that the form of assessment is critical, arguing. that teachers inevitably teach to the test but that doing so can be a good. Teachers To what extent does test preparation contribute to educational. inequities of the mathematics standards stated that students would “develop the abil-. The Ambiguity of Teaching to the Test [William A. Firestone, Roberta Y. Testing is one of the most controversial of all state and federal educational policies. on educational reform, policy implementation, school culture, and leadership the impact of large scale assessment on teaching practice and educational equity.

In the s 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 s, just as in the s, for educators to say, "We teach children, not subject matter.

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 s, 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. In the s 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. The basic skills of these military personnel should have been learned in the public schools but were not. Nevertheless, by the mids, 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. 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 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. 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 s, 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 s, 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 characterization of "students of education," he wrote in his book Madly They Teach:.

Progressive education was forced into retreat in the s, and even became the butt of jokes and vitriol. From to 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.

The "New Math" period came into being in the early s and lasted through the decade of the s. 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. Beberman's group published a series of high school math textbooks, and drew financial support from the Carnegie Corporation and the U.

Office of Education. In , the College Entrance Examination Board established a Commission on Mathematics to investigate the "mathematics needs of today's American youth. R launched Sputnik , the first space satellite, in the fall of The American press treated Sputnik as a major humiliation, and called attention to the low quality of math and science instruction in the public schools. Congress responded by passing the National Defense Education Act to increase the number of science, math, and foreign language majors, and to contribute to school construction. Beg1e, then at Yale University, to develop a new curriculum for high schools.

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. 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 In the late s, individual high school and college teachers started to write their own texts along the lines suggested by the major curriculum groups.

One of the contributions of the New Math movement was the introduction of calculus courses at the high school level. 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.

As a result public criticisms increased. A substantial number of mathematicians had already expressed serious reservations relatively early in the New Math period. The letter criticized New Math and offered some general guidelines and principles for future curricula. By the early s 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. Progressive education had recovered from its doldrums of the s, and by the late s and early s, it had regained its momentum. Niell's book Summerhill , published in , 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 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. 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 s, 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.

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 s 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 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.

In the early s, there was widespread recognition that the quality of math and science education had been deteriorating. A report by a presidential commission pointed to low enrollments in advanced mathematics and science courses and the general lowering of school expectations and college entrance requirements. The different points of view and prescriptions for change expressed in these two reports characterize to some extent the opposing factions in the math wars of the s.

The report called for new directions in mathematics education which would later be codified in in the form of national standards. An Agenda for Action recommended that problem solving be the focus of school mathematics in the s, along with new ways of teaching. The report asserted that "Requiring complete mastery of skills before allowing participation in challenging problem solving is counterproductive, " and "Difficulty with paper-and-pencil computation should not interfere with the learning of problem-solving strategies.

According to the report, "All students should have access to calculators and increasingly to computers throughout their school mathematics program. Perhaps the boldest and most far reaching recommendation of An Agenda for Action was its proposal for "Mathematics educators and college mathematicians" to "reevaluate the role of calculus in the differentiated mathematics programs.

The so-called "integrated" high school math books of the s contributed to this tendency. While those books contained parts of algebra, geometry, and trigonometry, the developments of these traditional subjects were not systematic, and often depended on student "discoveries" that were incidental to solving "real world problems. It was largely eclipsed by the report, A Nation At Risk. Secretary of Education, at that time. Unlike previous education reform efforts and reports by prestigious governmental bodies, this one captured the attention of the public.

A Nation At Risk warned, "Our nation is at risk A Nation at Risk addressed a wide variety of education issues, including specific shortcomings in mathematics education. Regarding remedial mathematics instruction, the report found that:. Business and military leaders complain that they are required to spend millions of dollars on costly remedial education and training programs in such basic skills as reading, writing, spelling, and computation.

A Nation at Risk described high school course offerings as a "curricular smorgasbord" and reported, "We offer intermediate algebra, but only 31 percent of our recent high school graduates complete it; we offer French I, but only 13 percent complete it; and we offer geography, but only 16 percent complete it. Calculus is available in schools enrolling about 60 percent of all students, but only 6 percent of all students complete it. The importance of student assessment was also addressed. The report envisioned a role for standardized tests that foreshadowed a movement toward accountability in the late s 49 :.

With public opinion in support of a strong focus on basic skills and clear high standards, the NCTM took steps to recast its own agenda under the label of standards. The Curriculum and Evaluation Standards for School Mathematics was developed during the summer of and revised in by four working groups whose members were appointed by John Dossey, the president of the NCTM at that time.

During the school year, input was sought from classroom teachers across the country. The project was coordinated by Thomas A. The final document was published in , and during the following decade it was commonly referred to as the NCTM Standards , or as the Standards. However, the NCTM successfully promoted the Standards as if they were developed through a grass-roots, bottom-up process. Harold Stevenson, a psychologist at the University of Michigan, described them as follows:. The NCTM standards list goals with which no one would be likely to disagree. Of course we want children to value mathematics, to be mathematics problem solvers, to be confident of their ability, and to be able to reason and communicate mathematically.

Certainly students must develop a number sense, have concepts of whole number operations, and the other kinds of skills and knowledge indicated under NCTM's curriculum standards. But the published standards do not integrate these two important components: the general attitudes and mathematical skills. Included on the list for decreased attention in the grades K-4 were "Complex paper-and-pencil computations," "Long division," "Paper and pencil fraction computation," "Use of rounding to estimate," "Rote practice," "Rote memorization of rules," and "Teaching by telling.

The following were included on the list to be de-emphasized: "Relying on outside authority teacher or an answer key ," Manipulating symbols," "Memorizing rules and algorithms," "Practicing tedious paper-and-pencil computations," "Finding exact forms of answers. On page 8, the Standards proclaimed, "The new technology not only has made calculations and graphing easier, it has changed the very nature of mathematics The NCTM Standards reinforced the general themes of progressive education, dating back to the s, by advocating student centered, discovery learning.

The utilitarian justification of mathematics was so strong that both basic skills and general mathematical principles were to be learned almost invariably through "real world" problems.

We have children who were crack babies, whose parents did drugs when they were pregnant. Supportive principals offer teachers advice on how to improve their practice by encouraging participation in learning opportunities even though they rarely know a great deal more about the subjects taught than do teachers. Another alternative is the use of performance items. Tell us if something is incorrect. A major concern with the use of educational assessments is the overall validity, accuracy, and fairness when it comes to assessing English language learners ELL.

Mathematics for its own sake was not encouraged. The term "constructivism" was adapted from cognitive psychology by educators, and its meaning in educational contexts is different from its use in psychology. Hirsch Jr. Mathematics education leaders drew support for educational constructivism from the writings of Jean Piaget and Lev Semenovich Vygotsky.

Piaget's ideas about developmental stages of learning, and Vygotsky's concept, "Zone of Proximal Development," seemed to be consistent with the child-centered, cooperative learning approaches to education long favored by colleges of education. In the fall of , President George H. Bush, then in his first year of office, was invited by the nation's governors to an education summit in Charlottesville, Virginia.

A bipartisan call went out for national standards. Participants at the Education Summit made a commitment to make U. Political leaders in the late s were motivated by employers' complaints about the costs of teaching basic skills to entry level workers, and by the low standing of U. The nation was looking for benchmarks that could improve education. The NCTM Standards had just been published, and by default they became the national model for standards.

Within a few years, the NCTM produced two additional documents as part of its standards. One published in was narrowly focused on pedagogy and the other, published in , was focused on testing. The NSF proceeded purposefully. The EHR developed a series of Systemic Initiative grants to promote fundamental changes in science and mathematics education in the nation's schools.

The Statewide Systemic Initiatives were launched in These grants were designed in part to encourage state education agencies to align their state mathematics standards to the NCTM Standards. This program allowed renewals of awards made under the USI program. At first, the Systemic Initiative grants were awarded to proposals generally aligned to the educational views of the NSF, but awardees were allowed substantial freedom to develop their own strategies for reform. As the program evolved, so did the guidelines. By , the NSF clarified its assumptions about what constitutes effective, standards-based education and asserted that 61 :.

In the decade of the s, the National Science Foundation sponsored the creation of the following mathematics programs for K An important component of the Systemic Initiatives was the aggressive distribution of NCTM aligned curricula for classroom use. The NCTM Standards were vague as to mathematical content, but specific in its support of constructivist pedagogy, the criterion that mattered most to the NSF.

It should be noted that the Systemic Initiatives sometimes promoted curricula not on the list above, such as College Preparatory Mathematics, a high school program, and MathLand, a K-6 curriculum. In addition to aligning state math standards to the NCTM standards and creating and distributing math books and programs aligned to those standards, the NSF attempted with considerable success to push these approaches up to the university level. Most notable in this regard was the NSF's funding of a "reform calculus" book, often referred to as "Harvard Calculus," that relied heavily on calculators and discovery work by the students, and minimized the level of high school algebra required for the program.

The NSF also funded distribution centers to promote the curricular programs it had helped to create. For example, an NSF sponsored organization created in called, "The K Mathematics Curriculum Center," had a mission statement "to support school districts as they build an effective mathematics education program using curriculum materials developed in response to the National Council of Teachers of Mathematics' Curriculum and Evaluation Standards for School Mathematics.

K education collectively was a multi-billion dollar operation and the huge budgets alone gave public education an inertia that would be hard to overcome. Even though the millions of dollars at its disposal made the EHR budget large in absolute terms, it was miniscule relative to the combined budgets of the school systems that the NSF sought to reform. It would not be easy to effect major changes in K mathematics and science education without access to greater resources. To some extent private foundations contributed to the goal of implementing the NCTM Standards through teacher training programs for the curricula supported the by the NSF, and in other ways.

Others such as the W. Keck Foundation and Bank of America contributed as well. However, the NSF itself found ingenious ways to increase its influence. The strategy was to use small grants to leverage major changes in states and school districts. Yet, the LASI project exerted almost complete control over mathematics and science education in the district. In addition to Title II funds, LASI gained control of the school district's television station and its ten science and technology centers.

According to Luther Williams' July Summary update, "[LASI] accountability became the framework for a major policy initiative establishing benchmarks and standards in all subject areas for the entire school system. All four sets of standards were adopted by the school district in The Los Angeles School district math standards were so weak and vague that they were a source of controversy. One typical standard, without any sort of elaboration, asked students to "make connections among related mathematical concepts and apply these concepts to other content areas and the world of work.

The word "triangle" did not even appear in the standards at any grade level. By design, trigonometry and all Algebra II topics were completely missing. The LASI annual report explained:. El Paso, Texas serves as an example. El Paso is geographically removed from other U. This made the effectiveness of the K and university programs easier to assess.

It also made the entire education system easier to control. During the s, the K education system in El Paso was highly coordinated and focused on implementing constructivist math and science education programs. For this reason, it became a model center for educators from other parts of the country to visit and study. Dana Center in Austin. The recommended criteria for selecting K-8 mathematics curricula included:.

The El Paso Collaborative for Academic Excellence created a confidential student evaluation questionnaire to monitor teaching methods used in high school math classrooms in all of EL Paso's public high schools. The evaluation included the following questions to students:. To understand the public backlash against the NCTM math programs of thes, one needs to understand some of the mathematical shortcomings of these programs. The mathematics books and curricula that parents of school children resisted shared some general features.

Those programs typically failed to develop fundamental arithmetic and algebra skills. Elementary school programs encouraged students to invent their own arithmetic algorithms, while discouraging the use of the superior standard algorithms for addition, subtraction, multiplication, and division. Calculator use was encouraged to excess, and in some cases calculators were even incorporated into kindergarten lesson plans. Student discovery group work was the preferred mode of learning, sometimes exclusively, and the guidelines for discovery projects were at best inefficient and often aimless.

Topics from statistics and data analysis were redundant from one grade level to the next, and were overemphasized. Arithmetic and algebra were radically de-emphasized. Mathematical definitions and proofs for the higher grades were generally deficient, missing entirely, or even incorrect. Some of the elementary school programs did not even provide books for students, as they might interfere with student discovery.

Written and published criticisms from many sources, including mathematicians, of specific mathematics programs were widespread in the s and reinforced the convictions of dissatisfied parents. But not everyone viewed the near absence of the standard algorithms of arithmetic in NCTM aligned books as a shortcoming. Some prominent educational researchers were explicit in their opposition to the teaching of algorithms to children. Citing earlier education research, the authors wrote, "By the s, some researchers were seriously questioning the wisdom of teaching conventional algorithms," and then listed examples of such research.

Tracing the history of this line of inquiry they added, "Some investigators went further in the s and concluded that algorithms are harmful to children," with examples provided. Elaborating, they wrote:. Opposition to conventional arithmetic algorithms was not restricted to academic researchers. Similar convictions were held by teacher trainers with substantial influence.

Sifting through the claims and counterclaims, journalists of the s tended to portray the math wars as an extended disagreement between those who wanted basic skills versus those who favored conceptual understanding of mathematics. The parents and mathematicians who criticized the NCTM aligned curricula were portrayed as proponents of basic skills, while educational administrators, professors of education, and other defenders of these programs, were portrayed as proponents of conceptual understanding, and sometimes even "higher order thinking.

The parents leading the opposition to the NCTM Standards, as discussed below, had considerable expertise in mathematics, generally exceeding that of the education professionals. This was even more the case of the large number of mathematicians who criticized these programs. Among them were some of the world's most distinguished mathematicians, in some cases with mathematical capabilities near the very limits of human ability. By contrast, many of the education professionals who spoke of "conceptual understanding" lacked even a rudimentary knowledge of mathematics.

More fundamentally, the separation of conceptual understanding from basic skills in mathematics is misguided. It is not possible to teach conceptual understanding in mathematics without the supporting basic skills, and basic skills are weakened by a lack of understanding. The essential connection between basic skills and understanding of concepts in mathematics was perhaps most eloquently explained by U. The obstacles faced by parents opposed to the NCTM programs for their children were formidable.

The events leading to the creation of the Princeton Charter school illustrate some of the generic difficulties. In a group of about parents of school children in Princeton, New Jersey petitioned the board of education for a more systematic and challenging math program. They found the one in use to be vague and weak. Many of the teachers did not even use textbooks. When parents asked about what was being taught in the classrooms, they were told that the curriculum was not very important, that "one size does not fit all," and, repeating the dictum of s Progressivists, that the teachers were there to "teach children, not curricula.

These responses have been reported by parents in many other school districts as well. Test scores in Princeton were among the highest in the state, but that was not the result of a well designed academic program. Many highly educated parents, including Princeton University faculty, were providing tutoring and enrichment for their own children.

Other children with limited resources in the Princeton Regional School system did not fare well in this highly progressivist environment. Hardback pagine. Testing is one of the most controversial of all state and federal educational policies. The effects of testing are quite ambiguous. The same test may lead to different consequences in different circumstances, and teachers may use very different strategies to prepare students for tests.

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Although most experts agree that mandatory testing leads to teaching to the test, they disagree about whether it leads to meaningless drill, wasted time, deprofessionalizing teachers, and demotivating students, or to more challenging and thoughtful curricula, more engaging teaching, increased student motivation, and increased accountability. To help sort through this ambiguity and provide a firmer basis for decisions, The Ambiguity of Teaching to the Test: Standards, Assessment, and Educational Reform offers a hard look at the effects of state testing, and thoroughly examines the ambiguity of test preparation and how test preparation practices are influenced by what teachers know and the leadership coming from the school and district.

Drawing on data from a three year study of New Jersey's testing policy in elementary mathematics and science, it helps to explain the variety of ways that teachers modify their teaching in response to state tests, raises important questions, and offers useful guidance on how state policymakers and local and district school administrators can implement policies that will improve educational equity and performance for all students. It also offers an indepth analysis of classroom practices that should inform teachers and teacher educators whose goal is to meaningfully implement conceptually based teaching practices.

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This comprehensive look at the statewide variation in testing practice features: a databased, nonideological treatment of how testing affects teachers, in a field characterized by ideologically driven beliefs and by anecdotes; an extensive and well integrated combination of qualitative and quantitative data sources that provide a statewide overview, as well as an indepth analysis of teachers and classrooms; a careful analysis of the variety of forms of teaching to the test; and a multilevel exploration of how a variety of personal and leadership factors can influence teaching to the test. This is an important book for researchers, professionals, and students in educational testing, educational policy, educational administration, mathematics and science education, educational reform, and the politics and sociology of education.