A brief History of American k-12 Mathematics Education in the 20th Century



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Public Resistance to the NCTM Standards

To understand the public backlash against the NCTM math programs of the1990s, 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 1990s and reinforced the convictions of dissatisfied parents.70

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. An article in the 1998 Yearbook of the NCTM entitled, The Harmful Effects of Algorithms in Grades 1-4 by Constance Kamii and Ann Dominick provides examples. Citing earlier education research, the authors wrote, "By the 1980s, 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 1990s and concluded that algorithms are harmful to children," with examples provided. Elaborating, they wrote:
 

Piaget's constructivism, and the more than sixty years of scientific research by him and others all over the world led Kamii to a compelling hypothesis: Children in the primary grades should be able to invent their own arithmetic without the instruction they are now receiving from textbooks and workbooks. This hypothesis was amply verified...

Kamii co-authored another article in the 1999 Yearbook of the NCTM in which similar conclusions were reached about the algorithms for the arithmetic of fractions.71

Opposition to conventional arithmetic algorithms was not restricted to academic researchers. Similar convictions were held by teacher trainers with substantial influence. In a 1994 article entitled, Arithmetic: The Last Holdout, Marilyn Burns wrote:


 

I am a teacher who has embraced the call for change completely. I've made shifts in my teaching so that helping children learn to think, reason, and solve problems has become the primary objective of my math instruction...I do not give timed tests on basic facts. I make calculators available for students to use at all times. I incorporate a variety of manipulative materials into my instruction. I do not rely on textbooks because textbooks, for the most part, encourage "doing the page" rather than "doing mathematics."72

Parents of school children in the 1990s were directly confronted by policies based on these ideas. For example, the Los Angeles Times reported in 1997:
 

One missionary in the Reform cause is consultant Ruth Parker, who rejects long division and multiplication tables as nonsensical leftovers from a pre-calculator age. She urges audiences to "let kids play with numbers," and they will figure out most any math concept. Parker has spoken before 20,000 people over the last six months at the behest of school districts.73

Parents who worried that their children were getting unsound educations from NCTM aligned mathematics programs did not give much credence to education research findings or the advice of education experts, and most mathematicians didn't either. Perhaps the general attitude of parents was best captured by Jaime Escalante, the nationally famous mathematics teacher immortalized in the film Stand and Deliver, when he said, "whoever wrote [the NCTM Standards] must be a physical education teacher.'' 74

Sifting through the claims and counterclaims, journalists of the 1990s 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." This dichotomy is implausible. 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.C. Berkeley mathematician Hung-Hsi Wu in his paper, Basic Skills Versus Conceptual Understanding: A Bogus Dichotomy in Mathematics Education.75

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 1991 a group of about 250 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 1930s Progressivists, that the teachers were there to "teach children, not curricula." When parents complained of deficiencies in what little curriculum even existed, they were treated as if their cases were new and unrelated to other complaints. 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.

Finding their requests ignored, the "Curriculumists," those parents favoring an organized coherent curriculum for all students, concentrated on winning school board seats. One of them, Chiara Nappi, a theoretical physicist at the Institute for Advanced Study in Princeton, won a seat in 1993. By 1994 the Curriculumists held a majority of positions on the school board. However, even with formal political power, the Curriculumists were unable to make substantive changes in the district. They eventually turned their attention to creating a charter school for grades K-8 whose focus would be the fundamental academic disciplines, and which would provide an atmosphere that affirmed academic achievement. However, even this effort was resisted by Progressivists in the district. Nevertheless, after considerable effort, the Princeton Charter School came into existence in 1997 and provided a genuine alternative to the educational philosophy of the school district.76

Parents in California were also alarmed by the mathematics programs their children were getting in school. California was ahead of the rest of the nation in implementing the approach to mathematics education envisioned in the NCTM Standards and An Agenda for Action. The 1985 California Model Curriculum Standards, Grades Nine Through Twelve already had prescriptions that closely resembled those in the NCTM Standards such as:

The mathematics program must present to students problems that utilize acquired skills and require the use of problem-solving strategies. Examples of strategies that students should employ are: estimate, look for a pattern, write an equation, guess and test, work backward, draw a picture or diagram, make a list or table, use models, act out the problem, and solve a related but simpler problem. The use of calculators and computers should also be encouraged as an essential part of the problem-solving process. Students should be encouraged to devise their own plans and explore alternate approaches to problems.

As a consequence, mathematics reform along the lines of the weak 1989NCTM Standards was well underway in California in the early 1990s. California was one of the first states to embrace the NCTM Standards, producing a state mathematics framework in 1992 that closely resembled the NCTM Standards. By 1994, the California State Board of Education had approved math curricula for grades K-8 aligned to the 1992 California mathematics framework, and by extension, the NCTM Standards.

The first significant parental rebellion in California occurred in Palo Alto, a highly educated community that included Stanford University faculty and business leaders. In May 1994, more than 600 parents signed a petition asking that the school district retain a traditional pre-algebra curriculum at one of the middle schools in the Palo Alto Unified School District. The district was about to replace the remaining traditional courses with a math program aligned to California's 1992 math framework. Finding the district uncooperative, 25 parents in Palo Alto formed "Honest and Open Logical Debate," or HOLD in February 1995, put up a website the next month, and within a short period of time there were nearly 500 households on the HOLD mailing list. The already considerable math credentials of HOLD members were increased by the support and participation of Henry Alder, a professor of mathematics at UC Davis, a former president of the Mathematical Association of America, and a former member of the California State Board of Education. Alder had long been advocating themes similar to those of HOLD.

HOLD criticized the 1992 California math framework and the NCTM Standards, and pointed to a decrease in Stanford Achievement Test scores coinciding with the implementation of "whole math" in district schools. From 1992 to 1994 the average overall student score for 8th grade math students had decreased from the 91st national percentile rank to the 81st. The decrease was more dramatic on the portion of the exam that tested computation. On that portion the scores dropped from the 86th percentile in 1992 to the 58th percentile in 1994. Parents took steps to compensate for the lack of computational skills taught to their children in school. According to Bill Evers, one of the cofounders of HOLD:


 

Palo Alto School District parents are sufficiently discontented with the district's math performance that in massive numbers they are resorting to outside math tutoring programs. Forty-eight percent of parents report providing outside help in math for their children (in the middle schools, this number rises to 63 percent). The math-basics group HOLD's own informal survey of the best-known commercial math programs shows that Palo Alto parents are spending at least $1 million a year for math tutoring.77

With the extra tutoring, the district scores partially rebounded the following school year.

At the southern end of the state, four parents, Paul Clopton, Larry Gipson, Mike McKeown, and Martha Schwartz came together in the Autumn of 1995 to form "Mathematically Correct." Their common nemesis was fuzzy math and in particular, College Preparatory Mathematics (CPM), a secondary, integrated math program. Martha Schwartz had just participated in a group of parents that had collected more than 1,000 signatures for a petition to a school district in Torrance, California asking for a traditional alternative to CPM. This same program had been introduced in San Diego schools in 1993, and the founding parents found common cause in confronting the problems this curriculum and others like it were causing school children.

The founders of Mathematically Correct had credentials in science and mathematics that could not easily be dismissed. Gipson was a professional engineer; Clopton a statistician working for the Department of Veterans Affairs in San Diego; Schwartz was finishing up a Ph.D. in geophysics; McKeown was a faculty member at the Salk Institute for Biological Studies in San Diego (a few years later, McKeown accepted a professorship in the Division of Biology and Medicine at Brown University). They were soon joined by others, notably Wayne Bishop, a professor and former chair of the Mathematics Department at California State University, Los Angeles, and Frank Allan, a former president of the NCTM. Both had many years of experience dealing with mathematics education issues, and both were critics of the 1989NCTM Standards.

Organized for the explicit purpose of assisting parents dissatisfied with "fuzzy math" in their children's schools, Mathematically Correct attracted a large number of supporters (including the author of this chapter). Like other groups of its type, Mathematically Correct charged no dues, had no annual budget, and there was no formal membership. Mathematically Correct was fueled entirely by the energy and dedication of its supporters, especially its webmaster, Paul Clopton. In the decade of the 1990s, Mathematically Correct emerged as the most influential and effective organization to challenge the NCTM agenda. It served as a national clearing house for information and advice on K-12 mathematics education. Its supporters entered the political process, met with reporters and politicians, served on California government panels and commissions related to mathematics education, and testified before national boards and the U.S. Congress. Mathematically Correct and HOLD played important roles in establishing the California mathematics standards in 1997, a topic taken up in the next section.

Mathematically Correct also came into contact with other like-minded parent organizations, such as Parents Raising Educational Standards in Schools (PRESS), based in Milwaukee, Wisconsin; Concerned Parents of Reading, Massachusetts; Concerned Parents in Petaluma, California; Mountain View Achievement; Santa Monicans Working for Equity and Excellence in Public Schools; as well as many others. All of these grassroots parents' organizations were opposed to NCTM aligned math curricula in the schools and had information or websites linked from the Mathematically Correct website. The Internet was a powerful organizing tool for parents of school children during the 1990s.

A parent group in Plano, Texas took the unusual step of suing the school district in order to find an alternative to one of the NSF funded math programs. In 1996, Plano Independent School District (PISD) began piloting Connected Math in four of its nine middle schools. By the summer of 1998, some parents were objecting to the program. One parent who criticized Connected Math was removed from a textbook advisory committee in the Fall of 1998. Another parent was prevented from passing out information critical of Connected Math at PISD informational meetings, and was also prevented from collecting signatures to a petition asking for an open discussion with parents about the merits of the program. As a result, parents formed the organization, MathChoice, in January 1999. Frustrated that the district continued to ignore parental complaints about the program, MathChoice started another petition drive in May 1999. The petition was really just a one page form that parents could fill out requesting an alternative math class for their children. Each form began with the sentence, "This petition is for the addition of a specific, traditional/conventional academic class in the course of study of math for the parent or guardian's child named: ..." The district responded by sending letters to parents in the school district that countered the petition, effectively putting an end to the petition drive. However, by the end of May, 521 signatures had already been collected.

Finding their petitions ignored, the Plano parents turned to litigation. In October 1999, MathChoice incorporated as the Plano Parental Rights Council. They attained non-profit status from the IRS the following spring and elected Susan Sarhady as president. Seeking class certification, six parents filed suit in federal court "against the Plano Independent School District for violations of the parentís constitutionally protected rights of free speech/expression, equal protection and the fundamental right as parents to direct the education and upbringing of their children."78 In May 2000, a federal judge ruled that "Plano Independent School District cannot be compelled to offer an alternative middle-school math program despite the objections of some parents to the new Connected Math approach...." However, the judge "also found that certain allegations by the parents should go forward to trial ... The lawsuit claim[ed] that the First Amendment rights of several parents were violated when they were prohibited from distributing or displaying materials opposing the Connected Math program at several meetings."79

Another important parents' organization emerged in 1999 in New York City. The New York City school system had been awarded an Urban Systemic Initiative grant from the NSF in 1994, and New York state had a Statewide Systemic Initiative grant. The New York Urban Systemic Initiative reported training 4,200 teachers in inquiry-based curricula, and more than 700 teachers in the use of calculators for high school mathematics courses. According to posted reports to the NSF, the USI also implemented "exemplary curricula" in over 5,000 classrooms in New York City.

New York City Schools are grouped into 32 community school districts. Each has its own school board and superintendent. Community School District 2 consists of about 42 schools serving 22,000 racially diverse students. The district included relatively affluent neighborhoods as well as neighborhoods with substantial concentrations of lower income families and recent immigrants.

Beginning in 1993, teachers were trained in materials created by Marilyn Burns, a prominent teacher trainer cited earlier in this section. From 1995 to 1998 pilot programs in TERC's Investigations in Number, Data and Space and Connected Mathematics gradually expanded in District 2. By 1999 TERC and CMP were used districtwide. The NSF funded curriculum, Mathematics: Modeling Our World (ARISE), was scaled up for use in all of the high schools in 2000 and 2001.

In May 1999, Elizabeth Carson, a concerned parent of a middle school student, began a search for allies to try to reverse the districtwide implementation of weak NCTM aligned mathematics programs. The result was an alliance consisting of parents, teachers, City University of New York mathematics professors, and a substantial portion of the faculty of the math department of the Courant Institute of New York University. They named themselves "New York City HOLD" (NYC HOLD) after the Palo Alto group. Allies of NYC HOLD communicated with each other largely through the Internet, but many of them met weekly at New York University for planning sessions or discussions with interested visitors, including education journalists. On June 6, 2001, NYC HOLD held an open forum for parents and teachers in an auditorium in the New York University Law school. Approximately 350 people attended, and plans were subsequently made for other projects to challenge the nearly exclusive use of NCTM-aligned curricula in the schools.

In the decade of the 1990s, the parent organizations in California, especially Mathematically Correct, experienced the greatest successes, not only in blocking the use of dubious classroom materials, but also in implementing coherent, effective mathematics policies at the state level. The California program at the end of the 20th century included high quality mathematics textbooks and a testing system aligned to the California standards. However, parent organizations did not accomplish these changes unilaterally. Many other sectors of society and prominent individuals played critical roles. They included classroom teachers and principals, university mathematicians, legislators, state school board members, journalists, and two successive governors.



Mathematicians, California, and the Nation

No state had so great a national impact as California on mathematics education during the 1990s. This was due in part to the fact that California was the most populous state, and as a consequence, the demands placed on textbook publishers to sell to the California market influenced what was available to the rest of the nation. But the effect of California's new educational policies during the middle and late 1990s went deeper. Perhaps the clearest indications of the importance of California's choices were the harsh public denunciations by both the NCTM and the NSF of California's 1997 mathematics standards immediately following their release. This will be contrasted with the strong support given by university mathematicians and parent groups, later in this section. By the end of the decade, it was clear that California's mathematics program threatened a century of progressivist domination in K-12 mathematics education.

Not since the New Math period of the 1960s had university mathematicians played such important roles in K-12 education as in California during the 1990s. Mathematicians were involved in developing the state mathematics standards, the California mathematics framework, and in evaluating textbooks and professional development programs for teachers in California. Some mathematicians also helped to write and develop textbooks for the textbook adoption process of 2001. During the decade of the 1990s, at the national level, there were extended discussions about K-12 mathematics among research mathematicians through their professional meetings and magazines.80 The result was a greater participation by university mathematicians nationally in matters related to mathematics education, including interactions with parent organizations.

In California, by the mid-1990s, the dramatic failures of "whole language learning" in teaching primary grade students to read had already cost the education establishment substantial credibility with the public. Analogous failures in mathematics education were opening opportunities for critics of constructivist education policies to make changes. Mathematicians and parent activists displaced, to a considerable degree, the education professionals and college of education faculty who would normally be entrusted to work out the policy details for K-12 mathematics education. As a consequence, mathematicians were naturally drawn into educational and political debates.

An early example was the participation of Professor Wayne Bishop on a Mathematics Task Force formed by the state Superintendent of Schools, Delaine Eastin, in 1995. The 25 member Task Force was charged with recommending ways to improve mathematics instruction in California. Bishop publicly resigned from the Task Force in order to make known his disagreement with the weak recommendations the Task Force was making.

Following the release of the Task Force report, Professor Henry Alder addressed the California State Board of Education in December 1995. He articulated the views of the emerging parent organizations in California and indirectly reinforced Bishop's symbolic resignation. Alder recommended "a revision or perhaps even a complete rewriting of the 1992 California Mathematics Framework rather than issue a supplement." Paving the way for broader participation in mathematics education policy making, Alder also recommended that a:


 

new task force to be charged with your Board's directives be appointed in consultation with all affected constituencies, with an appropriate mix of expertise from all segments interested in and involved in mathematics education. This means, in particular, that its membership should not be dominated by those who prepared the 1992 California Mathematics Framework and those who constituted the Mathematics Task Force.81

The State Board of Education agreed with the critics and scheduled a rewrite of the 1992 Framework two years ahead of the normal time table. By this time there was considerable public pressure to improve the teaching of reading and mathematics in the schools. The legislature had just passed a bill that required school districts to include the teaching of basic skills in reading and math as part of their curriculum. Governor Wilson signed this "ABC Bill" in October 1995, and it became law in January 1996. The ABC laws had virtually no effect on school districts, which were generally run by committed constructivists, but political leaders felt compelled to do something about the mounting failures in education. Whole language and whole math--the math programs aligned to the NCTM Standards and 1992 California framework--were widely viewed as responsible for depriving children of fundamental skills.

At the national level during the mid-1990s much attention was focused on international comparisons of student mathematics achievement. The first available results of the Third International Mathematics and Science Study (TIMSS) were released in November 1996. U.S. 8th graders scored slightly below the international average in mathematics. The second TIMSS report comparing 4th grade students in math was released in June 1997. U.S. fourth grade students were slightly above the international average among the participating countries. The final report compared students at the end of high school and was released in 1998. The mathematics achievement of U.S. 12th graders was among the lowest of the participating nations. The TIMSS data contained valuable information, but it had relatively little political impact on the ensuing debates, as both sides cited the studies to reinforce their respective positions. However, TIMSS researchers did express support for the NCTM Standards. The eighth grade study found that82:


 
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