Ninetyfive percent of U.S. teachers stated that they were either "very aware" or "somewhat aware" of current ideas about teaching and learning mathematics. When asked to list titles of books they read to stay informed about current ideas, one third of U.S. teachers wrote down the names of two important documents by the National Council of Teachers of Mathematics, Curriculum and Evaluation Standards and Professional Teaching Standards [bold in original].
U.S. teachers believe that their lessons are already implementing the reform recommendations, but the findings described so far in this chapter suggest that their lessons are not. When asked to evaluate to what degree the videotaped lesson was in accord with current ideas about teaching and learning mathematics, almost 75 percent of the teachers respond either "a lot" or "a fair amount." This discrepancy between teachers' beliefs and the TIMSS findings leads us to wonder how teachers themselves understand the key goals of the reform movement, and apply them in the classroom.
The report suggests, without any experimental support, that if the U.S. teachers had properly followed the constructivist NCTM Standards, then U.S. students would have performed better in the study. However, it is possible that the teachers were correct in asserting that they were following "current ideas about teaching and learning mathematics," and there was no "discrepancy." The report went on to say:
Over 80 percent of the teachers in the study referred to something other than a focus on thinking, which is the central message of the mathematics reform movement. The majority of the teachers cited examples of handson math or cooperative learning, which are techniques included among the reform recommendations. However, these techniques can be used either with or without engaging students in real mathematical thinking. In fact, the videotape study observed many examples of these techniques being conducted in the absence of highquality mathematical content.
The authors of the report did not consider the possibility that the NCTM reform movement itself was a contributing cause of poor student performance. Even a cursory examination of the NCTM aligned math curricula would show a disturbing lack of "high quality mathematical content." Nevertheless, the TIMSS report prescribed still more of the same reform:
These findings suggest that the instructional habits and attitudes of U.S. mathematics teachers are only beginning to change in the direction of implementation of mathematics reform recommendations. Teachers' implementation of the reform still concentrates on isolated techniques rather than the central message, which is to focus lessons on highlevel mathematical thought. The finding that almost 20 percent of the teachers believed that they had implemented this focus on mathematical thinking, despite experts' judgments that a highquality sequence of mathematical ideas was virtually absent in their lessons, suggests that teachers may not yet understand what the reform movement means by this term [bold in original].
The growing criticisms of NCTM aligned reform curricula coming from professional mathematicians raised the possibility that the real focus of the reform movement was constructivist classroom techniques rather than "highlevel mathematical thought." This possibility was not considered by the authors of the TIMSS reports.
Results from National Assessment of Educational Progress (NAEP) were released in February 1996. While the nation as a whole made some improvements, California's fourth graders scored below their peers in 40 states and came out ahead of only those in Mississippi. A careful analysis of NAEP trends for the nation as a whole was published by the Brookings Institution later in September 2000, but California's relative downward slide reinforced the political will toward writing explicit mathematics standards and rectifying the 1992 framework to include more attention to basic skills.^{83} Adding to California's concerns was a steady increase in remedial math courses on the 23 campus California State University (CSU) system. The percentage of entering freshmen failing an entry level math test used by the CSU, and requiring remedial courses, steadily increased from 23% in 1989 to 54% in each of 1997 and 1998. While there was no proof that the decrease in math skills was caused by the constructivist math programs in the schools, school mathematics seemed to be getting worse rather than better as the NCTM reform agenda expanded.
In January 1997, a committee called the Academic Content and Performance Standards Commission (Standards Commission) was charged with writing mathematics (and other subject matter) standards for California and submitting its draft to the State Board of Education for final approval. The committee consisted of non expert citizens appointed through a political process. The majority of the Standards Commissioners were largely in agreement with the constructivist policies of the past. The result was a set of standards submitted to the Board in the Fall of 1997 that not only embraced the constructivist methods that California was trying to escape, but was also incoherent and full of mathematical errors.
Members of the State Board asked for help from Stanford University mathematics professors Gunnar Carlsson, Ralph Cohen, Steve Kerckhoff, and R. James Milgram. In a few short weeks they rewrote the standards, corrected more than 100 mathematical errors, and eliminated all pedagogical directives, leaving the standards pedagogically neutral. The final revisions, including those made by the State Board itself, resulted in a document that would allow teachers to use constructivist methods or direct instruction, or whatever classroom techniques worked for them, so long as they taught all of the grade level content standards. The mathematics framework was regarded as the proper document for discussions of pedagogy, but not the standards themselves. This was what the State Board was looking for, and the mathematicians' standards were adopted by California in December 1997. These standards were clear, coherent, and met the criteria set by the California legislature to be competitive with math standards of the highest performing countries.
Professor HungHsi Wu did a careful analysis of the California standards, that the board adopted, in comparison to the draft submitted by the Standards Commission which the Board rejected. Wu found numerous mathematical errors and lack of clarity and cohesion in the rejected standards, in contrast to an overall soundness and clarity in the adopted standards.^{84} In 1998, the Fordham Foundation conducted an independent review of the mathematics standards from 46 states and the District of Columbia, as well as Japan. California's new board approved mathematics standards received the highest score, outranking even those of Japan.^{85}
The NCTM responded to the new California mathematics standards with denunciations. The cover story of the February 1998 News Bulletin of the NCTM began with:
Mathematics education in California suffered a serious blow in December. Over protests from business, community, and education leaders, California's state board of education unanimously approved curriculum standards that emphasize basic skills and deemphasize creative problem solving, procedural skills, and critical thinking.
NCTM President Gail Burrill used strong words in a letter to the president of the California Board of Education. She wrote, "Today's children cannot be prepared for tomorrow's increasingly technological world with yesterday's content...The vision of important school mathematics should not be one that bears no relation to reality, ignores technology, focuses on a limited set of procedures ...California's children deserve more.''
The NSF also condemned California's deviation from constructivism. Luther Williams, the National Science Foundation's Assistant Director for Education and Human Resources, wrote a letter to the Board on NSF letterhead stationary. Williams' letter, dated December 11, 1997, explained that the Boardís decision to adopt the mathematics standards "vacates any serious commitment to elevating problem solving and critical thinking skills..." Williams added, "The Board action is, charitably, shortsighted and detrimental to the longterm mathematical literacy of children in California." Speaking for the National Science Foundation, he chastised, "We view the Board action in California with grave disappointment and as a lost opportunity for the cities we support?indeed, for the entire state."
The condemnations of the new California math standards by non mathematicians turned into an avalanche. Judy Codding, a vice president of the National Center on Education and the Economy (NCEE), had served on California's Standards Commission. She made no secret of her opposition to the new standards. Speaking at an NCEE conference, she declared "I will fight to see that California math standards are not implemented in the classroom." California Superintendent of Schools Delaine Eastin also denounced the math new standards written by the Stanford mathematicians as being "dumbed down." According to Eastin, the California standards represented "a decided shift toward less thinking and more rote memorization.'' Eastin also complained that with the new standards, "we're not even going to let [students] use a calculator before the sixth grade."^{86} The statewide chairs of the Academic Senates of the UC, CSU, and California Community College systems, none of whom were mathematicians, also joined the chorus. They issued a joint statement condemning the adoption of California's math standards and falsely declared that "the consensus position of the mathematical community'' was in opposition to the new standards, and generally in support of the rejected, standards, written by the Standards Commission.
California mathematicians put an end to the rumor that there was any consensus in the mathematics community against the new California standards. More than 100 mathematics professors from colleges and universities in California added their names to an open letter in support of the California mathematics standards. The signatories included chairs of the math departments at Caltech, Stanford, several UC and CSU campuses, as well as community college faculty. Jaime Escalante also added his name in support.^{87} One of the flash points in the disagreement about the standards was whether long division should be taught in K12 beyond the case of single digit divisors, and this was indicated in the open letter. A detailed explanation of the importance of the division algorithm by two mathematicians was later provided for the benefit of teachers.^{88}
The criticisms of the California standards in the press diminished after a few months, and work proceeded on developing the Mathematics Framework for California Schools. R. James Milgram and HungHsi Wu played fundamental roles in the many mathematical portions of the final document. Important contributions were also made by others, including cognitive psychologist David Geary of the University of Missouri and educational researcher Douglas Carnine together with other members of the National Center to Improve the Tools of Educators (NCITE) at the University of Oregon. The State Board of Education contracted with NCITE to perform a study "to locate high quality research about achievement in mathematics, review that research, and synthesize the findings...From a total of 8,727 published studies of mathematics education in elementary and secondary schools, the research team identified 956 experimental studies. Of those, 110 were deemed high quality research because they met tests of minimal construct and internal and external validity."^{89}
The Framework was adopted by the California State Board of Education in December 1998. A system was developed for textbook adoptions in California which included panels of mathematicians, as well as different panels whose membership consisted primarily of classroom teachers. The panels of mathematicians were charged with evaluating mathematics curricula submitted for statewide adoption, on the basis of the quality of mathematical content. This screening process by mathematicians contributed important voices to California's 1999 and 2001 textbook adoption process. Most of the panel members came from California universities, but not all. Richard Askey of the University of Wisconsin at Madison and Ralph Raimi of the University of Rochester participated on the 1999 panels.
Even after California identified textbooks aligned to its new state standards, resistance to the California standards at the local school district level was significant. Decisions at the district level were largely under the control of administrators who looked for guidance from the NCTM, the NSF, and sometimes the NCEE. The new content standards of California would not easily be accepted. In one case which received front page coverage in the Los Angeles Times, a critic of the California math standards threatened a hunger strike in order to increase the chances of classroom use of NCTM aligned math programs.^{90} Nevertheless, as early as 1999 some school districts were coming to grips with the new guidelines. The Los Angeles Unified School District included Paul Clopton, HungHsi Wu, Ze'ev Wurman, one of the cofounders of HOLD, and Barry Simon, the mathematics department chair at Caltech, on a textbook selection committee. While the recommendations of these highly knowledgeable participants were largely ignored, the mere fact of their participation was a departure from the past.
One of the signal events of 1999 was the release of Liping Ma's book, Knowing and Teaching Elementary Mathematics.^{91} Ma compared answers to elementary school math questions by 23 U.S. elementary school teachers to those by 72 Chinese elementary school math teachers. Of the U.S. teachers, 12 were participating in an NSF sponsored program whose "goal was to prepare excellent classroom mathematics teachers to be inservice leaders in their own school districts or regions."^{92} The remaining U.S. teachers were interns, each with one year experience teaching. The interns were to receive Masters Degrees at the end of the summer during which interviews took place. By contrast, most Chinese teachers had only 11 or 12 years of formal education, completing only the ninth grade in high school followed by two or three years of normal school. In spite of their fewer years of formal education, the Chinese teachers demonstrated much greater understanding of fundamental mathematics than did their U.S. counterparts. Ma masterfully explained the interrelationships of pedagogy and content at the elementary school level and drew important lessons from her investigations. Liping Ma's book was embraced by all sides in the math wars. That unique distinction offered at least some hope that the warring factions could at some point find substantive issues upon which to agree.
Other events in 1999 were less unifying. In October, the U.S. Department of Education released a list of ten recommended math programs, as indicated at the beginning of this chapter. The programs were designated as either "exemplary" or "promising," and those programs are listed in the appendix to this chapter. The Open Letter to United States Secretary of Education Richard Riley was published on November 18, 1999 as a full paid ad in the Washington Post, paid for by the Packard Humanities Institute. The authors of the letter were David Klein, Richard Askey, R. James Milgram, and HungHsi Wu. Descriptions of some of the shortcomings of the "exemplary" and "promising" curricula were later published in the American School Board Journal.^{93} The NCTM responded to the open letter by explicitly endorsing all ten of the "exemplary" and "promising" programs (see appendix).
The ten "exemplary" and "promising" math programs were chosen by an "Expert Panel" designated by the U.S. Department of Education . The one mathematician on the Expert Panel, Manuel Berriozabal, publicly distanced himself from its decisions. The Christian Science Monitor reported that "Berriozabal abstained or voted against all 10 programs," and:
"The panel was a good idea," Dr. Berriozabal says, "but we made some bad judgments. From the best I could tell, none of the programs we selected as 'promising' or 'exemplary' had any kind of longterm track record of achievement." After Berriozabal arrived in Washington, the panel began debating the criteria to determine a successful program. Berriozabal thought that longterm proof of achievement should top the list. Most others on the panel wanted to require programs to conform to NCTM standardsthen gauge achievement.^{94}
Not all mathematicians were in agreement with the Open Letter. The most prominent critic of the Open Letter was Hyman Bass, the incoming president of the American Mathematical Society. Bass posted a message on a national listserve that denounced the Open Letter.^{95} The only program he defended in his message was Connected Math, though he did acknowledge that this grade 68 "exemplary" program did not include any treatment or explanation of division of fractions, as pointed out by Richard Askey. Bass complained that the Open Letter politicized the discussion. As reported in the Notices of the American Mathematical Society:
Bass disagrees with many of the conclusions in the letter, but his main objection is that the letter has inserted the debate over mathematics curricula "into the world of journalism and politics, whereÖserious and balanced discussion will no longer be possible." He also expressed concern that "What appear to be very sensible reservations about what the Department of Education did [have] become in fact part of a veiled and systematic assault on the professional education community."^{96}
In his email message, Bass expanded on his political objections:
Mathematically Correct, an important agent in promoting this Open Letter, has been politically active around the country. In Massachusetts it is allied with efforts of the Deputy Commissioner of Education, Sandra Stotsky, to review proposed revisions to the State Framework. Her ideological and uninformed opposition to "constructivist ideas" has reached the incredible state where she is opposed to inclusion of discussion of "Classical Greek constructions" as being "constructivist pedagogy." Is this what serious mathematicians want to associate themselves with?
Formerly a research associate at Harvard and an expert on children's reading, Dr. Sandra Stotsky was one of a handful of education leaders at the state or national level who endorsed the Open Letter. Chester Finn, a former U.S Assistant Secretary of Education, and Lisa Graham Keegan, the Superintendent of Public Education of Arizona also endorsed the Open Letter to U.S. Secretary Riley. Bass' accusation that Stotsky was opposed to "Classical Greek constructions" in geometry was completely without basis, as she later informed him; Bass had unwittingly misinterpreted another person's sarcastic comments. Indeed, Stotsky was on record as wanting a strong set of high school geometry standards in the revision of the mathematics curriculum framework for Massachusetts and sought the advice of Harvard mathematics professor Wilfried Schmid. Schmid provided generous assistance in the development of the new mathematics framework for Massachusetts, which suffered from similar opposition as the one in California. The Massachusetts math framework, much like California's, deviated from the constructivist prescriptions of the NCTM.^{97} Schmid, who was critical of NCTM aligned curricula, also signed the Open Letter.^{98}
Several months after the publication of the Open Letter to Secretary Riley, the U.S. Department of Education designated two more curricula as "promising": I Can Learn and Growing With Mathematics. The Department of Education praised these two programs, for their alignment to the NCTM Standards, among other reasons.
At the state level, California all but ignored the U.S. Department of Education recommendations. Of the 12 "exemplary" and "promising" math programs, only the UCSMP grade 7 and 8 textbooks were adopted in 1999 in California, and none were accepted for statewide adoption in 2001. Several NSF sponsored math curricular programs were submitted for statewide adoption in California in 1999 and 2001, but due to deficiencies in mathematical content, none were adopted in either year.
Given the size of the California textbook market, it is not surprising that there were heated debates between mathematicians, on the one hand, and the mathematics education community, on the other, about specific curricula and the influence of the California standards. As an illustration, the creators of one of the "exemplary" programs, CorePlus, posted an article on their website from Western Michigan University that included as part of a rebuttal of criticisms of CorePlus:
... Mr. Milgram also has a strong antireform agenda and was a leader in the campaign that led to the new California Mathematics Standards that have been widely criticized as retrograde by the mathematics education community.^{99}
The culminating event for mathematics education of the 1990s occurred in April 2000 when the NCTM released a new document entitled, Principles and Standards for School Mathematics (PSSM).^{100} PSSM was a revision of the 1989 NCTM Standards intended to address some of the criticisms of the earlier document. The writing teams for the year 2000 national standards began work on the PSSM in 1997, and many organizations were solicited for suggestions. The PSSM is a 402 page document organized into eight chapters, and it is similar in many respects to its predecessor, the 1989 NCTM Standards. Some of the more radical declarations from the 1989 NCTM Standards were eliminated, and slightly greater emphasis was given to the importance of arithmetic algorithms and computational fluency in the new document. The PSSM provided guidelines for spans of grades: prekindergarten to grade 2, 35, 68, and 912. As explained by Ralph Raimi who served on a committee of the American Mathematical Society to make recommendations for the new standards, the revisions fell short of what many of the critics would have preferred:
As Joan FerriniMundy, its principal editor, explained in her September Notices [of the American Mathematical Society] article, NCTM this time commissioned the commentary of many mathematicians, including committees of AMS, MAA, and SIAM, upon an earlier draft prepared for us. I myself served on the AMS committee and (by commission) as an individual too. NCTM solicited public advice at large, and I know several who also attempted to link the mathematical world with the new document, but the effort was to little avail; the messagethe "vision" of PSSMremains, in my vision, much the same as that of the original 1989 Standards.
PSSM continues to abhor direct instruction in, among other things, standard algorithms, Euclidean geometry, and the uses of memory. Though like its predecessor it has the word "standards" in its title, it is not a set of standards in the usual meaning of the term, for it refuses to say what exactly a child should learn in thirteen years of schooling. Long division? Quadratic formula? How to compute the quotient of two fractions? (See p. 218 of PSSM for an enlightening discussion.) Proof of a theorem on inscribed angles? Trigonometric identities? PSSM will neither affirm or deny, lest it seem to dictate content.^{101}
Concluding Remarks
At the end of the 20th century, mathematics education policies in U.S. public schools were in a state of flux. Disagreements between parents and mathematicians, on the one hand, and professional educators, on the other, continued without clear resolution. Wilfried Schmid described the disagreements at the end of the 1990s succinctly:
The disagreement extends over the entire mathematics curriculum, kindergarten through high school. It runs right through the National Council of Teachers of Mathematics (NCTM), the professional organization of mathematics teachers. The new NCTM curriculum guidelines, presented with great fanfare on April 12 [2000], represent an earnest effort at finding common ground, but barely manage to paperover the differences.
Among teachers and mathematics educators, the avantgarde reformers are the most energetic, and their voices drown out those skeptical of extreme reforms. On the other side, among academic mathematicians and scientists who have reflected on these questions, a clear majority oppose the new trends in math education. The academics, mostly unfamiliar with education issues, have been reluctant to join the debate. But finally, some of them are speaking up.
Parents, for the most part, have also been silent, trusting the expertsthe teachers' organizations and math educators. Several reform curricula do not provide textbooks in the usual sense, and this deprives parents of one important source of information. Yet, also among parents, attitudes may be changing...
The stakes are high in this argument. State curriculum frameworks need to be written, and these serve as basis for assessment tests; some of the reformers receive substantial educational research grants, consulting fees or textbook royalties. For now, the reformers have lost the battle in California. They are redoubling their efforts in Massachusetts, where the curriculum framework is being revised. The struggle is fierce, by academic standards.^{102}
The stakes are high not only for mathematics education in the public schools, but also for the nation's colleges and universities. Through a domino effect that begins in the elementary school grades and works its way up the educational ladder, the socalled reforms promoted by the NCTM, and other education organizations, are sure to affect university level mathematics education. Without adequate foundations in arithmetic skills and concepts from elementary school, entering middle school students will be unable to progress to algebra. Without strong foundations in algebraic skills and ideas, the doors to subsequent meaningful mathematics courses will be closed. University mathematicians are worried. As HungHsi Wu explained in 1997:
This reform once again raises questions about the values of a mathematics education ...by redefining what constitutes mathematics and by advocating pedagogical practices based on opinions rather than research data of largescale studies from cognitive psychology.
The reform has the potential to change completely the undergraduate mathematics curriculum and to throttle the normal process of producing a competent corps of scientists, engineers, and mathematicians. In some institutions, this potential is already a reality.^{103}
In an era of international competition, it is unlikely that the public will tolerate such trends indefinitely. It was the broad implementation of the NCTM reforms themselves that created the resistance to them. Ironically, the extraordinary success in disseminating progressivist mathematics programs may, in the long run, be the principal reason for the demise of progressivism in mathematics education.
