This is a preprint of an article that appeared in the BSHM Bulletin: Journal of the British Society for the History of Mathematics, Volume 22, Issue 1, p.  22-33 (2007), (c) Taylor & Francis.  The definitive version is available at: http://dx.doi.org/10.1080/17498430601148762  The BSHM Bulletin may be found online at: http://www.tandf.co.uk/journals/titles/17498430.asp

 

 

A quarter century of US ‘math wars’ and political partisanship

 

David Klein

 

California State University, Northridge, USA

 

This article traces the history of the US ‘math wars’ from 1980, and discusses the political polarizations that fuelled and resulted from the disagreements.

 

Keywords: math wars; progressive education; politics

 

2000 Mathematics Subject Classifications: 01A60, 01A61, 97-03, 97B99

 

Introduction

 

Treatises on education and its social implications span at least two millennia. Since the18th century, two major strands within this genre may be identified: progressive education and classical education.[1] The latter traces its origins to Plato, who argued that education for a just society requires the reinforcement of the rational over the instinctive and emotional aspects of human nature. Systematic instruction and practise are part of the classical tradition. They are also essential components of Asian and other non-Western educational systems.

 

The hallmarks of progressive education, by contrast, are naturalistic, child-centred instruction and discovery learning. Progressive education is an outgrowth of the Romantic Movement with roots going back to Jean Jacques Rousseau. John Dewey and William Heard Kilpatrick were instrumental in ensuring the dominance of progressive education theory in teachers colleges through most of the 20th century.[2] In the variant promoted by Kilpatrick, who was especially influential in mathematics education, subjects would be taught to students based on their direct practical value, or if students independently wanted to learn them.[3]

 

The history of US mathematics education of the past quarter century cannot be separated from these historical strands, nor from contemporary political and economic influences. A distinctive feature of the 1990s and early years of the 21st century was the association of right and left wing political ideologies with competing mathematics education programmes and their advocates.[4]

 

Particular textbooks and curricular programmes were the focal points of disagreement. Examples have been identified and described elsewhere.[5] Sometimes referred to as ‘constructivist’,[6] those textbooks and programmes were aligned with, promoted by, and in some cases endorsed in writing[7] by the National Council of Teachers of Mathematics (NCTM), the leading organization for pre-collegiate mathematics education in the US. In opposition to the use of these progressivist school programmes, organizations of parents sprang up across the US, and worked in collaboration with university mathematicians and other academics. The resulting ‘math wars’ of the 1990s often fractured along political lines. But it was the events of the 1980s that spawned the controversies of the succeeding decade.

 

Seeds of Reform: the 1980s

 

In a long series of documents published by the NCTM, three have been especially influential: An Agenda for Action (1980), Curriculum and Evaluation Standards for School Mathematics (1989), and Principles and Standards for School Mathematics (2000). The latter two are referred to respectively as the 1989 NCTM Standards and the 2000 NCTM Standards, or just Standards when the context is clear.

 

An Agenda for Action paved the way for major trends of the 1990s.[8] It recommended that problem solving be the focus of school mathematics. It asserted that ‘difficulty with paper-and-pencil computation should not interfere with the learning of problem-solving strategies’. Technology would make problem solving available to students without basic skills. According to the report, ‘All students should have access to calculators and increasingly to computers throughout their school mathematics program’, including elementary school students. The report called for ‘decreased emphasis on such activities as…performing paper and pencil calculations with numbers of more than two digits’.

 

An Agenda for Action also argued that ’emerging programs that prepare users of mathematics in non-traditional areas of application may no longer demand the centrality of calculus. . .’ The de-emphasis of calculus would later support the move away from the systematic development of its prerequisites: algebra, geometry, and trigonometry. The ‘integrated’ high school mathematics books of the 1990s contributed to this tendency. While those books contained parts of algebra, geometry, and trigonometry, these traditional subjects were not developed systematically, and often depended on student ‘discoveries’ that were incidental to solving ‘real world problems’.

 

The 1989 NCTM Standards described general standards for the bands of grades: K-4, 5-8, and 9-12. It promoted the views of An Agenda for Action, but with greater elaboration. The grade level bands included lists of topics that were to receive ‘increased attention’ and others for ‘decreased attention’. The K-4 band called for greater attention to ‘Operation sense’, ‘Use of calculators for complex computation’, ‘Collection and organization of data’, ‘Pattern recognition and description’, ‘Use of manipulative materials’, and ‘Cooperative work’.

 

Slated for decreased attention were ‘Long division’, ‘Paper and pencil fraction computation’, ‘Rote practice’, and ‘Teaching by telling’. Topics listed for decreased attention in grades 5-8 included: ‘Relying on outside authority (teacher or an answer key)’, ‘Manipulating symbols’, ‘Memorizing rules and algorithms’, and ‘Finding exact forms of answers’.

 

As in An Agenda for Action, the Standards put strong emphasis on the use of calculators throughout all grade levels. 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 therefore recommended that, ‘appropriate calculators should be available to all students at all times’.

 

The Standards reinforced the general themes of progressive education by advocating student centred, 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. Mathematics for its own sake was not encouraged.

 

The arguments in support of these changes took two major themes: social justice in the form of challenging racial and class barriers on the one hand, and the needs of business and industry on the other.[9] The following passage from Alan Schoenfeld’s, The Math Wars, (page 255) is representative:

 

… lack of access to mathematics is a barrier – a barrier that leaves people socially and economically disenfranchised. For these reasons, noted civil rights worker Robert Moses declared that “the most urgent social issue affecting poor people and people of color is economic accessÉ. I believe that the absence of math literacy in urban and rural communities throughout this country is an issue as urgent as the lack of registered Black voters in Mississippi was in 1961.”

 

The needs of business were described on page 3 of the 1989 NCTM Standards:

 

Traditional notions of basic mathematical competence have been outstripped by ever-higher expectations of the skills and knowledge of workers . . . employees must be prepared to understand the complexities and technologies of communication, to ask questions, to assimilate unfamiliar information, and to work cooperatively in teams. Businesses no longer seek workers with strong backs, clever hands, and “shopkeeper” arithmetic skills.

 

Arguments for social justice and business needs were often conjoined, as on page 9 of the Standards:

 

If all students do not have an opportunity to learn this mathematics, we face the danger of creating an intellectual elite and a polarized society. The image of a society in which a few have the mathematical knowledge needed for the control of economic and scientific developments is not consistent either with the values of a just democratic system or with its economic needs.

 

Everybody Counts, a 1989 report of the National Research Council, not only restated the same themes, but offered the 1989 NCTM Standards as the solution:

 

Through the Standards, parents and teachers will be able to understand in concrete terms what a school mathematics program might look like if it is to serve our national objectives adequately. (page 89)

 

The confluence of social justice themes, attendance to the needs of business, and the promise of conceptual understanding of mathematics for all students gave the NCTM’s agenda the momentum it needed. Business, government, and labour unions could all find something to like in the proposal. By the beginning of the final decade of the 20th century, the NCTM’s vision for mathematics education was unstoppable.

 

NCTM Reform and Counter-Reform: the 1990s and Beyond

 

Even though the NCTM was not a governmental agency, its standards played the role of national standards. Virtually all state standards were modelled on the 1989 NCTM Standards. California’s 1992 Mathematics Framework was, if anything, even more extreme. For example, it instructed, ‘Calculators are the “electronic pencils” of today’s world…In every grade calculators can be issued to students just as textbooks are. A reasonable goal is to make calculators available at all times for in-class activities, homework, and tests.’ California’s progressive education policies would lead to public opposition later in the decade.

 

The California Mathematics Council, an affiliate of the NCTM, sent a letter to the California Board of Education dated April 17, 1996 citing an ordered list of 13 basic skills ‘desired by Fortune 500 companies’ with computation in 12th place. It praised the 1992 California Framework for addressing the needs of business:

 

Equally impressive is that these changes in the way we teach mathematics are supported by the business community. What Work Requires of Schools: A SCANS Report for America 2000 concludes that students must develop a new set of competencies and new foundation skills. It stresses that skills must be learned in context, that there is no need to learn basic skills before problem solving, and that we must reorient learning away from mere mastery of information toward encouraging students to solve problems. . . Learning in order to know must never be separated from learning to do.

 

 

Through the 1990s, funding for textbooks aligned to the Standards flowed from the National Science Foundation (NSF) and corporate foundations.[10] Parents were the first to object, especially in California,[11] and they formed grassroots organizations[12] to pressure schools to use other textbooks, or allow parental choice. NCTM aligned books and programmes were criticized for diminished content and lack of attention to basic skills. The elementary school programmes required students to use their own invented arithmetic algorithms in place of the standard algorithms of arithmetic. Calculator use was encouraged to excess and integrated even into kindergarten lessons. Student discovery group work, at all grade levels, was the preferred pedagogy, but in most cases, projects were aimless or inefficient. Statistics and data analysis were overemphasized repetitiously at all grade levels at the expense of algebra and more advanced topics. Mathematical definitions and proofs for the higher grades were deficient, missing entirely, or even incorrect.[13] Critics openly derided constructivist programmes in their schools as ‘dumbed-down’, and described the genre as, ‘fuzzy math’, ‘new-new math’, or ‘whole math’, in analogy with the failed reading pedagogy known as ‘whole language learning’.

 

Some of the widely used programmes aligned to the Standards did not even include textbooks, since books might interfere with student discovery. MathLand, a K-6 programme, was one such example, and it was nearly devoid of mathematical content.[14] Nevertheless, by 1997 MathLand was adopted by 60% of the state’s public elementary schools, according to its publishers,[15] and was one of many NCTM aligned programmes in use across the US. Promotional materials for MathLand cited the same SCANS report and Fortune 500 ordered list of topics.

 

The views of most American mathematicians involved in these debates were similar to those of their British counterparts, who were simultaneously confronted by the same issues in public education.[16] Criticisms by mathematicians on both sides of the Atlantic were well articulated in a 1995 report entitled Tackling the Mathematics Problem, commissioned jointly by the London Mathematical Society, the Institute of Mathematics and its Applications, and the Royal Statistical Society, as seen in the following excerpts:

 

In recent years English school mathematics has seen a marked shift of emphasis, introducing a number of time-consuming activities (investigations, problem-solving, data surveys, etc.) at the expense of ‘core’ technique. In practise, many of these activities are poorly focused; moreover, inappropriate insistence on working within a context uses precious time and can often obscure the underlying mathematics…

 

[W]e have also seen implicit ‘advice’ … that teachers should reduce their emphasis on, and expectations concerning, technical fluency. This trend has often been explicitly linked to the assertion that “process is at least as important as technique”. Such advice has too often failed to recognise that to gain a genuine understanding of any process it is necessary first to achieve a robust technical fluency with the relevant content…

 

In parallel with these changes in emphasis, evidence that many English pupils were unable to solve standard problems involving, for example, decimals, fractions, ratio, proportion and algebra … was interpreted by many curriculum developers and those responsible for defining national curricula, as meaning that such topics were ‘too hard’ for most English pupils in the lower secondary years.

 

As public opposition increased in the US, debates became more polarized along political lines. Mushrooming press accounts of bizarre classroom projects and deficient textbooks led to open ridicule by columnists, especially conservatives, and involvement of politicians. Generally, the politicians most sympathetic to the criticisms were Republican. However, as part of the Congressional debate on education legislation, Democratic Senator Robert Byrd made searing criticisms of the mathematics education reform movement from the Senate floor. He focused on a particular textbook,[17] scorned by critics as ‘Rainforest Algebra’. The Congressional Record of June 9, 1997 includes the following passages from his speech:

 

Mr. President, over the past decade, I have been continually puzzled by our Nation’s failure to produce better students despite public concern and despite the billions of Federal dollars… I took algebra instead of Latin when I was in high school. I never had this razzle-dazzle confusing stuff…

 

This odd amalgam of math, geography and language masquerading as an algebra textbook goes on to intersperse each chapter with helpful comments and photos of children named Taktuk, Esteban, and Minh. … I still don’t quite grasp the necessity for political correctness in an algebra textbook. Nor do I understand the inclusion of the United Nations Universal Declaration of Human Rights in three languages or a section on the language of algebra which defines such mathematically significant phrases as, “the lion’s share,” the “boondocks,” and “not worth his salt.”

 

… From there we hurry on to lectures on endangered species, a discussion of air pollution, facts about the Dogon people of West Africa, chili recipes and a discussion of varieties of hot peppers…what role zoos should play in today’s society, and the dubious art of making shape images of animals on a bedroom wall, only reaching a discussion of the Pythagorean Theorem on page 502.

 

Falling back on the theme of social justice, progressive educators cast the critics as politically right wing, and presented the disagreements to journalists as a conflict between conservative traditionalists who demanded basic skills and progressive reformers who advocated conceptual understanding.

Many conservatives did indeed rally in opposition to the progressive maths of the NCTM, for example, Lynne Cheney, Chester Finn, Rush Limbaugh, Phyllis Schlafly, and Thomas Sowell. However, not all opponents were conservative. Three of the four founders of the parents group, Mathematically Correct, described themselves as liberal Democrats as did Elizabeth Carson, the leader of NYC HOLD, which by the beginning of the 21st century had emerged as the leading opposition group to NCTM aligned programmes. Following a presentation to the California Board of Education, Abigail Thompson, a mathematics professor at the University of California at Davis

 

. . .was invited to speak at a local Republican convention. A liberal Democrat, Thompson was stunned. Mathematicians tend to jump into such issues with both feet, she says, “and then they find themselves labeled as right-wing conservatives. And it’s pretty hilarious. I don’t know any mathematicians who are right-wing conservatives.”[18]

 

Alfie Kohn, an advocate of progressive education, criticized parental opposition in an April 1998 Phi Delta Kappan article entitled, Only for My Kid: How Privileged Parents Undermine School Reform. He observed, ‘It is common knowledge that the Christian Right has opposed all manner of progressive reforms’, but he also identified the subversive role played by liberal parents opposed to the reform curricula:

 

Jeannie Oakes, author of Keeping Track, calls them “Volvo vigilantes,” but that isn’t quite accurate – first, because they work within, and skillfully use, the law; and second, because many of them drive Jeeps. They may be pro-choice and avid recyclers, with nothing good to say about the likes of Pat Robertson and Rush Limbaugh; yet on educational issues they are, perhaps unwittingly, making common cause with, and furthering the agenda of, the Far Right.

 

Like other progressives, Kohn reinforced the dichotomy of basic skills versus conceptual understanding, but he also advanced another dichotomy: doing what is best for learning versus doing what is best for admission to universities:

 

. . . more than one observer of the “math wars” has wondered whether we are witnessing a debate over pedagogy or about something else entirely. Are parents really trying to deny that encouraging students to figure out together what lies behind an algebraic formula is more valuable than getting them to memorize algorithms or slog through endless problem sets? Do they seriously doubt that such an approach is better preparation for higher math in college? Or does parental opposition really just reflect the fear that more sophisticated math instruction might be less useful for boosting SAT scores and therefore for getting students into the most elite colleges? Math reformers who counterpose merely doing arithmetic with really understanding (and being able to apply) mathematical principles may be missing the more pertinent contrast, which is between doing what is best for learning and doing what is best for getting my child into the Ivy League.

 

The arguments were strained. The ‘traditional curriculum’ was accused of being too focused on basic skills at the expense of understanding, or more concisely, of being ‘dumbed-down’. Thus, progressive programmes were putatively ‘better preparation for higher math in college’. Yet, elite universities expected a traditional curriculum as preparation for admission. The unstated implication is that elite universities favoured a lower level of understanding over the concept-rich programmes that the NCTM claimed to offer all students. But this was hard to reconcile with criticisms of the NCTM reform by top university mathematicians. After all, mathematicians devote their lives to mathematical concepts. How could they be opposed to conceptual understanding of mathematics?

 

In The Math Wars Schoenfeld also describes the traditional curriculum as elitist and portrays the math wars as a battle between equality and elitism:

 

. . . the traditional curriculum bore the recognizable traces of its elitist ancestry: The high school curriculum was designed for those who intended to pursue higher education. (page 267)

 

The Standards, buttressed by NCTM’s call for “mathematics for all” and the equity agenda in Everybody Counts, clearly sat in the education-for-democratic-equality [camp]. . . In contrast, . . . the traditional curriculum was a vehicle for . . . the perpetuation of privilege. . .Thus the Standards could be seen as a threat to the current social order. (page 268)

 

. . . the traditional curriculum, with its filtering mechanisms and high dropout and failure rates (especially for certain minority groups) has had the effect of putting and keeping certain groups “in their place.” (page 281)

 

These powerful indictments demanded radical solutions. Mathematics reform was social reform, and that meant redefining K-12 mathematics to make it more accessible. The resulting decline of K-12 mathematical content had obvious repercussions for universities. Hung-Hsi Wu expressed concerns of mathematicians when he wrote:

 

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 large-scale 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.[19]

 

The term ‘traditional’ was never clearly defined in the debates. The NCTM aligned programmes were easy to define simply by listing them, and it is true that some specific ‘traditional’ mathematics programmes favoured by parents and mathematicians could also be identified, but it is unclear what tradition, if any, they followed. Some of the secondary school mathematics books favoured by ‘traditionalists’ dated back to the ‘new math’ period of the 1950s and 60s, at which time they were considered anything but traditional. Others, like Saxon Math, included innovations such as review of previous topics within each problem set. The strongest tradition in US education is progressivism itself, not the challenges to it.

 

The conflict between reformers and the California public reached a turning point in 1997. Under a Republican governor, the California Board of Education rejected the reform-oriented draft standards from one of its advisory committees, the Academic Standards Commission. The Commission majority, knowing the Board was opposed to their NCTM aligned draft,

 

delayed presenting it … for the required Board approval until the very last minute allowed by law, foreclosing effective debate within the Commission at its final meeting and expecting that the deadline of January 1, 1998 for the Board’s final decision would also prevent the Board’s changing much of what they were springing in October… But the success of this hardball politics was foiled by the rapid response of the (forewarned) Board, which appointed four mathematicians [Gunnar Carlsson, Ralph Cohen, Steve Kerckhoff, and R. James Milgram] at Stanford University [to rewrite the draft]. [20]

 

In a few weeks the Stanford mathematicians rewrote the standards, correcting more than 100 mathematical errors, and eliminating all pedagogical directives. The document was approved in December 1997 by the State Board. California’s new standards were clear, coherent, and met the criteria set by the California legislature to be competitive with mathematics standards of the highest performing countries in mathematics education.[21]

 

The reaction from mathematics reformers was swift. The lead story in the February 1998 News Bulletin of the NCTM, New California Standards Disappoint Many, charged, ‘Over protests from business, community, and education leaders, California’s state board of education unanimously approved curriculum standards that emphasize basic skills and de-emphasize creative problem solving, procedural skills, and critical thinking’. Joining the educational progressives, the state-wide chairs of the Academic Senates of the public colleges and universities in California issued a joint statement condemning California’s standards and claimed that ‘the consensus position of the mathematical community’ was against them.

 

California mathematicians put a stop to the rumour of a consensus in the mathematics community against the state’s standards. More than 100 mathematics professors from colleges and universities in California added their names to an open letter in support of the California standards. The signatories included chairs of the mathematics departments at the California Institute of Technology, Stanford, and several state universities. Jaime Escalante, portrayed in the movie Stand and Deliver, also added his name in support.[22] The conflict was more than just a theoretical disagreement. At stake was the use of NCTM aligned textbooks in California, the biggest market in the nation.

 

California proceeded to develop state-wide tests and a system for textbook adoptions that included review panels of mathematicians and classroom teachers. Thus, California became the national base for opposition to the NCTM reform movement.

 

In October 1999, the US Department of Education released a report designating 10 mathematics programmes as ‘exemplary’ or ‘promising’. Several of the programmes on the list, including MathLand, had been sharply criticized by mathematicians and parents for much of the decade. The imprimatur of the US government carried by these controversial programmes threatened not only to undermine California’s new direction in mathematics education, it could marginalize criticisms of the NCTM aligned textbooks nationwide.

 

Within a month of the release of the Education Department’s report, more than 200 university mathematicians added their names to an open letter to Secretary Riley calling upon him to withdraw those recommendations.[23] The list of signatories included seven Nobel laureates and winners of the Fields Medal, the highest international award in mathematics, as well as mathematics department chairs of many of the top universities in the US, and a few state and national education leaders. The open letter was published on 18 November 1999 as an ad in the Washington Post, paid for by the Packard Humanities Institute.[24]

 

Within days the NCTM responded to the mathematicians’ open letter with its own letter to Secretary Riley in which the organization explicitly endorsed all of the ‘exemplary’ and ‘promising’ programmes.[25] Nevertheless, in the following years, the mathematicians’ letter continued to be a useful tool for parents opposed to NCTM aligned textbooks. Recognizing its utility, NCTM President Johnny Lott in January 2004 posted a denunciation of the open letter on the NCTM website, under the title, “Calling Out” the Stalkers of Mathematics Education, in which he wrote:

 

Consider people who use half-truths, fear, and innuendo to control public opinion about mathematics education. As an example, look at Web sites that continue to use a public letter written in 1999 to then Secretary of Education Richard Riley by a group of mathematicians and scientists defaming reform mathematics curricula developed with National Science Foundation grants. . . A small group continues to use the letter in an attempt to thwart changes to mathematics curricula.[26]

 

Resistance through the end of the century to abandoning the NCTM style textbooks in some California school districts was considerable. One case resulted in front page newspaper coverage. A critic of the California standards, Guillermo Mendieta, threatened a hunger strike on behalf of NCTM aligned mathematics programmes. Mendieta was the Director of Mathematics Education for the Achievement Council, a non-profit organization that addressed educational inequities. At stake was whether the Los Angeles Unified School District (LAUSD) would use California state approved textbooks or continue with ‘integrated math’ in secondary schools, along with MathLand and similar programmes in elementary schools. Mendieta was supported by a coalition that included the NCTM, Center X within the University of California at Los Angeles, and various Latino and African American Organizations. LAUSD School Board President Genethia Hayes also extended her support to Mendieta and declared, ‘I will advocate as hard as I can with my colleagues to make sure this particular door never gets shut for children of color. I really do see this as an issue of social justice’.[27]

 

One of the advocates for California approved textbooks was Barry Simon, the mathematics chair and IBM Professor of Mathematics and Theoretical Physics at the California Institute of Technology. Simon asserted that basic skills are essential to mathematics and counselled against redefining algebra via the NCTM aligned programmes in order to increase passage rates. Confronting the social justice arguments of the progressivists, he countered, ‘If anyone is racist or sexist, it is those who claim that women and minorities are unable to deal with traditional mathematics’.[28] Others gave similar advice. Nevertheless, Mendieta’s arguments carried the day. LAUSD’s use of state approved textbooks was thus delayed until 2001.

 

Social justice arguments in support of NCTM programmes continued into the new century. The introduction to the book, Rethinking Mathematics: Teaching Social Justice by the Numbers, published in 2005, argued that,

 

Teachers cannot easily do social justice mathematics teaching when using a rote, procedure-oriented mathematics curriculum. Likewise a text-driven, teacher centered approach does not foster the kind of questioning and reflection that should take place in all classrooms, including those where math is studied. (page 4)

 

The release of the 2000 NCTM Standards had little impact on nationwide disagreements, beyond affirming the direction of the 1989 Standards. The revisions were primarily rhetorical and the document did not differ substantially from its predecessor. More significant was the ‘No Child Left Behind Act’, signed into law with bipartisan support in 2002. Although flawed in many respects, it asked for challenging academic standards, high quality teachers, and imposed annual testing requirements on US schools. Perhaps even more significant was the creation in April 2006 of the ‘National Mathematics Advisory Panel’ charged with making policy recommendations to the President and Secretary of Education for the improvement of mathematics achievement of students. Of the 17 expert panellists appointed by the conservative Bush administration, five were signatories of the 1999 open letter to former Education Secretary Riley. The Expert Panel that recommended the ‘exemplary’ and ‘promising’ mathematics programmes in 1999 was appointed by a Democratic administration. The stark difference between the two expert panels reflects the political divide in the maths wars.

 

Concluding Remarks

 

Why did disagreements about school mathematics books in the US diverge according to left and right politics?

 

Part of the answer is historical. The roots of progressive education are intertwined with anti-authoritarian ideals from the Romantic Era. In addition, progressive educators, including a former NCTM president, argued that women and members of ethnic minority groups learn mathematics differently from white males.[29] Such views were harmonious with the politically liberal ethnic identity ideologies popular during this period, especially in universities. Taking into account the anti-elitism and social justice arguments surrounding constructivist mathematics programmes, it is then not surprising that multiculturalists and liberals would be attracted to the NCTM vision, even if they did not understand the mathematical issues involved. As those groups constituted parts of the electoral base for Democratic politicians, the latter would be reluctant to challenge the use of constructivist maths in schools.

 

The NCTM reform was an attempt to redefine mathematics in order to correct social inequities. To make mathematics more accessible to minority groups and women, progressive educators argued for programmes that eliminated basic skills and the intellectual content that depends on those skills. Ironically, progressivists’ advocacy of such mathematics programmes for the supposed benefit of disenfranchised groups contributed to racial stereotyping, in contradiction to core progressive values.

 

In the course of the math wars, parents of school children and mathematicians who objected to the dearth of content were dismissed as right wing, but there is nothing inherently left wing about the NCTM aligned mathematics programmes. Neither the former Soviet Union nor other socialist countries participated in education programmes remotely like those promoted by the NCTM. Progressive maths is a purely capitalist phenomenon. Indeed, one of the promotional themes of the NCTM was to prepare students for the needs of business.

 

Ultimately, the injection of left and right ideologies into mathematics education controversies is counterproductive. The math wars are unlikely to end until programmes espoused by progressives incorporate the intellectual content demanded by parents of school children and mathematicians.

 

Acknowledgments. The author would like to thank Elizabeth Carson, Harry Hellenbrand, Ralph Raimi, Mary Rosen, and Sandra Stotsky for critical readings and suggestions.

 

Bibliography

 

Congressional Record of the US Senate, Robert Byrd, ‘A Failure to Produce Better Students,’ Senate, June 9, 1997, p S5393.

 

Hirsch E D, The Schools We Need: Why We Don’t Have Them, New York, New York, Double Day, 1996.

 

Hirsch E D, Romancing the Child, Education Next (Spring 2001) 34-39

www.educationnext.org/2001sp/34.html. Accessed on 22 August 2006.

 

Jackson, A, The math wars: California battles it out over mathematics education (Part II). Notices of the American Mathematical Society, 44(7), (1997) 817-823.

 

Klein, D, ‘Big business, race, and gender in mathematics reform’, (appendix) in Steven Krantz, How to Teach Mathematics, American Mathematical Society (1999) 221-232.

 

Klein, D, Math problems: why the US Department of Education’s recommended math programs don’t add up. American School Board Journal, Volume 187, No. 4, (2000) 52-57. www.mathematicallycorrect.com/usnoadd.htm. Accessed on 22 August 2006.

 

Klein, D, et al, The state of the state math standards 2005, Washington, D. C., Thomas B. Fordham Foundation, 2005, www.edexcellence.net/foundation/publication/publication.cfm?id=338. Accessed on 22 August 2006.

 

Klein, D, ‘A brief history of American K-12 mathematics education in the 20th century’, in James Royer (ed) Mathematical cognition: a volume in current perspectives on cognition, learning, and instruction, Information Age Publishing, 2003, 175 – 225

www.csun.edu/~vcmth00m/AHistory.html. Accessed on 22 August 2006

 

McKeown, M, Klein, D, Patterson, C, ‘National science foundation systemic initiatives: how a small amount of federal money promotes ill-designed mathematics and science programs in k-12 and undermines local control of education’, in Sandra Stotsky (ed), What’s at stake in the k-12 standards wars: a primer for educational policy makers, New York, Peter Lang Publishing, Inc., 2000, 313 – 369.

 

National Research Council, Everybody Counts: A Report to the Nation on the Future of Mathematics Education, National Academy Press, Washington, D.C., 1989

 

Raimi, R, Uncivil war, in Education Next (Hoover Institution, Stanford, CA, Summer 2004). This is a review of California Dreaming: Reforming Mathematics Education by Suzanne M. Wilson (Yale University Press, 2003, xvi+303 pages), and may be found in expanded form at http://www.educationnext.org/unabridges/20042/raimi.pdf. The review as printed is at http://www.educationnext.org/20042/81.html. Accessed on 22 August 2006.

 

Rethinking mathematics: teaching social justice by the numbers, (ed) Eric Gutstein and Bob Peterson, copyright 2005, Rethinking Schools Ltd.

 

Martin Scharlemann Open Letter on MathLand, 11 October 1996. http://mathematicallycorrect.com/ml1.htm. Accessed on 22 August 2006.

 

Schoenfeld, A. The math wars, Educational Policy, Vol. 18 No. 1, (2004) 253-286.

 

Wu, H, The mathematics education reform: why you should be concerned and what you can do, American Mathematical Monthly 104 (1997), 946-954.

 

Wu, H, Basic skills versus conceptual understanding: a bogus dichotomy in mathematics education American Educator, American Federation of Teachers, Fall 1999.

 

 

[1] For an elaboration of this dichotomy, see Hirsch, Romancing the Child

[2] See Hirsch, The Schools We Need, p. 71-79

[3] Klein, A Brief History of Mathematics Education in the 20th Century (A Brief History)

[4] For the sake of transparency, I identify myself as a socialist and a registered member of the Green Party.

[5] See Klein, Math Problems and A Brief History

[6] Constructivism in this context is a variant of progressivism. See Hirsch, The Schools We Need, p. 245

[7] See the appendix of A Brief History

[8] A Nation at Risk, another important document from this period, is discussed in A Brief History.

[9] These same themes appeared in A Nation at Risk.

[10] For more details, see A Brief History

[11] However, a group of parents of school children in Princeton, New Jersey, including Princeton University faculty, in 1991 objected to progressivist programs and eventually founded their own Charter school. See A Brief History.

[12] Two of the most important were “Mathematically Correct” (www.mathematicallycorrect.com) and New York City HOLD (www.nychold.com)

[13] See for example, Klein, Math Problems

[14] See Scharlemann, An Open Letter on MathLand

[15] Time, August 25, 1997, Suddenly, Math Becomes Fun And Games. But Are The Kids Really Learning Anything?

[16] See for example Wu, Basic Skills Versus Conceptual Understanding: A Bogus Dichotomy in Mathematics Education

[17] Focus on Algebra, Addison-Wesley Secondary Math, An Integrated Approach, Addison-Wesley, Menlo Park, CA, 1996.

[18] Quoted from Jackson, The math wars: California battles it out over mathematics education (Part II).

[19] Wu, The Mathematics Education Reform

[20] Quoted from Raimi, Review of the book California Dreaming

[21] See Klein et al, The State of State Math Standards 2005

[22] See A Brief History

[23] I am a co-author of that letter.

[24] It is posted at: mathematicallycorrect.com/nation.htm

[25] The letter appears in the appendix to A Brief History

[26] For a rebuttal, see www.mathematicallycorrect.com/rebutlott.htm

[27] Quoted from Richard Colvin, Debate Over how to Teach Math Takes Cultural Turn, Los Angeles Times, March 17, 2000

[28] Ibid

[29] See Big Business

My Turn: Forget the Fads—The Old Way Works Best

What will fix public education? A teacher, a chalkboard and a roomful of willing students

By Evan Keliher
NEWSWEEK

Mathematically Correct presents

The Pythagorean Theorem

G. D. Chakerian and Kurt Kreith

---

At a recent school meeting, a group of Davis parents and teachers used the Pythagorean theorem to illustrate the difference between a constructivist vs. traditional approach to teaching. Their goal was to provide other parents with a basis for responding to a recent decision by the Davis Board of Education. For in Fall, 1996 Davis junior high schools will offer a choice between two different courses in Algebra 1, one emphasizing constructivist pedagogy and the other relying on a more traditional deductive approach.

As set forth in the currently used text Themes, Tools and Concepts, one constructivist approach to the Pythagorean theorem is based on the use of geoboards. A traditional approach appropriate to Algebra 1, one found in many algebra texts, is based on the dissection of a square.

 

   The former calls on students to use rubber bands to build a right
   triangle on a geoboard, use rubber bands to enclose the squares
   defined by the triangle's legs and hypotenuse, and then look for
   patterns in the areas of the squares so generated.  

   The latter asks the student to visualize two different dissections
   of a square of size (a+b) x (a+b).  Using the usual notation of a,
   b, and c for the legs and hypotenuse of a right triangle, one such
   dissection corresponds to a^2 + b^2 + 2ab and the other corresponds
   to c^2 + 2ab .  Equating these two expressions yields the usual
   symbolic representation of the Pythagorean theorem: a^2 + b^2 = c^2.
   
   [^2 notation indicates squared terms]

Laudable as the use of experimentation as a prelude to mathematics may be, there are serious dangers hidden in this constructivist approach to the Pythagorean theorem. For instance, while it is easy to construct right triangles on a geoboard by orienting the legs of the triangle along the horizontal and vertical axes, the example given in Themes, Tools and Concepts suggests that the student should use more general orientations. The question that then arises is, “how is the student to know whether a triangle with such general orientation is, or is not, a right triangle?” (It is very easy to construct geoboard triangles with one angle imperceptibly close, but not quite equal, to a right angle.)

The only mathematical solution is to use the Pythagorean theorem itself (or more precisely, its converse) to confirm that such a triangle is in fact a right triangle. However, this is precisely the knowledge that the student is being urged to construct! Thus, implicit in this particular constructivist approach to the Pythagorean theorem is the notion that the student should build his or her own knowledge by “eyeballing” right angles.

Training a generation of carpenters to rely on “eyeballing” right angles would be a national disaster (none of us would let such a carpenter touch our house). Yet, in the name of constructivism, we seem to be encouraging a generation of children to erect this pillar of mathematical knowledge on just such a basis.

Another problem arising in this constructivist approach to the Pythagorean theorem is that of calculating the areas of the squares built on the sides and hypotenuse of a geoboard triangle. The usual formula “Area = Side x Side” requires that we first determine the lengths of the sides of these squares. However, unless these squares are aligned with the geoboard’s vertical and horizontal axes, finding the lengths of their sides also requires the Pythagorean theorem!

An alternative way of finding the areas of “tilted squares” is to use an advanced mathematical result called Pick’s theorem. While children can be taught to use Pick’s theorem at an early age, any semblance of a mathematical understanding of this tool is well beyond the traditional high school curriculum. It may also be that students will be taught to calculate areas by counting unit squares, and pieces thereof.

Another possibility is to enclose such a “tilted square” within a larger square whose sides are parallel to the axes of the geoboard. While this provides an appropriate way of calculating areas, note that it corresponds to the dissection “(a+b)^2 = c^2 + 2ab” arising in the traditional proof of the Pythagorean theorem cited above. That is, this particular approach to implementing the “discovery process” takes the student half way to actually understanding the Pythagorean theorem. However, on ideological grounds, it stops short of conveying the gift of understanding.

Some may argue that it doesn’t really matter which method is used to teach the Pythagorean theorem – i.e., that both methods lead to the same result. However, this is not true. Experiments with the geoboard correspond to a cumbersome verification of the Pythagorean theorem in rather special circumstances (the geoboard’s discrete structure is well suited to experimentation, but it fails to represent the more general structure of the Euclidean plane). The traditional dissection approach corresponds to a proof of the theorem, providing an answer to the question “why.” One approach sets the stage for discoveries that lead to conjectures; the other emphasizes properties of area that lead to the understanding of an important truth.

Many parents believe that the most important end of education in any field is to raise the question “why,” to know when an answer might or might not exist, and to demand an answer when it can be given. Having children “discover” a hodge-podge of mathematical properties, without providing answers to which properties are true and why, is to deny them a real mathematical education.

While manipulatives can be powerful tools for leading students through a discovery process that reinforces mathematics, the haphazardly planned use of manipulatives can be destructive. An essential adjunct to “hands-on” mathematics is an effort to organize ideas and develop the capacity for mathematical thought and reason. Experiments performed under the tutelage of unskilled guides can lead students into a chaotic jungle, one in which their minds become entangled in an underbrush of mismatched concepts to which they, their parents, and their future teachers will be hard pressed to bring order.

---

NYC MATH WARS
NYC Honest Open Logical Debate (NYC HOLD)On Math Reform
Elizabeth Carson

November 27, 2000

New York, NY. NYU mathematicians speak out against controversial new math programs being taught in NYC’s premiere School District 2 in Manhattan. Community School Board 2 will hear the professors’ preliminary report at the calendar meeting on Tuesday, November 28, 6:30 pm at 333 7^{th Ave, Seventh Floor.}

^{A groundswell of parent concern over how math is being taught in District 2 schools led to a front page New York Times article last spring, “The New, Flexible Math Meets Parent Rebellion.” CBS Weekend News brought national attention to the local struggle last May in the segment, “New, New Math = Controversy.”}

^{District 2 parents have now gained the support of mathematicians at NYU and the University of Rochester in grieving their concerns to District officials. The professors are scheduled to take part in a Math Forum sponsored by Community School Board 2 scheduled for March 1, 2001.}

^{The controversy in District 2 has gained the attention of Schools Chancellor Harold O. Levy and members of his recently established Commission on Mathematics Education charged with investigating how math is being taught in schools across the city.}

^{District 2 pilots the latest wave of experimental math programs, called the “new, new math,” echoing an earlier failed reform in the 60’s called “new math.” Memorization of math facts is no longer emphasized. Children are encouraged to use language to describe solutions and the way they feel about math.}

^{Community response in NYC and across the country has erupted in what have become known as the “math wars.” Critical parents, joined by mathematicians and scientists advocate clarity and balance in math reform: urging the inclusion of grade by grade goals, explicit teaching of standard procedures, basic skill building and rigor along with the inclusion of some of the creative exercises in the new programs. The pendulum has swung too far and must be corrected.}

^{One parent, Mark Schwartz, in testimony last year before the House Education and Workforce Committee stated: “If medical doctors experimented with our kids in the same fashion school districts do they would be in jail.” The hearings were held to review the US Department of Education’s endorsement of 10 of the experimental programs.}

^{Over 200 of the nation’s top mathematicians, including seven Nobel Laureates,} winners of the Fields Medal ^{the department heads at more than a dozen universities including Caltech, Stanford and Yale, responded to the federal endorsements with an open letter of protest to Secretary of Education Richard Riley, published in the Washington Post in November, 1999.}

^{“These programs are among the worst in existence,” said David Klein, a Cal State Northridge professor who was one of the letter’s authors. “To recommend these books as exemplary and promising would be joke if it weren’t so damaging.” Several of the cited programs, Interactive Mathematics Project (IMP),Connected Mathematics Project (CMP) and Everyday Mathematics are now being used in NYC schools.}

^{The new wave of math reform is based on a “constructivist” teaching philosophy; emphasizing creative exercises, hands-on projects and group work, with far less attention given to basic skills. Students are asked to ‘construct’ their own solutions. The use of calculators is encouraged. Teachers are instructed to serve as “facilitators” and are discouraged from explaining to students the standard solutions of basic arithmetic. Practice and drill have been eliminated. In higher grades algebra is de-emphasized. Many of the programs have no textbooks.}

^{Members of Chancellor Levy’s Math Commission will consider the new programs, which are being used in over 60% of NYC schools, including roughly 50 of the city’s weakest, which comprise the Chancellor’s District . Plans are set to expand implementation into more schools. One of the most controversial programs, the Interactive Mathematics Project (IMP), will be mandated in Bronx High Schools beginning next year.}

^{Professor Richard Askey, who holds an endowed chair in math at the University of Wisconsin, explained his motives in co-authoring the protest letter to Secretary Riley, “I’m hoping to provide ammunition for teachers who are under pressure to adopt some of these programs.”}

^{High math scores remained stable in some of the privileged District 2 schools, last year; though some schools’ scores dropped significantly. Across the rest of the city, 75% of eighth grade students failed the state math test ; 66% failed the city math tests. Answers are critical at a time when graduation requires students pass a new math Regents exam.}

^{It was the introduction of Connected Mathematics Project (CMP) and Investigations in Number Data and Space (TERC), two of the District 2 programs, that sparked the initial parent revolt that led to the California Math Wars.} Six parents in Plano,Texas have filed suit in federal court against their local school district after parent requests for an alternative to CMP were denied. ^{A nuclear physicist in Okemos Michigan led the local campaign against CMP. The use of TERC in one school system in Massachusetts prompted members of the Harvard Mathematics Department to issue a public protest. Parents in Reading Massachusetts fought the adoption of Everyday Math.}

^{The “new, new math” programs are based on the 1989 National Council of Teachers of Mathematics (NCTM) Standards. Frank Allen, past president of the NCTM and Emeritus professor of Mathematics at Elmhurst College comments: “NCTM leaders must admit that they have urged the application, on a national scale of highly controversial methods of teaching before they have been adequately debated or understood and before researchers have verified them by well-controlled and replicated studies.”}

^{=============}

November 27, 2000

To: Harold O. Levy, Chancellor

Dr Judith Rizzo, Deputy Chancellor for Instruction

Burton Sacks, Chief Executive for Community School District Affairs

Dr Irving Haimer, Member, Board of Education, Manhattan Representative

Trudy Irwin, Director of Education, Office of the Manhattan Borough President

From: NYC HOLD, Steering Committee

Re: Community School Board 2 Calendar Meeting November 28, 2000

We respectfully invite you to attend the calendar meeting of Community School Board 2 in Manhattan scheduled for Tuesday November 28, beginning at 6:30 pm. During the public session the Board will hear comment on District 2’s K-12 math programs. Distinguished faculty of the Courant Institute of Mathematical Sciences at New York University and concerned parents plan to speak.

The District’s mathematics reform is one component of the systemic instructional reform initiated by Anthony Alvarado in the previous decade. District 2 prides itself in being a leader in education reform in New York City. Our District piloted the New Standards Performance Standards in Mathematics which were adopted city-wide last year.

The implications of the relative success of our District’s math reform for the entire NYC school system should be evident.

In our District, parent concern with their children’s progress in mathematics is escalating; worries are being strongly voiced about aspects of the new math programs just as full implementation takes hold. School year 1999-2000 marked the first year the new K- 8 programs, Investigations in Number Data and Space (TERC) and Connected Mathematics Project (CMP) were mandated.

Parents are tutoring in record numbers. For many of their children, the new constructivist programs fail to provide a sound foundation in basic arithmetic. Parents recognize the value of the new programs’ creative, hands-on classroom and homework activities. However, many question the extensive time devoted to such activities, ostensibly at the expense of explicit teaching of the standard procedures and practice of skills. The absence of textbooks has exacerbated the situation.

Concerned parents have reached out to the NYU mathematics community in District 2 for analysis and opinion of our programs in hopes of finding ways to amend or extend the implemented programs to reach a better balance in the math instruction.

There are clearly implications in the course the District 2 community takes for the work of the newly appointed Commission on Mathematics Education to review math programs city-wide. We hope our experiences in District 2 can contribute to the assessment of the Committee.

Parents, teachers and administrators in District 2 share a well earned pride in our exemplary schools and strong educational community. Parents see on a daily basis, through their children’s work and enthusiasm for learning, the results of the skill and commitment of the teaching staff and administrators in our schools. Parents appreciate the formidable challenges district administrators face in guiding systemic reform in a large and diverse community school district.

We look forward to an ongoing and meaningful partnership in District 2.

We sincerely hope you will take an interest in our concerns and efforts to advance the course of math education for our children.

Sincerely,

Steering Committee
NYC HOLD

Elizabeth Carson
Christine Larson
Garry Dobbins
Margaret Hunnewell
Mary Somoza, Member, CSB #2
Granville Leo Stevens, Esq
Maureen McAndrew, DDS
Michael Weinberg

Rocky Mountain News

Seebach: Race-gap study launches 3-stage rant about readers

April 30, 2005

Everybody knows that American blacks and Hispanics are at a disadvantage to whites and Asians both in education and income. Three economists have written a paper demonstrating that the patterns of disadvantages for blacks and for Hispanics are very different, raising questions about the explanations often given for those disadvantages.

In the authors’ own words, here are some take- away points:

• “For black males, controlling for an early measure of ability cuts the black-white wage gap in 1990 by 76 percent. For Hispanic males, controlling for ability essentially eliminates the wage gap with whites. For women the results are even more striking. Wage gaps are actually reversed, and controlling for ability produces higher wages for minority females.”

• “When we control for the effects of home and family environments on test scores, the Hispanic-white test score gap either decreases or is constant over time while the black-white test score tends to widen with age.”

• “Hispanic children start with cognitive and noncognitive deficits similar to those of black children. They also grow up in similar disadvantaged environments, and are likely to attend schools of similar quality. Hispanics have substantially less schooling than blacks. Nevertheless, the ability growth by years of schooling is much higher for Hispanics than blacks. By the time they reach adulthood, Hispanics have significantly higher test scores than blacks.”

• “Our analysis of the Hispanic data illuminates the traditional study of black-white differences and casts doubt on many conventional explanations of these differences since they do not apply to Hispanics who also suffer from many of the same disadvantages.”

I know this is contrary to just about everything you’ve heard or read, so you’re asking, “Who are these people?” They’re Pedro Carneiro, University College London; James J. Heckman, University of Chicago, American Bar Foundation and University College London (and winner of the 2000 Nobel Prize in economics for developing the kind of technical statistical analysis that undergirds this paper) and Dimitriy V. Masterov. The paper was written for the Institute for Labor Market Policy Evaluation, a part of the Swedish Ministry of Industry, Employment and Communications, in Uppsala, Sweden.

The paper is “Labor market discrimination and racial differences in premarket factors” and it’s at www.ifau.se/swe/ pdf2005/wp05-03.pdf on the Web.

They don’t argue against current policies on affirmative action – though they certainly could, based on their evidence – merely that policies addressing very early skill gaps are likely to do more good than additional affirmative action policies aimed at the workplace.

One possible explanation of persistent wage gaps is that there is “pervasive labor market discrimination against minorities.” Another, which they observe is equally plausible, is that “Minorities may bring less skill and ability to the market.” And of course both could be true in varying degrees, but I think this is the most important thing they say: “The two polar interpretations of market wage gaps have profoundly different policy implications.”

And how. So if you’re a policy-maker, Go Read The Whole Thing.

Now, since I have room for only a tiny bit of what’s significant in this paper anyway, I’m going to address a different issue that invariably comes up when I write about something so contrary to received opinion.

OK, (/turn rant on/) don’t waste your time writing me that I “haven’t considered” whatever particular bee is buzzing around your bonnet. You have no information about what I have considered; you know only what I have mentioned. And let me tell you, when I’m writing an 800-word summary of a highly technical 50-page paper bristling with statistical analysis, there’s a lot I’ve considered that I don’t mention. There’s even more data that the researchers have considered that they don’t mention – although the existence of Web appendices to scholarly papers has ameliorated that problem.

Next, don’t think you’re being erudite by citing some cliche about “lies, damned lies or statistics,” which I understand is properly credited to Benjamin Disraeli, but is often attributed to Mark Twain. Yes, it is possible to lie with statistics – there’s a charming and useful little book with that in the title – but it’s a lot harder to lie with statistics than without them.

Case in point, the current flap over the number of deaths statistically attributable to obesity. If you’re one of those people who fatuously asserts that “you can prove anything with statistics” I challenge you to find me a peer-reviewed journal article proving that smoking enhances longevity, or that women are taller than men.

Last, don’t talk about motives. You have no evidence about my motives aside from what I tell you – and I could be wrong about that; lots of people are. Even if we were both right about my motives, it would have no bearing on the cogency of my arguments, which do not adduce them; that’s the ad hominem fallacy. “Fallacy,” please note, which means that even if your premises are correct, your conclusion may be wrong. (/Turn rant off/)

Oh, I feel much better now. Excellent paper.

Linda Seebach is an editorial writer for the News. She can be reached by telephone at (303) 892-2519 or by e-mail at seebach@RockyMountainNews.com.

MORE SEEBACH COLUMNS »

Copyright 2005, Rocky Mountain News. All Rights Reserved.

NYC MATH WARS
NYC Honest Open Logical Debate (NYC HOLD)On Math Reform
Elizabeth Carson

November 27, 2000

New York, NY. NYU mathematicians speak out against controversial new math programs being taught in NYC’s premiere School District 2 in Manhattan. Community School Board 2 will hear the professors’ preliminary report at the calendar meeting on Tuesday, November 28, 6:30 pm at 333 7^{th Ave, Seventh Floor.}

^{A groundswell of parent concern over how math is being taught in District 2 schools led to a front page New York Times article last spring, “The New, Flexible Math Meets Parent Rebellion.” CBS Weekend News brought national attention to the local struggle last May in the segment, “New, New Math = Controversy.”}

^{District 2 parents have now gained the support of mathematicians at NYU and the University of Rochester in grieving their concerns to District officials. The professors are scheduled to take part in a Math Forum sponsored by Community School Board 2 scheduled for March 1, 2001.}

^{The controversy in District 2 has gained the attention of Schools Chancellor Harold O. Levy and members of his recently established Commission on Mathematics Education charged with investigating how math is being taught in schools across the city.}

^{District 2 pilots the latest wave of experimental math programs, called the “new, new math,” echoing an earlier failed reform in the 60’s called “new math.” Memorization of math facts is no longer emphasized. Children are encouraged to use language to describe solutions and the way they feel about math.}

^{Community response in NYC and across the country has erupted in what have become known as the “math wars.” Critical parents, joined by mathematicians and scientists advocate clarity and balance in math reform: urging the inclusion of grade by grade goals, explicit teaching of standard procedures, basic skill building and rigor along with the inclusion of some of the creative exercises in the new programs. The pendulum has swung too far and must be corrected.}

^{One parent, Mark Schwartz, in testimony last year before the House Education and Workforce Committee stated: “If medical doctors experimented with our kids in the same fashion school districts do they would be in jail.” The hearings were held to review the US Department of Education’s endorsement of 10 of the experimental programs.}

^{Over 200 of the nation’s top mathematicians, including seven Nobel Laureates,} winners of the Fields Medal ^{the department heads at more than a dozen universities including Caltech, Stanford and Yale, responded to the federal endorsements with an open letter of protest to Secretary of Education Richard Riley, published in the Washington Post in November, 1999.}

^{“These programs are among the worst in existence,” said David Klein, a Cal State Northridge professor who was one of the letter’s authors. “To recommend these books as exemplary and promising would be joke if it weren’t so damaging.” Several of the cited programs, Interactive Mathematics Project (IMP),Connected Mathematics Project (CMP) and Everyday Mathematics are now being used in NYC schools.}

^{The new wave of math reform is based on a “constructivist” teaching philosophy; emphasizing creative exercises, hands-on projects and group work, with far less attention given to basic skills. Students are asked to ‘construct’ their own solutions. The use of calculators is encouraged. Teachers are instructed to serve as “facilitators” and are discouraged from explaining to students the standard solutions of basic arithmetic. Practice and drill have been eliminated. In higher grades algebra is de-emphasized. Many of the programs have no textbooks.}

^{Members of Chancellor Levy’s Math Commission will consider the new programs, which are being used in over 60% of NYC schools, including roughly 50 of the city’s weakest, which comprise the Chancellor’s District . Plans are set to expand implementation into more schools. One of the most controversial programs, the Interactive Mathematics Project (IMP), will be mandated in Bronx High Schools beginning next year.}

^{Professor Richard Askey, who holds an endowed chair in math at the University of Wisconsin, explained his motives in co-authoring the protest letter to Secretary Riley, “I’m hoping to provide ammunition for teachers who are under pressure to adopt some of these programs.”}

^{High math scores remained stable in some of the privileged District 2 schools, last year; though some schools’ scores dropped significantly. Across the rest of the city, 75% of eighth grade students failed the state math test ; 66% failed the city math tests. Answers are critical at a time when graduation requires students pass a new math Regents exam.}

^{It was the introduction of Connected Mathematics Project (CMP) and Investigations in Number Data and Space (TERC), two of the District 2 programs, that sparked the initial parent revolt that led to the California Math Wars.} Six parents in Plano,Texas have filed suit in federal court against their local school district after parent requests for an alternative to CMP were denied. ^{A nuclear physicist in Okemos Michigan led the local campaign against CMP. The use of TERC in one school system in Massachusetts prompted members of the Harvard Mathematics Department to issue a public protest. Parents in Reading Massachusetts fought the adoption of Everyday Math.}

^{The “new, new math” programs are based on the 1989 National Council of Teachers of Mathematics (NCTM) Standards. Frank Allen, past president of the NCTM and Emeritus professor of Mathematics at Elmhurst College comments: “NCTM leaders must admit that they have urged the application, on a national scale of highly controversial methods of teaching before they have been adequately debated or understood and before researchers have verified them by well-controlled and replicated studies.”}

^{=============}

November 27, 2000

To: Harold O. Levy, Chancellor

Dr Judith Rizzo, Deputy Chancellor for Instruction

Burton Sacks, Chief Executive for Community School District Affairs

Dr Irving Haimer, Member, Board of Education, Manhattan Representative

Trudy Irwin, Director of Education, Office of the Manhattan Borough President

From: NYC HOLD, Steering Committee

Re: Community School Board 2 Calendar Meeting November 28, 2000

We respectfully invite you to attend the calendar meeting of Community School Board 2 in Manhattan scheduled for Tuesday November 28, beginning at 6:30 pm. During the public session the Board will hear comment on District 2’s K-12 math programs. Distinguished faculty of the Courant Institute of Mathematical Sciences at New York University and concerned parents plan to speak.

The District’s mathematics reform is one component of the systemic instructional reform initiated by Anthony Alvarado in the previous decade. District 2 prides itself in being a leader in education reform in New York City. Our District piloted the New Standards Performance Standards in Mathematics which were adopted city-wide last year.

The implications of the relative success of our District’s math reform for the entire NYC school system should be evident.

In our District, parent concern with their children’s progress in mathematics is escalating; worries are being strongly voiced about aspects of the new math programs just as full implementation takes hold. School year 1999-2000 marked the first year the new K- 8 programs, Investigations in Number Data and Space (TERC) and Connected Mathematics Project (CMP) were mandated.

Parents are tutoring in record numbers. For many of their children, the new constructivist programs fail to provide a sound foundation in basic arithmetic. Parents recognize the value of the new programs’ creative, hands-on classroom and homework activities. However, many question the extensive time devoted to such activities, ostensibly at the expense of explicit teaching of the standard procedures and practice of skills. The absence of textbooks has exacerbated the situation.

Concerned parents have reached out to the NYU mathematics community in District 2 for analysis and opinion of our programs in hopes of finding ways to amend or extend the implemented programs to reach a better balance in the math instruction.

There are clearly implications in the course the District 2 community takes for the work of the newly appointed Commission on Mathematics Education to review math programs city-wide. We hope our experiences in District 2 can contribute to the assessment of the Committee.

Parents, teachers and administrators in District 2 share a well earned pride in our exemplary schools and strong educational community. Parents see on a daily basis, through their children’s work and enthusiasm for learning, the results of the skill and commitment of the teaching staff and administrators in our schools. Parents appreciate the formidable challenges district administrators face in guiding systemic reform in a large and diverse community school district.

We look forward to an ongoing and meaningful partnership in District 2.

We sincerely hope you will take an interest in our concerns and efforts to advance the course of math education for our children.

Sincerely,

Steering Committee
NYC HOLD

Elizabeth Carson
Christine Larson
Garry Dobbins
Margaret Hunnewell
Mary Somoza, Member, CSB #2
Granville Leo Stevens, Esq
Maureen McAndrew, DDS
Michael Weinberg

---

MATH PROBLEMS
Why the U.S. Department of Education’s
recommended math programs don’t add up

By David Klein

---

What constitutes a good K-12 mathematics program? Opinions differ. In October 1999, the U.S. Department of Education released a report designating 10 math programs as “exemplary” or “promising.” The following month, I sent an open letter to Education Secretary Richard W. Riley urging him to withdraw the department’s recommendations. The letter was coauthored by Richard Askey of the University of Wisconsin at Madison, R. James Milgram of Stanford University, and Hung-Hsi Wu of the University of California at Berkeley, along with more than 200 other cosigners. With financial backing from the Packard Humanities Institute, we published the letter as a full-page ad in the Washington Post on Nov. 18, 1999, with as many of the endorsers’ names and affiliations as would fit on the page. Among them are many of the nation’s most accomplished scientists and mathematicians. Department heads at more than a dozen universities–including Caltech, Stanford, and Yale–along with two former presidents of the Mathematical Association of America also added their names in support. With new endorsements since publication, there are now seven Nobel laureates and winners of the Fields Medal, the highest award in mathematics. The open letter was covered by several newspapers and journals, including American School Board Journal (February, page 16).

Although a clear majority of cosigners are mathematicians and scientists, it is sometimes overlooked that experienced education administrators at the state and national level, as well as educational psychologists and education researchers, also endorsed the letter. (A complete list is posted at http://www.mathematicallycorrect.com.)

University professors and public education leaders are not the only ones who have reservations about these programs. Thousands of parents and teachers across the nation seek alternatives to them, often in opposition to local school boards and superintendents. Mathematically Correct, an influential Internet-based parents’ organization, came into existence several years ago because the children of the organization’s founders had no alternative to the now “exemplary” program, College Preparatory Mathematics, or CPM. In Plano, Texas, 600 parents are suing the school district because of its exclusive use of the Connected Mathematics Project, or CMP, another “exemplary” program. I have received hundreds of requests for help by parents and teachers because of these and other programs now promoted by the Education Department (ED). In fact, it was such pleas for help that motivated me and my three coauthors to write the open letter.

Common problems

The mathematics programs criticized by the open letter have common features. For example, they tend to overemphasize data analysis and statistics, which typically appear year after year, with redundant presentations. The far more important areas of arithmetic and algebra are radically de-emphasized. Many of the so-called higher-order thinking projects are just aimless activities, and genuine illumination of important mathematical ideas is rare. There is a near obsession with calculators, and basic skills are given short shrift and sometimes even disparaged. Overall, these curricula are watered-down math programs. The same educational philosophy that gave rise to the whole-language approach to reading is part of ED’s agenda for mathematics. Systematic development of skills and concepts is replaced by an unstructured “holism.” In fact, during the mid-’90s, supporters of programs like these referred to their approach as “whole math.”

Disagreements over math curricula are often portrayed as “basic skills versus conceptual understanding.” Scientists and mathematicians, including many who signed the open letter to Secretary Riley, are described as advocates of basic skills, while professional educators are counted as proponents of conceptual understanding. Ironically, such a portrayal ignores the deep conceptual understanding of mathematics held by so many mathematicians. But more important, the notion that conceptual understanding in mathematics can be separated from precision and fluency in the execution of basic skills is just plain wrong.

In other domains of human activity, such as athletics or music, the dependence of high levels of performance on requisite skills goes unchallenged. A novice cannot hope to achieve mastery in the martial arts without first learning basic katas or exercises in movement. A violinist who has not mastered elementary bowing techniques and vibrato has no hope of evoking the emotions of an audience through sonorous tones and elegant phrasing. Arguably the most hierarchical of human endeavors, mathematics also depends on sequential mastery of basic skills.

The standard algorithms

The standard algorithms for arithmetic (that is, the standard procedures for addition, subtraction, multiplication, and division of numbers) are missing or abridged in ED’s recommended elementary school curricula. These omissions are inconsistent with the mainstream views of mathematicians.

In our open letter to Secretary Riley, we included an excerpt from a committee report published in the February 1998 Notices of the American Mathematical Society. The committee was appointed by the American Mathematical Society to advise the National Council of Teachers of Mathematics (NCTM). Part of its report discusses the standard algorithms of arithmetic. “We would like to emphasize that the standard algorithms of arithmetic are more than just ‘ways to get the answer’–that is, they have theoretical as well as practical significance,” the report states. “For one thing, all the algorithms of arithmetic are preparatory for algebra, since there are (again, not by accident, but by virtue of the construction of the decimal system) strong analogies between arithmetic of ordinary numbers and arithmetic of polynomials.”

This statement deserves elaboration. How could the standard algorithms of arithmetic be related to algebra? For concreteness, consider the meaning in terms of place value of 572:

572 = 5 (10²) + 7(10) + 2

Now compare the right side of this equation to the polynomial,

5x² + 7x + 2.

The two are identical when x = 10. This connection between whole numbers and polynomials is general and extends to arithmetic operations. Addition, subtraction, multiplication, and division of polynomials is fundamentally the same as for whole numbers. In arithmetic, extra steps such as “regrouping” are needed since x = 10 allows for simplifications. The standard algorithms incorporate both the polynomial operations and the extra steps to account for the specific value, x = 10. Facility with the standard operations of arithmetic, together with an understanding of why these algorithms work, is important preparation for algebra.

The standard long division algorithm is particularly shortchanged by the “promising” curricula. It is preparatory for division of polynomials and, at the college level, division of “power series,” a useful technique in calculus and differential equations. The standard long division algorithm is also needed for a middle school topic. It is fundamental to an understanding of the difference between rational and irrational numbers, an indisputable example of conceptual understanding. It is essential to understand that rational numbers (that is, ratios of whole numbers like 3/4) and their negatives have decimal representations that exhibit recurring patterns. For example: 1/3 = .333…, where the ellipses indicate that the numeral 3 repeats forever. Likewise, 1/2 = .500… and 611/4950 = .12343434….

In the last equation, the digits 34 are repeated without end, and the repeating block in the decimal for 1/2 consists only of the digit for zero. It is a general fact that all rational numbers have repeating blocks of numerals in their decimal representations, and this can be understood and deduced by students who have mastered the standard long division algorithm. However, this important result does not follow easily from other “nonstandard” division algorithms featured by some of ED’s model curricula.

A different but still elementary argument is required to show the converse–that any decimal with a repeating block is equal to a fraction. Once this is understood, students are prepared to understand the meaning of the term “irrational number.” Irrational numbers are the numbers represented by infinite decimals without repeating blocks. In California, seventh-grade students are expected to understand this.

It is worth emphasizing that calculators are utterly useless in this context, not only in establishing the general principles, but even in logically verifying the equations. This is partly because calculator screens cannot display infinite decimals, but more important, calculators cannot reason. The “exemplary” middle school curriculum CMP nevertheless ignores the conceptual issues, bypassing the long division algorithm and substituting calculators and faulty inductive reasoning instead.

Steven Leinwand of the Connecticut Department of Education was a member of the expert panel that made final decisions on ED’s “exemplary” and “promising” math curricula. He was also a member of the advisory boards for two programs found to be “exemplary” by the panel: CMP and the Interactive Mathematics Program. In a Feb. 9, 1994, article in Education Week, he wrote: “It’s time to recognize that, for many students, real mathematical power, on the one hand, and facility with multidigit, pencil-and-paper computational algorithms, on the other, are mutually exclusive. In fact, it’s time to acknowledge that continuing to teach these skills to our students is not only unnecessary, but counterproductive and downright dangerous.”

Mr. Leinwand’s influential opinions are diametrically opposed to the mainstream views of practicing scientists and mathematicians, as well as the general public, but they have found fertile soil in the government’s “promising” and “exemplary” curricula.

Calculators

According to the Third International Mathematics and Science Study, or TIMSS, the use of calculators in U.S. fourth-grade mathematics classes is about twice the international average. Teachers of 39 percent of U.S. students report that students use calculators at least once or twice a week. In six of the seven top-scoring nations, on the other hand, teachers of 85 percent or more of the students report that students never use calculators in class.

Even at the eighth-grade level, the majority of students from three of the top five scoring nations in the TIMSS study (Belgium, Korea, and Japan) never or rarely use calculators in math classes. In Singapore, which is also among the top five scoring countries, students do not use calculators until the seventh grade. Among the lower achieving nations, however, the majority of students from 10 of the 11 nations with scores below the international average–including the United States–use calculators almost every day or several times a week.

Of course, this negative correlation of calculator usage with achievement in mathematics does not imply a causal relationship. There are many variables that contribute to achievement in mathematics. On the other hand, it is foolhardy to ignore the problems caused by calculators in schools. In a Sept. 17, 1999, Los Angeles Times editorial titled “L.A.’s Math Program Just Doesn’t Add Up,” Milgram and I recommended that calculators not be used at all in grades K-5 and only sparingly in higher grades. Certainly there are isolated, beneficial uses for calculators, such as calculating compound interest, a seventh-grade topic in California. Science classes benefit from the use of calculators because it is necessary to deal with whatever numbers nature gives us, but conceptual understanding in mathematics is often best facilitated through the use of simple numbers. Moreover, fraction arithmetic, an important prerequisite for algebra, is easily undermined by the use of calculators.

Specific shortcomings

A number of the programs on ED’s list have specific shortcomings–many involving use of calculators. For example, a “promising” curriculum called Everyday Mathematics says calculators are “an integral part of Kindergarten Everyday Mathematics” and urges the use of calculators to teach kindergarten students how to count. There are no textbooks in this K-6 curriculum, and even if the program were otherwise sound, this is a serious shortcoming. The standard algorithm for multiplying two numbers has no more status or prominence than an Ancient Egyptian algorithm presented in one of the teacher’s manuals. Students are never required to use the standard long division algorithm in this curriculum, or even the standard algorithm for multiplication.

Calculator use is also ubiquitous in the “exemplary” middle school program CMP. A unit devoted to discovering algorithms to add, subtract, and multiply fractions (“Bits and Pieces II”) gives the inappropriate instruction, “Use your calculator whenever you need it.” These topics are poorly developed, and division of fractions is not covered at all. A quiz for seventh-grade CMP students asks them to find the “slope” and “y-intercept” of the equation 10 = x – 2.5, and the teacher’s manual explains that this equation is a special case of the linear equation y = x – 2.5, when y = 10, and concludes that the slope is therefore 1 and the y-intercept is -2.5. This is not only false, but is so mathematically unsound as to undermine the authority of classroom teachers who know better.

College Preparatory Math (CPM), a high school program, also requires students to use calculators almost daily. The principal technique in this series is the so-called guess-and-check method, which encourages repeated guessing of answers over the systematic development of standard mathematical techniques. Because of the availability of calculators that can solve equations, the introduction to the series explains that CPM puts low emphasis on symbol manipulation and that CPM differs from traditional mathematics courses both in the mathematics that is taught and how it is taught. In one section, students watch a candle burn down for an hour while measuring its length versus the time and then plotting the results. In a related activity, students spend a whole class period on the athletic field making human coordinate graphs. These activities are typical of the time sacrificed to simple ideas that can be understood more efficiently through direct explanation. But in CPM, direct instruction is systematically discouraged in favor of group work. Teachers are told that as “rules of thumb,” they should “never carry or grab a writing implement” and they should “usually respond with a question.” Algebra tiles are used frequently, and the important distributive property is poorly presented and underemphasized.

Another program, Number Power–a “promising” curriculum for grades K-6–was submitted to the California State Board of Education for adoption in California. Two Stanford University mathematics professors serving on the state’s Content Review Panel wrote a report on the program that is now a public document. Number Power, they wrote, “is meant as a partial program to supplement a regular basic program. There is a strong emphasis on group projects–almost the entire program. Heavy use of calculators. Even as a supplementary program, it provides such insufficient coverage of the [California] Standards that it is unacceptable. This holds for all grade levels and all strands, including Number Sense, which is the only strand that is even partially covered.”

The report goes on to note, “It is explicitly stated that the standard algorithms for addition, subtraction, and multiplication are not taught.” Like CMP and Everyday Math, Number Power was rejected for adoption by the state of California.

Interactive Mathematics Program, or IMP, an “exemplary” high school curriculum, has such a weak treatment of algebra that the quadratic formula, normally an eighth- or ninth-grade topic, is postponed until the 12th grade. Even though probability and statistics receive greater emphasis in this program, the development of these topics is poor. “Expected value,” a concept of fundamental importance in probability and statistics, is never even correctly defined. The Teacher’s Guide for “The Game of Pig,” where expected value is treated, informs teachers that “expected value is one of the unit’s primary concepts,” yet teachers are instructed to tell their students that “the concept of expected value is nothing new … [but] the use of such complex terminology makes it easier to state complex ideas.” (For a correlation of lowered SAT scores with the use of IMP, see Milgram’s paper at ftp://math.stanford.edu/pub/papers/milgram.)

Core-Plus Mathematics Project is another “exemplary” high school program that radically de-emphasizes algebra, with unfortunate results. Even Hyman Bass–a well-known supporter of NCTM-aligned programs and a harsh critic of the open letter to Secretary Riley–has conceded the program has problems. “I have some reservations about Core Plus, for what I consider too shallow a coverage of traditional algebra, and a focus on highly contextualized work that goes beyond my personal inclinations,” he wrote in a nationally circulated e-mail message. “These are only my personal views, and I do not know about its success with students.”

Milgram analyzed the program’s effect on students in a top-performing high school in “Outcomes Analysis for Core Plus Students at Andover High School: One Year Later,” based on a statistical study by G. Bachelis of Wayne State University. According to Milgram, “…there was no measure represented in the survey, such as ACT scores, SAT Math scores, grades in college math courses, level of college math courses attempted, where the Andover Core Plus students even met, let alone surpassed the comparison group [which used a more traditional program].”

And then there is MathLand, a K-6 curriculum that ED calls “promising” but that is perhaps the most heavily criticized elementary school program in the nation. Like Everyday Math, it has no textbooks for students in any of the grades. The teacher’s manual urges teachers not to teach the standard algorithms of arithmetic for addition, subtraction, multiplication, and division. Rather, students are expected to invent their own algorithms. Numerous and detailed criticisms, including data on lowered test scores, appear at http://www.mathematicallycorrect.com.

How could they be so wrong?

Perhaps Galileo wondered similarly how the church of Pope Urban VIII could be so wrong. The U.S. Department of Education is not alone in endorsing watered-down, and even defective, math programs. The NCTM has also formally endorsed each of the U.S. Department of Education’s model programs (http://www.nctm.org/rileystatement.htm), and the National Science Foundation (Education and Human Resources Division) funded several of them. How could such powerful organizations be wrong?

These organizations represent surprisingly narrow interests, and there is a revolving door between them. Expert panel member Steven Leinwand, whose personal connections with “exemplary” curricula have already been noted, is also a member of the NCTM board of directors. Luther Williams, who as assistant director of the NSF approved the funding of several of the recommended curricula, also served on the expert panel that evaluated these same curricula. Jack Price, a member of the expert panel is a former president of NCTM, and Glenda Lappan, the association’s current president, is a coauthor of the “exemplary” program CMP.

Aside from institutional interconnections, there is a unifying ideology behind “whole math.” It is advertised as math for all students, as opposed to only white males. But the word all is a code for minority students and women (though presumably not Asians). In 1996, while he was president of NCTM, Jack Price articulated this view in direct terms on a radio show in San Diego: “What we have now is nostalgia math. It is the mathematics that we have always had, that is good for the most part for the relatively high socioeconomic anglo male, and that we have a great deal of research that has been done showing that women, for example, and minority groups do not learn the same way. They have the capability, certainly, of learning, but they don’t. The teaching strategies that you use with them are different from those that we have been able to use in the past when … we weren’t expected to graduate a lot of people, and most of those who did graduate and go on to college were the anglo males.”

Price went on to say: “All of the research that has been done with gender differences or ethnic differences has been–males for example learn better deductively in a competitive environment, when–the kind of thing that we have done in the past. Where we have found with gender differences, for example, that women have a tendency to learn better in a collaborative effort when they are doing inductive reasoning.” (A transcript of the show is online at (http://mathematicallycorrect.com/roger.htm.)

I reject the notion that skin color or gender determines whether students learn inductively as opposed to deductively and whether they should be taught the standard operations of arithmetic and essential components of algebra. Arithmetic is not only essential for everyday life, it is the foundation for study of higher level mathematics. Secretary Riley–and educators who select mathematics curricula–would do well to heed the advice of the open letter.

David Klein is a professor of mathematics at California State University at Northridge.

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Marks of a good mathematics program

It is impossible to specify all of the characteristics of a sound mathematics program in only a few paragraphs, but a few highlights may be identified. The most important criterion is strong mathematical content that conforms to a set of explicit, high, grade-by-grade standards such as the California or Japanese mathematics standards. A strong mathematics program recognizes the hierarchical nature of mathematics and builds coherently from one grade to the next. It is not merely a sequence of interesting but unrelated student projects.

In the earlier grades, arithmetic should be the primary focus. The standard algorithms of arithmetic for integers, decimals, fractions, and percents are of central importance. The curriculum should promote facility in calculation, an understanding of what makes the algorithms work in terms of the base 10 structure of our number system, and an understanding of the associative, commutative, and distributive properties of numbers. These properties can be illustrated by area and volume models. Students need to develop an intuitive understanding for fractions. Manipulatives or pictures can help in the beginning stages, but it is essential that students eventually be able to compute easily using mathematical notation. Word problems should be abundant. A sound program should move students toward abstraction and the eventual use of symbols to represent unknown quantities.

In the upper grades, algebra courses should emphasize powerful symbolic techniques and not exploratory guessing and calculator-based graphical solutions.

There should be a minimum of diversions in textbooks. Children have enough trouble concentrating without distracting pictures and irrelevant stories and projects. A mathematics program should explicitly teach skills and concepts with appropriately designed practice sets. Such programs have the best chance of success with the largest number of students. The high-performing Japanese students spend 80 percent of class time in teacher-directed whole-class instruction. Japanese math books contain clear explanations, examples with practice problems, and summaries of key points. Singapore’s elementary school math books also provide good models. Among U.S. books for elementary school, Sadlier-Oxford’s Progress in Mathematics and the Saxon series through Math 87 (adopted for grade six in California), though not without defects, have many positive features.–D.K.

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For more information

Askey, Richard. “Knowing and Teaching Elementary Mathematics.” American Educator, Fall 1999, pp. 6-13; 49.

Ma, Liping. Knowing and Teaching Elementary Mathematics. Mahwah, N.J.: Lawrence Erlbaum, 1999.

Milgram, R. James. “A Preliminary Analysis of SAT-I Mathematics Data for IMP Schools in California.” ftp://math.stanford.edu/pub/papers/milgram

Milgram, R. James. “Outcomes Analysis for Core Plus Students at Andover High School: One Year Later.” ftp://math.stanford.edu/pub/papers/milgram/andover-report.htm

Wu, Hung-Hsi. “Basic Skills Versus Conceptual Understanding: A Bogus Dichotomy in Mathematics Education.” American Educator, Fall 1999, pp. 14-19; 50-52.

Seebach: Race-gap study launches 3-stage rant about readers

April 30, 2005

Everybody knows that American blacks and Hispanics are at a disadvantage to whites and Asians both in education and income. Three economists have written a paper demonstrating that the patterns of disadvantages for blacks and for Hispanics are very different, raising questions about the explanations often given for those disadvantages.

In the authors’ own words, here are some take- away points:

 “For black males, controlling for an early measure of ability cuts the black-white wage gap in 1990 by 76 percent. For Hispanic males, controlling for ability essentially eliminates the wage gap with whites. For women the results are even more striking. Wage gaps are actually reversed, and controlling for ability produces higher wages for minority females.”

• “When we control for the effects of home and family environments on test scores, the Hispanic-white test score gap either decreases or is constant over time while the black-white test score tends to widen with age.”

• “Hispanic children start with cognitive and noncognitive deficits similar to those of black children. They also grow up in similar disadvantaged environments, and are likely to attend schools of similar quality. Hispanics have substantially less schooling than blacks. Nevertheless, the ability growth by years of schooling is much higher for Hispanics than blacks. By the time they reach adulthood, Hispanics have significantly higher test scores than blacks.”

• “Our analysis of the Hispanic data illuminates the traditional study of black-white differences and casts doubt on many conventional explanations of these differences since they do not apply to Hispanics who also suffer from many of the same disadvantages.”

I know this is contrary to just about everything you’ve heard or read, so you’re asking, “Who are these people?” They’re Pedro Carneiro, University College London; James J. Heckman, University of Chicago, American Bar Foundation and University College London (and winner of the 2000 Nobel Prize in economics for developing the kind of technical statistical analysis that undergirds this paper) and Dimitriy V. Masterov. The paper was written for the Institute for Labor Market Policy Evaluation, a part of the Swedish Ministry of Industry, Employment and Communications, in Uppsala, Sweden.

The paper is “Labor market discrimination and racial differences in premarket factors” and it’s at www.ifau.se/swe/pdf2005/wp05-03.pdf on the Web.

They don’t argue against current policies on affirmative action – though they certainly could, based on their evidence – merely that policies addressing very early skill gaps are likely to do more good than additional affirmative action policies aimed at the workplace.

One possible explanation of persistent wage gaps is that there is “pervasive labor market discrimination against minorities.” Another, which they observe is equally plausible, is that “Minorities may bring less skill and ability to the market.” And of course both could be true in varying degrees, but I think this is the most important thing they say: “The two polar interpretations of market wage gaps have profoundly different policy implications.”

And how. So if you’re a policy-maker, Go Read The Whole Thing.

Now, since I have room for only a tiny bit of what’s significant in this paper anyway, I’m going to address a different issue that invariably comes up when I write about something so contrary to received opinion.

OK, (/turn rant on/) don’t waste your time writing me that I “haven’t considered” whatever particular bee is buzzing around your bonnet. You have no information about what I have considered; you know only what I have mentioned. And let me tell you, when I’m writing an 800-word summary of a highly technical 50-page paper bristling with statistical analysis, there’s a lot I’ve considered that I don’t mention. There’s even more data that the researchers have considered that they don’t mention – although the existence of Web appendices to scholarly papers has ameliorated that problem.

Next, don’t think you’re being erudite by citing some cliche about “lies, damned lies or statistics,” which I understand is properly credited to Benjamin Disraeli, but is often attributed to Mark Twain. Yes, it is possible to lie with statistics – there’s a charming and useful little book with that in the title – but it’s a lot harder to lie with statistics than without them.

Case in point, the current flap over the number of deaths statistically attributable to obesity. If you’re one of those people who fatuously asserts that “you can prove anything with statistics” I challenge you to find me a peer-reviewed journal article proving that smoking enhances longevity, or that women are taller than men.

Last, don’t talk about motives. You have no evidence about my motives aside from what I tell you – and I could be wrong about that; lots of people are. Even if we were both right about my motives, it would have no bearing on the cogency of my arguments, which do not adduce them; that’s the ad hominem fallacy. “Fallacy,” please note, which means that even if your premises are correct, your conclusion may be wrong. (/Turn rant off/)

Oh, I feel much better now. Excellent paper.

 

 

Linda Seebach is an editorial writer for the News. She can be reached by telephone at (303) 892-2519 or by e-mail at seebach@RockyMountainNews.com.

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Where’s the Math?
– DEBRA J. SAUNDERS
Sunday, October 17, 1999

 

THIS MONTH, the U.S. Department of Education came out with a list of 10 “exemplary” or “promising” math education programs. Kings County fourth-grade teacher Doug Swords was shocked at the department’s bad choices.

Some three years ago, his school district adopted MathLand, a math curriculum that prefers not to give lessons with “predetermined numerical results.” The department of Educrats, oops, I mean, Education, rated MathLand as “promising.” Today, he said 14 out of 18 teachers use MathLand only as a supplement. “I stashed away my Addison-Wesley textbooks, as did a few other teachers,” he explained.

Do you teach your students how to multiply? I asked him. (You wouldn’t think that would be something I’d have to ask, but these days, it is.) Yes, he said. Is MathLand helpful in teaching kids to multiply? “No, quite frankly,” Swords answered.

UC Berkeley math professor Hung-Hsi Wu couldn’t believe the department described MathLand as “promising.” He’d describe MathLand as “execrable.”

Or how about: “I can’t believe it’s math class.” A second-grade MathLand exercise called Fantasy Lunch instructs students to think up their fantasy lunch, draw it on paper, then cut out the “food” and place their drawings into a bag.

A frantic teacher wrote to me two years ago, furious that she had spent 75 minutes on that exercise and there was no math in it. It was “like therapy,” she said. On more than one occasion, her students asked her, “Can we do some real math now?”

Wu had problems with the other nine picks as well. While there were things he liked about the high school programs, they lacked what he called “mathematical closure. You start something, you ought to finish it.”

He said almost all of his students took more traditional math classes not cited as “exemplary” or “promising” by the Department of Education. That wouldn’t surprise Melissa Lynn, who got As in high-school math, then placed in the bottom 1 percent in the University of Michigan math placement test. She blames the Core-Plus program which the department rated as “exemplary.” “It had very good intentions, and wanted you to apply real principles to real life scenarios,” she explained this spring, “but it was missing the crucial element of algebra.“

Wayne Bishop, a math professor at Cal State L.A. who is the Ralph Nader of math curricula, sees the department’s move as a reaction against California’s return to math sanity — after a mad fling when state educrats embraced “there is no right answer” new-new math curricula.

He’s right. The selection panel appoint ed by the department had as a main criterion that the math series ascribe to trendy standards put out by the National Council for Teachers of Mathematics (NCTM).

Don’t ask me why. Last year Bishop looked at the scores of some of the students subjected to the brilliance of new- new math wizards. In 1995, NCTM Chairman Jack Price boasted about a program on which he worked. Turns out, Price’s star school ranked in the bottom quartile nationally in the STAR test last year. Only 12 percent of the school’s eighth graders scored above the national average. Price called that a successful program.

The department cited data that show schools whose test scores improved with MathLand. Bishop isn’t impressed. “They appear to have excluded data where MathLand scores dropped,” he noted.

An administrator from an urban district that stopped using MathLand had just visited a school that had seen a 27 percent increase in its math scores after buying a traditional math series that didn’t rate in the department’s Top 10. Under ideal circumstances, he said, MathLand could work, but urban districts don’t have too many ideal circumstances.

Bill Evers of Stanford’s Hoover Institution called the department’s Top-10 picks “unconditional surrender to fuzziness.”

Fuzziness? The department praised one K-6 math program because, “Features include problem solving; linking past experience to new concepts; sharing ideas; developing concept readiness through hands-on explorations; cooperative learning through small-group activities; and home-school partnerships.”

Sounds more like marriage counseling than math class.

The problem: It’s not the kids who need counseling here. It’s the adults who care so little about children’s success that they would assert that Fantasy Lunch makes for a “promising” math program.

 

You can reach Debra J. Saunders on The Gate at sfgate.com.

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©2005 San Francisco Chronicle