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Reproduced with permission from the April 2000 issue
of American School
Board Journal.
Copyright © 2000, National School Boards Association.
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.
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 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:
Now compare the right side of this equation to the polynomial,
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.
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.
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.
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.
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.
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.
