State Mathematics Standards: An Appraisal of Math Standards in 46 States, the District of Columbia, and Japan
June 1, 1756.
Drank Tea at the Majors. The Reasoning of Mathematics is founded on certain and infallible Principles. Every Word they Use, conveys a determinate Idea, and by accurate Definitions they excite the same Ideas in the mind of the Reader that were in the mind of the Writer. When they have defined the Terms they intend to make use of, they premise a few Axioms, or Self evident Principles, that every man must assent to as soon as proposed. They then take for granted certain Postulates, that no one can deny them, such as, that a right Line may be drawn from one given Point to another, and from these plain simple Principles, they have raised most astonishing Speculations, and proved the Extent of the human mind to be more spacious and capable than any other Science.
State Mathematics StandardsAn Appraisal of Math Standards in 46 States, the District of Columbia, and Japan
Henry Alder, Ph.D., Professor Emeritus of Mathematics, University of California, Davis, California
Harold Stevenson, Ph.D., Professor of Psychology, University of Michigan, Ann Arbor, Michigan
National Report Card
Numerical Ratings for the States
The Thomas B. Fordham Foundation is pleased to present this appraisal of state mathematics standards by Ralph A. Raimi of the University of Rochester and Lawrence S. Braden of St. Paul's School.
This is the fourth such publication by the Foundation. In July 1997, we issued Sandra Stotsky's evaluation of state English standards. In February 1998, we published examinations of state standards in history and geography. Science follows.
Thus, we will have gauged the states' success in setting standards for the five core subjects designated by the governors and President Bush at their 1989 education "summit" in Charlottesville. The national education goals adopted there included the statement that, "By the year 2000, American students will leave grades four, eight, and twelve having demonstrated competency in challenging subject matter including English, mathematics, science, history and geography." Although other subjects have value, too, these five remain at the heart of the academic curriculum of U.S. schools.
All are critically important, to be sure, but mathematics has special significance in today's debates about boosting the performance of U.S. students by setting ambitious standards for their academic achievement.
Mathematics is, of course, the third of the "three R's." Practically nobody doubts its central place in any serious education, its intellectual significance, or its practical value. Math is ordinarily the second subject (after reading) that young children encounter in primary school. "Math aptitude" constitutes half of one's S.A.T. score. And it was in no small part the weak math performance of American youngsters on domestic and international assessments that led us to understand that the nation was at risk. (Because it is universal, because it is sequential and cumulative, and because its test questions are easy to translate, mathematics has long been the subject most amenable to illuminating cross-national comparisons of student performance.)
Math also blazed a trail into the maze of national standards. Even as the Charlottesville summit was convening, the National Council of Teachers of Mathematics (NCTM) was putting the finishing touches on its report entitled Curriculum and Evaluation Standards for School Mathematics. In the ensuing decade, that publication and its progeny have had considerable impact on U.S. education, not least on the state math standards reviewed in the following pages. I have no doubt that, of all the "national standards" set in the various academic subjects, these have been the most influential. Indeed, I have heard policy makers declare that what America needs in other academic subjects are counterparts to the "NCTM math standards."
It is vital to understand, however, that the NCTM's mission was not--and today is not--the codification of traditional school mathematics into clear content and performance standards. Rather, NCTM's main project was to transform the teaching and learning of mathematics in U.S. schools.
The effects of that hoped-for transformation on state math standards are abundantly clear in this appraisal. Some readers may judge that the states should go further still to transform their expectations for students and teachers in the direction set forth by NCTM. Others will judge that they have gone much too far already. In any case, it's noteworthy that today, nine years after it was unveiled, "NCTM math" no longer commands the public consensus that it once appeared to have. California, for example, recently adopted new statewide standards that could fairly be termed a repudiation of the NCTM approach.
The important thing to know about the present document is that we did not ask its authors--a distinguished university mathematician and a deeply experienced school math teacher--to grade the states on how faithfully their standards incorporate the NCTM's model for math education. Rather, we asked them to appraise state standards in terms of their own criteria for what excellent math standards should contain.
Advised by two other nationally respected scholars, the authors did precisely that. They developed nine criteria (under four headings) and then applied them with great care to the math standards of 46 states and the District of Columbia. (The remaining four states either do not have published standards or would not make their current drafts available for review.) For comparison purposes, the authors also describe Japan's math standards and apply their criteria to these.
The results are sobering. Only three states (California, North Carolina, and Ohio) earn "A" grades, and just nine get "B's." Those 12 "honor" grades must be set alongside 16 failing marks (and seven "C's" and 12 "D's").
The results differ markedly from those of the recent Council for Basic Education (CBE) appraisal of the "rigor" of state math standards at grades 8 and 12. The CBE study begins with a list of performance standards expressed in 51 clauses (or "benchmarks") for the 8th grade and 30 for the 12th. These clauses are largely drawn from the NCTM standards of 1989. The state documents under study were then scanned for those 81 demands, which, when present (and weighted by their closeness to the template clauses), were counted up for a total score.
The present document does not begin with a list of this kind, and similarity to the NCTM standards was not a desideratum. The criteria used by Braden and Raimi are well described within the report itself, and include not only analyses of the "academic content" expressed or implied, but also qualities of exposition and taste affecting the standards' usefulness.
In view of the ferment in American math education and the continuing lackluster performance of U.S. youngsters in this key discipline, we must take notice of the findings reported herein. While state math standards are in many cases too new for them fairly to be held responsible for pupil attainment in this discipline, it appears that these documents, which were supposed to improve the situation, in most cases will not help and in many instances appear to be symptoms of the very failure they were intended to rectify.
To be sure, excellent math education continues in some classrooms and schools. State standards are not supposed to place a ceiling on how much is taught and learned. But they are meant to serve as a floor below which schools and teachers and children may not sink. As we learn from Messrs. Raimi and Braden, in many states today that floor seems to have been confused with the muddy excavation that ordinarily precedes construction.
We are grateful indeed to both authors for the rare energy, thoroughness, and mathematical insight that they brought to this arduous project. Raimi is professor emeritus of mathematics at the University of Rochester and former chairman of the math department (and graduate dean) at that institution. His scholarly specialty is functional analysis, and he has had a lifelong interest in effective mathematics teaching. Braden has taught mathematics and science in elementary, middle, and high schools for many years in Hawaii, in Russia, and now in New Hampshire. He is a recipient of the Presidential Award for Excellence in Science and Mathematics Teaching. He holds a bachelor's degree in mathematics from the University of California and an M.A.T. in mathematics from Harvard.
We also thank the two distinguished scholars who advised the authors throughout. Henry Alder is professor emeritus of mathematics at the University of California and a former president of the Mathematical Association of America. He has been a member of the California State Board of Education and recently served on the committee to rewrite that state's mathematics framework. Harold Stevenson is professor of psychology at the University of Michigan, a 1997 recipient of the American Psychological Association's Distinguished Scientific Award, and can fairly be termed America's foremost authority on Asian primary/secondary education and its comparison with U.S. schools and students. Among many publications, he co-authored The Learning Gap, a pathbreaking analysis of elementary education in Asia and the United States. He has a particular interest in the standards, curricula, and pedagogy of mathematics, which discipline has been the focus of many of his comparative studies, and has been deeply involved with the Third International Mathematics and Science Study (TIMSS).
In addition to published copies, this report (and its companion appraisals of state standards in other subjects) is available in full on the Foundation's web site: http://www.edexcellence.net. Hard copies can be obtained by calling 1-888-TBF-7474 (single copies are free). The report is not copyrighted and readers are welcome to reproduce it, provided they acknowledge its provenance and do not distort its meaning by selective quotation.
For further information from the authors, readers can contact Ralph Raimi by writing him at the Department of Mathematics, University of Rochester, Rochester, N.Y. 14627, or e-mailing firstname.lastname@example.org. Lawrence Braden can be e-mailed at email@example.com. The Thomas B. Fordham Foundation is a private foundation that supports research, publications, and action projects in elementary/secondary education reform at the national level and in the vicinity of Dayton, Ohio. Further information can be obtained from our web site or by writing us at 1015 18th Street N.W., Suite 300, Washington, D.C. 20036. (We can also be e-mailed through our web site.) In addition to Messrs. Raimi and Braden and their advisors, I would like to take this opportunity to thank the Foundation's program manager, Gregg Vanourek, as well as staff members Irmela Vontillius and Michael Petrilli, for their many services in the course of this project, and Robert Champ for his editorial assistance.
Chester E. Finn, Jr. President
Almost every one of the 50 States and the District of Columbia have by now published standards for school mathematics, designed to tell educators and the public officials who direct their work what ought to be the goals of mathematics education from kindergarten through high school. They are generally given as "benchmarks" of desired achievement as students progress through the grade levels to graduation, though sometimes they include guides to pedagogy as well. The present report represents a detailed analysis of all such documents as were available, 47 in all, though it has only space to offer rather abbreviated judgment of their value and a rating of their comparative worth. Grades of A, B, C, D, or F were given to each state, based on an analysis of the contents according to criteria and grade levels as described early in the report. Some comments on each State conclude the report.
On the whole, the nation flunks. Only three states received a grade of A, and just nine others a grade of B. More than half receive grades of D or F, and must be counted as having failed to accomplish their task. The grading is described below, but it should be understood that anything less than an A should be unacceptable.
A state, after all, is not a child to be graded for promise or for effort; the failure of a state to measure up to the best cannot be excused for lack of sleep the night before the exam. The failure of almost every State to delineate even that which is to be desired in the way of mathematics education constitutes a national disaster.
Even if the states' standards documents were exemplary, there would remain a problem of implementation. The public usually hears of the problems of schools as questions of funding, of discipline, and even sometimes of teacher preparation or recruitment, but it generally imagines that their intellectual goals are clear. For basketball players and musicians the goals are indeed known. But for elementary and secondary education in the United States today, there are no such agreements in place regarding its essential core: its academic program. This is especially so in mathematics, as the standards under review here illustrate.
The authors of this report believe it unconscionable that, in writing these standards--these documents of pure intent, whose success depends only on the efforts of experts already in place--so many states are so remiss in their duty.
As we have seen it, the principal failures stem from the mathematical ignorance of the writers of these standards, sometimes compounded by carelessness and sometimes by a faulty educational ideology. We are convinced that the average math teacher can be led to a better grasp of both the material that should be taught at various grade levels and the manner in which it should be presented, than the writers and editors of these documents imagine.
Our criteria for judgment were four: Clarity of the document's statements, and sufficient Content in the curriculum described or outlined in the text, were our first two demands. Third, since deductive reasoning is the backbone of mathematics, we looked to see how insistently that quality (denominated Reason) was to be found threaded through all parts of the curriculum. Finally, we assessed whether the document avoided the negative qualities that we called False Doctrine and Inflation. These four major criteria, some of them broken down into subcriteria, were individually graded and the scores combined for a single total.
The most serious failure was found in the domain of Reason. There is visible in these documents a currently fashionable ideology concerning the nature of mathematics that is destructive of its proper teaching. That is, mathematics is today widely regarded (in the schools) as something that must be presented as usable, "practical," and applicable to "real-world" problems at every stage of schooling, rather than as an intellectual adventure.
Mathematics does indeed model reality, and is miraculously successful in so doing, but this success has been accomplished by the development of mathematics itself into a structure that goes far beyond obvious daily application. Mathematics is a deductive system, or a number of such systems related to one another and to the world, as geometry and algebra are related to each other as well as to statistics and physics; to neglect the systematic features of mathematics is to condemn the student to a futile exercise in unrelated rule memorization. Most of the standards documents we have read, for all that they claim to foster "understanding" above rote learning, lack the qualities that would lead their readers, America's teachers, in the desired direction.
This lack of logical progression, seen especially in what passes for geometry and algebra in the grades from middle school upward, is visible in the lack of clarity of the documents. It is also visible in their advocacy of the use of calculators and computers in the early grades, where arithmetic and measurement as ideas should rather be made part of the student's outlook, by his learning through much experience and practice the nature of the number system. Learning to calculate, especially with fractions and decimals, is more than "getting the answer"; it is an exercise in reason and in the nature of our number system, and it underlies much that follows later in life. Only a person ignorant of all but the most trivial uses of calculation will believe that a calculator replaces--during the years of education-- mental and verbal and written calculation. Adults have need of calculators, and indeed computer programs, for computing their income taxes and doing their jobs. But the educational needs of children are quite different.
Content was the most successful part of these documents. This country has a traditional curriculum from the point of view of content, and many states at least mention most of it, including such recent additions as statistics and probability. However, much has been lost, especially from the Euclidean geometry that was so large a part of a high school program 50 years ago; and the fragmentation of the curriculum into too many different "threads" has also diluted the traditional curriculum.
The enterprise of writing standards goes hand-in-hand with the improvement of classroom practice, and there is no doubt that teachers of the next few years, seeing the inadequacy of most of what we have surveyed, will themselves offer suggestions for improvement. Members of the public, too, are often dissatisfied with vague education, led by vague standards, and they, too, will be heard. We believe the exercise of writing these documents is worthwhile, and we wish more states took it seriously enough to put their best talent to work on them.
In particular, the "best talent" must include not only members of the school establishment and state departments of education, but also persons knowledgeable in the uses of mathematics and the creation of new mathematics. That is to say, scientists (including statisticians, engineers, and applied mathematicians) and research mathematicians from the mathematics departments of the universities. These two communities have been most noticeably absent from the first rounds of standards construction, and future improvement is not possible without them. There is visible in these documents a currently fashionable ideology concerning the nature of mathematics that is destructive of its proper teaching.
State Math Standards