The following is Chapter 2 of the symposium volume, What's at Stake in the K-12 Standards Wars, subtitled A Primer for Educational Policy Makers, edited by Sandra Stotsky and copyright 2000 by Peter Lang Publishing Company, New York. It is reprinted here by permission of Peter Lang.
Judging State Standards for K‑12 Mathematics Education
Ralph A. Raimi
University of Rochester
In the summer of 1997, Lawrence Braden and I were commissioned by Chester E. Finn, Jr., president of the Thomas E. Fordham Foundation, to conduct a survey of the mathematics standards published by each of the states of the Union. Our report was issued by Fordham in March of 1998, along with or following corresponding reports, written by corresponding specialists, for state standards in English, history, geography and science.
Lawrence Braden is a mathematics teacher at St. Paul's School in Concord, New Hampshire, and I am professor emeritus of mathematics at the University of Rochester. We are thus at first glance an unlikely pair for such a job, as neither of us has any credentials in "mathematics education." To understand our enterprise one must know that "math education" is not to be confused with mathematics on the one hand, nor with teaching on the other. Systematic mathematics is at least four thousand years old, and teaching is surely older than that, but "mathematics education" is strictly a 20th Century phenomenon. This specialty is the domain of professors in the education colleges, members of the State and Federal education departments, officers of the teachers' unions and professional associations, and writers, consultants, and editors of school textbooks and books designed for use in teachers' colleges as well. Many of those one calls "math educators" also conduct research in methods of mathematics education, but this is not the same as research in mathematics itself, which is also abundant, though necessarily arcane. There is some apparent overlap between the three categories (mathematicians, teachers of mathematics in the schools, and math educators) in that a given person might move from one to another during his lifetime, but that person cannot simultaneously be an active member of more than one of these groups. The jobs are simply different.
For the Fordham Foundation analysis of the state standards in Kû12 Mathematics, then, Braden, a teacher, and I, a mathematician, temporarily took on a task more usually associated with mathematics educators, those whose business it is, or should be, to compose and analyze such documents and their use. Our main results are only partly contained in the grades we assigned to this or that state, though these judgments were the primary purpose of our study. While the details will be given below, there is no need to conceal, even at the outset, that those grades were terrible.
More important for the long run, I believe, was our more general conclusion that the divorce of the math education community from that of the mathematicians, and to some degree of the teachers, too, has led to a disastrous decline in our traditionally low expectations for student performance in school mathematics, as evidenced in the state standards we read and judged; and that this division cannot help contributing to the miserable standing of American students on international assessments such as the recent "TIMSS" (Third International Mathematics and Science Study). We further concluded that to reverse this trend, the nation requires an unprecedented increase in participation of mathematicians in what is today regarded as the domain of math education specialists, both in the establishment of curricula for the schools and in the education of the cadre of teachers needed to understand and use such improved curricula.
As with every other problem, the improvement in school mathematics, that is, in the mathematical education of children in the Kindergarten through high school levels ("K-12"), is a problem with a history, which will be outlined in the first section. This history is more than background, for it helps explain the nature of the problem, not just its past, and it contains elements that are already illuminating its future. The present, too, is much misunderstood. The second section deals with today's math education community's influence on present‑day curricula and pedagogy, for this is essentially the subject of our actual findings concerning the state standards we studied. Those standards were largely written by math educators, and certainly inspired by the earlier writings of that group. To the attitudes implied by these writings, we oppose the criteria by which our own judgments were formed; and as we shall see, they are quite different. Examples given here should illustrate the points; many more will be found in the Fordham Foundation report itself. The third section will describe some parallel assessments of the state standards which were made by two other organizations, the Council for Basic Education (CBE) and the American Federation of Teachers (AFT), and will explain why their conclusions differed so from those we made for the Fordham Foundation. Fundamentally, the Fordham Foundation report reflects a view from outside the math education community, while the other two represent the currently predominant philosophy of the National Council of Teachers of Mathematics and the State Education Departments being judged. State standards judged by their own writers, in effect, are like the famous lawyer who argues his own case. His case is a winner with his family, but open court is another matter. In conclusion, the fourth will return to the recent history of the relationship between the two domains, of mathematics itself on the one hand, and school curricula and teaching on the other, and some consequent recommendations for the future.
From "New Math" to "Reform Math" (1951‑1989)
Before about 1900 there was nothing special about math education in the schools. One presumed that a teacher, whether in mathematics, history, or Latin, merely taught what he knew, using textbooks written by those who knew even more. In the case of mathematics in the American public schools, practical arithmetic was the norm, and only a small minority went to high school at all. For the upper levels, the universities' entrance requirements pretty well dictated what the high schools (most importantly, private "prep schools") were to teach. With the great changes due to immigration and technologically driven increases in standard of living in the new century, however, the high schools of this century were no longer for only a privileged few, and the earlier grades had to take account of a possible high school future for their grade 8 graduates. For mathematics this presented a special problem, since its more advanced levels --- algebra, geometry, trigonometry --- were not the stuff of daily life; nor was the average product of a normal school (as teachers colleges were once styled) equipped to teach it. Mathematicians (university professors mainly) were before World War II very few indeed, and remote from the schools in distance as well as interest.
Thus inevitably grew up was what William Duren, a mathematician influential during the middle of this century, called the "PEB," meaning Professional Educational Bureaucracy. Someone had to govern the teaching of mathematics in the schools, with due regard to the new constraints and demands, and this group, which had not existed before, emerged to perform that task. That they did not do it well is not a reflection on their labor and dedication, for the problem had many dimensions, and amounted to raising a nation by its own bootstraps.
By 1940 it became something of a public scandal that Army draftees knew so little of mathematics that the Army itself had to undertake their training in the arithmetic necessary for even the most mundane bookkeeping and gunnery; and by 1945 the deficiency became even more evident when the wartime developments in radar, navigation, operations analysis, cryptography, rockets and atomic weapons (among others) showed the extent of mathematical accomplishment needed for a modern society, at war or at peace. "Practical" mathematics was not just arithmetic and interest rates any more, and whatever it was, America didn't have it. About 1950 there arose the beginnings of an attempt at reform, generated by some previously indifferent mathematicians among others, an attempt that burgeoned when the Soviet Sputnik of 1957 plunged Congress into shock.
Thus there arrived the one brief period in the history of mathematics education when professional mathematicians did come to exert some influence, and that was this era of the "New Math," which can be placed between the establishment of Max Beberman's University of Illinois project in 1951 and the gradual discrediting of all "New Math" projects with loss of Federal grant support in the early 1970s. The (temporary) prestige of science and mathematics during this time emboldened mathematicians, who had always found school mathematics preparation of college students inadequate anyhow, to organize projects to rewrite curricula and train teachers in different ways; but these reform efforts would have had no more effect than earlier ones had not Sputnik goaded Congress into financing such projects on a hitherto undreamed‑of scale.
The exact nature of these failed reforms makes a long story; so do the reasons for that failure. For the present purpose it is enough to say that the intent of the reform was to introduce the essentials of mathematical reasoning, of logic and rigor, into all parts of the curriculum, on the hypothesis that this was necessary if the later teachings were to be more than a drill in mindless algebraic abracadabra. The main hypothesis underlying those attempts were that essentially all children could be taught these things and would ultimately learn more sophisticated mathematics and with less time and effort, than by any other means.
Another hypothesis, this one totally unrealistic, was that teachers could be trained overnight (as Congress seemed to expect) to accomplish all this, and for the whole nation, once a group of mathematicians could construct the correct curriculum and test it out. This last requirement was nearest the truth, for such curricula were indeed constructed, if not perfectly, at least by very competent mathematicians, and with the help of practicing and experienced school teachers. But though testing was as thorough as time permitted, it appears that this was mainly done by specially trained teachers who would have had a hard time failing whatever curriculum they used.
In some places and in some small part the mathematicians were able to accomplish a good bit, but it was still only experimental, and only a beginning, on a task destined for the generations. Nonetheless, the popular desire to be up with the times immediately generated a flood of commercial textbooks affecting to be "new math," and school boards and superintendents were swept along with a meretricious tide of puffery. The result was far from what the experimenters had arrived at, imperfect though that already was. The public made no distinctions, however; in a few years the entire enterprise was judged to have failed, and the country clamored for a return to the familiar: "Back to basics" became the motto of the 1970s.
Actually, the true attempts at reform hadn't traveled very widely, and despite the incessant journalistic debates concerning "new math" and its successes and failures, the great majority of students never saw more than the slogans. "Back to basics" in 1975 was therefore business as usual for most schools anyway, and for the others the residue of the experiment was rather slight, with the exception of some changes in the senior high schools for college‑preparatory students, changes that would have taken place anyway.
From 1970 to 1990, then, the schools were not far from 1950 in their math programs, with one sad exception, though an exception that had already been under way before the Sputnik‑induced revolutions: The total collapse of deductive Euclidean geometry, to which at least lip-service had been given before the New Math. Geometry was already only a carryover from 19th century high school practice, and in pre‑New‑Math times a last remaining beacon of reason in school mathematics. But as the tenets of "progressive education" gained currency in all fields in the early years of this century, the writers of geometry texts, too, were increasingly slighting what was geometry's principal value, and were themselves becoming downright ignorant of the logical structures of mathematics itself.
Like Latin, history, and rhetoric, geometry had been a liberal art in the 19th century, studied mainly by those intending a university education. Nobody then regarded Euclid as a practical guide to carpentry or navigation; it was an intellectual exercise. Abraham Lincoln was proud to have studied the first six books of Euclid during the years he was also studying for the bar, though he had no intention of becoming a carpenter. During the more egalitarian 20th century, however, geometry gradually became downgraded in favor of what was deemed more practical and more in alignment with the "felt needs" of school children.
Such a development was convenient for those who didn't understand Euclid anyway. Some Euclidean geometry, with some formal constructions and proofs, remained in some schools, but textbooks were increasingly taking the attitude that geometry was a sort of science of description of natural objects, with mensuration formulas its goal. By 1950 there were very few children getting the real thing, though even today one may sometimes meet an old‑timer (non‑scientist) who remembers Euclidean geometry as one of the thrills of an otherwise drab school experience of long ago. The New Math reforms did attempt to deal intelligently with geometry, but it was difficult, and the form in which the commercial world of publishers then took it up was a travesty. By 1975, Euclid was dead in the schools.
With the decline and fall of the New Math, the mathematicians went home to their universities and their researches, and the PEB tried to make sense of "basics." Parents might have been mollified for a while by the fact that their children were no longer bringing home weird tales of "set intersections" and "truth tables" as they had done in the 1960s. But their accomplishment in terms of such national assessments as the National Assessment of Educational Progress continued to show the American children no better off in mathematics --- even in "basics" --- than they had been in 1950. The federal government established a National Institute of Education in 1972, intending it to conduct research towards improvement; and little by little the professional educators, who had, by and large, tried to celebrate and participate in the New Math projects while they seemed popular, recovered their nerve and reassumed direction of school mathematics.
In 1980 the National Council of Teachers of Mathematics (NCTM), their principal professional organization, issued a brief report, "An Agenda for Action," announcing that school mathematics should have "problem solving" as its primary focus. Other features of this manifesto, which showed no signs of mathematicians' influence, are also of interest, but apart from its emphasis on the calculator or computer as a tool for the future, all of it could have been written before 1950. In particular, the Agenda did not portray mathematics in any way as an intellectual adventure, or as a preparation for scientific studies, or as a thing of beauty.
That the document urged "understanding" of mathematics over memorization and routine did not reflect an attachment to the sort of reasoning one finds in either Euclid or the New Math axiomatic systems for algebra. Earlier educators, both traditional and those attempting reform, had always exalted "understanding." Who would not? But the profession's understanding of "understanding" seems to shift with the times. The new "understanding" was to be the relationship of mathematics with the "real world" rather than with reason. And with the interpretation of "real world" now being given, not much mathematics could be made relevant. In short, a dumbing‑down was in process, mathematics to be made palatable by a diminution of content, accompanied by a sugarcoating labeled improved pedagogy.
Over the next decade the NCTM formed committees to fill in the specifics, culminating in the 1989 publication of the rather large and ambitious Curriculum and Evaluation Standards for School Mathematics. This volume was actually only one of three allied "standards" reports, the others more narrowly concerning pedagogy and testing, but the 1989 volume is the one commonly referred to as "the NCTM standards," and the one that was to have a decisive influence on the developments Lawrence Braden and I were to study in 1997.
The National Science Foundation, following publication of the NCTM standards (which had not been federally financed but was the product of NCTM alone), undertook to help finance a similar study on the part of every state that would itself create a similar document for use in that state. Some states, of course, had long had standards in one form or another, to guide school districts in their choice of textbooks and curricular emphases, and in many cases as syllabi for statewide examinations. But with the federal legislation of the 1980s came a wholesale production of such standards, many of them in avowed imitation of the one produced by NCTM, though generally much shorter, and different in other ways.
The American tradition of local control of education more or less forbids a national curriculum, except de facto in some degree. These state standards, though federally encouraged and supported, are supposed to be each state's vision of the future, of what mathematics education ought to be. Some were apparently written by enormous committees of teachers and math education specialists, but the final texts obviously were assembled and organized at the state education department level, sometimes with the help of one of the regional educational "laboratories" set up and financed by the U.S. Department of Education. Despite the regional differences, the influence of NCTM and these laboratories has imparted a certain sameness to many of the state standards we ended up studying. Almost all of them had publication dates of 1996 or 1997.
Braden and I got the documents, we read them, we graded them. We could grade only 47, counting D.C., as three were incomplete drafts we could not cite or quote, and Iowa on principle does not produce one. Our grades were divided by criteria of our own devising, of which there were four: their Clarity, their demands for mathematical Content, their demand for what we call Mathematical Reasoning throughout the curriculum, and their freedom from the Negative Qualities we called False Doctrine and Inflation. Each criterion had parts, that is, subcriteria, but our overall judgment gave equal weight to these four groupings, producing a grade of A,B,C,D, or F (for "failure").
In all, 16 states got an F and 12 more got a D, making a clear majority at the more melancholy levels. Not that the C and B states were admirable. There were only three grades of A: North Carolina, Ohio, and California.
Japan got an A, too, but not as a 51st state. When we decided to include the Japanese document, our copy of which was dated 1990, we hadn't yet known that any American ones would receive a good grade, and we wanted to make sure our report contained some good examples for comparison's sake. In truth, the Japanese document (despite its high grade) could not really be used verbatim by any of our states, even if it were better translated, because of cultural differences that allow Japan's standards a rather allusive, if not cryptic, style, one that would not easily explain itself to American school systems. Just the same, it is well worth study by every one of our states, and some of the best American standards were written by states that had avowedly considered the Japanese model during their deliberations.
To our minds, anything less than a grade of A should be unacceptable. Grades of "B" and "C" are counted respectable for college students and children, but states are different. States can and should hire their best talent to write their papers, something frowned on in rich and lazy undergraduates but which in a state is not plagiarism. Nor can a state be excused for lack of sleep the night before the exam, or an attack of mononucleosis. Under these circumstances, the overall failure of all but (generously) thirteen of our states, those graded A or B, to produce a sensible document that simply and adequately delineates even what it hopes for in its citizens' mathematical education, must be considered a national disaster.
We didn't have to look hard to find faults; they hit us in the face. First off was the bloated language, which even in mathematics can only be called Educationese. It was our duty to read it anyhow; perhaps it was only an unfamiliar jargon which, once penetrated, does in fact make sense. Every trade has its jargon, after all. We persevered, we penetrated, but we found no bottom to it. This sort of thing we downgraded as Inflation, a gentler word than our in‑house use of "blather". Now one can say that blather is really only a matter of style; was there anything really wrong about what was written in these standards? Well, yes, there was. Some of what we read did make sense, but in places we would rather it had not, for the sense it made was too often destructive of mathematical learning. Such things we called False Doctrine, a telling example of which we shall come back to later in some detail. For now, the principal false doctrines we found prevalent in state standards were these:
1. The denial of the value of memorizing anything, especially the basic definitions and facts of arithmetic, and their computational algorithms; also the concomitant notion that since the hand calculator and computer largely replace the computational skills of the past in daily life, they should do so during the arithmetic learning process as well.
2. The urging of "real‑life" experience as the touchstone of value in mathematical knowledge, i.e. that mathematical ideas without immediate physical realization should not be imposed on children, or cannot.
3. The idea that whatever cannot, or apparently cannot, be taught to all children should be taught to none.
Not every failing state exhibited all of these doctrines at every turn, and certainly never in so bald a statement of principle as here summarized, though the philosophy is plain enough: "rote learning" should give place to "real understanding of concepts," for example. Rather than say all mathematical truths of value must have a physical realization, the state will prescribe "manipulatives" such as "algebra‑tiles," and team projects such as measuring playgrounds, so insistently that mathematics can no longer be seen as an intellectual adventure. And, rather than say that the curriculum is to be limited to what average and below‑average students of previous generations seem to have been able to accomplish, states combine the language of inclusiveness and equity with the omission of everything difficult in order to accomplish that end quietly. Other false doctrines will be considered in their turn, below. ("Manipulatives" are objects designed to teach mathematical lessons by analogy. Kindergarten blocks are an example, and for the early grades many other devices are convenient and instructive, but the genre has recently passed all bounds, both intellectual and financial, with the enthusiastic advocacy of most states' standards. One state even mentions the possible purchase, for educational purposes, of a Pascal's Triangle; but we have not yet seen advertisements for Occam's razors.)
Overlying each false doctrine there is usually a fog of reference to high‑sounding psychological theories of cognition, learning‑style differences, and the like, making it sound as if the mathematical contents of a program are quite unproblematical, and that the only real problems are how to transmit or cause the student to discover a well-understood body of mathematical material. That the relevant body of mathematical material is not so well understood by the educators themselves can be seen from the multitude of mathematical solecisms and errors with which so many of the state standards are sprinkled, and the vagueness of much of the rest. A small selection of these are quoted, though with minimal explanation, in our report for the Fordham Foundation. Alas, the cited errors are not mere oversights or misprints, but represent a genuine reservoir of mathematical ignorance.
Our second criterion, Content, was split three ways: primary school, middle school, high school. There has long been a traditional content in American schools: arithmetic for primary schools, geometry and algebra for the high schools, and time‑wasting reviews, usually called "ratio and proportion," or "business applications," for the middle schools. Fitful and shifting reforms since 1950 have, we found, improved things somewhat in some directions, especially for college‑intending high school students, but have degraded geometry on the whole, and have unreasonably delayed the introduction of algebra in the middle schools in most States.
Reason was our third criterion of judgment. Euclidean geometry is not its only home in school mathematics, of course, and the New Math of the era 1955‑1975, led by mathematicians, had made the attempt to introduce rigorous reasoning throughout the curriculum, and genuine algebra earlier than high school. The "back to basics" movement of the 1970s and 1980s therefore generated a tragedy of its own, for it helped render the community of mathematicians suspect and even excluded by the mathematics educators of the generations since. And not just the mathematicians, but the lessons of mathematics itself.
In today's most widely used textbooks the logical structures of algebra have disappeared, though some few items of vocabulary remain. The emphasis on "problem‑solving" has exalted the linear equation and its uses to a veritable definition of algebra, with such things as the binomial theorem and the quadratic formula downplayed or omitted. "Euclidean geometry" is in practice regarded by most state standards as a sort of empirical study of the shape of the world around us, dimensions of playgrounds and soup cans, augmented by some exercises (aided by computer software) in naming symmetries and looking at things in mirrors. While geometry as a deductive system has had some small staying power in actual classrooms, present "math‑education" philosophy, as expressed in most state standards, would guarantee that in another generation no teachers will be left us who are competent to handle the Euclidean system.
Reason in other parts of the school curriculum, especially in algebra, became so discredited during the New Math era that it is only sporadically visible today. "Deductive and inductive reasoning" may be a mantra in standards‑land, but by "inductive reasoning" is mostly meant extrapolating number sequences and playing with geometric manipulatives. Most states in most contexts, moreover, confuse mathematical, i.e., deductive, reasoning with the process by which real-life problems are converted into algebraic equations, a process better described as mathematical modeling. So, led by the bellwether NCTM standards of 1989, most State standards lump "problem‑solving" with "mathematical reasoning" under a single rubric, which might be termed the burial shroud of the New Math.
Comparing 1998 with 1948 overall, as seen in the content of the state standards taken together, there are visible a few, but very few, noticeable effects or legacies of the intervening reform efforts. One of these is the presence of some of the ideas of statistics in the school math curriculum, and another is the recurrent attempt to leap‑frog the time‑wasting of the middle school by "acceleration," at least for the better students, leading to Advanced Placement calculus in the 12th grade. This is not much, and not entirely to the good, either, for it has often been accomplished by some curricular thinning in other directions.
Furthermore, the ignorant descriptions the new topics (along with the old) are so often given in the state standards betoken a national cadre of math educators unable to lead the nation's teachers to a suitable intellectual plane. For example, the definitions given mathematical terms in the glossaries that so many standards (unnecessarily) include are sometimes real howlers. This one from Tennessee, for example, exhibits the straining after "humanistic" values in math education that someone who didn't understand its really humanistic values was forced to invent:
Algebraic Thinking: thinking skills which are developed by working with problems which require students to describe, extend, analyze, and create a variety of oral, visual, and physical patterns (such as ones based on color, shape, number, sounds) from real life and other subjects such as literature and music.
Nobody will be particularly misled by this definition when preparing for examinations, of course, but a later definition from the same glossary is more characteristic:
Equation: two mathematical expressions joined by an equals sign.
While this describes the physical appearance of an equation, it omits the essence, that an equation is a sentence, or clause in a sentence, helping make a statement of a certain sort. Students and teachers who take this definition's attitude cannot learn to make use of equations, except to pass certain multiple choice examinations.
Today's NCTM‑led movement to a "reform" math runs along different lines from either "new math" or the "back‑to‑basics" that succeeded it. It would replace what it calls "mindless drill" by "real understanding of concepts." But "mindless drill" is a straw man and characterizes bad teaching rather than a bad curriculum. Meanwhile, today's touted "concepts" are little connected with mathematical reasoning and instead said to be connected with "problem‑solving" and the "real world." This new attitude, which has been relentlessly propagandized by the NCTM, underlies the main substantive failings in the standards we were reading and judging last fall.
One difficulty in invoking "the real world" as a touchstone of value in school math is that the real "real world" is one quite unknown to children and generally of little interest. A real‑world arithmetic problem might concern the amount of paint to order for the redecorating of an office building. What could be more tedious and unenlightening for a child learning to multiply numbers? Children are imaginative and in fact can become more interested in the decomposition of an integer into primes than in any amount of schoolyard measurement.
Other so‑called "real‑world" problems are too often strained attempts to attach names to old routine exercises. "A candy store sells n(n+1) boxes of candy on the nth day..." Why a candy store? Because the author can't think of any other real‑world application of the equation he plans to introduce. Any child will see this is not even sugar‑coating of the algebraic pill; it is saccharine. Genuine quadratic equations are actually more interesting than phony candy stores.
The 1989 NCTM standards document (which was not one of the documents graded in our Fordham Foundation report but is their progenitor) is longer than almost every set of state standards we read. It is not too hard to write state standards that resemble the NCTM document, and most States tried. The difficulty is this: NCTM is very vague about a lot of things and doesn't make it entirely clear what the content should be at each stage of schooling. One would think a set of standards should tell a new teacher what to teach in each grade, or at least what a student should know at the end of, say, grades 6, 9, and 12, if not year by year.
Instead, NCTM offers aspirations of a more general sort, though still classified by year or range of years. Here is an example (p.81):
In grades 5-8, reasoning shall permeate the mathematics curriculum so that students can:
* recognize and apply deductive and inductive reasoning;
* understand and apply reasoning processes, with special attention to spatial reasoning and reasoning with proportions and graphs;
* make and evaluate mathematical conjectures and arguments;
* validate their own thinking;
* appreciate the pervasive use and power of reasoning as a part of mathematics.
These points are typical of NCTM rhetoric in that however many times I read them I cannot remember all five, even in paraphrase, and would fail any test on the whole statement. I invite any amateur (or professional) actor reading these words to try to memorize them; they rival The Bald Soprano in their structureless difficulty, and most of the works of Gertrude Stein.
The quoted standard is followed by examples of problems ---classroom exercises meant to illustrate some of these demands; but it does not outline or even suggest a coherent program. Following the five points quoted above, one example concerns tiling a plane, a second concerns the search for prime integers, and a third shows how a graph can represent a plot of speed against time for a pictured roller coaster.
All worthy exercises but in context typical of an unworkable current dogma, which is to have all the possible threads of mathematics appear in some form in each year of schooling. Tiling a plane might be called geometry, study of the primes might be called number theory, graphing speed against time might be called analysis (or "pre‑calculus" in the schools). Placing three such diverse examples under one rubric is here a deliberate NCTM invitation to "integrate" the curriculum. But the straining for integration leads more often to mathematical incoherence. In an outline for a mathematics curriculum one should invoke some system, something memorable, as for example Euclidean geometry does by the very ordering of its theorems. The examples for Reason should, as in the Japanese standards, be organized by subject matter or by year, to exhibit a progression of skill in reasoning; it is not something to be isolated as a subject of its own, to be illustrated by random examples.
Almost all the state standards imitate NCTM in this regard, though usually rather more dangerously than the national organization. The result has been characterized by one critic as a curriculum that is "a mile wide and an inch deep." Some of the standards we read have something called "algebra" in every grade, K-12, but manage to end the series (as Connecticut does) without the quadratic formula, without geometric series, and without the binomial theorem: a triumph of nomenclature over substance. Such algebra should rather be called "algebra appreciation," unless, as in some "reform" textbooks, it is also tricked out with today's political virtues and may better be called, to quote an unfriendly but accurate critic, "rain‑forest algebra."
NCTM, and hence most states, have taken up another popular doctrine, which is that children should "discover," maybe even construct, knowledge for themselves. They should not be told things; the teacher is to be a "guide on the side" rather than a "sage on the stage." This Rousseauean idea has gone through many generations of educational theorists, including our own John Dewey and his followers. In the legendary case of Mark Hopkins and a log, it has some validity; but in the schools of middle America it leads to such absurdities as the "discovery" of the Pythagorean Theorem by children instructed exactly how to spend their hours communally cutting out appropriately shaped pieces of paper.
Certainly children do not learn by having knowledge laid upon them like a blanket, and if we do not participate in our own educations we will learn nothing. But this obvious insight has been elevated into a doctrine that in practice often excuses the teacher from ever bringing a lesson to a conclusion, or even knowing the answers.
As another example, New Jersey's 1996 Framework, avowedly "built on" the 1989 NCTM standards, contains, in addition to its own content standards, a great deal of amplifying educational philosophy and pedagogical advice, some of it revealing in this regard. Under Number Sense (Standard 6: "All students will develop number sense and an ability to represent numbers in a variety of forms and use numbers in diverse situations"), the Framework offers extra advice according to grade level. For grades 5 and 6, it explains,
Models are essential for the continued exploration of fraction meaning... Fraction circles and Fraction Bars help children... establish rudimentary meaning for fractions but have the drawback of using the same size unit for all the pieces. This is a fairly serious drawback leading to the misconception, for instance, that 1/3 is always less than 1/2 without regard to the units in which these fractions are expressed; students need to be aware, for example, that 1/3 of a large pizza is frequently larger than 1/2 of a small one.
This might sound like a deep insight to someone, but certainly not to a mathematician, for whom 1/3 has always been less than 1/2 and always will be. New Jersey is for some reason deliberately confusing the numbers 1/2 and 1/3 with the use of their names as adjectives in practical applications. If there is any point to the study of mathematics, it is exactly in the distinctions it draws between objects one might wish to eat and numerical abstractions which can partially describe them. Certainly mathematics does not take account of flavor and material, and only somewhat of shape and weight; but if a child is ever to make use of the properties of fractions, what is truly "essential" is not the use of manipulatives like "fraction circles," but rather the distinction between "1/3" and "1/3 of a large pizza."
The person who wrote about "the misconception that 1/3 is always less than 1/2" was suffering from an overdose of Piaget, the psychologist who most perceptively wrote of the development of number sense in small children. Children develop concepts of "large" and "small," "few" and "many," by stages, and their early perceptions sometimes lead them to believe, or say, e.g., that an apple divided in two parts is thereby diminished. There may even be a sense in which the division does lead a child to prefer the whole, or to call it "bigger" (or is it "smaller"?), and there is no denying that such perceptions slow the child's early understanding of the mathematician's vocabulary of halves and thirds. But this does not make 1/3 greater than the number 1/2 --- ever. The child will simply have to learn that while the third part of a very large pizza, or of a galaxy, still lacks two parts of the whole, we are not thereby insisting it is "small" compared to half a banana. We are duty bound to explain the meaning of the words --- a restricted meaning, to be sure, but a necessary one. It is for the child to learn the world's conventions and the world's science, and not for the world to defer to the child's initial imaginings, even though understanding the sources of a child's difficulties is of high importance in teaching. The notion that "in this case one third is greater than one half" is simply mischievous, an exaltation of psychological insight and sympathy over the demands of reason.
In passing, it is a pity that students at the sixth grade level are not yet, and not only in New Jersey, weaned from such material objects as the "fraction circles" prototypes of simple fractions. Of course, a child begins life with material experiences: in number, in vocabulary, in the study of ethics for that matter. But the genius of mathematics is that it organizes raw experience by means of a universe of ideals whose manipulation is for some purposes easier and more illuminating than living the experiences themselves. Yet, rather than giving life to the abstractions, a process essential in every science, the New Jersey standards counts it "essential" that the objects themselves be used in the classroom and indeed confounded with their numerical abstractions. Their only caution in the present case is that all the pieces marked "1/3" are unfortunately the same size, as if that were a hindrance to the truth of their representation of numbers. It is not a hindrance. The day will come when the mere symbol "1/3" will have to replace even the "fraction circles" in describing pizzas and galaxies alike.
Another example of a mathematically mischievous doctrine deriving from a pedagogical insight that has become all but forgotten in the transition is found under New Jersey's Standard #4, Reasoning: ("All students will develop reasoning ability and will become self‑reliant, independent, mathematical thinkers"). The "K-12 Overview" section under Reasoning contains the following philosophical note:
Multiple solutions. There is no single "best" solution; rather, there are many solutions, each with costs and benefits.
Solution to what, one might ask --- to a mathematical problem? Surely twice two hasn't become ambiguous in recent times? No, the examples show that no such thing is meant. New Jersey offers in this context a "vignette" of a second grade classroom exercise: "Can a dinosaur fit in this room?" Children are to make or find appropriate measurements, define "fit in the room," etc. It looks like a pleasant enough exercise, which teaches among other things that dinosaurs come in several sizes, as do classrooms; and it certainly has no single "best" solution. But the question is not a mathematical one. Even at the second grade level, the mathematical component of a problem should be separated from the empirical part; such separation is later essential in all science, though it is hard in a brief space to indicate its importance. That the "many solutions" doctrine is correct in many human endeavors is plain obvious, and nobody needs to be taught that. But this lesson is not mathematics.
Yet this false (mathematical) doctrine has a history in pedagogy that goes beyond today's unreasonable emphasis on "real‑life" problems. In the days of the Three R's, legend has it, children were lined up in rows and taught to shout viva voce, or write on their slates, the answers to an interminable list of identical exercises. "Drill and kill" is the derisory phrase used by today's pedagogy to describe this sort of teaching, and if one looks at a common textbook of the year 1910 one does see pages gray with routine exercises of this sort. Whether this implies that classroom activity was equally rigid is hard to say; certainly the other legend, that of the one‑room schoolhouse for all grades, implies otherwise. However that might be, the picture is certainly that of "one question, one answer," and the object of most mathematics reform efforts of the past century has been to avoid catechism and to open the mind.
One observable aspect of most non‑trivial reasoning is that there is often no single road to the answer. A proof of a Euclidean proposition should not be memorized as if it were a simple fact, for example, but should be analyzed and understood; and a student who finds an alternate proof is deserving of praise. The same is true of all sufficiently complicated problems, and not only in mathematics. But this idea of several roads to a (mathematical) solution of a (mathematical) problem has become conflated with the idea of several solutions to a non‑mathematical problem, resulting in the confused notion that mathematical problems may have several answers.
The NCTM insistence on this theme, visible in many state standards, has led to the creation of whole curricula where students are not only invited to create their own mathematical methods but are not told whether what they have created is right. And while the students struggle to create their own rivals to the algorithms mankind has developed over the centuries, they do not get to take advantage of the efforts of their ancestors. While it is true that a rote memorization may be unaccompanied by understanding, it does not follow that it is the cause of mindlessness. Beethoven didn't have to invent the well‑tempered scale --- it was taught to him; but his imagination survived this quite well. Learning how mankind has already arrived at the single correct solution to a certain well‑defined problem is not necessarily going to cripple the math student's imagination. To insist otherwise is false doctrine.
John Adams (1735‑1826), a schoolmaster before he took up the Law, wrote in his Diary the following entry:
June 1, 1756 Drank Tea at the Majors. The Reasoning of Mathematicians 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.
Adams was certainly aware that the law employs reason in somewhat the same way, and that a carpenter must measure and calculate elaborately from a simple set of tools. But he shows his astonishment at the extent to which a single line of reason can lead, in this case in the single science of Euclidean geometry. Such an appreciation is lost in a splintered curriculum of the current dispensation. NCTM's standards has no place for an edifice so great as that which astonished Adams and which no classroom full of cooperative children is likely to "discover," even when aided by a guide on the side. First a bit about tilings, then a bit about primes, and then some measurements in the schoolyard, all in the name of "integrating" the strands of a diverse mathematical education.
Well, Braden and I arrived at our list of grades, and the results have been published. Two other organizations have also produced comparable studies of state math standards: the American Federation of Teachers and the Council for Basic Education. Their grades differed from ours. Michigan got a C from AFT, a B+ from CBE, and an F from us (Fordham). A Michigan State official remarked wryly, for the newspapers, that standards seem to be in the eyes of the beholders, as if to say he had no cause for anxiety if the experts could not agree. New Jersey, which got from the CBE the only pure grade of A they gave, received only a C from us, and a D from AFT.
Thus our grades were not merely lower than the others on average, they showed a different spectrum entirely. For another example, where CBE graded New Jersey and New York A and B+, respectively, our Fordham report gave them C and B, respectively. CBE gave Alabama a C (and they gave very few grades that low) where we gave them a B, one of the few states scoring that high. AFT gave Alabama a B as well, though they were quite free with grades that high. Why? There is clearly a difference of "expert" opinion across the country; is it idiosyncratic, principled, careless?
In the case of the AFT we are unable to analyze the reasons for the grades they gave, except to say that they apparently valued most highly the seriousness with which each document was prepared, its definiteness and suitability as a guide to statewide evaluations, and so on, but have little to say about actual curricular or content choices, at least in mathematics. We may also note that the AFT evaluators were few and "in‑house," with mathematicians having no hand in the judgment.
In the case of the Council for Basic Education we can say much more because CBE has published its method of judgment in a separate report. CBE says it collected a panel comprising "subject specialists, teachers, parents, and business representatives to help to 'develop' CBE's definition of rigor in standards?" The definition was in fact written by an in‑house group apparently called "CBE." There were 81 items they ended with (51 at the eighth grade level and 30 at the twelfth), but rather than being criteria by which to judge standards, they were actual statements of content or pedagogical desiderata, of the sort one might expect to find as entries in a standards document. These were avowedly drawn from a combination of the two most authoritative sources in the country: the 1989 NCTM standards and the NAEP guidelines for the periodic national diagnostic tests at certain grade levels. The ultimate author of all these statements, for all that CBE winnowed them out a bit, is the math education community, not including mathematicians.
Some of the entries were exactly the sort of thing Braden and I considered unhelpful or vague, though others were reasonable. For example, from the eighth grade level: "Add, subtract multiply, and divide with rational numbers." This isn't really bad but avoids the question of how far the required algorithms are to be carried. Braden and I downgraded states that insisted on calculator calculations whenever one of the factors was of more than two digits; CBE had no way to notice such a subtlety and in fact found that almost all states satisfied the CBE standard in this case. (CBE's own explanatory text states that it will consider the use of calculators in its next such evaluation.) Others of CBE's 81 items were much more vague: "Ask clarifying and extending questions related to mathematics;" and (under "reasoning"), "Make and test conjectures."
As a four‑word summary of the purpose of all science and philosophy, this last demand, "Make and test conjectures," is not bad. As a guide to teachers and test‑makers it is impossibly broad; it won't do without further guidance. In the NCTM standards that phrase appears on page 143 (for grades 9-12) and (slightly varied) on page 61 (for grades 5-8). NCTM follows these headlines with a good deal of commentary, but the CBE evaluators, according to their explanation of method, would give full marks on that "reasoning," item for the four words alone, provided they appeared at the twelfth grade level. At the eighth grade level, also under "reasoning" the following words produce another four‑point (full marks) score: "Make and evaluate mathematical conjectures and arguments."
Their scoring method was this: CBE trained panels made up entirely of teachers or former teachers to judge the state standards, giving instructions designed to make the procedure almost automatic and in any case uniform in result when two different panelists graded the same document. The 81 statements, three of which have been quoted above, were taken as a template, and if a state contained one of them verbatim, or essentially verbatim though perhaps divided into parts appearing in different parts of the document, it received a full 4 points. If the corresponding statement, or combination of desiderata, in a state document asked a bit less than CBE did in its own formulation, the state got 3 points. And so on, down to zero if the state standards contained nothing mentioned in the template statement. Finally, the points were added and grades assigned.
Had the system worked perfectly, any state repeating the 1989 NCTM standards plus NAEP benchmarks would have got a near perfect score. CBE did not make any independent judgment second‑guessing the experts as to the desirability of these items, though it had necessarily to omit or combine some of the many items to be found among the NCTM and NAEP guidelines and standards. But, even so, the system didn't work perfectly. How imperfect the judgment was, even by template standards, we can only judge by the few examples the CBE report gave to illustrate the way 4, or 3, or 2, or 1, or no points might be given to a standard intended to cover a template item.
One example was this: Under the rubric Data Analysis, Statistics, and Probability is found the twelfth grade benchmark, "Model real‑world situations to determine the probabilities of dependent and independent events and compare these experimental probabilities with what would be expected based on theoretical models." Any state essentially repeating this benchmark gets 4 points. A state whose corresponding standard "omits one of the essential concepts or skills or a few minor components" or which "aligns with the framework benchmark, but is written at a lower degree of sophistication" receives a 3. Here the CBE document gives as an example of what would be worth 3 points rather than 4: "Model real world situations to determine the probabilities of dependent events and compare these experimental probabilities with what would be expected based on theoretical models."
Now the 3‑point benchmark differs from the template benchmark by omitting only the two words, "and independent." That's all. Anyone unacquainted with probability theory might be fooled into thinking something was thereby left out of the shorter statement, but this is not so. It is impossible to understand or use the phrase "dependent events" without also understanding and being able to use the phrase "independent events;" thus, while it is not usual to say it this way, to ask a student to understand and use formulas and insights concerning "dependent events" is asking no less than did the original formulation.
The degree of ignorance or carelessness demonstrated in considering the second of these two formulations less inclusive (or "rigorous") than the first is exhibited even more strikingly in a (doubtless hypothetical) example that CBE said it would have awarded 2 points: For example, the CBE document says, "Explain the difference between a dependent and an independent event and..."
Two points? There is no such thing as an independent event or a dependent event. Dependence is a property of sets of events, not of events singly. Hence any state writing "Explain the difference between a dependent and an independent event..." should be scored zero on grounds of ignorance, carelessness, false doctrine and pretence of understanding. In this case, having most of the words right is something like omitting a "not" when quoting a theater reviewer for advertising purposes. The conjectured 2 point "standard" would be a positive disservice to its users.
Thus the system of scoring employed by the CBE misses its target in two ways: First, in that it chooses as "benchmarks" those published by the authoritative mathematics education establishment of the day, rather than collecting its own judges and having them begin the process from the beginning, using criteria for genuine judgment rather than template exhortations for imitation; and second, in imagining that subjective standards of judgment can be eliminated by its "quantitative" evaluation of adherence, item by item, to its own model benchmarks. If the NCTM and NAEP model of standards is accepted without further debate there is no need for state standards at all, and every state would score well by reprinting the relevant parts of documents already in print. But even if this model of judgment were taken as informative, an ignorant judgment of the intellectual distance between a benchmark being judged and the template (as in the case of the notion of "an independent event") renders the numerical scale untrustworthy even as measured against the template. A judgment of whether a mathematical statement makes sense, or is in a correct context, or is worth teaching, can only be made by persons who understand the mathematics in question. The ultimate lesson of the CBE judgments is that they fail because the judges didn't know enough mathematics.
Knowledge of mathematics makes a difference, even in the prescriptions for the earliest grades. This statement needs defending, for it is hard for a member of the general public to believe, concerning such elementary things as the fourth grade rules for arithmetic or the sixth grade calculations with fractions, which everyone knows, that the educator's mathematical ignorance can be a factor in the educational program. Since the arithmetic of the primary grades apparently presents no great intellectual or philosophical challenge, one might imagine the problem of the schools to be a matter of good teaching style, small classes, discipline, and so on, but not a question of what is truth concerning fractions or multiplication.
There are two considerations allied here. As to "what is truth," I would like to assure the reader that fractions are more complicated than one might remember, and that calculators don't render their understanding and applications a whit easier, except for the mindless part of the computations. But more important than the mathematical content of such elementary lessons is their intellectual setting in the classroom. That will critically affect future understanding. In the earliest grades are developed, for example, the baneful reflexes of "getting the answer" from key words of standardized questions and explanations, reflexes that are actually generated by teaching that has no regard for future progress because it proceeds from its own ignorance of what that future can contain. A piano teacher who has never actually played Chopin (let alone Prokofiev) will give misdirected lessons in finger placement, arpeggios, and scales, and teach "shortcuts" that will return a year or two later to cripple the student's sincerest efforts. That analogy with arithmetic is precise.
Among other things, there is today, despite lip service to the contrary in the NCTM standards and most state standards, a sad misapprehension of the role of deductive reasoning in school mathematics. Teachers and their own mentors in the schools of education all too often consider rigorous mathematics, based on careful definitions and using logically structured arguments, to be "difficult," and a fortiori too difficult for the children they will be teaching. But people untaught in mathematical reasoning are not being saved from something difficult; they are, rather, being deprived of something that makes easy all that follows. And this observation applies to children having difficulty with their lessons in fractions even more than to children who begin their intellectual lives with every advantage of educated parents and peaceful environments.
That a proper intellectual setting is needed beyond the mere facts of mathematical information transmission is, of course, realized by the math education community, too; but its estimate of proper intellectual setting is in too many cases faulty through lack of a sufficient understanding of mathematics itself. The standards documents we awarded low grades could not possibly have been passed by any mathematician at the local state university, although their solecisms and inappropriate prescriptions were passed through a multitude of reviews (according to their Introductions) by certified math educators.
That the general public does not appreciate the difference between a mathematician and a math educator choosing curriculum for a school district is understandable, but the difference is enormous and should be understood by the educator and made use of. Mathematicians by definition don't teach in the schools, nor can they administer a school district; but to do either without the scrutiny of mathematicians is like building a house with skilled carpenters and no architect.
Every state has a state university with mathematicians in it, almost across the street from the state education department in many cases. Is it not unconscionable that a governor or a board of education wanting advice on a mathematics curriculum doesn't pick up the telephone and give it a call? They go, if at all, to the school of education, thinking this the obvious place where advice is to be had when in‑house experts in the department of education seem to want it.
But in the school of education, most math educators have retreated into a culture of their own, with a language of their own, devoted to pedagogy as if it were a skill that could be applied irrespective of subject. As evidenced in the standards they have been writing, this attention to just one part of the educational process will not do. That this division between the mathematicians and the math educators is partly the legacy of the failed intervention of mathematicians in the era of "the new math" doesn't make it correct. Every other course has failed, too, after all; that is why we are where we are today.
Since the publication of the Fordham Foundation report on mathematics standards, I have received several invitations to review a new edition of the standards whose earlier drafts were the subject of our study, or from states not included in our report. Some I answered, but as I could not participate in every case, and since there is nothing unique about my own expertise anyway, I answered a couple of the others in this way:
If your standards were composed without the significant participation of mathematicians, let me advise you to go down to your best state university and find a professor of mathematics, at least 40 years old, who is willing to help you. He need not have heard of Piaget and Bruner, and he might very well be of such a personality that you would never trust him in a fifth grade class, but he should be an English‑speaking American who himself has gone through our public school system, and he should be a genuine mathematician who has published at least a handful of research articles in the refereed professional journals of pure or applied mathematics. (Not journals of math education; you have such people in your department of education already.) Find out that this mathematician is willing to devote a few days to your project. Give him a copy of the Fordham Foundation report on the state standards to read, with particular attention to the criteria for judgment contained therein, and give him copies of the printed school mathematics standards of California, Ohio, North Carolina, and Japan. (These were the ones we counted best in our report.) Then give him a copy of your own state's draft standards and ask for a written commentary. Then use it.
It is not possible to put school mathematics education in the hands of the mathematicians. If we did, they would cease to be mathematicians. Theirs is another trade. But math education in the schools, while conducted by school teachers, must still be done in a climate of mathematical understanding. At the present time and for the foreseeable future this understanding is insufficient among those who need it. The reasons are complex and are not unique to the present generation, either; they constitute a whole story of their own, and they may never be satisfactorily repaired. But one should not deduce from this that the educators should be exterminated and replaced by mathematicians, or even that mathematicians' ideas of what and how to teach should automatically be credited and put into practice at once.
However that may be, the education community cannot get along without the advice of the mathematicians, and many more mathematicians than now do should learn something of the problems of school mathematics and stand ready to provide that advice. Among other things, the teaching of mathematics to future teachers is done in colleges or universities, but not enough of it is done by mathematicians, who today take more care with the teaching of future engineers and scientists than future teachers, and who today almost never see future elementary school teachers at all. College textbooks in algebra and geometry, and more advanced topics for undergraduates as well, are written by mathematicians. But not enough of them are written with sufficient attention to the needs of future teachers.
To repair all of this is the work of generations. What is most serious at the present time is that the math education community demonstrably needs more such advice now, today, than it thinks it needs. A quick mathematical fix, a Fordham Foundation Report, even a brilliant textbook, will of course not suffice, but beginnings are necessary if never sufficient.
In one of Edgar Allan Poe's stories there is recounted an anecdote about the famous 18th Century Scottish physician, John Arbuthnot. At a dinner party he was seated next to a lady who at some length described to him her symptoms, ending with, "Well, then, Doctor, what should I take?" "Take?" said Arbuthnot, "Why, take advice, of course."