Are the NCTM Standards Suitable for Systematic Adoption?

The fundamental characteristic of mathematics is its abstraction. Its idealizations have been found to give surprisingly accurate descriptions of phenomena in the real world. Thus mathematics is endowed with incredible power to serve a broad range of diverse areas. The same mathematical tools, differently interpreted, can be applied to seemingly unrelated subjects to solve problems in a great variety of disciplines. Any educational program must address these important features of mathematics in a balanced way. One of the serious problems with the NCTM standards is that the essence of the subject is lost. Problems are viewed as relevant and interesting to students only if they are related to familiar daily experiences. Unless they, are tied in with theory, however, isolated problems may well fragment the discipline. The power of mathematics vanishes when nothing occurs to show that the concrete is an illustration of the abstract. We have a magnificent opportunity to enlighten our students, leading them to the realization that mathematics is an intrinsically beautiful and exciting discipline in its own right. Its structure gives us substantial flexibility for teaching. We may use either the applications to develop the abstraction or, conversely, start with the abstraction as a lead to application. By turning at various times in either direction, we can provide great mathematical enrichment for our students.

The National Council of Teachers of Mathematics (NCTM) is currently carrying out an extensive review of its original standards documents (NCTM, 1989, 1991, 1995). It is gathering suggestions and comments from a wide range of organizations and individuals as it prepares to issue a new version in the year 2000. The initial NCTM materials generated substantial discussion and varied points of view among different constituencies. They are based on the following six fundamental premises.

● mathematical power for all in a technological society;

● mathematics as something one does—solve problems, communicate, reason;

● a curriculum for all that includes a broad range of content, a variety of contexts, and deliberate connections;

● the learning of mathematics as an active, constructive process;

● instruction based on real problems;

● evaluation as a means of improving instruction, learning, and programs.

In this article, I examine these views and address serious concerns expressed by many active mathematicians, school administrators, teachers, and parents. Note, for example, the web-site paper by a former NCTM president (Allen, 1996). Troubling to this group is the fact that these standards fall short of providing a reasonable balance. They highlight the applications of everyday experiences. On the other hand, they fail to emphasize adequately the theoretical aspects that make mathematics a unique and important discipline. In addition, they do not give enough attention to the development of sound basic skills. Misinterpretations and misrepresentations are some of the consequences of the vagueness of the language and of the general ambiguity of recommendations. Indeed, some teachers are omitting significant material altogether. Further, there is a marked reduction in expectation of student performance with resulting mediocre achievement. It is most disconcerting, too, that teaching techniques are limited to the one type depicted in most illustrations of the standards. Generally ignored, for instance, are the preferences of some teachers who may like different strategies. In their experience, they may have found these ultimately more effective.

Apart from indicating specific problems with the existing NCTM documents, I include some reports of unintended interpretations and implementations in the classrooms and discuss some of my own views of school mathematics. A revision of the standards must give careful consideration to all of these issues to avoid repeating the flaws of the initial documents. Above all, it is essential to design mathematics education standards in a way that provides a balanced foundation that will serve all students well throughout their lives. They must learn the importance of careful, sound, logical reasoning. In this way, they can analyze exaggerated claims and become informed and contributing citizens in whatever line of work they eventually pursue.

As at every grade level, those teaching in colleges or universities must ascertain that the overall mathematical education of all of their students is sound. Substantial information on the learning effectiveness of earlier courses is essential. In order to make any significant progress with more advanced material, it is vital to master much basic content. Indeed, those who expect to become teachers must be well grounded in the essence of their subject. In this way, they will eventually make certain that their own students have a solid knowledge and understanding of mathematics. Prospective teachers must have a genuine feeling for, and love of, the subject. They must be able to explain why it is necessary to perform certain operations. Further, they need to have a good idea of where the subject is heading. With such knowledge, they can then provide the leadership essential to set their classes in the right direction. Otherwise, their teaching will be mechanical, and they will stress little more than formalisms. Unhappily, their students then will not be learning to appreciate the nature of a fundamental and fascinating subject.

When teachers do not have important questions answered early in their careers, there may be serious consequences for their students. In particular, any replica of the following situation must never occur. An experienced high school teacher once asked one of my colleagues, “Tell me, I know you get two answers when you solve a quadratic equation, but which is really the correct one?” Clearly, this teacher had no sense of the topic he had been teaching for years. It is unfortunate that he continued to remain puzzled by this problem for so long, always hesitant to ask anyone about it, lest he disclose his uncertainty.

It is imperative that an atmosphere be created in all classrooms that allows everyone to raise any question. The prevailing attitude must be that the only “dumb” question is the one unasked. It is important to identify and resolve difficulties and misconceptions as soon as possible. Further, no one should have to worry about a loss of status in raising a question. Teachers should not hide behind an assignment of “mindless” drills. At the same time, they should not escape by having “groups” attempt to do what they themselves cannot. They must be a source of accurate information. No teacher should find it necessary to pretend to have a ready answer to every question. There may well be occasions requiring the admission that there is a need for everyone to seek a solution to some unexpected situation. Teachers must be particularly careful to see to it that no one is allowed to mislead any class. All students must be encouraged, indeed inspired, to replace their feelings of “hating mathematics” with expectations of gaining a solid knowledge and understanding of a consistent, comprehensible, exciting, but demanding, basic discipline. Above all, that must be the primary message conveyed.

With a sound program in mathematics, those students who join the work force immediately on graduation from high school may be able to take advantage of openings that provide potentially challenging opportunities. On the other hand, with a strong background in mathematics, students headed for college can explore the possibilities of the many fields available to them for furthering their formal education In either case, they will be in a far stronger position to meet the demands of modern life. They will have had ample opportunity to develop sound, logical reasoning. The results of their school mathematics will endure well beyond high school graduation. It is thus vital that all students be able to extend their mathematical understanding to the utmost of their abilities throughout their school years. Further, they must recognize that a technological society expects them to have benefited from their mathematical education. They thus must have mastered the basics, learned to think clearly, and acquired substantial flexibility in order to deal with any unexpected situation.

THE ESSENCE OF MATHEMATICS

One of the serious problems with the standards is the loss of the essence of the subject in the effort to reach everyone. The emphasis on “realistic” problems is viewed as relevant and interesting to students since it relates to their daily experiences. Also, giving so much prominence to specific, isolated problems may well fragment the discipline if they are not tied in with theory. Students often can benefit from having actual objects before them so the theoretical can become much more comprehensible. It is of vital importance, however, not to allow the power of mathematics to vanish by having teachers neglect to show that these problems are generally illustrations of the concrete leading to the abstract.

In other days, mathematics was described, on the one hand, as the Queen of the Sciences, and on the other, as the Handmaiden of the Sciences. It would thus appear that there is no question about the gender of mathematics, merely uncertainty about her social status. Actually, these designations attempt to encapsulate a broad, fundamental discipline and describe it in a terse, dramatic way. We may be inclined to dismiss the essence of these titles altogether as not worthy of attention. They may seem rather outmoded and politically unacceptable anyway. Despite any doubts we may harbor, however, let us examine these labels to see what they are attempting to connote. What, if any, is the justification for these characterizations? Are the titles really descriptive of the discipline? Are there, indeed, two roles that mathematics has, as these titles imply? If we reflect on the nature of mathematics, we cannot but conclude that mathematics does play both roles. Sometimes these merge, but each is important in different ways, and we must address each adequately and seriously.

In its abstractions and in its theoretical conclusions, mathematics reigns magnificently, perhaps even majestically, like a “queen.” It is a vital, dynamic, exciting discipline, where new results emerge continually as our knowledge expands. Periodically, long-standing problems that have challenged the most brilliant minds finally give way. Generations of determined mathematicians make constant attempts to find solutions. They never give up until someone is successful.

A recent example, for instance, is Fermat’s last theorem, (Aczel, 1996). The brilliant Frenchman, Pierre de Fermat, before his death at the age of sixty-four in 1665, believed that he had proved that the equation mp + np = kp has no solution for k, m, n, positive integers when p is an integer greater than 2. When p=2, we have the well-known Pythagorean theorem, where we know, for example, that 3² + 4² = 5² or that 5² + 12² - 13². On conjecturing his result, Fermat jotted down the following comment in the mathematics book he was reading: “I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain.” The difficulties encountered throughout the years by leading mathematicians seeking to solve the problem suggest, however, that Fermat did not have a proof at all, certainly not a rigorous one by current standards.

Only very recently, the untiring efforts of Andrew Wiles of Princeton University have led to the proof of the Fermat problem. Wiles’s work so ignited the imaginations even of ordinary citizens that it was featured in major daily newspapers. Further, this famous problem was presented to the public at the Exploratorium, a large San Francisco auditorium. Attendance required the purchase of tickets, but the demand was so great that all seats were quickly taken. This resulted in scalpers entering the scene. They sold tickets to the event at their usual highly inflated prices, a truly amazing development for an advanced mathematics program!

Its abstraction allows mathematics to play the all-important role of queen. On the other hand, its idealizations give surprisingly accurate descriptions of phenomena in the real world. Thus, through its applications, mathematics has incredible power to serve as “handmaiden” to a broad range of diverse areas. The same mathematical tools, differently interpreted, can be applied to seemingly unrelated subjects to solve problems in a variety of disciplines. This is a major contribution of the field that many frequently fail to recognize fully. Indeed, when those who are mathematically educated leave academic life for the commercial world, they find that breakthroughs in applied fields often may be given great prominence. At the same time, the fundamental tools that led to these breakthroughs are generally ignored. The importance of the role played by mathematics thus fails to be noted, let alone valued.

We have a magnificent opportunity to enlighten our students. We must lead them to the recognition, appreciation, and understanding that mathematics is an intrinsically beautiful and exciting discipline in its own right. In addition, it is such a basic subject that it can be applied in many ways to many other areas. Although the utilitarian role of mathematics is emphasized throughout the school curriculum, it is important not to lose sight of its “queenly” characteristic. We need to stress both roles as we strive for balance.

Since mathematics is a unique discipline, based on idealized premises, we are fortunate to have substantial flexibility for teaching. We may either use the applications to develop the abstraction or, conversely, we can start with the abstraction as a lead to applications. By turning at various times in either direction, we can provide great mathematical enrichment for our students. In contrast, when the teaching is restricted to the applications and the abstraction is never reached, students lose a great opportunity. They cannot step back and see the richness of the subject, learn of its great overall power, and appreciate the position of their particular problems within the discipline. Conversely, if no applications are introduced, students are deprived of recognizing the incredible opportunities that the subject provides for solving many problems whatever field they may ultimately adopt.

As we note from Fermat’s last theorem, some mathematical work appears to have no relevance to anything except the field itself. Sometimes, however, often unexpectedly and many, many years later, someone ingenious recognizes that mathematics holds the clue to some need in real life. An important application of mathematics is then born. It was not so long ago, for instance, that number theory, of which Fermat’s theorem is a famous example, was considered totally “pure.” How wrong most of us were to hold that view! It took centuries, but today, results of number theory affect us all in many ways in our daily lives. Examples include the great strides in cryptography made since World War II, as well as such ordinary matters as bank security. Indeed, we have ample reason to appreciate the important role of a “real-life” application of what had been viewed earlier as a totally abstract area of mathematics.

The rush for seats in San Francisco indicates that the public is not always concerned just with the practical. Number theory has many problems that are relatively easy to understand, such as Fermat’s conjecture, although they are difficult and far too deep to prove. Yet the nonprofessional public craves to know more. Many will seek out presentations of deep and difficult mathematical problems even when they cannot really fully follow the solution or totally appreciate it. This is clear from the quotations of some of the San Francisco attendees. Their obvious pleasure at hearing about a great achievement was not dampened in the least by their failure to understand much of the actual techniques of the proof.

While abstraction defines the nature of mathematics, there is the innate requirement for precision, a major characteristic that fails to be duly stressed in the standards. A question has an answer dependent on the hypotheses, whether they are given explicitly or are assumed to be implicit. That answer can change only as the assumptions do, and it cannot be accepted as valid until it has been firmly established. The solution may be found along many paths, but as long as assumptions remain unchanged, there is no variation in the answer. This aspect of mathematics should instill in students the need to question and to be skeptical of unsubstantiated statements in general. This is an important and useful consequence that is particularly vital today when so much misinformation is claimed to be valid.

Another feature of mathematics that is missing from the standards is conciseness. The assumptions on which a result depends are best if they are minimal, without any superfluous information. This characteristic can be translated into daily life in the discouragement of verbosity. Ignorance may sometimes be disguised by an attempt to overwhelm with vacuous and meaningless words and sentences. In addition to conciseness, it is important that the conditions given be consistent without contradicting one another in any way.

Over the years, a notation has evolved to express significant ideas as clearly and succinctly as possible. For example, one that has been universally adopted is that of subscripts and superscripts. This symbolism actually took a long time to develop. It enables us to obtain many new results and also to express old ones more simply and compactly. Students can take advantage of years of development of notation; by familiarizing themselves with the subject matter, they can express themselves clearly using their ordinary language, as well as the mathematical symbolism currently available. Both are important and students should learn to use these in both oral and written communication.

Designating mathematics as an art is the rather subjective characteristic of elegance. An answer to a problem may sometimes be obtained by applying brute force. If the same result can be derived instead by some technique that establishes it more simply and elegantly, that solution is generally more highly appreciated.

Experiments in mathematics cannot be considered as anything but a means of leading to conjectures. Conjecture can never replace sound theory. Before a result can become a part of mathematical theory, it must be proved completely and convincingly. Ambiguities are not acceptable. Experimentation may show that the result holds for a large number of special cases, perhaps even billions. Such examples exist, though not at the school level where the validity of conclusions already generally has been confirmed. Until a proof has been established, however, it is extremely important not to be swayed by an apparent pattern that merely suggests a conjecture, as was the case with Fermat’s results. Students may guess a form for generalizations, but if they do not yet have the mathematical maturity to provide an ironclad justification, they should be aware of the existence of a gap in their knowledge.

Thus, to understand and appreciate the nature of the subject, we must recognize abstraction as the fundamental characteristic of mathematics. We must take note of its emphasis on proofs, precision, and conciseness, and it’s striving for elegance. Further, we must realize that from this abstraction, the discipline acquires the powerful tool of specific application to a highly varied array of problems in many different fields. Any educational program must address these two important features of mathematics, as well as their interactions, in a balanced way. This is not done in the current version of the standards.

MATHEMATICS EDUCATION REFORM

Throughout my professional career, there have been many attempts by various groups to reform the teaching of mathematics. These have occurred periodically. It may happen after the publication of a serious report that decries the poor status of our educational system. On the other hand, it might follow the release of some international comparisons in which our students fail to show up well in the rankings. While far-reaching changes are generally proposed in the various reform programs, strong criticism always ensues for one reason or another. For example, a proposed change might come under attack for its failure to reach all students. On the other hand, it might be because of its omission of some pertinent group from initial direct involvement. In every case, the reform experiments were limited in their reach and invariably were short-lived.

Although many factors, not only inadequate instruction, are responsible for students’ unsatisfactory performance, our concern is with teaching. There is little doubt that poor teaching exists in the schools as well as in the colleges and universities. Teachers who demand mindless calculations in the traditional format fail to convey any semblance of the nature of the subject to their students. Similarly, those fall short who instruct students in the use of the new technology by indicating what buttons to push on their calculators without providing or eliciting essential explanations for the choice.

Despite the problems in our mathematics educational system, individual teachers have never ceased to look for effective approaches. They continually seek to improve their impact regardless of formal recommendations. Exceptionally dedicated teachers always have inspired and influenced their students to become leaders in the field. The mathematical community should use its expertise and influence to improve the teaching of mathematics. In making recommendations for change, however, it is vital that there be awareness of the history of mathematical achievement and instruction. Despite the interest in bringing about rapid change, we must preserve those aspects that have been most successful. Until there is clear evidence acceptable to all that the implementation of replacements will result in significant improvement, no general changes should be contemplated.

Any reform efforts with the goal of educating all students in mathematics must be constructed in such a way that they preserve the intellectual aspects of the field. In seeking relevance by highlighting mathematical problems that arise in daily activities, we must take measures not to destroy the essence of the field. Also, in ensuring that the subject is accessible to everyone, it is important that we keep our expectations high. We must provide a challenging experience to all.

CURRICULUM AND EVALUATION-STANDARDS FOR SCHOOL MATHEMATICS

The NCTM Curriculum and Evaluation Standards (1989) are different from other attempts at revision by virtue of the fact that they have been far more widely disseminated than any in the past. They are being seriously considered now for systemic adoption. Indeed, they are the closest we have come to the introduction of national standards. They thus merit most careful review. We must be as sure as can be that their provisions are sound. We must be certain that any new standards improve education for those who have been overlooked in the past while at the same time preserving what has been effective for others.

School mathematics is a most essentials part of the education of all our citizenry. In a highly technical society, we will need a skilled and competent workforce. In addition to the stars who lead this nation to preeminence in the world of mathematics, we must have many others, at all levels of achievement, to provide needed support. We can no longer afford to waste the talent in our midst. We also cannot accept the conventional wisdom that claims that the bright will succeed anyway and do not need any guidance. As we seek justice and equity, any practice of exclusion cannot be tolerated. We must make sure that everyone has a good opportunity to acquire a solid background in mathematics. This should last at least for as long as legally required, but preferably from kindergarten through high school. How much further will we grow as a nation if we take advantage of the talents of all our students.

Seeking such an end, the NCTM produced three sets of standards: the Curriculum and Evaluation Standards for School Mathematics (1989) the Professional Standards for Teaching Mathematics (1991), and the Assessment Standards for School Mathematics (1995). They created a vision- based on the six items listed at the beginning of this article. One of their laudable goals is reaching out to all students. To achieve their objective, they propose a curriculum that they believe students will find attractive. They expect it to give students a deeper mathematical understanding than they have received in the past. They try to instill in students enthusiasm for the subject, since mathematics is generally unpopular and students often attempt to drop the subject as soon as they can. Further, they want to generate in students a feeling of confidence in their own abilities. All are lofty objectives with which one can hardly take issue.

The problem with the standards is that they are so lengthy and vague that they provide far too much opportunity for misrepresentation and misinterpretation. Even when a statement seems reasonable, it can be, and has been, misconstrued. Too many teachers, for example, have eliminated topics that are listed for reduced emphasis, completely ignoring the actual instructions. In addition, teachers spend an inordinate amount of time in reviews of material that should have been mastered in earlier grades. As a result, in practice, there has been a serious downgrading of content, as well as expectations, of student achievement. Indeed, in some classes, the ablest students are ignored entirely.

For any progress to be made, all students must have basic mathematical tools that must be absorbed by everyone studying any part of mathematics. Introductory definitions and postulates, as well as computational skills, must be developed to a high degree. At every level, it is important to convey to students an accurate description of the essence of the subject in a comprehensible way. We must impart to them our own excitement and delight with mathematics as a fascinating, intriguing, and enticing area of study. Above all, we must be careful not to imply that the entire thrust of the educational mathematics experience is on relevance and the examination of “everyday” problems. In the “social constructivist” direction of the standards, however, there is a clear attempt to reshape the discipline by emphasizing only the utilitarian part. There is a total loss of reasonable balance between abstraction and application.

It is essential that all students have an equal opportunity to learn and to become mathematically literate. Learning any subject thoroughly enough to understand and appreciate its concepts fully, though, requires work. No one can seriously object to keeping students absorbed and interested, even entertained, if this can be achieved in the process. It is not constructive, however, merely to succeed in getting students to acquire a superficial and distorted glimpse of an important discipline. Indeed, it is counterproductive for students to develop an unwarranted feeling of competence. Although this result is not intended by those writing the standards, since confidence in mathematical ability is highly stressed, some teachers have construed this to mean that they should not correct any errors. This would not be a problem if they could then lead their students to recognize their mistakes and correct them themselves. If, however, the teachers are not firm in their knowledge nor clear in their directions, their students simply will end up not learning the subject. It is crucial that the standards be far more specific to avoid their being subject to so much misinterpretation.

One of the major problems with the NCTM standards and the current reform movement in mathematics education is the great emphasis on the relevance of mathematics and its application to everyday events. On the other hand, the innate beauty and elegance of the subject, even in these areas, is rarely mentioned. Any acknowledgment of this characteristic seems parenthetical, with the entire subject becoming distorted as a result.

Diane Ravitch (1995) states unequivocally that the purpose of having standards is “to signal students and teachers about the kind of achievement that is possible with hard work” (p. 5). There is no substitute for the requirement of diligence. The ancient admonition, attributed to Euclid, that there is “no royal road to geometry” (or any other subject in whatever area) still holds. Where in the standards is the indication that a student has to work in order to learn? A difficult, basic subject like mathematics requires nearly all of us to work hard in order to achieve understanding.

Teachers cannot do more than seek to arouse interest and to inspire. All too many students have little conception of their own need to exert some effort to learn. This has generally not been a requirement in their current educational experience. It is certainly not proposed anywhere in the standards. Quite the opposite is implied. Indeed, the entire thrust is on having teachers in the background. They are charged with creating an atmosphere for learning in which there are-no mistakes and everyone’s contribution is recognized, however wrong! Students are to be enticed into the subject by making it accessible. Indeed, since the level of comprehension required is effectively brought down to that of the poorest student, the standards basically eliminate any challenge even to moderately gifted students, eloquent disclaimers notwithstanding.

Let me introduce a not uncommon analogy, which is important as a means of bringing out a comprehensible comparison with mathematics. In the United States, there is a far too prevalent belief that people either are born with mathematical talent or not. Hard work and application to try to excel are not in general considered essential, Facility with mathematical ideas is perceived as a function of some inborn ability.

While we consider any effort expended to learn mathematics rather futile, our attitude toward athletics is quite different. Indeed, we accept the fact that in sports there are superstars. We judge their performance most critically. We expect them to excel at a very high level, not only by virtue of talent, but also by a rigorous and demanding regimen of dedicated hard work. We value the professionals who are outstanding, but we insist that they must work hard to achieve a high degree of competence. Further, there are the coaches whom we require to be familiar with every nuance of their particular activity and to have learned it well enough to inspire and teach others. In addition, we have the amateurs, who take their roles seriously, also work hard, and enjoy the level they can reasonably attain. Finally, there are the fans, who have learned enough and have absorbed the essence of the sport so that they understand what transpires and can appreciate the talent of the best.

Why do many view mathematics in an entirely different light? Why do we not expect our youth to expend the same, or indeed greater, energy and effort on their studies as we demand on the athletic field? In our concern for previously underrepresented groups, are we ignoring even the most talented among them by reducing our expectations for all in their study of serious academic subjects. We hold teachers accountable for their teaching performance but at the same time ask little of students, who are not supposed to struggle to learn, even when the material they must absorb is difficult to understand. Is there no realization that everyone meets obstacles and difficulties in mathematics? Only hard work enables one to reach one’s level of ability. Clearly some have more talent than others, but all have to expend effort to progress.

The standards misrepresent the essence of mathematics entirely when they portray it merely as the “foundation discipline for other disciplines [that] grows in direct proportion to its utility” (NCTM, 1989, p. 7). It is distressing to find such an unbalanced view being promulgated to our teachers and students. Despite the glowing descriptions and the references to the “value” of mathematics that all students are to learn (p. 5), the final result is to regard as important only the real-life applications. It is unfortunate that the thesis promoted throughout is that it is vital to engage primarily in problem-solving. The broad picture and abstract concepts play but a minor role in understanding the nature of the subject.

In attempting to be sure that “everybody counts” in mathematics, the NCTM standards highlight topics that are thought important enough to be emphasized; others are designated for de-emphasis. The result, of course, is that in practice poorly prepared teachers, as well as many others, take advantage of the situation. They interpret de-emphasis recommendations as meaning that they do not have to bother with material that is considered less important. The aim seems to be to empower students by having them construct for themselves even well-known mathematical results. Since such activities can be extremely time consuming, it has become routine in many classes to omit de-emphasized items entirely. As a consequence, virtually all students are required to learn only an unacceptable minimum. The low floor has become the norm. This can hardly be a challenge for any but the mathematically weak student. Our expectation of all our students’ performance has been substantially lowered. We are educating them far more poorly than their abilities would indicate.

No one would support the premise that all students need to be able to give careful proofs of every mathematical statement they encounter. Proofs are generally quite difficult to provide. They take substantial mathematical maturity to give accurately. Nonetheless, the goal of giving proofs has been implicitly challenged in the standards (Wu, 1997, p. 947). Indeed, the standards limit the construction-of proofs to “college-intending students.” Why? Since students who enter the workforce immediately on graduation from high school generally will not have a chance for further mathematical study, is not it important for them to have learned some logical reasoning?

It takes preparation and time to absorb, appreciate, and understand fully what is involved in providing a sound proof. Yet the standards confuse the issue by the introduction and sprinkling throughout of such words as confirm, justify argue, or judge. They do so without making it clear what is intended and why. How is a conjecture confirmed? Is not a counterexample a proof? Who determines whether an argument is logical or valid?

In the Curriculum and Evaluation Standards (NCTM, 1989), for example, there is a discussion of logarithmic properties. High school students then are asked to “confirm a generalization on their computers” by testing “several numerical values” (p. 44). Three are suggested. Is one to conclude that generalizations are valid despite the fact that only a very small, finite number of examples are even tried? Students are not given the opportunity to understand the difference between conjecture and proof or experimentation and ultimate validity. These are concepts that are very important in mathematics. Instead, students are asked to generalize results without any indication that these generalizations are mere conjectures. They may hold even for millions of specific cases and yet not be valid in general.

Although there is some mention of mathematical theory and structure, the standards generally omit difficulties. They fail even to address the need to fill gaps when students are mathematically ready. The need to focus some attention on the abstract nature of mathematics is disregarded. Where are the theoretical aspects of mathematics that form its core?

Although the idea is controversial among psychologists, there is evidence supporting the premise that abstract instruction may at times be even more effective than the concrete (Anderson, Reder, & Simon, in press). Learners are often able to transfer the abstract knowledge derived to another context far more readily than if they are exposed entirely to the specific. A reasonable balance is needed.

PROFESSIONAL STANDARDS FOR TEACHING MATHEMATICS

In the Professional Standards for Teaching Mathematics (NCTM, 1991), the fact is stressed that “each student’s knowledge of mathematics is uniquely personal,” [p. 2]. That observation may be correct. However each student learns, though, the final mathematical result must either agree with the prevailing structure or form the beginning of some extended new theory. If the hypotheses are sound, the conclusion is never in doubt unless it is an unproved conjecture. Ambiguity is foreign to mathematics.

Mathematics cannot be forced to be like other disciplines, nor should it be. Despite the current trend to regard mathematics as “social,” it is not a democratic discipline. Majority rule does not hold when an incorrect result is involved. Even if all the students and the instructor agree on the validity of some conclusion, they could be wrong. It is thus incumbent upon teachers to understand thoroughly much more than the material being taught and to know where it is headed. In addition, they must recognize that they may not always know just what the correct answer might be. If they encourage students to “brainstorm,” they must be prepared to direct the discussion when it gets off the track. They must also accept the fact that there will be times when they are unfamiliar with the correctness of a statement. They then must admit as much while they and their students seek a resolution of whatever problem is in question.

“Logic and mathematical evidence” (NCTM, 1991, p. 3) are certainly essential in confirming the validity of a result. It is the teacher, however, who should be the ultimate judge of what constitutes these characteristics. The teacher should have the background to know, or find out, whether the reasoning is sound or merely seems so. In that respect, the teacher must be an authority on the subject.

Romberg (1992) argues that “to improve teachers, one must improve teaching” (p. 799). It is hard to understand how this can be effected. In the proposed teaching approach, the teacher is asked primarily to be an expert at asking for explanations and establishing a discourse that allows everyone to feel confident, however misguided. Romberg also states that asking students to explain each step “consistently, irrespective of the correctness of student’s analysis, is an important part of establishing a discourse centered on mathematical reasoning” (p. 35). A skillful teacher, who recognizes when an error has been made and asks appropriate questions, may be able to get some students to realize their difficulty. They may change their answers when they try to explain their solutions. Unless incorrect statements, however, are not allowed to stand indefinitely, the damage done can be serious. Never, never, never should students be left with wrong answers without their being made very much aware that a correction is needed. Otherwise, they will form-destructive views, sometimes of basic mathematical concepts. They may be led so far astray that they may find it very difficult to recover lost ground.

Just as learning is very much an individual experience, so too is teaching. The interactions between student and teacher always come into play. Students react and learn very differently depending on who is teaching and how. Various forms of teaching are mentioned, and teachers are encouraged to use a variety of styles. Like much else in the standards, however, this has been misinterpreted so that one form has emerged as being primarily advocated. Teachers who may prefer to use traditional methods (such as, for example, direct instruction) quickly come to realize that these modes of teaching are regarded as having no value and appear in fact to be considered bad teaching.

There is also a strong emphasis on group learning. While mention is made of individual study, the thrust of the standards is that mathematics is a “social” activity. It is stressed that students need to cooperate in groups where they can learn best communally. With the ambiguities inherent in the document, the teaching standards have been interpreted as not allowing for any flexibility in deciding whether that approach is the best choice in a particular class. Merely glossed over are some of the serious problems with cooperative learning (see, for example, Wu, 1997, p. 950). It may be good for some students to develop the ability to work with others on problems. It is not clear, however, that all will benefit from that format. Indeed, there are those who would much prefer to work alone except when they themselves choose to discuss some issues with others. Should we ignore their wishes?

It is distressing to note that a teacher spends time to provide an illustrative problem to contradict “the image of mathematics as a domain of single right answers” (NCTM, 1991, p. 45). Since mathematics is a precise discipline, words are very carefully selected, and the word single is used in one very special way. Thus, if everyday language is intended, it is important that its meaning be clarified. This is not done here; the definition of single is never given. As indicated earlier, the solution to a problem depends on the hypothesis. In the vignette described on page 45, section 3.1, the teacher states that “the Wolverines scored 30 points in the first half of last night’s basketball game. The unusual thing is that they did it without scoring a single foul shot.” She then asks, “How did they score the 30 points?” Since the hypothesis here leads to a single linear equation in two variables, there will be multiple pairs of nonnegative integers that will constitute the single solution, unless some different definition has been introduced. It is important to differentiate between a complete and an incomplete answer of a single solution. If, however, a concept is to be used in some new way, it is vital to define it completely, making sure not to introduce a contradiction. Again, the details of a real-life problem are obscuring what should be opportunities to emphasize significant mathematical properties. Instead, misinformation about mathematics is disseminated. The teacher sought to disprove a reasonable property of mathematics, namely, its precision, by changing a definition without in any way clarifying that fact. How would that teacher deal with, say, a quadratic equation that is taught just a few grades later? Will she be in the same situation as the experienced high school teacher cited in the introductory material of this article? Is not the fundamental theorem of algebra important for teachers to know, not only at the high school level, but earlier as well?

Returning to the vignette described above (NCTM, 1991, p. 45), we note another aspect of the problem of the Wolverines. If all students are to be reached, is it the expectation that everyone knows how basketball games are scored? Why ignore issues of equity here? How could a student unfamiliar with scoring basketball games have solved that problem unless the information were given that two- or three-point shots only were involved?

ASSESSMENT STANDARDS FOR SCHOOL MATHEMATICS

Throughout the assessment commentary (NCTM, 1995) is the implication that the glowing description of the proposed changes will result in “[our having] high expectations for all students, envisioning a mathematics education that develops each student’s mathematical power to the fullest” (p. 15). We certainly aspire to reach that goal. Is there any valid evidence, however, to substantiate the premise that such will be the effect if we radically reform the curriculum as proposed? The drastic abandonment of every aspect of the “traditional” ways might sound good and has some theoretical appeal. It has not been shown, however, that it can produce students with greater understanding of mathematical concepts. The “old ways” at least have withstood the test of time. Their strengths as well as their flaws have become clear. Further, over time, some trouble spots have been eliminated and changes made. The NCTM and others have pointed to remaining weaknesses, and these must be addressed. On the other hand, describing the proposed reform as correcting all the ills of the past by replacing everything on all fronts with untried proposals is a dangerous route to follow. It must not be used as an overall experiment that involves all the nation’s children who attend public schools.

It is unreasonable to dismiss the traditional curriculum entirely without recognizing and preserving any of its strengths. While it can be criticized for some failings, such as not taking advantage of the potential talent of those who did not fit a given mold, there is little evidence to support the reform claims that are being presented as fact. Indeed, until we have a sound means to determine whether, overall, students exposed to the reform curriculum will outperform those of earlier times, there is no valid indication to substantiate systematic adoption.

It is hard to envision how teachers will have time for all the detailed assessments that are prescribed, even though they are supposed to be part of the learning process of students. Is there evidence that performance assessment or portfolio assessment provide valid, reliable information that justifies the time involved? Unless teachers have a real grasp of their subject matter, they may not be able to deal adequately with students who “respond in unanticipated ways” (NCTM, 1995, p. 15). The authors of the Assessment Standards seem to miss the whole point of mathematics when they state that “students may need to specify the assumptions they are making when they communicate the results of their work” (p. 15). Are not assumptions the heart of any result? And, should not a student know that results without specific assumptions are meaningless? Indeed, both definitions and assumptions must be clear to avoid the problems encountered in the example offered in the vignette.

At one time, the philosophy among some educators was that if one knew how to teach, it was not necessary to know the subject matter. Fortunately, that is no longer the case. In the NCTM Assessment Standards, on the other hand, there is too great a requirement that teachers carry out what appear to be unnecessarily demanding assessments. One cannot help feeling that students would benefit far more from teachers whose background in subject content was greater. Indeed, as has been the case in past experiments, those whose knowledge of the subject is strong will circumvent the flaws in the standards and will thus improve the mathematical education of their students.

The vignette in the Assessment Standards (NCTM, 1995, p. 35) is intended as an example of good, explicit, and prompt feedback from teacher to student. It is thus disappointing to note that the seventh-grade teacher involved misses the essence of the problem that the student, Stewart, submits. Moreover, it is unfortunate that the teacher does not use the occasion to point out that whatever generalization Stewart states, it is merely a conjecture. Although at this stage that is all that Stewart can do, there still remains a gap created by the requirement of a proof to confirm the validity of the result for all possible cases including any that Stewart has established. Stewart begins by providing a concrete representation to describe the problem. He is trying to show that the sum of a number of the first few consecutive positive odd integers is equal to the square of that number. To this end, he sketches his view of the problem by drawing a diagram of one square within another. Each square has a side one unit larger than its predecessor (see Figure 1). He then explains his work by stating that he “got the next squared number from 1 by adding 1 and 3 which are odd.” Then he indicates that it is necessary to “add the next odd number 5, and you get the next squared number 9.” He comments further, “See how the pattern goes: 16 = 1 + 3 +5 + 7. Every square will work like that.” He then concludes “all square numbers are the sum of odd numbers because of the picture I made of squared numbers.” The teacher’s written comments fail to recognize that the seventh grader’s picture is far more than an indication of his realization that 1 + 3 + 5 + 7 = 16. It also shows Stewart’s awareness that 1 + 3 = 4, 1 + 3 + 5 = 9, and 1 + 3 + 5 + 7 + 9 = 25. Indeed, he was on the way to a reasonable guess of a generalization, not only by his verbal explanation, but also by his visual one. The teacher’s remarks to Stewart are distracting as he misses his student’s major point. Just any odd numbers that add up to 16, like 11 + 5, as the teacher suggests, have nothing to do with this particular result. Indeed, Stewart brings out in his illustration, as well as in his clarification, that he adds the next odd number each time to get the squared number for the sum. With questions along the lines Stewart was pursuing, the teacher might well have succeeded in getting Stewart to perfect his answer to the point of stating it precisely. In abstract form, it can be written as 1+3+5+. . . + (2n-1) = n2 and be proved by induction, a more sophisticated procedure than most seventh-grades would know or be ready to understand and appreciate.

In the standards, memorization of any kind is derided, although no serious mathematician would propose mindless memorization or consider it productive. We all would favor having students who understand the concepts and appreciate the value of learning. Some simple arithmetic facts, however, must be memorized, at least for efficiency. It is unreasonable, for example, to have no recourse when a computer is down, with clerks unable to carry out primary-grade calculations without its use. Unfortunately, this is not a rare occurrence. It illustrates the justifiable fear of many, based on empirical and anecdotal evidence, that the availability of calculators and computers does indeed make students dependent on them for the simplest calculations. It is regrettable that basic principles essential for understanding mathematics are largely eliminated from the curriculum. The new technology interferes when introduced before young children have fully mastered important arithmetic facts. It is of concern also that those who have learned their arithmetic facts rely so heavily on the gadgetry that they make no effort to recall or work out what they once could do readily. It is wiser to reserve the technology for the problems encountered after the basics have been absorbed.

While emphasizing problem-solving of real-life problems on the one hand, those seeking to revise the assessment procedures completely and to eliminate comparisons among students, even at the high school stage, are ignoring the fact that such comparisons are a daily occurrence for everyone in the real world. There is a need for an evaluation of students’ knowledge that is useful, reliable, and simple to administer. Teachers must have some impartial way to assign grades to upper-level students that will be useful in predicting future performance. Without being able to ascertain the effectiveness of the proposed NCTM standards in actual practice, we can but consider them an untried experiment that will suffer the fate of previous reforms if they are left unchanged, yet are adopted.

CONCLUSION

Perhaps the greatest achievement of the standards is the extensive dialogue that has ensued within many groups. These include parents, who have failed to understand that “a return to basics” is not the answer; business and community leaders, who observed first-hand that their employees’ knowledge of mathematics needs substantial improvement; mathematicians, who have been aroused to give increased attention and time to this very important educational issue where their vision and their knowledge of where the subject is heading can serve as an invaluable resource; mathematics educators, who can provide criteria that can be implemented and who must thus be careful to ensure that these reflect tested theories about how students learn mathematics; teachers, who can best determine whether proposed changes will actually be workable in classrooms; and politicians of all sorts, who must satisfy their constituents.

It is not surprising, with such diverse constituencies holding a wide range of different views, that these reforms, as predicted (NCTM, 1989, p. 255), are meeting substantial opposition. That would be expected in any case. What seems reasonable and straightforward in theory often is a far cry from what happens in actual implementation in the classroom. Many mathematicians are concerned that the subject is being taught in so distorted a fashion that students are getting an unrealistic indication of their knowledge of mathematics. Indeed, the standards may well have succeeded in generating unwarranted feelings of confidence in mathematical competence.

Some groups may have to be convinced that the standards are a substantial improvement by more than the glowing descriptions of those directly involved in their creation. Objective, external, independent tests are needed to determine whether the current reforms do lead to better performance in mathematics than has been the case with students studying in previous programs. If there is no substantive evidence to support such a conclusion, the standards ultimately will join prior attempts at reform by becoming merely a temporary experiment that failed. The problem of finding a way to improve mathematical education will still have to be resolved.

On the whole, it seems unreasonable to propose such drastic changes on a systemic level when much of the material still has not been fully investigated. Let us hope that the next version of the standards avoids the flaws that occur in the initial one.

This article is a modification of a paper presented in. Madison, Wisconsin, in the summer of 1996 when the National Institute for Science Education (NISE) held an invitational conference to examine the issue of the possible systemic adoption of the NCTM standards.

REFERENCES

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