Comments: NCTM's Curriculum and Evaluation Standards

The comments in this article provide a background to the key notions underlying the National Council of Teachers of Mathematics (NCTM) Curriculum and Evaluation Standards for School Mathematics: first, the organization’s intent when it decided to prepare this document; second, the anticipated use of the standards by teachers, schools, and states to change how, mathematics has been organized and taught in American schools; third, the current process of preparing revisions to the standards by NCTM. The author ends by offering comments about unanticipated reactions to the document.

As former chair of the Commission on Standards for School Mathematics for the National Council of Teachers of Mathematics (NCTM) from 1986 to 1995, I have been asked to comment on NCTM’s attempt to lead the reform of school mathematics during the past decade.1 The vision of what mathematics students should have an opportunity to learn, how mathematics should be taught in classrooms, and how students and programs should be assessed and evaluated has been described in three documents prepared by NCTM: Curriculum and Evaluation Standards for School Mathematics (1989) Professional Standards for Teaching Mathematics (1991) and Assessment Standards for School Mathematics (1995). For this article, I have chosen to comment only on the Curriculum and Evaluation Standards because it has received the most attention during the past decade.

BACKGROUND

To put this work in perspective, let me go back to 1996, when the development of the Curriculum and Evaluation Standards was started, and summarize what NCTM was attempting to do. At the very beginning of the document, we wrote:

all students need to learn more, and often different, mathematics, and instruction in mathematics must be significantly revised. (NCTM, 1989, p. 1)

The key notions in that statement are:

1. Teaching mathematics to “all students” emphasizes the fact that all students need to be mathematically literate if they are to be productive citizens in the twenty-first century. In particular, this includes all underrepresented groups, not just “talented, white males.”

2. “More mathematics” implies that all students need to learn more than how to manipulate arithmetic routines. At present, nearly half of U.S. students never study any mathematics beyond arithmetic.

3. “Often different mathematics” refers to the fact that the mathematics all students need to learn includes concepts from algebra, geometry, trigonometry, statistics, probability, discrete mathematics, and even calculus.

4. “To learn” means more than to be shown or to memorize and repeat. Learning involves investigating, formulating, representing, reasoning, and using strategies to solve problems, and then reflecting on how mathematics is being used.

5. “Revised” instructions implies that classrooms must become discourse communities where conjectures are made, arguments presented, strategies discussed, and so forth.

To capture our thoughts on these ideas, I have chosen to discuss three of these issues: “all students,” “more and often different mathematics,” and “revised instruction.”

ALL STUDENTS

If students are to be mathematically literate and productive citizens in the twenty-first century, our belief is that all students need to have a good mathematics background. This concern about “mathematics for all” was in contrast to the reality of a decade ago. At that time, about 40 percent of U.S. students studied no more mathematics than what is typically covered up through Grade 8—shopkeeper arithmetic. These students were expected to learn only paper-and-pencil calculations and routines for whole numbers, common fractions, decimals, and percents. Furthermore, this 40 percent included a preponderance of minority students. Another 30 percent of the other students were expected to take an additional two years of high school mathematics—year courses in algebra and geometry. The assumption was that this was sufficient mathematics for general college entrance. This is the background that most elementary and middle school teachers have. Then another 20 percent of students were likely to be going to college to study in areas that required more mathematics. These students would take another year or two of mathematics in high school and, it was assumed, would then take some mathematics courses in college. Finally, about 10 percent of the students, including potential engineers and science and mathematics majors, in college would take more mathematics courses—four years of mathematics in high school (perhaps including advanced placement courses). This is the mathematical background of most high school teachers. These latter groups were composed predominantly of white males.

The long-term objective of the reform movement was to change both the percentages (40 percent, 30 percent, 20 percent, and 10 percent) and the racial and gender mix of students in these groups by focusing our work on providing the lower 90 percent of the population of American students with a reasonable opportunity to learn more mathematics.

This concern for expanding the opportunity to learn mathematics had been voiced on several occasions during the past century, but was rarely listened to by educational policymakers or the public at large. However, in 1984 the message was loudly shouted to the public in A Nation at Risk (National Commission on Excellence in Education, 1983) and Educating Americans for the 21st Century (National Science Board Commission, 1983). The authors of those documents claimed that competing in today’s global economic environment depends on a workforce knowledgeable about the mathematical, scientific, and technological aspects of the emerging information age. The belief expressed at that time was that

in most classrooms at all school levels, mathematics instruction is neither suitable nor sufficient to adequately equip our children with the mathematical concepts and skills needed for the 21st century. Furthermore, unless something is done to alter current schooling trends, conditions are likely to get worse in the coming decade. (Romberg, 1984, p. 1)

The mathematical sciences education community seized on this belief to initiate a long-term reform strategy for school mathematics. Two planning meetings were held and produced strikingly similar recommendations, including a call for a new content framework for school mathematics.2

MORE AND OFTEN DIFFERENT MATHEMATICS

The argument was that 8 years of arithmetic for 40 percent of the population and 2 years of high school mathematics for another 30 percent were simply no longer adequate. The basis of this argument was that jobs are changing. Today no one makes a living doing paper-and-pencil calculations. Calculators and computers have replaced shopkeeper calculations in business and industry. Additionally, these electronic tools are capable of doing massive calculation tasks quickly, displaying information in a variety of ways, and so forth—they have changed the skills that need to be emphasized in mathematics courses. Because the computer is a fast idiot that can carry out prodigious calculation feats, its impact on mathematics has been similar to the impact of the printing press on writing and reading. The printing press, which made certain skills (e.g., calligraphy) obsolete, also made texts universally available, vastly increasing the need for persons to write and read.

Similarly, today’s technology has made a certain range of skills (from simple to intricate paper-and-pencil calculations) obsolete, thus making it possible for persons to model complex problem situations, make predictions, and so on. Because of the pervasiveness of these technological advances, reflective knowledge about the power of mathematics has become important for every citizen in a democratic society. Furthermore, the jobs that use this technology are increasing rapidly and often require a real understanding of traditional mathematical topics, as well as some topics not in current school courses (e.g., discrete mathematics, mathematical modeling, statistics). In fact, although the discipline of mathematics should be understood as both an object of understanding and a means of understanding (Romberg & Kaput, in press), in too many mathematics classes, students are taught only the formal properties of the various mathematical domains while the applications of mathematics derived via mathematical modeling are overlooked. Students must study the mathematics used in such applications in order to grasp the power of mathematics to solve real-world problems and to reflect on the consequences of such uses.

REVISED INSTRUCTION

In most mathematics classrooms, daily instruction follows a five-step sequence: review of homework, explanation and illustration of a problem type by the teacher, work by students independently on a set of similar problems, summarization of work and responses to questions by the teacher, and assignment of homework, which consists of similar problems. We argued that this mechanistic sequence should be changed because to learn something involves more than to be shown and to be asked to repeat what was shown. Furthermore, the technology of traditional instruction includes a basal text, which is a repository of problem lists; a mass of paper-and-pencil worksheets; and a set of performance tests. Although a few of the books include things to read, there is very little that is interesting to read. Thus, workbook mathematics gives students little reason to connect ideas in “today’s” lesson with those of past lessons or with the real world. The tests currently used ask for answers that are judged right or wrong, but the strategies and reasoning used to derive those answers are not evaluated.

Israel Scheffler’s (1975) denunciation of the traditional mechanistic approach to teaching basic skills and concepts illustrates the difficulties with this perspective about school mathematics:

The oversimplified educational concept of a “subject” merges with the false public image of mathematics to form quite a misleading conception for the purposes of education: Since it is a subject, runs the myth, it must be homogeneous, and in what way homogeneous: Exact, mechanical, numerical, and precise—yielding for every question a decisive and unique answer in accordance with an effective routine. It is no wonder that this conception isolates mathematics from other subjects, since what is here described is not so much a form of thinking as a substitute for thinking. The process of calculation or computation only involves the deployment of a set routine with no room for ingenuity or flair, no place for guesswork or surprise, no chance for discovery, no need for the human being, in fact. (p. 184)

In a similar vein, George Polya (1957) argued:

A teacher of mathematics has a great opportunity. If he fills his allotted time with drilling his students in routine operations he kills their interest, hampers their intellectual development, and misuses his opportunity. But if he challenges the curiosity of his students by setting them problems proportionate to their knowledge, and helps them to solve their problems with stimulating questions, he may give them a taste for, and some means of, independent thinking. (p. v)

Traditional school mathematics fails to provide students with any sense of the discipline’s historical or cultural importance, nor with any sense of its usefulness: “We have inherited a mathematics curriculum conforming to the past, blind to the future, and bound by a tradition of minimum expectations” (Mathematical Sciences Education Board [MSEB, 1990, p. 4]).

In contrast, when we developed the standards, we thought instruction should take an approach involving investigating, formulating, representing, reasoning, reading, using strategies to solve problems, proving assertions, and reflecting on how mathematics is being used. Classrooms should become discourse communities where conjectures are made, arguments presented, and strategies discussed.

SUMMARY

NCTM’s intent was to set in motion a lengthy process to change the way in which mathematics has been organized and taught in American schools. At that time it was clear what we did not want: the routine, unimaginative instruction happening in most classrooms, which filters out too many students from further study of mathematics.

I must admit, however, that when the Curriculum and Evaluation Standards for School Mathematics was prepared, we did not have a clear image of what it was we wanted as an alternative or how reform could be achieved. Rhetoric about the importance of solving problems, or about the need for students to make conjectures and build arguments, or about doing something other than hours of routine calculation with little understanding does not make such changes actually happen. In fact, we did not have examples of what the implementation of such slogans would actually mean in U.S. classrooms, or how long it would take. But admitting our lack of clarity does not mean we had no hunches or conjectures about what might be done. The extensive experimentation with new technologies, the emerging psychological research from a cognitive perspective, and the host of examples of what and how mathematics was being taught in other countries provided a variety of ideas, examples, and research. What we hoped would happen was that teachers, developers, researchers, and others would take the NCTM ideas in the documents as a starting point for a reform movement involving creative development and trials of new materials and methods of instruction.

CURRICULUM AND EVALUATION STANDARDS

The forty curriculum standards that NCTM produced were grouped into three levels corresponding with Grades K-4, 5-8, and 9-12. Four of the standards at each level are common standards about the mathematical process of problem solving, communication, reasoning, connections; the other twenty-eight deal with mathematical content. Each standard is a relatively brief statement, and has the following format:

These statements are the core of the document. They indicate a vision of the mathematical content that we agreed that all students should have an opportunity to learn. Furthermore, these are the proposed criteria that schools and states are to use to judge the quality of their mathematics curriculum and their textbooks. Look, for example, at the following standard:

STANDARD 6

NUMBER SYSTEMS AND NUMBER THEORY

In Grades 5-8, the mathematics curriculum should include the study of number systems and number theory so that students can—

● understand and appreciate the need for numbers beyond the whole numbers;

● develop and use order relations for whole numbers, fractions, decimals, integers, and rational numbers;

● extend their understanding of whole number operations to fractions, decimals, integers, and rational numbers;

● understand how the basic arithmetic operations are related to one another;

● develop and apply number theory concepts (e.g., primes, factors, and multiples) in real-world and mathematical problem situations. (NCTM, 1989, p. 91)

Each of the standards was written to indicate elements of a mathematical content domain that ought to be in the curriculum in those grades. The bulleted items indicate the key elements in that domain appropriate at that level. We expected educators to use the standards when they examined a curriculum plan or textbooks.

INTRODUCTIONS, EXAMPLES, AND NEXT STEPS

In addition to the standards, there are three other parts in the Curriculum and Evaluation Standards for School Mathematics. First, there are the introductions: one to the whole document and then two to the K-4, 5-8, and 9-12 sections. The introductions were written to set the stage for the reader. They contain brief statements about goals, the need for change, and the assumptions on which the standards were based. One topic in the overviews to each level of the standards has been misunderstood: namely, the summary tables indicating “increased and decreased attention.” In particular, “decreased attention” does not mean omit. The message we were trying to convey was that the emphasis had to be shifted from students’ simply being proficient at using procedures to their understanding them. For example, many students (and most adults) can divide a fraction by a fraction by “invert and multiply” and can multiply a two-digit number by another two-digit number to produce a correct answer, but they are unable to give any reason as to why either procedure works other than to say “that’s what I was taught to do.”

Second, following each standard, we wrote two subsections: an explanation of the standard, titled “Focus,” and a “Discussion,” containing examples for lessons. Sometimes these subsections have proven problematic. In discussion, a reader will object to the examples because “you didn’t include my favorite example.” Also, some examples are good, and some, not so good. In fact, when I now read certain examples, I ask myself “Why didn’t we include something else?” Next, I get comments from persons who say, “I’ve read this example, and that meant. . . ”

Unfortunately, often that is not what we intended. We realize that teachers bring to any task their background, their experience, their use of language and terms, and so forth. Unfortunately, their understanding of terms does not always match ours. Most of the detailed comments and criticisms of the Curriculum and Evaluation Standards for School Mathematics are not about the standards, but about the examples.

Third, there is a final section to the document entitled “Next Steps.” We expected educators to use the document as a starting point for an open discussion about what would be included in the school mathematics curriculum. In fact, the Mathematical Sciences Education Board (MSEB), following the publication of the Curriculum and Evaluation Standards for School Mathematics, advocated a year of national dialogue. The strategy we expected would be followed was based on the notion that because we live in a supply-and-demand economy, a demand would have to be created for different texts, instructional procedures, and tests following a seven-step iterative strategy (see Figure 1).

The steps and the relationships between them are as follows:

1. Before any plan can proceed, a need for change must be established.

2. Vision is a key factor. To create a new program one must consider values, goals and standards. The NCTM standards documents were designed to fulfill this vision.

3. Planning includes involving everybody in a system or school in arriving at consensus about the details of long- and short-range plans (with timetables) for change. It is at this step that demand is created.

4. The next step involves identifying specific elements of the system to be targeted for change (curriculum material, instructional methods, assessment material, teachers, technology) and setting priorities.

5. Any system depends on suppliers. Schools must demand that textbook publishers, testing companies, staff developers, teacher education programs, and others contribute the ingredients necessary for the desired changes in elements.

6. Then it is time to make the new materials, procedures, and programs operational. Draft materials and procedures have to be tried out, feedback from this trial phase matched with the vision and the plan; and revisions made.

7. Finally, a product (a curriculum, an instructional procedure, assessment material) is developed. Quality then should be judged in terms of what students are able to do and whether what they do meets society’s needs.

Unfortunately, this did not happen as envisioned. The document was meant to be a background document to initiate a process for changing the content of school mathematics, not to be taken as the final word. The rational sequence of short-and long-term planning, targeting needed changes, and trial and re-trial of new products in many cases was cut short in the search for “quick fixes.” For example, publishers began to claim that their materials met the standards shortly after publication of the Curriculum and Evaluation Standards for School Mathematics. Such claims were a marketing tool with little substance behind them. They are what Preston (1996) called “puffery in advertising.” Such claims are legal, but often deceptive. To reflect the standards truly, one would have to prepare a content blueprint for an age span (K-4, 5-8, or 9-12); develop, try out, and revise instructional units designed to provide students an opportunity to learn with understanding the key ideas in such mathematical strands as number, algebra, geometry, statistics and probability, and so forth; create professional development programs; and so on. Only in the past year have new programs become available that were developed in the detailed manner just outlined and with funding from the National Science Foundation.

REVISED CONTENT STANDARDS

At this time, NCTM is in the process of developing an updated set of curriculum standards. Development of the current document began in 1986. The first draft was printed for review in 1987, revised in 1988, and published in 1989. Every ten years, NCTM said it would prepare a revision. The revisions being considered include the following:

● a reorganization of sections to fit more closely with school practices. There will be sets of standards for Grades pre-K-2, 3-5, 6-8, and 9-12.

● The four basic process standards (problem solving, communication, reasoning, and connections) will be retained, but their role in cutting across the content standards will be emphasized. They were not supposed to be considered independent of content. Some readers claimed they developed problem-solving activities, for instance. One does not just solve problems. One solves problems within particular mathematical domains.

● A fifth common standard with respect to procedures or routines will probably be included. In retrospect, although procedures and the need to understand procedures appear in many standards, they are buried in the bullets and examples.

● There will be an emphasis on content strands (number, algebra, geometry, statistics, and so forth) across levels. For example, at present there are several different standards on number in the K-4 and 5-8 standards, but nothing at 9-12. An examination of the coherence of ideas in a strand across the grades is needed.

And then of course, the examples that are included will be updated, and their quality and appropriateness verified.

THREE FINAL COMMENTS

First, parents and others have commented about the importance of the mathematics they were expected to learn, basic skills and math facts, and have expressed concern about their role in the reform efforts. The emphasis in the Curriculum and Evaluation Standards for School Mathematics on problem solving, communication, reasoning, and connections does not seem to reflect the importance of these skills. Of course we want all students to know how to assign numbers to things by counting and measuring. We want all students to learn how to write mathematical sentences, formulas, graphs, tables, and so on, to represent a variety of common situations (e.g., an addition sentence to represent a join situation; a functional equation to represent the relationship between distance and time; a graph to represent a batch of data). We want all students to understand procedures for calculating or manipulating mathematical symbols, to be able to decide on the appropriate procedure for a given situation, and to carry out that procedure so that an answer or prediction is found. We want all students to be able to build justifications for their mathematical assertions. All of these ideas are contained in the document, but we also expect students to be able to use these ideas to solve non-routine problems.

Second, when we finished writing the Curriculum and Evaluation Standards for School Mathematics, we were unsure of its impact. The pessimists among us thought the document would occasionally be read by graduate students, but would have little overall impact. After all, there have been documents like this produced fairly regularly in the history of education that had little real impact. Aware of this possibility, NCTM provided a copy free to all of its members, held press conferences and briefing meetings with a variety of audiences, and, with the help of MSEB, solicited support from a host of mathematical professional organizations. The effort paid off in that today there is wide acceptance of the Curriculum and Evaluation Standards for School Mathematics at a very general level. I have heard politicians say, “We need to do something in mathematics. This looks like a reasonable set of ideas . . . solve problems, communicate, reason . . . deal with numbers . . .” without really understanding the message. One reason it has succeeded is because it filled a political void. A second reason was that it was written in very general terms. It is a vision, not a recipe. It contains only ideas that one should think about when one considers the content in the school mathematics curriculum. A third reason has to do with the fact that it was written by NCTM, the group whose members are responsible for teaching students in classrooms. In the past, teachers have had little voice in what they teach. In fact, the organization has argued, teachers have long needed something that they can put in front of administrators and other policymakers to say, “Here is what we ought to consider in our curriculum or as we adopt textbooks, rather than being told by publishers what is available.” Teachers have found the document empowering. For me, this fact has made the effort truly worthwhile.

Third, the biggest surprise for me has been the unintended impact on both school mathematics in other countries and on other disciplines in the United States. None of us involved in preparing the Curriculum and Evaluation Standards for School Mathematics could have predicted the translation of this document into such languages as Spanish, Russian, and Japanese, and the parallel work in several other nations. Nor could we have anticipated that “standards-based” work by professional organizations would occur in almost every school subject in this country. It is both exciting and humbling to have led such a movement.

SUMMARY

In mathematics we did not prepare a recipe for teachers or schools to follow. Instead we prepared a vision of a school mathematics curriculum. This vision reflected our understanding of changes in society, changes in mathematics and the uses of mathematics, and changes in what is known about learning. We expected schools to establish a dialogue with constituents (parents, the community, students, other teachers, administrators, etc.) about this vision, to consider their current practices, to develop a plan within their community for change, and so forth. This is exactly what has happened in many schools during the past decade, and what we believe should be happening in all schools. Only then will all students have the opportunity to learn mathematics with understanding, which in turn will make it possible for them to be productive citizens in the next century.

REFERENCES

Conference Board of the Mathematical Sciences. (1983). New goals for mathematical sciences education. Washington, DC: Author.

Mathematical Sciences Education Board. (1990). Reshaping school mathematics. Washington, DC: National Academy Press.

National Commission on Excellence in Education. (1983). A nation at risk: The imperative for educational reform? Washington, DC: Government Printing Office.

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics (1995). Assessment standards for school mathematics. Reston, VA: Author.

National Science Board Commission on Precollege Education in Mathematics, Science, and Technology. (1983). Educating Americans for the twenty-first century: A plan of action for improving the mathematics, science, and technology education for all American elementary and secondary students so that their achievement is the best in the world by 1995. Washington, DC: Government Printing Office.

Polya, G. (1957). How to solve it. New York: Doubleday.

Preston, I. L. (1996) The great American blow-up: Puffery in advertising and selling. Madison: University of Wisconsin Press.

Romberg, T. (1984). School mathematics: Options for the 1990’s. Washington, DC: Department of Education.

Romberg, T., & Kaput, J. (in press). Mathematics worth teaching, mathematics worth understanding. In E. Fennema & T. Romberg (Eds.), Classrooms that promote mathematical understanding. Mahwah, NJ: Erlbaum.

Scheffler, I. (1975, October). Basic mathematical skills: Some philosophical and practical remarks. In The NIE Conference On basic mathematical skills and knowing. Vol. 1. Contributed position papers (pp. 182-189). Euclid, OH, and Los Alamitos, CA: SWRL Educational Research and Development.