Occasional Paper

Commentaries on Mathematics and Science Standards

William Clune, Deborah T. Haimo, Judy Roitman, Thomas Romberg, & John C. Wright and Carol S. Wright

Introduction

The four papers in this set of commentaries reflect the richness and energy of the issues surrounding national standards in mathematics and science as they begin to reach a wider audience and take shape in implementation. The dominant issue discussed and debated in the papers is the shift in emphasis from memorizing procedures (calculations) to problem solving and understanding. The authors themselves make an interesting group in this respect, comprising three university professors of mathematics or science with longstanding interest in educational reform (Haimo/mathematics, Roitman/mathematics, John Wright/chemistry), a professor of education and the Chair of the standards committee of the standard-setting National Council of Teachers of Mathematics (NCTM standards, Romberg), and a teacher/ administrator with hands-on experience implementing the new standards (Carol Wright).

A notable common theme across the papers is sympathy with the twin goals of the standards to (a) reach more students and (b) make mathematics and science more interesting and meaningful to teach and learn. Within this broad umbrella the papers differ in the degree of skepticism about how well the standards achieve the objectives. Most skeptical is D. T. Haimo who sees the potential for major distortion, confusion, and lowering of the quality of mathematics content and instruction. Judy Roitman, the other mathematics professor, gives a more favorable but still mixed review, seeing many of the same problems but also many strong points. With his long involvement in the development of the mathematics standards, Tom Romberg tends to view the criticisms as refinements if not quibbles and sees the reforms as a quantum improvement for students who historically received rudimentary and inferior instruction. But Romberg does concede the force of many of the criticisms in pointing to a new round of revisions of the mathematics standards that emphasize the importance of traditional procedures (calculations), the integration of problem solving with content, and the details of curriculum and instruction. The Wrights are unqualifiedly enthusiastic about the science standards (calling them "brilliant") but skeptical about the capacity of our current teaching force and social culture to implement and accept such lofty and ambitious goals—the education of an entire population capable of independent inquiry and critical reflection.

The papers are fascinating in themselves, and I will not attempt a summary or synthesis. Instead, I look across the papers for areas of agreement, disagreement, and a shared sense of the incomplete agenda. I conclude that many of the issues appropriately joined at the conceptual level over the standards can only be resolved in the context of real curricula and instruction, where specific tradeoffs between competing goals and risks are made and can be evaluated.

An interesting place to start a discussion of the goals of the standards is Romberg's description of the origins and impetus behind the mathematics standards—the highly stratified but overall low level of traditional mathematics instruction among the nation's students. A decade ago, Romberg tells us, about 40% of American students stopped at eighth-grade mathematics, another 30% at the second high school course, another 20% with enough to qualify for selective colleges, and 10% with enough to prepare for scientific training in college.

Our long-term objective was to change the percentages—40%, 30%, 20%, and 10%—by focusing our work on the needed changes for the 90% of American students who took the least mathematics. (p. 44)

In light of this comment, we might ask ourselves why reform should not consist of simple upgrading and acceleration—more students taking traditional courses. In fact, that is one common strand of reform, sometimes called "intensification," and one that is clearly responsible for some of the gains in student achievement that occurred during the 1980s. Let us be clear about this. Despite the protestations of constructivists, substitution of Algebra for General Mathematics and getting more students to reach Calculus are legitimate goals of reform.

But there is another side of reform captured in Romberg's disparagement of eighth-grade mathematics as "shopkeeper mathematics":

These students were expected to learn only paper-and-pencil calculations and routines for whole numbers, common fractions, decimals, and percents. (p. 43)

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Today no one makes a living doing paper-and-pencil calculations. Calculators and computers have replaced shopkeeper calculations in business and industry. . . . The fields that use new technologies are growing rapidly and often require a deep understanding of traditional mathematical topics as well as some topics not in current school courses. (p. 44)

This dislike of mindless calculation has broad support among reformers and the other authors of our papers. If the nightmare of traditionalists is kids who can't get the right answer, the nightmare of other reformers is kids who don't know what a right answer means. Roitman begins her essay with the example of students who, when asked how many buses at 23 children each would be required to carry 121 students, answered 5 6/23, or rounded down to 5. "Our kids could calculate, but they couldn't make sense" (p. 23). D. T. Haimo says that,

when teachers themselves have little feel for what they are doing, cannot really explain why they perform certain operations or where the subject is heading, they teach mechanically, stressing only formalisms. . . . [T]heir students suffer the consequences. (p. 7)

And for the Wrights, thinking, not calculation, is the essence of science: inquiry, self-confident discovery, disciplined criticism, cooperative problem-solving. This "making sense" of mathematics and science, sometimes called "teaching for understanding" has a special connection with equity. Disadvantaged students often have been taught the lowest level, most mindless version of basic skills and may have a special need for instruction in complex problem solving. For these kinds of reasons, the NCTM standards for mathematics included four "process standards": problem solving, communication, reasoning, and connections.

But the consensus on mathematical and scientific reasoning is deeper yet. Disapproval of "mindless calculations" is not the same as disapproving all calculations or precision. Approval of intuitive "making sense" does not imply disapproval of abstraction. Indeed, all of the authors

agree that some kind of powerful, exact conceptual framework or approach—reification as the articles call it—is the essential outcome of the whole learning process. For example, the Wrights say:

Technology such as computer animations can provide assistance in helping the students to form mental pictures that interrelate physical quantities, but they cannot substitute for the mental pictures that must form [in students’ minds] if reification is to occur. As one progresses, mathematics enters and the level of abstraction increases. (p. 57)

From the other side Roitman tells us, "My own work, for example, is in the very abstract world of set-theoretic applications to Boolean algebra and general topology, but I cannot think clearly without using things like dots, lines, and circles" (p. 26, note 2).

While the approval of making sense and teaching for understanding is almost universal, critics of the mathematics standards see the emphasis on applications, intuitive problem solving, and active learning by students as prone to serious errors. (The science standards have not been so heavily criticized, perhaps because they are much newer.) In the language of investment, the new elements have a greater downside risk because of vagueness, ideology, self-delusion, and unrealistic demands on the nation's teachers.

Haimo articulates the risks of wholesale reform in a way reminiscent of Edmund Burke's critique of the excesses of the French Revolution:

The drastic abandonment of every aspect of the "traditional" ways might sound good and be appealing theoretically, but it has not yet been shown that it can produce students with greater understanding of mathematics concepts. The "old ways" at least have withstood the test of time so that their strengths as well as their flaws have become clear. Further, over time, some trouble spots have been eliminated and some changes have been made. On the other hand, using the proposed reform to correct all the ills of the past by replacing everything on all fronts with untried proposals is a dangerous route to follow. (p. 18)

The papers cite examples of serious problems in the standards and real classroom practice, for example:

• problems and applications that are vague, overly complex, technically incorrect, and (surprisingly) needlessly technical

• teachers who obviously do not understand the underlying mathematical or scientific principles and who completely overlook both gross errors and powerful insights of their students

• an emphasis on applications and cross-disciplinary problem solving to the exclusion of core subject-matter content.

Such criticisms appear to be having an impact and may produce a new level of consensus. Romberg mentions that the following revisions are planned for the next set of NCTM curriculum standards: the integration of process standards like problem solving with mathematical content, the addition of a fifth process standard on "procedures or routines," increased emphasis on content strands across grade levels (e.g., number, algebra, geometry, statistics), and a careful review of all examples and applications.

The dominant impression I get from these papers is the need for developing and implementing specific curricula and teaching methods so that the debate can move from the general to the concrete. Ideological broadsides quickly lose their usefulness and become empty slogans: "applications vs. calculations," "problem-solving vs. traditional content," "basics vs. higher order thinking." Sometimes battles between rival camps like the "constructivists" and "mathematically correct" seem aimed more at winning symbolic political victories than real change in the teaching and learning for real children. Most of the ideological dichotomies are misleading or downright false. For example, traditional formal mathematics eventually lends itself to numerous applications in engineering. And constructivist curricula usually build around the framework of traditional mathematics. Development of real curricula forces the proponents of an educational philosophy to become specific about the tradeoffs of different educational goals and risks that are necessary in the limited time available for instruction. As the Wrights conclude, "The standards need to be transformed into workable teaching plans" (p. 65).

So, we desperately need development, implementation, and evaluation of alternative curricula, each involving an explicitly stated mix of alternative goals. Pieces of new curricula now exist, often developed under sponsorship of the National Science Foundation, including in the sites of the Systemic Initiatives (states, districts, schools). These curricula and methods of instruction put different emphasis on different social goals (preparation for advanced training vs. general vocational competence), the blend of student discovery and control by the teacher, the mix of intuitive applications and formal disciplinary content, and even the mix of enjoyment and hard work. Complete integrated curricula spanning multiple grades are harder to find, raising the disturbing possibility that multiple, disjointed curriculum reform may produce its own kind of "splintered curriculum."

In principle all curricula and programs of instruction can be evaluated in terms of some common set of criteria, such as the four questions recommended by Roitman for evaluating technology in the classroom (and adapted from the assessment standards of the NCTM). As paraphrased by me in terms of curriculum, the four questions are (1) What mathematical or scientific content is reflected? (2) What efforts are made to ensure that the content is significant and correct? (3) Does the curriculum engage the students in realistic and worthwhile mathematical and scientific activities? (4) Does the curriculum produce a deep understanding of aspects of the subject matter that are important to know and be able to do? To these, we might add the acid test for equity suggested by Haimo and Roitman: Regardless of rationale, does the curriculum lower expectations and constrict opportunities for students at any range of performance and achievement? Put positively, the goal of equity must always be twofold: increased access together with higher standards.

Under any set of fair criteria, it seems likely that different kinds of curricula can, in principle, be judged to possess high quality. Unfortunately, here we encounter a possible tension between curriculum diversity and systemic reform. All four papers agree that student achievement—not ideology or rhetoric—is the ultimate test for reform. Since systemic reform is aimed at change in the "whole system," there is a powerful tendency to adopt a single measure of student achievement implying a single authorized curriculum.

Consider, for example, the much publicized case of Michigan students who performed at the top levels of high school grades and the SAT and ACT tests but less well on a new state test emphasizing different kinds of knowledge and skills. My guess is that students who do well on either set of assessments will make good mathematicians and scientists. If that is not true, we certainly need to know. But if the truth is that alternative curricula produce alternatively accomplished students, why should systemic reform be used to champion one and subdue the other? The answer cannot be that all students must be rated on the same test to facilitate selective admission to higher education. American universities already admit students from many different states with various kinds of curricula, grading practices, and standardized tests.

Any new course or test must establish its credibility and quality with external audiences, such as the labor market and professionals in higher education; otherwise, high performance would not serve as a useful credential. But once quality has been established, why not let states, districts, schools, and even students select from among alternative curricula and tests? A 90th percentile on a test of mathematically correct mathematics ought to be as good as the 90th percentile on a test of deep-inquiry mathematics if they are both of high quality. Systemic reform means higher standards for all, not exactly the same standards or curriculum. In New York, the Regents program coexists with Advanced Placement, each with its own set of examinations. ChemCom is taught in the same schools as traditional Chemistry. Montana is developing a new mathematics curriculum for grades 9-10 that puts more emphasis on probability and statistics than algebra and geometry (as well as on technology); but students can take more traditional courses later in high school.

Somehow the legislated world of entrance requirements and tests must be made to conform to the real world of quality alternatives rather than destroying quality and diversity out of a divisive quest for uniformity. I'm not sure that there is any great harm done if a state does succeed in establishing a single high quality test as the sole measure of achievement in its schools, and perhaps states are the logical unit for experimentation and diversity. Most standardized tests do seem to change gradually in response to reform, by incorporating new items and testing formats (e.g., complex problem solving, actual speaking of foreign languages). Unfortunately, experience has taught us that, in the battle for total victory or defeat, the potential for divisive politics is also quite high. A quest for uniformity could turn ideological wars into real ones.

In conclusion, the debates and issues identified by these four papers tell us much about the national standards in mathematics and science and also about the process of implementation where the same issues reappear. The positive vision is of rigorous, meaningful, and useful content and achievement for all students realized through different instructional programs. The risks include no change, lower standards, and a repressive uniformity. The only way that we can be faithful to the vision and avoid the risks is to stay with the process and guide it in productive directions.

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