The "Standards Wars" in Perspective

In this conclusion, I want to explore further the areas of agreement and disagreement across the articles about the general issue of active learning, teaching for understanding, or, at the risk of raising a red flag, “constructivism.”

THE AREA OF CONSENSUS OVER GOALS—MORE THAN THE “RIGHT ANSWER”

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

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

In light of this comment, we might ask ourselves why reform should not consist of simple upgrading and faster pacing—more students taking traditional courses earlier. In fact, that is one common strand of reform, sometimes called “intensification,” and one that seems to be responsible for some of the gains in student achievement that occurred during the 1980s. For example, there is little doubt that substitution of algebra for general math, a nationwide trend and the goal of Equity 2000, increases scores on standardized tests, opens up opportunities in postsecondary education, and represents a definite step in the direction of higher order thinking (Gamoran, Porter, Smithson, & White, 1997; Lee, Croninger, & Smith, 1997). Commentators on all sides of the debate probably support such changes at least as intermediate goals.

But there is another side of reform captured in Romberg’s disparagement of eighth grade math as “shopkeeper math”:

These students were expected learn only paper-and-pencil calculations and routines for whole numbers, common fractions, decimals, and percents. (p. 8)

Today no one makes a living doing paper-and-pencil calculations. Calculators and computers have replaced shopkeeper calculations in business and industry. (pp. 9-10).

This dislike of mindless calculation has broad support among reformers and the other authors of our articles. If the nightmare of traditionalists is children who cannot get the right answer, the nightmare of other reformers is children who do not 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.

Roitman reinforces the need for combining-precision with insight when she says, “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. 41). This “making sense” of math and science, sometimes called “teaching for understanding” (Cohen, McLaughlin, & Talbert, 1993), 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 reasons, the National Council of Teachers of Mathematics (NCTM) standards for math include four “process standards”: problem solving, communication, reasoning, and connections.

Turning to science, for the Wrights, thinking, not calculation, is the essence of science: inquiry, self-confident discovery, disciplined criticism, cooperative problem solving. Note, however, that 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. The authors agree that some kind of powerful, exact conceptual framework or approach—reification as the Wrights call it—is the essential outcome of the whole learning process. The Wrights say:

Technology, such as computer animation, can provide assistance in helping students form mental pictures that interrelate physical quantities, but they cannot substitute for the mental pictures that must form if reification is to occur. As one progresses; mathematics enters and the level of abstraction increases. (p. 127)

Raizen notes that the Third International Mathematics and Science Study (TIMSS) identified a glaring weakness of science education in the United States at the most advanced level:

Even given the different incentives, it is noteworthy that only about 6.6 percent of U.S. students take AP exams, while roughly 25 percent to 50 percent of all students in the other nations take and pass these types of advanced exams. That other countries are bringing a considerable percentage of their students to this high a level of achievement generally comes as a shock to Americans, who still think of the European and Japanese systems as elite and exclusionary, but this has changed considerably in the last two or three decades. Indeed, “the great majority of college-bound students in countries other than the U.S. must (emphasis in original) take and pass some advanced subject-specific examinations. . . . In France, Germany, and Israel, academically oriented students who do not seek further education still take these examinations because passing them is a prestigious credential in their societies” (Britton & Raizen, 1996, p. 201). Moreover, in most of the countries except the United States, students must take these examinations in several fields, varying from three subjects in England/Wales to seven or eight subjects in France. (pp. 107-108

THE AREA OF DISAGREEMENT—APPLICATIONS, AMATEUR PROBLEM SOLVING, AND THE RISK OF JUNK MATH AND SCIENCE.

While the approval of making sense and teaching for understanding is almost universal, critics of the math 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, or, as Patrick Shields once said, “good constructivism is surely better than good drill and practice, but bad constructivism is probably worse than bad drill and practice.”

Examples of serious errors cited by the authors from the standards or real classrooms are:

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.

Haimo articulates the risks of wholesale reform in this way:

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 some changes have been made. . . . 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. (p. 59)

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 quality review of all examples and applications.

CONCLUSION TO THE STANDARDS DEBATE: REDUCING POLARIZATION, ENCOURAGING QUALITY, AND ASSESSING OUTCOMES

One reaction that seems appropriate to the-debate over the standards is that it is exaggerated and needlessly polarized. As pointed out in the Raizen article, with respect to science, most of the standards adopted by states, and the curricula implemented under them, are hybrids of the old and new. Uri Treisman (1997) makes the following comment about instruction in a set of high-performing, high-poverty schools in Texas:

In so many cases we observed mathematics teachers supplementing their textbooks in ways that compensated for perceived limitations. I took some pleasure in observing “Saxon” teachers assigning worksheets of interesting and challenging problems to prepare their students to do well on state and AP examinations. I also observed in several schools UCSMP (University of Chicago School Mathematics Project) teachers supplementing their texts with structured drill and practice.

In other words, real curricula often are less far apart than the worst-case models in the minds of critics.

Second, many of the problems with constructivist math and science probably can be solved through improved curricula, quality control of problem sets, and better professional development. The energies of critics would be better spent strengthening the new approaches than resisting them. 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) whether the curriculum engages the students in realistic and worthwhile mathematical and scientific activities; (4) whether the curriculum produces a deeper 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 for all.

Finally, there are strong empirical claims made on behalf of active learning and teaching for understanding, and these claims deserve to be evaluated. The research question might be put this way: To what extent can a “less-is-more” curriculum (fewer topics, more active learning) yield the following benefits:

more advanced content sooner for more students

conceptual, deep understanding

no loss of basic skills

comfort and skill with applications and problem solving

enthusiasm for the subject matter, advanced study, and careers

lifetime “habits of mind” (e.g., quantitative sense and scientific inquiry)

higher access and achievement for underserved students

no substantial increase in teaching costs (e.g., tutoring, professional development)

Such claims may be difficult to evaluate in practice because of some of the points made earlier in this essay. Hybrid curricula making few departures from tradition would offer a weak test. No one would expect low-quality and poorly taught curricula to produce such benefits. On the other hand, ambitious, high-quality curricula are being implemented around the country, and they should be evaluated, because verification of such benefits would go a long way toward resolving the debate. For example, is it true that initially low achieving elementary school students exposed to “inquiry” learning will achieve a higher level of conceptual understanding sooner and be better prepared for both academic and applied courses in later grades? Proof of that proposition could have a powerful impact.

One essential requirement to keep in mind in conducting such evaluations is the need to assess a full range of relevant outcomes over time (see Ridgway, 1998, on “balanced assessment”). Standardized tests that measure basic skills are not a fair test of instruction that aims at deeper understanding, additional practical skills (e.g., communication), higher attainment in later grades, and change in attitudes and habits of mind. Also, the fact that the same historically small group of students is still succeeding in the academic fast track does not diminish the need for major advances by other students. What we need are evaluations of traditional and inquiry-based instruction for different groups of students, each measuring the same full range of outcomes. Until then, the standards debate will remain frustratingly ideological and starved for facts.

REFERENCES

Cohen, D., McLaughlin M., & Talbert, J. (Eds.). (1993). Teaching for understanding: Challenges for policy and practice. San Francisco: Jossey-Bass.

Gamoran, A., Porter, A., Smithson, J., & White, P. (1997). Upgrading high school mathematics instruction: improving learning opportunities for low-achieving, low-income youth, Educational Evaluation and Policy Analysis, 9(4), 325-338.

Lee, V., Croninger, R., & Smith, J. (1997). Course-taking, equity, and mathematics learning: Testing the constrained curriculum hypothesis in U.S. secondary schools, Educational Evaluation and Policy Analysis, 19(2), 99-121.

Ridgway, J. (1998). From barrier to lever: Revising roles for assessment in mathematics education (Brief, vol. 2, no. 1). Madison: University of Wisconsin-Madison, National Institute for Science Education.

Shields, P. (1997, March 13). Author’s notes of oral presentation at a conference on evaluation of systemic reform in Madison, Wisconsin.

Treisman, U. (1997). What have we learned? What do we need to know? In W. H. Clune et al. (Eds.), Research on systemic reform: What have we leaned! What do we need to know? (pp. 16-21) (Workshop Report No. 4, vol. 2). Madison: University of Wisconsin-Madison, National Institute for Science Education.