Overview

With President Clinton's call for voluntary national tests of reading in fourth grade and of mathematics in eighth grade, the debate about the role of the government in establishing standards and assessments has reached new heights on the political and educational landscapes. At the state level, debates about standards have been particularly heated in California, especially during the process of adopting state content and performance standards for all students. Throughout the country, much of the debate about standards has taken the form of dichotomies growing from language that positions opponents at extreme ends of the spectrum, arising particularly from opposing views about how people learn mathematics. For example, must automaticity with procedural skills precede any problem solving, or should thinking and reasoning permeate all aspects of the discipline, even before focusing on skill development? Although resolving these issues is beyond the scope of this document, finding common ground that transcends these dichotomies is a challenging but necessary part of the process of developing standards.

     From any perspective in the standards debate, and from any political position, the call for standards is born, in part, out of parents' concerns for their children's futures: What should my child be learning? What should my child know in order to be admitted to a good college or university? What should my child know in order to get a good job? One of the main themes of this document--that problems from the workplace and everyday life can enhance the mathematical education of all students--implies the fortunate conclusion that the answers to these questions need not conflict.

     Discussion of national standards has a long history. The release of A Nation at Risk (National Commission on Excellence in Education, 1983) had an effect like that of national standards, for many high schools responded by increasing their course requirements for graduation. Since then, many organizations concerned with different aspects of education have released documents delineating standards. In 1989, the National Council of Teachers of Mathematics (NCTM) published Curriculum and Evaluation Standards for School Mathematics. More recently, many states have produced or are producing frameworks describing new goals for K-12 performance and instruction in mathematics as well as other disciplines. Some of those states have also produced state-wide assessments that are explicitly aligned with those frameworks. At the national level, the Secretary's Commission on Achieving Necessary Skills (SCANS) described competencies needed for careers in its 1991 report, What Work Requires of Schools. The Goals 2000: Educate America Act established the National Skill Standards Board in 1994 to serve as a catalyst in the development of a voluntary national system of skills standards, assessments, and certifications for business and industry. In science, the American Association for the Advancement of Science (AAAS) developed Benchmarks for Science Literacy (AAAS, 1993). After four years of development, consensus-building, and extensive formal review, the National Research Council (NRC) contributed the National Science Education Standards (NRC, 1996). Also in 1995, the American Mathematical Association of Two-Year Colleges published Crossroads in Mathematics: Standards for Introductory Mathematics Before Calculus.

     With so many voices contributing standards and recommendations, teachers are faced with difficult challenges. Despite the fact that the meaning of the word "standard" varies greatly among and even within the above documents--from statements about values to visions of the future; from statements about goals or expectations to criteria for evaluation--nevertheless, some common themes emerge. The standards for mathematics and those for science, for example, have many commonalities, as Jane Butler Kahle discusses in her essay, including ideas such as problem solving and communication.

     The SCANS requirements for the workplace, described in Arnold Packer's essay, also emphasize problem solving and communication, but these skills are embedded in a framework that is hard to reconcile with the traditional division of schooling into subject areas. In particular, the SCANS requirements for Planning Skills, Interpersonal Skills, and Personal Qualities, do not fall under any of the traditional grades 9-12 subject headings and are rarely explicitly discussed as part of the curriculum, especially in mathematics. How might such skills might be developed in a mathematics class? Part of an answer lies in the SCANS Commission's belief that the requirements should be taught in context. Throughout the nation, calls for new standards, new pedagogy, or new curricula often meet with a challenge: But how will students do on standardized tests? Discussions of student achievement as measured by standardized tests often misses two important questions: For what purpose was the test designed? And what is the relationship between this test and the educational goals that I value? The NCTM Assessment Standards (NCTM, 1995) describe four purposes of assessment: monitoring students' progress, making instructional decisions, evaluating students' achievement, and evaluating programs. Though the relationship between standards and assessments often goes unarticulated in public discussions, a constructive discussion of changes in mathematics education must carefully consider the roles of the assessment tools in place, some of which function as implicit standards because they influence expectations about what counts mathematically. Because the SAT and the ACT have come to serve gatekeeping functions in college admissions and in scholarship decisions, they influence the ideas of parents and policy makers alike about what constitutes desirable mathematical performance. The ACT purports to measure the academic skills that students will need to perform college-level work. Based upon what is taught in the high school curriculum, and requiring integration of knowledge from a variety of courses, the ACT Assessment tests are designed to determine how well students solve problems, grasp implied meanings, draw inferences, evaluate ideas, and make judgments. The SAT aims to measure verbal and mathematical reasoning abilities. Both tests are intended as predictors of success in college (in particular, in the first year of college); they attempt to fulfill a necessary function: providing colleges with a metric that allows for comparison of students from different schools when grades might not be comparable.

     But how do SAT and ACT scores compare with other measurements of mathematical performance? Can a test like the SAT adequately assess learning when the curriculum emphasizes extended, open-ended, or collaborative tasks? William Linder-Scholer poses these questions in his essay and suggests a data-analysis task for parents: assessing the SAT as a measurement of student achievement and school quality, and as a basis for comparing education in different states.

     Though Linder-Scholer's essay provides some understanding of the difficulties in comparing average test scores for schools or states, many important questions are left unaddressed. Serious consideration of the role of college entrance exams depends upon answers to two perennial questions: What do colleges, parents, and scholarship organizations do with the individual scores from these tests? And does the SAT measure what colleges value? Both the ACT and the SAT have evolved over the years to better serve the needs of parents, colleges, and schools. Nonetheless, these questions require ongoing research in response to changing curricula in high schools and colleges, and in response to the changing needs of colleges and the workplace. Constructive evolution of the role of college entrance exams requires collective deliberations among parents, teachers, college representatives, and testing experts around what these tests should be and how they may be used most effectively.

     In his essay, John Dossey provides an international perspective on the character of the examinations given to students between high school and college. Looking at such examinations in the United States, England, Wales, and four European countries, he finds that contextualized, extended-response tasks are not routinely included. Such tasks are included, however, on the National Assessment of Educational Progress (NAEP), which provides national and state indicators about mathematics education but no individual scores (see, e.g., Reese et al., 1997). The NAEP results indicate that if such tasks are to assess individual student learning, "students will need a great deal of support in formulating, solving, and communicating their results."

     The tasks that accompany these essays serve as examples and illustrations of tasks that might help provide that support. Well-chosen tasks are in no way to be construed as standards, but instead provide opportunities for building understanding of the mathematical ideas that are embodied in standards.

     Mental Mathematics (p. 83) suggests ways that mental arithmetic and algebra might each contribute to the other. Though Buying on Credit (p. 87) is about finance and Drug Dosage (p. 80) is about pharmacology, the mathematics behind the tasks is quite similar, involving rates, series, and recursion. Despite the similar mathematical content of the two tasks, both are included to suggest an opportunity for students to engage in mathematical thinking: to describe and understand whole classes of tasks by noticing commonalities among their procedures or representations. The similarity, after all, can be clear only with sufficient mathematical understanding of multiple contexts.

References

American Association for the Advancement of Science. (1993).

Benchmarks for science literacy. Washington, DC: Author.

American Mathematical Association of Two-Year Colleges (1995).

Crossroads in mathematics: Standards for introductory college mathematics before calculus. Memphis, TN: Author.

National Commission on Excellence in Education. (1983).

A nation at risk: The imperative for educational reform. Washington, DC: Author.

National Council of Teachers of Mathematics. (1989).

Curriculum and evaluation standards for school mathematics. Reston, VA: Author.

National Council of Teachers of Mathematics. (1995).

Assessment standards for school mathematics. Reston, VA: Author.

National Research Council. (1996).

National science education standards. Washington, DC: National Academy Press.

Reese, C. M., Miller, K. E., Mazzeo, J., & Dossey, J. A. (1997).

NAEP 1996 mathematics report card for the nation and the states. Washington, DC: National Center for Education Statistics.

United States Department of Labor. Secretary's Commission on Achieving Necessary Skills. (1991).

What work requires of schools: A SCANS report for America 2000. Washington, DC: Author.