Students in secondary school face choices and decisions that will determine the course of their lives. As they approach the end of required schooling, they must have the opportunity to explore their career interests—which may change during high school and later—and their options for postsecondary education. To ensure that students will have a wide range of career and educational choices, the secondary school mathematics program must be both broad and deep.

The high school years are a time of major transition. Students enter high school as young teenagers, grappling with issues of identity and with their own mental and physical capacities. In grades 9–12, they develop in multiple ways—becoming more autonomous and yet more able to work with others, becoming more reflective, and developing the kinds of personal and intellectual competencies that they will take into the workplace or into postsecondary education.

These Standards describe an ambitious foundation of mathematical ideas and applications intended for all students. Through its emphasis on fundamental mathematical concepts and essential skills, this foundation would give all students solid preparation for work and citizenship, positive mathematical dispositions, and the conceptual basis for further study. In grades 9–12, students should encounter new classes of functions, new geometric perspectives, and new ways of analyzing data. They should begin to understand aspects of mathematical form and structure, such as that all quadratic functions share certain properties, as do all functions of other classes—linear, periodic, or exponential. Students should see the interplay of algebra, geometry, statistics, probability, and discrete mathematics and various ways that mathematical phenomena can be represented. Through their high school experiences, they stand to develop deeper understandings of the fundamental mathematical concepts of function and relation, invariance, and transformation. »

In high school, students should build on their prior knowledge, learning more-varied and more-sophisticated problem-solving techniques. They should increase their abilities to visualize, describe, and analyze situations in mathematical terms. They need to learn to use a wide range of explicitly and recursively defined functions to model the world around them. Moreover, their understanding of the properties of those functions will give them insights into the phenomena being modeled. Their understanding of statistics and probability could provide them with ways to think about a wide range of issues that have important social implications, such as the advisability of publicizing anecdotal evidence that can cause health scares or whether DNA "fingerprinting" should be considered strong or weak evidence.

Secondary school students need to develop increased abilities in justifying claims, proving conjectures, and using symbols in reasoning. They can be expected to learn to provide carefully reasoned arguments in support of their claims. They can practice making and interpreting oral and written claims so that they can communicate effectively while working with others and can convey the results of their work with clarity and power. They should continue to develop facility with such technological tools as spreadsheets, data-gathering devices, computer algebra systems, and graphing utilities that enable them to solve problems that would require large amounts of computational time if done by hand. Massive amounts of information—the federal budget, school-board budgets, mutual-fund values, and local used-car prices—are now available to anyone with access to a networked computer (Steen 1997). Facility with technological tools helps students analyze these data. A great deal is demanded of students in the program proposed here, but no more than is necessary for full quantitative literacy.

All students are expected to study mathematics each of the four years that they are enrolled in high school, whether they plan to pursue the further study of mathematics, to enter the workforce, or to pursue other postsecondary education. The focus on conceptual understanding provides the underpinnings for a wide range of careers as well as for further study, as Hoachlander (1997, p. 135) observes:

Most advanced high school mathematics has rigorous, interesting applications in the work world. For example, graphic designers routinely use geometry. Carpenters apply the principles of trigonometry in their work, as do surveyors, navigators, and architects.... Algebra pervades computing and business modeling, from everyday spreadsheets to sophisticated scheduling systems and financial planning strategies. Statistics is a mainstay for economists, marketing experts, pharmaceutical companies, and political advisers.

p. 288

With the experience proposed here in making connections and solving problems from a wide range of contexts, students will learn to adapt flexibly to the changing needs of the workplace. The emphasis on facility with technology will result in students' ability to adapt to the increasingly technological work environments they will face in the years to come. By learning to think and communicate effectively in mathematics, students will be better prepared for changes in the workplace that increasingly demand teamwork, collaboration, and communication (U.S. Department of Labor 1991; Society for Industrial and Applied Mathematics 1996). Note that these skills are also needed increasingly by people who will pursue careers » in mathematics or science. With its emphasis on fundamental concepts, thinking and reasoning, modeling, and communicating, the core is a foundation for the study of more-advanced mathematics. Consider, for example, the recommendations for precalculus courses generated at the Preparing for a New Calculus conference (Gordon et al. 1994, p. 56):

Courses designed to prepare students for the new calculus should:

• cover fewer topics ... with more emphasis on fundamental concepts.

• place less emphasis on complex manipulative skills.

• teach students to think and reason mathematically, not just to perform routine operations....

• emphasize modeling the real world and develop problem-solving skills.

• make use of all appropriate calculator and computer technologies....

• promote experimentation and conjecturing.

• provide a solid foundation in mathematics that prepares students to read and learn mathematical material at a comparable level on their own.

A central theme of Principles and Standards for School Mathematics is connections. Students develop a much richer understanding of mathematics and its applications when they can view the same phenomena from multiple mathematical perspectives. One way to have students see mathematics in this way is to use instructional materials that are intentionally designed to weave together different content strands. Another means of achieving content integration is to make sure that courses oriented toward any particular content area (such as algebra or geometry) contain many integrative problems—problems that draw on a variety of aspects of mathematics, that are solvable using a variety of methods, and that students can access in different ways.

High school students with particular interests could study mathematics that extends beyond what is recommended here in various ways. One approach is to include in the program material that extends these ideas in depth or sophistication. Students who encounter these kinds of enriched curricula in heterogeneous classes will tend to seek different levels of understanding. They will, over time, learn new ways of thinking from their peers. Other approaches make use of supplementary courses. For instance, students could enroll in additional courses concurrent with the program. Or the material proposed in these Standards could be included in a three-year program that allows students to take supplementary courses in the fourth year. In any of these approaches, the curriculum can be designed so that students can complete the foundation proposed here and choose from additional courses such as computer science, technical mathematics, statistics, and calculus. Whatever the approach taken, all students learn the same core material while some, if they wish, can study additional mathematics consistent with their interests and career directions.

These Standards are demanding. It will take time, patience, and skill to implement the vision they represent. The content and pedagogical demands of curricula aligned with these Standards will require extended and sustained professional development for teachers and a large degree of administrative support. Such efforts are essential. We owe our children no less than a high degree of quantitative literacy and mathematical knowledge that prepares them for citizenship, work, and further study.