The high school years are a timeof major transition. Students enter high school as young teenagers, grapplingwith issues of identity and with their own mental and physical capacities. Ingrades 9–12, they develop in multiple ways—becoming more autonomous and yetmore able to work with others, becoming more reflective, and developing thekinds of personal and intellectual competencies that they will take into theworkplace or into postsecondary education.
These Standards describe anambitious foundation of mathematical ideas and applications intended for allstudents. Through its emphasis on fundamental mathematical concepts andessential skills, this foundation would give all students solid preparation forwork and citizenship, positive mathematical dispositions, and the conceptualbasis for further study. In grades 9–12, students should encounter new classesof functions, new geometric perspectives, and new ways of analyzing data. Theyshould begin to understand aspects of mathematical form and structure, such asthat all quadratic functions share certain properties, as do all functions ofother classes—linear, periodic, or exponential. Students should see theinterplay of algebra, geometry, statistics, probability, and discretemathematics and various ways that mathematical phenomena can be represented.Through their high school experiences, they stand to develop deeperunderstandings of the fundamental mathematical concepts of function andrelation, invariance, and transformation. »
In high school, students shouldbuild on their prior knowledge, learning more-varied and more-sophisticatedproblem-solving techniques. They should increase their abilities to visualize,describe, and analyze situations in mathematical terms. They need to learn touse a wide range of explicitly and recursively defined functions to model theworld around them. Moreover, their understanding of the properties of thosefunctions will give them insights into the phenomena being modeled. Theirunderstanding of statistics and probability could provide them with ways tothink about a wide range of issues that have important social implications,such as the advisability of publicizing anecdotal evidence that can causehealth scares or whether DNA "fingerprinting" should be consideredstrong or weak evidence.
Secondary school students needto develop increased abilities in justifying claims, proving conjectures, andusing symbols in reasoning. They can be expected to learn to provide carefullyreasoned arguments in support of their claims. They can practice making andinterpreting oral and written claims so that they can communicate effectivelywhile working with others and can convey the results of their work with clarityand power. They should continue to develop facility with such technologicaltools as spreadsheets, data-gathering devices, computer algebra systems, andgraphing utilities that enable them to solve problems that would require largeamounts of computational time if done by hand. Massive amounts ofinformation—the federal budget, school-board budgets, mutual-fund values, andlocal used-car prices—are now available to anyone with access to a networkedcomputer (Steen 1997). Facility with technological tools helps students analyzethese 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 tostudy 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 conceptualunderstanding provides the underpinnings for a wide range of careers as well asfor further study, as Hoachlander (1997, p. 135) observes:
Most advancedhigh school mathematics has rigorous, interesting applications in the workworld. For example, graphic designers routinely use geometry. Carpenters applythe principles of trigonometry in their work, as do surveyors, navigators, andarchitects.... Algebra pervades computing and business modeling, from everydayspreadsheets to sophisticated scheduling systems and financial planningstrategies. Statistics is a mainstay for economists, marketing experts,pharmaceutical companies, and political advisers.
With the experience proposed herein 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 toadapt to the increasingly technological work environments they will face in theyears to come. By learning to think and communicate effectively in mathematics,students will be better prepared for changes in the workplace that increasinglydemand teamwork, collaboration, and communication (U.S. Department of Labor1991; Society for Industrial and Applied Mathematics 1996). Note that theseskills are also needed increasingly by people who will pursue careers » in mathematics or science. With its emphasis onfundamental concepts, thinking and reasoning, modeling, and communicating, thecore is a foundation for the study of more-advanced mathematics. Consider, forexample, the recommendations for precalculus courses generated at the Preparingfor a New Calculus conference (Gordon et al. 1994, p. 56):
Coursesdesigned 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 Principlesand Standards for School Mathematics is connections. Students develop amuch richer understanding of mathematics and its applications when they canview the same phenomena from multiple mathematical perspectives. One way tohave students see mathematics in this way is to use instructional materialsthat are intentionally designed to weave together different content strands.Another means of achieving content integration is to make sure that coursesoriented toward any particular content area (such as algebra or geometry) containmany integrative problems—problems that draw on a variety of aspects ofmathematics, that are solvable using a variety of methods, and that studentscan access in different ways.
High school students withparticular interests could study mathematics that extends beyond what isrecommended here in various ways. One approach is to include in the programmaterial that extends these ideas in depth or sophistication. Students whoencounter these kinds of enriched curricula in heterogeneous classes will tendto seek different levels of understanding. They will, over time, learn new waysof thinking from their peers. Other approaches make use of supplementarycourses. For instance, students could enroll in additional courses concurrentwith the program. Or the material proposed in these Standards could be includedin a three-year program that allows students to take supplementary courses inthe fourth year. In any of these approaches, the curriculum can be designed sothat students can complete the foundation proposed here and choose fromadditional courses such as computer science, technical mathematics, statistics,and calculus. Whatever the approach taken, all students learn the same corematerial while some, if they wish, can study additional mathematics consistentwith their interests and career directions.
These Standardsare demanding. It will take time, patience, and skill to implement the visionthey represent. The content and pedagogical demands of curricula aligned withthese Standards will require extended and sustained professional developmentfor teachers and a large degree of administrative support. Such efforts areessential. We owe our children no less than a high degree of quantitativeliteracy and mathematical knowledge that prepares them for citizenship, work,and further study.
摘自NCTM