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There is consensus that throughout the world societies are evolving from "The Industrial Age" into "The Information Age". The most important features of the Information Age are: first, it represents a profound switch from physical energy to brain power as its driving force; and second, a change from concrete products to abstractions as its primary products. Information is the new capital and the new raw material, the ability to communicate is the new means of production, and the communication network provides the relations of production. The assumption is that our citizens need a better understanding of mathematics, science, and technology if our society is to prosper in the emerging Information Age.
As a consequence for over a decade the public has demanded action (e.g. National Commission on Excellence in Education, 1983; National Science Board Commission on Precollege Education in Mathematics, Science, and Technology, 1983). Because of these demands, the National Council of Teachers of Mathematics (NCTM) in 1986 formed a Commission on Standards for School Mathematics with the mission to present a vision of the mathematics curriculum, methods of instruction, techniques of assessing student performance, and procedures for evaluating programs that would be needed. 1 Here I only address issues related to the school mathematics curriculum.
The task of developing a new school mathematics curriculum must be viewed as a design task. The emphasis on design implies that current materials are inadequate. Thus, simple alterations of existing programs would not suffice. Instead, the fundamental way in which mathematics programs are organized and developed must be changed so that a radically new program is created. In fact, curriculum development is seen as more than a change in content and method; it is an effort to change the instructional culture of schools. By curriculum, it also should be understood that what was to be developed is a total instructional package, not just a curriculum guide or a basal textbook series. The message I want understood is that the vision NCTM presented is for a mathematics program that is significantly different from current practice, and the emphasis in the vision is that mathematics "makes sense." It should make sense to students, and students should use mathematics to help them make sense of the world.
Systemic Reform Strategy
The strategy which underlies the curriculum reforms was based on the notion that since we live in a supply-and-demand economy, if the mathematics communit y wanted different texts and tests, a demand would have to be created. To respond to this challenge the mathematical sciences community has followed a seven step iterative strategy (see Figure 1). The steps and the relationships between them are as follows:
Figure 1.
Principle 1. Conceptual domains should be specified.
The mathematical domains that we expect students to engage must be identified, and then a curriculum must be built around those conceptual domains. The domains should be selected because of their generality and ability to subsume more specialized components of the curriculum deemed desirable for the development of problem-solving ability and quantitative reasoning. These domains should not be considered independent of each other. While it is true that each domain has some unique properties (signs, symbols, rules), I would rather think of them as the roots of a tree whose trunk involves problem solving, communication, and reasoning.
Principle 2. Curriculum units, each of which takes two to four weeks to teach and each of which tells a story, need to be constructed so that they provided students with an opportunity to investigate increasingly complex problem situations within (and occasionally across) domains.
Students should be expected to construct meanings, interrelate concepts and skills, and use those meanings in a variety of problem situations. Each unit should be similar to a chapter in a Dickens' novel. It should introduce or reintroduce the characters to the reader, and there should be a new problem situation to be explored that involves conflict, suspense, crisis, and resolution.
Principle 3. Students should be exposed to the major features of a mathematical domain as they arise naturally in problem situations.
Ideas are best introduced when students see a need or a reason for their use. Promoting the development of understanding requires an integrated curriculum. Problem situations may include the historic reasons for the development of a mathematical domain, the relationship of that domain to other domains, or the uses of ideas in that.
Principle 4. The activities within each unit should he related to how students come to understand a domain.
The activities embedded in these units should include having students:
Modeling, or representing phenomena in the world by means of a system of theoretically specified objects and relations (often mathematical relations). In classrooms, we find it fruitful to consider modeling as a cycle including model construction, model exploration, and model revision.
Argument and standards of evidence, with an emphasis on promoting students' skills for generalization in mathematics.
Big ideas and technologies, reflecting that student work needs to be about important mathematical and scientific ideas, with enabling technologies that have pedagogical promise--those generating new forms of teaching and learning.
Equity, consistent with contemporary views of mathematics that stress that this discipline is developed in social contexts rather than the isolated pursuit of "truth." This view suggests that the very nature of mathematics is defined communally, making participation by all not only a fundamental civil right, but also important to the continued vitality of mathematics and science to the nation.
Assessment, reflecting that classrooms, schools, and parents need ways of ensuring that classroom practices that foster understanding are, in fact, improving student achievement.
Furthermore, each unit should provide review of prior concepts and skills and lay foundations for concepts and skills to be learned later. Activities used to teach algorithms should differ from those used to teach problem solving, and activities requiring assimilation should differ from those requiring accommodation. Furthermore, the method of instruction is likely to differ. Students might be addressed as a large group when being exposed to new information and work in small groups when inventing, proving, or applying. Assimilation may require exercises requiring little prior knowledge, while accommodation may demand a dissimilar array of problem situations involving varying cognitive structures. A higher degree of teacher-imposed structure and control may be desirable for lower-level cognitive outcomes, while a greater degree of group autonomy may aid higher-level cognitive outcomes.
Principle 5. Curriculum units should always be considered as problematic.
All curriculum sequences need to be adapted and modified in light of what knowledge the students bring to the unit and the context in which instruction takes place. The difference between the intended and the actual curriculum should be apparent. What actually occurs will differ among classrooms. The program can not be "teacher proof." Instead, the program should assist each teacher in making reasonable adaptations so that the prior knowledge and interests of the students are taken into account in instruction.
SUMMARY
The design of new mathematics curricula should be seen as a critical ingredient in the current systemic reform movement. New curriculum materials are a necessary, but not sufficient, component in changing schooling practices.
Notes
1 - The three standards documents published by the National Council of Teachers of Mathematics are: Curriculum and Evaluation Standards for School Mathematics (1989), Professional Standards for Teaching Mathematics (1991), and Assessment Standards for School Mathematics (1995).
2 - The development was funded by the National Science Foundation and created with the help of staff from the Freudenthal Institute at the University of Utrecht, The Netherlands.
References
Carpenter, T., & Lehrer, R. (in press). Teaching and learning mathematics with understanding. In E. Fennema & T. A. Romberg (Eds.), Classrooms that promote understanding. Book in preparation.
National Center for Research in Mathematical Sciences Education & Freudenthal Institute. (Eds.) (in press). Mathematics in context: A connected curriculum for grades 5-8. Chicago: Encyclopedia Britannica Education Corporation.
National Commission on Excellence in Education. (1983). A nation at risk: The imperative for educational reform. Washington, DC: U.S. Government Printing Office.
National Council of Teachers of Mathematics (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics. (1991). Professional standards for teaching mathematics. Reston, VA: Author.
National Council of Teachers of Mathematics (1995). Assessment standards for school mathematics. Reston, VA: Author.
National Science Board Commission on Precollege Education in Mathematics, Science and Technology. (1983). Educating Americans for the 21st century: A plan of action for improving the mathematics, science and technology education for all American elementary and secondary students so that their achievement is the best in the world by 1995. Washington, DC: National Science Foundation.