Problem Solving must be the Focus of School Mathematics in the 1980s
The development of problem-solving ability should direct the efforts of mathematics educators through the next decade. Performance in problem solving will measure the effectiveness of our personal and national possession of mathematical competence.
Problem solving encompasses a multitude of routine and commonplace as well as nonroutine functions considered to be essential to the day-to-day living of every citizen. But it must also prepare individuals to deal with the special problems they will face in their individual careers.
Problem solving involves applying mathematics to the real world, serving the theory and practice of current and emerging sciences, and resolving issues that extend the frontiers of the mathematical sciences themselves.
This recommendation should not be interpreted to mean that the mathematics to be taught is solely a function of the particular mathematics needed at a given time to solve a given problem. Structural unity and the interrelationships of the whole should not be sacrificed.
True problem-solving power requires a wide repertoire of knowledge, not only of particular skills and concepts but also of the relationships among them and the fundamental principle that unify them. Each problem cannot be treated as an isolated example. This recommendation looks toward the need to solve problems in an uncertain future as well as here and now. As such, mathematics needs to be taught as mathematics, not as an adjunct to its fields of application. This demands a continuing attention to its internal cohesiveness and organizing principles as well as to its uses.
1.1 The mathematics curriculum should be organized around problem solving.
- The current organization of the curriculum emphasizes component computational skills apart from their application. These skills are necessary tools but should not determine the scope and sequence of the curriculum. The need of the student to deal with the personal, professional, and daily experiences of life requires a curriculum that emphasizes the selection and use of the skills in unexpected, unplanned settings.
- Mathematics programs of the 1980s must be designed to equip students with the mathematical methods that support the full range of problem solving, including:
- the traditional concepts and techniques of computation and applications of mathematics to solve real-world problems, the rational and real number systems, the notion of function, the use of mathematical symbolism to describe real-world relationships, the use of deductive and inductive reasoning to draw conclusions about such relationships, and the geometrical notions so useful in representing them;
- methods of gathering, organizing, and interpreting information, drawing and testing inferences from data, and communicating results;
- the use of the problem-solving capacities of computers to extend traditional problem-solving approaches and to implement new strategies of interaction and simulation;
- the use of imagery, visualization, and spatial concepts.
- Mathematics programs should give students experience in the application of mathematics, in selecting and matching strategies to the situation at hand. Students must learn to:
- formulate key questions;
- analyze and conceptualize problems;
- define the problem and the goal;
- discover patterns and similarities;
- seek out appropriate data;
- transfer skills and strategies to new situations;
- draw on background knowledge to apply mathematics.
- Fundamental to the development of problem-solving ability is an open mind, an attitude of curiosity and exploration, the willingness to probe, to try, to make intelligent guesses.
- The curriculum should maintain a balance between attention to the applications of mathematics and to fundamental concepts.
1.2 The definition and language of problem solving in mathematics should be developed and expanded to include a broad range of strategies, processes, and modes of presentation that encompass the full potential of mathematical applications.
- Computational activities in isolation from a context of application should not be labeled "problem solving."
- The definition of problem solving should not be limited to the conventional "word problem" mode.
- As new technology makes it possible, problems should be presented in more natural settings or in simulations of realistic conditions.
- Educators should give priority to the identification and analysis of specific problem-solving strategies.
- Educators should develop and disseminate examples of "good problems" and strategies and suggest the scope of problem-solving activities for each school level.
1.3 Mathematics teachers should create classroom environments in which problem solving can flourish.
- Students should be encouraged to question, experiment, estimate, explore, and suggest explanations. Problem solving, which is essentially a creative activity, cannot be built exclusively on routines, recipes, and formulas.
- The mathematics teacher should assist the student to read and understand problems presented in written form, to hear and understand problems presented orally, and to communicate about problems in a variety of modes and media.
- The mathematics curriculum should provide opportunities for the student to confront problem situations in a greater variety of forms than the traditional verbal formats alone; for example, presentation through activities, graphic models, observation of phenomena, schematic diagrams, simulation of realistic situations, and interaction with computer programs.
1.4 Appropriate curricular materials to teach problem solving should be developed for all grade levels.
- Most current materials strongly emphasize an algorithmic approach to the learning of mathematics, and as such they are inadequate to support or implement fully a problem-solving approach. Present textbook problems tend to be easily categorized and stylized and often bear little resemblance to highly diversified, real-life problems. They do not permit the full range of strategies and abilities actually demanded in realistic problem contexts.
- The potential of computing technology for increasing problem-solving ability should be explored and exploited by the development of creative and imaginative software.
1.5 Mathematics programs of the 1980s should involve students in problem solving by presenting applications at all grade levels.
- Applications should be presented that use the student's growing and changing repertoire of basic skills to solve a multitude of routine and commonplace problems essential to the day-to-day living of every citizen.
- Applications of mathematics to other disciplines such as the social sciences, business, engineering, and the natural sciences should be presented.
- The enormous versatility of mathematics should be illustrated by presenting as diversified a collection of applications as possible at the given grade level
- At the college level, courses in mathematics and the mathematical sciences should give prospective teachers experiences that develop their capacities in modeling and problem solving.
1.6 Researchers and funding agencies should give priority in the 1980s to investigations into the nature of problem solving and to effective ways to develop problem solvers.
- Support should be provided for:
- the analysis of effective strategies;
- the identification of effective techniques for teaching;
- new programs aimed at preparing teachers for teaching problem-solving skills;
- investigations of attitudes related to problem-solving skills;
- the development of good prototype material for teaching the skills of problem solving, using all media.
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