In grades 9-12, the mathematics curriculum should include the refinement and extension of methods of mathematical problem solving so that all students can--

• use, with increasing confidence, problem-solving approaches to investigate and understand mathematical content;
• apply integrated mathematical problem-solving strategies to solve problems from within and outside mathematics;
• recognize and formulate problems from situations within and outside mathematics;
• apply the process of mathematical modeling to real-world problem situations.

Focus

Mathematical problem solving, in its broadest sense, is nearly synonymous with doing mathematics. Thus, whereas it is useful to differentiate among conceptual, procedural, and problem-solving goals for students in the early stages of mathematical learning, these distinctions should begin to blur as students mature mathematically. In grades 9-12, the problem-solving strategies learned in earlier grades should have become increasingly internalized and integrated to form a broad basis for the student's approach to doing mathematics, regardless of the topic at hand. From this perspective, problem solving is much more than applying specific techniques to the solution of classes of word problems. It is a process by which the fabric of mathematics as identified in later standards is both constructed and reinforced.

Discussion

One consequence of students' increasing mathematical sophistication is that problem situations, which for younger students necessarily arise from the real world, now often spring from within mathematics itself. Thus. mathematical problem solving serves not only to answer questions raised in everyday life, in the physical and social sciences, and in such professions as business and engineering but also to further extend and connect mathematical theory itself. A student who proves a theorem in order to extend knowledge in an axiomatic system and one who solves an application involving an optimal production and marketing decision have each engaged in varying levels of mathematical problem solving.

Problems and applications should be used to introduce new mathematical content, to help students develop both understanding of concepts and facility with procedures, and to apply and review processes they have already learned. For example, a situation such as finding the maximum height of the path of a projectile might be posed for which students have no readily available solution techniques. The learning process would require them to analyze the situation in light of their existing knowledge, develop appropriate mathematical techniques, and subsequently apply those techniques to solve the problem. "Looking back" over the problem situation and the whole problem-solving process also can provide a springboard from which even more efficient solution methods or problem extensions can be developed in ways that mathematically enrich the students' experience. This scenario may take place over a few days or even a few weeks; it often may be appropriate for students to work cooperatively in groups. It is the intent of this standard that this process be repeated across the curriculum on a regular and sustaining basis and that it entail appropriate student use of calculator and computer technology.

Students in grades 9-12 should also have some experience recognizing and formulating their own problems, an activity that is at the heart of doing mathematics. For example, an exploration of the perimeters of various rectangles with area 24 cm squared by means of models or drawings, with data as recorded in table 1.1, could lead to student recognition and formulation of such problems as the following: Is there a rectangle of minimum perimeter with the specified area? What are its dimensions?

      TABLE 1.1



      Rectangle Data



       Area                 Length        Width        Perimeter



      24 cm squared          1 cm         24 cm          50 cm



      24 cm squared          2 cm         12 cm          28 cm



      24 cm squared          3 cm          8 cm          22 cm



      24 cm squared          4 cm          6 cm          20 cm



      24 cm squared          6 cm          4 cm          20 cm



      24 cm squared          8 cm          3 cm          22 cm

Instructional settings that encourage investigation, cooperation, and communication foster problem posing as well as problem solving. In addition, all students can profit from discussions of specific problem-posing techniques. Forming the "dual" of the problem above leads to the question, Is there a rectangle of maximum area with a specified perimeter? Other useful techniques include relaxing conditions in, or generalizing from, problem situations and considering the converse of mathematical statements.

Another important component of mathematical thinking is the process of mathematical modeling as illustrated in figure 1.1.

Fig. 1.1. Mathematical modeling

The various stages in building and using a mathematical model are exemplified in the following solution of a famous problem first posed in the ninth century and finally solved 800 years later in 1654 by the famous French mathematicians Pierre Fermat and Blaise Pascal. Observe that the problem comes from probability, but the mathematical model is geometric. (Note: We have modified the content but not the nature of the original problem setting.)

Real-world problem situation. In a two-player game, one point is awarded at each toss of a fair coin. The player who first attains n points wins a pizza. Players A and B commence play; however, the game is interrupted at a point at which A and B have unequal scores. How should the pizza be divided fairly? (The intuitive division, that A should receive an amount in proportion to A's score divided by the sum of A's score and B's score, has been determined to be inequitable.)

Problem formulation. Consider the situation with the following data: The winning score is n = 10; when the interruption occurs, the score is A:8 and B:7. The pizza will be divided in proportion to each player's probability of winning the game.

Mathematical model. See figure 1.2. At each turn, P(A wins a point) = P(B wins a point) = 1/2. A's share = P(A wins 10 points) x area of pizza; B's share = total pizza - A's share. Let a square region represent the original game state with the score A:8 to B:7 as indicated. At each turn, the square or interior rectangles are halved to represent P = 1/2 for winning (or losing) a point. Thus, in this model the resulting fraction of the original area also represents the probability of reaching that game state.

Fig. 1.2. A geometric probability model

Solution within the model. See figure 1.3.

Fig. 1.3. Solution using the model

Interpretation of solution in original problem formulation

A's share = 11/16 of pizza
B's share = 5/16 of pizza

Validation in original real-world problem situation. Empirical evidence gained from actually playing out the game many times or, more easily, from computer simulation (using random numbers to represent coin tosses) confirms this solution. Simulation techniques are further illustrated in the standards on mathematical connections and probability.

The importance of problem solving to all education cannot be overestimated. To serve this goal effectively, the mathematics curriculum must provide many opportunities for all students to meet problems that interest and challenge them and that, with appropriate effort, they can solve.