In grades 5-8, the mathematics curriculum should include numerous and varied experiences with problem solving as a method of inquiry and application so that students can--

• use problem-solving approaches to investigate and understand mathematical content;
• formulate problems from situations within and outside mathematics;
• develop and apply a variety of strategies to solve problems, with emphasis on multistep and nonroutine problems;
• verify and interpret results with respect to the original problem situation;
• generalize solutions and strategies to new problem situations;
• acquire confidence in using mathematics meaningfully.

Focus

To solve a problem is to find a way where no way is known off-hand, to find a way out of a difficulty, to find a way around an obstacle, to attain a desired end, that is not immediately attainable, by appropriate means. (G. Polya in Krulik and Reys 1980, p. 1)

Problem solving is the process by which students experience the power and usefulness of mathematics in the world around them. It is also a method of inquiry and application, interwoven throughout the Standards to provide a consistent context for learning and applying mathematics. Problem situations can establish a "need to know" and foster the motivation for the development of concepts.

In grades 5-8, the curriculum should take advantage of the expanding mathematical capabilities of middle school students to include more complex problem situations involving topics such as probability, statistics, geometry, and rational numbers. Situations and approaches should build on and extend the mathematical language students are acquiring and help them to develop a variety of problem-solving strategies and approaches. Although concrete and empirical situations remain a focus throughout these grades, a balance should be struck between problems that apply mathematics to the real world and problems that arise from the investigation of mathematical ideas. Finally, the mathematics curriculum should engage students in some problems that demand extended effort to solve. Some might be group projects that require students to use available technology and to engage in cooperative problem solving and discussion. For grades 5-8 an important criterion of problems is that they be interesting to students.

Computers and calculators are powerful problem-solving tools. The power to compute rapidly, to graph a relationship instantly, and to systematically change one variable and observe what happens to other related variables can help students become independent doers of mathematics.

The curriculum must give students opportunities to solve problems that require them to work cooperatively, to use technology, to address relevant and interesting mathematical ideas, and to experience the power and usefulness of mathematics.

Discussion

The nonroutine problem situations envisioned in these standards are much broader in scope and substance than isolated puzzle problems. They are also very different from traditional word problems, which provide contexts for using particular formulas or algorithms but do not offer opportunities for true problem solving. Real-world problems are not ready-made exercises with easily processed procedures and numbers. Situations that allow students to experience problems with "messy" numbers or too much or not enough information or that have multiple solutions, each with different consequences, will better prepare them to solve problems they are likely to encounter in their daily lives.

The exploration of problem situations can provide a context in which students further their knowledge about the interrelationships of mathematical ideas, for example:

Maria used her calculator to explore this problem: Select five digits. Use the five digits to form a two-digit and a three-digit number so that their product is the largest possible. Then find the arrangement that gives the smallest product.

Students can be encouraged to generalize their solutions to any five digits and to any number of digits. This problem helps students deepen their understanding of place value, multiplication, and number sense.

Students should model many problems concretely, gather and organize data in tables, identify patterns, graph data, use calculators to simplify computations, and use computers to assist in generating and analyzing information. The power of computers to store, generate, and depict--in various ways--vast quantities of information makes them a valuable source of interesting problems. Data bases and computer programs can engage students in posing and solving problems. Students sample data, analyze and make predictions on the basis of their samples, make conjectures, discuss and validate their conclusions, and prepare arguments to convince others of their conclusions. Students also should experience problem situations rich in opportunities to formulate and define problems, determine the information required, decide on methods for obtaining this information, and determine the limits of acceptable solutions. The following is one example of such an open-ended problem situation:

The teacher demonstrates a pendulum constructed from string and a weight. Students work in small groups to construct a pendulum, investigate how it functions, and formulate questions that arise. These questions might include, How long does it take to make one complete cycle? How does the length of the string affect the cycle? How does the weight affect the cycle? How does the height from which the pendulum begins its swing affect the cycle? How long does it take before it comes to rest?

Groups might first share their questions with the whole class and then, in small groups, decide which questions they wish to investigate. Such situations allow students to formulate questions based on their own interests.

Students should frequently work together in small groups to solve problems. They can discuss strategies and solutions, ask questions, examine consequences and alternatives, and reflect on the process and how it relates to prior problems. Students must verify results, interpret solutions, and question whether a solution makes sense. They should verify their own thinking rather than depend on the teacher to tell them whether they are right or wrong. Such experiences develop students' confidence in using mathematics.

Instruction should also help students develop their ability to understand and apply a variety of strategies (e.g., guess and check, make a table, look for patterns). These strategies should be explored in the context of solving problems. Through group and classroom discussions, students can examine a variety of approaches and learn to evaluate appropriate strategies for a given situation. The instructional goal is that students will build an increasing repertoire of strategies, approaches, and familiar problems; it is the problem-solving process that is most important, not just the answer. The following problem illustrates how students might share their approaches in solving problems:

How many handshakes will occur at a party if every one of the 15 guests shakes hands with each of the others?

Some students will choose to act out the problem. Some might draw a picture of a simpler case to approximate the situation (fig. 1.1). Other students might start with a simpler problem and look for a pattern (fig. 1.2):

Fig. 1.1. Drawing a picture of a simpler code

Fig. 1.2. Looking for patterns

Class discussions in which students share their approaches enrich and expand their repertoire of strategies for solving problems.

In addition to cooperative effort, real-world problems often require a substantial investment of time. Students should be encouraged to explore some problems as extended projects that can be worked on for hours, days, or longer. For example, students concerned about the traffic congestion at an intersection near their school decided to design a method to study the situation. Their study required that they answer such questions as what constituted "traffic," when to count it, how to record data, what the data meant, how to alleviate the congestion, who was responsible for dealing with such problems, and how to construct a convincing argument that the situation should be remedied. After several weeks of study, the results were presented to the city council, which accepted the students' recommendation and installed a traffic signal.

Not all problems require a real-world setting. Indeed, middle school students often are intrigued by story settings or those arising from mathematics itself. For example, which number from 1 to 100 has the most factors? What matters is that students experience mathematics in situations in which they come to view it as personally empowering. For example, the ability to see and use similar triangles to solve a nonroutine indirect measurement problem is empowering in two ways. It gives students confidence in their ability and perception to solve such problems in their own lives. Further, they see that their mathematical thinking contributes to their mastery of other mathematical ideas, such as their understanding of the relationship between ratio and fraction concepts.

The 5-8 standards contain numerous examples of how problem situations can serve as a context for exploring mathematical ideas. Through these situations, students have opportunities to investigate problems, apply their knowledge and skills across a wide range of situations, and develop an appreciation for the power and beauty of mathematics.