Through problem solving, students can experience the power and utility of mathematics. Problem solving is central to inquiry and application and should be interwoven throughout the mathematics curriculum to provide a context for learning and applying mathematical ideas. Middle-grades students whose curriculum is based on the Standards in this document will benefit from frequent opportunities for both independent and collaborative problem-solving experiences. They will engage profitably in complex investigations, perhaps occasionally working for several days on a single problem and its extensions.

What should problem solving look like in grades 6 through 8?

Problem solving in grades 6–8 should promote mathematical learning. Students can learn about, and deepen their understanding of, mathematical concepts by working through carefully selected problems that allow applications of mathematics to other contexts. Many interesting problems can be suggested by everyday experiences, such as reading literature or using cellular telephones, in-line skates, kites, and paper airplanes.

Instruction in grades 6–8 should take advantage of the expanding mathematical capabilities of students to include more-complex problems that integrate such topics as probability, statistics, geometry, and rational numbers. Situations and approaches should build on and extend the mathematical understanding, skills, and language that students have acquired.

Well-chosen problems can be particularly valuable in developing or deepening students' understanding of important mathematical ideas. Consider the following problem that might be used by a teacher who wants her students to think about various ways to use ratios and proportions:

A baseball team won 48 of its first 80 games. How many of its next 50 games must the team win in order to maintain the ratio of wins to losses?

Students can solve this problem in many ways. One student might express the ratio of wins in the first 80 games as 48/80 and note that the ratio is a little more than one-half; that is, the team wins a little more than half the time. She might then estimate that in the next 50 games » the team should win about 28 games. She could compare the resulting ratio of 28/50 to the given ratio of 48/80 and adjust her estimate until the two ratios are equivalent. Another student might look at the ratio of wins to losses, 48:32, and simplify it to 3:2. Restating this result as "3 wins in every 5 games" and noting that there are 10 sets of 5 games in the 50 games to be played, he could conclude that 30 games is the solution. Yet another student might use a proportion, 48/80 = x/50, to find the solution. A fourth student might use percents or decimals (as newspapers do when reporting the "standings" of baseball teams). This student might divide 48 by 80 and represent the ratio as 60 percent and then find 60 percent of 50 games, or represent it as 0.600 and multiply by 50, to determine that 30 games must be won to maintain the success rate. Such problems help students develop and use a variety of problem-solving strategies and approaches, and sharing these methods within the classroom affords students opportunities to assess the strengths and limitations of alternative approaches to considering them.

After students have had similar experiences with ratios, rates, and proportions in grades 6 and 7, a teacher who wanted to extend and deepen her eighth-grade students' understanding of the topics might use a problem like the following:

Over the past few weeks, the American Movie Corporation has introduced two new kinds of candy at the concession stands in movie theaters in town. For three weeks, two theaters have offered Apple Banana Chews. For two weeks, five other theaters sold Mango Orange Nips. Only one of the two types of candy was sold at each theater, and all the theaters showed the same movies and had roughly the same attendance each week during the introductory period. During that period, 660 boxes of Apple Banana Chews and 800 boxes of Mango Orange Nips were sold. Suppose you have been hired by the company to help them determine which candy sold better. Use the information to decide which type of candy was more popular, and carefully and completely explain the basis for your answer.

This problem can help students see the need to go beyond superficial approaches and to dig deeply into their understanding of ratios and rates. For some students, an initial response will be that Mango Orange Nips (MONs) were more popular because more were sold. In an early class discussion, other students might point out to them that such a direct comparison is misleading because the two types of candy were sold at different numbers of theaters and for different amounts of time. From this discussion, a need to probe more deeply into the relationships in the problem will be apparent. Students are likely next to consider ratios or rates, which they have learned to use to express quantitative relationships. By inviting individuals or groups of students to present possible solutions, the teacher can initiate a lively discussion of competing approaches and arguments. Some students will consider the average number of candies sold for each theater—330 boxes of ABCs (Apple Banana Chews) per theater versus 160 boxes of MONs per theater—and conclude that ABCs were more popular. Other students will consider a different rate, namely, the average number of candies sold each week—220 boxes of ABCs per week versus 400 boxes of MONs per week—and conclude that MONs were more popular. Because each of these answers is different and seems to be based on a sensible approach, neither answer can be argued to be "better" than the other. So students can see that they must go beyond these simple rates to answer the question. » The teacher can then help them develop a more complex rate—the average number of boxes of candy per theater per week—that incorporates all the information in the problem and yields a defensible solution: the rates are 110 for ABCs and 80 for MONs, so ABCs are the better sellers.

Teachers should regularly ask students to formulate interesting problems based on a wide variety of situations, both within and outside mathematics. Teachers should also give students frequent opportunities to explain their problem-solving strategies and solutions and to seek general methods that apply to many problem settings. These experiences should engender in students important problem-solving dispositions—an orientation toward problem finding and problem posing; an interest in, and capacity for, explaining and generalizing; and a propensity for reflecting on their work and monitoring their solutions. They should be expected to explain their ideas and solutions in words first, and then teachers can help them learn to use conventional mathematical symbols or their own forms of representations, as appropriate, to convey their thinking.

The availability of technology—in the form of computers and scientific or graphing calculators—allows middle-grades students to deal with "messy," complex problems. The technology can alleviate much of the drudgery that until recently often constrained middle-school mathematics to using only problems with "nice numbers." Computers, calculators, and electronic data-gathering devices, such as calculator-based laboratories (CBLs) or rangers (CBRs), offer means of gathering or analyzing data that in years past might have been considered too troublesome to deal with. Similarly, classroom Internet connections make it possible for students to find information for use in posing and solving a wide variety of problems. For example, students might be interested in investigating whether it is cost-effective to recycle aluminum cans at their school, or they might explore weather patterns in different regions. Graphing calculators and easy-to-use computer software enable students to move between different representations of data and to compute with large quantities of data and with messy numbers, both large and small, with relative ease. As a result, problems in the middle grades can and should respond to students' questions and engage their interests.

What should be the teacher's role in developing problem solving in grades 6 through 8?

Make a sum of 1000, using some eights (8s) with some plus signs (+s) inserted. »

Because this problem can be solved in more than one way, students could find several solutions (888 + 88 + 8 + 8 + 8 = 1000 is one solution, and 888 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 + 8 = 1000 is another). They could then analyze these solutions and discuss whether others exist.

Teachers motivate students by encouraging communication and collaboration and by urging students to seek complete solutions to challenging problems. Recognizing students' contributions can add to their motivation. Some teachers, for example, find it effective to name a problem, conjecture, or solution method after the student who proposed it (e.g., Tamela's problem).

Research suggests that an important difference between successful and unsuccessful problem solvers lies in their beliefs about problem solving, about themselves as problem solvers, and about ways to approach solving problems (Kroll and Miller 1993). For example, many students have developed the faulty belief that all mathematics problems could be solved quickly and directly. If they do not immediately know how to solve a problem, they will give up, which supports a view of themselves as incompetent problem solvers. Furthermore, many students believe there is just one "right" way to solve any mathematics problem. Not only do these students become dependent on the teacher or an answer key for a verification of their solution, but they also fail to appreciate the excitement and insight that can come from recognizing and connecting very different ways to solve a problem. To counteract negative dispositions, teachers can help students develop a tendency to contemplate and analyze problems before attempting a solution and then persevere in finding a solution.

The essence of problem solving is knowing what to do when confronted with unfamiliar problems. Teachers can help students become reflective problem solvers by frequently and openly discussing with them the critical aspects of the problem-solving process, such as understanding the problem and "looking back" to reflect on the solution and the process (Pólya 1957). Through modeling, observing, and questioning, the teacher can help students become aware of their activity as they » solve problems. For example, consider how the following problem might be used to develop students' skill in problem solving (Schroeder and Lester 1989, p. 40):

On centimeter graph paper outline all the shapes that have an area of 14 square cm and a perimeter of 24 cm. For each shape you draw, at least one side of each square must share a side with another square.

Students may initially assume that the shapes referred to are rectangles. Under that assumption, students can discover that the problem has no solution. A teacher might allow this line of thinking to surface early so that it can be addressed. Once students understand that shapes other than rectangles are possible, they might approach the problem by experimenting with a few shapes, using graph paper or 14 square cutouts. If students do not begin to recognize that haphazard experimentation is not likely to produce a complete solution, questioning by the teacher might help them. The teacher can help students develop a systematic way to keep track of the shapes that have been tried. The teacher can also ask provocative questions that encourage students to find all possibilities: What makes two shapes "different"? Are shapes different if one is a flip of the other? What strategies can be used to create new shapes from old ones in a way that preserves both the area and the perimeter? In this problem, students draw on their knowledge of various geometric ideas, such as area, perimeter, and congruence and transformations that preserve area and perimeter. Moreover, they engage in a process that is applicable to a wide variety of problems: gradually understanding a problem more deeply and then working systematically to determine all possible solutions. As research has shown, effective problem solvers move flexibly among aspects of the problem-solving process as they work through a problem (Kroll and Miller 1993).

Although it is not the main focus of problem solving in the middle grades, learning about problem solving helps students become familiar with a number of problem-solving heuristics, such as looking for patterns, solving a simpler problem, making a table, and working backward. These general strategies are useful when no known approach to a problem is readily apparent. These processes may have been used in the elementary grades, but middle-grades students need additional experience and instruction in which they consider how to use these strategies appropriately and effectively.

Students also should be encouraged to monitor and assess themselves. Good problem solvers realize what they know and don't know, what they are good at and not so good at; as a result they can use their time and energy wisely. They plan more carefully and more effectively and take time to check their progress periodically. These habits of mind are important not only in making students better problem solvers but also in helping students become better learners of mathematics.

For several reasons, students should reflect on their problem solving and consider how it might be modified, elaborated, streamlined, or clarified: Through guided reflection, students can focus on the mathematics involved in solving a problem, thus solidifying their understanding of the concepts involved. They can learn how to generalize and extend problems, leading to an understanding of some of the structure underlying mathematics. Students should understand that the problem-solving » process is not finished until they have looked back at their solution and reviewed their process.

An important aspect of a problem-solving orientation toward mathematics is making and examining conjectures raised by solving a problem and posing follow-up questions. For example, according to the Pythagorean relationship, if squares are built on the legs and the hypotenuse of any right triangle, then the areas of the squares on the legs will together sum to the area of the square on the hypotenuse. This well-known relationship, summarized with the formula a² + b² = c², where a and b are the lengths of the triangle's legs and c is the length of its hypotenuse, is used frequently to solve numerical and algebraic problems. It can be the source of much interesting problem posing and generalization for middle-grades students. A teacher might orchestrate a discussion in which students pose a variety of "what if" questions about variants and extensions of the Pythagorean relation (Brown and Walter 1983), for example, Would the area relationship hold if we built something other than squares on the sides of right triangles, say for equilateral triangles? Or regular hexagons? Or semicircles? Will the areas still sum in the same way? Such conjectures can easily be examined by using interactive geometry software, which can also facilitate students' search for a counterexample to disprove a conjecture. Although formal proof of a generalization of the Pythagorean relationship may be beyond the reach of most students in the middle grades, some students might be able to use their developing understanding of proportionality and similarity to argue that the generalization holds because the areas of similar figures are proportional to the square of the lengths of their corresponding sides.

By reflecting on their solutions, such as in this extension of the Pythagorean relationship, students use a variety of mathematical skills, develop a deeper insight into the structure of mathematics, and gain a disposition toward generalizing. The teacher can ensure that classroom discussion continues until several solution paths have been considered, discussed, understood, and evaluated. It should become second nature for students to talk about connections among problems; to propose, critique, and value alternative approaches to solving problems; and to be adept in explaining their approaches.