In grades K-4, the study of mathematics should emphasize problem solving so that students can--

• use problem-solving approaches to investigate and understand mathematical content;
• formulate problems from everyday and mathematical situations;
• develop and apply strategies to solve a wide variety of problems;
• verify and interpret results with respect to the original problem;
• acquire confidence in using mathematics meaningfully.

Focus

Problem solving should be the central focus of the mathematics curriculum. As such, it is a primary goal of all mathematics instruction and an integral part of all mathematical activity. Problem solving is not a distinct topic but a process that should pe rmeate the entire program and provide the context in which concepts and skills can be learned.

This standard emphasizes a comprehensive and rich approach to problem solving in a classroom climate that encourages and supports problem-solving efforts. Ideally, students should share their thinking and approaches with other students and with teachers, and they should learn several ways of representing problems and strategies for solving them. In addition, they should learn to value the process of solving problems as much as they value the solutions. Students should have many experiences in creating pro blems from real-world activities, from organized data, and from equations.

In the early years of the K-4 program, most problem situations will arise from school and other everyday experiences. When mathematics evolves naturally from problem situations that have meaning to children and are regularly related to their environment, it becomes relevant and helps children link their knowledge to many kinds of situations. As children progress through the grades, they should encounter more diverse and complex types of problems that arise from both real-world and mathematical contexts.

When problem solving becomes an integral part of classroom instruction and children experience success in solving problems, they gain confidence in doing mathematics and develop persevering and inquiring minds. They also grow in their ability to communica te mathematically and use higher-level thinking processes.

Discussion

Classrooms with a problem-solving orientation are permeated by thought-provoking questions, speculations, investigations, and explorations; in this environment, the teacher's primary goal is to promote a problem-solving approach to the learning of all mat hematics content. The following two examples illustrate this meaning.

A lesson designed to develop the characteristics of parallelograms can be approached from a problem-solving perspective. The teacher, who has a collection of quadrilaterals like the ones shown in figure 1.1, has the children discove r the teacher's rule for sorting the shapes. One rule is to have all parallelograms in one loop and all nonparallelograms in the other.

Fig. 1.1

In turn, each child picks a shape and decides in which loop to put it. The teacher says yes or no as each student places a shape. Throughout the process, the children are asked to think about the common characteristics of the shapes in each loop; after al l shapes are placed, these common characteristics are discussed. As a result, the children learn to define parallelograms and name their characteristics in the context of a thought-provoking activity.

Basic subtraction facts also can be presented in a problem-solving setting.

Each child is asked to put 13 small counters under one hand and, without looking, to move 6 of them into view. The teacher asks, "Can you figure out how many counters are still under your hand?" The children are invited to share their solutio n strategies. Responses might include the following:

There are six over here [outside]; six more would be twelve, so there must be seven left.

You have six here. Four more make ten and three more make thirteen; four and three are seven. Seven are left.

The class then discusses solving subtraction problems by "adding on."

Once again, the mathematical ideas have originated with the children rather than the teacher, in an inquiry-oriented manner.

Computer software also is a significant component of a comprehensive problem-solving program. Many excellent software packages enable children to develop and apply problem-solving strategies in geometry, logical reasoning, classification, measurement, fra ctions and decimals, and other mathematical content.

A major goal of problem-solving instruction is to enable children to develop and apply strategies to solve problems. Strategies include using manipulative materials, using trial and error, making an organized list or table, drawing a diagram, looking for a pattern, and acting out a problem.

Consider the following problem:

I have some pennies, nickels, and dimes in my pocket. I put three of the coins in my hand. How much money do you think I have in my hand?

This problem leads children to adopt a trial-and-error strategy. They can also act out the problem by using real coins. Children verify that their answers meet the problem conditions. Follow-up questions can also be posed: "Is it possible for me to h ave four cents? Eleven cents? Can you list all the possible amounts I can have when I pick three coins?" The last question provides a challenge for older or more mathematically sophisticated children and requires them to make an organized list of pos sible coin combinations, perhaps like the one in figure 1.2.

Fig. 1.2

The initial conditions can be altered to include quarters:

I have six coins worth 42 cents; what coins do you think I have? Is there more than one answer?

A vital component of problem-solving instruction is having children formulate problems themselves. Children can write variations for problems previously explored, word problems that correspond to a number sentence, or a question that can be answered by in vestigating data in a menu, advertisement, or chart, like the one in figure 1.3, which lists children's eye colors. A primary feature of this context is that it lends itself to the use of calculators. Using calculators in problem-so lving settings to perform tedious calculations enables children to focus on the problem-solving processes rather than on the calculations.

Fig. 1.3

Children might pose such questions as, "How many children are in each school? How many more blue-eyed children than brown-eyed students are in grade 1? In all? Why do you think school 2 is different from school 1?"

Project problems, which often require several days of class time, provide an opportunity for children to become immersed in problem-solving activity. Some situations allow children to be particularly creative in their formulation of problems. Here is one such situation:

The class is given the opportunity to plan and participate in an all-school "Estimation Day." The children, in pairs or threes, are to design estimation activities to be completed by children in other classes. Each group will supply all the n ecessary materials and monitor the activities. The activities might include guessing children's heights, the number of candies in a jar, the lengths of various pieces of string, the weight of a bag of potatoes, the length of the room, the number of times they can write their names in a minute, or the length of time required for an ice cube to melt.

Participation in project problems allows children to acquire confidence in their problem-solving ability. In working with others, they become a vital part of a team and find that their contributions are essential to the success of the project.

When problem solving is an integral part of the curriculum, beginning with a child's earliest encounters with mathematics, children develop a point of view about what it means to learn mathematics and solve problems in mathematics.