By Buschman, Larry
Teaching Children Mathematics , Vol. 9, No. 9
NCTM's Principles and Standards for School Mathematics describes problem solving as a "hallmark of mathematical activity and a major means of developing mathematical knowledge" (NCTM 2000, p. 116). However, many children in my classroom have experienced difficulty when solving problems and often commented, "I don't get it!" or "It's too hard!" Now my students enjoy solving problems. My classroom is filled with comments such as 'That was a good problem!" (Micah, age 8) and "I like the way Shanna solved the problem--it's cool because it's easier to get the answer" (Stephanie, age 7). Like many other teachers who have changed their students' attitudes toward problem solving, I began by turning to professional resources for guidance. But I discovered another valuable resource in my classroom: the children I teach. By asking children questions such as "Why is problem solving difficult?" I learned that they can offer amazing insights into the intricacies of the problem-solving process and how they learn to become pr oblem solvers.
What Children Say about Problem Solving
Over the past ten years, I have documented the comments that children made as they solved a wide variety of problems and as they shared their solutions with their peers in a multiage classroom of first, second, and third graders (see fig. 1). I also have used one-on-one interviews to record children's comments and reactions to various classroom practices during problem-solving activities. What follows are my findings.
Young children want to solve problems, and their enjoyment of problem solving increases when children can solve problems in ways that make sense to them. Jose, Amber, and other children have taught me the importance of independence in the process of becoming a problem solver. Seven-year-old Jose said, "I know what I know. I get messed up when I try to do problems the way you [the teacher] want." Amber, age 8, said, "The ways on the [problem-solving strategies] chart are hard to remember, and most of the time I get mixed up and I can't decide which one to use, especially when it's a hard problem."
In the past, I tried to teach problem solving by showing children how to solve particular types of problems in a prescribed manner with computation algorithms or traditional problem-solving strategies such as guess and check, work backward, look for a pattern, and so on. Now I encourage children to solve problems in ways that make sense to them, and I try to help them acquire the habits, behaviors, and dispositions of a problem solver, such as patience, perseverance, and a positive attitude. As I learned to trust in children's natural problem-solving abilities, the children learned to trust in themselves and developed the self-confidence they needed to become successful problem solvers.
When children are given the opportunity to solve problems "their way," they take great pride and pleasure in developing their own strategies, instead of simply practicing strategies that adults have shown them. Children can not only create solutions to problems that resemble traditional problem-solving strategies but also invent ways of solving problems that reflect the ingenuity and creativity of their young minds. One day while exploring various relationships between different pattern blocks, I posed the following problem: "How many rhombi will it take to balance four hexagons?" Most children in the classroom solved the problem in the following manner, shown in figure 2:
* First, they placed three rhombi on top of a hexagon so that the surface area of the rhombi matched the surface area of the hexagon.
* Next, they repeated this action three times.
* Finally, they counted the total number of rhombi used: twelve.
One second grader, however, took two trapezoids and said, "Well, these two make a hexagon, but a trapezoid is really one rhombus and one triangle hooked together like this [see fig. …