The impact of context on children's performance in solving everyday mathematical problems with real-world settings.

Abstract. This study aimed to investigate children's performance and perception of problem difficulty in solving everyday mathematical problems with familiar versus unfamiliar contexts. In addition, the ways that children identify the similarities in problem-solving approaches between problem settings and everyday shopping were also examined. Forty-eight 4th-grade children participated in this study. Both quantitative and qualitative analyses were used. The results demonstrated that the familiar contexts neither enhance children's problem-solving performance nor decrease problem difficulty. More than half of the children did not identify the similarity in problem-solving approaches between problem settings and real shopping. When making judgments, even good problem-solvers were found to be distracted by superficial features in outward appearance and global mapping in similarity. This research highlights that children in the process of solving problems tend to be distracted by non-mathematical features in the problem settings. Some implications for instruction are given.

**********

In recent years, there has been an increasing emphasis within the field of mathematics education on the application of mathematics, such as the solving of problems posed in realistic contexts (Beishuizen, Gravemeijer, & van Lieshout, 1997; Gravmeijer, 1994). Many believe that integrating children's everyday knowledge and school mathematics would enable them to develop their understanding of mathematics by applying it to textually represented realistic problems; others argue for the application of mathematics to real life. Because cognitive activity in everyday life is socially defined, interpreted, and supported (Rogoff & Lave, 1984), it is plausible to suggest that a stronger connection between school mathematics and everyday life will enhance school learning.

It is well-known that problem solvers have less difficulty with problems in which the context makes it possible to retrieve learned knowledge and apply similar problem-solving experiences to formulate solutions (Bernardo, 1994). The term "context" is used here to describe the non-mathematical meanings present in the problem statement. Kulm (1984) stated that context helps to give meaning to the mathematical content. The verbal context or setting of a problem often provides a connection between mathematical content and its application. The context categories may include those that make the setting of the problem more or less relevant to the problem solvers' experience and interests.

The familiar/unfamiliar dichotomy is dependent upon the problem solvers themselves--that is, their backgrounds and experiences (Caldwell, 1984). Previous studies found that the level of familiarity with the context of a problem will affect the problem-solving process (Rogoff & Lave, 1984). Although prior research has shown that children have difficulty in unfamiliar contexts, little is known about how the familiarity of contexts and various quantities involved in problems affects children's performance and perception of difficulty in solving problems within real-world settings. This study seeks to address these issues by using mathematical problems with isomorphic structures involving multiplication in familiar versus unfamiliar contexts.

The Impact of Familiarity of Problem Context on Children's Performance in Solving Mathematical Problems

Contextual similarity was found to facilitate recall in a variety of earlier problem-solving studies (Bassok & Holyoak, 1989; Gick & Holyoak, 1983). Furthermore, familiarity with question terms matters for the feelings of knowing and facilitates understanding (Reder & Ritter, 1992). The hypothesis is that a familiar context provides a less abstract and more directly experienced grounding for the new domain, thus enhancing the use of particular strategies or activating known structures that allow for more efficient processing. Since some problem-specific information of a problem context is sometimes correlated with structural information (Bassok & Holyoak, 1989; Bernardo, 1994), it follows that familiar information in problem contexts would enhance solvers' performance (Donnelly & McDaniel, 1993). The findings in the above studies support the benefit of more familiar contexts, and imply that anything that concretizes the problems will improve solvers' inference-level learning.

Besides the familiarity of context that is presumed to be associated with the difficulties children have in solving arithmetic problems, the quantities involved and the appropriate arithmetic operation also can affect children's problem-solving performance (Caldwell, 1984; Stern, 1997). With respect to the numerical quantity involved, a problem may be more difficult if it contains whole numbers in the hundreds or above (Collis, 1975) or if it contains decimals (Bell, Swan, & Taylor, 1981). Furthermore, Battista, Clements, Arnoff, Battista, and Borrow (1998) found that the multiplication structure of rectangular arrays is not intuitively obvious to children. In studying the strategies that children use to solve covering tasks, Outhred and Mitchelmore (2000) pointed out that the shape of the surface to be covered may affect problem difficulty. Given these findings, it is questionable that providing real-world settings with familiar contexts will be important when the problem-solving task involves specific numerical data as mentioned, with which children find it difficult to deal.

Similarity Judgment on the Problem-solving Approach Between Problems in Real-world Settings and Everyday Life

When people are given problems to solve, they often use previously learned knowledge or similar problem-solving experiences to structure and formulate solutions for the new problems (Bernardo, 1994). The major requirement for ensuring successful transfer is to enhance the problem solver's ability to apply relevant prior knowledge, according to the problem-type schemata. Gick and Holyoak (1983) and Bernardo (1994) pointed out that problem-type schemata are acquired through some inductive or generalization process involving comparisons among similar or analogous problems of one type. Therefore, recognizing the shared structural features in a problem-solving approach across different problem contexts would be the crucial process towards applying relevant prior knowledge.

Some studies (e.g., Smith, 1989) concentrated on children's classifications of multidimensional stimuli, with the result that young children are more likely to classify by overall similarity. Children are less likely to analyze a stimulus into its components; instead, they respond in terms of overall similarity, tending to judge similarity on the basis of surface features or perceptual information presented in the context. Another observed trend is that, as children mature, they tend to base their similarity judgments on more abstract, more relational, and less superficial properties (Gentner, 1989).

Regarding the similarity judgment of problem solvers with different ability, Chi, Feltovich, and Glaser (1981) and Hembree (1992) demonstrated that good solvers tend to organize efficient solutions according to their discovery of a familiar problem pattern. They pointed out that the difference in similarity judgment between good and poor solvers appears to be quite general. Good problem-solvers with high achievement are known to be highly sensitive to the structural features of a problem and are thereby able to use structural features of a problem to make an appropriate adaptation. On the other hand, poor problem-solvers with lower achievement tend to concentrate more on information of salient superficial features, which prevents them from making an appropriate adaptation (Huang, 2003).

Several studies have demonstrated that children are able to apply acquired knowledge when solving everyday problems. They further reveal that children construct their …