For the past twenty years, problem solving has been touted as a primary focus for mathematics instruction at all grade levels (National Council of Supervisors of Mathematics, 1978; National Council of Teachers of Mathematics, 1980, 1989, 2000; Mathematical Sciences Education Board, 1989). Despite this persistent call for problem-solving as an instructional approach, teachers across the United States struggle with helping children solve problems. National and international assessments consistently reveal that U.S. students do not perform well on problem-solving tasks that require more than one step (DosKey, 1993; Dossey, 1994; Kenney and Silver, 1997; Silver, 1998; Stigler and Hiebert, 1997). Furthermore, studies of mathematics teachers at all grade levels indicate that teachers have difficulty in planning and implementing lessons that build students' problem-solving skills. (Cooney, 1985; Silver, 1985; Thompson, 1989) Clearly, teachers need instructional strategies that help students become better problem solvers.
The literature on the teaching of problem solving over the past twenty years promotes building students' metacognitive skills--planning, monitoring, and evaluating ones own thinking--as a means to improve their problem-solving skills (Costa, 1991; Perkins, 1992, 1995; Fogarty, 1994; Marzano et al, 1997; Swartz and Parks, 1994; Tishman, Perkins, and Jay, 1995). As adults, we see some evidence of children planning, monitoring, and evaluating. Parents listen as their children explain what candy they will select at the movie theatre. Teachers hear children stop what they are doing in a group and tell another child he is not doing the assignment correctly. A grandparent may hear a grandchild comment confidently about the quality of a drawing he insists be displayed on the refrigerator. When given a novel task in school, however, children are very likely to jump into the problem with one strategy, continue the strategy without "looking back," and finish without reexamining the solution. Often, the result can be a misunderstood problem, or an ineffective strategy, and/or a solution that does not work. According to Perkins (1995), metacognition, or reflective intelligence as he calls it, "particularly supports coping with novelty" (p. 112). Perkins also suggests that reflective intelligence "supports thinking contrary to certain natural trends" (p. 113), thus contributing to breaking mental sets and exploring new ideas.
If students across the United States are to become proficient mathematical problem solvers, teachers at all grade levels must learn how to develop and assess metacognitive skills in their students. Teachers are searching for instructional strategies that will help students plan, monitor, and evaluate their own thinking during problem solving. This study seeks to determine the effectiveness of a set of year-long instructional strategies designed to improve metacognitive skills, thereby improving the problem-solving skills of eight- and nine-year old students.
Metacognition emerged as an important mental activity for solving problems when researchers began to study children's intelligence and problem solving. Early analyses of problem-solving performance revealed that good or expert problem solvers tended to plan, monitor, and evaluate their thinking during problem solving more often and more efficiently than did poor or novice problem solvers (Flavell, 1976). More recent studies have confirmed this finding for students at middle school, high school, and college levels (Bookman, 1993; Cai, 1994; Lucangeli, Coi, and Bosco, 1997). Flavell (1976) defined metacognition broadly as "one's knowledge concerning one's own cognitive processes and products ... [and] the active monitoring and consequent regulation and orchestration of these processes" (p. 232).
Based on this definition, Garofalo and Lester (1985) identified three types of metacognitive knowledge related to mathematical problem solving: person knowledge; task knowledge; and strategy knowledge. …