Academic journal article Journal of Developmental Education

Comprehension Monitoring: An Aid to Mathematical Problem Solving

Academic journal article Journal of Developmental Education

Comprehension Monitoring: An Aid to Mathematical Problem Solving

Article excerpt

Mathematics students traditionally have great difficulty in solving word problems. They frequently state that they just don't know what to do. Their real problem may well be that they don't know what it is they don't understand. If they are not aware of what is wrong with their attempts to solve a problem, they can never take any kind of corrective action that might lead them to success. This is not a new idea. Brown (1978) has observed that there is a "distinction between knowledge and the understanding of that knowledge" (p. 157) and has found that "a very basic form of self-awareness involved in all ... problem-solving tasks is the realization that there is a problem of knowing what you know and what you do not know" (p. 82). One way to teach students how to determine what they don't understand is through the use of comprehension monitoring techniques.

Comprehension monitoring is generally defined as the awareness and control of one's understanding or lack of understanding. It is part of the self-regulatory aspect of metacognition. Baker and Brown (1984) describe comprehension monitoring as "keeping track of the success with which one's comprehension is proceeding, ensuring that the process continues smoothly, and taking remedial action if necessary" (p. 355). It is this author's belief that comprehension monitoring is a critical part of the problem-solving process, and one that can be taught to developmental mathematics students to help them to improve their problem-solving abilities.

Background

Problem Solving

Problem-solving strategies have been the focus of mathematics education research for several decades. George Polya, in his now classic text, How to Solve It (1945), set in motion the concepts and strategies that have become the basis for teaching and learning how to solve mathematical problems. This work has become the cornerstone for research in mathematical problem solving, and it was here that he first coined the term heuristics, which is generally meant to include all strategies that, if used properly, will enhance one's ability to solve mathematical problems. But Polya's model by itself does not guarantee success in solving problems.

As researchers investigated the use of heuristics and other procedures, it became clear that there was no consistent answer. Heuristics seemed to enhance performance in some cases, but not in others. Other factors, in addition to having a comprehensive plan or strategy, seemed to be playing an important role. "Lack of expertise with mathematical content was a more frequent impediment to solving algebra problems than problem solving deficiencies" (Kelly, 1993, p.2).

Meta-cognition

In 1976, Flavell proposed that the concepts of the newly-emerging field of metacognition might provide significant insights into the process of solving problems. In particular, it might explain why students were unable to solve problems even though they had all the required information and knew the necessary solution procedures. He defined metacognition as follows:

"Metacognition" refers to one's knowledge concerning one's own cognitive processes and products or anything related to them, e.g., the learning-relevant properties of information or data .... Metacognition refers, among other things, to the active monitoring and consequent regulation and orchestration of these processes in relation to the cognitive objects or data on which they bear, usually in the service of some concrete goal or objective. (Flavell, 1976, p. 232)

The use of metacognitive techniques by problem solvers provides two very important mathematical purposes: First, it allows students to keep track of what they have done and plan to do next, and, secondly, it allows them to make connections between their problem-solving work and their knowledge of subject matter content and mathematical procedures (Finkel, 1996).

One of the major foundational studies for examining the role of metacognition in mathematical problem solving was the study of two 7th-grade classes in a middle school in Bloomington, Indiana conducted by Lester, Garofalo, and Kroll (1989). …

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