# Metacognition and Mathematical Problem Solving: Helping Students to Ask The Right Questions

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1 Metacognition and Mathematical Problem Solving: Helping Students to Ask The Right Questions Shirley Gartmann and Melissa Freiberg The acquisition of problem solving, reasoning and critical thinking skills has been identified by the National Council of Teachers of Mathematics (NCTM, 1989) as a critical goal. Lester (1985) defines this goal as helping students to think within a mathematical context: The primary purpose of mathematical problem solving instruction is not to equip students with a collection of skills and processes, but rather to enable them to think for themselves. The value of skills and processes instruction should be judged by the extent to which the skills and processes actually enhance flexible, independent thinking (p. 66). Students abilities to think flexibly can be developed and enhanced by teachers modeling their own thinking processes, giving students opportunities to problem solve, and helping students become aware of their own thought processes as they solve mathematical problems. This process of analyzing our own thought processes is called metacognition and includes thinking about how we are approaching a problem, the strategies we choose to use to find a solution, and the questions we ask ourselves about the problem are all part of metacognition. Schoenfeld (1985) has characterized metacognitive skills as aspects of mathematical understanding that extend beyond the mastery of routine facts and procedures (p. 361) and noted that these skills do not usually develop in mathematics instruction because of the focus on factual and procedural knowledge. Campione, Brown and Connell (1989) state that successful learners can reflect on their own problem solving activities, have available powerful strategies for dealing with novel problems, and oversee and regulate those strategies efficiently and effectively (p. 94). They also indicate that assessing this type Shirley Gartmann is an associate professor in the Department of Curriculum and Instruction at the University of Wisconsin- Whitewater. She is interested in the use of manipulatives to develop primary students problem solving skills and is involved in programs on distance and technology education. Melissa Freiberg is an assistant professor in the Department of Curriculum and Instruction at the University of Wisconsin- Whitewater. Her research interests are in problem solving in mathematics education and induction of beginning teachers into the profession. of learning requires dynamic rather than static measures. While static measures test knowledge and process, dynamic measures depend on determining how a student uses knowledge and skills to progress beyond a starting point. Dynamic measures are better predictors of gains in performance and are significantly more diagnostic than learning scores from static tests. In light of these considerations, this study examined how students used their thinking skills to complete a problem solving task and how those thinking skills change given practice. Additionally, the study sought to determine if guidance in the form of hints given to subjects would make a difference in the manner in which they attended to the task. Subjects College students enrolled in a mathematics content course required of all prospective elementary teachers served as subjects for this study. This group was chosen with the assumption that older subjects would be more likely to be able to reflect on their own thought processes and analyze their own performances. All students in this class were required to have taken an introductory algebra course or tested out of it, and the present course would be the terminal college mathematics course for most subjects, except for a few mathematics minors. Subjects were randomly assigned to one of two groups, either a treatment or control group. Each group contained seventeen members with the treatment group having 2 men and 15 women and the control group 3 men and 14 women. Examination of the two groups after group assignment showed them to be comparable in range of abilities, age, mathematical background, and success in previous math courses. The Task A public domain computer program by Arlene Cram entitled PHANTNUM (Apple translation by Preston Marsh) served as the problem solving task. A computer game was used for four reasons. First, subjects can work in a computer lab with minimal supervision. Second, this game provides instant responses to student clues and allows for individuals to work at their own pace. Third, the game provides feedback as to incorrect guesses without Volume 6 Number 1 9

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