Towards efficient measurement of metacognition in mathematical problem solving

Abstract

Metacognitive monitoring and regulation play an essential role in mathematical problem solving. Therefore, it is important for researchers and practitioners to assess students' metacognition. One proven valid, but time consuming, method to assess metacognition is by using think-aloud protocols. Although valuable, practical drawbacks of this method necessitate a search for more convenient measurement instruments. Less valid methods that are easy to use are self-report questionnaires on metacognitive activities. In an empirical study in grade five (n039), the accuracy of students' performance judgments and problem visualizations are combined into a new instrument for the assessment of metacognition in word problem solving. The instrument was administered to groups of students. The predictive validity of this instrument in problem solving is compared to a wellknown think-aloud measure and a self-report questionnaire. The results first indicate that the questionnaire has no relationship with word problem solving performance, nor the other two instruments. Further analyses show that the new instrument does overlap with the thinkaloud measure and both predict problem solving. But, both instruments also have their own unique contribution to predicting word problem solving. The results are discussed and recommendations are made to further complete the practical measurement instrument.

The role of epistemic emotions in mathematics problem solving

“…Given that deep cognitive and metacognitive strategies were positive predictors of students' problem solving achievement, it is imperative to design learning environments that reduce boredom or that can provide scaffolds for students when boredom arises to shift their learning strategies or regulate that boredom. As previous research in mathematics problem solving has shown, central to successful problem solving is employment of these kinds of deep cognitive and metacognitive strategies (Jacobse & Harskamp, 2012;Muis, 2008;Schoenfeld, 1985). Our results provide support for this.…”

“…In this situation, the individual is more likely to be vigilant while solving the problem, ensuring to understand the problem and its various components, generating relevant prior knowledge, coordinating informational sources, making comparisons, and generating inferences, which are considered deep processing strategies. The individual is also more likely to continually evaluate progress, and check answers against set goals, which are key metacognitive processes for successful problem solving (Jacobse & Harskamp, 2012). The importance placed on correctly solving the problem may also result in anxiety if issues arise when solving the problem, confusion or frustration if the issue cannot be resolved, or enjoyment if there is resolution.…”

Analysis of Sixth Grade Students’ Think-Aloud Processes While Solving a Non-routine Mathematical Problem

“…Therefore, lack of paraphrasing indicates that students omit this important stage. It was also observed in Jacobse and Harskamp's (2012) study that students did not use paraphrasing during mathematical problem solving while thinking aloud. One of the reasons for this is tendency to solve the problems as quickly as possible, probably due to pressure of succeeding in national exams; hence it may not be very surprising that all groups skipped this part quickly.…”

“…The subject is encouraged to give a concurrent account of their thoughts and avoid interpretation or explanation of what they are doing; they only have to concentrate on the task (Ericsson & Simon, 1993;Someren, Bardnard, & Sandberg, 1994). One of the major advantages of using the think-aloud method is being able to capture the individual's immediate thoughts (Ericsson & Simon, 1993;Jacobse & Harskamp, 2012;Wade, 1990;Young, 2005), which is important in understanding students' problem solving processes. The thinkaloud method is considered a valid form of verbal data collection (Ericsson, 2006;Ericsson & Simon, 1993) and researchers in the field of social sciences have been using this method for quite some time.…”

A Case Study on Pre-service Secondary School Mathematics Teachers' Cognitive-metacognitive Behaviours in Mathematical Modelling Process

“…Think-aloud Protocols: Think aloud protocols are on-line measuring techniques which ask individuals tell-describe their thoughts verbally while they are working on a task (problem) [3,48]. Think-aloud protocols provide the richest data which can be obtained about metacognitive processes throughout learning, as they are related to gathered metacognitive behaviours [3] during a learning task [52].…”