Combining Cognition and Metacognition During the Problem Solving Process

By Dawn Woods, RME Elementary Mathematics Coordinator

Mathematical problem solving extends beyond the application of mathematics skills and concepts to include the semantics and syntax of language and the situations that the language represents within social-cultural contexts. Sometimes when students consider word problems, they rely on coping strategies such as using key words or apply general strategies such as “draw a picture”, which can limit the student’s problem solving abilities (Clements & Sarama, 2009). However research is showing that when students are engaged in metacognition, or thinking about their thinking, that their problem solving competency increases through the awareness of their reasoning (Cambell & White, 1997; Goos, Galbraith & Renshaw, 2002; Caswell & Nisbet, 2005).

Intervention Central, an online RtI resource, outlines a research-based strategy designed to engage struggling students in the problem solving process. Based on Montague’s work, students apply a “Say-Ask-Check” routine to stimulate metacognition as they work through the cognitive steps of the problem solving process (1992). During each step of the problem solving process, students are taught to “say” or self-instruct by stating the purpose of the step; “ask” or self-question what he or she plans to do to complete the step; and “checks” by self-monitoring the successful completion of the step. This “Say-Ask-Check” routine with close teacher support during instruction can increase the likelihood of student success.

Following is an example of what the “Say-Ask-Check” routine could look like when applied to George Polya’s four-step mathematical problem solving techniques (1945; Wright, 2011).

Problem Solving Steps	"Say-Ask-Check" Routine
Understanding the Problem	Say (Self-Instruction):  “I will read the problem until I can restate the problem in my own words.” Ask (Self-Question):  “Do I understand the problem?” Check (Self-Monitor):  “I understand the problem.”
Devise a Plan	Say (Self-Instruction):   “I will create a plan to solve the problem.” Ask (Self-Question):  “What is my first step? What is the next step, etc.?” Check (Self-Monitor):  “My plan has the right steps to solve the problem.”
Carrying Out the Plan	Say (Self-Instruction):  “I will solve the problem” Ask (Self-Question):  “Is my answer reasonable?” Check (Self-Monitor):  “I carried out my plan to solve the problem.”
Looking Back	Say (Self-Instruction):  “I will check my work.” Ask (Self-Question):  “Did I check each step in my calculation?” Check (Self-Monitor):  “The problem appears to be correct.”

Combining cognition and metacognition through using the problem solving process and the Say-Ask-Check routine increases a students’ awareness in his/her reasoning thereby increasing the likelihood of his/her academic success.

References

Campbell, P., & White, D. (1997). Project IMPACT: Influencing ad supporting teacher change in predominately minority schools. In E. Fennema & B.Nelson (Eds.), Mathematics teachers in transition (pp 309-355). Mahway, NJ: Erlbaum.

Caswell, R., Nisbet, S. (2005). Enhancing mathematical understanding through self-assessment and self-regulation of learning: he value of meta-awareness. Building Connections: Research, Theory and Practice. Retrieved from http://www98.griffith.edu.au/dspace/handle/10072/2482 .

Clements, D. & Sarama, J. (2009). Learning and teaching early math: the learning trajectories approach. New York: Routlege.

Goos, M., Galbraith, P., & Renshaw, P. (2002). Socially mediated metacognition: Creating collaborative zones of proximal development in small group problem solving. Education Studies in Mathematics, 49 (2), 193-223.

Montague, M. (1992). The effects of cognitive and metacognitive strategy instruction on the mathematical problem solving of middle school students with learning disabilities. Journal of Learning Disabilities, 25, 230-248.

Polya, G. (1945). How to solve it. Princeton, NJ: Princeton University Press

Wright, J. (2011). Math problem solving: Combining cognitive and metacognitive strategies. Retrieved from http://www.interventioncentral.org/academic-interventions/math/math-problem-solving-combining-cognitive-metacognitive-strategies.