Internal and External Dialogue as Support for Metacognition

The research summarized in How People Learn and Adding It Up and the professional experience summarized in the standards of the National Council of Teachers of Mathematics all emphasize how important it is for students to communicate about mathematics and for teachers to help them learn to do so. Students can learn to reflect on and describe their mathematical thinking. They can learn to compare methods of solving a problem and identify the advantages and disadvantages of each. Peers can learn to ask thoughtful questions about other students’ thinking or help edit such statements to clarify them. Students can learn to help each other, sometimes in informal, spontaneous ways and sometimes in more organized, coaching-partner situations. The vignette in Box 5-3 illustrates such communication about mathematical thinking after it has been developed in a classroom. Experience in the Children’s Math Worlds Project indicates that students from all backgrounds can learn to think critically and ask thoughtful questions, reflect on and evaluate their own achievement, justify their points of view, and understand the perspectives of others. Even first-grade students can learn to interact in these ways.

Of course, teachers must help students learn to interact fruitfully. To this end, teachers can model clear descriptions and supportive questioning or helping techniques. In a classroom situation, some students may solve problems at the board while others solve them at their seats. Students can make drawings or use notations to indicate how they thought about or solved a problem. Selected students can then describe their solution methods, and peers can ask questions to clarify and to give listeners a role. Sometimes, pairs of students may explain their solutions, with the less-advanced partner explaining first and the other partner then expanding and clarifying. Students usually attend better if only two or three of their fellow students explain their solution method for a given problem. More students can solve at the board, but the teacher can select the methods or the students for the class to hear at that time. It is useful to vary the verbal level of such explainers. Doing so assists all students in becoming better explainers by hearing and helping classmates expand upon a range of explanations. The goal in all of this discussion is to advance everyone’s thinking and monitoring of their own understanding and that of other students rather than to conduct simple turn taking, though of course over time, all students can have opportunities to explain.

Seeking and Giving Help

Students must have enough confidence not only to engage with problems and try to solve them, but also to seek help when they are stuck. The



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