• Questioning and discussion should elicit students’ thinking and solu tion strategies and should build on them, leading to greater clarity and precision.

  • Discourse should not be confined to answers only but should include discussion of connections to other problems, alternative representations and solution methods, the nature of justification and argumentation, and the like.

Linking Experience to Abstraction

Students acquire higher levels of mathematical proficiency when they have opportunities to use mathematics to solve significant problems as well as to learn the key concepts and procedures of that mathematics. Although mathematics gains power and generality through abstraction, it finds both its sources and applications in concrete settings, where it is made meaningful to the learner. There is an inevitable dialectic between concrete and abstract in which each helps shape the other. Exhortations to “begin with the concrete” need to consider carefully what is meant by concrete. Research reveals that various kinds of physical materials commonly used to help children learn mathematics are often no more concrete to them than symbols on paper might be. Concrete is not the same as physical. Learning begins with the concrete when meaningful items in the child’s immediate experience are used as scaffolding with which to erect abstract ideas. To ensure that progress is made toward mathematical abstraction, we recommend the following:

  • Links among written and oral mathematical expressions, concrete problem settings, and students’ solution methods should be continually and explicitly made during school mathematics instruction.

Assigning Independent Work

Part of becoming proficient in mathematics is becoming an independent learner. For that purpose, many teachers give homework. The limited research on homework in mathematics has been confined to investigations of the relation between the quantity of homework assigned and students’ achievement test scores. Neither the quality nor the function of homework has been studied. Homework can have different purposes. For example, it might be used to practice skills or to prepare the student for the next lesson. We believe that independent work serves several useful purposes. Regarding independence and homework, we make the following recommendations:

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