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We traced the emerging relations between children's understanding of multidigit numbers and their computational skill and investigated how instruction influenced these relations. We followed about 70 children over the first 3 years of school while they were learning about place value and multidigit addition and subtraction in 2 different instructional environments. By interviewing the students several times each year, we found that understanding and skill were closely related on tasks for which students had not yet received instruction as well as on more difficult tasks even after instruction. Students appeared to apply specific understandings to invent new procedures and modify old ones. The alternative instruction, which encouraged students to develop their own procedures and to make sense of procedures presented by others, appeared to facilitate higher levels of understanding and closer connections between understanding and skill.
Hiebert et al. (Sun,) studied this question.