Traditional pen-and-paper vs mental approaches to computation: the lesson of Adrien

New mathematics syllabi are facing the issue of whether to discontinue the emphasis on traditional pen-and-paper algorithms and replace it with a focus on self initiated written algorithms, mental computation and number sense. Efficient and effective strategies for mental computation differ markedly from those that underlie traditional algorithms. They tend to be more wholistic and less reliant on separation into place values. These strategies often reflect the strategies required for estimation, and are more closely related to the spontaneous computational activity of children. This paper discusses traditional and mental approaches to computation in relation to the mental strategies for multiplication and division word problems employed by a child, Adrien, over a three-year period from 1993 to 1995 (Years 4 to 6). Although he was considered to be a higher ability student, Adrien was not a "lightning calculator", nor was he capable of such calculative feats as products of two eight-digit numbers. However, he was successful at multiplying and dividing two and three-digit numbers before such calculations were taught because he employed his own efficient and (it could be argued) advanced strategies that exhibited more number sense than the classroom taught traditional algorithms. His strategies exhibited both change and consistency and showed associated understandings. His performance highlighted the possibilities for computation syllabi where children are allowed to develop their own spontaneous strategies and indicated the disadvantages for syllabi, such as that still existing in Queensland, where traditional algorithms are still a major component.