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Summary of Reference

Steffe, Leslie

Children’s multiplying schemes

Bibliographic data:
In G. Harel & J. Confrey (Eds.) (1994) The development of multiplicative reasoning in the learning of mathematics (pp. 3-39). Albany, NY: SUNY.

In this chapter Steffe discusses problems arising when children use their existing schema based on counting methods. This is particularly a problem when the answer is correct. Using neo-Piagetian ideas Steffe makes the theoretical assumption that children assimilate new knowledge into their existing schemes. His work is also based on a second theoretical assumption that students actively build up or construct new understandings. Steffe’s point is that the teacher needs to know how a child is thinking so as to guide the development of these understandings. He discusses social context as a primary setting for accommodation or the generation of new as opposed to using existing structures or schemes.

Steffe’s work is relevant to the New Zealand context as his theorising about counting and multiplication underpins approaches currently being introduced as part of the numeracy projects. Much of the language used to describe counting schemes is technical and more detailed explanations can be found in his earlier work with Paul Cobb. Through his work with Cobb and von Glaserfeld the complexities arising in classroom instruction in multiplication and division are further discussed. Key ideas featured in this chapter include repetition or iteration often referred to in instructional material under the heading of “multiplication as repeated addition”. Steffe stresses that the broad category of repeated addition includes both counting-based and collection-based schemes and that it may not always be obvious to the teacher which the child is drawing on. Insights into children’s thinking, such as might be gained through a diagnostic interview, are important if the teacher wants to guide the child in developing more sophisticated strategies.