This thesis explores children’s conceptions of the equals sign from the vantage point of notating task design. The existing literature reports that young children tend to view the equals sign as meaning “write the result here”. Previous studies have demonstrated that teaching an “is the same as” meaning leads to more flexible thinking about mathematical notation. However, these studies are limited because they do not acknowledge or teach children that the equals sign also means “can be exchanged for”. The thesis explores the “sameness” and “exchanging” meanings for the equals sign by addressing four research questions. The first two questions establish the distinction, in terms of task design, between the two meanings.\ud Does the “can be exchanged for” meaning for the equals sign promote attention to statement form? Are the “can be exchanged for” and “is the same as” meanings for the equals sign pedagogically distinct? The final two research questions seek to establish how children might coordinate the two meanings, and connect them with their existing implicit knowledge of arithmetic principles.\ud Can children coordinate “can be exchanged for” and “is the same as” meanings for the equals sign?\ud Can children connect their implicit arithmetical knowledge with explicit transformations of notation?\ud The instrument used is a specially designed notational computer-microworld called Sum Puzzles. Qualitative data are generated from trials with pairs of Year 5 (9 and 10 years), and in one case Year 8 (12 and 13 years), pupils working collaboratively with the microworld toward specified task goals.\ud It is discovered that the “sameness” meaning is useful for distinguishing equality statements by truthfulness, whereas the “exchanging” meaning is useful for distinguishing statements by form. Moreover, a duality of both meanings can help children connect their own mental calculation strategies with transformations of properly formed notation
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