The Equals Sign: Operations, Relations and Substitutions

By Jones, Ian; Inglis, Matthew et al. | Mathematics Teaching, September 2011 | Go to article overview

The Equals Sign: Operations, Relations and Substitutions


Jones, Ian, Inglis, Matthew, Gilmore, Camilla, Mathematics Teaching


Ian Jones, Matthew lnglis, and Camilla Gilmore with new software designed to help understanding of the equals sign.

Most primary children interpret the equals sign as an instruction to perform an arithmetic operation, rather than as symbolising an equivalence relationship. Research to date has concerned itself with how children move from an operational to a relational conception of the equals sign. The operational conception involves viewing the symbol ' = ' as a signal to perform an arithmetic calculation, and always expecting number sentences to have an answer after ' = '. The relational conception is the understanding that the symbol ' = ' means 'is the same as', and knowing number sentences can have answers and expressions on either side of ' = '. Our recent work has considered whether a substitutive conception might also play a role in understanding equivalence. The substitutive conception involves understanding that the expression on one side of the equals sign can be used to replace the other.

Children's conceptions of the equals sign

We investigated how children think about the equals sign in Britain and China. We were interested in making a comparison - because in China a different approach is taken to introducing this important mathematical symbol. Previous research has shown that, by the end of primary school, Chinese children are more fluent at solving number sentences than Western children. This had been attributed to Chinese children having a more relational view, and Western children having a more operational view of the equals sign. We discovered that in fact children from both countries view the equals sign relationalry to the same extent, and while children in Britain have a strongly operational view, Chinese children have a strongly substitutive view.

Teaching the substitutive conception

If the substitutive conception is important how might it be taught? We explored this using specially designed arithmetic software called Sum Puzzles. The software lets children use number sentences to make substitutions of arithmetic notation in order to solve puzzles.

An example puzzle is shown in Figure 1. The challenge is to change the boxed expression at the top, 40 + 34, into its answer, 74, using the provided number sentences. To start we can select 34 = 30 + 4 and use it to substitute 34 for 30+4 in the boxed expression, thereby changing the expression to 40 + 30 + 4. We can next select the expression 40 + 30 = 30 + 40 and make another substitution, this time changing the boxed expression to 30 + 40 + 4.

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