Teaching the substitutive conception of the equals sign

Abstract

A cumulative body of research has shown that children typically shift from an operational to a relational conception of the equals sign as they move through schooling. Jones (2008) argued that a truly relational conception of the equals sign comprises a substitutive component and a sameness component. Here we present two studies that build on this argument. The first investigated how the equals sign is typically presented to primary children in England, and we report that in the main an operational conception seems to be promoted. The second study measured the impact of a specially designed intervention on early secondary children's conceptions of the equals sign. Pre- and post-test data revealed that the intervention promoted substitutive and sameness components of symbolic equivalence. We consider the theoretical and pedagogical implications of the results.

Keywords:

• substitution
• equivalence relations
• equals sign

Acknowledgements

This project was funded by a grant from the Esmée Fairbairn Foundation. Ian Jones is supported by a Royal Society Shuttleworth Educational Research Fellowship, Matthew Inglis. is supported by a Royal Society Worshipful Company of Actuaries Educational Research Fellowship, and Camilla Gilmore is supported by a British Academy Postdoctoral Fellowship.

Notes

1. Referred to simply as the relational conception in the literature. We use the term sameness-relational to distinguish it from substitutive-relational, described above.

2. The Sum Puzzles software can be accessed online at www.sumpuzzles.org.

3. “Write the correct sign <,>or =” followed by three statements with empty boxes between two expressions, as in 10+5−9 □ 10+9−5. Our analysis was a count of printed equals signs and so this was coded as one occurrence.

4. Unfortunately, the rural school that participated in the study was closed due to adverse weather conditions for the week immediately following the intervention. Thus, the post-test took place 10 days later than planned in this school. Nevertheless, there was not a main effect for school on any of the analyses described in this section, and nor did school interact with any other factor, so we do not believe that this influenced the results of the study in any significant fashion.

5. We also conducted an ANOVA, which included school as a between-subjects factor. But as there was no main effect of school, and neither did it interact with any other factor, we do not discuss it further.