### Abstract:

Equality is a key concept in the mathematics curriculum at primary school. Conceptions of equality of primary students have been identified by mathematics education researchers. The relationships among students’ conceptions of equality and students’ achievement in mathematics, however, have not received the same degree of attention. These relationships are generally characterised dichotomously as either correct structural conceptions or incorrect misconceptions. To explore these relationships further, national assessment data were examined from a perspective that emphasised the mathematical structure expressed by primary students as they solved three additive arithmetic missing number problems. As expected, a greater proportion of older students expressed appropriate conceptions of equality than younger students and these conceptions were associated with higher levels of achievement. At both year levels, however, students’ conceptions of equality were also found to be contingent on a student’s mathematical achievement and the mathematical structure of problem. Therefore, the relationships among students’ conceptions of equality and student’s mathematical achievement appear to be more diverse and complex than previously documented and theorised. In particular, the framework has been expanded to include procedural, competing, structural-but-tacit, and structural-and-explicit conceptions of equality. The findings of this study can be used by mathematics education researchers to expand the theoretical framework that describes the relationships among students’ conceptions of equality and achievement. Findings will also be of interest to educators because they may wonder why so many students’ appear to struggle with the concept of equality. Although it appears that certain conceptions of equality act as barriers for students to interpret the mathematical structure of problems appropriately, other conceptions appear to act as gateways for students to appreciate advanced relationships that are possible, given the mathematical structure of the problems. Findings suggest that it is not only the quantity of mathematical knowledge a student has, but it is the quality of the connections between procedural and conceptual knowledge that allows them to solve problems with the concept of equality successfully.