Misconceptions of the order of operations and associativity use

Background: Conceptual knowledge of arithmetic is essential for progressing in mathematics. Associativity is one concept, which permits some problems to be solved in different ways; for example, solving ‘a + b – c’ by ‘b – c’ then ‘+a’ is a ‘shortcut’ strategy derived from associativity. However, individuals struggle to apply associativity and misconceptions of the order of operations may be one factor that is responsible.

Aims: To investigate whether misconceptions of the order of operations hinder associativity shortcut use.

Samples: 76 (Study 1) and 130 (Study 2) adults aged 18–60 years participated.

Method: In Study 1, we developed a novel instrument that quantitatively measures how people interpret the order of operations. In Study 2, we conducted a well-powered, pre-registered experiment to investigate whether reminding individuals of the correct order of operations improved a) knowledge of the order of operations and b) associativity shortcut use.

Results: We found that only 16% of adults fully understood the order of operations and almost 50% had specific ‘literal’ and ‘left-to-right’ misconceptions of acronyms used to teach it. Those with misconceptions were less likely to use associativity shortcut. Reminding individuals of the order of operations reduced misconceptions of the order of operations but did not improve associativity shortcut use.

Conclusions: Misconceptions of the order of operations hinder the application of associativity shortcut strategies. Our findings have theoretical impact on the relationship between procedural and conceptual knowledge of arithmetic and have practical benefits for teachers who could use our instrument to identify misconceptions.