The use of cross multiplication and other mal–rules in fraction operations by pre-service teachers

•This study reports on mal–rules (mainly related to cross-multiplication) used by PST’s in fraction operations..•The interviews revealed the PST’s strong conviction of whole number operations applied separately to parts of the fraction.•The use of theorems-in-action provides useful insights into the rules as well as properties about some of the rules. Many studies show that prospective teachers often have misconceptions about fractions. In this case study, we report on some of the mal–rules used by a group of 60 prospective South African primary school teachers. The students’ written responses to two items focusing on addition and multiplication of fractions which formed part of an assessment, were analyzed. Semi-structured interviews were also used to elicit the reasoning used in the students’ calculations. Less than half of the participants completed both items correctly, and many of the other students displayed various mal–rules. To interpret the pre–service teachers’ misconceptions, we studied the rules used by the participants, and expressed them as theorems–in–action. An interesting mal–rule governing the multiplication of fractions was the widespread ‘cross multiplication’ rule which after some mutations led to other mal–rules, illustrating how students’ misconceptions can persist many years after their initial learning.