This paper reports on an exploratory study designed to determine and enhance the conceptual understanding of a group of pre-service mathematics teachers at one Irish university utilizing an established framework for understanding mathematics. 23 students on a one year Professional Diploma in Mathematics Education participated in the study, which involved the distribution of a pre- and post-test and engagement in a ten week intervention designed to enhance their subject matter knowledge
(SMK) and pedagogical content knowledge (PCK). The findings
highlight that although there was an improvement in overall conceptual understanding across the entire cohort at the end of the intervention, within certain mathematical topics there was a
statistically insignificant improvement and many deep-rooted issues remain. In this paper we focus on the pre-service teachers’ understanding of elementary algebra, in particular, how to solve a linear equation.