The omnipresence of the Letter-as-Object misconception: a multilevel Latent Transition Analysis on understanding variables

Abstract When students first encounter variables in algebra, they often develop the Letter-as-Object misconception, interpreting letters as labels for objects rather than references to numerical values. This study analysed the response patterns of 2220 German Year 7 and 8 students (aged 12–14) using the SMART test ‘Meaning of Letters’ , designed to assess various subtypes of this misconception. Employing multilevel Latent Transition Analysis, we identified five latent classes and tracked shifts in students’ classifications over 4 weeks of algebra instruction. These classes revealed persistent patterns in how the Letter-as-Object misconception and its subtypes manifest. While the majority of students exhibited this misconception, their responses varied depending on the equation type in the test items. Encouragingly, the class of students with mostly correct responses tripled in size after instruction, though it remained small, with a majority of students staying in their original classes. This study highlights the potential of multilevel Latent Transition Analysis to track misconceptions and conceptual changes over time and is among the first to account for the multilevel data structure of students nested within teachers in a Latent Transition Analysis.