Same variable, different values? A multilevel Latent Transition Analysis of students’ thinking about values of variables

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Abstract
Understanding the values that variables can take is a central challenge in students’ early encounters with algebra. While previous research has documented a range of numerical preconceptions related to variable values, much less is known about how they co-occur and develop. This study examines German Year 7/8 students’ (mis)understandings of variable values before and after four weeks of regular algebra teaching. Data were collected from 2205 students who completed the SMART ‘Values for Letters’ test. The test targets well-documented numerical preconceptions, including difficulties with substituting given values, alphabet-based value assumptions, treating multiple occurrences of the same letter as referring to independent values, and assuming that different letters must represent different numbers. To capture qualitative differences in students’ response patterns and their development over time, we applied multilevel Latent Transition Analysis (LTA), accounting for students nested within teachers. The analysis identified four latent classes representing distinct ways of reasoning about variable values. Although overall response rates changed only modestly, the LTA revealed substantial movement between latent classes. Alphabet-based value assumptions were rare, whereas treating multiple occurrences of the same letter as referring to independent values emerged as a persistent, intermediate bottleneck in students’ learning. Moreover, the assumption that different letters must refer to different values appeared as an overgeneralization rather than a foundational difficulty, occurring primarily among students in the highest-performing latent class. These findings show how LTA reveals learning trajectories invisible in aggregate analyses and supports diagnostic work in algebra instruction. Limitations include attrition and the short period between timepoints.

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