How to influence and assess whether students engage in conceptual thinking are longstanding methodological problems in mathematics education. Recently, eye-tracking technology has fueled a discussion on whether eye movement analysis can support valid inferences about mathematical thinking. This study investigates whether eye movement analysis can distinguish between conceptual and non-conceptual thinking in a geometric classification task where both modes of thinking lead to identical responses. Participants were asked to classify geometric shapes while we tracked their eye movements and to report their thinking verbally. Our findings indicate that self-reported conceptual thinking is characterised by fewer eye movements between task shapes and response shapes, and that self-reported non-conceptual thinking involves comparing the shapes' similarity directly. A logistic regression model correctly classified the self-reported ways of thinking in 80.3% of the cases. We conclude that eye-tracking can contribute to making inferences about mathematical thought processes and facilitate research on how to engage students in conceptual thinking and development.