Proof comprehension, proof validation, & proof construction: All the same or different skills? An answer based on an empirical analysis

Mathematical proof is essential for mathematics as a discipline. Thus, mathematics students are expected to learn to handle proofs proficiently in multiple situations, for example, when constructing, comprehending, or validating proofs. However, there is only limited evidence on whether performances in proof construction, comprehension, and validation can better be described by one overall mathematical proof skill or multiple skills that may be highly related but are still separable. In particular, prior theoretical conceptions and empirical research provide no conclusive evidence in either direction. The present study empirically addresses this issue by comparing five different structural equation models (SEM), ranging from a 1-dimensional overall skill to 2-dimensional models to a 3-dimensional model with individual skills for each activity. Evidence from 153 undergraduate mathematics students showed that a 2-dimensional model corresponding to a skill for proof construction and a combined second skill for proof comprehension and validation matches the data best and is superior to the other models based on multiple criteria for SEM model fit and comparison. While both skills are highly related, data still suggests that they should be conceived as separate skills. The finding has implications for ways how validating and comprehending proofs could be incorporated into learning how to handle mathematical proofs.