ARE STUDENTS BETTER AT VALIDATION AFTER AN INQUIRY-BASED TRANSITION-TO-PROOF COURSE? Annie Selden John Selden New Mexico State University New Mexico State University We present the results of a study of the observed proof validation abilities and behaviors of sixteen undergraduates after taking an inquiry-based transition-to- proof course. Students were interviewed individually towards the end of the course using the same protocol that we had used earlier at the beginning of a similar course (Selden and Selden, 2003). Results include a description of the students’ observed validation behaviors, a description of their proffered evaluative comments, and the, perhaps counterintuitive, suggestion that taking an inquiry- based transition-to-proof course does not seem to enhance validation abilities. We also discuss distinctions between proof validation, proof comprehension, proof construction and proof evaluation and the need for research on their interrelations. Key words: Transition-to-proof, Proof, Validation We present the results of a study of the observed proof validation behaviors of 16 undergraduates after taking an inquiry-based transition-to-proof course emphasizing proof construction. Students were interviewed individually towards the end of the course employing the same protocol used in our earlier study (Selden & Selden, 2003). Here, as in our earlier study, we regard proofs as texts that establish the truth of theorems and use our previous description of proof validation as the reading of, and reflection on, proofs to determine their correctness. Indeed, A validation is often much longer and more complex than the written proof and may be difficult to observe because not all of it is conscious. Moreover, even its conscious part may be conducted silently using inner speech and vision. Validation can include asking and answering questions, assenting to claims, constructing subproofs, remembering or finding and interpreting other theorems and definitions, complying with instructions (e.g., to consider or name something), and conscious (but probably nonverbal) feelings of rightness or wrongness. Proof validation can also include the production of a new text—a validator- constructed modification of the written argument—that might include additional calculations, expansions of definitions, or constructions of subproofs. Towards the end of a validation, in an effort to capture the essence of the argument in a single train-of-thought, contractions of the argument might be undertaken. (p. 5). In this paper, we provide detailed descriptions of the observed validation behaviors that our 16 undergraduates took – something either not done, or only partially done, in prior validations studies and perhaps not at all for this level of student. Past validation studies include: first-year Irish undergraduates’ validations and evaluations (Pfeiffer, 2011); U.S. undergraduates’ validations at the beginning of a transition-to-proof course (Selden & Selden, 2003); U.S. mathematics majors’ validation practices across several content domains (Ko & Knuth, 2013); U.S. mathematicians’ validations (Weber, 2008); and U.K. novices’ and experts’ reading of proofs, using eye-tracking, to compare their validation behaviors (Inglis & Alcock, 2012). Our ultimate goal is to understand both the process of proof construction and the process