Learners’ beliefs about the functions of proof: building an argument for validity

Abstract

Learners’ difficulties with proof have been ascribed to their lack of understanding of functions that proof performs in mathematics, namely, verification, explanation, communication, discovery, and systematization. However, the extant mathematics education literature on validation of instruments designed to measure learners’ beliefs about the functions of proof is scant. The purpose of this mixed methods study was to use the Standards for Educational and Psychological Testing (as discussed by the American Educational Research Association/American Psychological Association/National Council on Measurement in Education (AERA/APA/NCME), 2014) as a theoretical and methodological platform to support arguments for the validity of the learners’ beliefs about the functions of proof (BAFP) instrument. Scale items were generated from de Villiers’ model and a review of the literature, panel of experts, and key informants. The resulting instrument, comprising 28 Likert-scale items and 5 open questions that assessed the five functions of proof, was administered to 87 grade 11 learners in one high school in the Pinetown Education District, KwaZulu-Natal, South Africa. The results put the validity of the scores on the BAFP instrument into question. The study was an exploration of validity arguments based on the evidence from various data sources. Confirmatory factor analysis with a larger sample in a different context needs to be conducted to warrant the use of the BAFP instrument. The key contribution of this study to the field is that it sheds light on the complexities of instrument validation; the scope of the effort may explain why comprehensive validation efforts have not been documented extensively in literature.