Acceptance criteria for validating mathematical proofs used by school students, university students, and mathematicians in the context of teaching

Abstract

Although there is no generally accepted list of criteria for the acceptance of proofs in mathematical practice, judging the acceptability of purported proofs is an essential aspect of handling proofs in daily mathematical work. For this reason, school students, university students, and mathematicians need to hold certain acceptance criteria for proofs, which are closely tied to their epistemology, notion of proof, and concept of evidence. However, acceptance criteria have so far received little academic attention and results have often been related to mathematicians’ acceptance criteria in a research context. Still, as the role of proof changes during mathematical education and differs from research, it can be assumed that acceptance criteria differ substantially between educational levels and different communities. Thus, we analyzed and compared acceptance criteria by school students, university students, and mathematicians in the context of teaching. The results obtained reveal substantial cross-sectional differences in the frequency and choice of acceptance criteria, as well as major differences from those acceptance criteria reported by prior studies in research contexts. Structure-oriented criteria, regarding the logical structure of a proof or individual inferences, are most often referred to by all groups. In comparison, meaning-oriented criteria, such as understanding, play a minor role. Finally, social criteria were not mentioned in the context of teaching. From an educational perspective, the results obtained underline university students’ need for support in implementing acceptance criteria and suggest that a more explicit discussion of the functions of proof, their evidential value, and the criteria for their acceptance may be beneficial.