Students' interpretations of mathematical statements involving quantification

Mathematical statements involving both universal and existential quantifiers occur frequently in advanced mathematics. Despite their prevalence, mathematics students often have difficulties interpreting and proving quantified statements. Through task-based interviews, this study took a qualitative look at undergraduate mathematics students' interpretations and proof-attempts for mathematical statements involving multiple quantifiers. The findings of this study suggest that statements of the form "There exists ... for all ..." (which can be referred to as EA statements) evoked a larger variety of interpretations than statements of the form "For all ... there exists ..." (AE statements). Furthermore, students' proof techniques for such statements, at times, unintentionally altered the students' interpretations of these statements. The results of this study suggest that being confronted with both the EA and AE versions of a statement may help some students determine the correct mathematical meanings of such statements. Moreover, knowledge of the structure of the mathematical language and the use of formal logic may be useful tools for students in proving such mathematical statements.