How intuition and language use relate to students' understanding of span and linear independence in an elementary linear algebra class

A possible contributing factor to students' difficulty in learning advanced mathematics is the conflict between students' "natural" learning styles and the formal structure of mathematics, which is based on definitions, theorems, and proofs. Students' natural learning styles may be a function of their intuition and language skills. The purpose of this study was to explore the possible relationships between students' intuition, language use, and quality of learning. The participants in the study were seven students from one elementary linear algebra class. Quality of learning was assessed by interpreting the nature of students' understanding of span and linear independence. The findings indicated there was an association between students' intuition, language use, and understanding. Students with stronger understanding had lower levels of interfering intuitions and stronger writing skills, while students with weaker understanding had higher levels of interfering intuitions and weaker writing skills. Students' conceptual understanding of definitions varied more among the students than did their procedural understanding. Students whose interfering intuitions were more grounded in beliefs than perceptions had weaker understanding. Vocabulary use and completeness in writing were distinguishing factors between stronger and weaker writers. These findings suggest that one approach to improving student learning in advanced mathematics may be to use instructional practices that remediate students' interfering intuitions and encourage the development of stronger language skills.