Student Conceptions of Functions: How Undergraduate Mathematics Students Understand and Perceive Functions

Functions are an integral element in mathematics. They are essential from secondary mathematics, where students first learn the definition, all the way through graduate school, where students use them in their dissertations, and beyond. Yet, at a time in which more content is being introduced earlier into students’ mathematical experiences, less time is spent with the fundamental concepts of functions. As a result, many students enter undergraduate mathematics with only a procedural understanding despite our expectations for deeper comprehension. This study, therefore, explored the disconnect observed between students’ use of functions procedurally and how they demonstrate their understanding of functions. Using a mixed methods approach of surveys and interviews, students enrolled in a variety of first semester calculus courses (including Business Calc and Bio Calc), Calculus II, and Discrete Math responded to prompts about mathematics and functions and solved problems involving functions and related applications. After examining students’ perceptions, various tasks and interview techniques were used to assess their understanding of functions, including the use of different function representations and more formal, generalizable statements. This study found that students in contextually-driven Calculus I courses tend to focus less on the generalizability of their statements, but they did show evidence of forming connections between various mathematical ideas. Additionally, there is a relationship that students who demonstrated evidence of both generalizability and forming connections also tended to fluently switch between function representations, which often exhibits a higher level of understanding