What does ‘y is defined as an implicit function of x’ mean?: An application of APOS-ACE

Abstract Implicit functions is a basic and important topic in calculus and yet very little research has been done on students’ understanding of this notion. In a previous study we had documented students’ misconceptions and lack of understanding, and proposed a conjecture of mental constructions (called a genetic decomposition; GD) students may do in order to understand this topic. In the present study the proposed GD is tested with student interviews and refined. The study starts by using semi-structured interviews of fourteen students who had just finished a traditional one-variable calculus course, further exploring students’ understanding of implicit functions. Then using the GD, a cycle of Activities on computer, Classroom discussion, and Exercises (ACE cycle) was designed and implemented to help students do mental constructions found to be lacking. Finally, the fourteen students were again interviewed after they participated in the activities of the ACE cycle. The students’ answers in the second interview revealed that their understanding of implicit functions had improved. The results of this article are of practical importance as they underscore that activities based on a GD have the potential of improving student understanding of implicit functions. Further, the study also suggested refinements to the genetic decomposition for implicit functions.