Calculus students\u27 understandings of the concepts of function transformation, function composition, function inverse and the relationships among the three concepts

Researchers investigating students\u27 understanding of functions have found that many students, including pre-service teachers, have a limited understanding of functions. While much research has been conducted on students\u27 understanding of functions, little attention has been paid to students\u27 understanding of function transformation, function composition, function inverse, and the students\u27 perceptions of how these three concepts are related. This study sought to extend prior research on students\u27 understanding of function ideas by investigating calculus students\u27 understandings of the related concepts of function transformation, function inverse, and function composition, and the students\u27 perceptions of the relationships among the three concepts. This study employed both qualitative and quantitative methods, and was influenced by a constructivist theory of learning and a framework of flexibility. In the first phase of the study, I administered a 20-question questionnaire to 70 students in the first course of the calculus sequence --40 AP Calculus students in high school and 30 college students taking Calculus I. The questionnaire gathered data about the students\u27 mathematical background, demographic information, as well as their understanding of the concept of function. Summary statistics of the students\u27 scores in the questionnaire were used to select 17 of the 70 respondents to participate in individual task-based interviews. The participants\u27 work, including transcripts of the interview, was collected and analyzed inductively using a combination of grounded theory and a framework of flexibility. The results of this study confirmed findings of other studies before it that showed students\u27 limited understandings about the three concepts. Most of the students exhibited an understanding of the inverse of a function, as a reversal of the correspondence that defines the original function. On the other hand, the participants had a limited understanding of function composition and function transformation with many treating compositions or transformations as processes that produced new functions with no ties to the original function(s). These students\u27 understandings of the three concepts seemed to be affected by factors such as function representation, wording used in the item/task presentation, and the students\u27 function schema. In addition, a major contribution of this study was in the students\u27 perception of the relationships among the three concepts. I found that there were differences in the abilities of the participants to see relationships among the three concepts. As expected, the students\u27 abilities to perceive the relationships among the three concepts seemed to depend on a good understanding of each of the three concepts. Most of the participants responded to the items/tasks using memorized formulas or strategies that were called upon whenever the participants encountered the concept words (or notation for) transformation, composition, or inverse. This seemed to inhibit the students from seeing relationships among the three concepts. There were some exceptions to this pattern with a few students showing great insight about their understanding of the three concepts in any of the three representations used, and exhibiting an understanding of relationships among the three concepts through their responses