The purpose of this study was to gain insight into students' understanding of the vertex of the quadratic function in connection to the concept of derivative by use of the think-aloud method as a means of data collection. Thirty students enrolled in a Calculus of One Variable I course participated. By analyzing students' comprehension of the vertex of a quadratic function and the derivative not only during think-aloud sessions, but also during follow up interviews and on written work, this

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... ibuted to a better understanding of how students relate the concepts. Several different theoretical frameworks were used to analyze student comprehension. This gave multiple viewpoints to further explore students' thoughts as they worked either aloud individually or in a group setting. First, APOS theory (Asiala et al., 1996) was used to analyze students' understanding of the concept of vertex of the quadratic function in relation to the derivative on certain tasks. Students' personal meaning of the vertex and its impact on the understanding of the derivative was noted as well as students' lack of connection between explicit and real world problems. Obstacles of misconception of the vertex, trouble with the free fall formula, and problems with graphing due to a weak schema of quadratic functions were all identified as barriers to student understanding of real world problems. Next, Skemp's (1976) relational and instrumental understanding framework was used to explain how students think-aloud individually. Trends in the thought process while working alone as well as students' ability to identify and correct mistakes were analyzed. Lastly, Vygotsky's (1978) concept of zone of proximal development was used to describe the difference in ability of students working by themselves versus in a group setting. In a group setting, some students worked within their zone of proximal development as they were influenced by peers to fix incorrect solutions. Based on APOS, several suggested activities pertaining to the quadratic function [...]