Date of Award

Summer 8-2019

Level of Access Assigned by Author

Open-Access Thesis

Degree Name

Master's of Science in Teaching (MST)

Department

Science and Mathematics Education

Advisor

Natasha Speer

Second Committee Member

John Thompson

Third Committee Member

Robert Franzosa

Abstract

Challenges that students face in first semester calculus have been found to be a factor in high attrition rates of students from science, technology, engineering, and mathematics (STEM) majors. With an increase in the demand for STEM graduates, an attempt must be made to remedy this issue. Research has shown that students have difficulties with many topics in the realm of calculus. Of these, students have been found to struggle with the concept of derivative and ideas related to it. However, some derivative topics have not been examined as thoroughly as others. Implicit differentiation, a technique that allows us to differentiate equations that are not explicit functions, is one such topic. The goal of the study was to examine student understanding of and ability to carry out implicit differentiation and to identify whether student work on such problems is influenced by the same factors as other derivative topics or if there are additional challenges that arise for students in the context of implicit differentiation.

Written surveys were collected from 136 first semester calculus students. Clinical interviews were then conducted with five calculus students. For the surveys, students were tasked with explaining what they believed implicit differentiation to be and asked to solve problems using this technique. The interviews were similar in nature, but students were additionally asked to explain their reasoning and thought processes.

Findings suggest that implicit differentiation is challenging for students. Approximately 50% of the survey responses to implicit differentiation problems were correct. The interviews suggested that some students had a strong understanding of implicit differentiation and others did not. Students who have a strong understanding of the idea of implicit differentiation appear to be more successful in carrying out the procedures. For other calculus topics, researchers have found that students can show skill with procedures without an understanding of the ideas. This appears to be less so with implicit differentiation. The interviews also suggested that students do not have a strong understanding of what the symbols of differentiation represent.

Student difficulties with implicit differentiation appeared to be influenced by what they knew about function and derivative. Prior research has identified that understanding of function influences students’ abilities to recognize when and how to use derivative rules. In the case of the chain rule, students need a solid understanding of composition of functions. For the product rule, students need to be fluent in function notation and identifying the product of two functions. Difficulties identified in student work on implicit differentiation problems are similar to those found in other areas of calculus. Though, in addition to these areas of difficulty, there appear to be other ideas about function that influence performance. In particular, function equality and the idea of applying operators to both sides of an equation may be sources of difficulty. These ideas have been examined in algebra, but not in the context of calculus. Implications for educators, as well as opportunities for future research, are proposed to address these points of discussion.

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