Research
My current research interest include the fields of complex analysis, specifically iteration theory, and mathematics education, specifically undergraduate mathematics education.
For my doctoral disseration I investigated the properties of the iterated Newton map of complex rational functions. In addition, I classified all degree two Netwon maps through conjugation and am currently trying to generalize these ideas to the degree three (and higher) cases.
For my master's thesis I conducted a quantitative study investigating the correlations between student motivation and instructor immediacy behavoirs and teaching styles, as well as student effort attributions and instructor immediacy behaviors and teaching styles.
Papers
Categorization of all Newton maps of rational functions conjugate to quadratic polynomialsAbstract: Newton's method is a well-known iterative method to find roots of a function. It has primarily been studied for polynomials, but more recently transcendental functions have also been studied. However, according to our literature search, nothing has been studied for Newton's method for rational functions. Here we extend the known properties of the degree and fixed points of Newton maps of polynomials to Newton maps of rational functions. We also categorize all Newton maps conjugate to quadratic polynomials and show the complications in extending this to degree three Newton maps.
The Influence of Teacher Practice on Calculus Students MotivationAbstract: This quantitative study investigated two correlations among Calculus II students' perceptions of their instructors' teaching styles. The first was between student motivation and instructor immediacy behaviors and teaching styles. The second correlation was between student effort attributions and instructor immediacy behaviors and teaching styles. Instructor immediacy behaviors include nonverbal and verbal; teaching styles include instructor affect -- how positive the teacher appears to be with the class and mathematics -- assigned group work and application problems. Students in a large Southwestern research university were surveyed at the start and end of a semester. A significant positive correlation was found between student motivation and instructor behaviors and teaching styles. The three strongest correlations were between student motivation and teacher affect as well as verbal and nonverbal immediacy behaviors. In addition, there was not a significant correlation between student effort attributions and instructor behaviors and teaching styles. These results have implications for instructors teaching a Calculus II type class. Instructors may be advised to exhibit a more positive immediacy behavior as well as have a positive attitude towards the class and mathematics. Further studies may involve other subject levels of mathematics and within different types of institutions.