#### Date of Award

5-2011

#### Document Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Legacy Department

Curriculum and Instruction

#### Committee Chair/Advisor

Switzer, Debi

#### Committee Member

Switzer , Debi

#### Committee Member

Hazari , Zahra

#### Committee Member

Benson , Lisa

#### Committee Member

Cook , Michelle

#### Committee Member

Che , Megan

#### Abstract

There is abundant evidence that many students in the United States are not adequately prepared for college calculus. How to design and implement instruction to adequately prepare secondary students for college calculus is a concern to both college mathematics professors and secondary mathematics teachers. While both groups agree that rigorous instruction promotes mathematical understanding, they hold different opinions about how to optimally prepare high school students for single variable college calculus. This is important because readiness for success in college calculus is a known gatekeeper for success in STEM majors. The data used in this study was drawn from the Factors Influencing College Success in Mathematics (FICSMath) project, which focuses on finding evidence for effective strategies that prepare students for college calculus success. Funded by the National Science Foundation (NSF award #0813702), FICSMath is a large-scale study from the Science Education Department at the Harvard-Smithsonian Center for Astrophysics, which surveyed a nationally representative sample of college students who were enrolled in single variable college calculus courses in the fall semester of 2009. The purpose of the FICSMath study was to gain insight into what high school

teachers did that had a significant effect on single variable college calculus performance. The development of the FICSMath survey was informed by several components in order to establish content validity. One particularly informative source was the open-ended responses gathered from mathematics professors and secondary mathematics teachers across the nation, via an online survey. The mathematics professors were asked, 'What do high school teachers need to be doing to prepare their students for college calculus success?' and the mathematics teachers were asked, 'What are you doing that you think prepares students for college calculus success?' An unequal status concurrent mixed method design was used to analyze the data. Phenomenographical analysis compared the variation between the mathematics professors' and secondary mathematics teachers'

responses. The quantitative data came from students who were in two and four-year large, medium, and small colleges and universities across the nation who completed the FICSMath survey. Participating schools of higher-ed administered the 61-item FICSMath survey in the beginning of the fall semester of 2009. The professors held the surveys until the end of the semester, at which time they recorded the grades earned, and then returned the surveys to Harvard University. The surveys included questions on students' demographics, academics, and pedagogical practices from their last high school

mathematics course. The sample included in the analysis were pre-calculus, non-AP Calculus, AP Calculus AB, and AP Calculus BC students who moved directly from secondary mathematics to single variable college calculus where the FICSMath survey was administered. The dependent variable was performance in college calculus and the independent variables were pedagogical variables that aligned with components of the Four Component Instructional Design (4C/ID) model. This model was designed from cognitive load theory and has four distinct components. The support, procedure, learning task, and part-task components were placed together by van Merriënboer and other cognitive load researchers in order to assist with the instruction of complex tasks and to enhance transfer of learning. Using multiple-regression analysis, two models were

created that are predictive of college calculus performance, one for pre-calculus students, and one by combining all three levels of secondary calculus students together. The precalculus model (n=964) had four significant pedagogical variables, one with positive

effect on performance, and three with negative effects. The predicted difference for those experiencing positive verses negative predictors was a total predicted difference of 19.9 points earned in students' final college calculus grade. Because there was no significant

difference between the mean high school performance across the three levels of secondary calculus, or in the mean performance in college calculus, all three levels of secondary calculus were grouped together to create the calculus model. The calculus model (n=1,999) revealed 11 significant pedagogical variables. Four variables had a

positive effect on performance and seven had negative effects. The predicted difference for those experiencing positive verses negative predictors was 25.29 points in college calculus performance. Seven of the eleven categories from the phenomenography aligned with significant variables from the two multiple regression models. The

triangulation of findings led to functions being classified as a learning task variable, meaning working with functions is a specific complex task for students. Triangulation also revealed that even though professors and teachers believed that connecting mathematics to the real world is important, such pedagogy was not predictive of future

performance in college calculus. Chapter 6 addresses the change in variability in the data that is explained when student affect variables are added to the models, and also provides an implications section for teachers. The 4C/ID model was modified to the 3C/ID Pre-

Calculus Model for College Calculus Performance and the 3C/ID Secondary Calculus Model for College Calculus Performance

#### Recommended Citation

Wade, Carol, "SECONDARY PREPARATION FOR SINGLE VARIABLE COLLEGE CALCULUS: SIGNIFICANT PEDAGOGIES USED TO REVISE THE FOUR COMPONENT INSTRUCTIONAL DESIGN MODEL" (2011). *All Dissertations*. 745.

https://tigerprints.clemson.edu/all_dissertations/745