Engineering-Based Problem Solving Strategies In AP Calculus: An Investigation Into High School Student Performance On Related Rate Free-Response Problems by John Thieken A Dissertation Presented in Partial Fulfillment Of the Requirements for the Degree Doctor of Philosophy Approved June 2012 by the Graduate Supervisory Committee: Tirupalavanam Ganesh, Chair Finbarr Sloane James Middleton ARIZONA STATE UNIVERSITY August 2012 ABSTRACT A sample of 127 high school Advanced Placement (AP) Calculus students from two schools was utilized to study the effects of an engineering design-based problem solving strategy on student performance with AP style Related Rate questions and changes in conceptions, beliefs, and influences. The research design followed a treatment-control multiple post-assessment model with three periods of data collection. Four high school calculus classes were selected for the study, with one class designated as the treatment and three as the controls. Measures for this study include a skills assessment, Related Rate word problem assessments, and a motivation problem solving survey. Data analysis utilized a mixed methods approach. Quantitative analysis consisted of descriptive and inferential methods utilizing nonparametric statistics for performance comparisons and structural equation modeling to determine the underlying structure of the problem solving motivation survey. Statistical results indicate that time on task was a major factor in enhanced performance between measurement time points 1 and 2. In the experimental classroom, the engineering design process as a problem solving strategy emerged as an important factor in demonstrating sustained achievement across the measurement time series when solving volumetric rates of change as compared to traditional problem solving strategies. In the control classrooms, where traditional problem solving strategies were emphasized, a greater percentage of students than in the experimental classroom demonstrated enhanced achievement from point 1 to 2, but showed decrease in achievement from point 2 to 3 in the measurement time series. Results from the problem solving motivation i survey demonstrated that neither time on task nor instruction strategy produced any effect on student beliefs about and perceptions of problem solving. Qualitative error analysis showed that type of instruction had little effect on the type and number of errors committed, with the exception of procedural errors from performing a derivative and errors decoding the problem statement. Results demonstrated that students who engaged in the engineering design-based committed a larger number of decoding errors specific to Pythagorean type Related Rate problems; while students who engaged in routine problem solving did not sustain their ability to correctly differentiate a volume equation over time. As a whole, students committed a larger number of misused data errors than other types of errors. Where, misused data errors are the discrepancy between the data as given in a problem and how the student used the data in problem solving. ii ACKNOWLEDGEMENTS Thank you to my committee for your continued support and patience throughout my studies. Dr. Ganesh, I cannot express how much I appreciate all your support over my graduate career. You have given me so much by including me in your research and your life. I will never forget the lessons and opportunities you have shared with me. Dr. Sloane, without you I would never have made it this far. I am very fortunate that you were in your office that summer day. You have taught me the subtleties and integrity of good research, while grounding me in the reality of the process. I am forever in your debt for always being there for me. Dr. Middleton, your passion and enthusiasm for teaching are addictive. Thanks for all your help and direction on both my comprehensive exams and my dissertation. Your ability to conceptualize the larger picture has shown me that the “big” idea is always within reach. iii TABLE OF CONTENTS Page LIST OF TABLES................................................................................................ vii LIST OF FIGURES ............................................................................................... ix CHAPTER 1 INTRODUCTION Overview................................................................................................1 Problem Statement.................................................................................2 Research Questions................................................................................3 Broader Impact.......................................................................................4 2 LITERATURE REVIEW Problem Solving: A Brief Overview......................................................6 The Problem Solving Process ................................................................8 Problem Solving Competencies.............................................................9 Problem Solving Influences.................................................................13 Conceptual Understanding and Inquiry ...............................................17 Assessing Problem Solving .................................................................19 Bridging the Gap..................................................................................20 3 METHODS Participants...........................................................................................23 Measures ..............................................................................................24 Procedures............................................................................................30 Differences and Similarities Among the Three Instructors .................40 iv CHAPTER Page 4 DATA ANALYSIS AND RESULTS Assumptions and Correction Techniques ............................................42 Skills assessment..................................................................................44 Related Rate.........................................................................................50 Pre-Assessment Descriptive Statistics ...........................................52 Post1-Assessment Descriptive Statistics........................................53 Post2-Assessment Descriptive Statistics........................................54 Within-Class Differences...............................................................55 Between-Class Differences............................................................56 Longitudinal Analysis..........................................................................58 Relating Two Variables .................................................................60 Differentiating an Equation............................................................61 Indentifying Key Information........................................................61 Error Analysis ......................................................................................62 Misused Data .................................................................................65 Misinterpreted Language ...............................................................68 Logically Invalid Inference............................................................71 Distorted Theorem Or Definition...................................................73 Problem Solving Motivation Survey ...................................................75 Exploratory Factor Analysis ..........................................................76 Latent Mean Differences................................................................82 5 CONCLUSIONS AND RECCOMENDATIONS ....................................85 v Page REFERENCES ......................................................................................................95 APPENDIX A IRB Letters of Approval ..........................................................................101 B Consent and Assent Letters......................................................................103 Student Assent ...................................................................................104 Parental Consent ................................................................................105 C MEASURES ............................................................................................106 Skills assessment................................................................................107 Related Rate Pre-Assessment ............................................................108 Related Rate Post1-Assessment.........................................................109 Related Rate Post2-Assessment.........................................................110 Problem Solving Motivation Survey .................................................111 D CURRICULAR MATERIALS...............................................................113 Instructor 1 Materials.........................................................................114 Instructor 3 Materials.........................................................................117 E Assessment Rubrics.................................................................................120 Skills assessment................................................................................121 Related Rate Pre-Assessment ............................................................123 Related Rate Post1-Assessment.........................................................125 Related Rate Post2-Assessment.........................................................128 F