Engineering-Based Problem Solving Strategies In AP Calculus

156 Pages·2012·2.71 MB·English

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
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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.
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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.

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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
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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
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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 STATISTICAL TABLES........................................................................130
G Graphs.....................................................................................................140
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LIST OF TABLES
Table Page
1. Sample Demographics.....................................................................................24
2. Skills assessment, Question 4..........................................................................45
3. Skills assessment, Question 5..........................................................................46
4. Skills assessment:
Number Correct by Question Total............................................................46
5. Related Rate Pre-Assessment: Percent Correct by Total.................................48
6. Related Rate Post1-Assessment: Percent Correct by Total.............................48
7. Related Rate Post2-Assessment: Percent Correct by Total.............................49
8. Follow-Up Comparisons: Related Rate Assessment.......................................57
9. Longitudinal Achievement: Determining Relationships
Over Time..................................................................................................60
10. Longitudinal Achievement: Ability to Perform Derivatives.........................61
11. Longitudinal Achievement: Identifying Key Information.............................61
12. Error Analysis: Total Number of Misused Data Errors.................................66
13. Error Analysis: Student Examples of Misused Data Errors...........................67
14. Error Analysis: Total Number of Misunderstood Language Errors..............69
15. Error Analysis: Student Examples of Misunderstood Language Errors........70
16. Error Analysis: Total Number of Logically Invalid Inference.......................72
17. Error Analysis: Student Examples of Logically Invalid Inference.................73
18. Error Analysis: Total Number of Distorted Theorem or Definition...............74
19. Error Analysis: Student Examples of Distorted Theorem or Definition.........75
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Table Page
20. Standardized Factor Loadings, Initial Exploratory Factor Analysis...............77
21. Standardized Factor Loadings, Follow-Up Exploratory Factor Analysis.......79
22. Problem Solving Motivation Survey:
Significant Within-Class Results...............................................................82
23. Problem Solving Motivation Survey:
Significant Between-Class Results............................................................84
24. Skills assessment:
Number Correct by Question Part............................................................131
25. Related Rate Assessment: Within-class Wilcoxon Results.........................131
26. Related Rate Assessment: Adjusted Significant Between-class
Kruskal-Wallis Results............................................................................133
27. Related Rate Assessment: Non-adjusted Significant Between-class
Kruskal-Wallis Results............................................................................134
28. Problem Solving Motivation Survey: Within-class
Wilcoxon Results.....................................................................................135
29. Problem Solving Motivation Survey: Between-class
Kruskal-Wallis Results............................................................................135
30. Problem Solving Motivation Pre-Survey Correlations
With Means and Standard Deviations......................................................136
31. Problem Solving Motivation Post-Survey Correlations
With Means and Standard Deviations......................................................138
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LIST OF FIGURES
Figure Page
1. Standardized Parameter Estimates..................................................................80
2. Skills Assessment: Average Class Score.......................................................141
3. Related Rate Assessment: Average Class Score............................................142
4. Longitudinal Analysis Results: Relating Two Variables...............................143
5. Longitudinal Analysis Results: Correctly Taking the Derivative..................144
6. Longitudinal Analysis Results: Identifying Key Information.......................145
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