FUNCTION CONCEPTIONS OF AP CALCULUS STUDENTS by Adam R. Vrabel BA, Mathematics ...

186 Pages·2014·2.58 MB·English

by  Vrabel, Adam Robert

FUNCTION CONCEPTIONS OF AP CALCULUS STUDENTS
by
Adam R. Vrabel
BA, Mathematics and Statistics, Miami University, 2002
MS, Mathematics, Miami University, 2004
MEd, Mathematics Education, University of Pittsburgh, 2009
Submitted to the Graduate Faculty of
the School of Education in partial fulfillment
of the requirements for the degree of
Doctor of Education in Mathematics Education
University of Pittsburgh
2014
UNIVERSITY OF PITTSBURGH
SCHOOL OF EDUCATION
This dissertation was presented
by
Adam R. Vrabel
It was defended on
February 10, 2014
and approved by
Dr. Ellen Ansell, Associate Professor, Instruction and Learning
Dr. James Greeno, Emeritus Professor, Stanford University
Dr. Douglas Ward, Professor, Miami University
Dissertation Advisor: Dr. Margaret S. Smith, Professor, Instruction and Learning
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Copyright © by Adam R. Vrabel
2014
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Function Conceptions of AP Calculus Students
Adam R. Vrabel, EdD
University of Pittsburgh, 2014
Functions are one of the most important topics in secondary school mathematics,
especially for students who wish to take higher-level mathematics courses beginning with
calculus. The prerequisites for Advanced Placement Calculus state that a thorough
understanding of functions is needed for those who wish to succeed in the course and pass the
AP Calculus Exam. However, research has shown that students’ struggles with calculus
concepts could be traced to inadequate prerequisite knowledge of functions. Since thousands of
students take and pass the AP Calculus Exam every year, it is important to investigate which
aspects of functions AP Calculus students generally do and do not understand, and to determine
how well their understandings of functions relate to their performance on the exam.
In order to explore this, students from AP Calculus classes in three different schools were
tested on their understandings of functions at the end of the course, after they had already taken
the AP Calculus Exam. Additionally, some of the participants were also interviewed to further
ascertain their understandings of functions. Finally, the AP Calculus Exam scores for all
participants were collected and compared to their function understandings. It was found that 1)
most participants’ understandings of functions were less than sufficient for an AP Calculus
course, and 2) there was a high positive correlation between function understandings and scores
on the AP Calculus Exam. These results suggest that more work must be done in developing
students’ understandings of functions in the secondary mathematics curriculum, and greater
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measures should be taken to ensure that students entering AP Calculus have a sufficient
understanding of the prerequisite knowledge.
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TABLE OF CONTENTS
ACKNOWLEDGEMENTS .................................................................................................... XIV
1.0 STATEMENT OF THE PROBLEM ......................................................................... 1
1.1 INTRODUCTION ............................................................................................... 1
1.2 BACKGROUND .................................................................................................. 3
1.2.1 The Big Ideas of Functions ............................................................................. 3
1.2.2 APOS Theory ................................................................................................... 5
1.2.3 Functions, Calculus, and AP Calculus Students ........................................... 6
1.3 PURPOSE AND RESEARCH QUESTIONS ................................................... 8
1.4 WHY CALCULUS? ............................................................................................ 8
1.5 SIGNIFICANCE OF THE STUDY ................................................................... 9
1.6 LIMITATIONS OF THE STUDY ................................................................... 10
1.7 OVERVIEW ....................................................................................................... 11
2.0 REVIEW OF THE LITERATURE .......................................................................... 12
2.1 THE FUNCTION CONCEPT .......................................................................... 12
2.1.1 Development of the Function Concept ......................................................... 13
2.1.2 Views of Functions ......................................................................................... 14
2.1.3 Definitions of Functions ................................................................................ 20
2.1.4 Difficulties and Misconceptions .................................................................... 22
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2.2 THE OTHER BIG IDEAS ................................................................................ 25
2.2.1 Families of Functions..................................................................................... 26
2.2.2 Covariation and Rate of Change .................................................................. 27
2.2.3 Multiple Representations .............................................................................. 31
2.2.4 Combining and Transforming Functions .................................................... 37
2.3 FUNCTIONS AND CALCULUS STUDENTS ............................................... 41
2.4 SUMMARY ........................................................................................................ 45
3.0 METHODOLOGY ..................................................................................................... 48
3.1 PARTICIPANTS ............................................................................................... 49
3.1.1 The Schools and Teachers ............................................................................. 50
3.1.2 Anonymity and Confidentiality of the Participants ................................... 55
3.2 DATA COLLECTION ...................................................................................... 56
3.2.1 The Precalculus Concept Assessment .......................................................... 56
3.2.2 Interviews ....................................................................................................... 60
3.2.3 The AP Calculus Exam ................................................................................. 64
3.3 DATA CODING AND ANALYSIS .................................................................. 65
3.3.1 Scoring the PCA Items .................................................................................. 65
3.3.2 Coding Interview Responses ......................................................................... 69
3.3.3 Coding Function Understanding .................................................................. 74
3.3.4 Coding Function View and Understandings of the Big Ideas .................... 78
3.3.5 Reliability Coding .......................................................................................... 85
3.3.6 Analysis ........................................................................................................... 86
4.0 RESULTS ................................................................................................................... 91
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4.1 AP CALCULUS STUDENTS’ UNDERSTANDINGS OF FUNCTIONS .... 91
4.1.1 Function Understandings on the PCA ......................................................... 91
4.1.2 Function Understandings on the Interviews ............................................... 94
4.1.3 Function View ................................................................................................ 97
4.1.4 Understandings of the Big Ideas................................................................. 104
4.2 COMPARISONS TO AP CALCULUS EXAM PERFORMANCE ........... 110
4.2.1 Function Understandings and the AP Calculus Exam ............................. 111
4.2.2 Function View and the AP Calculus Exam ............................................... 115
4.2.3 The Big Ideas and the AP Calculus Exam ................................................. 116
4.3 PCA VS. INTERVIEWS ................................................................................. 119
4.4 SUMMARY ...................................................................................................... 123
5.0 DISCUSSION ........................................................................................................... 125
5.1 AP CALCULUS STUDENTS’ UNDERSTANDINGS OF FUNCTIONS .. 126
5.1.1 General Function Understandings ............................................................. 126
5.1.2 The Participants’ View of Functions ......................................................... 128
5.1.3 Difficulties and Misconceptions .................................................................. 129
5.1.3.1 Identifying functions and nonfunctions in algebraic form ............ 129
5.1.3.2 Transforming functions .................................................................... 131
5.1.3.3 Misinterpreting function inverse notation ...................................... 131
5.1.3.4 Viewing graphs as iconic representations of phenomena .............. 132
5.1.3.5 Moving between real-world situations and functional
representations ................................................................................................. 133
5.1.4 Summary ...................................................................................................... 135
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5.2 FUNCTION UNDERSTANDINGS NEEDED FOR SUCCESS IN AP
CALCULUS ...................................................................................................................... 136
5.3 OTHER CONSIDERATIONS ....................................................................... 140
5.3.1 Demographic Splits...................................................................................... 140
5.3.2 On Big Idea 4 and Object View .................................................................. 144
5.3.3 PCA Score Adjustments for No Work Shown .......................................... 145
5.4 CONCLUSIONS, IMPLICATIONS, AND FURTHER RESEARCH ....... 146
APPENDIX A ............................................................................................................................ 151
APPENDIX B ............................................................................................................................ 159
BIBLIOGRAPHY ..................................................................................................................... 166
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LIST OF TABLES
Table 1.1. Alignment of the Confrey and Smith (1991) framework and the CCSS with the Big
Ideas. ............................................................................................................................................... 4
Table 2.1. Mental Actions of Carlson, et al.’s Covariation Framework. ..................................... 30
Table 3.1. Demographic information of study participants ......................................................... 51
Table 3.2. Collected data for analysis. .......................................................................................... 57
Table 3.3. PCA Taxonomy. .......................................................................................................... 59
Table 3.4. PCA Items as aligned with each Big Idea................................................................... 60
Table 3.5. PCA Item 1 Scoring Rubric ........................................................................................ 67
Table 3.6. PCA items that do and do not require work to be shown for partial credit. ............... 69
Table 3.7. General coding rubric for interview responses. .......................................................... 70
Table 3.8. Rubric for what makes a correct answer and good explanation for interview tasks. . 72
Table 3.9. Coding Function Understanding based on the PCA. .................................................. 76
Table 3.10. Coding Function Understanding based on interview response code configurations. 77
Table 3.11. Overall coding of action vs. process view. ............................................................... 81
Table 3.12. Coding of object view for interview items. .............................................................. 83
Table 3.13. Rubric for coding item sets for each Big Idea on the PCA. ...................................... 84
Table 3.14. Overall understanding of each Big Idea for interview participants. ......................... 85
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