College algebra students' understanding and algebraic thinking and reasoning with functions

258 Pages·2007·1.87 MB·English

AN ABSTRACT OF THE DISSERTATION OF
Lakana Nilkiad for the degree of Doctor of Philosophy in Mathematics Education
presented on April 5, 2004.
Title: College Algebra Students' Understanding and Algebraic Thinking and Reasoning
with Functions.
Abstract approved:
Margaret L. Niess
The purpose of study was to investigate college algebra students' understanding
of function concepts. In addition, their solution strategies and algebraic thinking and
reasoning were explored. Twenty-four volunteer students from one college algebra
recitation class participated in the study to access their understanding of functions. Five
out of 24 volunteer students were selected to participate in problem-solving interview
sessions to provide a rich description of their understandings of functions and their
algebraic thinking and reasoning.
A Function Understanding Questionnaire was administered to gather these
college students' understandings of functions after they completed the college algebra
course. The questionnaire consisted of four questions asking students to identify their
understanding of: (1) the definition of function, (2) multiple representations of
functions, (3) the use of functions in doing mathematics, and (4) the use of functions in
the real-world situations. Formal interviews prior to, during, and after instruction on
functions with the five students were conducted, and their work on homework
problems, quizzes and tests were explored to clarify these college students'
understanding of functions and to explore their solution strategies and algebraic
thinking and reasoning while solving problems.
Overall, instruction supported students' understanding of functions. The
students' definitions of a function improved toward a more formal definition. In
addition, students had a better understanding of multiple representations, function
transformations, and the application of functions to new mathematical situations and to
real-world situations after completing the course.
Algebraic reasoning includes the ability to approach and solve mathematical
problems in multiple ways. The students in this study were able to use different
methods to solve mathematical problems when they were encouraged to do so.
However, the instruction did not encourage this activity. Perhaps for this reason, their
algebraic thinking and reasoning abilities did not seem to progress as much.
In concert with the recommendation of the several mathematics education
organizations, more research needs to deal with the development of algebraic thinking
and reasoning. In addition, research involving the communication of mathematical ideas
and connection of mathematical understanding, thinking, and reasoning to other
mathematics disciplines, to different subject areas, and to real-world situations are
recommended.
College Algebra Students' Understanding and
Algebraic Thinking and Reasoning with Functions.
By
Lakana Nilklad
A DISSERTATION
Submitted to
Oregon State University
in partial fulfill of
the requirements for the
degree of
Doctor of Philosophy
Presented April 5, 2004
Commencement June 2004
Doctor of Philosophy dissertation of Lakana Nilkiad
presented on April 5, 2004
APPROVED:
Major Professor, representing Mathematics Education
Chair of the bepartment of Science and Mathematics Education
Dean of uate School
I understand that my dissertation will become part of the permanent collection of
Oregon State University libraries. My signature below authorized release of my
dissertation to any reader upon request.
Lakana Nilklad, Author
ACKNOWLEDGEMENTS
I would like to express my appreciation to several people who have provided
help, support, and endless encouragement throughout this study. I am perpetually
grateful to my major advisor, Dr. Margaret Niess, for her wisdom, guidance,
patience, and belief in me. Without her assistance I would not have achieved this
success.
Special thanks are also extended to my other committee, Dr. Dianne
Erickson, Dr. Barbara Edwards, Dr. Larry Flick, and Dr. Bruce Cobentz who have
given me valuable advice and support. Also, I would like to thanks Dr. Norman
Lederman, Dr. Larry Enoch, and Dr. Edith Gummer for their instruction and
direction.
In addition, my deepest appreciation goes to the faculty, staff, and fellow
graduate students in both the Department of Science and Mathematics Education
and the Department of Mathematics. Appreciation is also extended to the instructor,
the Graduate Teaching Assistant, and the College Algebra students who were
involved in this study. Without them this study would have been impossible.
I express gratefulness to my family and friends for their support and
encouragement throughout my life. My special appreciation goes to two of my best
friends, Mary Skoda and Brian Sullivan, who gave me endless help, support, and
encouragement.
Finally, acknowledgement is made to the Royal Thai Government for
providing me the opportunity and scholarship for my studies at Oregon State
University.
TABLE OF CONTENTS
CHAPTER I: THE PROBLEM
1
Statement of the Problem 5
Significance of the Study 8
CHAPTER II: REVIEW OF THE LITERATURE 10
Introduction 10
Development of Function Understanding 12
Students' Mathematical and Algebraic Reasoning 23
Summary 28
CHAPTER III: MATERIALS AND METHODS 30
The Participants 31
The Instructor Staffs 31
The Instructor 31
The Graduate Teaching Assistant 32
Student Participants 33
Data Collection Instruments 34
Background Information Questionnaire 35
Function Understanding Questionnaire 35
Thinking and Reasoning Interview Problems 36
Classroom Observations 39
The Researcher 40
Researcher's Fieldnotes and Journals 41
Data Analysis 41
Summary 43
CHAPTER IV: RESULTS 44
Course Overview 44
Research Question 1: Understanding of Functions 48
Instructional Episodes 48
Episode 1: Function Definition 45
Episode 2: Multiple Representations 52
Episode 3: Transformations of Functions 54
TABLE OF CONTENTS (Continued)
Episode 4: One-to-One and Inverse Functions 57
Student Profiles Concerning Understanding of Functions 60
Amy 60
Ross 63
Emma 82
Lindsey 93
Kyle 103
Questionnaire: College Students' Understanding of Functions 112
Analysis of Students' Understanding of Functions 119
Research Question 2: Solution Strategies and algebraic Thinking and 122
Reasoning
Instructional Episodes 122
Episode 1: Identifying Functions 122
Episode 2: Multiple Representations 121
Episode 3: Transformations of Functions 123
Episode 4: One-to-One and Inverse Functions 125
Episode 5: Applications to Real-World Situations 128
Summary of the Instructor and GTA's Approaches to the
Problems 135
Student Profiles: Solution Strategies and Algebraic Thinking and
Reasoning 136
Amy 136
Ross 146
Emma 161
Lindsey 176
Kyle 187
Analysis of Student Profiles: Solution Strategies and Algebraic
Thinking and Reasoning 199
CHAPTER IV CONCLUSION AND DISCUSSION 203
College Students' Understanding of Functions 203
Solution Strategies and Algebraic Thinking and Reasoning 206
Limitations of the Study 210
Implications for College Level Algebra Curriculum and Instruction ... 211
TABLE OF CONTENTS (Continued)
Recommendations for Future Research 208
REFERENCES 216
APPENDIX A: Letter to Students 225
APPENDIX B: Instructor and Graduate Teaching Assistance Consent Form 230
APPENDIX C: Students Interview Protocol (Pre-Instructional Interview) 232
APPENDIX D: Student Interview Protocol (During Instructional Interview) 233
APPENDIX E: Student Interview Protocol (Post-Instructional Interview) 235
APPENDIX F: Function Understanding Questionnaire 236
APPENDIX G: Pre- Instructional Interview Problems 237
APPENDIX H: Instructional Interview Problems 240
APPENDIX I: Post-Instructional Interview Problem 247
LIST OF FIGURES
Figure
1. Graphs of questionnaire used in Tall and Bakar's study 20
2. Graph of speed vs. time of two cars 25
3. Instructor's example of a table representation of a function 48
4. A tabular representation of a non-function created by the instructor 49
5. Table representation of a function in midterm exam 51
6a. An arrow diagram defining a function 52
6b. An arrow diagram defining a non-function 52
7. A graph and a table of a function y =2x —1 displayed by the instructor 53
8. The example of finding zeros of a function using symbolic manipulation and
graphical representation 54
9. The horizontal transformation of y = x2, two units to the left and two units
totheright
55
10. The vertical transformation of y = x2,up and down 3 units 56
11. Summary of horizontal and vertical transformation presented by the
instructor 56
12. An example of the use of the vertical line test for a function 58
13. Instructor's example of a function that was not one-to-one 58
14. Graph of y = x3 that is a one-to-one function
15. Pre-instructional interview graphs for the popcorn problem 62
16. Graphical representation for Instructional Interview No. 2 64
17. Graphical representation for Instructional Interview No. 3 a 64
18. Graphical representation for Instructional Interview No. 3b 67
19 Amy's examples of one-to-one and not one-to-one functions 69
20. Kyle's example of a function 104
21. A quadratic function used for discussing vertical and horizontal
transformation 127
22. An absolute value function used for discussing vertical and horizontal
transformation 127
LIST OF FIGURES (Continued)
Figure
23. A graph used for a ball thrown example problem 132
24. Graphicalrepresentation of eating popcorn over time 137
25. Graphs of popcorn remaining over a period of time 148
26. Ross's diagonal line showing the car average speeds 153
27. Emma's graph for the lawn-mowing situation 164