Student Difficulties with Linearity and Linear Functions and Teachers' Understanding of Student Difficulties by Valentina Postelnicu A Dissertation Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy Approved April 2011 by the Graduate Supervisory Committee: Carole Greenes, Chair Victor Pambuccian Finbarr Sloane ARIZONA STATE UNIVERSITY May 2011 ABSTRACT The focus of the study was to identify secondary school students' difficulties with aspects of linearity and linear functions, and to assess their teachers' understanding of the nature of the difficulties experienced by their students. A cross-sectional study with 1561 Grades 8-10 students enrolled in mathematics courses from Pre-Algebra to Algebra II, and their 26 mathematics teachers was employed. All participants completed the Mini-Diagnostic Test (MDT) on aspects of linearity and linear functions, ranked the MDT problems by perceived difficulty, and commented on the nature of the difficulties. Interviews were conducted with 40 students and 20 teachers. A cluster analysis revealed the existence of two groups of students, Group 0 enrolled in courses below or at their grade level, and Group 1 enrolled in courses above their grade level. A factor analysis confirmed the importance of slope and the Cartesian connection for student understanding of linearity and linear functions. There was little variation in student performance on the MDT across grades. Student performance on the MDT increased with more advanced courses, mainly due to Group 1 student performance. The most difficult problems were those requiring identification of slope from the graph of a line. That difficulty persisted across grades, mathematics courses, and performance groups (Group 0, and 1). A comparison of student ranking of MDT problems by difficulty and their performance on the MDT, showed that students correctly identified the problems with the highest MDT mean scores as being least difficult for them. Only Group 1 students could identify some of the problems with lower MDT mean scores as being difficult. i Teachers did not identify MDT problems that posed the greatest difficulty for their students. Student interviews confirmed difficulties with slope and the Cartesian connection. Teachers' descriptions of problem difficulty identified factors such as lack of familiarity with problem content or context, problem format and length. Teachers did not identify student difficulties with slope in a geometric context. ii ACKNOWLEDGMENTS Foremost, I would like to express my deepest gratitude to my advisor Professor Carole Greenes for her generous and unwavering support throughout this journey. From the beginning, she set high standards and guided me with immense knowledge and patience, page after page, chapter after chapter, draft after draft, towards the final draft of my thesis. This dissertation would not have been possible without her help and encouragement. Her determination and passion for improving mathematics education for every student continue to motivate me and provide an inspirational model for me as a mathematics educator. I am forever grateful to Professor Finbarr Sloane who helped me every step of the way with his intimate knowledge of Statistics. He has always offered generous and critical feedback, and ideas for new directions regarding the interpretation of my results. I shall always remember his warm encouragement throughout the years. I am indebted to Professor Victor Pambuccian who helped me with the mathematics and the history of mathematics content of my thesis. I feel privileged to have had access to his vast reservoir of knowledge, and I am grateful for his constant input during the course of this thesis. It is an honor for me to thank all my professors from Arizona State University. I learned so much from all of them. I would also like to pay tribute to Freudenthal and von Glasersfeld. Reading their books gave me a feeling of déjà vu, and generated the belief that there might be a place for me in mathematics education. iii Lastly, I would like to thank my husband Florin, my daughters Iris and Eveline, and my mother for their unconditional love and trust. The research reported here was supported in part by a grant from the National Science Foundation (NSF 0833760). iv TABLE OF CONTENTS Page LIST OF TABLES ...................................................................................................... ix LIST OF FIGURES .................................................................................................. xiv CHAPTER 1 OVERVIEW OF THE STUDY ............................................................ 1 1.1 Introduction .................................................................................. 1 1.2 Definition of Terms ...................................................................... 3 1.3 Rationale ....................................................................................... 6 1.3.1 The Need for More Scientists ........................................... 6 1.3.2 Algebra, the Primary Course in the Pipeline to STEM Careers ........................................................................................ 7 1.3.3 Addressing Failure with Algebra I .................................. 9 1.3.4 Studies of Student Difficulties with Linearity and Linear Functions in Grades 8-10 ......................................................... 11 1.3.5 Studies with other Populations ....................................... 14 1.3.6 Studies of Teacher Understanding ................................. 18 2 REVIEW OF LITERATURE ............................................................. 21 2.1 Introduction ................................................................................. 21 2.2 Algebra as a Body of Knowledge: Historical Perspective ....... 21 2.3 Algebra in the School Curriculum ........................................ 24 2.3.1 Algebra in the School Curriculum: Historical Perspective .................................................................................................... 24 v CHAPTER Page 2.3.2 School Algebra from a Curriculum Perspective ........... 28 2.4 Research on Student Understanding of Linearity and Linear Functions ......................................................................................... 30 2.4.1 Studies on Student Understanding of Linearity and Linear Functions ................................................................................... 30 2.4.2 Teaching Experiments on Student Learning of Linearity and Linear Functions ................................................................. 42 2.4.3 Studies on Student Understanding of Slope or Rate of Change in Other Content Areas ................................................ 50 2.5 Research on Teacher Knowledge of Student Understanding of Linearity and Linear Functions ........................................................ 52 2.6 Student and Teacher Identification of Problem Difficulty ....... 65 2.7 Concluding Remarks ................................................................. 76 3 DATA COLLECTED AND METHODS OF ANALYSES .............. 79 3.1 Introduction ................................................................................. 79 3.2 Study Instruments ....................................................................... 80 3.3 Theoretical Framework .............................................................. 84 3.4 Pilot Study ................................................................................... 86 3.4.1 Scoring Guide (SG) for the MDT ................................... 87 3.4.2 Preliminary Coding Categories for Ranking Questionnaires .......................................................................... 95 3.4.3 Preliminary Coding Categories for Interviews .............. 97 vi CHAPTER Page 3.5 Participants ........................................................................... 104 3.5.1 Student Demographics .................................................. 105 3.5.2 Teacher Demographics ................................................. 109 3.6 Timeline for Instrument Development, Recruitment of Participants, Data Collection and Analyses ................................... 111 3.7 Methods of Analyses ................................................................ 113 4 DATA ANALYSES .......................................................................... 117 4.1 Introduction ............................................................................... 117 4.2 Research Question 1 ............................................................... 119 4.2.1 Inter-Rater Agreement and Instrument Reliability ..... 119 4.2.2 Student Performance on the MDT by Grade Level .... 122 4.2.3 Student Performance on the MDT by Current Mathematics Course ............................................................... 125 4.2.4 MDT Student Scores by Performance Group .............. 130 4.2.5 Student Performance on the MDT by Problem ........... 134 4.3 Research Question 2 ............................................................... 157 4.4 Research Question 3 ............................................................... 161 4.5 Research Question 4 ................................................................. 165 4.5.1 Coding and Identifying the Nature of Student Difficulties from RQS Written Comments ................................................ 165 4.5.2 Coding and Identifying Students‟ Mathematical Difficulties from Interviews ...................................................