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High School Student Representational Adaptivity and Transfer

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High School Student Representational Adaptivity and Transfer HIGH SCHOOL STUDENT REPRESENTATIONAL ADAPTIVITY AND TRANSFER IN MULTIPLYING POLYNOMIALS By CAMERON STARR SWEET A dissertation submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY WASHINGTON STATE UNIVERSITY Department of Mathematics and Statistics DECEMBER 2018 © Copyright by CAMERON STARR SWEET, 2018 All Rights Reserved © Copyright by CAMERON STARR SWEET, 2018 All Rights Reserved To the Faculty of Washington State University: The members of the Committee appointed to examine the dissertation of CAMERON STARR SWEET find it satisfactory and recommend that it be accepted. Libby Knott, Ph.D., Chair Sandra Clement Cooper, Ph.D. Nairanjana Dasgupta, Ph.D. Shiv Smith Karunakaran, Ph.D. ii ACKNOWLEDGMENT So do not throw away your confidence; it will be richly rewarded. You need to persevere so that when you have done the will of God, you will receive what he has promised. For in just a very little while, “He who is coming will come and will not delay. But my righteous one will live by faith. And if he shrinks back, I will not be pleased with him.” But we are not of those who shrink back and are destroyed, but of those who believe and are saved. Hebrews 10:35-39 My words are not adequate to express my appreciation for the life and calling God has given me to teach his Word, outdoor and life skills, mathematics and statistics. The life experiences and recommendations from others God provided me ten years ago were direction in answer to prayer for a calling. My talents and values have since been used to help students understand mathematics necessary for their vocations and life decisions. I am grateful to Ms. Gady, Dr. Cochran and the rest of the Whitworth University community for commissioning me and praying for me in my vocation to teach. I look forward to continuing to pray for and encourage students to seek God’s calling on their lives as well. I am thankful to my family for their love, care and support. Most directly related to this study, I appreciate my parents allowing me to frequently stay with them during the Fall 2017 semester when I was collecting data at the high school and returning to Pullman to teach late afternoons three days a week. I am not sure if helping them care for others provided relief from research or if research allowed me to escape the struggles of life, but I appreciate them allowing me to help carry their burdens. No one else has prayed for God to be glorified in my life and work more than my parents have. I am also thankful to my brother and sister-in-law, Casey and iii Anna Sweet, for letting me stay with them while attending conferences and on-campus interviews. Conducting a research project with high school students was a daunting task to complete in the time constraint of a Ph.D. program. My committee warned me of many of the hurdles I would face when I began to prepare my dissertation proposal and I am grateful for their guidance and support in my first experiences with research of grade school students. I appreciate Dr. Knott agreeing to be my committee chair when there were no mathematics professors at Washington State University with more experience in grade school mathematics education research. She allowed me to work independently to a greater extent than I am used to and continued to support me through obstacles both of us experienced in and out of academia. Dr. Cooper provided me with many varied teaching opportunities to grow professionally in her role as the department’s associate chair. Through her experience overseeing the remedial mathematics and algebra classes at Washington State University, she also contributed to the rationale and task format of this study. I am convinced that I had Jesus Christ and Dr. Dasgupta looking out for me during my graduate studies at Washington State University. In addition to finding me a Master’s project, collaborating with me to get it published, advising me on the quantitative portions of this study, and helping me prepare for interviews, she and Dr. Johnson have welcomed me into the life of their family. I have greatly missed the quick conversations of my research with Dr. Karunakaran in which he would suggest reading about metarepresentational competence by diSessa, Billing’s discussions of transfer, or obstacles by Brousseau, each of which sparked further reading that formed the basis for this study and my research trajectory. His extensive knowledge of research literature, familiarity with the academic job market and care for students was an enormous asset to the mathematics education graduate students at Washington State University. iv I was blessed with many wonderful fellow graduate students at Washington State University who were my colleagues, coworkers and friends. Those I enjoyed working alongside most closely were my Neill 318 officemates, Abby, Casey, J.E., Spencer and Ian, all of which were connected to the mathematics education program to some extent. I am also grateful to Yingzi Li for her help in using the repolr package in R to run generalized estimating equations for ordinal logistic regression of the data in this study. I am thankful to the Department of Mathematics and Statistics administration and staff for their help in my graduate school experience. The addition to the department of Dr. Moore as chair has benefitted graduate students through his organization of GQE preparation sessions, support for American Mathematical Society activities and conference funding. I am humbled that he has retained me as a teaching assistant longer than I deserve. Our graduate chair, Dr. Tsatsomeros, has been supportive of all the mathematics and statistics graduate students as a teacher, mentor and friend. Ms. Bentley and Ms. Lewis, the graduate coordinators I have worked most closely with, have also been instrumental in guiding me through the graduate school and mathematics and statistics degree requirements. This study was inspired by two of my teachers who taught multiplication using multiple representations. My elementary school mathematics coach, Mr. Ude, taught the third-grade classes to multiply large integers and rational numbers in decimal form using both place value and lattice multiplication. While my first-grade and music teachers, as well as many of my classmates, verbalized that they viewed me as nothing but trouble, Mr. Ude saw a bored student who needed a challenge and spent time teaching me and a few classmates about areas, volumes and computer maintenance so we could help our teachers with the classroom computers. When I moved, I discovered that many of my fifth-grade classmates had difficulty multiplying integers v using place value and shared lattice multiplication with them. Dr. Garraway also encouraged my group theory class at Eastern Washington University to teach students place value multiplication of polynomials in introductory and intermediate algebra as an alternative to standard distribution. I am indebted to the high school teachers and students who participated in this study. Without their collaboration and insights into struggles and achievements, this project would not have been possible. I wish I could thank them here by name. The school administrators, parents and communities were also gracious in allowing me to observe and interview their students. Finally, I am grateful to the many teachers, classmates, friends, family members, Scouts, Scouters, church youth volunteers and children from Bible studies I worked with who recognized my calling to teach before I did. In this age of information, many of them have remained in contact with me over the last eight years to continue praying with and encouraging me in developing as a mathematics educator. May this work and my further teaching and research continue to help students and teachers. vi HIGH SCHOOL STUDENT REPRESENTATIONAL ADAPTIVITY AND TRANSFER IN MULTIPLYING POLYNOMIALS Abstract by Cameron Starr Sweet, Ph.D. Washington State University December 2018 Chair: Libby Knott While there is an extensive amount of research on representations for solving problems involving functions, there are few studies on high school student use of multiple representations for multiplying polynomials. This study contributes to current mathematics education literature by focusing on the appropriateness of high school student choices of representations for multiplying polynomials and the extent of their transfer of knowledge from multiplication of integers. Study participants are 85 students enrolled in four high school algebra classes in which multiplication of polynomials is covered and the teacher has been observed to encourage students to use multiple methods for problem solving. Choice/no-choice assessments were administered to determine student representational fluency and adaptivity with standard distribution, lattice and place value multiplication of polynomials. Semi-structured task-based interviews were also conducted with ten students from the study to examine student choices of representation for multiplying polynomials and components transferred from integer multiplication. The results of generalized estimating equations for ordinal logistic regression reveal that students are more likely to accurately use lattice than standard distribution to obtain accurate solutions for vii polynomial multiplication tasks. Students also tended to transition from choosing standard distribution to the lattice as the number of terms in the polynomials to be multiplied increased, stating that standard distribution was more efficient for solving simple multiplication tasks while the lattice made it easier to organize multiplication of polynomials with many terms. Students selected place value for fewer choice assessment polynomial multiplication tasks than standard distribution, and though they used it to obtain nearly as many accurate solutions on the no-choice assessment, the place value representation was used less accurately as many students forgot to align place values in the factors. However, more students interviewed recognized a larger number of symbolic components transferred from integer multiplication to place value multiplication of polynomials than the other two representations.
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