High School Student Representational Adaptivity and Transfer

HIGH SCHOOL STUDENT REPRESENTATIONAL ADAPTIVITY AND TRANSFER

IN MULTIPLYING POLYNOMIALS

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

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. The value of teaching polynomial multiplication with multiple representations is then introducing students to adaptive choices for differing tasks and preferences as well as relating the representations to familiar integer multiplication.

viii TABLE OF CONTENTS

Page

ACKNOWLEDGMENT...... iii

ABSTRACT ...... vii

LIST OF TABLES ...... xii

LIST OF FIGURES ...... xiii

CHAPTER ONE: RATIONALE ...... 1

Research Questions ...... 7

CHAPTER TWO: LITERATURE REVIEW ...... 8

Algebra in K-12 Curriculum ...... 8

Representations and Symbol Systems ...... 12

Representational Fluency, Flexibility and Adaptivity ...... 19

Transfer ...... 23

Researcher ...... 27

CHAPTER THREE: METHODOLOGY ...... 32

Sequential Mixed Method Design ...... 32

Subjects and Setting ...... 33

Choice/No-Choice Assessments ...... 36

Quantitative Analysis ...... 37

Semi-Structured Task-Based Interviews ...... 42

Statement Analysis...... 44

CHAPTER FOUR: RESULTS ...... 46

Representational Fluency ...... 46

ix Representational Adaptivity...... 54

Statement Analysis of Justifications ...... 59

Selection of Participants ...... 60

Representation Choices for Integer Multiplication Tasks ...... 61

Representation Choices for Polynomial Multiplication Tasks ...... 63

Justifications for Representation Choices ...... 65

Standard Distribution ...... 67

Lattice ...... 70

Place Value ...... 72

Multiplying Polynomials Missing Terms or in Nonstandard Form ...... 75

Transfer ...... 80

CHAPTER FIVE: DISCUSSION ...... 88

Conclusions ...... 88

Limitations ...... 92

Ethical Consideration ...... 93

Implications for Research ...... 94

Implications for Instruction...... 95

REFERENCES ...... 97

APPENDIX

APPENDIX A: STUDENT USE OF MULTIPLE REPRESENTATIONS IN BEGINNING ALGEBRA TO RELATE MULTIPLICATION OF POLYNOMIALS TO MULTIPICATION OF INTEGERS ...... 103 APPENDIX B: R REPOLR GEE CODE FOR REPRESENTATIONAL FLUENCY ...118

APPENDIX C: IN-CLASS WORK SHEET ...... 126

x APPENDIX D: CHOICE ASSESSMENT ...... 130

APPENDIX E: NO-CHOICE ASSESSMENT ...... 131

APPENDIX F: INTERVIEW SCRIPT ...... 134

APPENDIX G: STUDENT INTERVIEW INSTRUCTIONS ...... 139

APPENDIX H: INTERVIEW WITH EVE ...... 140

APPENDIX I: INTERVIEW WITH BEN ...... 153

APPENDIX J: INTERVIEW WITH PABLO ...... 172

APPENDIX K: INTERVIEW WITH TOM ...... 188

APPENDIX L: INTERVIEW WITH KEREN ...... 205

APPENDIX M: INTERVIEW WITH REBEKAH...... 227

APPENDIX N: INTERVIEW WITH SARAH...... 243

APPENDIX O: INTERVIEW WITH JOHN ...... 257

APPENDIX P: INTERVIEW WITH CALEB ...... 277

APPENDIX Q: INTERVIEW WITH SAMUEL ...... 292

xi LIST OF TABLES

Page

Table 3.1: Description of Variables Used in GEE for Ordinal Logistic Regression Analysis ...... 38

Table 4.1: Ordinal Responses from No-Choice Assessment ...... 46

Table 4.2: Ordinal Responses Based on Representation from No-Choice Assessment ...... 47

Table 4.3: Results of Generalized Estimating Equations for Ordinal Logistic Regression ...... 52

Table 4.4: Student Representational Choice for Each Task from Choice Assessment ...... 56

Table 4.5: Summary Statistics of Adaptivity Scores ...... 57

Table 4.6: Student Selected Representations to Solve Interview Tasks ...... 63

Table 4.7: Student Statements for Representational Choices ...... 67

xii LIST OF FIGURES

Page

Figure 2.1: Examples of Expanded Place Value Symbol Systems for Multiplication ...... 16

Figure 2.2: Examples of Standard Distribution Symbol Systems for Multiplication ...... 18

Figure 2.3: Examples of Lattice Symbol Systems for Multiplication ...... 19

Figure 4.1: Accurate Solution with Incomplete Representation Using Lattice ...... 48

Figure 4.2: An Accurate Solution with Inaccurate Representation Using Standard Distribution .49

Figure 4.3: Another Accurate Solution with Inaccurate Standard Distribution Representation....49

Figure 4.4: Accurate Solution with Inaccurate Representation Using Place Value Hybrid ...... 50

Figure 4.5: Hybrid with Place Value When Using Standard Distribution ...... 51

Figure 4.6: Interactions Among Representation and Task Type ...... 53

Figure 4.7: Ben’s Integer Multiplication ...... 62

Figure 4.8: Keren’s Integer Multiplication ...... 62

Figure 4.9: John’s Hybrid of Standard Distribution and Place Value Multiplication ...... 65

Figure 4.10: Tom’s Hybrid of Standard Distribution and Place Value Multiplication ...... 76

Figure 4.11: John’s Lattice Multiplication for Missing Terms ...... 77

Figure 4.12: Sarah and Eve’s Place Value Multiplication for Missing Terms ...... 77

Figure 4.13: Ben and Pablo’s Standard Distribution Multiplication for Nonstandard Form...... 78

Figure 4.14: Caleb and Rebekah’s Lattice Multiplication for Nonstandard Form ...... 79

Figure 4.15: Sarah and Eve’s Place Value Multiplication for Nonstandard Form ...... 79

Figure 4.16: Pablo’s Standard Distribution Multiplication of 12 and 15 ...... 81

Figure 4.17: Keren’s Multiplication Examples of Transfer ...... 82

xiii

Dedication

To our Lord and Savior Jesus Christ,

who has called me to teach, and to the teachers and students of Washington State

he has called me to serve.

xiv

CHAPTER ONE: RATIONALE

The Common Core State Standards for Mathematical Practice (Common Core State

Standards Initiative, 2010a) have set forth goals for enhancing mathematics in the grade school classroom. These standards are currently measured in 15 states using the Smarter Balanced

Assessment to determine student level of college and career readiness (State of Washington

Office of Superintendent of Public Instruction (OSPI), 2018). Schools and districts also use the

Smarter Balanced Assessment to identify curriculum and instruction gaps, strengths and weaknesses (State of Washington OSPI, 2018).

The State of Washington began using the Smarter Balanced Assessment and End-of-

Course Exams in 2015 to measure students' critical thinking and problem-solving skills with respect to the Common Core State Standards (State of Washington OSPI, 2018). In 2015, 62% of grade 11 students in Washington taking the Smarter Balanced Assessment did not attain the minimum score necessary to meet the state graduation requirement in mathematics. The following year, 55% of grade 11 students had scores below the necessary minimum, as did 63% in 2017 (State of Washington OSPI, 2018). If these assessments are indicators of college and career readiness, most high school students are not ready.

I reviewed the Common Core State Standards for Mathematical Practice to compare these standards with current curriculum. From the high school algebra standards students are held to, the one that I do not recall from my experience as a product of thirteen years in the Washington

State Public Schools is the following:

“Understand that polynomials form a system analogous to the integers, namely, they are

closed under the operations of addition, subtraction, and multiplication”

(Common Core State Standards Initiative, 2010a, para. 1).

While the concept of similarities between operations on integers and polynomials was not unfamiliar to me in secondary school algebra studies, analogies were not made explicit in my classes. It was not until my first term of graduate school when a professor presented a place value representation for multiplying polynomials during a group theory lesson involving products of multiple complex numbers that I considered the advantages of this technique for multiplying integers and extension to polynomials.

However, performance on specific standards are not readily available from reports on the

Smarter Balanced Assessment nor Trends in International Mathematics and Science Study

(TIMSS) assessments (National Council for Education Statistics, 2018). I conducted a pilot study included as Appendix A to examine student analogies between systems for multiplying polynomials and multiplying integers to focus on this standard and take a step back from the research focus on functions (Duvall, 1987; Even, 1998; Smith, 2003; Gagatsis & Shiakalli, 2004;

Carraher, Schliemann, & Brizuela, 2005; Carraher, Schliemann, Brizuela, & Earnest, 2006;

Acevedo Nistal, Van Dooren, Clarebout, Elen, & Verschaffel, 2010; Acevedo Nistal, Van

Dooren, & Verschaffel, 2012, 2013) and to consider the skill of operating on polynomials which is frequently used when graphing, analyzing and solving functions. It was found in this pilot study of college students in an introductory quantitative reasoning course that many of these students struggle with multiplication of polynomials. When asked to explain how their method for multiplying polynomials related to multiplication of integers, only two of the 28 students who participated in this study attempted to do so. To enroll in this college level course, students were required to demonstrate intermediate algebra proficiency through a placement exam or a satisfactory grade in a remedial algebra course. Despite this requirement, the pilot study demonstrated a possible gap in the above-mentioned standard.

One consideration for students' performance could be that students may find representations confusing. Students often learn new content, such as how to multiply polynomials, from representations while learning about how these representations depict information. When students are learning new content they do not yet understand from a representation they do not yet fully understand, this creates a representational dilemma (Rau,

2016). Rau's work describes how students gain representational competencies while learning content. This study instead focuses on student multiplication of polynomials based on prior representational competencies in multiplying integers. The pilot study also confirmed that college students both chose and accurately used a place value symbol system for multiplying integers. Using a similar symbol system to introduce algebra students to multiplication of polynomials may enhance their performance at this skill as well as their articulation of how multiplication of integers and polynomials are analogous through addressing this representational dilemma (Rau, 2016). Acevedo Nistal et al. (2010) call for research efforts directed at equipping student to make appropriate representational choices.

I provide definitions here that I am using from research literature. A broad definition of a representation is something that should stand for something else with a correspondence between the two (Kaput, 1987a). A mathematical symbol system is then a collection of characters with rules for identifying and combining these characters with a correspondence for representing a mathematical field of reference, such as polynomial multiplication (Kaput, 1987b).

Representational fluency is the accuracy of solutions (Acevedo Nistal et al., 2010) and use of symbol systems selected to solve problems. Making appropriate representational choices for a particular problem is known as representational adaptivity (Acevedo Nistal et al., 2010). Billing

(2007) defines transfer as change in the performance of a task based on prior performance of a

different task, the degree of which is determined by the number of symbolic components recognized between tasks.

Intentional practice in high school algebra of representations for multiplying polynomials may help fill the gap in student understanding of similarities between operations on polynomials and integers. The Common Core Standards for Mathematical Practice (Common Core State

Standards Initiative, 2010a) adopted the use of representations for developing mathematical ideas from the National Council of Teachers of Mathematics (NCTM) Principles and Standards for

School Mathematics (NCTM, 2000), which were formed for the purpose of improving students' experiences with school mathematics. Students are expected to both use and translate among representations when problem solving, which aligns with researchers' claims that clear recognition of two or more representations of a mathematical idea are required to gain understanding of that idea (Duvall, 1987; Gagatsis & Shiakalli, 2004).

Research discussing extension of the familiar place value multiplication symbol system of integers to multiplication of polynomials is not readily available, though texts such as

Messersmith's Intermediate Algebra (2013) present this method. As extending distribution and lattice multiplication of integers to polynomial multiplication has been discussed elsewhere

(O’Neill, 2006; Nugent, 2007), I will outline the discussion on extending the place value symbol system for multiplying integers to polynomials in the literature review. While there is an extensive amount of research demonstrating that the ability to relate one representation of a function to another is necessary for understanding the concept of function (Duvall, 1987; Even,

1998; Gagatsis & Shiakalli, 2004), there are few studies on using representations to help high school algebra students relate multiplication of polynomials to multiplication of integers.

As students from my first pilot study, Appendix A, reported that they chose standard distribution to multiply polynomials because this was how they had been taught to do so in high school, I conducted a second pilot study in spring 2017 with introductory high school students whose teachers were observed to teach linear equation problem solving and polynomial multiplication with multiple representations during the previous school year to study representational choices of students who had been introduced to multiple representations for multiplying polynomials. I worked with the teachers to prepare a worksheet, Appendix C, on using standard distribution, lattice and place value multiplication to supplement their instruction of these representations for multiplying integers and polynomials. They also collaborated with me to format assessments, Appendices E and F, patterned after their in-class quizzes. Of the 36 students from both classes, all five who chose lattice accurately multiplied (x + 5)2. However, only four of the 31 students who chose standard distribution multiplied (x + 5)2 accurately.

Additionally, one student was only able to use place value multiplication to accurately multiply polynomials.

I find the study successful if it helps at least one student, as it may have already helped the student from my pilot study. A place value symbol system for multiplying polynomials will not likely be the optimal choice for all students, but it has the potential to be used as a familiar tool by a student who would otherwise struggle in algebra or to pique the interest of a student who is bored with the usual distributive property. While there is limited research on representations for multiplying polynomials, O’Neill (2006) found that students who struggle with algebra could solve difficult polynomial multiplication problems using a symbol system that he described as similar to lattice multiplication of integers. Reflection can also be especially beneficial to low-achieving students when transferring knowledge from an initial domain to a

target domain, as noted by Billing (2007). My aim was to discover whether recognizing similar symbolic components between representing multiplication of integers and multiplication of polynomials may help students who are struggling in algebra.

The skill to multiply polynomials is useful for graphing, analyzing and solving functions, factoring polynomials, applications of mathematics and mathematical modeling, in much the same way skill at multiplying integers is necessary for dividing integers. This study takes a step back from work in algebra involving functions to view representations for multiplying polynomials that may help students generalize their content knowledge from arithmetic to work with functions in algebra. A familiar symbol system for introducing the new concept of multiplying polynomials could also help alleviate the representational dilemma. Work in both of these gaps may additionally contribute to higher achievement when preparing students for college and careers.

In narrowing the goals of this study, there are limitations to consider. This study does not explore representations constructed by students. The focus is instead on how students adapt to symbol systems presented by their teacher. This study also does not conjecture about internal representations, focusing instead on more readily observable external symbol systems. The intention of this study is not to critique standards, but to address possible curriculum to meet standards and evaluate how students attempt to meet standard expectations. Translating between representations will not be a focus of this study as research exists demonstrating that this ability aids in problem solving (Duvall, 1987; Even, 1998; Gagatsis & Shiakalli, 2004).

The goal of this study is to examine the appropriateness of students’ choices of representations when multiplying polynomials. Additionally, this study explores whether presenting multiplication of polynomials using the same methods in which integers are

multiplied may be beneficial to students' representational fluency. The research questions for this study are:

Research Questions 1. Does high school students' representational fluency in multiplying polynomials differ based on symbol systems used to teach multiplication of polynomials? That is, does accuracy of solutions and use of symbol systems differ based on symbol systems students use to solve polynomial multiplication tasks of similar type? 2. What does representational adaptivity for high school students multiplying polynomials look like? That is, which symbol systems do high school students select for multiplying polynomials and what reasons do they provide for their choices? 3. In what ways are introductory algebra students able to transfer knowledge of multiplying integers to multiplying polynomials?

CHAPTER TWO: LITERATURE REVIEW

Algebra in K-12 Curriculum

Research and standards for mathematics call for algebra to be a theme throughout K-12 education (Schoenfeld, 1995; Kaput, 1999; Smith, 2003; Carraher et al., 2005). The purpose for further preparation and practice of algebra throughout mathematics education came about from the view that algebra plays a role as a gateway to higher education and full participation in citizenship (Moses, 1994; Kaput, 1999; Smith, 2003). In proposing that algebra should pervade the curriculum, Schoenfeld (1995) recommends that algebra experiences begin early for students and increase gradually through eighth grade for a total of at least one year of work. Researchers have formed and tried various methods for incorporating algebraic concepts throughout grade school, a few of which will be discussed in this section (Schoenfeld, 1995; Smith, 2003; Carraher et al., 2005; Carraher et al., 2006). Along these lines, this study provides a possible intervention to help students utilize a symbol system they are familiar with for multiplying integers in early education to multiply polynomials.

One possibility for integrating algebra throughout curriculum is to focus on the mathematics of change. Within grade school curriculum, Smith (2003) builds a framework for using stasis and change to guide students in making generalizations to solve problems involving patterns, functions and algebra. In this framework, stasis is the current structure of a pattern or function while change is used to describe how a pattern could be extended or repeated. For the most part, Smith uses examples from work with functions to consider stasis and change in algebra, citing the importance of functions to algebra. While polynomials can be used in functions to demonstrate change in dependent variables, the emphasis on patterns in the

framework of stasis and change does not explain how to guide students in manipulating polynomials using operators such as multiplication so that they can recognize such patterns.

Situating function concepts in early education curriculum on arithmetic operations has also been researched as a means to prepare students earlier for algebra (Carraher et al., 2005;

Carraher et al., 2006). Also, citing the major role of functions in algebra, Carraher et al. (2006) use the number line to introduce a view of addition and subtraction as functions to 9-10 year old students. After discussing problem situations and allowing students to form their own representations for solving problems, instructors introduced conventional representations and connected them to students' representations and the original problem. The high school teacher involved in my study has in contrast been observed to guide students in assimilating formal formula, table and graph representations for solving linear equations and determining slopes and intercepts. Addition and multiplication are treated as functions in another study by Carraher et al., (2005) where tables are used to represent rules to determine the constant value combined with an independent variable to obtain a dependent sum or product. While the studies conducted by Smith (2003), Carraher et al., (2005), and Carraher et al. (2006) focus on functions, a question

I still have is: what opportunities have been overlooked for the potential to incorporate algebraic characteristics into K-12 curriculum? The research focus on functions leaves a gap as to how to guide students in manipulating the polynomials often used to represent functions within equations.

There are also differing ideas on when algebra should be taught. Kaput (1999) recommends that algebra teaching and learning should begin early, then provides examples of K-

8 lessons which promote algebraic reasoning without stating an ideal timeline for incorporating these lessons. It is also argued by Schoenfeld (1995) that students should have at least one year

of work in algebra by the completion of eighth grade, that this should be accomplished by beginning algebra experiences early for students, and that such experiences should be equivalent to current ninth grade algebra. According to Schoenfeld (1995), beginning to learn a language such as algebra is better done earlier so it can occur over a longer period of time. Schoenfeld

(1995) also notes that there is less tradition of separating students in earlier grades, though evidence is not provided that such sorting has not occurred due to differences in performance of algebra apart from tradition. On the other hand, Carraher et al. (2005, p. 12) contend that, “early algebra is about teaching arithmetic and other topics of early mathematics more deeply, not about teaching algebra earlier.” These studies indicate that skills and concepts mastered in arithmetic, such as multiplication, should be useful to students studying algebra, regardless of when it formally enters the curriculum. As my study is focused on student use of symbol systems learned in arithmetic for algebra instead of early algebra opportunities in arithmetic to prepare students, participants in my study will be high school students, which is currently when algebra courses are traditionally taken.

Many standards for school mathematics require algebra to be integrated throughout K-12 curriculum as well. The NCTM Standards (2000) view algebra as a strand in the curriculum from prekindergarten on, with which teachers can help students build a foundation and prepare for work in high school algebra. The Algebra Standard for high school assumes that this foundation for algebra is in place by the end of eighth grade. In regard to this standard, I expect the high school students in my study to be familiar with multiplication of integers as well as symbol systems for performing this operation. Among the Common Core State Standards for

Mathematical Practice,

“Each standard is not a new event, but an extension of previous learning. Coherence is

also built into the standards in how they reinforce a major topic in a grade by utilizing

supporting, complementary topics”

(Common Core State Standards Initiative, 2010b, para. 2).

The Common Core State Standards for Mathematical Practice clearly state that they do not dictate curriculum or content delivery and in introducing the standards to parents, the U.S.

Department of Defense Education Activity Director Brady explains,

“These standards reflect what a student is taught, not how they are taught. The

individuality and creativity teachers bring to lesson plans and instruction will be

preserved” (U.S. Department of Defense Education Activity, 2016, video).

Both standards then leave room for teachers and teacher educators to develop and evaluate curriculum to help students use the foundation built in earlier grades for algebra work in high school. One of these standards that research has yet to address is that high school algebra students must,

“Understand that polynomials form a system analogous to the integers, namely, they are

closed under the operations of addition, subtraction, and multiplication”

(Common Core State Standards Initiative, 2010a, para. 1).

Determining student use of symbol systems to relate multiplication of polynomials to multiplication of integers is then the focus of my study.

Two concepts common throughout the literature on integrating algebra throughout K-12 curriculum are algebra as generalized arithmetic and the power of symbols and representations in algebra (Schoenfeld, 1995; Kaput, 1999; Smith, 2003, Carraher et al., 2005; Carraher et al.,

2006). Both ideas are illustrated by Schoenfeld (1995) in noting that the general pattern for the

order of adding two numbers, a + b = b + a, is the same as the more specific, 2 + 3 = 3 + 2, and contains no unnecessary symbols, making it shorter than a verbal or visual representation. To infuse algebra throughout school mathematics curriculum, Kaput (1999) recommends beginning with more use of formal language and generalization in arithmetic. Smith (2003) discusses a view of algebra as generalized arithmetic where symbols are one linguistic tool for making generalizations. One suggestion by Carraher et al. (2005) to increase focus on algebra preparation in early mathematics is generalizing arithmetic, which they do by using tables to allow students to experience addition and multiplication rules for integers before introducing algebraic function symbols. Algebra as generalized arithmetic is also the focus of Carraher et al.

(2006) in using the number line to allow students to form representations of addition and subtraction problems before presenting conventional representations of functions. Similarly, studying representations for multiplication of polynomials from generalizations of symbol systems for multiplying integers will be the focus of my work.

Representations and Symbol Systems

The use of symbols has become customary throughout algebra, though verbal descriptions were and still can be used to generalize arithmetic. Thus, while Smith (2003) describes symbols as one linguistic tool for making generalizations, it is a prevalent tool for representing and manipulating ideas due to its power to fit ideas into manageable “compact chunks.” The main focus of work in the first eight years of school is also about representation systems for numbers, such as the base ten place holder system or systems for the rational numbers, and properties of these systems as opposed to focusing on numbers and properties of numbers. Numerical algorithms for manipulating symbols then provide the power to quickly and accurately work with the representations of numbers to produce a symbol that represents the

correct answer (Kaput, 1987a). However, the usefulness and power of representations as tools for algebra are limited when representations are taught and learned as if mastering the tools were the goal instead of generalizing arithmetic (NCTM, 2000). Goldin (2003) argues that students can still gain mathematical power by developing a variety of representations with appropriate translations between representations. The following discussion of research on mathematical representations and symbol systems will then connect this topic to my study on curriculum to help students translate multiplication of integers to multiplication of polynomials.

Mathematical representations are both the actions involved in communicating an idea and the form used to articulate the concept. To effectively support student reasoning using representations, teachers must help students recognize intricacies, advantages and limitations of representations, elicit conditions for relating multiple representations to each other and using these representations to construct generalizations, and scaffold student work on representations

(Mitchell, Charalambous & Hill, 2014). Teachers can help students relate mathematical ideas to concepts students are already familiar with using visual representations, speech and gesture

(Alibali et al., 2014). While there are many factors involved in problem solving, Gagatsis and

Shiakalli (2004) concluded that the ability to translate between representations is one factor correlated to problem solving ability. For example, it has also been demonstrated that the ability to relate one representation of a function to another is necessary for understanding the concept of function (Duvall, 1987; Even, 1998; Gagatsis & Shiakalli, 2004).

In order to help students translate between representations when multiplying polynomials, teachers must have conceptual understanding of why algorithms they teach students work in order to lead students toward computational fluency. After discussing ways in which the place value multiplication algorithm and lattice multiplication relate to the distributive property when

multiplying integers, Nugent's (2007) preservice teachers were able to explain how lattice multiplication could be extended to multiplication of polynomials and how this algorithm relates to the distributive property when multiplying polynomials. O'Neill (2006) also reported that a method similar to this lattice multiplication of polynomials aided students solving difficult polynomial multiplication problems. Both sources cited student difficulties with the FOIL method for multiplying polynomials, as well as its limitations in relationship to distribution and restriction to multiplication of two binomials. I have not discovered research discussing extension of the place value multiplication algorithm of integers to multiplication of polynomials, though recent editions of algebra texts (Miller, O’Neill, & Hyde, 2007; Lial,

Hornsby, & McGinnis, 2011; Messersmith, 2013; Bittinger, 2014; Martin-Gay, 2016) present this method. While there is an extensive amount of research demonstrating that the ability to relate one representation of a function to another is necessary for understanding the concept of function (Duvall, 1987; Even, 1998; Gagatsis & Shiakalli, 2004), studies on using representations to help high school algebra students relate multiplication of polynomials to multiplication of integers are not readily available.

There are many definitions of representations in research literature for understanding different uses of the term representation (Janvier, 1987; Smith, 2003; etc.), but a definition of representation should describe context for what is being represented, what is doing the representing and how they are related (Janvier, 1987). A broad definition of a representation is something that should stand for something else and should describe (1) the represented world,

(2) the representing world, (3) what aspects of the represented world are being represented, (4) what aspects of the representing world are doing the representing, and (5) the correspondence between the two worlds (Kaput, 1987a). Kaput goes on to state that the types of representations

described by this definition include (1) cognitive and perceptual representation, (2) explanatory representation involving models, (3) representation within mathematics, and (4) external symbolic representation. As the focus of my study is on symbolic representation in standard school algebra curriculum, I have chosen to use Kaput's (1987a) above definition of representation.

Symbolizing with representations is then a passage between the symbol and what it symbolizes, or the representing world and the represented world. In defining a symbol system,

Kaput (1987b) describes the representing world as a symbol scheme S, which is a collection of characters with rules for identifying and combining these characters. A field of reference F is the represented world that gives meaning to a symbol scheme. A symbol system is then defined to be a symbol scheme S together with its field of reference F and the correspondence c between them, sometimes written as an ordered triple S = (S, F, c). A mathematical symbol system is further a symbol system in which the field of reference is a mathematical structure.

An example of a familiar symbol system that will be used in this study is the standard place value symbol system for multiplying two integers. The mathematical structure FZ represented by this system is the represented field of reference for multiplying two integers together to form another integer, for which I have chosen to denote this specific field of reference with the subscript Z to refer to integer multiplication. A symbol scheme SZV to represent this structure contains the Hindu-Arabic numerals used to symbolize base-ten integers where two integer symbols are stacked vertically in either order with a horizontal line beneath the bottom one and the symbol × to the left of either of the represented integers to indicate multiplication, such as in Figure 2.1. Note that the chosen label for this symbol scheme, SZV, again contains the subscript Z for integer multiplication as well as V for vertical algorithm or

place value symbolic representation. The rules for the symbol scheme continue with each base ten addend of one integer symbol (10 and 3 in Figure 2.1) multiplied by each base ten addend of the other (100, 50 and 2) and appropriate symbols for the partial products between two addends stacked beneath the line with place values vertically aligned for the purpose of quickly and accurately combining these values. After the addends of both factors have been distributed, the partial products are summed by combining integer symbols to form a single integer symbol for the solution, as the integers are closed under multiplication. The correspondence cZV is described here between the factors, product, operations and rules of FZ and the symbols for each in SZV.

Place value multiplication of Place value multiplication of 152 and 13 (푥2 + 5푥 + 2) and (푥 + 3)

152 (푥2 + 5푥 + 2) x 13 x (푥 + 3) 456 152 × 3 3푥2 + 15푥 + 6 (푥2 + 5푥 + 2) ⋅ 3 + 1520 152 × 10 + 푥3 + 5푥2 + 2푥 (푥2 + 5푥 + 2) ⋅ 푥 1976 푥3 + 8푥2 + 17푥 + 6 Figure 2.1. Examples of Place Value Symbol Systems for Multiplication

Now that the familiar symbol system for multiplying integers has been established, an analogous place value symbol system SPV can be demonstrated to represent the mathematical structure for multiplying two polynomials, FP. I have selected the subscript P for the elements of this symbol system in reference to polynomial multiplication, with V again referring to a vertical algorithm for representing multiplication. As SPV is a generalization of SZV in that unknown values are represented by letters, usually from the English alphabet, additional rules apply for adding and multiplying polynomials though these operators are represented by the same symbols from SZV. For instance, superscripts following letters to the right are used to symbolize repeated multiplication of an unknown and unknowns next to each other or integers symbolize

multiplication of adjacent symbols instead of place value. A convenient ordering of monomial symbols is usually chosen, such as a descending sum by order of each monomial or alphabetically by letters used to symbolize unknowns. When representing multiplication of two polynomials using SPV, two polynomials are again stacked vertically in either order with a line beneath and the same multiplication symbol used in SZV, such as in Figure 2.1. The rules for the symbol scheme SPV continue to mirror those of SZV in that each monomial symbol of one polynomial (x and 3 in Figure 2.1) is multiplied by each monomial symbol of the other (x2, 5x and 2) and appropriate symbols for the partial products between two monomials are stacked beneath the line with monomials with like terms vertically aligned for quickly and accurately combining partial products. Finally, the coefficients of like terms are summed by combining integer symbol coefficients for like terms, then including the addition symbol between monomial symbols to represent a single polynomial symbol for the solution since polynomials are also closed under multiplication. As the symbol schemes SZV and SPV are here demonstrated to be similar, so will be their correspondences cZV and cPV to the mathematical structures they represent.

Providing a symbol system such as SPV that is generalized from the familiar place value symbol system for multiplying integers may help students solve algebra problems relying on multiplication of polynomials. Use of a place value symbol system for multiplying polynomials could aid in filling the representational dilemma gap with use of a familiar representation from arithmetic for introducing students to the new concept of multiplying polynomials. However, making representations available to students is not sufficient to ensure learning of content

(NCTM, 2000; Zbiek, Heid, Blume, & Dick, 2007). Students should also interact with representations in meaningful ways, such as constructing and translating among representations

(Duvall, 1987; Schoenfeld, 1995; Even, 1998; Kaput, 1999; Gagatsis & Shiakalli, 2004; Carraher et al., 2005; Carraher et al., 2006). As extensive research exists on student construction and ability to link representations, it is expected that students who are able to do so when multiplying polynomials will also have a better understanding of this concept. This study instead focuses on the accuracy, frequency and reported justifications of student choices for symbol systems when multiplying polynomials.

The standard distribution symbol system is outlined in many introductory and intermediate algebra texts, including Miller et al. (2007), Lial et al. (2011), Messersmith (2013),

Bittinger (2014) and Martin-Gay (2016). Examples of this representation for multiplying integers and polynomials is provided in Figure 2.2.

Standard distribution multiplication of Standard distribution multiplication of 152 and 13 (푥2 + 5푥 + 2) and (푥 + 3)

(100 + 50 + 2) (10 + 3) (x2 + 5x + 2)(x + 3) = 100(10 + 3) + 50(10 + 3) +2(10 + 3) = x2 (x + 3) + 5x (x + 3) + 2 (x + 3) = 1000 + 300 + 500 + 150 + 20 + 6 = x3 + 3x2 + 5x2 + 15x + 2x + 6 = 1976 = x3 + 8x2 + 17x + 6

Figure 2.2. Examples of Standard Distribution Symbol Systems for Multiplication

O’Neill (2006) and Nugent (2007) also describe the lattice multiplication symbol system for both integers and polynomials. Figure 2.3 contains examples of lattice multiplication in each of these domains.

Lattice multiplication of Lattice multiplication of 152 and 13 (푥2 + 5푥 + 2) and (푥 + 3)

= 1000 + 500 + 20 + 300 + 150 + 6 = x3 + 5x2 + 2x + 3x2 + 15x + 6 = 1976 = x3 + 8x2 + 17x + 6 Figure 2.3. Examples of Lattice Symbol Systems for Multiplication

Representational Fluency, Flexibility and Adaptivity

Three measures of students’ ability to solve problems using representations are representational fluency, representational flexibility and representational adaptivity.

Representational fluency describes student efficiency at interpreting, constructing, translating and switching among external representations (Acevedo Nistal et al., 2010). Even (1998) and

Acevedo Nistal et al. (2010) use accuracy and speed to determine efficiency when observing student representations for fluency. As this study seeks to gain insight on student choices of representations for multiplying polynomials, I intend to measure accuracy of choices without urging students to perform tasks quickly. I have also narrowed this study to focus on student interpretation of representations presented and practiced in their algebra class as mentioned above. When investigating student problem solving, representational fluency can be viewed as outcomes, conditions or stages of development (Zbiek et al., 2007). While observing efficiency to categorize students’ development in problem solving could be of interest for further study, I again intend to narrow the focus here to accuracy when solving problems.

However, many students in my pilot studies were observed to make mistakes such as arithmetic or exponentiation errors. In such cases students could use the representation accurately to obtain an inaccurate product. Other students were observed to misuse a representation or to combine components of multiple representations for a hybrid method to obtain an accurate product. As this study is focused on representations students use to obtain accurate products when multiplying polynomials, both the solution and the use of the representation are important. I will then view representational fluency as the accuracy of solutions and use of symbol systems used to solve tasks. Representational fluency will also be used to compare use of representations and to inform the study of high school student representational adaptivity in multiplying polynomials.

Conditions for student choices and their accuracy will also be investigated. Making appropriate representational choices for a particular problem is often known interchangeably in mathematics education research as representational flexibility or representational adaptivity

(Acevedo Nistal et al., 2010; Acevedo Nistal et al., 2012, 2013). However, Verschaffel, Luwel,

Torbeyns and Van Dooren (2009) found that flexibility usually references the use of multiple strategies while representational adaptivity specifically denotes appropriate choice of representation. As I am interested in how representations help students solve polynomial multiplication tasks, this study will mainly focus on representational adaptivity. Students who use all three representations from this study well are considered flexible in their use of representations, meaning that any representation they choose is both a flexible and adaptive choice.

One way to classify a choice as appropriate is if it matches the expected rational choice based on demands of the task to be performed. Given a set of tasks, a researcher would make

judgments as to the choice or choices that would best aid in solving each task, then classify a matching student choice as appropriate. Another method that I will use instead to measure representational adaptivity is to compare a student’s choice for a task to characteristics of the student and the context of their choice. Acevedo Nistal et al. (2010) studied representational adaptivity in this way by comparing representational fluency in solving comparable linear function tasks under choice- and no-choice-conditions.

The choice/no-choice method was developed by Siegler and Lemaire (1997) for investigating strategy choices in mathematics and has been demonstrated to be useful in determining student representational adaptivity for problem solving in linear functions (Acevedo

Nistal et al., 2010; Acevedo Nistal et al., 2012). Under the choice-condition, a student is instructed to solve a problem using a single representation of their choice to do so. The frequency and accuracy of this choice is then compared to the student’s accuracy at solving similar tasks when they are instructed to use a specific representation for comparison under a no- choice-condition. While this method and its strengths and limitations will be reviewed further in the methodology section, a few results will be mentioned here. Both conditions revealed that participants solved slope and intersection problems more accurately using graphs or tables than with formulas (Acevedo Nistal et al., 2010). Intercept problems were solved more accurately and faster with graphs. Comparing representational fluency of algebra students multiplying polynomials could similarly inform teachers of symbol systems for helping students solve such problems more efficiently when preparing for college and careers. However, formulas were chosen more frequently to solve all three problem types under the choice-condition than graphs or tables, which contributed to lack of representational adaptivity (Acevedo Nistal et al., 2010).

A strong correlation was found between representational adaptivity and accuracy under the

choice-condition when solving linear function problems (Acevedo Nistal et al., 2012), indicating that students who are able to make appropriate representational choices will solve problems more efficiently, as described at the beginning of this section. This study similarly focuses on representational fluency and representational adaptivity, but in the context of helping students multiply polynomials.

Justifications reported by students for representational choices in solving linear function problems have also been explored in research literature. Acevedo Nistal et al. (2013) interviewed students while they chose a representation to solve problems, then formed categories for the justifications students provided for the representations they chose. Any mention of the problem that was being solved as reason for choosing a representation was classified as task-related.

Subject-related justifications were mentions of the student’s characteristics or experience with representations. It was found that context-related justifications, such as instruction to use a specific representation in class, were also provided by students. Properties of the chosen representation were mentioned and categorized as representation-related justifications. These included the information provided by a representation and its simplicity. Acevedo Nistal et al.

(2013) noted that factors for justifying choices were combined at times and that it was difficult to separate representation-related justifications from the other three factors. The reasoning provided for the strong interaction of representation-related factors with other justifications is that the choice of a representation to solve a problem is dependent on the specific task performed by a specific student in a specific context.

While student justifications for choosing a particular representation are insightful, I am most interested in representation-related factors provided by students. What information does a representation chosen by a student provide about distribution when multiplying two

polynomials? How is this distribution among polynomials similar to distribution when multiplying integers? Do students recognize a correspondence between representations for multiplying polynomials and representations for multiplying integers and which representations facilitate this recognition? Exploring representation-related justifications, or representation- related factors in combination with other factors, may begin to reveal the nature of student understanding of similarities between polynomial and integer multiplication.

Acevedo Nistal et al. (2013) also found that some students were overwhelmed by the instruction to choose a representation while others were unable to provide justification for their choice. When interviewing students for this study, I reassured them that expressing difficulty choosing or justifying their choice is acceptable and encouraged students to describe why they experienced difficulty. I also chose algebra classes in which I observed the teacher frequently encourage students to use and try multiple representations to solve problems or check their work on a single problem. Students in my study should then have been familiar with making choices between multiple symbol systems when solving algebra problems.

Transfer

In considering representations as cultural tools, Rau (2016) states that understanding involves the ability to map representations to prior knowledge of what is being represented. Prior knowledge may be transferred from a separate domain that shares similarities to what is being represented. Billing (2007) defines transfer as change in the performance of a task based on prior performance of a different task. Processes of low road to transfer, in which a skill is practiced extensively and in varied situations, and high road to transfer, which depends on mindful abstraction of a principle by a learner, are also discussed by Billing (2007). While I may observe high road transfer while interviewing students, I mainly expect low road transfer as significant

time has been spent practicing multiplication of integers before taking a course in algebra and practicing polynomial multiplication with various representations during this course.

Many sources are cited by Billing (2007) in evidence that whether transfer occurs is dependent on the conditions (Catrambone & Holyoak, 1989; Perkins & Salomon, 1989; Bereiter,

1995; Alexander & Murphy, 1999; Bransford, Brown, & Cocking, 1999; Hatano & Greeno,

1999). Among favorable conditions for transfer to occur, Billing (2007) discusses ways in which transfer is promoted through learning in social context. Learning actively in social practices, such as assimilation of representations for solving problems in the society of a classroom, involving reflection on and comparison of these social practices promotes transfer (Billing,

2007). Transfer promoted from active learning in social practices also brings about justifications and explanations of the same social practices (Billing, 2007). Hatano and Greeno (1999) conclude that ability of learners to use prior knowledge to solve new problems is dependent on situations in which new problems are presented and that if transfer is supported socio-culturally it will occur often. Billing (2007) also found that transfer is more likely when transfer between domains is encouraged in the learning environment by demonstrating how problems resemble each other and encouraging students to discover this themselves.

Hatano and Greeno (1999) suggest that research on transfer should focus more on using prior learning in novel situations. I find analogies between multiplication of integers and polynomials a novel situation in that there are many similar symbolic components for solving problems in both domains. Billing (2007) reasons that teaching by analogy can help students improve transfer in problem solving as transfer and reasoning about analogies are related processes. However, transfer of prior knowledge will be best facilitated if a learner has similar representations for tasks in the initial and target domains and is encouraged to reflect on

similarities between problems (Billing, 2007). According to Billing (2007), the extent of transfer depends on the number of symbolic components a student recognizes between domains.

Analogies and similarities recognized by students between representations are used interchangeably to describe extent of transfer in this study.

Motivation and beliefs can also influence transfer. A factor in the occurrence of transfer is whether a student has a mental set to achieve transfer (Billing, 2007), that is, whether a student has the disposition to believe that prior knowledge from earlier tasks can aid solving new tasks.

Beliefs may sway students to resist change necessary for transfer and affect ways in which knowledge is recognized from an initial domain. Thus, the organization of a student’s belief systems can affect their ability to reason and think critically when solving problems (Billing,

2007).

In determining the occurrence of transfer, Schoenfeld (1982) developed a measure which considers students’ plausible approaches adapted from familiar tasks to solving related mathematical problems. Plausible approaches within this measure were not required to produce a solution but did need to be relevant to the task. Three types of problems categorized as closely related, somewhat related, and unrelated to instruction were posed to students, where the degree of analogy to problems solved in class determined the category the problem belonged to.

Schoenfeld (1982) then scored the number of plausible approaches and degree of success in solving problems and finding more approaches and more complete solutions to closely related and somewhat related problems than to unrelated problems determined transfer had occurred. In my study, all polynomial multiplication problems could be considered closely related to tasks performed in class. I am instead investigating whether students recognize similar symbolic components between domains.

Transfer may also occur from new material to previously learned concepts. Hohensee

(2014) defines backward transfer as the influence new knowledge has on a student’s mathematical activities with related concepts they have encountered previously. In his study,

Hohensee (2014) discovered that quadratic function instruction could be used to positively influence student reasoning about linear functions. While this study instead focuses on forward transfer, backward transfer of student reasoning about multiplication of integers based on instruction of polynomial multiplication may be a topic for further study.

Most of Billing’s (2007) conditions for transfer are favorable in my study. Students in these algebra classes have acquired a familiar place value symbol system for multiplying integers within a society and culture which includes a teacher who encourages them to use this prior knowledge and multiple representations to solve problems. This teacher also demonstrates similar components of multiplication of integers and polynomials and encourages students to reflect and compare analogous representations through classroom discussion and activities.

However, some students may still have difficulty believing knowledge of integer multiplication can be used to multiply polynomials or lack the motivation to use such information.

Students transfer knowledge by using prior knowledge and components of their symbol systems for solving problems in an initial domain to solve problems in a target domain. I agree with Billing (2007) in determining the extent of transfer from the number of symbolic components a student recognizes between domains. Therefore, transfer can occur with student recognition of a single component, though recognition of multiple symbolic components would provide evidence for a greater extent of transfer. Justification of transfer could occur from a low road process based on extensive practice or a high road process of abstraction from one domain

to another. Symbol systems from both domains are cultural tools developed for thinking, communicating and problem solving within society (Rau, 2016).

In addition to the insights of Hatano and Greeno (1999), Billing (2007) and Rau (2016), transfer is useful in research for determining aspects of a practiced activity from one domain to another across a context. For example, Schoenfeld (1982) studied transfer of problem solving approaches from problems solved in class to exam problems across the degree of analogy between problems. Boaler (1993) researched transfer of fraction and number sense from typical school tasks to real world tasks across content and context. This study reports transfer of multiplication components from arithmetic to algebra across representation components.

Researcher

It is important to understand the philosophical assumptions in a research study because these assumptions shape the way in which the researcher thinks about the problem and forms research questions, are required by scholarly communities with practiced traditions, though the philosophical assumptions of an individual may change over time, and the audience has their own philosophical assumptions, which are anticipated by the writer (Creswell, 2012). My philosophical assumptions as the researcher in this study are formed from my education and experiences, a few of which I share here to position these assumptions. I was a student in

Washington State public schools for all thirteen years of grade school. The mathematics coach at the elementary school I attended taught the third-grade classes place value and lattice multiplication of large integers by relating these representations to each other and multiplication of single digit integers. I continued to practice both representations for checking my work when a calculator was unavailable. In secondary school, standard distribution and lattice multiplication of polynomials were demonstrated to my algebra classes, though I do not recall the teachers

explicitly relating their operations to multiplication of integers through concepts or examples. I do recall helping many of my classmates who were struggling with algebra concepts from class and in preparation for the Washington Assessment of Student Learning (WASL). Through tutoring classmates during class and after school, I discovered that alternate representations for solving tasks such as multiplying polynomials frequently helped students who struggled with misusing algorithms commonly presented in class and text books. I also observed that not every student uses the same representation well and that classmates who used more than one representation effectively to solve tasks often experienced less difficulty with assignments and assessments on concepts related to such tasks.

After completing my undergraduate studies in mathematics, I began teaching remedial algebra at Eastern Washington University where there was need for algebra graduate teaching assistants to work with the hundreds of students admitted each year without mastering high school algebra. I entered graduate school to become a mathematics teacher educator and specialize in mathematical content knowledge to help pre-service and in-service teachers consider different methods to develop more students in problem solving. During my first term of graduate school, my group theory professor presented the class with a place value representation that he extended from integer multiplication for multiplying complex numbers and polynomials.

As many of the enrolled graduate students were also algebra teaching assistants, this professor encouraged us to try using this representation in addition to standard distribution when working with students. I continued to do so when teaching Exploring Mathematics, Calculus for Life

Sciences, Statistical Methods, and Algebra Methods courses at Washington State University.

While most students continued using standard distribution, stating this as the representation they were familiar with from grade school, a few adopted place value multiplication and appreciated

its similarities to their familiar representation for multiplying integers. These student experiences and continued student misuse of standard distribution multiplication led to my belief that initial exposure to concepts greatly impacts student beliefs and understanding and that multiple representations for problem solving will be most effectively grasped by students if presented when a topic is introduced. I also believe that mathematical concepts are better taught and learned when they build on prior knowledge and was excited to have a place value representation to readily demonstrate analogies between integer multiplication in arithmetic and polynomial multiplication in algebra.

In reading literature to determine what had already been discovered about student use of representations in algebra, I found that researchers have determined that students who are able to use and translate between representations demonstrate better understanding of function concepts

(Duvall, 1987; Even, 1998; Gagatsis & Shiakalli, 2004). Researchers also found that students struggle when new material is presented with unfamiliar representations (Rau, 2016) but can excel when taught to transfer knowledge from one domain to another (Billing, 2007). These findings, along with inspiration to conduct research on making algebra prevalent throughout grade school curriculum (Schoenfeld, 1995) and the appropriateness of student choices when problem solving (Acevedo Nistal et al., 2010) further strengthened my beliefs on how students use multiple representations and when they should be taught.

While I practice teaching with multiple representations and could have facilitated the lessons reviewing multiplication of polynomials for this study, I view mathematical symbol systems as cultural tools learned in the society of a classroom (Confrey, 1995; Forman, 2003;

Smith, 2003). Taking this socio-cultural stance, I asked the teacher of the classes for this study to guide the lessons on multiple representations for multiplying polynomials and analogies to

integer multiplication so that I could act as an observer and interviewer instead of disrupting the teacher’s role in the class.

I have adopted a postpositivist realist ontological perspective (Creswell, 2012) for this study as I believe there is an objective reality describing the practice of all people, but that it can only be approximated since people are individually different and any study will be limited by the individuals participating and the extent participants can be studied. My education and training as both a mathematics education and statistics graduate student inform me that quantitative methods and analysis answer questions of what populations are doing while qualitative research studies how and why events happen. I believe that quantitative study is often limited in determining what is happening by not examining the actions of groups and individuals to discover to what extent findings are valid. I also believe that qualitative research can be misdirected when descriptive studies are conducted without first verifying that something is occurring to further investigate or considering variation or outliers. Both my statistics and education research training have emphasized that research methods should be built on existing theory and research. For these reasons, this project is based on studies of student representational choices for problem solving

(Siegler & Lemaire, 1997; Verschaffel, 2009; Acevedo Nistal et al., 2010; Acevedo Nistal et al.,

2012, 2013) and first asks what representations students are using and how accurately they are using them, and then considers why they are selecting different representations and how they use representations to relate multiplication of polynomials to integer multiplication.

I selected the lens of socio-cultural theory for the epistemological perspective of this study because of my agreement with research on representational fluency as a social activity

(Billing, 2007; Rau, 2016), symbols as a linguistic tool to view algebra as generalized arithmetic

(Forman, 2003; Smith, 2003), and the perspective of education as an active process in society

(Davydov, 1995; Vygotsky, 1978). However, not all research on problem solving takes a socio- cultural perspective to learning. When considering representations for problem solving, many researchers (Schoenfeld, 1995; diSessa, 2004; diSessa & Cobb, 2004; Carraher et al., 2005;

Carraher et al., 2006) consider ways in which knowledge is constructed by students. However, these studies are focused on student constructed representations, which in these cases are later compared to representations commonly used in society to guide students in discussing further problem solving. I have instead chosen socio-cultural theory to investigate links between instructional goals for students to discover symbol systems to accurately multiply polynomials and learning outcomes (Forman, 2003). As opposed to considering representations constructed or reconstructed by students, I am interested in how students use representations provided by their teacher. Analysis of the data collected from this study using a constructivist perspective with the same research questions and methodology may provide interpretation of how individual students reconstruct symbol systems for multiplying polynomials, justify their representations, and transfer knowledge from multiplication of integers as opposed to the social and cultural experiences that could be observed to inform choices from a socio-cultural perspective.

CHAPTER THREE: METHODOLOGY

Sequential Mixed Method Design

This is a mixed methods study to gain an understanding of whether presenting multiplication of polynomials using the same methods in which integers are multiplied may be beneficial to students' representational fluency, adaptivity and transfer. In this study, quantitative methods are used to determine differences in student performance based on symbol systems used to teach multiplication of polynomials and which symbol systems high school students select for multiplying polynomials. Qualitative methods are also used to investigate reasons algebra students provide for their symbol system choices and ways in which introductory algebra students are able to transfer knowledge of multiplying integers to multiplying polynomials. The purposes for using mixed methods in this study are for the analysis and results of these methods to inform each other and provide breadth and scope to the project. (Tashakkori & Teddlie, 1998).

Data is triangulated using classroom observations to provide background, choice/no-choice assessments to describe the general practice of high school students, and task-based interviews for understanding different choices students make when multiplying, why they make these choices, and how they transfer knowledge of components for multiplying integers to multiplying polynomials. Methods are also triangulated among regression analysis, adaptivity scores and statement analysis to provide different perspectives for a fuller view of students’ representational adaptivity when multiplying polynomials.

This study has a sequential mixed method design (Tashakkori & Teddlie, 1998; Creswell

& Clark, 2007). Quantitative methods for collecting data regarding representational fluency and adaptivity are performed first, followed by qualitative methods to determine why students make representational choices and how they transfer knowledge from multiplication of integers to

multiplication of polynomials. While I mainly rely on qualitative data to determine extent of transfer and justifications for representational choices, I also reflect on analysis of quantitative data to reflect on the appropriateness of student selection of symbol systems in considering choices as adaptive and reasons for student accuracy with different representations. This is not unusual as Tashakkori and Teddlie (1998) note that data collection and analysis may go through several cycles in a sequential mixed method design. Additionally, Schoenfeld (1982) collected qualitative responses of plausible approaches to solving problems which were then quantitatively coded to determine that transfer had occurred. Acevedo Nistal et al. (2013) also reflected on their quantitative study (Acevedo Nistal et al., 2010) when discussing qualitative analysis of student justifications for representational choices.

Subjects and Setting

The four intermediate algebra classes studied were taught by one teacher at a public high school in the state of Washington. These classes consist of 85 students from 14-18 years old. The four classes ranged in size from 19-23 students with a mix of three students in grade 9, 40 in grade 10, 38 in grade 11, and four in grade 12. Forty-one students were male and 44 were female. Students were required to pass a one-year introductory algebra class in which polynomial multiplication was covered and another year of high school geometry for admission to intermediate algebra. When meeting with the teacher before beginning the study, he informed me that the teacher at the same high school where many of these students had taken introductory algebra incorporated classroom discussion involving the three representations from this study for multiplying polynomials. High school algebra classes were chosen because high school is when introductory and intermediate algebra is traditionally taught in the United States (Schoenfeld,

1995) and college students in my pilot study, included as Appendix A, have been observed to

continue misusing representations when they have been previously introduced to multiplication of polynomials.

All 85 students took part in the choice/no-choice assessment. Ten students were then selected to participate in the task-based interviews. Six students were selected with two each who had accurately used one of the three representations to obtain accurate solutions in both the choice and no-choice assessments. Two students were also selected who had used all three representations well in the no-choice assessment and had used more than one representation in the choice assessment. Another two students were selected who struggled with all three representations in the assessments. The selection of these ten students provides diverse perspective on the choices of high school students when multiplying polynomials.

The reason for choosing these specific classes was that the algebra teacher had been observed to encourage students to use multiple representations for solving problems and reflect on their choices. While I was searching for introductory algebra classes to study students who were working with polynomial multiplication for the first time, introductory algebra classes in

Washington do not typically cover this topic until the spring semester of the school year as foundational work with variables and exponential operations must be covered first. Due to time constraints for this project, I needed participants for the fall semester and was fortunate to find a high school algebra teacher with four intermediate algebra classes that were reviewing polynomial operations in November 2017.

Additionally, the classes exhibited many of the favorable conditions for transfer noted by

Billing (2007). The teacher facilitated class discussions and activities comparing representations while covering multiplication of polynomials as well as earlier topics. Formulas, graphs and tables were used when solving problems involving linear equations. Both the standard

distribution and lattice representations were demonstrated for multiplying complex numbers. The standard representation and place value representation for adding and subtracting polynomials was provided both in class lecture and the text book (Miller et al., 2007), with some students rewriting text book problems in place value form and stating that it allowed them to better organize their work. The text book, lecture and work sheet I co-created, included as Appendix C, also included standard distribution, lattice multiplication and place value representations, with the lecture and work sheet demonstrating similar representations for integers. The teacher also encouraged students to try using more than one representation to check their work when solving tasks, though the only time they were required to do so on their own when multiplying polynomials was on the work sheet and no-choice assessment. Another of Billing’s (2007) favorable conditions for transfer is a familiar representation, which the students were excited to return to when using the place value symbol system to multiply large integers at the beginning of the lecture. Finally, students were asked to reflect on their prior knowledge of multiplying integers and exponential operations when multiplying polynomials.

The classes in this study covered the chapter involving operating on polynomials from

November 13 – December 15, 2017, typically meeting five days a week for one-hour class sessions each day. Students were already familiar with each other and the teacher from the first two months of the semester. Two class sessions were spent in discussion and activity reviewing multiplication of polynomials before students were assessed on this topic. In-class written assessments were conducted the following week on November 20 with students completing the choice assessment before the no-choice assessment. Adaptivity scores were then analyzed and students were selected to participate in task-based interviews conducted December 4-12.

Choice/No-Choice Assessments

The two-part assessment contained tasks for the purpose of using the choice/no-choice method (Siegler & Lemaire, 1997; Acevedo Nistal et al., 2010; Acevedo Nistal et al., 2012) to answer the quantitative questions for my study. Under the choice-condition, a student is instructed to:

“Show your work while solving all problems. Simplify answers if possible.

The accuracy of your work and solution will both be assessed.

For each problem, choose one method and multiply the factors. If you use another

method to check your work, circle your work for your first method or write the name of

the method you used.”

The frequency and accuracy of this choice is then compared to the student’s accuracy at solving similar tasks when they are instructed to use a specific representation for comparison under a no- choice-condition. The choice-condition tasks were on the first assessment to avoid bias from carry-over of representations provided in the no-choice-condition (Siegler & Lemaire, 1997).

No-choice tasks on the second quiz immediately following the first were used to determine accuracy with each representation for comparison to choice-condition accuracy. Responses for these assessments were measured both for accuracy in using the representation as well as obtaining an accurate solution.

Students solved six tasks under a choice-condition and four tasks under each of three no- choice-conditions, corresponding to symbol systems using the standard distributive property

(Rep 1), lattice multiplication (Rep 2), and place value multiplication (Rep 3). In each of these four conditions, students solved four problems: multiplication of a monomial and a binomial

(Type 1), two binomials (Type 2), a binomial and a trinomial (Type 3), and two trinomials (Type

4). The additional two tasks in the choice assessment were collected to pilot further study not researched here. Tasks 2, 3, 5 and 6 from the choice assessment in Appendix D were each compared to problems of similar type in the no-choice assessment in Appendix E. The number of terms in the polynomials for these tasks were chosen to reveal student choices to transition from one representation to another based on the choices of students in my pilot study at another high school. The tasks were formed in collaboration with one of the teachers from my pilot study and the assessments formed to pattern his quizzes. The teacher in this study agreed to use these assessments as well. My committee agreed that the tasks for each of the four types of polynomial products described here were similar enough for comparison in this study. Additionally, students were asked to multiply polynomials such as 3x + 3 and x + 7 instead of (3x + 3) (x + 7) so as not to impose the standard distribution representation. However, each task is also presented in the form of the required representation in the no-choice assessment to deter choosing a representation alternate to the instructions. Negative coefficients are not included in the tasks to prevent confounding difficulties students may have in operations involving negatives with the polynomial multiplication representations of interest.

Quantitative Analysis

Logistic regression analysis has been used to determine factors related to representational fluency for similar studies of an ordered assessment score response with multiple continuous and categorical predictors (Acevedo Nistal et al., 2010; Acevedo Nistal et al., 2012), however each of the students in my study was assessed on twelve tasks in the no-choice assessment. These twelve responses will then have dependent predictors, which violates one of the assumptions of logistic regression (Hosmer & Lemeshow, 2000). One method for dealing with observations from the same subject is generalized estimating equations (GEE). Regression parameters are estimated by

treating the observations as independent, then estimating correlations between observations from the same subject to produce new estimates (Olsen, 1997). GEE repeats this process until the difference between new parameter estimates is small.

I used GEE for ordinal logistic regression to determine how representational fluency differs among representation used, as well as the length of polynomials multiplied, class hour, gender and grade level. The pseudo-ordered response variable allows for accurate use of the representation and accurate solution to be measured simultaneously. A subject variable to denote which responses came from the same student is required for GEE and is here denoted as

TotalCount. The no-choice assessment data is less biased to use for representational fluency as all students were required to use all three representations to solve the same tasks of all four types

(Siegler & Lemaire, 1997). Table 3.1 provides a description of the variables used in the analysis.

Table 3.1: Description of Variables Used in GEE for Ordinal Logistic Regression Analysis

Variable Name Description Response Variable: Response Ordered score where: Accurate use of representation, accurate solution = 4, Accurate use of representation, inaccurate solution = 3, Inaccurate use of representation, accurate solution = 2, Inaccurate use of representation, inaccurate solution = 1 Explanatory Variables (Type): TotalCount Indicator from 1-85 for each student in the study (Polytomous) Class Category for class periods 1, 2, 3, 6 (Polytomous) Gender Male = 0, Female = 1 (Dichotomous) Grade High school grade level 9, 10, 11, 12 (Polytomous)

Rep Standard distribution = 1, Lattice = 2, Place Value = 3 (Polytomous) Type Product of a monomial and a binomial=1, Product of two binomials=2, (Polytomous) Product of a binomial and a trinomial=3, Product of two trinomials=4

A significance level of α = 0.10 was used to determine factors related to the response as opposed to the traditional 0.05 level of significance because of the exploratory nature of this study. GEE provides coefficients for each factor, which are log-odds ratios between two factor levels. Odds ratios can then be computed by exponentiating the log-odds ratios. Odds ratios are often easier to interpret since they are the ratio of favorable odds at one factor level and favorable odds at another. However, coefficients for interaction effects are differences in log- odds ratios of two levels of one factor at each level of the other factor. Associated odds ratios for interactions are often comparably larger or smaller than odds ratios for factors because of the exponentiation step, and no more interpretable than the log-odds ratios. For these reasons, log- odds ratios are used to interpret interactions while factors are used to describe odds ratios in the

Results.

Chi-Square Tests of Homogeneity (Ott & Longnecker, 2010) were also conducted using

Pearson Chi-Square to determine whether frequencies of chosen symbol systems are significantly different. For this test, the null hypothesis is that all three symbol systems are chosen with the same frequency. The comparison of frequencies produced by the Chi-Square

Tests is mainly used to interpret representational adaptivity, though it also informs analysis of both the GEE and the task-based interviews in determining representational fluency and components transferred from one symbol system to the other.

Representational adaptivity will further be explored using individual adaptivity scores

(Siegler & Lemaire, 1997). These scores compare the actual accuracy under the choice-condition

(Ac_actual) to a simulated accuracy (Ac_sim) based on the frequency of each representation in the choice-condition (Fc_standard, Fc_lattice, Fc_pv) and the accuracy of each representation in the no- choice-condition (Anc_standard, Anc_lattice, Anc_pv). The simulated accuracy is then:

Ac_sim = (Fc_standard * Anc_standard) + (Fc_lattice * Anc_lattice) + (Fc_pv * Anc_pv)

An individual’s adaptivity score is Ac_actual - Ac_sim (Siegler & Lemaire, 1997). Two adaptivity scores are computed for each student with one to determine adaptivity at accurately solving the task and another to describe adaptivity at accurately using the representation. A score of 1 is assigned for each task with a correct solution and 0 for tasks with an incorrect solution in determining solution accuracy for each representation. Similarly, a score of 1 is assigned for each task in which a student properly uses the conventions of the representation toward a solution and

0 for violating a convention or combining components of representations for the adaptivity score describing accurate use of the representation.

An adaptivity score near 0 indicates that an individual did not do much different when given a choice than if they had chosen a symbol system at random. A positive adaptivity score indicates ability to choose a symbol system that will help an individual multiply polynomials more accurately while a negative adaptivity score indicates that an individual may have been better off selecting a representation randomly. One limitation of adaptivity scores is that they may not tell the full story. For instance, a student with an accuracy rate of 1 in both the choice and no-choice-conditions is flexible and can adaptively choose any representation while a student with 0 accuracy in both conditions cannot choose any representation to help them.

However, both receive an adaptivity score of 0 (Acevedo Nistal et al., 2010). This highlights the need for the Chi-Square Test to compare frequency of choices and qualitative analysis on justifications for representational choices.

Accurate use of representations will be assessed using the following criteria. Guidelines for standard distributive property and lattice are discussed in O’Neill (2006) and Nugent (2007) while components for place value multiplication are described in (Miller et al., 2007). The use of these representations was discussed with the teacher, who included them in his lecture and discussion in the initial class session reviewing polynomial multiplication.

Standard Distributive Property Criteria -each term of one polynomial multiplied by every term of other polynomial -product represented as sum -like terms added -neither factors nor product need to be in standard form Lattice Criteria -one polynomial factor written horizontally with column of boxes beneath for each term -other polynomial factor written vertically with row of boxes for each term to side beneath columns of first polynomial factor -terms of polynomials multiplied in corresponding boxes -product represented as sum with like terms from boxes added -neither factors nor product need to be in standard form -place holders are not necessary for factors or product Note: use of previous two allows addition of partial terms along diagonals of lattice Place Value Criteria -two polynomials written horizontally, one beneath the other -factors, partial sums and product must be in standard form to quickly add partial sums -place holders used for missing terms -each term of second polynomial multiplied by first polynomial and written as partial sum in its own row -designated columns align like terms in factors, partial sums, and final product -like terms added from columns of partial sums

Semi-Structured Task-Based Interviews

While a narrative study might tell the story of the quantitative questions I would like to answer, my qualitative research questions might better be answered by a case study for the purpose of gaining further understanding of student practices based on different representations taught for multiplying polynomials. Case studies are used throughout the social sciences and are thought to have been formed from research in anthropology and sociology (Creswell, 2012). One goal of using a case study approach for this study is to cite lessons learned from this case. A case study is useful for gaining in-depth understanding of reasoning about multiplying polynomials within the context of a classroom. The case for this study was all the students enrolled in intermediate algebra at a single high school in Washington during the Fall 2017 semester. This is a single instrumental case study focused on extent of transfer and representational adaptivity between multiplication of integers and polynomials. This case study is not intrinsic because students are learning to multiply polynomials in algebra classes throughout the country. As discussed earlier in the Methodology, the sequential mixed methods design of this study includes multiple, extensive data sources and analysis to provide thorough insight on student use of representations for multiplying polynomials.

The method used to collect data toward answering my qualitative questions is semi- structured task-based interviews (Goldin, 2000). Goldin (2003) offers principles for planned task-based interviews that I incorporated to make empirical research on representation in mathematics education scientifically reliable and reproducible. First, interviews should be designed to address research questions (Goldin, 2000), which I have formed. While the goal of the interview is to address research questions, I have chosen a semi-structured interview to allow flexibility to ask clarifying and follow-up questions. Tasks were chosen that could be

meaningfully represented by interviewees using varied and meaningful representations (Goldin,

2003).

Tasks for the interview are included in Appendix F. The first four tasks involved multiplication of integers, which algebra student should be familiar with representing, to provide background on representation in the initial domain. These tasks were strategically chosen to lend themselves to various methods, such as techniques for placing zeros behind a factor multiplied by a power of ten for multiplying 53 * 10 = 530 or regrouping to familiar terms when multiplying 102 * 97 = (100 + 2) (100 – 3). The remaining tasks involved multiplication of two polynomials of varying length and form, which students in the classes participating in this study practiced representing using the standard distributive property, lattice multiplication and place value symbol systems. Four of these tasks, 5, 7, 8 and 9, were similar to the four types of tasks from the choice/no-choice assessments for consistency in triangulating data to determine representational adaptivity. Task (x + 5)2 will be used to pilot further study. The last two tasks include polynomials in non-standard form to determine how students use or misuse representations for multiplying.

Semi-structured task-based interviews should also be described explicitly from a script accounting for major contingencies (Goldin, 2000). To do so, students were instructed verbally and in writing to solve a set of multiplication problems and for each task to choose one method and multiply each of the terms provided. They were also asked to think aloud while deciding which method to use and while showing work to solve each task. If they had difficulty expressing reasoning for their choice, they were asked to say so and the interview continued with scripted follow-up questions. After completing polynomial multiplication tasks, students were asked, “Does your chosen method relate to how you multiplied integers?” with scripted follow-

up questions for either response. Students were then asked how they distributed terms from one polynomial into the other. Students were asked if they knew other representations for multiplying factors in each task, not to assess representational flexibility but to prompt further justifications for their initial choice of representation.

While I had considered providing representations for students to choose from in the multiplication tasks, Goldin (2003) recommends encouraging free problem solving if possible. I decided to take this recommendation by asking students to state and choose a method for multiplying. To prevent imposing a representation, students were not provided these tasks in writing. As the interviewer, I read the factors to be multiplied while students wrote the product on a sheet of paper I provided with instructions for the interview at the top, as seen in Appendix

G. The semi-structured nature of this interview permitted students and I to determine that we were talking about the same products, though I was careful to validate the factors and not the representations. In preparing this study, high school students that I conducted this interview with found this method of only receiving products verbally unfamiliar, but manageable in that they and I could ask each other clarifying questions. To record as much of the interview as possible

(Goldin, 2000), I audio recorded responses while video recording the student’s written work.

Statement Analysis

The semi-structured task-based interviews were analyzed using statement analysis

(Acevedo Nistal et al., 2013). A statement is a single, verbal justification for a particular choice of representation by a student while an utterance is all of the statements a student provides for a single choice of representation in a task (Crookes, 1990; Acevedo Nistal et al., 2013). Statement analysis has been used by Acevedo Nistal et al. (2013) to categorize the types of factors students report as influencing their choices of representation when solving linear function problems. They

then used utterance analysis to study tensions students reported in choosing between representations. I used statement analysis to categorize student justifications for representations to determine representational adaptivity and symbolic components a student recognizes between domains to investigate occurrences of transfer between multiplication of integers and polynomials.

After determining all student justifications from the interviews, I categorized these justifications, making comparisons to existing factor categories for student representational choices from research literature. I then considered all statement categories related to each representation. While the Chi-Square Tests and adaptivity scores tell which representations students choose, reporting student justifications for choosing a representation provided perspective on why they choose certain representations. Statements regarding transfer are student references to prior knowledge or work in an earlier task that influence their representational choice (Billing, 2007). A student’s degree of transfer for their chosen representation was determined by the number of symbolic components they recognized from multiplication of integers.

CHAPTER FOUR: RESULTS

Representational Fluency To answer the first research question, if high school students’ accuracy of solutions and use of symbol systems differ based on symbol systems students use to solve polynomial multiplication tasks of similar type, I refer to the summaries of the no-choice assessment and analysis in Tables 4.1, 4.2 and 4.3. Table 4.1 describes the outcomes of the 1020 tasks from the

85 students performing twelve no-choice tasks. Note that the cells of this table are each for one of the ordered responses 1-4 described in the Methodology. As a whole, the students were able to complete 85% of the tasks accurately and used the required representations properly in 88% of the tasks. Only 6% of the tasks demonstrated lack of knowledge in multiplying polynomials by a handful of students. It is not surprising that these classes of students did so well, as they have already completed a year of introductory algebra in which they learned how to multiply and factor polynomials and apply this knowledge.

Table 4.1: Ordinal Responses from No-Choice Assessment

Use of Representation Solution Accurate Inaccurate All Tasks Accurate 810 58 868 (79%) (6%) (85%) Inaccurate 89 63 152 (9%) (6%) (15%) All Tasks 899 121 1020 (88%) (12%) (100%)

Table 4.2 describes the same outcomes, but in terms of the representation required for the tasks in the no-choice assessment. Students successfully used lattice multiplication to obtain an accurate solution on more than 90% of the tasks with only one student unable to use the lattice due to adding instead of multiplying polynomials in all twelve tasks. This was far better than use

of standard distribution or place value representations. Students obtained an accurate solution to

83% of tasks using standard distribution and a comparable 80.9% of tasks using place value.

However, the requirements for like terms to be aligned in factors, partial sums and product when using the place value symbol system left students unable to use it properly for 25% of tasks while only 8.2% of tasks were completed using standard distribution improperly. Students were also unable to obtain an accurate solution or misused place value on 38 tasks, which is far more than either of the other two representations. It initially appears that students are better able to use the lattice than standard distribution or place value, and that while students are comparably accurate at obtaining solutions using standard distribution and place value, they are more likely to misuse the place value representation. Note that Table 4.2 presents assessment tasks, outcomes of which were often replicated by individual students on the twelve no-choice tasks.

Table 4.2: Ordinal Responses Based on Representations from No-Choice Assessment

Ordinal Response Representation 1 2 3 4 All Tasks Standard 21 7 37 275 340 Distribution (6.1%) (2.1%) (10.9%) (80.9%) (100%) Lattice 4 4 25 307 340 (1.2%) (1.2%) (7.3%) (90.3%) (100%) Place Value 38 47 27 228 340 (11.2%) (13.8%) (7.9%) (67.1%) (100%) All Tasks 63 58 89 810 1020

The two most common errors for which students accurately used the required representation but obtained an inaccurate solution were arithmetic errors and difficulties with properties of exponents. Work for which students received an ordinal score of 1 for inaccurate solution and representation included adding or dividing factors.

In addition to the student who added polynomials, two students received an ordinal score of 2 for lattice multiplication tasks because they did not add like terms, as in Figure 4.1. They

had an accurate product but missed this portion of the representation. The accuracy value is then

1 for solution adaptivity and 0 for representation adaptivity for such work.

Figure 4.1. Accurate Solution with Incomplete Representation Using Lattice

There were seven tasks in which students inaccurately used standard distribution to first obtain the accurate solution, then continued to operate on the solution inaccurately. In one of the errors as observed in Figure 4.2, the student appears to be trying to solve for x in the expression.

Another error was what seems to be factoring or simplifying the product as seen in Figure 4.3.

Figure 4.2. An Accurate Solution with Inaccurate Representation Using Standard Distribution

Figure 4.3. Another Accurate Solution with Inaccurate Standard Distribution Representation

When using place value to multiply polynomials, students inaccurately used the representation to obtain an inaccurate solution on 13.8% of the tasks. This was because they did not keep like terms for partial sums and products aligned in columns with the factors, as seen in

Figure 4.4. They instead used the horizontal sum of terms from the standard distributive property. As students were able to obtain accurate solutions by this method, I classify this as a hybrid of the place value and standard distribution representations.

Figure 4.4. Accurate Solution with Inaccurate Representation Using Place Value Hybrid

However, a hybrid of standard distribution and place value by aligning like terms in multiple rows of partial sums and the final product does not violate the guidelines of standard distribution. Figure 4.5 illustrates one student’s example of this hybrid that many students used for the standard distribution tasks containing polynomials with numerous terms. This hybrid

representation was awarded an ordinal score of 4 and corresponding accuracy values of 1 for both solution and representation adaptivity when using standard distribution.

Figure 4.5. Hybrid with Place Value When Using Standard Distribution

The results of generalized estimating equations for ordinal logistic regression more fully answer the question of whether students’ accuracy of solutions and use of symbol systems differ based on symbol systems students use to solve polynomial multiplication tasks of similar type.

To determine factors related to accuracy of solutions and use of representation, I chose the significance level to be α = 0.10 instead of 0.05 because of the exploratory nature of the study.

Parameters significantly related to the ordinal response for accuracy then have p-values less than or equal to 0.10. Descriptions of the parameters in Table 4.3 can be found in the Methodology chapter and Table 3.1. Estimates for coefficients are provided and interpreted for significant interaction effects. Factors are instead interpreted using odds ratios from the exponential function of the coefficient for the factor as discussed in the Methodology.

Table 4.3: Results of Generalized Estimating Equations for Ordinal Logistic Regression

Parameter Coefficient p-value Class 2 v. 1 -0.0838 0.7141 Class 3 v. 1 -0.1637 0.5560 Class 6 v. 1 0.0999 0.7305 Gender 1 v. 0 -0.6413 < 0.0001 Grade 10 v. 9 -0.0218 0.9704 Grade 11 v. 9 0.8035 0.1472 Grade 12 v. 9 0.8562 0.1647 Rep 2 v. 1 -0.7342 0.1078 Rep 3 v. 1 0.4685 0.1063 Type 2 v. 1 0.2166 0.5204 Type 3 v. 1 0.3054 0.3477 Type 4 v. 1 1.0021 0.0005 Rep 2 * Type 2 1.8833 0.1168 Rep 3 * Type 2 0.1361 0.8377 Rep 2 * Type 3 1.0245 0.3942 Rep 3 * Type 3 -0.3525 0.5062 Rep 2 * Type 4 2.1436 0.0512 Rep 3 * Type 4 -0.7137 0.1882

Checking first for interaction effects, the only significant interaction was between

Representation 2 for task Type 4 and Representation 1 for task Type 1. The coefficient for this interaction indicates that the difference between the log-odds ratio comparing Type 4 and Type 1 polynomial products using lattice multiplication and the log-odds ratio comparing Type 4 and

Type 1 polynomial products using the standard distribution representation is greater than 2. As can be seen in Figure 4.6, this is because the difference in mean responses for Type 1 and Type 4 tasks using standard distribution was about 0.25 while the difference between the same tasks using lattice multiplication was about 0.3. This indicates that students are better able to use standard distribution to multiply tasks with fewer terms than longer polynomials using the lattice.

The remaining first and higher order interactions are not significant but were checked using the

R code in Appendix B. While not significant, the interaction between Representation 2 for task

Type 2 and Representation 1 for task Type 1 is notable in that it has a p-value close to 0.10. The interaction plot in Figure 4.6 further indicates little difference in ordinal scores for tasks Type 1 and 2 using standard distribution while the difference in mean scores for these tasks is about 0.1 using lattice. This is notable in that students may be better able to use standard distribution to solve Type 1 and 2 tasks than lattice multiplication. The significant interaction between

Representation and problem Type also means that factors reported with the interaction model are not interpretable and should be tested without the interaction. These parameter estimates for

Representation and Type alone are reported in Table 4.3.

Figure 4.6. Interactions Among Representation and Task Type

Neither Grade level 9-12 nor Class period 1, 2, 3 or 6 are found to be significantly related to accuracy. The student’s Gender is related, with the proportional odds ratio interpreted as male students being 0.5266 times lower than females to use a representation properly to obtain an

accurate solution verses the combined response categories 1, 2 and 3. Task Type is also related to accuracy, with students 2.7240 times more likely to accurately solve a Type 1 task than a Type 4 task while using a representation accurately than to not have an accurate solution or to misuse the representation. This means there is little difference at using representations to solve tasks among

Grade levels or Class periods, though female students are more likely to accurately use a representation in accurately solving a task and it is confirmed that high school students are more likely to multiply a monomial and a binomial accurately than two trinomials.

More importantly to the question of representational fluency, representation is also related to accuracy with p-values equal to the significance levels. Students are 0.4799 times less likely to use standard distribution accurately to get an accurate solution as compared to the alternative combined categories 1, 2 and 3 than with the lattice and 1.5976 times more likely to properly use standard distribution to correctly solve a task than with place value multiplication.

So high school students’ representational fluency in multiplying polynomials does depend on representations used from symbol systems they have been taught. Students are more likely to accurately use lattice multiplication to obtain an accurate solution when multiplying polynomials than standard distribution or place value multiplication. This is especially true for polynomials with many terms, as was inferred from the significant interaction. While students are comparably likely to obtain an accurate solution from standard distribution and place value, they are better able to adopt the guidelines of standard distribution than place value when multiplying polynomials.

Representational Adaptivity

While the representational fluency GEE findings describe high school students in general, lattice multiplication may not be the representation each individual student is able to use most

efficiently to multiply polynomials. For this reason, I am also studying which symbol systems high school students select for multiplying polynomials and what reasons they provide for their choices. Frequencies of student choices are compared using Chi-Square Tests for Homogeneity for each type of task. Adaptivity scores summarize individual student choices by comparing their accuracy with each representation in the no-choice assessment to their representation in the choice assessment. Finally, statement analysis of student interviews provides justifications for why students select certain representations.

The relationship between frequency of students’ choice of representation and the length of polynomials they are multiplying is demonstrated in Table 4.4. While there are more students using standard distribution than either of the other two representations for all tasks in the choice assessment, this difference is most prevalent when they are multiplying a monomial and a binomial or two binomials. When multiplying a binomial and a trinomial, there is only one more student using standard distribution than lattice multiplication. More students used the lattice to multiply two trinomials than either of the other two representations. The number of students using place value to multiply polynomials increased with the number of terms in the polynomials as well. As the polynomial multiplication tasks became more challenging, the number of students choosing to use standard distribution decreased while the number of students using lattice and place value representations increased.

Table 4.4: Student Representational Choice for Each Task from Choice Assessment

Polynomial Product Type Representation 1 2 3 4 All Tasks Standard 59 46 39 35 179 Distribution (69%) (54%) (46%) (41%) (53%) Lattice 22 34 38 39 133 (26%) (40%) (45%) (46%) (39%) Place Value 4 5 8 11 28 (5%) (6%) (9%) (13%) (8%) All Students 85 85 85 85 340

While it could be beneficial to test if there is a significant relationship between frequency of students’ choices in representation and types of polynomial products, the observations in this data are dependent since each student answered all four tasks. This means the results of a Chi-

Square Test for Independence would be invalid. Chi-Square Tests for Homogeneity can be conducted to assess whether the frequency of representational choices is the same for each task since students only solved each task once. For Task 1, multiplying 4x and x + 5, there was a significant difference from equally proportioned student choices for at least one representation

(χ2 = 55.5, p < 0.0001). Contributions to the chi-square test statistic indicate that standard distribution was chosen more than average while lattice and place value were selected less than average. When multiplying 3x + 3 and x + 7 in Task 2, standard distribution and place value are selected more than expected (χ2 = 31.4, p < 0.0001) and place value is selected less than average.

Place value continues to be selected less than equally (χ2 = 21.9, p < 0.0001) when multiplying x

+ 4 and x2 + 3x + 5, though lattice and standard distribution are used by about the same number of students to work Task 3. Lattice and standard distribution are both used more than equally for

Task 4 (χ2 = 16.2, p < 0.0001), while fewer than expected students use place value to multiply

2x2 + 3x + 5 and x2 + 2x + 1.

These results demonstrate that high school students have group-wise representational adaptivity when multiplying polynomials. Considering that lattice multiplication was found to be the most representationally fluent choice for most students, more students selected the lattice to solve tasks 2, 3 and 4 than if students had equally chosen to use each representation at random.

This is even more noteworthy since the interactions between lattice multiplication and task

Types 2 and 4 indicated that this was the best representation for multiplying these polynomials.

Fewer students used place value multiplication than either of the other representations for all four tasks, which was also found to be the representation students had the most difficulty using properly. While more students used standard distribution than either of the other representations, this choice was most notable in Tasks 1 and 2 and may be a more appropriate choice for some students.

Individual student representational adaptivity is measured using Siegler and Lemaire’s

(1997) adaptivity score as discussed in the Methodology. Accuracies are determined using the same guidelines as the GEE method, but separately for solution and use of representation with a value of 1 for accuracy and 0 for inaccuracy. These adaptivity scores are summarized in Table

4.5.

Table 4.5: Summary Statistics of Adaptivity Scores

Score Students Mean SD Minimum Q1 Median Q3 Maximum Solution 85 -0.0691 0.2128 -0.875 -0.25 0 0 0.250 Representation 85 -0.0515 0.0145 -0.500 0 0 0 0.125

While there were only three positive representation adaptivity scores and fifteen positive solution adaptivity scores, none of which were significantly greater than zero, this data still tells a story. Of the 85 students, 43 had an adaptivity score of 0 for their choices of representation

based on solution accuracy and 68 had an adaptivity score of 0 for accurate use of representation, which accounts for the means and medians at or near 0 with low standard deviations and interquartile ranges. This means that more than half of the students chose a representation that is just as likely to provide them an accurate solution than if they were assigned a specific representation. Similarly, 80% of the students chose a representation they can use with the same accuracy as if they are not given a choice. Considering that students accurately used a representation to find an accurate solution for 79% of tasks in the no-choice assessment, as seen in Table 4.1, and that these 810 accurate tasks are distributed among students who were not all accurate on all twelve tasks, many students could not have performed better on the choice assessment and thus have an adaptivity score of 0.

However, some of the 0 adaptivity scores also came from students who made the same representation or arithmetic errors on both assessments, including the student who added factors in all tasks. Looking at the significant minimum values, the student with the -0.875 adaptivity score for solution accuracy also had one of the four -0.5 representation accuracy adaptivity scores. They had inaccurate solutions for all four choice assessment tasks, misusing standard distribution for the first two and accurately using place value with exponent errors for the last two. They then had accurate solutions and work for all four tasks using both lattice and place value and an exponent error on one of the standard distribution tasks for the no-choice assessment. From their work in the no-choice assessment, it appears that this student could have more appropriately used lattice multiplication, or even place value, and had more accurate solutions than when they were given a choice of representation.

The student with a 0.125 adaptivity score for choice of representations used standard distribution accurately for the first task in the choice assessment, then added factors using

standard distribution for one task and lattice for two more. This student also added factors for all standard distribution tasks and three of the lattice multiplication tasks in the no-choice assessment. Not factored into the adaptivity scores since this student did not choose to use the other representation is that they also used the conventions of place value multiplication perfectly for all four no-choice tasks while adding exponents for like terms in two of the tasks. For this student, place value is their most representationally fluent choice. This example also demonstrates that the only way to obtain a positive adaptivity score is to perform more poorly with the representations in the no-choice assessment that were chosen in the choice assessment.

Chi-Square Tests for Homogeneity demonstrate that high school students are more likely to choose standard distribution when multiplying simpler polynomials while more students transition to using lattice multiplication as the number of terms in the polynomials increase. This is a representationally adaptive choice as lattice multiplication was found by the GEE method to be the symbol system that students most accurately use to come to accurate solutions. Adaptivity scores for both accurate solutions and use of representations did not indicate any students that make highly adaptive choices, but the representational fluency analysis leaves reason to believe that adaptivity scores would be at or below 0 since the students successfully used representations to accurately solve the majority of tasks on the no-choice assessment. Knowing what choices of representations students make when multiplying polynomials and the accuracy with which they use those choices, statement analysis will illuminate why students choose different representations.

Statement Analysis of Justifications

In this section, I provide a description of how students were selected to participate in the task-based interviews and summarize their choices of representations for multiplying integers

and polynomials to provide background for the statement analysis. The factors students report for their justifications of choices are then discussed for all multiplication representations.

Justifications of choices for each representation are then analyzed to determine why students choose specific representations for multiplying. Finally, student work for multiplying polynomials that are not in standard form or are missing terms is further analyzed to inform study of representational adaptivity.

Selection of Participants

Instead of interviewing all students, a small representative sample was selected to describe the choices of the population. I interviewed six students, two each of which had used one of the three representations predominantly well in the choice/no-choice assessments.

Additionally, I interviewed two students who had used all three representations well and two who struggled with representations for multiplying polynomials. The reported names for the ten students interviewed for this study are pseudonyms.

Eve and Ben were chosen because they both used all three representations well. Eve used standard distribution for the first two tasks and lattice for the last two on the choice assessment.

In addition to successfully solving these tasks, she also obtained accurate solutions to all of the no-choice tasks, though she did not keep place values aligned when multiplying 4x + 6 and x + 2 using place value. Ben used standard distributive property for all four choice assessment tasks but also used the representations for the no-choice assessment effectively to get accurate solutions.

Pablo and Tom both used standard distribution effectively to solve all four choice tasks and the four standard distribution tasks on the no-choice assessment. While Keren solved all

sixteen tasks well, she chose to use lattice multiplication for the four choice tasks. Rebekah chose standard distribution to multiply 4x and x + 5 but used the lattice to accurately multiply the other three choice tasks and successfully used both representations for the no-choice tasks. Sarah accurately used place value to solve the four choice tasks and the no-choice tasks. John chose to use place value for two of the tasks and standard distribution for the other two, which he did successfully as well as accurately using both representations in the no-choice assessment.

Caleb had arithmetic and exponent errors on three of the choice tasks using lattice multiplication. Similar errors were made on one of the standard distribution tasks and two of the lattice tasks in the no-choice assessment. Caleb’s solutions were accurate when using place value, but he did not keep the like terms aligned. Samuel used all three representations accurately, choosing standard distribution for the choice tasks. However, he made mistakes involving exponents on all four of the tasks for multiplying two trinomials and two of the tasks for multiplying a binomial and a trinomial. While I attempted to recruit more students who had misused the representations, including the student with the -0.875 adaptivity score for representation accuracy, they were not interested in participating in the task-based interviews.

Representation Choices for Integer Multiplication Tasks

On integer multiplication tasks 1, 3 and 4, place value was the first choice of representation for nine of the students, though Keren and John were also familiar with repeated addition and Keren drew and described multiplying arrays of seven rows and 26 columns of tally marks. Ben did not use place value for any of the integer multiplication tasks. Instead he would break one of the numbers into a sum, not always by base ten multiples, then multiply the addends and add these partial sums. Ben’s justification for this choice was:

45 Interviewer Why are you multiplying 53*2? 46 Ben It's essentially the same reasoning as this (points back to 7*20 and 7*6), getting it into simpler terms and adding up these solutions. 47 Interviewer Why did you choose 2? 48 Ben (writes 106 after each 53*2=) Because it's the easiest to do in my head.

Figure 4.7. Ben’s Integer Multiplication

Figure 4.8. Keren’s Integer Multiplication

All ten students used the property for multiplying by powers of ten to write a 0 after 53 when multiplying 53 and 10 for task 2. Seven of the students also used place value to multiply these factors and Ben added the partial sum of 53 and 2 five times. With the exception of Ben and the cases of multiplying by a power of ten, students showed a preference for using a place value representation to multiply integers.

Representation Choices for Polynomial Multiplication Tasks

The representations each student chose to solve the polynomial multiplication tasks in the interview were what was to be expected for the selection of each student, as demonstrated in

Table 4.6. The column denoting students contains their most fluent representation from the choice/no-choice assessments in parentheses. While Eve and Ben could choose and use any of the three representations well, they both selected standard distribution first for each task. Caleb and Samuel, who encountered mainly arithmetic and exponent errors using all three representations, used standard distribution for tasks 5, 6 and 7, then alternate representations for the longer tasks. The class hour and interview ended before Samuel started the last task.

Table 4.6: Student Selected Representations to Solve Interview Tasks

Interview Tasks Student 5 6 7 8 9 10 11 (Choice) Eve Standard Standard Standard Standard Standard Standard Standard (Flexible) Ben Standard Standard Standard Standard Standard Standard Standard (Flexible) hybrid PV hybrid PV hybrid PV Pablo Standard Standard Standard Standard Standard Standard Standard (Standard) Tom Standard Standard Standard Synthetic? Standard Standard Synthetic? (Standard) hybrid PV hybrid PV Keren Standard Standard Lattice Lattice (Lattice) (Lattice) Rebekah Standard Standard Lattice Lattice Lattice Standard Lattice (Lattice) Sarah PV PV PV PV PV PV PV (PV) John Standard Standard PV PV Standard Standard Standard (PV) hybrid PV Caleb Standard Standard Standard Lattice Lattice Lattice Lattice (None) Samuel Standard Standard Standard PV Lattice Standard (None)

Pablo selected standard distribution for all the polynomial multiplication tasks, as did

Tom for most, though he encountered an obstacle at trying to use synthetic division to multiply polynomials in tasks 8 and 11. The algebra classes had been taught synthetic division after the assessments and before the interviews, and in Tom’s excitement to use synthetic division he did not realize that he was dividing polynomials instead of multiplying until I discussed this with him after the interview. Though Keren and Rebekah showed preference for lattice multiplication in the choice assessment, they were also flexible in their use of the other two representations on the no-choice assessment and selected standard distribution for the first two tasks and lattice multiplication for many of the later tasks. Keren first selected standard distribution for task 6 to

“find the square of the quantity of x plus five,” describing it as a short cut (line 182) but missed the middle terms in the product until she used lattice multiplication. Keren’s excitement to use and describe multiple representations to solve multiplication tasks took an entire class hour for the first 8 tasks, at which time the school day ended and she had an extracurricular activity obligation. As she had already provided numerous justifications and transfer of components from integer multiplication to polynomial multiplication, I elected to end the interview with the plethora of data she provided instead of continuing another day when the continuity of the interview tasks would be broken. Sarah chose to use place value for all the interview tasks, while

John used place value for a few of the longer tasks but mainly chose standard distribution. John,

Ben and Tom each used a hybrid of standard distribution and place value in which each partial sum from standard distribution was written in a separate row with place values aligned between rows on some tasks.

Figure 4.9. John’s Hybrid of Standard Distribution and Place Value Multiplication

Justifications for Representation Choices

From the multiplication task-based interviews, 140 statements were found. As expected, the factors related to these justifications aligned with those reported by Acevedo Nistal et al.

(2013). The categories of task, subject, context, and representation-related justifications for representational choices are discussed in the Methodology. While statements were analyzed, I occasionally provide utterances here to provide background. Also notably similar to Acevedo

Nistal et al.’s (2013) context-related justifications is that Ben, Pablo, Keren and Samuel stated that they would use multiple representations to check their solution from a primary choice.

123 Interviewer Why did you choose the second way after your first way? 124 Samuel Just to kinda, maybe like, kinda, you could also do this way (points to PV) to check your work if you're kinda unsure of the first way you did it. 125 Samuel So you could always to this way (points to PV) to check your work. … 138 Samuel And then another method, you could just check your work (writes x^2 5x +2) would probably be the easiest one, 139 Samuel is called the box method (writes 2 by 3 boxes below x^2 5x +2) this one's pretty easy (writes x and 3 each right of boxes)

A justification that differed from what I had expected from these categories was student reliance on memory. Eight of the students stated that they chose the first representation that came to mind

or the representation they remember working with. As these responses were characteristics of the student related to their choices, I categorized such statements as subject-related.

108 Pablo but this is just the one that I know I'm never gonna miss it. 109 Pablo Yeah, I don't know if that's a reason or not. 110 Interviewer What's a reason? 111 Pablo I don't know, it's just since I remember that I learned this, I use this one, you know, yeah.

The students interviewed occasionally found it difficult to justify their preferences and choices of representations. Even though students are instructed to “think aloud while deciding which method to use and while showing your work to solve each problem,” and prompted with semi-structured questions about their choices, they were not always able to give a clear reason.

Sometimes when asked to explain their choices, they would instead explain what they did as illustrated here:

56 Interviewer And why did you write these like this? 57 Samuel This is the distributive method, distributive method, so with the 5, you're just going to multiply it in to both of members (writes curves from 5 to x and 2) 58 Samuel and you'll get (writes) 5 x plus 10.

Of the 140 interview statements, 18.6% were task-related, 29.3% were subject-related,

15.7% were context-related, and 36.4% were representation-related, with distribution among representations presented in Table 4.7. A look at each of these categories within their chosen symbol systems provides an assortment of justifications for choosing these representations and how each may be an appropriate choice for students when solving multiplication tasks.

Table 4.7: Student Statements for Representational Choices

Justification Representation Task Subject Context Representation All Standard 6 16 12 23 57 Distribution (10.5%) (28.1%) (21.0%) (40.4%) Lattice 14 7 1 7 29 (48.3%) (24.1%) (3.5%) (24.1%) Place Value 6 18 9 21 54 (11.1%) (33.3%) (16.7%) (38.9%) All 26 41 22 51 140 (18.6%) (29.3%) (15.7%) (36.4%)

Standard Distribution

Representation-related statements were provided for 40.4% of the justifications for choosing standard distribution. The majority of these statements were that standard distribution was the easiest to use, though most students were unable to explain why.

66 Tom But why would anyone do that? (crosses off PV) This is so much easier (points to standard dist.)

A few of Pablo’s statements explained that standard distribution was more compact or took up less space when solving tasks.

45 Pablo And why'd I do that, cause it occupies less space, usually when I'm doing homework 46 Pablo instead of doing the, kinda like the other method, like doing like this, you know 47 Pablo (writes x+2, next line 5, PV aligned, equal bar below) 48 Pablo I just think it occupy less space (points to standard dist.) than using the other method (points to PV)

Representation-related justifications were also given by Ben, Tom and John for using a hybrid of standard distribution and place value representations.

194 Interviewer This task is to multiply x^2+3x+5 and x^2+x+2. 195 John (writes (x^2+3x+5)(x^2+x+2)) I'm going to factor this out, so x squared times x squared, you add the exponents, so would be x^4 196 John (next line writes x^4+x^3+2x^2) 3 x times x squared, 3 x cubed (next line writes 3x^3 directly below x^3) 197 Interviewer Why did you write the 3 x cubed there? 198 John So it would line up with x cubed up here (points to x^3) so that when I add it down, it will line up.

170 Tom And then I'm still going to stack them because it'd just be really difficult to write them all out in a big, big line and then try to find the ones that match.

Many of the subject-related justifications for choosing standard distribution were given by Pablo.

While he acknowledged representational advantages to the other two representations, he felt more comfortable and confident with his ability to use standard distribution.

108 Pablo but this is just the one that I know I'm never gonna miss it.

200 Pablo It's nice, the bad part about this type of way I do it is the like terms are always in different places. 201 Pablo It's not like the other two, that they're just separated for me and I just have to add them. 202 Pablo I have to look for them in this type that I like to do, which takes up quite a bit more time, 203 Pablo but I'm just more comfortable doing it, and it's just more fluid I think.

Most of the context-related justifications cited that standard distribution is usually used to represent classroom, text book or exam problems. However, multiplication problems such as 26

* 7 are often presented in books and exams this way, yet nine of the students used place value when asked verbally to multiply these factors.

38 Eve Cause that's how we've been setting up problems in the class, just where my brain went.

In the interview with Pablo, he realized that even without being provided a representation he has adopted the notation of his text books.

161 Pablo usually the equation will be given in this format (points to (x^2+5x+2)(x+3)) 162 Pablo let's say in the book, the equation will be given in this format, most of the time at least, you know. 163 Interviewer But I didn't give you a format, I just read you what you were to multiply. 164 Pablo You're right, you're right, you didn't give me a format, that's right. 165 Interviewer You put it in that format because? 166 Pablo Because, I don't want to say because the book always give me that way, 167 Pablo but it does usually give me that way, so that's how I imagine in my mind.

The six task-related statements referenced standard distribution as the easiest way to multiply polynomials that students did not find challenging to work with.

151 Interviewer Why are you going to do the distributive property? 152 Keren Because it's easier and it's easier for this one because if you did it in any other way then it would probably, not really take more work

52 Interviewer This task is to multiply 5 and x+2. 53 Rebekah (writes (5)(x+2)=) 54 Interviewer And why did you write those like that? 55 Rebekah Cause I think when multiplying polyn, er like, things with x's, it's easier to do distributive property unless it's a big one, then I'd rather use the lattice.

Eve described standard distribution as the easiest and fastest way to multiplying x + 2 and x + 5, which she considered a basic task.

70 Interviewer Why did you set this up and choose this method this way? 71 Eve Cause since it's really basic, it seemed like the easiest and fastest way.

In summary, students who chose to use standard distribution found it easy to use when multiplying polynomials, possibly because it can take less space when solving tasks. Such

students have more confidence with standard distribution than other representations for multiplying polynomials. This representation is also familiar to students from class, text books and exams. Standard distribution may be an adaptive choice, not only for students that prefer it for the previous reasons, but also in solving tasks that students find simple. This agrees with the

Chi-Square Tests of Homogeneity comparing number of students choosing this representation to multiply a monomial and a binomial or two binomials.

Lattice

Nearly half of the justifications for choosing lattice multiplication to multiply polynomials were task-related. Rebekah and Caleb both expressed that while they would use standard distribution to multiply simpler polynomials, they found that the lattice helped them to better organize products and like terms when multiplying polynomials with many terms.

63 Interviewer Do you know any other methods for multiplying 5 and x+2? 64 Rebekah Yeah (writes 1 by 2 boxes, 5 left of boxes, x and 2 above) the box method (writes 5x and 10 in boxes, + on line between boxes) 65 Rebekah That's 5x+10 (writes 5x+10 below boxes) It's just kind of weird if it's not a lot of numbers, to do it that way, I think. 66 Interviewer Why's it weird? 67 Rebekah It's just a lot quicker to do it this way (points to standard dist.) or this way (points to PV) I think.

134 Rebekah (writes 2 by 3 boxes) I'm going to do the box method, because I find it easier if there's more numbers, to keep track of my thoughts.

198 Rebekah I would use the box method because when it's more than just two terms time two terms, that's my preference.

158 Caleb Now I'm going to set up the lattice method again. 159 Interviewer Why are you using the lattice method for this one?

160 Caleb Because there's a lot of stuff going on here. There's three integers with variables in this box and an integer and a variable in this box (points at (3x^2+x+2)(x+3)).

Representation-related statements for choosing lattice multiplication also highlighted the organization and visual appeal students found in this representation. Rebekah specifically noted that the lines between products in the lattice allowed her to better separate terms.

179 Rebekah because it has lines in between all the numbers, and I don't have the best handwriting, so I can tell them apart easier, yeah.

154 Interviewer Why are you doing the boxes here? 155 Rebekah Because I find it easier to keep track of things, and then cause you'll have to the fourth power here (points to upper left box row, column: 1,1)

105 Caleb I like this method more (points to lattice) cause I can visually see it.

Students also stated that they found lattice multiplication simpler than the other two without providing justifications.

219 Keren And then there's I think maybe two or three ways that you can do this, which is box method which I use a lot (writes boxes) because it's just kind of simpler.

178 Rebekah I guess I like this one (points to boxes) better than this one (points to stacks) even though they're pretty much the same,

A stronger subject-related statement was that Keren felt that she could consistently use the lattice representation to obtain an accurate solution.

192 Keren Ok. This is the method I usually use (points to boxes) I don't know, cause I feel like it's always right.

The only context-related justification for choosing lattice multiplication was provided by Ben, who used the lattice representation to check his solution using standard distribution.

190 Interviewer Why did you choose your second method after your first method? 191 Ben Um, I guess as a way as double checking my first one.

The findings of the GEE logistic regression and Chi-Square Test were that students are more likely to choose and accurately use lattice multiplication to find products of polynomials with many terms. Statement analysis further shows that students choose to use lattice multiplication for visually organizing multiplication of polynomials with many terms, while some students find the lattice unnecessary for polynomial products they find less challenging.

Students who use the lattice for their primary multiplication representation express more confidence at solving tasks using it than the other two representations. Lattice multiplication can also be used as a backup for checking answer from other representations.

Place Value

The nine students that used place value to multiply integers stated that they did so because that was how they were taught to multiply integers, or this was the way they had been multiplying big numbers all their life.

1 Interviewer Your first task is to multiply 26 and 7. 2 Rebekah (writes 26, next line 7 directly below 6 with PV aligned, equal bar below) 3 Interviewer So why did you write those like that? 4 Rebekah Because that's what I was first taught (next line writes 2 directly below 7, writes 4 above 2 in 26, writes 18 left of 2 below equal bar, PV's aligned)

As for the other representations, representation-related justification for use of place value multiplication included individual preference for the representation without explanation.

39 Sarah I just think it's more easier and I'm comfortable doing it this way.

Visual display was a popular justification for choosing place value as well. The organization and vertical alignment of like terms were stated as reasons for choosing place value by Sarah and

John, who both chose place value on the assessments and interviews.

68 Sarah Because when you have like an extra number or letter (points to x in second factor x+5) I like to line them up as like the same letter as like the first (points to factors) 69 Interviewer Why do you like to line them up like that? 70 Sarah So it would be easier to add them.

166 Interviewer And why did you choose this method first? 167 John It seems a little more organized, maybe, I guess, I don't know.

John even noticed similarities between place value representations for multiplying integers and polynomials, which will be discussed further when answering the question of transfer.

100 John (right of work writes x+2, next line x+5, PV aligned, equal bar below) This way's a little bit better in this situation. 101 Interviewer Why? 102 John It's more like doing a normal multiplication problem, like 5 times 10 or something like that.

Subject-related statements included personal preference, similar to justifications for the other two representations. John also stated that he relied on memory of how to multiply polynomials when selecting a representation.

137 John I kind of forgot how to do that method (points to standard dist.) before this one (points to PV)

175 Interviewer Why not? 176 John Well, I think these ways are faster than the box, the box is just kind of like over too much, you don't need a whole box to figure it out.

Sarah provided context-related statements for choosing place value, including comfort using the representation and familiarity with it from use in class the previous year.

41 Sarah Since I learned the multiplication how to do big numbers, I like to set it up this way, so I feel like comfortable doing.

51 Sarah I just been doing this, like for, since last year I think I learned this (points to PV), so I like to do it this way.

Place value was also used to check work from other representations, as noted by Samuel.

94 Interviewer So why did you pick this way after your first way? 95 Samuel Just to kinda show that you can get both the same answers if you use different methods. 96 Samuel You can use different methods and for sure try to come out with the same answer. You should get the same answer?

Difficulty in multiplying polynomials with many terms was provided as a task-related justification for choosing place value. Caleb also noted that he uses place value for both integers and polynomials that are difficult to multiply in his head.

98 Interviewer This task is to multiply x+2 and x+5. 99 John (writes (x+2)*(x+5)) For this, I think I can do x times x, so x squared (next line writes x^2) then, I don't know, I want to set this up differently. 100 John (right of work writes x+2, next line x+5, PV aligned, equal bar below) This way's a little bit better in this situation.

78 Caleb That's how I normally multiply integers that are a little bit more difficult than ones I can do in my head. 79 Interviewer Why did you choose that method last? 80 Caleb The only reason I would choose this method is if it was a more difficult number problem, but since it's so easy I can do it in my head.

The organization of the place value symbol system in aligning like terms was the most prominent justification reported by students for its selection in multiplying polynomials. As with the other representations, students who regularly rely on place value express more confidence in using it than other techniques. Only one student mentioned the context of learning this representation for multiplying polynomials in school, though nine of the students expressed place value multiplication of integers as the socially expected choice of representation when multiplying integers. Similarities between multiplication of integers and polynomials were stated for choosing place value, which will be analyzed to determine extent of student transfer between the two.

Multiplying Polynomials Missing Terms or in Nonstandard Form

Tasks 10 and 11 were included in the interview to determine how students would use representations to multiply polynomials missing terms or in nonstandard form. As outlined in the

Methodology, the place value representation requires polynomial factors to be in standard form with place holders for missing terms so that factor, partial sum and product like terms align in columns. This allows partial sums to be added quickly for obtaining the final product. The requirements of factors in standard form with necessary place holders do not apply to accurate use of standard distribution or lattice multiplication, though adopting them may organize work toward finding products.

Of the nine students who multiplied x2 + 10 and x + 4, seven students initially used standard distribution. In doing so, Tom used the hybrid with place value in which he wrote partial sum in separate rows with none of the place values aligned since this product contained no like terms. Caleb used the lattice while Sarah chose the place value representation. Place holders were not used in standard distribution by any students.

Figure 4.10. Tom’s Hybrid of Standard Distribution and Place Value Multiplication

Four students chose to use lattice multiplication to check their work for this task, in addition to Caleb. John was the only student who wrote a 0 as a place holder in the factor x2 + 0

+ 10 so he could add like terms along the diagonal.

277 John When you do box method, you're supposed to, or you have to add zero and 10 (writes x^2+0+10 right of boxes) 278 John and the reason is because, with this it's x squared plus 10 x (10 ?), and there needs to be x, but since there isn't an x, it means just x equals zero, 279 John so you have to put zero for this method, you multiply diagonal and you can't multiply diagonal with that, cause it doesn't work, 280 John that's why you have to put that (points to 0) right there. 281 John And so, I'll redo it over here (writes 2 by 3 boxes below x^2+0+10, x and 4 each right of boxes) 282 John I forgot about that part of it. I don't use this method too often. 283 John (writes partial sums in boxes) Now it will make sense, cause those don't combine (points to boxes with 4x^2 and 0)

Figure 4.11. John’s Lattice Multiplication for Missing Terms

In addition to Sarah’s use of place value for task 10, three more students used place value to check their work. While none of the students used place holders or aligned like terms in the factors, all four obtained accurate solutions by spacing their partial sums and final product so that like terms aligned. Eve also included 0’s and a space in the partial sums as place holders, though she did not do so in the factors. Though not completely accurate use of the place value symbol system, students made use of the components of the representation that allow them to find and add like terms.

Figure 4.12. Sarah and Eve’s Place Value Multiplication for Missing Terms

For task 11, multiplying x + 3x2 + 2 and 3 + x, four students chose standard distribution, though Ben used a by putting factors in standard form, writing partial sums in separate rows with place values aligned, then keeping place values aligned in the product as well. Pablo also wrote

factors in standard form, though Eve and John did not. All four, as well as Rebekah who used standard distribution to check her work, had accurate solutions.

Figure 4.13. Ben and Pablo’s Standard Distribution Multiplication for Nonstandard Form

Rebekah and Caleb selected lattice multiplication for task 11. Caleb wrote the factors in standard form before using the lattice, stating that his teacher instructed him to write polynomials in this form so like terms could be added along the diagonals of the lattice. Ben and John also wrote factors in standard form before using the lattice to check their work on this task. Caleb had a minor arithmetic error, though used this representation as well as the other three students.

152 Caleb So here I notice, like I mix it around. So my teacher told me, is that you should always have to have them in equation form. 153 Caleb So that's the highest power in the front, plus the next highest, and then the next highest (writes (3x^2+x+2)). 154 Caleb Then the same for the other one (writes (x+3) after (3x^2+x+2)). 155 Interviewer Your teacher told you that you always have to have it in that form? 156 Caleb Well, you don’t have to have it in that form, but if you want it, say, how it was in this (points to lattice for (x^2+3x+5)(x^2+x+2)), 157 Caleb can add it up like that (points to circled diagonals) then you want to have it in this form (points back to (3x^2+x+2)(x+3)). 158 Caleb Now I'm going to set up the lattice method again.

Figure 4.14. Caleb and Rebekah’s Lattice Multiplication for Nonstandard Form

Sarah again chose place value for solving this task. She was sure to write the factors in standard form and keep polynomial terms in the factors, partial sums and product aligned. Pablo and Rebekah did likewise when using place value to check their work. While Eve left factors in nonstandard form, she again aligned like terms when writing partial sums and product.

Figure 4.15. Sarah and Eve’s Place Value Multiplication for Nonstandard Form

Even though students interviewed were able to provide accurate solutions when multiplying polynomials missing terms or in nonstandard form using all three representations, students would occasionally miss some of the conventions of place value multiplication. More students continued to choose standard distribution first when multiplying polynomials in these cases as well. Student difficulties at accurately using place value further illustrates why students

were least fluent at using place value, as determined by the GEE method, and less likely to select place value when given a choice of representation. However, when used consistently by students such as Sarah, place value can be selected and performed properly as an adaptive choice for multiplying polynomials in nonstandard form. Components of place value may also be adopted to find or align like terms, as seen with the hybrid representations used by Ben, Tom and John.

Transfer

Nine of the students interviewed chose place value as their primary representation for multiplying integers. However, students only selected place value for 8% of polynomial multiplication tasks on the choice assessment. Based on Billing’s (2007) definition of transfer as change in the performance of a task based on prior performance of a different task, as well as infusion of algebra throughout curriculum (Schoenfeld, 1995; NCTM, 2000) so that students should understand how addition, subtraction and multiplication of polynomials are similar to the same operations for integers (Common Core State Standards Initiative, 2010a), it would appear that students are not generally able to transfer representations for multiplication from integers to polynomials. Little evidence of transfer is to be expected from students who collectively choose place value to multiply integers but standard distribution or lattice to multiply polynomials, though individual students who adaptively select place value or flexibly use all three symbol systems may provide insights. The extent of transfer, dependent on the number of symbolic components a student recognizes between integer multiplication and polynomial multiplication

(Billing, 2007), will be examined for students from the interviews.

Most of the student statements about transfer reference place value multiplication.

However, Pablo, John, Keren and Ben attempted to relate integer multiplication representations to standard distribution multiplication of polynomials as well. Pablo, who consistently chose

standard distribution when multiplying polynomials, stated that he had not considered similarities between these domains. He used multiplication of 12 and 15 as an example, but unsuccessfully attempted to distribute the 1 and 2 in 12 to the 1 and 5 in 15, treating the 1’s as 1 instead of 10.

Figure 4.16. Pablo’s Standard Distribution Multiplication of 12 and 15

The only similar component between integer multiplication and standard distribution multiplication of polynomials John and Keren identified was distribution. John noted that both require distribution from one factor to the other, though terms are distributed vertically in place value integer multiplication and horizontally in standard distribution multiplication of polynomials. Keren created the example of 2 (3 + 2) to illustrate and explain how distribution could be similar when multiplying integers and polynomials.

285 Interviewer Does this method relate to how you multiply integers? 286 Keren Um… aren't integers like numbers that are negative, right? 287 Interviewer So, I guess I could rephrase the question, does this relate to how you multiply numbers. 288 Keren Oh! Ha, yes. 289 Interviewer How? 290 Keren Because when you're multiplying numbers like, you can use the distributive property if it was 2 times 3 (writes 2(3+2)) plus 2. 291 Keren You would combine these (points to 2+3) you could either distribute first or you could combine them first. 292 Interviewer How is that similar to how you multiply polynomials? 293 Keren Polynomials, because you'll still be doing, you'll still be doing, you'll still be lining like, I don' know how to explain while doing it. 294 Keren You'll still be distributing it, like how you would distribute this (points from x^2 to (x+3) along curves) 295 Keren and even though there's like, you won't have it like an answer like this points to (points to x^3+8x^2+17x+6) 296 Keren you'll still get an answer like (writes 2*3+2*2 below 2(3+2)) which would be (next line writes 6+4=10) 297 Keren which is the same thing that you did over here (points to standard dist.) but it's just longer because the bigger equation 298 Keren I don't know, I think that's right (points to 6+4=10)

Figure 4.17. Keren’s Multiplication Examples of Transfer

Keren was the only student to relate multiplication of integers to lattice multiplication of polynomials. She again used her example multiplying 2 and 3 + 2 to demonstrate that like terms

of partial sums in lattice multiplication boxes can be combine when multiplying integers or polynomials.

304 Keren You could do the box method, but it's really no use I guess, you could use it, but it's basically the same thing as everything else. 305 Keren (writes 1 box by 2 boxes with 2 left of boxes, 3+2 above) I don't usually use plus but you can if you want 306 Keren (writes 6 and 4 in boxes) Then those are like terms so you combine them and that would be 10 (writes 10 below boxes) so that's the only, yeah.

Recall that Ben did not use place value for any of the integer multiplication tasks. He was able to recognize that the addends he used to multiply integers before combining partial sums were similar to polynomial terms from the standard distribution representation in that they made it easier for him to keep track of his multiplication. This was the only representational component Ben transferred from multiplication of integers to polynomials.

98 Interviewer Does you chosen method relate to how you multiplied integers? 99 Ben Mmm… kind of. 100 Interviewer How? 101 Ben Um… I guess it would be for the fact that that these were in simpler terms (points to 5(x+2)) 102 Ben and in the fact that when I was doing these (points back to integer multiplication) I was trying to put them into simpler terms. 103 Interviewer What do you mean by simpler terms? 104 Ben Terms that are easier to do mentally. 105 Interviewer Are there any ways that your method for multiplying 5 and x+2 is different from how you were multiplying integers? 106 Ben Other than the fact that this (points to 5(x+2)) I don't really see much of a difference.

Seven students recognized a total of four symbolic components between place value representations for multiplying integers and multiplying polynomials that were similar. When asked if his representations for multiplying 5 and x + 2 relate to how he multiplies integers,

Caleb responded that place value does but was unable to explain how or why beyond a description of when he would use place value.

78 Caleb That's how I normally multiply integers that are a little bit more difficult than ones I can do in my head.

Eve, Tom, Rebekah (lines 62, 107-108), Keren (lines 315, 319), Sarah (lines 45, 47),

John (lines 106-109) and Samuel (lines 106, 91) all recognized both that factors are set up vertically and terms for one factor are distributed to the other factor when using place value to multiply integers or polynomials. Rebekah even referenced how the set up and act of distribution in using place value to multiply x + 2 and x + 5 were similar to how she used place value to multiply 26 and 7, providing evidence that she recognized these actions as similar from earlier tasks.

45 Interviewer Does your first method relate to how you multiply integers? 46 Eve Um… kind of, but not really? 47 Interviewer Do any of your other methods relate to how you multiply integers? 48 Eve Yeah, this one (points to PV) 49 Interviewer How? 50 Eve Cause you set it up the same way. 51 Interviewer What's the same about the way you set it up? 52 Eve You do like one set (points to x+2) then another set (points to 5) then you put the line underneath and just multiply like that (points from 5 to each term in x+2). 53 Eve Between these two (points from 5 to x) and the other two (points from 5 to 2).

103 Interviewer Does your second method relate to how you multiply integers? 104 Tom Absolutely. 105 Interviewer How? 106 Tom The way that they're set up and that you take one thing (points to 5 in factor x+5 of PV and places finger over 5) you just act like the x isn't there, 107 Tom And then you multiplying this (points from 5 to x+2 in PV) and then you go to that one (covers 5 in x+5, points from x to x+2) and go to these. 108 Tom They're all going to relate: they're all multiplying.

104 Interviewer So what did you do here? 105 Rebekah I just wrote one on top of the other (points to rows x+2, x+5) and multiplied, like how I did in the first question. 106 Interviewer And by the first question you mean…? 107 Rebekah The 26 times 7. Just 2 times 5, and then 5 times x, and then x times 2, and then x times x (motions PV multiplication) 108 Rebekah and then I got 5x+10 for the first two (points from 5 in row x+5 to x+2) and then x^2+2x for my second two 109 Rebekah and then I combined my like term again.

Rebekah, Keren, Sarah, and John (line 115) additionally noticed that like terms are vertically aligned when using place value multiplication for both integers and polynomials. Keren and

Sarah specifically mention the usefulness of this alignment in combining like terms.

322 Keren And then all the terms, everything is lined up to where it will just simply add, it's easier to add down all the like terms.

86 Sarah but you would still get the same answer, the same as the multiplication of the integer ones. 87 Sarah And you have to line it up to get the same. 88 Interviewer You have to line what up? 89 Sarah Line the (points to lines with factors) if you have multiple numbers on the bottom (points to x+5) you have to line, like, 90 Sarah So there's two (points to x+5) you have to have two on the bottom as well (points to two rows of partial sums) that line up

The use of place holders to keep like terms aligned was also noted by Keren, Samuel and John.

John used his place value representation from task 4, multiplying 102 and 97, to illustrate the similar functions of place holders when using place value to multiply integers and polynomials.

321 Keren And then I put the place holder (next line writes 0) which would be zero, and then (left of 0 writes x^3+5x^2+2x+ with equal bar below)

88 Samuel And then another place holder (slashes 5 in "x x+5", writes 0 below 25) zero, then I'll just do 5 x again (writes 5x below 5x on last line, left of 0) 89 Interviewer Why do you need a place holder? 90 Samuel Cause you've already done that number (points to 5 in "x x+5") so you need to like move it over one, 91 Samuel cause you need to times the x one by (point to x in "x x+5") by x plus 5 (points to top factor x+5)

122 John and you still have your place holders like I did here (points to 0 in x^2 2x 0) and you just multiply in out by each (points to x and 5 in x+5) 123 John so I mean it's kind of the same with having one number, some numbers here (points to x+2) and some numbers (points to x+5) and multiplying them out. 124 Interviewer Do the place holders do similar things or different things in each case? 125 John Similar I'd say. 126 Interviewer How are they similar? 127 John They make it so, you have to use them in the same cases, like as soon as you go to that number (points to x in x+5) 128 John your second number that you're multiplying by, you have to put one, so is that one there (points to 9180 in multiplying 102 and 97)

While Sarah identified three components as similar between place value multiplication of integers and polynomials and John found four similar components, they also noted that carry values to the next place value when multiplying polynomials since place values represent different polynomial terms with their coefficients. Sarah referenced the ten’s places in the integer tasks to demonstrate this difference.

94 Sarah Well for the integer one (points to integer multiplication tasks) is kind of different just because this one has x in it (points to current work), 95 Sarah so you can't really have the whole number (points to 10's in partial sum and product) because it's not a like term, 96 Sarah so you can't have the whole number like on the other side (points to 7x in product) of the x, so you can't add the, 97 Sarah if this is like ten (points to 10 in product) it should be like the zero and like one carried over (points to 7x in product) but you can't add that because it's not a like term.

108 John So 5 times 2 would be 10 (writes 0 below equal bar in one's place) then add the 1 up there (writes 1 above x in x+2) I think 109 John 5 times x, uh… that's different, maybe I put the 10 right there (writes 1 left of 0) then 5x (finishes writing 5x+10, PV's aligned) right there

As seen from the statement analysis of these task basked interviews, more students were able to transfer their knowledge of integer multiplication to polynomial multiplication using place value that standard distribution or lattice multiplication. Similarities in distribution of integer quantities and polynomial terms as well as multiplication of addends and terms was recognized by two separate students comparing integer multiplication and standard distribution of polynomials. One student also realized that like terms from lattice multiplication boxes could be added when multiplying integers or polynomials. However, seven students recognized similar vertical placement of factors, distribution of terms, alignment of like terms and place holders between place value representations for integers and polynomials. John and Keren both described all four of these components. Sarah and John, the two students selected as having chosen and properly used place value to multiply polynomials on the choice/no-choice assessments, also realized that values cannot be carried to different terms when working with coefficients of polynomial sums. While place value may not be students’ most popular or accurate choice of representation, it is an important representation for helping students recognize similar symbolic components between a familiar representation for multiplying integers and polynomial multiplication.

CHAPTER FIVE: DISCUSSION

The goal of this study was to explore the adaptivity of high school student choices of representations for multiplying polynomials and the extent of their transfer of knowledge from multiplication of integers. In conclusion, I will summarize the results of this study and its limitations. Implications for instruction and research of mathematical symbol systems will also be discussed.

Conclusions

Generalized estimating equations for ordinal logistic regression of a no-choice assessment were used to answer the question of whether symbol systems students use to multiply polynomials are related to the accuracy of solutions and accuracy at using each symbol system. It was found that students are more likely to accurately use lattice than standard distribution to obtain accurate solutions for polynomial multiplication tasks (p = 0.1078), especially when multiplying polynomials with many terms (p = 0.0512). Students are also more likely to use standard distribution accurately than place value when multiplying polynomials (p = 0.1063), though students obtained solutions on a comparable number of tasks using standard distribution

(282, 83%) as place value (275, 80.9%). Additionally, female students were more likely to use representations to solve polynomial multiplication tasks accurately (p < 0.0001) and students were more likely to properly use representations to multiply a monomial and a binomial than two trinomials (p = 0.0005), illustrating the difference in difficulty of these tasks.

As illustrated from student work in both the assessments and interviews, the difference in student use of representations is because of the additional requirements of place value multiplication. Several students did not rewrite polynomials in standard form or write like terms from factors in a single column, though they usually did so for partial sums and products. I

classify this inaccuracy as a hybrid of place value and standard distribution in which factors are written vertically, one above the other, to conveniently organize distribution for each term represented by a row. Like terms can then be combined from each row as they would be when using standard distribution, though not from a single column. A more accurate hybrid observed was writing partial sums from each term in a separate row with place values aligned using standard distribution notation for the initial factors. This hybrid of standard distribution and place value may have been the result of instruction on multiple representations for multiplying polynomials.

To answer the question of which representations high school students select when multiplying polynomials and what reasons they report for making their choices, Chi-Square

Tests for Homogeneity and statement analysis of semi-structured task-based interviews were conducted. While students chose standard distribution more frequently in the choice assessment than each of the other two representations, this was mainly for multiplying polynomials that they described in the interviews as simple or easy to multiply, particularly when students wanted to save space on assignments. As a group, students made the adaptive choice of lattice multiplication more than would be expected if they had chosen a representation at random for the latter three of the four polynomial multiplication tasks on the choice assessment (p < 0.0001).

They also tended to move from choosing standard distribution to selecting an alternate representation, particularly the lattice, as the number of terms in the polynomials to be multiplied increased (see Table 4.4), which was again in agreement with task-related statements from the interviews that standard distribution was more efficient for solving simple multiplication tasks.

Adaptivity scores did not show individual students to be particularly adaptive at choosing representations, with means and medians at or near 0 and low standard deviations and

interquartile ranges for both solution accuracy and accurate use of representation. This is not unexpected as students solved 85% of tasks accurately and used the required representation accurately on 88% of tasks for the no-choice assessment. In these numerous cases, students have a flexible choice of any representation on the choice assessment and would be unable to obtain positive adaptivity scores. Positive adaptivity scores can only come from students who do poorly on the no-choice assessment, for which there were no significant cases.

Individual student preference, regular practice and confidence using a symbol system were expressed by students as subject-related justifications for choosing all three representations.

General representation-related statements about finding different symbol systems easy to use were also expressed for all three. Other common justifications for choosing standard distribution were context-related, citing this representation as familiar from class, text books and exams.

Most of the statements for choosing lattice multiplication were task-related, as students found that the lattice made it easier for them to multiply polynomials with many terms. When selecting place value for multiplying polynomials, students made representation-related justifications about the organization it provided when problem solving and context-related statements about its similarity to the familiar place value multiplication of integers.

As most students frequently selected place value to multiply integers and standard distribution to multiply polynomials they found easy to work with or lattice when multiplying more difficult polynomials, there may not be a general case for student transfer of representation from multiplication of integers to polynomials. However, more students were able to recognize similar components between place value symbol systems of multiplying integers and polynomials than they did with the other two representations combined. Seven of the ten students interviewed recognized four similar components between place value representations for

multiplying integers and polynomials. Two of these students individually recognized the vertical placement of factors, how digits and terms are distributed from the bottom factor to the top factor, alignment of place values for combining like terms, and use of place holders to keep place values aligned. Three students were each able to identify one similar component between integer multiplication and standard distribution and one student recognized that like terms from lattice boxes can be combined when multiplying integers or polynomials.

While place value multiplication of polynomials was neither the most fluent or adaptive choice of students in this study, I find it useful for instruction in that it was the representation with which students recognized the most components from integer multiplication. Student transfer of symbolic components from integer multiplication using the place value representation for multiplying polynomials helps bridge the representational dilemma recognized by Rau (2016) through a symbol system students recognized from arithmetic. Familiarity with place value multiplication from arithmetic may make this representation the most adaptive choice for students such as Sarah or the high school student from my pilot study who was unable to accurately multiply polynomials using standard distribution or the lattice. The place value representation also allowed students to draw more analogies between multiplication of integers and polynomials than standard distribution or lattice multiplication, making instruction of place value multiplication of polynomials valuable to algebra instruction for meeting the following

Common Core State Standard for Mathematical Practice:

“Understand that polynomials form a system analogous to the integers, namely, they are

closed under the operations of addition, subtraction, and multiplication”

(Common Core State Standards Initiative, 2010a, para. 1).

Without the place value representation for multiplying polynomials, students may be limited to recognizing that they are multiplying terms from factors and combining like terms at best or not recognize any similarities between integer multiplication and polynomial multiplication as was seen from students in this study. More emphasis should be placed on the guidelines of place value multiplication when it is taught, particularly the requirement that like terms be aligned in factors, partial sums and product, so that students can use this representation more accurately.

Presenting and practicing place value multiplication of polynomials in an introductory algebra class where more time is spent introducing this topic may help alleviate this concern.

Standard distribution and lattice multiplication should also continue to be taught in algebra classes as students were more fluent at using these representations and chose them more frequently than place value. While lattice multiplication was used the most accurately for all tasks in this study, standard distribution was chosen more frequently for less demanding tasks such as multiplying a monomial and a binomial or two binomials. I recommend teaching multiplication of polynomials using all three representations to provide students choices for solving tasks of differing demands and to meet the diverse learning preferences of individual students.

Limitations

There were limitations to this study and its corresponding results that should be noted.

Both this study and my pilot study with college students in Appendix A involved students who had taken algebra courses before. Subjects from these studies referenced familiarity with polynomial multiplication and choosing representations based on instruction from previous algebra courses that did not necessarily emphasize multiple representations for multiplying polynomials or similarities to symbol systems for integer multiplication. While introductory

algebra classes may have been preferable for studying students with no prior experience multiplying polynomials, intermediate algebra classes were selected due to time constraint necessitating data collection in the fall before this topic is covered in introductory algebra curriculum. When I reported students’ greater accuracy of solutions and representation use with lattice multiplication to the teacher in the study, he shared that the high school teacher that had taught many of these students introductory algebra often chooses lattice multiplication of polynomials. I do not know the extent of this effect on the study as I did not observe these earlier classes while the participating students were enrolled. However, none of the students interviewed mentioned the introductory teacher’s preference or instruction when asked about their choices.

The intermediate algebra classes also spent less time reviewing polynomial multiplication and use of representations than the introductory algebra classes from my pilot study did introducing this topic. The classes for this study spent two days reviewing polynomial multiplication for a chapter on operations with polynomials, while the introductory algebra classes from the previous semester worked on polynomial multiplication for four weeks. The increased time spent on any mathematical topic could also be a confounding factor in research.

Students in this study had also passed an introductory algebra course and would be expected to have a better understanding and performance at multiplying polynomials. This could account for the large percentage of adaptivity scores close to 0 and high accuracies at using representations and obtaining accurate solution to polynomial multiplication tasks. Introductory algebra students may have provided a wider range of experiences with these representations.

Ethical Considerations

While I did not interfere with the study by acknowledging student errors during the interviews, I also did not want students to come away with inaccurate understandings of

representations for multiplication. I offered to discuss work with students after each interview either at that time or during a latter class session. This gave students the opportunity to ask me questions about their work that I did not answer during the interview and allowed me to discuss some of the obstacles they encountered when multiplying. For example, Ben and I revisited his initial representation for multiplying 26 and 7 in which he multiplied 26 by a factor of 2 three times before adding a final 26, providing the product of 26 and 8 before the final 26 was added.

He agreed that his second representation better combined 26 seven times. I reminded Tom that his synthetic division representation is for dividing one polynomial by another. Tom expressed his interest in synthetic division but admitted that he had forgotten that this representation is not used for multiplying polynomials. I discussed the place value multiplication guidelines with John and Eve, reminding them to keep place values aligned in the factors, partial sums and product even when polynomial factors are not presented in standard form or are missing terms. I also discussed the results of the assessments with the teacher and provided my opinion that more emphasis on representation guidelines should be accentuated when teaching the place value representation for multiplying polynomials.

Implications for Research

Extensive research of student use of representations has focused on solving problems involving linear equations (Duvall, 1987; Even, 1998; Gagatsis & Shiakalli, 2004; Smith, 2003;

Carraher et al., 2006; Acevedo Nistal et al., 2010; Acevedo Nistal et al., 2012, 2013).

Comparatively few studies are available on student use of representations for multiplying polynomials (O’Neill, 2006; Nugent, 2007). While the place value representation for multiplying polynomials is being included in recent texts (Miller et al., 2007; Lial, Hornsby, & McGinnis,

2011; Messersmith, 2013; Bittinger, 2014; Martin-Gay, 2016), I am unaware of other research

studies examining student use of this symbol system. This study furthers research of student representational fluency by taking a step back from solving function problems to examine student use of standard distribution, lattice and place value representations for multiplying polynomials.

My work also provides varied perspective to the necessity expressed by Acevedo Nistal et al. (2010) for research of student representational adaptivity. Many studies of the appropriateness of students’ representational choices focus on accuracy of their solutions (e.g.

Acevedo Nistal et al., 2010; Acevedo Nistal et al., 2012, 2013). This study additionally considers student choices with regard to the accuracy of their use of conventions for the selected representation.

This is also the first study I am aware of that explores students’ transfer of multiplication components from integers to polynomials across representation components. Ben, Pablo and

Keren’s examples of applying standard distribution components to multiply integers may indicate that further study of backward transfer from polynomial multiplication may reveal change in student understanding of integer multiplication (Hohensee, 2014).

Implications for Instruction

This study offers evidence that providing students multiple representations for solving problems can help prepare them to make adaptive choices, as was seen when students chose standard distribution to multiply less involved polynomials such as a monomial and a binomial but were better able to choose lattice multiplication when working with two trinomials. Multiple representations also cater to a larger number of students, such as Rebekah who regularly used lattice to keep her work organized or students such as Sarah who appreciated the similarities of place value multiplication to her integer multiplication representation. This study included

classes in which the teacher presented multiple representations for solving problems and encouraged students to explore advantages to these representations for different tasks and contexts. However, these students came to this class with different experiences at multiplying polynomials that could influence their representational adaptivity, for their benefit or detriment.

A similar study of student choices for multiplying polynomials after they have been introduced to this topic with multiple representations could be of interest for further study.

Presenting multiple representations for multiplying polynomials, especially place value, also seemed to bridge the didactical obstacle of student unfamiliarity with similarities between integer and polynomial multiplication that was observed in my pilot study, provided as Appendix

A. Inclusion of place value polynomial multiplication with discussion of analogies to integer multiplication then contributes to Schoenfeld’s (1995) proposal that algebra should pervade the curriculum. Another study of polynomial multiplication could examine whether students initially introduced to multiple representations in introductory algebra classes where more time is spent covering this topic are more accurately able to use place value multiplication of polynomials.

Further study of the history of mathematics may also provide insight to obstacles encountered in developing the concept and representations for multiplying polynomials, rooting student difficulty with this topic as an epistemological obstacle (Brousseau, 1997; Fischbein, 1994;

Gallardo & Rojano, 1994; Radford, 1997; Gallardo, 2001).

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APPENDIX A: STUDENT USE OF MULTIPLE REPRESENTATIONS

IN BEGINNING ALGEBRA TO RELATE

MULTIPLICATION OF POLYNOMIALS

TO MULTIPLICATION OF

INTEGERS

by Cameron Sweet Washington State University Department of Mathematics and Statistics Spring 2015

Abstract

While there is an extensive amount of research demonstrating that the ability to relate one representation of a function to another is necessary for understanding the concept of function, there are few studies on using multiple representations to help high school algebra students relate multiplication of polynomials to multiplication of integers. This study found that college students in an introductory mathematics course tend to use the vertical algorithm for multiplying integers and distribution to multiply polynomials, reporting that these are the methods they were taught or learned in grade school. After a class session of discussing how the distributive property is used in the vertical algorithm, lattice multiplication, and distribution of both integers and polynomials, these same students continued to use distribution to multiply binomials because they had been taught or learned to do so in grade school. However, nearly all of these students transitioned to using the vertical algorithm or lattice multiplication when multiplying polynomials with more than two monomials, citing difficulties with distribution and more organization than distribution when using these methods to solve more challenging problems.

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Introduction

The Common Core State Standards for mathematics have set forth goals for enhancing mathematical practice in the grade school mathematics classroom. Among high school algebra standards students are held to, they must, “Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication” (Common Core State Standards Initiative, 2010). The Common Core Standards for Mathematical Practice also adopt the use of representations for developing mathematical ideas from The Principles and Standards of School Mathematics (National Council of Teachers of Mathematics, 2000), which were formed for the purpose of improving students' experiences with school mathematics. Students are expected to both use and translate between representations when problem solving, which is great, considering that researchers claim that clear recognition of two or more representations of a mathematical idea is required to gain understanding of that idea (Duvall, 1987; Gagatsis & Shiakalli, 2004).

I would like to gain an understanding of whether presenting multiplication of polynomials using the same methods in which integers are multiplied may be beneficial to student understanding. I hope to either improve curriculum or verify that current curriculum on the multiplication of polynomials is sufficient for learning in introductory algebra. Due to time constraints, I was unable to perform research at a local high school and ran my study on the

Exploring Mathematics course I am teaching this semester. This is an introductory college course covering a flexible arrangement of topics for the purpose of helping students build and use quantitative reasoning skills. It therefore seemed appropriate and convenient to discuss relationships between multiple representations for multiplying integers and multiplying polynomials with this class. The research questions I focus on are:

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Which methods for multiplying polynomials do Exploring Mathematics students select and

what reasons do they provide for their choices?

In what ways are students able to relate their chosen method of multiplying polynomials to

multiplication of integers?

Literature Review

Mathematical representations are both the actions involved in communicating an idea and the form used to articulate the concept. It is the role of teachers to guide student experiences and help students describe their mathematical ideas (Von Glasersfeld, 1987). To effectively support student reasoning using representations, teachers must help students recognize intricacies, advantages and limitations of representations, elicit conditions for relating multiple representations to each other and using these representations to construct generalizations, and scaffold student work on representations (Mitchell, Charalambous & Hill, 2014). Teachers can help students relate mathematical ideas to concepts they are already familiar with using visual representations, speech and gesture (Alibali et al., 2014). While there are many factors involved in problem solving, Gagatsis and Shiakalli (2014) concluded that the ability to translate between representations is one factor correlated to problem solving ability. For example, it has also been demonstrated that the ability to relate one representation of a function to another is necessary for understanding the concept of function (Duvall, 1987; Even, 1998; Gagatsis and Shiakalli, 2014).

Being able to translate between representations when multiplying integers may also be helpful to student problem solving when multiplying polynomials. Teachers must have conceptual understanding of why algorithms they teach students work in order to lead students toward computational fluency. After discussing ways in which the vertical multiplication algorithm and lattice multiplication relate to the distributive property when multiplying integers,

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Patricia Nugent's (2007) preservice teachers were able to explain how lattice multiplication could be extended to multiplication of polynomials and how this algorithm relates to the distributive property when multiplying polynomials. Michael O'Neill (2006) also reported that a method similar to this lattice multiplication of polynomials aided students solving difficult polynomial multiplication problems. Both sources cited student difficulties with the FOIL method for multiplying polynomials, as well as its limitations in relationship to distribution and restriction to multiplication of two binomials. I have not discovered research discussing extension of the vertical multiplication algorithm of integers to multiplication of polynomials, though texts such as Messersmith's Intermediate Algebra (2013) present this method.

Theoretical Perspective

To determine what reasons Exploring Mathematics students provide for selecting particular methods for multiplying polynomials, it seems appropriate to analyze student responses through social constructivism. This lens takes into consideration the effects of society and culture on students as well as students' constructions of knowledge (Cobb & Yackel, 1996).

Social constructivism builds on the work of Vygotsky, who described activity as the basis of human cognition through psychological tools such as literacy and number systems (Confrey,

1995). The goal of this paper is to observe how use of multiple representations in classroom discussion may help Exploring Mathematics students relate multiplication of polynomials to multiplication of integers.

While a narrative study might tell the story of the quantitative questions I would like to answer, my research questions might better be answered by a case study for the purpose of gaining further understanding of student practices based on different representations for multiplying polynomials taught. Case studies are used throughout the social sciences, and are

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thought to have been formed from research in anthropology and sociology (Creswell, 2012). One goal of using this approach for this pilot study is to cite lessons learned from this case. A case study is useful for gaining in-depth understanding of reasoning about multiplying polynomials, within the context of a classroom. This is a single instrumental case study focused on student recognition of relationships between multiplication of polynomials and integers within an

Exploring Mathematics classroom at Washington State University.

Methods

The Exploring Mathematics course studied is taught by the author through the Washington

State University Department of Mathematics at the Pullman campus during the Spring 2015 semester. Twenty-eight students from this course participated in this study. Washington State

University (WSU) has 28,686 students enrolled, 19,756 of which are studying at the Pullman campus. The average age of undergraduates at this university is 23 years old, though exactly 50% of the students in this class are at freshman standing. The gender balance for WSU is 49% female and 51% male students. Eighty-three percent of students come from within the state, and international students coming from 90 countries consist of 7% of the student population. On the

Pullman campus, 26% of students are listed as multicultural (http://www.wsu.edu).

This study takes place during three subsequent, one hour class sessions, consisting of a pretest, intervention, and post test respectively. The pretest contains six exercises involving multiplication of integers, four where a scalar integer or variable is multiplied by a binomial, and four in which polynomials with two or three monomials are multiplied. Each exercise asks students to choose a method and multiply two terms, explain why they chose this method, describe other ways they know to multiply these terms, and use one of the alternate methods they know. The integer multiplication exercises are included in the pretest for the purpose of

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determining which methods Exploring Mathematics students choose to use and why. As students are prompted to use more than one method to solve each problem, these problems were strategically chosen to lend themselves to various methods, such as techniques for placing zeros behind a factor multiplied by a power of ten for multiplying 53 * 10 = 530, or regrouping to familiar terms when multiplying 102 * 97 = (100 + 2)(100 - 3).

The second class session, the author lead a class discussion utilizing Mitchell et al.’s

(2014) tasks for effectively teaching multiple representations for multiplying integers and multiplying polynomials. Representations chosen by students on the pretest for multiplying both integers and polynomials were related to the distributive property, and a scaffold to further representations was then built upon the ideas from this discussion. While some of the less frequently used multiplication methods were briefly discussed, the conversation mainly focused on extending distribution, lattice multiplication, and the vertical algorithm for multiplication for integers to polynomials. As extending distribution and lattice multiplication have been discussed elsewhere (O’Neill, 2006; Nugent, 2007), I will outline the discussion on extending the vertical algorithm for multiplying integers to polynomials.

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Example 1 Example 2 Using the distributive property in Using the distributive property in multiplying152 × 13 multiplying(푥2 + 5푥 + 2)(푥 + 3)

152 (푥2 + 5푥 + 2) x 13 x (푥 + 3) 6 2 × 3 150 50 × 3 6 2 ⋅ 3 300 100 × 3 15푥 5푥 ⋅ 3 20 2 × 10 3푥2 푥2 ⋅ 3 500 50 × 10 2푥 2 ⋅ 푥 + 1000 100 × 10 5푥2 5푥 ⋅ 푥 1976 + 푥3 푥2 ⋅ 푥

푥3 + 8푥2 + 17푥 + 6

The vertical algorithm for multiplying two integers begins by multiplying each addend of one factor by every addend of the other factor. In Example 1, each base ten addend of 13 (10 and

3) is multiplied by each base ten addend of 152 (100, 50, and 2). After the addends of both factors have been distributed, the partial products are summed to form a single integer, as the integers are closed under multiplication. Partial products of similar base ten digits such as ones, tens and hundreds are traditionally aligned vertically for the purpose of quickly and accurately combining these values. When multiplying two polynomials using an analogous vertical algorithm, the first step is to multiply each monomial of one polynomial by every monomial of the other polynomial. In Example 2, each monomial of x + 3 (x and 3) is multiplied by each monomial of x^2 + 5x + 2 (x^2, 5x, and 2). Monomials of the same variables and degrees are aligned vertically, in a similar manner to the vertical algorithm for integers, to quickly and accurately sum partial products formed using the distributive property.

The post test consists of four exercises in which two polynomials are multiplied, where the

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number of monomials in at least one of the polynomials increases with each subsequent problem, beginning with multiplying two binomials and building up to multiplying two polynomials containing six and five monomials each. Each exercise again asks students to choose a method and multiply two terms, explain why they chose this method, and use other ways they know to multiply these terms for the purpose of determining reasons students provide for continuing to use methods used on the pretest or choosing new methods after the class discussion. In addition, each exercise asks how the first method a student chooses for multiplying polynomials relates to multiplying integers.

Results

Note that reasons for using a method are not mutually exclusive, as some students provided more than one answer for their choice of method used, such as having learned a specific method and believing it will provide an accurate solution. The same holds for using a second method to multiply, as some students used more than one method to multiply integers, such as lattice multiplication and regrouping of factors into integers they are familiar with multiplying after using the vertical algorithm.

On the pretest, 26 of the students studied used the vertical algorithm as their first choice for multiplying two integers. Most of the explanations these students provided referred to having learned to multiply this way, having been taught this method in grade school, or familiarity with this method for multiplying. The second most common response for choosing the vertical algorithm was that students found it easy to use, with only one student describing, “it’s easier to visualize it.” Only three students told me that they were confident this method would provide an accurate answer, and two others stated that this is the only way that they knew how to multiply.

Three students did not provide an explanation for their choices. Of the other two students in the

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study, one regrouped factors into integers they were familiar with multiplying, such as counting

25 sevens then adding a seven to multiply 26 and 7, because they found, “it easier to multiply

25’s.” The other student appeared to be lost without a calculator, both in their computations and explanations.

Students used a larger variety of methods when asked to perform other ways of solving the same problems. Fifteen students recognized that multiplying 53 and 10 would result in 530, using a rule to, “add a 0 to the end of 53,” on one of the exercises. Eleven students regrouped factors into integers they were familiar with multiplying, especially on the problems with smaller integers. Lattice multiplication was used by seven students to solve integer multiplication problems, and four students drew cells or rows of tally marks to provide visual solutions.

When multiplying polynomials on the pretest, 27 students used distribution or FOIL while one student left these problems and explanations blank. The majority of the students sampled again told me that this was the way in which they had been taught or had learned to multiply polynomials in grade school. Seven students said that distribution was an easy or simple method for multiplying polynomials without telling me why they thought so, and three other students stated that they chose this method because it was the only way they knew how to multiply polynomials. Reasons for using distribution were not provided by seven students. After using the distributive property to multiply, one student attempted to solve for variables in the exercises.

Only six students used a second method to multiply any of the polynomials. Three of these students used the lattice method, and one used the vertical algorithm for multiplying the trinomial and binomial only. Two other students extended their use of regrouping integers to multiplication of a scalar and a binomial, though they did not generalize this to multiplying a variable and a binomial or two polynomials.

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All 28 students used distribution for their first choice when multiplying the first two pairs of polynomials on the post test. The majority again cited that distribution was the way they had learned or been taught to multiply polynomials. Eleven students stated that they chose to use the distributive property because it was easy or simple, with only two explaining that this method made it, “easier to visualize what I’m doing,” and another stating that it kept their work, “more organized.” On these first two problems, eighteen students were able to use an alternate method to solve the same polynomial multiplication problems. Fifteen of these students used the lattice method and the other three used the vertical algorithm.

On the last two exercises of the post test, involving multiplication of two trinomials and two polynomials with six and five monomials each, only three students decided to use the distributive property. Of these three students, one calculated an accurate solution to the last problem using only distribution and another corrected their distribution error after using the lattice method. Eleven students used the vertical algorithm to solve these problems, stating,

“when you can’t FOIL, this is the next best and easiest method,” “easier than foiling and take less time,” and, “it’s more organized.” The lattice method was used by another nine students who described it as, “the easiest way to multiply trinomials,” “more organized than other methods,” and, “way easier than FOIL. With FOIL it would be so messy, if you lost track of where you were, then it would be hard to start over.” Two students who used the lattice method were also able to solve these problems with the vertical algorithm, and one who used the vertical algorithm first used the lattice method as well.

When asked how their chosen method for polynomial multiplication relates to multiplying integers, most students in this study reiterated my question in their response or stated, “I’m not sure,” or, “I used multiplication.” One student told me that the vertical algorithm is, “just like

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standard multiplication.” Another seemed to recall from class that, “If you change variables to numbers (integers) you can multiply the integers.” The last two students mentioned appeared to begin to grasp a relationship between multiplying integer and multiplying polynomials.

Conclusion

On the pretest, it was found that nearly all of the students participating in this study used the vertical algorithm for multiplying integers and distribution to multiply polynomials. The majority of these students stated that they chose these methods because this is how they were taught or learned to multiply in grade school, while other students found these methods easy to use. After a class session of discussing how the distributive property is used in the vertical algorithm, lattice multiplication, and distribution of both integers and polynomials, the same students continued to use distribution to multiply binomials because they had been taught or learned to do so in grade school or found multiplication of polynomials by distribution an easy or organized method to use. However, nearly all of these students transitioned to using the vertical algorithm or lattice multiplication on the post test when multiplying polynomials with more than two monomials, citing difficulties with distribution and more organization than distribution when using these methods to solve more challenging problems.

Only two students were able to relate their chosen method for multiplying polynomials to multiplication of integers, despite discussion the previous class session which made such relations explicit. It appears that while the Exploring Mathematics students studied were able to adapt to solving more challenging polynomial multiplication problems by using multiple representations, they continued to have difficulty relating their methods to multiplication of integers.

For this study, I hypothesized that students would be more likely to report using a method

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for multiplying polynomials that they were able to relate to a method familiar to them for multiplying integers. While I asked open ended questions that gave students the opportunity to explain their choice of method for multiplying polynomials, their responses were on a written assessment. It is quite possible that I could misinterpret students' responses by either determining that a response fits within my hypothesis when a student may not have considered their work to relate to their knowledge of integers, or by missing the opportunity to further question students to gain understanding of whether they might have related multiplying polynomials to multiplying integers when this may not be apparent from their written assessment responses. When asking students why they chose a particular method, it was particularly difficult to interpret student responses that were not particularly well justified, such as, “This method is the easiest for me to understand.” Such responses hopefully have deeper reasoning, which may contribute to or refute my hypothesis.

Initially I had designed a study to determine how high school students are able to relate multiplication of polynomials to multiplication of integers, based on representations used to teach multiplication of polynomials. Due to time constraints, I was unable to perform research at a local high school and ran my study on the Exploring Mathematics course I am teaching this semester. As opposed to high school beginning algebra students, my college students were already familiar with methods for multiplying polynomials that they had been introduced to and practiced in high school. Further study could inquire as to which methods high school algebra students select for multiplying polynomials, why they select these methods, and how performance at solving such problems differs based on method selected. High school algebra students learning to multiply polynomials for the first time using multiple representations may also be able to more clearly relate multiplication of polynomials to multiplication of integers,

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since distribution is commonly taught for multiplying polynomials while the vertical algorithm is more frequently used for multiplying integers.

For a larger study, student interviews could be helpful in better interpreting student meaning. By discussing students' work and decisions with them, either while they are working or as a follow-up to assessment responses, I may have opportunity to better understand the meaning of students' responses through observation and questioning. Interviews may also diversify the data collected for further study.

A larger study would also benefit from a control group for making comparisons. Student responses and trends in these responses could be compared between a class which is taught multiple representations for multiplying polynomials and a class which is taught a single method for doing so, before intervention of learning other representations. Such a comparison may provide more evidence to demonstrate that teaching multiple representations contributes to students' understanding of relationships between multiplying polynomials and multiplying integers, as opposed to an alternate factor.

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REFERENCES

Cobb, P. & Yackel E. (1996). Constructivist, emergent, and sociocultural perspectives in the context of developmental research. In Classics in Mathematics Education Research(pp. 208-226). Reston, VA.

Common Core State Standards Initiative. (2010). Common core state standards for English language arts and literacy in history/social studies, science, and technical subjects. Washington, DC: National Governors Association Center for Best Practices and the Council of Chief State School Officers. Retrieved April 1, 2015, from http://www.corestandards.org/Math/Content/HSA/APR/

Confrey, J. (1995). A theory of intellectual development: Part II. For the Learning of Mathematics, 15(1), 38-48.

Creswell, J. W. (2012). Qualitative inquiry & research design: Choosing among five approaches. Thousand Oaks, CA: Sage Publications, Inc.

Duval, R. (1987). The role of interpretation in the learning of mathematics. Diastasi, 2, 56-74.

Even, R. (1998). Factors involved in linking representations of functions. Journal of Mathematical Behavior, 17(1), 105-121.

Gagatsis, A. & Shiakalli, M. (2004). Ability to translate from one representation of the concept of function to another and mathematical problem solving, Education Psychology: An International Journal of Experimental Educational Psychology, 24(5), 645-657.

Messersmith, S. (2013). Intermediate Algebra with Power Learning. Boston: McGraw-Hill.

Mitchell, R., Charalambous, C.Y., & Hill, H. C. (2014). Examining the task and knowledge demands needed to teach with representations. Journal of Teacher Education, 17, 37-60.

National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA

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Nugent, P. M. (2007). Lattice multiplication in a preservice classroom. Mathematics in the Middle School, 13(2), 110- 113.

O'Neill, M. C. (2006). Multiplying Polynomials. Mathematics Teacher, 99(7), 508-510.

Von Glasersfeld, E. (1987). Learning as a constructive activity. In C. Janvier (Ed.), Problems of representation in the teaching and learning of mathematics. Hillsdale, NJ: Lawrence Erlbaum.

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APPENDIX B: R REPOLR GEE CODE FOR REPRESENTATIONAL FLUENCY install.packages("repolr") library(repolr)

TotalCount <- c(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34, 35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65, 66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15 ,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46, 47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77, 78,79,80,81,82,83,84,85,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27 ,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58, 59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,1,2,3,4,5,6,7, 8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39, 40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70, 71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20 ,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51, 52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82, 83,84,85,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32 ,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63, 64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,1,2,3,4,5,6,7,8,9,10,11,12,13 ,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44, 45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75, 76,77,78,79,80,81,82,83,84,85,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25 ,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56, 57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,1,2,3,4 ,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37, 38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68, 69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18 ,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49, 50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80, 81,82,83,84,85,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30 ,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61, 62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,1,2,3,4,5,6,7,8,9,10,11 ,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42, 43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73, 74,75,76,77,78,79,80,81,82,83,84,85)

118

Class <- c(1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3 ,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 ,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6,6 ,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2 ,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,1,1,1,1 ,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3 ,3,3,3,3,3,3,3,3,3,3,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2 ,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6,6,6,6,6,6,6 ,6,6,6,6,6,6,6,6,6,6,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2 ,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,1,1,1,1,1,1,1,1,1 ,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3 ,3,3,3,3,3,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2 ,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6,6,6,6,6,6,6,6,6,6,6,6 ,6,6,6,6,6,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3 ,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,1,1,1,1,1,1,1,1,1,1,1,1,1,1 ,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3 ,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2 ,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6 ,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3 ,3,3,3,3,3,3,3,3,3,3,3,3,3,3,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6)

Gender <- c(0,1,0,0,1,0,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,0,0,1,0,1,0,0,1,0,1,1,0,1,0,1,0,0,0,0,0,0 ,1,0,1,0,0,1,1,1,0,0,0,0,1,1,0,1,1,1,0,0,1,1,1,1,1,1,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0,1,1,0,1,0 ,0,1,0,1,0,0,1,1,0,1,0,0,0,0,1,0,1,0,0,1,0,1,1,0,1,0,1,0,0,0,0,0,0,1,0,1,0,0,1,1,1,0,0,0,0,1,1,0,1,1,1,0 ,0,1,1,1,1,1,1,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,0,0,1,0,1,0,0 ,1,0,1,1,0,1,0,1,0,0,0,0,0,0,1,0,1,0,0,1,1,1,0,0,0,0,1,1,0,1,1,1,0,0,1,1,1,1,1,1,1,0,0,1,0,1,1,1,0,1,0,0 ,1,0,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,0,0,1,0,1,0,0,1,0,1,1,0,1,0,1,0,0,0,0,0,0,1,0,1,0,0 ,1,1,1,0,0,0,0,1,1,0,1,1,1,0,0,1,1,1,1,1,1,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0 ,0,1,1,0,1,0,0,0,0,1,0,1,0,0,1,0,1,1,0,1,0,1,0,0,0,0,0,0,1,0,1,0,0,1,1,1,0,0,0,0,1,1,0,1,1,1,0,0,1,1,1,1 ,1,1,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,0,0,1,0,1,0,0,1,0,1,1,0 ,1,0,1,0,0,0,0,0,0,1,0,1,0,0,1,1,1,0,0,0,0,1,1,0,1,1,1,0,0,1,1,1,1,1,1,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1 ,0,1,0,0,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,0,0,1,0,1,0,0,1,0,1,1,0,1,0,1,0,0,0,0,0,0,1,0,1,0,0,1,1,1,0,0 ,0,0,1,1,0,1,1,1,0,0,1,1,1,1,1,1,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1 ,0,0,0,0,1,0,1,0,0,1,0,1,1,0,1,0,1,0,0,0,0,0,0,1,0,1,0,0,1,1,1,0,0,0,0,1,1,0,1,1,1,0,0,1,1,1,1,1,1,1,0,0 ,1,0,1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,0,0,1,0,1,0,0,1,0,1,1,0,1,0,1,0,0 ,0,0,0,0,1,0,1,0,0,1,1,1,0,0,0,0,1,1,0,1,1,1,0,0,1,1,1,1,1,1,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0,1 ,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,0,0,1,0,1,0,0,1,0,1,1,0,1,0,1,0,0,0,0,0,0,1,0,1,0,0,1,1,1,0,0,0,0,1,1,0 ,1,1,1,0,0,1,1,1,1,1,1,1,0,0,1,0,1,1,1,0,1,0,0,1,0,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,0,0,1

119

,0,1,0,0,1,0,1,1,0,1,0,1,0,0,0,0,0,0,1,0,1,0,0,1,1,1,0,0,0,0,1,1,0,1,1,1,0,0,1,1,1,1,1,1,1,0,0,1,0,1,1,1 ,0,1,0,0,1,0,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0,1,0,0,1,1,0,1,0,0,0,0,1,0,1,0,0,1,0,1,1,0,1,0,1,0,0,0,0,0,0,1 ,0,1,0,0,1,1,1,0,0,0,0,1,1,0,1,1,1,0,0,1,1,1,1,1,1,1,0,0,1,0,1,1,1)

Grade <- c(10,10,11,11,11,10,10,11,10,11,10,10,9,11,10,11,11,10,10,10,9,10,9,11,10,11,10,10,11,10,11,1 1,10,11,12,10,11,11,11,11,10,11,11,10,10,11,11,10,11,11,10,10,11,10,10,10,11,11,10,11,11,11,1 0,11,12,10,12,11,12,10,11,10,11,10,11,10,10,11,11,10,10,11,10,10,10,10,10,11,11,11,10,10,11,1 0,11,10,10,9,11,10,11,11,10,10,10,9,10,9,11,10,11,10,10,11,10,11,11,10,11,12,10,11,11,11,11,10 ,11,11,10,10,11,11,10,11,11,10,10,11,10,10,10,11,11,10,11,11,11,10,11,12,10,12,11,12,10,11,10, 11,10,11,10,10,11,11,10,10,11,10,10,10,10,10,11,11,11,10,10,11,10,11,10,10,9,11,10,11,11,10,1 0,10,9,10,9,11,10,11,10,10,11,10,11,11,10,11,12,10,11,11,11,11,10,11,11,10,10,11,11,10,11,11,1 0,10,11,10,10,10,11,11,10,11,11,11,10,11,12,10,12,11,12,10,11,10,11,10,11,10,10,11,11,10,10,1 1,10,10,10,10,10,11,11,11,10,10,11,10,11,10,10,9,11,10,11,11,10,10,10,9,10,9,11,10,11,10,10,11 ,10,11,11,10,11,12,10,11,11,11,11,10,11,11,10,10,11,11,10,11,11,10,10,11,10,10,10,11,11,10,11, 11,11,10,11,12,10,12,11,12,10,11,10,11,10,11,10,10,11,11,10,10,11,10,10,10,10,10,11,11,11,10, 10,11,10,11,10,10,9,11,10,11,11,10,10,10,9,10,9,11,10,11,10,10,11,10,11,11,10,11,12,10,11,11,1 1,11,10,11,11,10,10,11,11,10,11,11,10,10,11,10,10,10,11,11,10,11,11,11,10,11,12,10,12,11,12,1 0,11,10,11,10,11,10,10,11,11,10,10,11,10,10,10,10,10,11,11,11,10,10,11,10,11,10,10,9,11,10,11, 11,10,10,10,9,10,9,11,10,11,10,10,11,10,11,11,10,11,12,10,11,11,11,11,10,11,11,10,10,11,11,10, 11,11,10,10,11,10,10,10,11,11,10,11,11,11,10,11,12,10,12,11,12,10,11,10,11,10,11,10,10,11,11, 10,10,11,10,10,10,10,10,11,11,11,10,10,11,10,11,10,10,9,11,10,11,11,10,10,10,9,10,9,11,10,11,1 0,10,11,10,11,11,10,11,12,10,11,11,11,11,10,11,11,10,10,11,11,10,11,11,10,10,11,10,10,10,11,1 1,10,11,11,11,10,11,12,10,12,11,12,10,11,10,11,10,11,10,10,11,11,10,10,11,10,10,10,10,10,11,1 1,11,10,10,11,10,11,10,10,9,11,10,11,11,10,10,10,9,10,9,11,10,11,10,10,11,10,11,11,10,11,12,10 ,11,11,11,11,10,11,11,10,10,11,11,10,11,11,10,10,11,10,10,10,11,11,10,11,11,11,10,11,12,10,12, 11,12,10,11,10,11,10,11,10,10,11,11,10,10,11,10,10,10,10,10,11,11,11,10,10,11,10,11,10,10,9,1 1,10,11,11,10,10,10,9,10,9,11,10,11,10,10,11,10,11,11,10,11,12,10,11,11,11,11,10,11,11,10,10,1 1,11,10,11,11,10,10,11,10,10,10,11,11,10,11,11,11,10,11,12,10,12,11,12,10,11,10,11,10,11,10,1 0,11,11,10,10,11,10,10,10,10,10,11,11,11,10,10,11,10,11,10,10,9,11,10,11,11,10,10,10,9,10,9,11 ,10,11,10,10,11,10,11,11,10,11,12,10,11,11,11,11,10,11,11,10,10,11,11,10,11,11,10,10,11,10,10, 10,11,11,10,11,11,11,10,11,12,10,12,11,12,10,11,10,11,10,11,10,10,11,11,10,10,11,10,10,10,10, 10,11,11,11,10,10,11,10,11,10,10,9,11,10,11,11,10,10,10,9,10,9,11,10,11,10,10,11,10,11,11,10,1 1,12,10,11,11,11,11,10,11,11,10,10,11,11,10,11,11,10,10,11,10,10,10,11,11,10,11,11,11,10,11,1 2,10,12,11,12,10,11,10,11,10,11,10,10,11,11,10,10,11,10,10,10,10,10,11,11,11,10,10,11,10,11,1 0,10,9,11,10,11,11,10,10,10,9,10,9,11,10,11,10,10,11,10,11,11,10,11,12,10,11,11,11,11,10,11,11 ,10,10,11,11,10,11,11,10,10,11,10,10,10,11,11,10,11,11,11,10,11,12,10,12,11,12,10,11,10,11,10, 11,10,10,11,11,10,10,11,10,10,10)

Rep <-

120

c(1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 ,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 ,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 ,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 ,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 ,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 ,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2 ,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2 ,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2 ,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2 ,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2 ,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2 ,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2 ,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3 ,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3 ,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3 ,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3 ,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3 ,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3 ,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3)

Type <- c(1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 ,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2 ,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2 ,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3 ,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4 ,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4 ,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 ,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 ,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2 ,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3 ,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3 ,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4 ,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4 ,4,4,4,4,4,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1 ,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2 ,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2 ,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3 ,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3,3

121

,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4 ,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4)

Response <- c(4,4,2,4,4,4,4,4,4,1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,1,4,4,4,4,4,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4 ,4,1,4,2,4,4,1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,2,4,4,4,4,4,4,1,4,3,4,4,4,4,4,4,4,4,4,3,4,4,1,4,4,4,4,4,4,2,4 ,4,4,4,4,4,4,4,4,4,4,1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,1,2,4,4,1,4,2,4,4,3,4,4,4,4,4,4,4,4,4,4,4,4 ,4,4,4,4,4,4,4,4,4,1,4,4,4,4,4,4,4,3,4,3,4,4,4,4,1,4,4,4,4,3,4,4,4,4,3,4,4,4,4,4,4,4,4,1,4,4,4,4,4,3,4,4 ,4,4,4,4,3,4,4,4,4,4,4,4,4,1,4,1,4,3,4,4,3,4,4,4,3,4,4,4,4,4,4,4,4,4,4,4,4,4,3,4,2,4,1,4,4,4,4,4,4,4,3,3 ,4,3,3,4,4,1,4,4,4,4,3,4,3,3,4,4,4,4,4,4,4,4,4,4,1,4,4,4,3,3,3,3,4,4,3,3,4,3,3,3,4,4,3,4,4,4,1,4,1,3,3,4 ,4,1,4,4,3,4,4,4,4,4,4,4,4,4,4,4,3,4,3,4,4,4,4,1,4,3,4,4,4,4,4,4,4,4,4,4,4,4,1,4,4,4,4,4,4,4,4,4,4,4,4,4 ,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4 ,4,4,4,4,4,4,4,4,4,4,4,4,2,4,4,4,4,4,4,1,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,2,4,4,4,4,4,3,4,4,4,3,4,4,4 ,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,3,4,4,4 ,1,4,4,4,4,4,4,3,4,4,4,4,4,4,4,4,4,4,4,2,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4 ,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,3,4,4,3,4,3,4,4,4,1,4,4,4,4,4,4,4,3,3,4,4,3,4,4,4,3,4,4 ,2,4,4,4,4,4,3,4,4,4,3,3,4,3,4,3,4,3,4,4,4,4,4,4,3,4,4,4,4,3,4,4,4,4,4,4,4,4,4,4,3,4,3,4,4,4,4,4,4,4,3,3 ,4,4,3,4,4,4,4,2,4,3,4,2,2,4,1,4,4,4,4,4,4,2,4,4,4,3,4,2,4,4,4,4,4,4,4,4,4,4,4,3,4,4,4,4,4,4,4,2,1,4,2,1 ,4,1,4,4,4,1,2,4,3,4,4,4,4,4,4,4,4,4,4,4,2,2,2,4,1,4,4,4,4,4,4,4,1,2,2,4,4,4,4,4,4,4,4,2,2,1,2,1,4,4,4,4 ,4,4,1,4,4,4,4,4,2,4,4,4,4,4,3,4,4,4,4,4,3,3,4,4,3,4,4,4,2,4,3,2,1,4,1,4,4,4,1,2,1,4,4,3,4,4,4,4,4,4,4,4 ,4,2,2,2,4,1,4,4,4,4,4,4,4,1,2,2,4,4,4,4,4,4,4,4,1,2,2,4,1,4,4,4,4,4,4,1,4,4,4,4,4,2,3,4,4,4,4,3,4,4,3,4 ,3,3,4,4,4,4,4,4,3,2,1,4,1,1,4,1,4,4,4,1,2,4,4,4,4,4,4,4,4,4,4,4,4,4,2,2,2,3,1,4,4,4,4,4,4,4,1,2,2,4,4,4 ,4,4,3,3,4,2,2,1,4,1,4,4,4,4,4,4,1,4,4,4,4,4,2,4,4,3,4,4,3,4,4,3,4,3,3,2,4,4,1,1,4,3,2,1,4,2,2,4,1,4,4,4 ,1,2,4,4,4,3,4,4,4,4,4,4,4,4,4,2,2,1,4,1,4,4,3,4,4,4,4,1,2,2,4,1,4)

NC = data.frame(TotalCount = factor(TotalCount),

Class = factor(Class), Gender = factor(Gender), Grade = factor(Grade),

Rep = factor(Rep), Type = factor(Type), Response)

mod.c <- repolr(Response ~ Class, data = NC,