high school student representational adaptivity and transfer

HIGH SCHOOL STUDENT REPRESENTATIONAL ADAPTIVITY AND TRANSFER

IN MULTIPLYING POLYNOMIALS

By

CAMERON STARR SWEET

A dissertation submitted in partial fulfillment of

the requirements for the degree of

DOCTOR OF PHILOSOPHY

WASHINGTON STATE UNIVERSITY

Department of Mathematics and Statistics

DECEMBER 2018

© Copyright by CAMERON STARR SWEET, 2018

All Rights Reserved

© Copyright by CAMERON STARR SWEET, 2018

All Rights Reserved

ii

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.

iii

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

iv

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.

v

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

vi

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.

vii

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

viii

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.

ix

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

x

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

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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

xii

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

xiii

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

xiv

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.

1

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).

2

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.

3

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

4

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.

5

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

6

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

7

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?

8

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

9

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

10

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,

11

“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

12

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

13

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

14

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

15

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

16

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

152 and 13

152

x 13

456 152 × 3

+ 1520 152 × 10

1976

Place value multiplication of

(𝑥2 + 5𝑥 + 2) and (𝑥 + 3)

(𝑥2 + 5𝑥 + 2) x (𝑥 + 3) 3𝑥2 + 15𝑥 + 6 (𝑥2 + 5𝑥 + 2) ⋅ 3

+ 𝑥3 + 5𝑥2 + 2𝑥 (𝑥2 + 5𝑥 + 2) ⋅ 𝑥

𝑥3 + 8𝑥2 + 17𝑥 + 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

17

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

18

(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

152 and 13

(100 + 50 + 2) (10 + 3)

= 100(10 + 3) + 50(10 + 3) +2(10 + 3)

= 1000 + 300 + 500 + 150 + 20 + 6

= 1976

Standard distribution multiplication of

(𝑥2 + 5𝑥 + 2) and (𝑥 + 3)

(x2 + 5x + 2)(x + 3) = x2 (x + 3) + 5x (x + 3) + 2 (x + 3) = x3 + 3x2 + 5x2 + 15x + 2x + 6

= 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.

19

Lattice multiplication of

152 and 13

= 1000 + 500 + 20 + 300 + 150 + 6

= 1976

Lattice multiplication of

(𝑥2 + 5𝑥 + 2) and (𝑥 + 3)

= x3 + 5x2 + 2x + 3x2 + 15x + 6

= 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.

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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

21

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

22

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

23

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

24

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

25

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.

26

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

27

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

28

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

29

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

30

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

31

(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.

32

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

33

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

34

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

35

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.

36

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

37

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

38

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

(Polytomous)

Indicator from 1-85 for each student in the study

Class

(Polytomous)

Category for class periods 1, 2, 3, 6

Gender

(Dichotomous)

Male = 0, Female = 1

Grade

(Polytomous)

High school grade level 9, 10, 11, 12

39

Rep

(Polytomous)

Standard distribution = 1, Lattice = 2, Place Value = 3

Type

(Polytomous)

Product of a monomial and a binomial=1, Product of two binomials=2,

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

40

(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.

41

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

42

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

43

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-

44

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

45

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.

46

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

(79%)

58

(6%)

868

(85%)

Inaccurate 89

(9%)

63

(6%)

152

(15%)

All Tasks 899

(88%)

121

(12%)

1020

(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

47

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

Distribution

21

(6.1%)

7

(2.1%)

37

(10.9%)

275

(80.9%)

340

(100%)

Lattice 4

(1.2%)

4

(1.2%)

25

(7.3%)

307

(90.3%)

340

(100%)

Place Value 38

(11.2%)

47

(13.8%)

27

(7.9%)

228

(67.1%)

340

(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

48

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.

49

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.

50

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

51

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.

52

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

53

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

54

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

55

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.

56

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

Polynomial Product Type

Representation 1 2 3 4 All Tasks

Standard

Distribution

59

(69%)

46

(54%)

39

(46%)

35

(41%)

179

(53%)

Lattice 22

(26%)

34

(40%)

38

(45%)

39

(46%)

133

(39%)

Place Value 4

(5%)

5

(6%)

8

(9%)

11

(13%)

28

(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.

57

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

58

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

59

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

60

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

61

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:

62

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.

63

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

(Choice)

5 6 7 8 9 10 11

Eve

(Flexible)

Standard Standard Standard Standard Standard Standard Standard

Ben

(Flexible)

Standard Standard Standard Standard

hybrid PV

Standard

hybrid PV

Standard Standard

hybrid PV

Pablo

(Standard)

Standard Standard Standard Standard Standard Standard Standard

Tom

(Standard)

Standard Standard Standard Synthetic? Standard

hybrid PV

Standard

hybrid PV

Synthetic?

Keren

(Lattice)

Standard Standard

(Lattice)

Lattice Lattice

Rebekah

(Lattice)

Standard Standard Lattice Lattice Lattice Standard Lattice

Sarah

(PV)

PV PV PV PV PV PV PV

John

(PV)

Standard Standard PV PV Standard

hybrid PV

Standard Standard

Caleb

(None)

Standard Standard Standard Lattice Lattice Lattice Lattice

Samuel

(None)

Standard Standard Standard PV Lattice Standard

64

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.

65

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

66

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.

67

Table 4.7: Student Statements for Representational Choices

Justification

Representation Task Subject Context Representation All

Standard

Distribution

6

(10.5%)

16

(28.1%)

12

(21.0%)

23

(40.4%)

57

Lattice 14

(48.3%)

7

(24.1%)

1

(3.5%)

7

(24.1%)

29

Place Value 6

(11.1%)

18

(33.3%)

9

(16.7%)

21

(38.9%)

54

All 26

(18.6%)

41

(29.3%)

22

(15.7%)

51

(36.4%)

140

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.

68

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.

69

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

70

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?

71

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.

72

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.

73

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)

172 John Yeah, it really depends.

173 Interviewer What does it depend on?

174 John Kind of what I'm feeling, I like them both, I think they both do, I think

they're both better than the box. I don't like the box too much.

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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?

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.

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.

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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.

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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)

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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

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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.

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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

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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

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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.

130 Interviewer Does your method for multiplying polynomials here, your first one,

relate to how you multiply integers?

131 Pablo Integers like the first four ones that we did, um, yeah, I mean…

132 Pablo I never stopped to think about it, but say that I was going to do like 12

times 15, you know.

133 Pablo (writes 12 x 15) I don't know if you can do this, but if I separate these

two numbers right here

134 Pablo (writes curves from 1 in 12 to 1 and 5 in 15 and from 2 in 12 to 1 and 5

in 15)

135 Pablo and go 1 times 1 and 1 times 5, and 2 times 1 and 2 times 5,

136 Pablo I don't think it equals the same number 12 times 15 would, that's why I

separate them,

137 Pablo like this (writes 12, next line 15, PV aligned, equal bar below)

138 Pablo you know, so I can see what, like I don't even know. Does it?

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.

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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

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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.

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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.

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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)

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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.

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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.

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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

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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

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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

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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).

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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

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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

94

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

95

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

96

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).

97

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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

107

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

108

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.

109

Example 1

Using the distributive property in

multiplying152 × 13

152

x 13

6 2 × 3

150 50 × 3

300 100 × 3

20 2 × 10

500 50 × 10

+ 1000 100 × 10

1976

Example 2

Using the distributive property in

multiplying(𝑥2 + 5𝑥 + 2)(𝑥 + 3)

(𝑥2 + 5𝑥 + 2) x (𝑥 + 3)

6 2 ⋅ 3

15𝑥 5𝑥 ⋅ 3

3𝑥2 𝑥2 ⋅ 3

2𝑥 2 ⋅ 𝑥

5𝑥2 5𝑥 ⋅ 𝑥

+ 𝑥3 𝑥2 ⋅ 𝑥

𝑥3 + 8𝑥2 + 17𝑥 + 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

110

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

111

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.

112

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

113

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

114

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,

115

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.

116

REFERENCES

Alibali, M. W., Nathan, M. J., Wolfgram, M. S., Church, R. B., Jacobs, S. A., Martinez, C. J., &

Knuth, E. J. (2014). How teachers link ideas in mathematics instruction using speech and

gesture: A corpus analysis. Cognition and Instruction, 32(1), 65-100.

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.

118

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)

119

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

120

,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 <-

121

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

122

,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4,4

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

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,

categories = 4, subjects = "TotalCount",

times = c(1:12), corr.mod = "uniform", alpha = 0.5)

summary(mod.c)

mod.g <- repolr(Response ~ Gender, data = NC,

123

categories = 4, subjects = "TotalCount",

times = c(1:12), corr.mod = "uniform", alpha = 0.5)

summary(mod.g)

mod.a <- repolr(Response ~ Grade, data = NC,

categories = 4, subjects = "TotalCount",

times = c(1:12), corr.mod = "uniform", alpha = 0.5)

summary(mod.a)

mod.r <- repolr(Response ~ Rep, data = NC,

categories = 4, subjects = "TotalCount",

times = c(1:12), corr.mod = "uniform", alpha = 0.5)

summary(mod.r)

mod.t <- repolr(Response ~ Type, data = NC,

categories = 4, subjects = "TotalCount",

times = c(1:12), corr.mod = "uniform", alpha = 0.5)

summary(mod.t)

mod.rt <- repolr(Response ~ Rep * Type, data = NC,

categories = 4, subjects = "TotalCount",

times = c(1:12), corr.mod = "uniform", alpha = 0.5)

summary(mod.rt)

124

mod.ag <- repolr(Response ~ Gender * Grade, data = NC,

categories = 4, subjects = "TotalCount",

times = c(1:12), corr.mod = "uniform", alpha = 0.5)

summary(mod.ag)

mod.gr <- repolr(Response ~ Gender * Rep, data = NC,

categories = 4, subjects = "TotalCount",

times = c(1:12), corr.mod = "uniform", alpha = 0.5)

summary(mod.gr)

mod.gt <- repolr(Response ~ Gender * Type, data = NC,

categories = 4, subjects = "TotalCount",

times = c(1:12), corr.mod = "uniform", alpha = 0.5)

summary(mod.gt)

mod.ar <- repolr(Response ~ Grade * Rep, data = NC,

categories = 4, subjects = "TotalCount",

times = c(1:12), corr.mod = "uniform", alpha = 0.5)

summary(mod.ar)

mod.at <- repolr(Response ~ Grade * Type, data = NC,

categories = 4, subjects = "TotalCount",

times = c(1:12), corr.mod = "uniform", alpha = 0.5)

summary(mod.at)

125

mod.grt <- repolr(Response ~ Gender * Rep * Type, data = NC,

categories = 4, subjects = "TotalCount",

times = c(1:12), corr.mod = "uniform", alpha = 0.5)

summary(mod.grt)

mod.art <- repolr(Response ~ Grade * Rep * Type, data = NC,

categories = 4, subjects = "TotalCount",

times = c(1:12), corr.mod = "uniform", alpha = 0.5)

summary(mod.art)

mod.agrt <- repolr(Response ~ Gender * Grade * Rep * Type, data = NC,

categories = 4, subjects = "TotalCount",

times = c(1:12), corr.mod = "uniform", alpha = 0.5)

summary(mod.agrt)

mod.acgrt <- repolr(Response ~ Class * Gender * Grade * Rep * Type, data = NC,

categories = 4, subjects = "TotalCount",

times = c(1:12), corr.mod = "uniform", alpha = 0.5)

summary(mod.acgrt)

126

APPENDIX C: IN-CLASS WORK SHEET

Multiple Ways to Multiply

Name:

1.) Examine three ways (lattice, distributive property and vertical system) to multiply 12 and 13,

then use each of these methods to multiply x + 2 and x + 3.

Note: 12 = 10 + 2 x + 2 and x + 3

13 = 10 + 3

Lattice: Lattice:

10 2

100 20 10

30 6 3

= 100 + 20 + 30 + 6

= 156

Distributive Property: Distributive Property:

(10 + 2) (10 + 3)

=10 ⋅ 10 + 10 ⋅ 3 + 2 ⋅ 10 + 2 ⋅ 3

= 100 + 30 + 20 + 6

= 156

Place Value: Place Value:

12

13

36

+12

156

127

2.) Examine three ways (lattice, distributive property and vertical system) to multiply 152 and

13, then use each of these methods to multiply x2 + 5x + 2 and x + 3.

Note: 152 = 100 + 50 + 2 x2 + 5x + 2 and x + 3

13 = 10 + 3

Lattice: Lattice:

100 50 2

1000 500 20 10

300 150 6 3

= 1000 + 500 + 20 + 300 + 150 + 6

= 1976

Distributive Property: Distributive Property:

(100 + 50 + 2) (10 + 3)

=100 ⋅ 10 + 100 ⋅ 3 + 50 ⋅ 10 + 50 ⋅ 3 + 2 ⋅ 10 + 2 ⋅ 3

= 1000 + 300 + 500 + 150 + 20 + 6

= 1976

Place Value: Place Value:

152

13

456

+152

1976

128

3.) Use all three methods to multiply (x + y)2.

Hint: multiply x + y and x + y

Lattice:

Distributive Property:

Place Value:

4.) When multiplying large polynomials, the place value system is usually the best because like

terms are already aligned.

Multiply (x + y)3.

Hint: multiply your product from (3.) by x + y

129

5.) Do changes need to be made when multiplying polynomials that are not in standard form?

Multiply 3𝑥 + 𝑥2 − 5 and 𝑥 + 2

6.) Do changes need to be made when multiplying polynomials with missing terms?

Multiply 𝑥2 + 5𝑥 + 2 and 𝑥2 + 3

7.) Which method would you use to multiply the following?

𝑥5 + 8𝑥4 + 12𝑥3 + 𝑥2 + 7𝑥 + 10 and 5𝑥4 + 4𝑥3 + 3𝑥2 + 𝑥 + 8

130

APPENDIX D: CHOICE ASSESSMENT

Choose-Your-Own-Adventure Quiz

Name:

Grade (Circle One): 9, 10, 11, 12

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.

1.) 2 and x+8 2.) 4x and x + 5

3.) 3x + 3 and x + 7 4.) (x+3)2

5.) x + 4 and x2 + 3x + 5 6.) 2x2 + 3x + 5 and x2 + 2x + 1

131

APPENDIX E: NO-CHOICE ASSESSMENT

Multiplication Quiz

Name:

Grade (Circle One): 9, 10, 11, 12

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

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

Use only the method given to multiply expressions.

Another method may only be used to check work.

1.) Use the distributive property to multiply 5x and x + 3.

5x(x + 3)

2.) Use the distributive property to multiply 3x + 5 and x + 8.

(3x + 5)(x + 8)

3.) Use the distributive property to multiply x + 3 and x2 + 2x + 7.

(x + 3)(x2 + 2x + 7)

4.) Use the distributive property to multiply 3x2 + 2x + 6 and x2 + 5x + 1.

(3x2 + 2x + 6)(x2 + 5x + 1)

132

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

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

Use only the method given to multiply expressions.

Another method may only be used to check work.

5.) Use lattice multiplication to multiply x and x + 2.

x + 2

x

6.) Use lattice multiplication to multiply 2x + 4 and x + 3.

2x + 4

x

3

7.) Use lattice multiplication to multiply x2 + 6x + 4 and x + 5.

x2 + 6x + 4

x

5

8.) Use lattice multiplication to multiply 4x2 + 3x + 5 and x2 + 4x + 2.

4x2 + 3x + 5

x2

4x

2

133

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

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

Use only the method given to multiply expressions.

Another method may only be used to check work.

9.) Use the place value method to multiply 3x and x + 4.

x + 4

3x _

10.) Use the place value method to multiply 4x + 6 and x + 2.

4 x + 6

x + 2

11.) Use the place value method to multiply x2 + 5x + 2 and x + 4.

x2 + 5x + 2

x + 4

12.) Use the place value method to multiply 5x2 + 3x + 2 and x2 + 4x + 4.

5x2 + 3x + 2

x2 + 4x + 4

134

APPENDIX F: INTERVIEW SCRIPT

You will be asked to solve a set of multiplication problems. For each problem, choose one

method and multiply the factors. Think aloud while deciding which method to use and while

showing your work to solve each problem. If you find it challenging to express your reasoning

for a particular choice, please say so and the interview will continue.

1.) 26 and 7

2.) 53 and 10

3.) 311 and 129

4.) 102 and 97

135

You will be asked to solve a set of multiplication problems. For each problem, choose one

method and multiply the factors. Think aloud while deciding which method to use and while

showing your work to solve each problem. If you find it challenging to express your reasoning

for a particular choice, please say so and the interview will continue.

5.) 5 and 𝑥 + 2

6.) square of the quantity of x plus five: (𝑥 + 5)2

7.) 𝑥 + 2 and 𝑥 + 5

8.) 𝑥2 + 5𝑥 + 2 and 𝑥 + 3

9.) 𝑥2 + 3𝑥 + 5 and 𝑥2 + 𝑥 + 2

10.) 𝑥2 + 10 and 𝑥 + 4

11.) 𝑥 + 3𝑥2 + 2 and 3 + 𝑥

136

You will be asked to solve a set of multiplication problems. For each problem, choose one

method and multiply the factors. Think aloud while deciding which method to use and while

showing your work to solve each problem. If you find it challenging to express your reasoning

for a particular choice, please say so and the interview will continue.

Tasks 1-4 prompts:

Why did you choose this method?

If difficulty choosing occurs:

Which choices are you deciding between?

What are benefits of each choice?

What are difficulties or limitations of each choice?

After tasks 1-4 are completed:

Do you know any other methods for multiplying factors in any of these problems?

Use any other methods you know to multiply these factors.

Can you think of any reasons why you chose (last method) last?

137

You will be asked to solve a set of multiplication problems. For each problem, choose one

method and multiply the factors. Think aloud while deciding which method to use and while

showing your work to solve each problem. If you find it challenging to express your reasoning

for a particular choice, please say so and the interview will continue.

Tasks 5-11 prompts:

Why did you choose this method?

If difficulty choosing occurs:

Which choices are you deciding between?

What are benefits of each choice?

What are difficulties or limitations of each choice?

After each task is completed:

Does your chosen method relate to how you multiplied integers?

(If “Yes”)

How does your chosen method relate to how you multiplied integers?

In what ways is your method for multiplying similar to how you multiplied integers?

In what ways is your method for multiplying different from how you multiplied integers?

(If “No)

Could you relate your chosen method to how you multiplied integers?

How did you distribute terms from one polynomial into the other polynomial?

Where did you demonstrate this distribution in your work?

138

Use any other methods you know to multiply these factors.

Does this method relate to how you multiplied integers?

(If “Yes”)

How does this method relate to how you multiplied integers?

In what ways is this method for multiplying similar to how you multiplied integers?

In what ways is this method for multiplying different from how you multiplied integers?

(If “No)

Could you relate this method to how you multiplied integers?

Can you think of any reasons why you chose (last method) last?

139

APPENDIX G: STUDENT INTERVIEW INSTRUCTIONS

You will be asked to solve a set of multiplication problems. For each problem, choose one

method and multiply the factors. Think aloud while deciding which method to use and while

showing your work to solve each problem. If you find it challenging to express your reasoning

for a particular choice, please say so and the interview will continue.

140

APPENDIX H: INTERVIEW WITH EVE

Interviewer: Cameron Sweet

1 Interviewer The first set of factors I have for you are to multiply 26 and 7

2 Eve (writes PV) 26 and 7?

3 Interviewer Correct.

4 Eve Ok. So just do basic (writes 2 below equal bar and carries 4 above 2 in 26)

like that (writes 18 below equal bar in front of 2).

5 Interviewer Why did you choose this method?

6 Eve Cause that's how I learned how to multiply numbers that are big.

7 Interviewer Do you know any other methods for multiplying 26 and 7?

8 Eve Um… no.

9

10 Interviewer Ok. The next factors I have for you are to multiply 53 and 10.

11 Eve (writes PV, then 530 to the right of PV) Ok.

12 Interviewer What did you do?

13 Eve Since it's 10, you just add the zero after the 53.

14 Interviewer Do you know any other methods for multiplying 53 and 10.

15 Eve Yeah, just do it this way (completes PV partial sums and product) and

that's 530.

16 Interviewer Why did you choose one method before or after the other?

17 Eve Cause it's a lot faster (points at initial 530 to right of PV).

18

19 Interviewer The next task I have for you is to multiply 311 and 129.

20 Eve (writes PV) 129?

21 Interviewer Yes.

22 Eve (completes PV for product of 12119)

23 Interviewer Why did you choose this method?

24 Eve Because it's the best way to multiply big numbers.

25 Interviewer Do you know another method for multiplying big numbers?

26 Eve Honestly, no.

27

28 Interviewer Last task for multiplying integers, multiply 102 and 97.

29 Eve (writes PV and completes product of 9894)

30 Interviewer Do you know any other way to multiply 102 and 97?

31 Eve No.

32

33 Interviewer This one is to multiply 5 and x+2.

34 Eve (writes 5(x+2), then 5x+10 below)

35 Interviewer What did you do?

36 Eve I just set it up, and then I did 5 times x and 5 times 2 (motions curve from

5 to each term in second factor).

141

37 Interviewer Why did you set it up that way.

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

my brain went.

39 Interviewer Do you know another method for multiplying 5 and x+2?

40 Eve I do (writes PV) like that (writes 5x+10 below equal bar).

41 Interviewer Do you know another method for multiplying 5 and x+2?

42 Eve Yeah, the box one (writes two boxes with x+2 above and 5 to the right)

and 5x plus 10 (writes 5x in one box and 10 in other).

43 Interviewer Why did you choose that method last?

44 Eve Cause it seems unnecessary with that many.

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).

54

55 Interviewer This task is multiply the square of the quantity x+5.

56 Eve So, (writes (x+5)^2) x plus 5?

57 Interviewer Correct.

58 Eve So (writes (x+5)(x+5) on next line, then x^2+25+10x on next line)

59 Interviewer Where did those values come from?

60 Eve I multiplied like this (points in curve motions to indicate standard dist.)

and I've been doing it for a while, so I kind of know what's going to

happen.

61 Eve And have the x square and these two (motions from 5 to other 5) and since

it's both plus you just add the 5x plus 5x.

62 Interviewer Do you know any other methods for multiplying the quantity of x+5

squared?

63 Eve Yeah, (writes PV, completes partial sums and product x^2+10x+25 with

like terms aligned)

64

65 Interviewer This one is multiply x+2 and x+5.

66 Eve (writes (x+2)(x+5), then x^2 on next line)

67 Interviewer Where did the x square come from?

68 Eve Cause I multiplied x times x (motions from x to other x) like that.

142

69 Eve (writes after x^2) 2x plus 5x plus 10. (writes on next line) x square plus 5x

plus 10.

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.

72 Interviewer Do you know another method for multiplying x+2 and x+5?

73 Eve Yeah (writes PV).

74 Interviewer How did you set this up?

75 Eve Like how I was doing with the normal numbers, you do one (points to x+2

on top) and then the other (points to x+5 below) and then you multiply.

76 Eve So (writes partial sums and product x^2+7x+10 with like terms aligned)

77 Interviewer Why did you choose this method after your first method?

78 Eve Cause it's kind of like the more extra way to go about it, with like these

numbers (points to to standard dist. work) I guess, and the other way was

just easier.

79

80 Interviewer This task is multiply x^2+5x+2 and x+3.

81 Eve (writes ( x^2+5x+2)(x+3)) ok, so (writes on next line) x cubed plus 3x…

82 Interviewer Why the x cubed and 3x?

83 Eve I'm multiplying like this (writes curves from x^2 to x and 3 in second

factor) and then x with that (writes curve from 5x to x).

84 Eve (finishes writing partial sums, then on next line writes x^3+8x^2)

85 Interviewer Where did the 8x^2 come from?

86 Eve Adding 5x^2 and 3x^2. Now I'm just putting together like terms (finishes

writing product x^3+8x^2+17x+6).

87 Interviewer How did you distribute terms from one polynomial into the other

polynomial?

88 Eve I took the first one (points to x^2), and then I just multiplied over (points to

x and 3) like to each of the ones, and then I went on to the next one (points

to 5x) and did the same.

89

90 Interviewer This task is multiply x^2+3x+5 and x^2+x+2.

91 Eve (writes (x^2+3x+5)(x^2+x+2)) ok (writes on next line) x^4 plus x^3 plus

2x^2 (writes slash through x^2 in first factor).

92 Eve (writes) plus 3x^3 plus 3x^2 plus 6x plus (writes slach through 3x in first

factor).

93 Eve (writes) 5x^2 plus 5x plus 10, ok.

94 Interviewer Why did you put the cross through 3x?

95 Eve So that way I would know that I did that one already. (writes on next line

product x^4+4x^3+ then slashes x^3 and 3x^3 on previous line)

96 Interviewer How did you get that 4x^3?

97 Eve I added the like terms (points to x^3 and 3x^3 on previous line)

98 Interviewer Ok.

143

99 Eve (slashes 2x^2, 3x^2 and 5x^2 on previous line, continues writing) 10x^2

plus 11x plus 10.

100 Interviewer Does this method for multiplying polynomials relate to how you multiply

integers?

101 Eve Um, no, it's a different way.

102 Interviewer You told me, not really, earlier, so I was just checking.

103 Eve Yeah.

104 Interviewer I Are you up for doing two more?

105 Eve Sure.

106 Interviewer Thank you.

107

108 Interviewer This task is multiply x^2+10 and x+4.

109 Eve (writes (x^2+10)(x+4)) ok (writes on next line) x^3 plus 4x^2 plus 10x

plus 40.

110 Interviewer Do you know another method for multiplying x^2+10 and x+4?

111 Eve (writes PV with partial sums 4x^2+0+40 and x^3+0+10x, keeping like

terms aligned)

112 Interviewer What are these zeroes here for?

113 Eve They're to like, cause there's no number there, so I can line up what I'm

adding together, to keep it like, better understand it I guess.

114 Eve (writes product x^3+4x^2+10x+40)

115 Interviewer You told me this one doesn't relate to how you multipy integers (points to

standard dist.) does this one relate to how you multiply integers (points to

PV)?

116 Eve Yeah.

117 Interviewer How?

118 Eve It's like the same way, except more and you do it a little differently, but it's

still the same concept.

119 Interviewer How would these zeroes show up if you were multiplying numbers?

120 Eve They wouldn't show up. Since you're just multiplying, they're all like the

same and you don't have to have the zeroes.

121

122 Interviewer This last task is multiply x+3x^2+2 and 3+x.

123 Eve (writes (x+3x^2+2)(3+x) ok (writes on next line) 3x plus x^2 (slashes x in

first factor) plus 9x^2 plus 3x^3 (slashes 3x^2 in first factor) plus 6 plus

2x.

124 Eve (writes on next line) 3x^3 plus 10x^2 (slashes x^2 and 9x^2 on previous

line) plus 5x plus 6.

125 Interviewer Why did you choose this method?

126 Eve Cause it's like what I'm used to doing and the easiest way to do it I guess.

127 Interviewer Do you know another method for multiplying x+3x^2+2 and 3+x?

128 Eve Yeah, do (writes PV with x+3x^2+2 on top and 3+x below)

144

129 Eve (writes x^2+3x^3+2x below equal bar, slashes x in 3+x, writes 6 below

and to right of 2x, +3x below 2x and +9x^2 below x^2, keeping like terms

aligned in partial sums)

130 (class bell rings)

131 Eve (writes 10x^2+3x^3+5x 6 below equation bar and partial sums)

145

146

147

148

149

150

151

152

153

APPENDIX I: INTERVIEW WITH BEN

Interviewer: Cameron Sweet

1 Interviewer The first task I have for you is to multiply 26 and 7.

2 Ben (writes 26*7=) I'm going to try to do the simple way first, so I go (writes

26*2 on next line) 26 times 2, which is 26 plus 26, and it's two (writes 52).

3 Ben Multiply that by two (writes *2) that would equal, essentially times four,

so that's (writes =104) 104, then times two again, equals six (writes *2).

4 Ben So (writes =208) 208, then plus 26 (writes +26=234) 234.

5 Ben (second line is 26*2 = 52*2 = 104*2 = 208+26 = 234)

6 Interviewer Why did you multiply 52 times 2?

7 Ben Because multiplying this by 2 equals 52, and multiplying it again by 2

would make it times 4, then times 2 is times 6, that's the last one (points to

+26)

8 Interviewer Do you know another method for multiplying 26 and 7?

9 Ben Mmm… (writes seven 26's vertically below previous work with + to the

left and equal bar below)

10 Ben So that's… (writes ]12 three times to right of first six 6's in groups of two,

then groups two 12's with ]24, then 24 and last 6 with ]30)

11 Ben (writes 2 below equal bar aligned with units column for 26's above, and 4

above ten's column above)

12 Ben (writes 4[ three times to left of first six 26's in groups of two, then groups

two 4's with 8[, then 8 and last 2 with 10[, ten and 4 with 14[, and 14 and

carried 4 with 18)

13 Ben (writes 18 in front of 2 below equal bar) I fell like I made a mistake

somewhere.

14 Interviewer What do you feel like you did wrong?

15 Ben I think I either messed up here (points to 234) or I messed up here (points

to 182). I think I messed up here (points to 234).

16 Interviewer Why?

17 Ben Because multiplying this by 2 (points to 26*2) would not be the same as

doing this (points to 26*7).

18 Ben It would be more like, cause this times 2 (points to 26*2) times 2, wait… I

don't know why it feels wrong, but it feels like I did something wrong

there.

19 Interviewer I have the statement there that you can tell me you don't know why you are

unsure.

20 Ben I don't know why I'm unsure.

21 Interviewer But you feel better about this one? (points to repeated PV addition)

22 Ben Yes, I feel better about this one than that one (points to 234).

23 Interviewer Can you describe why?

24 Ben Because while doing this one I was able to see (motions up PV addition)

everything, instead of this one's (points to 234) a little bit more, I don't

want to say vague, but that's kind of what it feels like.

154

25 Ben It's easier for me to like see everything written out.

26 Interviewer Why did you choose this method first? (points to 26*7)

27 Ben Because it was the first one that came to mind.

28 Interviewer Do you know another method for multiplying 26 and 7?

29 Ben (writes 7*20=140, on next line 7*6=42 with place values for 140 and 42

aligned vertically, then + in front of 42 and equal bar below)

30 Interviewer Why did you multiply 7 and 20?

31 Ben (writes 182 below equal bar) So what I did was I essentially took these

(points at original 26*7) into the most basic of each digit number, so ten's

and one's.

32 Ben Since 7 times 20 is 140, that would be 7 times 20, and then 7 times 6 is

essentially the same as multiply 7 by 26 if I add the solutions together.

33 Ben So now I feel more confident about this (points to 182 below repeated

addition) than I do this (points to 234) since I got the same answer.

34 Interviewer Do you know any other method for multiplying 7 and 26?

35 Ben Not that I can think of off the top of my head.

36

37 Interviewer The next task is to multiply 53 and 10.

38 Ben (writes 53*10 = 530)

39 Interviewer What did you do?

40 Ben I added the zero. For ten's, it's simpler to see it as adding the zero because

in science right now, we're doing scientific notation, so I am used to seeing

how many zeroes there are. That's how many you add to the end of the

number.

41 Interviewer What science is this?

42 Ben Chemistry.

43 Interviewer Do you know another method for multiplying 53 and 10?

44 Ben (writes 53*2= five times vertically) It's a little longer, but…

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.

49 Ben (writes ]12 twice to right of first four 6's in groups of two then groups two

12's with ]24)

50 Ben (writes 0 below equal bar aligned with units column for 106's above, and 3

above ten's column above)

51 Ben (writes 3 below equal bar aligned with ten's column, then 1 in front of 3

aligned with hundred's column)

52 Interviewer Do you know another method for multiplying 53 and 10?

53 Ben Not that I can think of right now.

54 Interviewer Why did you choose this one last (points to second method)?

55 Ben Because I already knew how to do it this way, which is the simpler way.

This is just another way of doing it that I could think of.

155

56

57 Interviewer This task it to multiply 311 and 129.

58 Ben (writes 311*129=, then on next line 311*100=31,100, on next line

311*20=6,220, on next line 311*9= keeping PV aligned in 311's, in 100,

20 and 9, and partial sums)

59 Ben (writes to right 311 +311 in PV, then 622*2= below equal bar, then 1,244)

60 Interviewer Why did you multiply 622 by 2?

61 Ben To get the 4, so I'm up to 311 times 4. So then, if I double that, it will be

times 8 (writes *2 below 1,244, then 2,488 below equal bar)

62 Ben (writes +311 below 2,488 in PV, then 2,799 below equal bar. Writes 2,799

to right of 311*9)

63 Interviewer Why did you add that last 311?

64 Ben To get to 9, cause right at here was only times 8 (points to 2,488), plus the

311 gets it to be times 9.

65 Ben I'll rewrite that, it's not all lined up (writes partial sums with better

vertically aligned PV below earlier work with + to left)

66 Interviewer Why do you want it to be lined up?

67 Ben It makes doing the addition easier. (writes 40,119 below equal bar)

68 Interviewer Why did you choose this method?

69 Ben Because it was simplest for me to do without having a calculator.

70 Interviewer Do you know another method for multiplying 311 and 129?

71 Ben Um… technically yes, but basically, it's the inverse of just doing this,

which would be (writes) 300 times 129 (next line) 10 times 129 (next line)

1 times 129.

72 Interviewer And you already explained that it would be similar but different to how

you did this, right?

73 Ben Yeah, it would get the same answer as this, it's just these numbers here

(points to partial sums in previous method) would be different.

74 Interviewer Do you want to go through that, or try another multiplication task?

75 Ben Um, I'll do another task.

76

77 Interviewer This one is to multiply 102 and 97.

78 Ben (writes 102*97, next line 100*97=, next line 2*97= keeping PV aligned in

102, 100 and 2, as well as all three 97's)

79 Ben (writes 9,700 to right of 100*97=, then + 194 to right of 2*194 keeping PV

aligned, then equal bar under + 194, then 9,894 under equal bar)

80 Interviewer Why did you choose this method?

81 Ben Same reason for the other times, it's the easiest to do in my head because

it's simple numbers, 100 add two zeroes, times two, add this (points to 97)

to itself.

82 Ben Double check that's 194 (writes to right of work 97, next line +97 with PV

aligned, next line 194) Yep, ok.

83 Interviewer Do you know another method for multiplying 102 and 97?

84 Ben Not that wouldn't be really sloppy.

156

85

86 Interviewer This task it to multiply 5 and x+2.

87 Ben You said to do five times what?

88 Interviewer x plus 2.

89 Ben (writes 5(x+2), next line 5x+10) Do you want me to solve for x?

90 Interviewer I didn't ask you to solve for x.

91 Ben I know, um…

92 Interviewer So, why did you set this up this way when I asked you to multiply 5 and

x+2?

93 Ben Cause I assumed that when you said 5 times x+2 you were assuming, I was

assuming you meant in parentheses.

94 Interviewer Why did you assume that?

95 Ben I actually don't know.

96 Interviewer Do you know another method for multiplying 5 and x+2?

97 Ben It's gonna look basically the same, just the 5 would be on this side (points

to the right of 5(x+2))

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.

107 Interviewer Where was distributive property used here?

108 Ben 5 times x (writes curve from 5 to x) and 5 times 2 (writes curve from 5 to

2)

109

110 Interviewer This one is to multiply the square of the quantity x+5.

111 Ben (writes (x+5)^2) So I can square it (points to (x+5)) right?

112 Interviewer Correct.

113 Ben (writes x^2+25 on next line)

114 Interviewer What did you do to get the x square and 25?

115 Ben One of our rules is that if there is an exponent on the outside, you square

everything on the inside,

116 Ben So x squared would be x squared since there's no term, and x is undefined

at the moment, 5 squared is 25.

117 Interviewer Do you know another method for multiplying x+5 quantity squared?

157

118 Ben (writes (x+5)(x+5) to right of work, writes curves from first x to second x

and 5 and from first 5 to second x and 5)

119 Interviewer So what did you do to set this one up?

120 Ben I set it up in terms of FOIL, which is first, (motions) outer, (motions)

inner, last. And the reason I did this was cause x plus 5 squared is the same

as saying x+5 times x+5.

121 Ben So working it out, it would get (writes x^2 on next line) first, outer which

is (writes +) 5x, then after that it would go inner, which is (writes +5x +25

for x^2+5x+5x+25))

122 Ben (writes on next line x^2+10x+25)

123 Interviewer And where did the 10x come from?

124 Ben That came from this times that (motions from first 5 to second x on first

line) and this times that (motions from first x to second 5)

125 Ben I feel like I forgot something over here (points to x^2+25)

126 Interviewer What do you feel like you forgot

127 Ben (circles 10x) The 10x

128 Interviewer How would you have gotten it in your first method?

129 Ben Seeing… oh, that's what I forgot to do (cross off x^2+25) so a way of

seeing it that I simply do but forgot to do this time was, I do (writes) x^2,

130 Ben and then since it's the same number (points to 5 in (x+5)^2) then I just

automatically just double this (points to to in (x+5)^2)

131 Interviewer Why do you double that?

132 Ben Because essentially what you're doing over here (points to (x+5)(x+5)) is

you're adding it to itself, which is the same as multiplying by 2, and then

add the x to it,

133 Ben (writes +10x after x^2) and then squared (writes +25 after x^2+10x) that

way you don't have to go through this long process (points to

x^2+5x+5x+25)

134 Interviewer Do you know another method for multiplying x+5 quantity squared?

135 Ben Mmm… not that I can think of.

136

137 Interviewer This task is to multiply x+2 and x+5.

138 Ben (writes (x+2)(x+5), then on next line x^2+5x)

139 Interviewer So why did you set this up this way?

140 Ben Similar to the last option I did in the last one, is when you multiply a

binomial by binomial in parenthesis, then setting up in FOIL is an easy

way of doing it.

141 Interviewer Where did the x^2 come from?

142 Ben x times x (writes curve from first x to second x in (x+2)(x+5)) and then the

5x is 5 times x (writes curve from first x to 5),

143 Ben plus 2x (writes curve from 2 to second x) plus 10 (writes curve from 2 to

5) then combine like terms (writes on next line x^2+7x+10)

144 Interviewer Do you know another method for multiplying

145 Ben (writes four boxes, 2-by-2, with x and 2 above and x and 5 to the right)

158

146 Interviewer So what did you do here?

147 Ben I set up a Punnett square, or the box method for multiplying, which is you

multiply column by row (writes x^2 in upper left box)

148 Ben So column x times its row is x times x, column x times 5 row is (writes in

lower left box) 5x, (writes in upper right box) 2x, (writes in lower right

box) 10.

149 Ben Then you go this way (motions diagonally and circles x^2) to determine

like terms (circles 2x and 5x, then circles 10) to add together.

150 Ben So (writes below boxes) x^2+7x+10.

151 Interviewer How did you distribute terms from one polynomial into the other

polynomial in this method?

152 Ben That would be multiplying row by column.

153 Interviewer Do you know another method for multiplying x+2 and x+5?

154 Ben Not on the top of my head.

155 Interviewer Using the second method, does this method relate to how you multiply

integers?

156 Ben Not the second method,… well it does, but I don’t like typically using it.

It's the same as doing this (points to standard dist.),

157 Ben simplifying in the fact that for the first it's FOIL, so first (motions from

first x to second x) is x times x, first is this (points to box with x^2)

158 Ben then second is 5 times x (motions from first x to 5 in standard dist.), that's

the way I normally do this, I just go straight down (motions from x above

boxes down both boxes below)

159 Ben I just go x, then 2 times x (points from 2 above boxes to x to right of

boxes) then 2 times 5 (motions from 2 above boxes to 5 to right of boxes)

160 Ben Which is the same order in which I do FOIL (points back to standard dist.)

161 Interviewer So you were able to relate it back to this method (points to standard dist.),

you told me that it doesn't relate to how you multiply integers?

162 Ben Mmm, it does, but I don't like using this method (points to boxes), I prefer

this (points to standard dist.)

163 Interviewer So it does relate, or it doesn't relate?

164 Ben It does.

165 Interviewer How does it relate to multiplying integers?

166 Ben Well, in this one (points to boxes) again similar to what I said before it's

simple terms, easy for me to multiply together, kind of like I was trying to

do with the other method,

167 Ben which was trying to put it in simple terms.

168 Interviewer And by simple terms, remind me again…

169 Ben Ones that are easy to multiply in my head.

170

171 Interviewer This next task is to multiply x^2+5x+2

172 Ben (writes x^2) Could you say that again please?

173 Interviewer x^2+5x+2 and x+3.

174 Ben (writes +5x+2)*(x+3))

159

175 Interviewer And why did you set this up this way?

176 Ben Quantity (points to (x^2+5x+2)) times quantity (points to x+3)) (writes on

next line x^3+5x^2)

177 Interviewer Where did the x cubed come from?

178 Ben I'm doing distributive property, (writes +2x after x^3+5x^2) so I do x

times everything (points from second x to (x^2+5x+2)) first,

179 Ben and then 3 times everything (points from 3 to (x^2+5x+2)) and then add

like terms.

180 Ben (writes on next line 3x^2+15x+6 with PV aligned for partial sums, then

writes + to left of work and equal bar below)

181 Ben (writes below equal bar x^3+8x^2+17x+6 with PV aligned with partial

sums)

182 Interviewer So, any reason for your placement of your products here? (points to partial

sums and final product)

183 Ben I put them in line with each other so that the cubes were together (motions

to both x^3's) the squares (motions straight through 5x^2, 3x^2, 8x^2)

184 Ben the single x's (motions straight through 2x, 15x, 17x) and then the no x's

(motions straight through both 6's)

185 Interviewer Do you know another method for multiplying these factors?

186 Ben Yes (writes x 3 then boxes below, then x^2, 5x, 2 to the right of boxes.

Writes partial sums in boxes, circles like terms, writes x^3+8x^2+17x+6

below boxes)

187 Interviewer Why did you choose this method after your first method?

188 Ben Because out of the three ways I know how to solve it, these are the two I

remember the most how to solve it, distributive and box method,

189 Ben Place holder method I don't like too much to use, so I don't know it well.

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.

192

193 Interviewer This one is multiply x^2+3x+5 and x^2+x+2.

194 Ben (writes (x^2+3x+5)(x^+2+2)) You said x^2+3x+5 times x^+2+2 ?

195 Interviewer Yes.

196 Ben (writes on next line x^4+3x^3+5x^2, on next line x^3+ below 3x^3)

197 Interviewer Where did that x cubed come from?

198 Ben Do you mean x^4 ?

199 Interviewer This x cubed (points to x^3 in third line)

200 Ben Oh, that came from x^2 plus, times x

201 Interviewer And why did you put it here? (points to x^3 below 3x^3)

202 Ben To put it under the similar power. (points from 3x^3 to x^3)

203 Interviewer Ok.

204 Ben (writes 3x^2+5x with PV aligned, on next line writes 2x^2+6x+10 with PV

aligned with previous two lines, then + to left and equal bar below)

205 Ben (writes x^4+4x^3+10x^2+11x+10 below equal bar with PV aligned with

previous three lines)

160

206

207 Interviewer Are you up for doing two more?

208 Ben Sure.

209 Interviewer Thank you

210

211 Interviewer This one is to multiply x^2+10 and x+4.

212 Ben (writes (x^2+10)(x+4), on next line x^3, writes curves from x^2 to x and

x^2 to 4, then +4x^2, then writes curve from 10 to x, then 10x, then writes

curve from 10 to 4, then +40)

213 Interviewer Why did you choose this method?

214 Ben Because it's the easiest to do?

215 Interviewer Do you know another method for multiplying x^2+10 and x+4 ?

216 Ben (writes x^2 10 then boxes below, then x, 4 to right of boxes. Writes partial

sums in boxes, then below boxes writes x^3+4x^2+10x+40)

217

218 Interviewer Last one here is to multiply x+3x^2+2 and 3+x.

219 Ben (writes (x+3x^2+2)(x+x))

220 Interviewer 3 plus x

221 Ben Oops (writes line through (x+x), then (3+x) to right, then writes

(3x^2+x+2)(x+3) on next line)

222 Interviewer What did you do here?

223 Ben Writing it so that the terms of x are in front in order of descending power.

224 Interviewer Why did you do that?

225 Ben Because it makes it easier to work with (writes 3x^2+x^2+2x on next line,

then on next line 9x^2+3x+6 with PV aligned with previous line, then + to

left and equal bar below)

226 Ben (writes on next line 3x^3+10x^2+5x+6 with PV aligned with previous two

lines)

227 Interviewer Do you know another method for multiplying these factors?

228 Ben (writes box method with 3x^3+10x^2+5x+6 below)

161

162

163

164

165

166

167

168

169

170

171

172

APPENDIX J: INTERVIEW WITH PABLO

Interviewer: Cameron Sweet

1 Interviewer The first set is multiply 26 and 7.

2 Pablo (writes 26, next line 7 directly below 6, PV aligned, equal bar below) 26

and 7.

3 Pablo So I would just try to remember 7 times 6, which is 42 (next line writes

2, 4 above 2 in 26, circles 4)

4 Pablo Right, and then 14, 7 times 2, and then add 4 to 14 (writes 18 left of 2

below equal bar)

5 Pablo That's 182, that's it.

6 Interviewer Why did you choose the method that you did for multiplying 7 and 26.

7 Pablo That was the method that I learned first, and that I used my whole life,

8 Pablo so comes the most natural to me.

9 Interviewer Do you know any other ways for multiplying 7 and 26 ?

10 Pablo Haha, no. That would be the easiest I think.

11

12 Interviewer The next two are multiply 53 and 10.

13 Pablo This one, since it's 10, I would just move one decimal place.

14 Interviewer Move one decimal place? Would you show me please?

15 Pablo Yeah. Was it 52 and 10, right?

16 Interviewer 53 and 10.

17 Pablo 53 and 10 so it should be like this (writes 53 x 10)

18 Pablo and then I would just like count the signs (right of 53 x 10 writes =530

19 Pablo that's what I would do, yeah, so just kind of imagine in my head the

number 53,

20 Pablo move one decimal place to the right, and get the answer (points to 530)

21 Interviewer Do you know any other ways to multiply 53 and 10 ?

22 Pablo I would do this (points to PV 26 and 7) the first method, if I didn't know

this (points to 53 x 10)

23 Pablo quicker way, I would just do the first one.

24

25 Interviewer This is multiply 311 and 129.

26 Pablo (writes 311, next line 129, PV aligned, equal bar below)

27 Pablo So I would just do, like the first method (points to PV 26 and 7)

28 Pablo (next line writes 2799, next line 6220, next line 31100, PV's aligned,

equal bar below)

29 Pablo I would just add them (writes + left of 31100, next line 40119, PV's

aligned) and 40,119

30 Pablo and why I used this, this is probably the only way I know how to do it

when it is big like this.

31 Interviewer You anticipated my question already.

32 Pablo Yep. I did.

33

173

34 Interviewer One more on multiplying numbers, multiply 102 and 97.

35 Pablo (writes 102, next line 97, PV aligned, equal bar below)

36 Pablo So (next line writes 114, next line +9180, PV's aligned, equal bar below)

37 Pablo 4, 9, 2 and 9 (next line writes 9294) hope it's right.

38 Pablo Yeah, I just did this cause it's the only way I know it. It's easiest I think,

I don't know.

39 Pablo It's right (checks work) yeah, it's right.

40

41 Interviewer First multiplication problem here is multiply 5 and x+2.

42 Pablo So 5 and x+2, so I would just (writes x+2) x+2 and I would (finishes

writing (x+2)5) just kinda 5

43 Pablo (left of (x+2)5 writes =5x+10) 5 x plus 10

44 Pablo I don't know how you guys say it, but the, this (writes curved arrows

from 5 to x and 2) 5 x plus 10

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)

49 Pablo So, yeah, that's why I choose to do this one (points ot standard dist.)

50 Interviewer Can you multiply it using the other method as well?

51 Pablo Yes, I can. So I would just go (below equal bar writes 5x+10, PV's

aligned with factors) that's it.

52 Pablo Yeah, I can use both, I just prefer to use this one (points to standard

dist.)

53 Pablo I think when a number's really, really huge, I use this one (points to PV)

I think so, I don't remember

54 Interviewer Did your first method relate to how your multiply numbers?

55 Pablo Does it relate? Like, how did I think in my head?

56 Interviewer Does it relate to how you multiply numbers?

57 Pablo Yeah.

58 Interviewer How?

59 Pablo I don't know.

60 Interviewer you don't know. Ok.

61 Pablo I'm just trying to answer the question, what do you mean like does it

relate,

62 Pablo like the way I do this way (points to standard dist.) right here,

63 Pablo does it relate to the way I do this way right here (points to PV 311 and

129)

64 Interviewer Yes.

65 Pablo Um, yeah, it's kinda weird. I don't know why I did this way here (points

to (x+2)5)

174

66 Pablo It's just easier, it just comes, like naturally, and I don't know.

67 Pablo In my head I just see 5 times 2 and them 5 times x, it just happens, I

guess.

68 Interviewer How about your second method? Does it relate to how you multiply

numbers?

69 Pablo No, when you have like x and y's, I barely, like never use this method

(points to PV x+2 and 5)

70 Pablo I don't remember ever using this (points to PV) I just use this one (points

to (x+2)5)

71 Pablo Just kind of multiply one number, and then the next number, and then

the next number.

72

73 Interviewer This one is multiply the square of the quantity x+5)

74 Pablo So x plus 5 (writes (x+5)) and then square? (writes exponent 2)

75 Interviewer Correct.

76 Pablo Ok, so here I would go, so I would probably put this into two (next line

writes (x+5)(x+5))

77 Pablo It's just easier for me to work with. And then I would do the same thing

78 Pablo (writes curves from first x to second x and 5 and from first 5 to second x

and 5)

79 Pablo And then you get (next line writes x^2+5x+5x+25) right, yeah,

80 Pablo and then (next line writes x^2+10x+25) yep, that would be my first

instinct.

81 Pablo It's been quite a while, but if it was that week of Thanksgiving, I would

just probably know the answer

82 Pablo because I just squared the x, add the first number to each other (points to

5 in (x+5)^2)

83 Pablo and then I squared the number again, I just didn't remember right now.

84 Pablo And why'd I do this, is because I know I'm never going to get it wrong.

85 Pablo It doesn't matter how big the number is (points to 2 in (x+5)^2)

86 Pablo like cubic or by 4, I just know if I put all of them (points to (x+5)(x+5))

87 Pablo and just kinda go one by one like this, I'm never gonna get it wrong.

That's why I do it.

88 Interviewer Do you know any other method to multiply these?

89 Pablo You have the kind of method like this, right? (writes x+5, next line x+5,

PV aligned, equal bar below)

90 Pablo And then you have this method like this too, you go like this

91 Pablo (writes 2 by 2 boxes, x+5 above, x and +5 each right of boxes)

92 Pablo I don't remember the names now of them, I know we learned them,

yeah.

93 Pablo This one here would be x squared (writes x^2 in upper left box, writes

partial sums in boxes)

94 Pablo And then, you go diagonally, so you're gonna go (circles box with x^2)

95 Pablo (then circles boxes with +5x, writes below boxes x^2+10x+25)

175

96 Pablo and then you just, I don't like using this method.

97 Pablo And this one here's the same thing (points to PV, writes 5x+25, PV's

aligned below factors)

98 Pablo (next line writes x^2+25 0, equal bar below, PV's aligned)

99 Pablo (next line x^2+10x+25, factors, partial sums and product PV aligned)

100 Pablo Yeah, I just like this one better (points to standard dist.)

101 Pablo It's just more compact, it doesn't have to go like vertically (points to PV)

102

103 Interviewer Ok, multiply x+2 and x+5.

104 Pablo (writes (x+2)(x+5)) x plus 2, x plus 5. So again I will do the same thing,

go like this

105 Pablo (writes curves from first x to second x and 5, writes curves from 2 to

second x and 5)

106 Pablo multiply those numbers (next line writes x^2+5x+2x+10) and then just

add them

107 Pablo (next line writes x^2+7x+10) and yes, I can do the other methods too,

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.

112

113 Interviewer This one's multiply x^2+5x+2 and x+3.

114 Pablo Ok (writes (x^2+5x+2)(x+3)) so I'll do the same thing and I'll go like

this, so it would be

115 Pablo (next line writes x^3+3x^2+5x^2+15x+2x+6)

116 Pablo I would just add the like terms here now (next line writes

x^3+8x^2+17x+6)

117 Pablo And yes, I can do the other ways too. Would you like me to show them?

118 Interviewer If you would like to.

119 Pablo I mean, I can do them, it's just not as… sometimes I can miss them, you

know,

120 Pablo When I'm doing stuff fast, just homework or…

121 Pablo (writes x^2+5x+2, next line x+3, PV aligned, equal bar below)

122 Pablo (next line writes 3x^2+15x+6, next line x^3+5x^2+2x 0, PV aligned,

equal bar below)

123 Pablo (next line writes x^3+8x^2+17x+6)

124 Pablo And the other one was, the way how you guys call that thing, I forgot.

125 Pablo (writes 2 by 3 boxes, x^2+5x+2 above boxes, x and +3 right of boxes)

126 Pablo (writes partial sums in boxes) and then I would go diagonally again

127 Pablo (circles box with x^3, circles boxes with +5x^2 and 3x^2, circles boxes

with +2x and 15x)

128 Pablo (below boxes writes x^3+8x^2+17x+6)

129 Pablo Yep, and I would still choose this one (points to (x^2+5x+2)(x+3))

176

130 Interviewer Does your method for multiplying polynomials here, your first one,

relate to how you multiply integers?

131 Pablo Integers like the first four ones that we did, um, yeah, I mean…

132 Pablo I never stopped to think about it, but say that I was going to do like 12

times 15, you know.

133 Pablo (writes 12 x 15) I don't know if you can do this, but if I separate these

two numbers right here

134 Pablo (writes curves from 1 in 12 to 1 and 5 in 15 and from 2 in 12 to 1 and 5

in 15)

135 Pablo and go 1 times 1 and 1 times 5, and 2 times 1 and 2 times 5,

136 Pablo I don't think it equals the same number 12 times 15 would, that's why I

separate them,

137 Pablo like this (writes 12, next line 15, PV aligned, equal bar below)

138 Pablo you know, so I can see what, like I don't even know. Does it?

139 Pablo If I do 1 times 1 and 1 times 5, and 2 times 1 and 2 times 5, and add

them,

140 Pablo would I have the same number? I have no idea.

141 Interviewer Do you want to try it and see?

142 Pablo Sure, I want to try it. So 1 times 1 would be…

143 Pablo (continues writing 12 x 15 = 1+5+2+10)

144 Pablo six, seven, eight,… uh, yeah, it's not going to be the right answer, will

it?

145 Interviewer What would the right answer be?

146 Pablo (below PV 12, next line 15, writes 60, next line 120, equal bar below,

next line 180, PV's aligned)

147 Pablo 180, right, yes that would be the answer.

148 Pablo Maybe, yeah, this is just not right (points to 12 x 15 = 1+5+2+10)

149 Pablo I just, I never used this, I just don't know.

150 Pablo This here (points to (x^2+5x+2)(x+3)) just makes more sense because of

the variables

151 Pablo cause don't have any variables (points to 12 x 15) they have like

separations here,

152 Pablo in my mind, just small numbers, you know.

153 Pablo I do here (points to PV 12 and 15) because 5 is a smaller number than

15 to multiplying

154 Pablo and 2 is a smaller number than multiplying 12, so when I multiply 5 and

2

155 Pablo is easier than multiplying the whole number, if that makes sense,

hopefully, yeah.

156 Interviewer Does your second method here relate to how you multiply whole

numbers?

157 Pablo Uh, yeah, but, since you know how here I wanted them (points to PV 12

and 15)

158 Pablo these numbers to be super small so I could multiply them easier?

177

159 Pablo Here they're already like as small as they can be, like in my mind at

least.

160 Pablo They're already separated and kind of like beautifully good, you know.

And since they're already,

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.

168 Pablo and I said I don't use this way (points to PV)

169 Pablo because I don't have to compact it more to make it a smaller number to

multiply,

170 Pablo because it's already compacted and it's already smaller,

171 Pablo so I don't need to use this number

172 Pablo because it's not like I'm multiplying a medium times two mediums here,

173 Pablo It's just x^2+5x+2, so yeah. That’s my reason.

174 Interviewer Are you ready to try another one?

175 Pablo Yes, sir.

176

177 Interviewer This is x^2+3x+5 multiplied by x^2+x+2.

178 Pablo (writes (x^2+3x+5)(x^2+x+2)) Ok, I'll do the same thing, go one-by-

one, until I can,…

179 Pablo (next line writes x^4+x^3+2x^2+3x^3+3x^2+6x+5x^2+5x+10)

180 Pablo And then try and find like term but this one's big, so (next line writes

x^4+)

181 Pablo I would kind of cross it out, so (writes slash through x^3 and 3x^3 in

previous line)

182 Pablo This is 4 x cubed (continues writing x^4+4x^3+)

183 Pablo And then would be 2, 5, 10, equals 10 x squared

184 Pablo (writes slash through 2x^2, 3x^2 and 5x^2 in previous line, continues

writing x^4+4x^3+10x^2+)

185 Pablo And then 6 and 5 equals 11x, plus 10 (finishes writing

x^4+4x^3+10x^2+11x+10)

186 Pablo The same thing, it's just they are ready, it's not like it's a huge number

and another huge number,

187 Pablo they're already like separated, kinda like it was in that other method, you

know, and I just do it.

188 Interviewer Do you know other methods for multiplying these?

189 Pablo Yes, the same one's that I, should I do them to, or is this fine?

178

190 Interviewer If you want to. I have seen them already.

191 Pablo Yeah, it's fine, I'm not going to do them, it's kind of big.

192

193 Interviewer This is x^2+10 multiplied by x+4.

194 Pablo (writes (x^2+10)(x+4)) Ok, same way, so

195 Pablo (next line writes x^3+4x^3+10x+40) Oh, that's already…

196 Interviewer That's already what?

197 Pablo That's already done.

198 Interviewer What do you mean by done?

199 Pablo I mean there's no like terms to add, which keep the equation going.

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.

204 Interviewer Anything else you want to do with that one, or are you ready for another

one?

205 Pablo I'm ready for the next one.

206

207 Interviewer This one is multiply x+3x^2+2 and 3+x.

208 Pablo (writes (x+3x^2+2)(3+x)) So here, what I would do, I would just put,

since they're highest,

209 Pablo I don't know how to say it, they're highest x is not in the right place,

210 Pablo like I don't think there's a right place where they're supposed to be,

211 Pablo but in my mind I just have to put them in front,

212 Pablo instead of x+3x^2+2 and 3+x I would just put, I would change them,

213 Pablo put (next line writes (3x^2+x+2)(x+3)) instead of 3 plus x.

214 Pablo I don't think it matters in the end result since they're all scrambled when

I add the like terms.

215 Pablo And then I would just go again (next line writes

3x^3+9x^2+x^2+3x+2x+6)

216 Pablo Then I'll add the like terms (next line writes 3x^3+10x^2+5x+6) yep.

217 Pablo I don't think it would matter, having this format, like all kind of

scrambles things.

218 Pablo When it's here (points to 3x^3+9x^2+x^2+3x+2x+6) they're usually not

all together.

219 Pablo This one was just luck. I would just have to do this in my mind.

220 Pablo I guess it goes back to the other one, that you have to be like this, you

know, I guess.

221 Interviewer So you're talking about the other one.

222 Interviewer Could you show me how you would multiply this using the other one

that you're describing?

179

223 Pablo Yes. So should I do this way (points to (x+3x^2+2)(3+x)) or this way

(points to (3x^2+x+2)(x+3))?

224 Pablo I would do this way (points to (3x^2+x+2)(x+3)) I would never do this

way (points to (x+3x^2+2)(3+x))

225 Interviewer You would reorder it first and then you would do your other method?

226 Pablo Yes. I always reorder it. You know. So I would go

227 Pablo (writes 3x^2+x+2, next line x+3, PV aligned, equal bar below)

228 Pablo I would reorder it specially, I mean, I don't know, the way some

teachers, you know are like,

229 Pablo when you bring the answer, always put the highest number or highest x

or y on the left side.

230 Pablo That's why I reorder it, it just makes more sense in my mind. And then I

would just do it again.

231 Pablo (next line writes 9x^2 3x 6, next line 3x^3+x^2 2x 0, PV's aligned,

equal bar below)

232 Pablo (next line writes 3x^3+10x^2+5x+6, PV's for factors, partial sums and

product PV aligned)

233 Pablo Yep. They're the same, I was just checking.

234 Interviewer Anything else you want to do or ask me about with this problem?

235 Pablo No, I mean…

236 Interviewer Ok. That's all the tasks I have for you.

180

181

182

183

184

185

186

187

188

APPENDIX K: INTERVIEW WITH TOM

Interviewer: Cameron Sweet

1 Interviewer The firs task is to multiply 26 and 7.

2 Tom (writes 26 x 7) So you do this way instead (on next line writes "x (7)"

with 7 below 6, then equal bar below)

3 Interviewer Why are you doing that way?

4 Tom It's easier to add the second digit, the ten's (next line writes 6 below 7, 3

above 2 in 26) um, 14 plus 3, 17 (writes 17 left of 6) that's 176.

5 Interviewer Do you know another method for multiplying 26 and 7?

6 Tom I don't. I've never been taught. Well actually I know a really, really weird

method where you do like lines this way (motions left to right)

7 Tom then lines this way (motions up to down) then count the dots, but rarely

ever taught.

8 Interviewer Can you show me? If you don't want to that's fine.

9 Tom I don't really remember. I did it in third grade like twice. So no. But do

you know what I'm talking about? It's very rarely used.

10 Interviewer I don't know that I do, so I was hoping that you would show me.

11

12 Interviewer The next task is to multiply 53 and 10.

13 Tom 53 times (writes 530)

14 Interviewer What did you do? I see a number there.

15 Tom When you multiply by ten, you just need to add a zero at the end. There's

really, right, am I wrong here?

16 Interviewer You said when you multiply by ten you add a zero, right?

17 Tom Yeah. If you multiply by 100 it would be 5300.

18 Interviewer Do you know another way to multiply 53 and 10?

19 Tom No.

20 Interviewer No, ok.

21 Tom This is a study on how to show like, you're trying to figure out if knowing

multiple methods is helping or know if one method is helping, right?

That's kind of what this is for?

22 Interviewer I'm looking at how you multiply polynomials and how you are

multiplying integers. We'll see what comes out of it.

23

24 Interviewer The next task is multiply 311 and 129.

25 Tom (writes PV 311 above 129 with PV's aligned) 129

26 Interviewer Why did you set these up this way?

27 Tom Because it's the easiest way I know how to do this,

28 Tom and even though it's going to be really difficult because there's multiple of

them, it's still the only way that I've been taught, so that's what I have to

do.

29 Tom I don't even know if I remember how to do that. I guess, let's see if I

remember how to do this.

189

30 Tom (writes 2799 below equal bar, next line 6220, next line 31100 with PV's

aligned for partial sums and factors, equal bar below partial sums, next

line 40,119)

31

32 Interviewer The last task for multiplying integers is to multiply 102 and 97.

33 Tom (writes PV 102 above 97 with PV's aligned) I put the bigger on at the top

so I won't have to do as much addition.

34 Tom (writes 714 below equal bar, next line +9180 with PV's aligned for partial

sums and factors, equal bar below partial sums, next line 9,894, circles

9,894)

35 Interviewer Do you know any other ways to multiply 102 and 97?

36 Tom Um,… yes and no. It wouldn't be fun, but you could do 9,700 by 2, or no

you couldn't. It would be 9,700 plus 97 times 2.

37 Tom cause you could take…

38 Interviewer Could you show me what you're doing?

39 Tom (right of work writes 97x100, next line writes 97x2) And then you would

take both of them so

40 Tom (right of 97x100 writes =9700) um, geez, a number (right of 97x2 writes

194 with only partial sums PV's aligned) I don't know

41 Tom I don't know how to do it, but I know that there's a way that you can

separate the 100 and 2.

42 Interviewer What would you do with each of those?

43 Tom You would add them (writes + left of 194, equal bar below, next line

9,894, circles 9,894)

44 Interviewer Why did you choose this method after your earlier method? you already

told me why you chose your first method.

45 Tom I would never choose this method to be honest with you, unless it was just

an easy number like 100 or 10.

46 Tom Just cause you asked me to show it to you is why I showed it to you.

47 Interviewer Well I appreciate it.

48 Tom It's a pretty crappy method to be honest with you.

49

50 Interviewer This task is to multiply 5 and x+2. Again, please talk through your work

as you are doing it.

51 Tom (writes 5) 5 times x… ?

52 Interviewer x plus 2.

53 Tom (finishes writing 5 x x+2) So like not like that, I prefer this (writes

parentheses over first x for (5)(x+2))

54 Interviewer Why do you prefer that?

55 Tom I don't know. It's because usually when you have a times the dot (points to

" )( " over x) makes it look like a decimal when I see it,

56 Tom and the x, like the multiplication symbol, reminds me of an x,

57 Tom and so I hate that they decided to make both an x and a dot also other

things in math because it just gets really confusing.

190

58 Tom Even on tests sometimes I'll see the multiplication symbol and I'll go why

is there an x there?

59 Tom So it's (on next line writes) 5x+10 cause you take this and multiply it

there (writes curved arrows from 5 to x and 2)

60 Interviewer Does your chosen method relate to how you multiply integers?

61 Tom Come again?

62 Interviewer Does your method you used to multiply these polynomials relate to how

you multiplied integers?

63 Tom Not really. I guess this kind of goes in the opposite direction (points to

integer multiplication work)

64 Tom But it's still basically the same thing because you're multiplying one

number by two different numbers (points from 5 to x+2)

65 Tom Just at a different angle. I wonder if you could do that (right of work

writes PV with PV's aligned)

66 Tom But why would anyone do that? (crosses off PV) This is so much easier

(points to standard dist.)

67 Interviewer Why is it easier?

68 Tom Because when you do it like this (points to PV) you don't put in the plus

sign initially.

69 Tom Your brain just doesn't think to do that cause you're so used to not putting

in a plus sign for that type of multiplying up there (points to integer

multiplication work)

70 Tom And so you just are like, oh it's just five x ten. Yeah, it wouldn't be nice.

71 Interviewer Do you know any other operation for, any other methods for multiplying

5 and x+2?

72 Tom Nope.

73

74 Interviewer This task is to multiply the square of the quantity x+5.

75 Tom (writes (x+5)^2) So you square them both, so you just get (next line

writes x^2+5^2) x squared plus 5 squared.

76 Tom I usually don't show step, but gee I will, so I'll just take x square cause

you can't solve it and 25 cause that's squared (next line writes x^2+25)

77 Interviewer Why do you usually choose this method?

78 Tom It's the easiest and I don't know any other method. So I guess it could not

be the easiest but I wouldn't know. I can't imagine an easier way of doing

it.

79 Interviewer You said you don't know another method?

80 Tom No. I actually prefer not to solve that equation because I don't see why

anyone would want to.

81 Tom This (points to (x+5)^2) almost looks simpler than this (points to x^2+25)

82

83 Interviewer This task is to multiply x+2 and x+5.

84 Tom (writes (x+2)(x+5)) Let's see if I remember how to do this. I still do the

same thing I did in the other one because that's what I'm used to.

191

85 Tom (writes curved arrows from first x to second x and 5) I know there's

another way to do it,

86 Tom but I just forget it because it's hard to remember multiple ways of doing

things to be honest with you.

87 Tom (next line writes x^2+5x) I always mix those up, so luckily I didn't do that

this time, I usually write x five.

88 Tom And then you take (writes +) 2x+10 and add your (points from 5x to 2x)

things that are the same.

89 Tom (next line writes x^2+7x+10) Uh-oh, there are many math terms.

90 Interviewer Does your chosen method relate to how you multiply integers?

91 Tom I don't know, does it? I wouldn't say so, but that's mostly because of how

they're positioned.

92 Tom Yet again, I'm sure there is a way of how you could solve it by putting

them on top of each other and multiplying. It just wouldn't be nice.

93 Interviewer If it's not nice, would you be willing to show me how you think that

might go?

94 Tom How I think that might go? This is not going to go well. (right of work

writes x+2, next line "x x+5" with place values aligned and equal bar

below)

95 Tom And then multiplying them would be (below equal bar writes 5x+10, PV's

aligned with factors)

96 Tom And I would do it over here (next line writes 2x below 5x) just to keep

everything in line (writes +0 right of 2x)

97 Tom And (writes x^2 left of 2x, equal bar below partial sums) and I would add

those guys together (writes + left and right of x^2)

98 Tom (next line writes x^2+7x+10)

99 Interviewer And you said that wasn't nice?

100 Tom No, it's not nice, I, this has so many multiple steps (points at rows of

factors, partial sums and product in PV)

101 Tom whereas this one (points at rows of standard dist. work) you can just do

three,

102 Tom and I feel like it's just a lot more work to get done, and this is what we're

use to (points to standard dist.) when multiplying things in parentheses.

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.

109 Interviewer Do you know any other methods for multiplying x+2 and x+5?

192

110 Tom No. Maybe if I sat here long enough I could figure out another way to do

it, but it might only work for this equation.

111

112 Interviewer This task is to multiply x^2+5x+2…

113 Tom (writes x^2+5x+2)

114 Interviewer and x+3

115 Tom (writes parentheses around x^2+5x+2) so multiplying that times x+3?

116 Interviewer Correct.

117 Tom (finishes writing (x^2+5x+2)(x+3))

118 Tom This is far easier than any other version I've ever seen and I just recently

was taught it. (next line writes 3 1 5 2, then equal bar well below)

119 Tom (writes arrow pointing down below 1, writes 1 below equal bar, writes 3

below 5, 8 below equal bar)

120 Tom 8 times 3 is 24 (writes 24 below 2, 26 below equal bar)

121 Tom (first line: 3 1 5 2; second line: down arrow 3 24; third line blow equal

bar: 1 8 26)

122 Interviewer So what are you doing?

123 Tom I don't know what the method's called, because they give weird names to

all the methods, I think it's called like synthetic.

124 Tom I'm pretty much taking the number for each one of these (points at

coefficients in (x^2+5x+2)(x+3)) in the polynomial

125 Tom this one doesn't have an x (points to 2) but before x you just have to infer

that this is a one (writes 1 left of x^2)

126 Tom and I bring them all down (points at synthetic division? Coefficients) drop

the first one (motions from 1 to down arrow to 1 below equal bar)

127 Tom I don't know how it works, but it does every time. And then you would

take this (points to 1 below equal bar)

128 Tom and you take one less than the original exponent (circles 2 in x^2) so it's

just one x, or x (writes x below 1 below equal bar)

129 Tom plus 8 and then plus (writes +8+ right of x) 26 over this initial equation

(circles (x+3)) that you just took the…

130 Tom Oh, is this supposed to be negative 3? (points to 3 in first line of

synthetics division?) Rats! (writes - left of 3) negative 3.

131 Tom You know what I was talking about though. (writes - left of 3 in second

line of synthetic division?)

132 Tom Um (scratches 8 below equal bar, writes 2 below) I think it's 2, negative 3

(scratches 24, writes -6 right) negative 6

133 Tom (scratches 26) this is going to be negative 4 (writes -4 below)

134 Tom So this is going to be (scratches x+8+, writes below x+2+) x minus, or

plus 2,

135 Tom and then you're going to take your negative 4 and put it over the expency

(points to (x+3), right of x+8+ writes -4 with fraction bar below, x+3

below)

136 Tom (writes box around x+2+ -4 / x+3) and that's how I would do it.

193

137 Interviewer And why did you choose this method?

138 Tom I find it the easiest, it just has the simplest way of doing multiplication,

139 Tom and you're basically just adding which is what you first learn so it's

cemented in my head

140 Tom and you get rid of all the x's (points to (x^2+5x+2)(x+3) so you don't have

to worry about putting in the wrong exponents,

141 Tom and you just bring them back at the very end, which is nice.

142 Interviewer Do you know of any other method for multiplying these factors?

143 Tom Yes, I do. You can take this (points to (x+3)) and I prefer putting it in the

front when I do this method because I don't know why.

144 Tom (below work writes (x+3)(1x^2+5x+2)) and I kept the 1 by accident

because I wrote it up there.

145 Tom And I would take the x and multiply it by each one of these (writes

curved arrows from x to 1x^2, 5x and 2) first

146 Tom and then I would do the 3 (writes curved arrows from 3 to 1x^2, 5x, 2)

and then you would add all of that together

147 Tom so it would be like, I don't know if you want me to show all of this to you

but here you go

148 Interviewer Thank you.

149 Tom (next line writes x^3+5x^2+2x) and then you would take the other ones,

so you have (next line writes 3x^2 below 5x^2, equal bar below)

150 Interviewer Why did you put 3 x squared there?

151 Tom I'm going to add them together after I'm done and I'm just trying to keep

everything aligned (writes + left of 3x^2)

152 Tom (right of 3x^2 writes +15x+6) and then (below equal bar writes

x^3+8x^2+17x+6)

153 Interviewer You added them all together, ok. Why did you pick this second method

after your first?

154 Tom I have different answers for each of them, that's not good. I know there's a

third way of doing it, and I've never understood the third way and so that

third way sucks.

155 Interviewer But why did you pick this after your first method?

156 Tom Yet again, this just required a lot of multiplication (points to standard dist.

arrows) just constant multiplication with the x's involved,

157 Tom and I suck at keeping my exponents in the order that they need to be,

158 Tom and I feel like this answer (points to x+2+ -4 / x+3) is more right than this

one is (points to x^3+8x^2+17x+6)

159 Interviewer Which one's more right?

160 Tom this one (points to x+2+ -4 / x+3). I could be wrong, it's a bit complex

right here (points to -4 / x+3) but that's fine.

161

162 Interviewer This task is to multiply x^2+3x+5 and x^2+x+2.

194

163 Tom (writes (x^2+3x+5)(x^2+x+2)) See now you get rid of my option to do

what I like, which is the synthetic because you have to only have one of

these in here to do that.

164 Interviewer One of what?

165 Tom Only x plus or minus whatever (covers second x^2 points to x+2) and so

that sucks

166 Tom and so now I have to go with my second option, and I don't remember the

third option, so that's what I'm going to do.

167 Tom Now this one has (points to x^2+x+2) lower numbers, smaller numbers,

whatever you want to say,

168 Tom and so I'm going to multiply these (points to x^2+x+2) over to these ones

(points to x^2+3x+5) rather than the other way around

169 Tom which means, I would like to rewrite this, whatever (next line writes

x^4+3x^3+5x^2)

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.

171 Tom I'll do it sometimes, and when I do that I always have to go through and

underline them in special way,

172 Tom so like all the ones that are to the power of 3 I would go through and

underline with a squiggly line,

173 Tom all the ones that are to power of two I would do a jagged line or

something, so it kind of makes it easier, but I'm not doing that, so

174 Tom and that's usually just when they don't give me enough room on my paper.

175 Tom (next line writes x^3+3x^2+5x with PV's aligned with previous line)

176 Interviewer If you need more room let me know.

177 Tom (next line writes 2x^2+6x+10, equal bar below, + to left, below equal bar

writes x^4+4x^3+10x^2+11x+10 with PV's aligned with partial sums)

178 Tom That is hideous. What do you call equations that have power of 4 and

they're set up like quadratic state from highest to lowest?

179 Interviewer It's a quartic polynomial.

180 Tom It's disgusting, I hate quartic polynomial. We're probably going to have to

learn how to solve those, aren't we?

181 Interviewer Do you know any other methods for multiplying these factors?

182 Tom No. There's a way that you could probably take out the x squared here

(points to x^2+x+2)

183 Tom then do it the synthetic version and add the x squared back in later, but I

wouldn't know.

184 Interviewer Does your method relate to how you multiplied integers?

185 Tom Not so much, but it relates closely to the way that we were multiplying on

like the second one.

186 Interviewer The second what?

187 Tom The second paper that we did that you took from me.

195

188 Tom It relates very closely because we were just multiplying each one by each

one (motions standard dist. from terms of x^2+3x+5 to terms of x^2+x+2)

189 Interviewer That one? (points to paper with work for multiplying 5 and x+2)

190 Tom Yeah. These just keep getting larger and larger as we go.

191 Tom This is also why a lot of kids I think choose this method because you

learned it earlier on. It's just adding more.

192 Interviewer Are you up for doing two more multiplication tasks?

193 Tom Yep.

194 Interviewer If you're done, you can let me know.

195 Tom I feel like it's taking longer than, than average person should.

196 Interviewer That's ok.

197 Tom What's the average time that people take in here?

198 Interviewer About twenty minutes give or take. We're still under that.

199

200 Interviewer This task is to multiply x^2+10 and x+4.

201 Tom (writes (x^2+10)(x+4) and x plus 4.

202 Tom Ok, so I would still do my method of multiplying each one (motions

curves for standard dist.) even though it is slowly getting, because you

have opposite things and that's fine.

203 Tom (next line writes x^3+4x^2, next line below and to right writes 10x+4)

And they don't overlap at all, so that's fun,

204 Tom so I could have technically kept that up there (points to x^3+4x^2) and

added a plus (writes + between lines x^3+4x^2 and 10x+4)

205 Interviewer Why did you choose this method?

206 Tom It's the only method that I knew how to do with these two and it was

fairly simple, especially because they didn't need any sort of adding at the

end.

207

208 Interviewer Last one here, if you would multiply x+3x^2+2 and 3+x.

209 Tom (writes (x+3x^2+2)(3+x)) Ok, well first I'm going to order them because

it's just weird to see them out of order,

210 Tom and somehow it makes things easier, especially because I'm about to do

my synthetic operation, if that's the right word,

211 Tom I just keep on saying it, even thought it's probably wrong (next line writes

(3x^2+x+2)(x+3))

212 Interviewer You said you're going to reorder this cause it will make it easier for you

to do synthetic operation?

213 Tom Yeah, what is that called? Is that the correct word? (writes -3 below work)

214 Interviewer I'm not sure. I'm not worried about what it's called so much as that you

are able to describe what you are doing, to me.

215 Tom Well I'm taking the answer to this, I imagine that this equals zero (right of

(3x^2+x+2)(x+3) writes =0) so x would be negative 3. I put that up in a

box.

196

216 Tom I put all these (points to 3 in 3x^2) down here (right of -3 writes 3 1 2,

then equal bar well below)

217 Tom Then I bring down the first one (writes down arrow below 3, 3 below

equal bar and down arrow)

218 Tom and I do negative 9 (right of down arrow writes -9) eight (writes 8 below -

9 and equal bar)

219 Tom negative 24 (right of -9 writes -24 below 2) then negative 22 (writes -22

below -24 and equal bar)

220 Tom Then I just take half of whatever this was (circles 2 in 3x^2) down here

(writes arrow from circled 2 to 3 below equal bar)

221 Tom And I take (next line below 3 8 -22 writes 3x+8+) yeah plus cause you

don't want to get rid of that negative here (points to -22)

222 Tom (right of 3x+8+ writes -22 with fraction bar below, x+3 below, box

around 3x+8+ -22 / x+3 )

223 Interviewer Do you know any other methods for multiplying these factors?

224 Tom Yes, you could use the methods I was using earlier, but like I said, I just

recently was taught how to do what I just did here,

225 Tom and I prefer it very much over the whole multiply, multiply, multiply,

multiply, and then add.

226 Tom I almost actually enjoy this one because it's so easy.

197

198

199

200

201

202

203

204

205

APPENDIX L: INTERVIEW WITH KEREN

Interviewer: Cameron Sweet

1 Interviewer The first task I have for you is to multiply 26 and 7.

2 Keren 26 and 7 (writes PV) So I'm doing just the regular multiplication you do

when you're younger, like yeah.

3 Interviewer And why did you decide to set theses factors up like this?

4 Keren It's just easier to see the numbers when they're like lined up (points from

6 down to 7 in unit's place) So (writes 2 below equal bar, 4 above 2 in

ten's place)

5 Interviewer And what are you doing here?

6 Keren I'm just multiplying the basic factors. So (writes 18 to left of 2 below

equal bar with PV aligned)

7 Keren I add, so I, when I multiplied these two (points from 7 to 6) I put the first

digit, 2, and then I like, it's 42, 6 times 7 is 42, so I kept the 2 down there

(points to 2 in 182)

8 Keren and put the 4 (points to 4 above 2) and added it to what 7 times 2 is,

which is 14, so that would be 182 (points to 182)

9 Interviewer Do you know any other methods for multiplying 26 and 7?

10 Keren Well, there's like longer ways, you could do like crosses and, like, I don't

know what it's called.

11 Interviewer Can you show me?

12 Keren So like, if you have (writes and counts 27 small vertical lines in a row to

right of work) I think there's 27.

13 Interviewer What's the 27 for?

14 Keren For how many there are (points to 26 in PV)

15 Interviewer How many what there are?

16 Keren How many number, like what the number is, like how many units, like

this, so that's like 26 units or something, depending on what you're

talking about.

17 Keren And then if you're multiplying by 7, you would use one of these units that

you already have for 26 (makes first small line bold)

18 Keren And you would start counting down (writes six more lines in rows below

first small bold line, counts to seven)

19 Keren And when you fill this in (points between row of 27 lines and column of

7 lines) you count all those up, which is the longest way I feel like.

20 Keren When you count all those up, that would get you 182 (points to 182 in

PV)

21 Interviewer Why did you choose this after your first method?

22 Keren Oh, I was just showing you a different method, this was my go to method

(points to PV)

23 Keren this is just another way (points to lines) for like younger kids to do when

they're younger, like whenever I'm tutoring,

206

24 Keren this is how I would show them how to do it because it's easier than, just

count it, except for like, this big of a number (points to 182) so like yeah

25 Keren And so, I would just, you would just fill in the rest (writes three more

lines in second row) so like yeah.

26 Interviewer But why did you choose this one (points to lines) after this one (points to

PV) ?

27 Keren It's just whatever came to mind first because, like yeah.

28 Interviewer Do you know any other methods for multiplying 26 and 7?

29 Keren You could also just do (writes in column PV aligned column) 26 plus 26

plus 26 plus (counts seven 26's, writes equal bar below) and just like add

them all up.

30 Keren And that would get you the same answer, which is

31 Interviewer Why would it be the same?

32 Keren Because 26 is like, well you could do seven 26 (points to PV) but I just

used the bigger number because then it would be less (points to repeated

addition)

33 Keren Because it's basically just adding like, I don't know, because it's like

seven 26's, so like, say like, I don't know, like seven apples.

34 Keren So if you were thinking this as a thing (points to first 26 in repeated

addition) if you have 7 of them, then you would add them up.

35 Keren So these are like the seven things (points to column of +26's) so that

would still get you 182 (writes 182 below equal bar)

36 Keren So yeah, I think that's all the methods I know, that I can remember right

now.

37 Interviewer Why did you choose this one last?

38 Keren Because I think this is mostly forgotten (points to repeated addition)

because, I don't know, because I use these two a lot (points to previous

two methods)

39 Keren Well I don't really use this one (points to lines) but I, I don't know, I just

remembered that we could use that too (points to repeated addition)

40 Keren If I could choose the order of which way I would do it first, if I couldn’t

do this way (points to PV) I would go this way (points to repeated

addition)

41 Keren And then I would go this way (points to lines) but I did this way (points

to line) because I forgot this way (points to repeated addition),

42 Keren whenever I forgot this method (points to repeated addition) before that

one (points to lines) so yeah.

43

44 Interviewer The next task I have for you is to multiply 53 and 10.

45 Keren So (writes 53, *10 in PV) it's just a basic one, that would be (writes) 530.

46 Keren And I did this like, so I didn't really multiply it, like you know how zero,

like, and then you like add, I just looked at it and knew this would be

behind there.

47 Keren This would go behind there (writes arrow from 0 in ten to right of 53) and

then times that by one (points to 53) so that would be 530.

207

48 Keren So I don't know if that's different than…

49 Interviewer Different than what?

50 Keren Different than the regular multiplication.

51 Interviewer Can you show me what you mean by regular multiplication?

52 Keren Well like so if I was to write it all out like (writes PV again to right of

work) usually what you're supposed to do is like zero-zero times (writes

00 below equal bar)

53 Keren times (writes x below right 0) that's an x, not a number.

54 Interviewer What is that x?

55 Keren It rep, it's a place holder for (writes 53 to left of x) yeah, it will get you

the same answer (writes 530 below equal bar) it's just longer if you have

like 12, if you're multiplying 12.

56 Interviewer Why did you need a place holder?

57 Keren Because it's like double digits (points to 53) so, I don't know, you have to

move it over… well.

58 Keren You have to line up the um (points to partial sums, 3 below left zero) I

really don't know, but I'm pretty sure it's because you have to line them

up, if I am correct. But I'm probably not.

59 Interviewer Line what up?

60 Keren Line up the right numbers (points to partial sums, 3 below left zero) like

the rights place values. So I'm pretty sure that's why.

61 Interviewer Do you know any other method for multiplying 53 and 10?

62 Keren Well, the same methods where you do the lines, but I don't really want to

do that one.

63 Keren And then you could add 53 ten times. You could also just…

64 Interviewer And both of those are just the same methods that you showed me on the

previous task, right?

65 Keren Yeah.

66 Interviewer Just making sure.

67 Keren And then you could just, you don't even need like to, if you were just

looking at 53 time 10 on like a test (writes 53*10 below work) or

something

68 Keren you don't really have to multiply at all, I usually just cross it out and then

take out this (scratches off work) Aah!

69 Interviewer Do you want to start over?

70 Keren Yes (writes 53x10 to right of scratched off work) 53 times 10, then I

would take out the one and the x (slash through x and 1)

71 Keren And I would get 530. I don't know if…

72 Interviewer Why did you take out the one and the x?

73 Keren It's just a simpler way for me because one time 53 and then you have to

have that zero at the end of it, cause it's the same thing as that (points to

first PV)

74 Keren Well like doing the same idea.

208

75 Interviewer And you said that that's the way you would usually multiply these on a

test?

76 Keren But I wouldn't cross it out, I would just do it in my head. Yeah, so there is

other ways probably I don't know of, other than the other two that I did.

77 Interviewer I just need to know the ones that you know of.

78 Keren So this one (points to last method) and then the counting one that will

take forever, and then the adding one.

79 Interviewer So why did you choose this method last (points to last method) ?

80 Keren Because it's not normal I guess, and this is the way that everybody else

know how to do it (points to PV), so I did that way first.

81 Keren And then this is the way that I do it, and it's not typically what happens

because it's only for special cases with anything multiplied by 10.

82 Keren So like, yeah, so I don't want to change problem yet… Say if it was like 4

times 10 you could just do that too.

83 Keren (writes 4x10=40) It would be 40, but you can't do that with… other…

84 Interviewer Oh, by change problem you mean give me another example?

85 Keren Yeah.

86 Interviewer That works great.

87 Keren That's what I usually do (scratches out x and 1) but I do that in my head, I

just was showing you, so, yeah.

88 Interviewer Do you know any other method for multiplying 53 and 10? Or 4 and 10?

89 Keren Um, no.

90

91 Interviewer This is multiply 311 and 129.

92 Keren (writes 311) And what?

93 Interviewer 129

94 Keren (writes x129 after 311) Ok, so.

95 Interviewer Why did you write these like this?

96 Keren Oh, I was just looking at it right now, but I'm (writes PV to right of

expression) I was just writing it so I could remember it, cause that's the

way I usually see it in real life.

97 Interviewer And you usually see this where?

98 Keren I usually see it this way, cause that's how the problem is usually when

you get it on a test, so I usually write it out like that sometimes.

99 Keren So I'm gonna multiply (points from 9 in PV to digits on line above it) 9's

across everything, so it would be (writes 2799)

100 Keren And then put my place holder (writes x below right 9 in 2799) one place

holder right now because you're going to the next one that would be

(writes 622 in front of x)

101 Keren Going to the next one (writes two x's on next line below x and right 2 in

previous line) because you're in the third, your in the hundred's place

instead of the ten's place.

102 Keren Now you're in the hundred's place, so (writes 311 in front of xx) and then

you add this all up (writes + to left and equal bar below partial sums)

209

103 Keren You get (writes 40,119) And you should get, I think if I'm right. Yeah.

104 Keren And you can also do the distributive property, because you're distributing,

you could also kind of do distributing I think.

105 Interviewer Can you show me what you mean by doing distributing?

106 Keren Like, now that I think about it, like 3 and like (writes 9(311) to right of

work) you were doing the 9

107 Keren and you would do (writes 2(311) to right of 9(311)) and then (writes

1(311) to right of 2(311))

108 Keren And then when you multiply that (writes 2799 below 9(311)) and then

plus (writes + after 2799)

109 Keren But you would have to put a…, yes you would have to put a place holder

still I think whenever you do the thing you would have to put a place

holder.

110 Interviewer Why do you need a place holder?

111 Keren To line it up still, because this is in the ten's place (writes 0 after 2 in

2(311)) kind of like a 20, but not really.

112 Keren I would usually multiply by 2, but there's still the 9 which is right there

(points to 9(311)) so you'd still need a zero there (points after 2799+)

113 Keren (writes 6220 right of 2799+) or an x, doesn't matter (writes x over 0 in

6220, then +)

114 Keren You need two place holders again, because (writes 31100 right of

2799+622x+) then you get the same answer (writes = 40,119).

115 Keren And then there's obviously the super long one (motions writing many

lines)

116 Interviewer What one would be the super long one?

117 Keren The one where, I don't know what it's called, but it's just like through

pictures, like not really pictures, but little dots or something you could

count a lot of dots, obviously.

118 Keren And then there's adding again.

119 Interviewer You said count a lot of dots. Do you know about how many dots you

would be counting?

120 Keren This many (writes arrow to 40,119 below PV) because it would be 300 by

129, so you would be counting all the dots in the area of that.

121 Interviewer Do you know any other method for multiplying 311 and 129?

122 Keren No.

123

124 Interviewer This one is to multiply 102 and 97.

125 Keren (writes 102x97, then PV to right) Ok, so (writes 714 below equal bar, on

next line writes 918x, keeping PV aligned between factors and partial

sums)

126 Keren (writes + left of partial sums, equal bar below, on next line 9,894 keeping

PV aligned) let me check, I'm pretty sure that's right.

127 Keren Yeah, so that would be the answer for that, and this is just the regular

multiplication method that most people use for bigger numbers,

210

128 Keren unless they can do it in their head like 2 times 3.

129 Interviewer Do you know any other methods for multiplying 102 and 97?

130 Keren You could do the distributive kind of (writes 7(102) right of PV) which is

kind of basically the same thing.

131 Interviewer How is it the same?

132 Keren As this, (points to PV) because like you're just singling out one number

and it looks easier to some people.

133 Interviewer What looks easier?

134 Keren The, like the distributive kind of, but you have to remember that this is

the second (points to 9 in 97 in PV) and there will be one more zero that

you'd have to add when you do your adding.

135 Interviewer Why do you need to add a zero?

136 Keren Because it's a place holder for like, so like (writes below PV) 90 plus 7, it

kind of like a place holder (points to 0 in 90)

137 Keren It's where 7 went out, but you don't multiply the 90, because it has 90,

you just multiply it as 9. (scratches out 90+7)

138 Keren So (writes +9(102) right of 7(102))

139 Keren (on next line writes 714+9180=9894) Yeah, and it's basically the same

thing, except, yeah, you would just add a place holder when you add it, so

yeah.

140 Keren And then there's also the counting method and the adding like, adding

that many (points to 102x97) but that's a lot.

141 Keren So like adding 102 97 times, which would be long and tedious, but if you

only know how to add then, yeah.

142

143 Interviewer This task is to multiply 5 and x+2

144 Keren (writes 5) x plus 2?

145 Interviewer Correct.

146 Keren (writes (x+2) after 5)

147 Interviewer Why did you write these like this?

148 Keren Because x is like added with 2, so it's like in it's own little, cause you

single it out, I don't know, because like it's adding together and then you

have to multiply what that is.

149 Keren Cause you don't know what x is, so you have to have it along with 2, so

when you distribute it.

150 Keren And I'm going to do the, well I mean you could do the box method, but

it's not really needed but I'm just going to do the distributive property.

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

153 Keren but, unless you think more work's (writes curves from 5 to x and 2, on

next line 5x+10) and that would be the answer when you multiply.

154 Keren And then there's also, you don't really have to do it for this one because

it's the lattice method, which you don't really need to do it for this one.

211

155 Keren (writes two boxes with x and 2 above and 5 to left, then 5x and 10 in

boxes) and you get the same answer.

156 Interviewer What's your answer?

157 Keren 5x plus 10 (points to boxes containing 5x and 10, then draws box around

5x+10 in standard dist.)

158 Keren There's also, you could also just do (writes to right of work) x plus 2

(writes on next line) times 5 in this way.

159 Keren But I don't, like this is regular multiplication but with adding and

subtracting, it's not really confusing for me, it's just, doesn't look normal

for me.

160 Keren (writes 5x+10 below equal bar with PV aligned in product and factors)

161 Keren And then, I don't think you can do the dot one for this one, it would

probably be too confusing for me,

162 Keren and I think this is the last way that I know to do this with this kind of

equation, so yeah.

163 Keren So just distributive, and the box, the lattice method, and just regular

multiplication, I forgot what this is called.

164 Interviewer Does distributive relate to how you were multiplying integers?

165 Keren I don't know.

166 Interviewer You don't know? Reminder you can say you're unsure or you don't know.

167 Keren Yeah.

168 Interviewer Do either of your other two methods relate to how you multiply integers?

169 Keren Um… I don't think so.

170

171 Interviewer This task is to multiply the square of the quantity x+5.

172 Keren (writes (x+5)^2)

173 Interviewer Why did you write this like this?

174 Keren Because, well, I don't know, this is the correct way to write it whenever

you, like you just said it different than I usually hear,

175 Keren because a quantity is x plus 5 so you just square that and it could also be

written like (writes (x+5)(x+5)) x plus 5 times x plus 5, I'm pretty sure.

176 Interviewer How do you usually hear this?

177 Keren It's just x plus 5 squared, you usually hear.

178 Keren And then the way, there's I think four ways you can solve this, which is x,

so like so if you're finding, I don't know if it's a 'p' or a 'q', but this is one

of the things

179 Keren (next line writes (x)^2) and this is 5 (next line writes (5)^2)

180 Keren And so you would square the 'a' and the 'b' (points to second (x+5))

because it would be the same for each one of them (points from first

(x+5) to second)

181 Keren So it would be (writes to right of (5^2)) x^2+25 as your answer (writes

box around x^2+25)

212

182 Keren And there's also the box method, which is, could be used for this one

(writes box to right of work) this is the short cut way (points to standard

dist.)

183 Keren This is also if you don't remember that way (still pointing to standard

dist.) for other ones

184 Keren (writes x and 5 above boxes, x and 5 each to left of boxes) If you were

subtracting it would be harder to, you'd just have to use a little bit

different things to remember it.

185 Keren (writes partial sums in boxes) Actually no, this is not the answer

(scratches out x^2+25 in standard dist.)

186 Interviewer Why is it not the answer?

187 Keren Because I totally forgot that it's only, for this one I'm pretty sure you can

only use it if it is subtracting (writes bold - over + in (x+5)^2)

188 Keren or like the short cut I think, because if one of them is not negative (points

to x and 5 above boxes) like one of the fives

189 Keren like if it's x minus 5 it would be, I'm pretty sure that I did it wrong

because, I don't know, I just feel like, where did I miss something?

190 Keren You said x plus 5, right?

191 Interviewer Mm-hm.

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.

193 Keren So whenever I check my answers, I usually know if I'm right or not

(writes x^2+10x+25 below boxes)

194 Keren So this is the correct answer, I'm pretty sure (writes box around

x^2+10x+25) Don't hold me on it.

195 Keren And then there's also just the multiplying way of down (writes PV to

right of work) which is (writes 5x+25 below equal bar with PV's aligned)

196 Keren And then place holder (writes x below 25) because you wouldn't have

another one, like another number without an x so, because it's an x,

197 Keren Then you do that plus (writes 5x+ in front of x)

198 Interviewer So your place holder is an x?

199 Keren Um, yeah, no it's not, it's like (scratches x and writes 0) I usually use x,

but it's not an x, you can say it's a zero if that confuses you,

200 Keren It's just a place holder, it doesn't matter what it is, like I do crosses, or an

x if you want to say, but it's not an actual x, I curve my x's.

201 Keren Let's say it's a zero cause x is, there's already x in there

202 Keren Then (writes x^2+ in front of 5x+ 0 with PV's aligned, then equal bar

below, next line x^2+10x+25 with PV's aligned) Oh, I remember why

this doesn't work.

203 Interviewer Why what doesn't work?

204 Keren (points to (x+5)(x+5) in standard dist.) This one, it doesn't work for this

one because it has to be subtracting 5 I'm pretty sure,

205 Keren Because for this one it can only work if it was (writes) (x minus 5)

squared then you could put 5 times 5 (next line writes (x)^2 ,next line

(5)^2)

213

206 Keren And then when you square that you would just do (writes x^2-25) x

squared minus 25,

207 Keren and that would be different because you have one x plus 5 and one x

minus 5 (writes (x+5)(x-5))

208 Keren And the 5's that made the 10 would cancel out (points to x^2+10x+25

below boxes)

209 Keren So yeah, that's why I got this one wrong the first time (points to standard

dist.) cause you're not supposed to use short cut unless it's subtract

210 Keren And then, I'm pretty sure that's all the ways that I know. This is not right,

so I'm gonna

211 Keren (writes Right below x^2+10x+25 below boxes, Not right below x^2-25

below standard dist.)

212 Keren And I think that's all the ways I know how to do it, that makes sense

213

214 Interviewer This one is to multiply x+2 and x+5.

215 Keren (writes (x+2)(x+5))

216 Interviewer And why did you write these like this?

217 Keren Because x plus 2 and x plus 5 are two separate like problems that you

need to multiply together to get like one answer,

218 Keren so and you see it like this whenever it's heard like that, so that's why I

wrote it like that.

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.

220 Keren And it doesn't really matter which one goes on the bottom and which one

goes on the top (writes x and 2 above boxes, x and 5 each to left of

boxes)

221 Keren (writes partial sums in boxes) ok and then it would be x squared (writes

x^2 to right of boxes)

222 Keren and then you would have to add these two (writes circle aound 2x and 5x

in boxes) together to combine like terms (writes +7x right of x^2)

223 Keren (finishes writing x^2+7x+10 right of boxes) and then you get that as your

answer (writes box around x^2+7x+10)

224 Keren And then you can also do distributive property, which this one is longer if

you want to show your work for it (writes (x+2)(x+5) right of work)

225 Keren which is what I forgot the last one, so (next line writes x*x+x*5+2*x)

226 Interviewer What are you doing here?

227 Keren I'm distributing them, so like x time x, (writes curve from first x to x and

5 in (x+2)(x+5)) times x times 5

228 Keren And then I'm doing 2 times x, (writes curve from 2 to second x and 5)

times 2 times 5

229 Keren And then when you add all those up and combine like terms you should

get the right answer. (writes +2*5 after x*x+x*5+2*x)

214

230 Keren And then combine them (next line writes x^2+5x+2x+10) and that would

combine to (next line writes x^2+7x+10, box around it)

231 Keren And I think there's one other way that I remember which is just where

you line up all the like terms (writes PV below boxes)

232 Keren (Below equal bar writes 5x+10 with PV's aligned with factors)

233 Keren And then I'll put the zero as a place holder (writes 0 below 10) cause

there's not going to be another number that's the same.

234 Keren It will just be easier to line up the terms so that it's easier to add it down

faster (writes +x^2+2x+ left of 0 with PV's aligned)

235 Keren (writes equal bar below, next line x^2+7x+10 with PV's aligned, box

around it)

236 Keren And yeah, through all of these you should be able to figure out that this is

right (points to x^2+7x+10 below PV)

237 Interviewer Why did you choose this method last?

238 Keren Because for this method, for this way of setting it up, it's just, it's not

confusing for me, it's just, I don't like seeing it like this because,…

239 Keren I usually see numbers without any terms in it, and it's just odd for me

seeing it like this, even though it lines it up perfectly it's just not normal

for me. So yeah.

240 Interviewer Does your first method relate to how you multiply integers?

241 Keren Umm… yeah.

242 Interviewer How?

243 Keren Because it, I probably should have said that for the other ones, but yeah,

because it won't matter if…

244 Keren are you talking about if it was an integer or like if it was an integer, like

the number, because it would work if it was an integer.

245 Interviewer What would work?

246 Keren The multiplication would still be the same if it was an integer.

247 Interviewer How would it be the same?

248 Keren Because it's like, you can have an integer right here (points to 5 to left of

boxes, 2 above boxes)

249 Keren and it will still come out as the right answer because it's just the same like

method and it will work for like anything.

250 Interviewer Would you do that when you're multiplying integers?

251 Keren Yeah and I said no cause I was like, huh

252 Interviewer Where? Was it one of these? (directs Keren back to integer

multiplication)

253 Keren Umm… I believe it might have been this one. I don't know. I don't

remember. Was it one of the one's with the boxes. I know it was one of

our, the first one.

254 Interviewer Here's the first one.

255 Keren I think it was like the second or third one. It was one of the first one's. It

was probably this one. I don't remember which one it was.

256 Interviewer Do either of your other methods relate to how you multiply integers?

215

257 Keren Umm… yeah, cause you can multiply integers either, like either one of

these (points to each method one at a time) cause it will just work.

258 Keren If that's what you're asking, maybe, sure.

259

260 Interviewer This is to multiply x^2+5x+2…

261 Keren (writes x^2+5x+2) Like that?

262 Interviewer Ok, and x+3.

263 Keren (finishes writing (x^2+5x+2)(x+3)) So that's my problem that I

interpreted it,

264 Keren because this is all in its own bracket (points to x^2+5x+2) cause you're

adding all these together,

265 Keren and you're adding these to these (points to x+3) numbers, even though

this is not a number yet (points to x)

266 Keren and you're not multiplying yet, so that's why you put them in different

brackets.

267 Keren For this one I would do lattice method first, which would be the box

method, although it's uneven boxes.

268 Interviewer (writes 2 boxes by 3 boxes with x^2, 5x and 2 above and x and 3 each to

left, then writes partial sums in boxes)

269 Keren As you can see there's (points from box with 15x to 2x) it should match

up when you go across (points from box with 3x^2 to 5x^2)

270 Keren to where it's easier to combine the same exponents.

271 Keren (writes x^3+8x^2 to right of boxes)

272 Interviewer And where did 8 x squared come from?

273 Keren Because you add these two (writes + on intersection of boxes with 3x^2

and 5x^2, then circles boxes) numbers because they're the same when

combining the like terms.

274 Keren And then you would add these two (writes circle around boxes with 15x

and 2x, then + on intersections of boxes) to get (writes +17x+6 after

x^3+8x^2)

275 Keren And that would be your answer and 6 you just, it's the same because you

can't find anything (writes box around x^3+8x^2+17x+6)

276 Interviewer Does this method relate to how you multiply integers?

277 Keren I want to say the same thing that I said before, but I think that's wrong, so

I don't know.

278 Interviewer Do you know another method for multiplying these factors?

279 Keren Yeah, use the distributive property. You'd multiply these across (writes

curves from x^2 to x and 3) then multiply these (writes curves from 5x to

x and 3)

280 Keren until you get all of them (writes curves from 2 to x and 3)

281 Keren So it would be (writes below boxes x^2*x+x^2*3+5x*x+5x*3+2*x+2*3)

282 Keren And you just multiply, I usually multiply and then add, combine the like

terms at the end cause it's easier.

216

283 Keren (next line writes x^3+3x^2+5x^2+15x+2x+6) Then I'm gonna combine

like terms, these are the like terms (writes circle around 3x^3 and 5x^2,

15x and 2x)

284 Keren (next line writes x^3+8x^2+17x+6) to get the same answer (writes box

around x^3+8x^2+17x+6)

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)

299 Interviewer Do you know another way to multiply 2 and the quantity 3+2 ?

300 Keren I'm pretty sure, I think you can do (writes PV, 3+2 next line "x 2" with

equal bar below)

301 Keren Just multiplying down should be (writes 6+4 below equal bar) which

would equal (writes =10 right of 6+4)

302 Keren Since there's no value, like x or anything (points to 17x in answer to

standard dist.) you can just read the answer and you would simplify to get

10 again.

303 Interviewer Do you know another way to multiply the quantity 2 and 3+2 ?

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.

307 Interviewer I thought you had said something about combining earlier.

217

308 Keren Oh, yeah, and you can also, if you have this (below work writes 2(3+2))

if you added these since these are like terms you can have (next line

writes 2(5))

309 Keren and then that would just be 2 times 5 (next line writes 2*5) which equals

10 (next line writes 10) that's another way.

310 Interviewer I thought I heard you say that. I hope I'm not remembering something that

wasn't there (it was there)

311 Keren Yeah, I did, and then I was thinking of just like ways of doing without

combining. So yeah.

312 Keren These are if you don't combine like terms (points to previous three ways

to multiply 2(3+2)) and then this one's if you do (points to last way)

313 Keren But in this case (points to standard dist. for (x^2+5x+2)(x+3)) you can't

really combine like terms cause there's not any in the equation that we

have.

314 Interviewer Do you know any other methods for multiplying x^2+5x+2 and x+3?

315 Keren Yes, there is just the up and down way for like the um, regular

multiplication, I really don't know what it's called.

316 Keren I just call it regular cause it's what most people say.

317 Interviewer Ok, I don't need a name.

318 Keren (below standard dist. writes PV with PV's aligned)

319 Keren (below equal bar writes +6) and it will always be addition cause there's

no negative sign, which you just have to watch for the negative sign.

320 Keren (finishes writing 3x^2+15x+6 with PV's aligned with factors)

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)

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.

323 Keren (below equal bar writes x^3+8x^2+17x+6) so yeah, and that's the last one

where I did.

324 Keren And I don't use this one a lot. It's the last one that comes to mind, so

yeah.

325 Interviewer Does this method relate to how you multiply integers?

326 Keren Yes.

327 Interviewer How?

328 Keren Because you just… I can only explain it with an example, but it you

like…

329 Interviewer You could give me an example or you could use one of the ones that you

worked on with the tasks at the beginning.

330 Keren Like this one (points to 2(3+2)) like right here where it's regular, it could

be written like 2 times 3 (writes 2(3)) that's how it could be written

331 Keren But you could multiply it like this (below writes 2, next line "x 3")

332 Keren which is the same way as this (writes arrow from PV for x^2+5x+2 and

x+3 to work) except for there's no x's and you'll only get 6.

333 Keren But that's a different problem, so yeah.

218

334

(class bell rings)

335 Interviewer How is this similar?

336 Keren Because, like you'll, these are lined up (points from 2 to 3) just like these

(points to PV for x^2+5x+2 and x+3) yeah.

337 Keren You're multiplying down, so yeah.

338 Interviewer Do have time for a few more tasks, or do you need to get going?

339 Keren I have practice.

340 Interviewer Ok. Thank you.

219

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227

APPENDIX M: INTERVIEW WITH REBEKAH

Interviewer: Cameron Sweet

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)

5 Rebekah I got 182, I have to say what my answer was, right?

6 Interviewer Sure. Do you know any other methods for multiplying 26 and 7?

7 Rebekah I bet there is, but I wouldn't rather do them.

8 Interviewer So you do know other methods, what are those methods?

9 Rebekah Oh, I mean, off the top of my head, no.

10 Interviewer No, you don't know other methods?

11 Rebekah No, but I'm sure there is.

12 Interviewer Ok.

13

14 Interviewer Your next task is to multiply 53 and 10.

15 Rebekah Ok (writes 53, next line 10 PV aligned, equal bar below, next line 530)

16 Interviewer So what did you do here?

17 Rebekah Same thing as the last one. Oh, hold on, suppose I should show my work

(scratches work, right of work writes 53, next line 10 PV aligned, equal

bar below again)

18 Rebekah (next line writes 00, next line writes 530, PV aligned) there, there's work

now.

19 Interviewer So what did you do differently here than what you did the first time?

20 Rebekah Oh, I just knew that 53 times 10 was 530, and on the second time I thought

that I would just do it the traditional way,

21 Rebekah and show that when zero's multiplied by a number it's just zero.

22 Interviewer Do you know any other methods for multiplying 53 and 10?

23 Rebekah Just move the decimal over one (points to lower right of 53)

24 Interviewer Is that any different from what you did the first time, or is it the same?

25 Rebekah It is.

26 Interviewer It is different? Can you show me?

27 Rebekah Like when you multiply something by 10 you just add a zero, no matter

what, and if it's 100 you add two zeroes.

28 Interviewer I think I understand. Can you write this for me?

29 Rebekah Sure (writes 20, next line 10 PV aligned, equal bar below)

30 Rebekah 20 times 10 is 200 (writes 200 below equal bar) zero (writes 0 right of 20,

scratches factors, leaving 200)

31 Interviewer So you add a zero on to the end of 53 similar to what you did at the end of

20 because you're multiplying it by 10?

32 Rebekah Cause you're multiplying it by 10, yeah.

33

228

34 Interviewer The next task is to multiply 311 and 129.

35 Rebekah (writes 311, next line 129, equal bar below, next line 2799, next line 6200,

next line 31100, equal bar below, next line 39,119 PV's aligned)

36 Interviewer Do you know any other methods for multiplying 311 and 129?

37 Rebekah Could be, I don't know.

38 Interviewer Ok.

39

40 Interviewer So then the last integer multiplication task I have is to multiply 102 and

97.

41 Rebekah (writes 102, next line 97, equal bar below, next line 714)

42 Interviewer What did you do so far?

43 Rebekah Multiplied 7 times 2 (points from 7 in 97 to 2 in 102) got 14 (points to 14

in 714) 7 times 0, and I got 0, but I dropped the 1 down (points to 1 in

714)

44 Rebekah and then I did 7 times 1 (points from 7 in 97 to 1 in 102) got 7 (points to 7

in 714) then put a space, or place holder or whatever (writes 0 below 4)

45 Rebekah and I'm going to do 9 times 2 (points from 9 in 97 to 2 in 102, writes 8

below 1 in 714, 1 above 0 in 102) and 9 times 0 and a 1 (writes 1 below 7

in 714)

46 Rebekah and 9 times 1 (writes 9 right of 18, equal bar below, next line 9,894)

47 Interviewer So you mentioned a place holder or space holder, what was that for?

48 Rebekah When you, like go, when you move on to the next section of ten's or

hundred's or thousand's or whatever, you put a zero.

49 Interviewer Why?

50 Rebekah Cause it's a whole 'nother, you know, like space up, I don't know what you

call that, just like a ten's place because you moved over into the next

column.

51

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.

56 Rebekah (next line writes 5x+10)

57 Interviewer Where did you use the distributive property in this multiplication

problem?

58 Rebekah (writes curved arrows from 5 to x and 2) I did 5 times x and I did 5 times

2, and I added them together (circles + in x+2)

59 Interviewer Does your method for multiplying 5 and x+2 relate to your method for

multiplying integers?

60 Rebekah Yeah, the, I don't know what it's called, where you like stack it (writes

x+2, next line 5 below 2, equal bar below, next line 5x+10, PV's aligned)

61 Interviewer How is this the same?

229

62 Rebekah You're multiplying the same things, so it's just written one on top of the

other instead of beside each other, I suppose.

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.

68 Interviewer That's what I want to hear, what you think.

69

70 Interviewer This next task is to multiply the square of the quantity x+5.

71 Rebekah (writes sqrt(x+5))

72 Interviewer This is square, I think you have square root.

73 Rebekah Ok, oops. Wait, x squared plus 5 ?

74 Interviewer x plus 5, and then you're going to square that quantity.

75 Rebekah Oh, ok (scratches work) let me redo this (writes (x+5)^2, next line

(x+5)(x+5))

76 Interviewer And what did you do here?

77 Rebekah Distribution again, but you could do it the other two ways I've shown you.

78 Interviewer What did you distribute?

79 Rebekah Oh, the square is just the same thing again (points to second (x+5))

80 Rebekah and then you do the (writes curves from first x to second x and 5, curves

from first 5 to second x and 5) that thing.

81 Rebekah (next line writes x^2+5x+5x+25) and then I'm gonna combine my like

terms (next line writes x^2+10x+25, underlines)

82

83 Interviewer This task is to multiply x+2 and x+5.

84 Rebekah (writes (x+2)(x+5)) and just for the sake of switching it up, I'll do the box

method (writes 2 by 2 boxes, x and 2 left of each row, x and 5 above each

column)

85 Interviewer Why are you switching it up?

86 Rebekah Why not, I guess? I like both ways. I'm not really fond of the one where

you stack it. (writes partial sums in boxes)

87 Rebekah That's weird (writes x^2+7x+10 below boxes)

88 Interviewer What was weird?

89 Rebekah Oh, I wrote the square big (points to box with x^2) and so I thought I

wrote 2 x just weird, but I didn't.

90 Interviewer Thank you for the clarification. Does this method relate to how you were

multiplying integers?

91 Rebekah I don't know, I never thought about that. I guess I could try it. What was it,

twenty something?

230

92 Interviewer We still have your work here if you would like to take a look at it (points

to integer multiplication work)

93 Rebekah 26 and 7 (writes 2 by 1 boxes, 2 and 6 left of each row, 7 above boxes, 14

in top box, 42 in bottom box)

94 Rebekah (below boxes writes 14, next line +42, equal bar, next line 56, PV's

aligned)

95 Rebekah Did I get 56 ? (looks back at previous PV work) No I did not (scratches

14, +42, 56) Hmm…

96 Interviewer You used a different method for multiplying 26 and 7 earlier when you

told me that you didn't know any other methods for multiplying integers,

right?

97 Rebekah Yes. Is it supposed to work for that as well?

98 Interviewer I'm asking what you're able to recognize between the two of them, I don't

know, maybe it could work.

99 Rebekah I bet I'm just doing something wrong, having a brain fart, I bet it does

work.

100 Interviewer Do you know any other methods for multiplying x+2 and x+5?

101 Rebekah Yeah (writes x+2, next line x+5, equal bar below, next line 5x+10 PV's

aligned)

102 Rebekah (next line writes x^2+2x, PV's NOT aligned)

103 Rebekah (next line writes x^2+5x) oops, should be a 7 (scrates x^2+5x, next line

writes x^2+7x+10, underlines)

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.

110 Interviewer So does this method relate to how you multiplied integers?

111 Rebekah Yes. Is it supposed to work for that as well?

112 Interviewer And I'm convinced you already told me how when you were explaining all

the pieces you were mutlpying.

113 Rebekah Yes.

114 Interviewer Do you know any other methods for multiplying x+2 and x+5?

115 Rebekah The distribution one (writes (x+2)(x+5), next line x^2)

116 Interviewer And what did you do to get the x squared?

117 Rebekah (writes curved arrows from first x to second x and 5) multiplied by first,

then, yeah, it's FOIL, right? First, outer, inner, last.

118 Rebekah And then I'm going to do x time 5 (right of x^2 writes +5x)

119 Rebekah And then I'm going to do 2 times x (right of x^2+5x writes +2x, writes

curved arrows from 2 to second x and 5) 2 times 5 get 10 (writes +10)

231

120 Rebekah And then combine like terms again, get 7x (next line writes x^2+7x+10)

121 Interviewer Does this method relate to how you were multiplying integers?

122 Rebekah Yes, just side by side instead of on top of each other, I suppose.

123 Interviewer So that's how they're a little bit different, but how are they similar to how

you were multiplying integer?

124 Rebekah Well I'm doing everything in, uh, not the same order, I'm just multiplying

the same things, just in different order.

125 Rebekah Like here (points to PV factors) I did 2 and 5 first, and here (points to

standard dist. factors) I did it last.

126 Rebekah Then my x squared's first here (points to PV factors) I mean my x times x

to get x squared last here, and first here (points to standard dist. factors)

127 Interviewer Do you know any other methods for multiplying x+2 and x+5?

128 Rebekah No.

129

130 Interviewer This task is to multiply x^2+5x+2 and x+3.

131 Rebekah (writes (x^2+5x+2)(x+3)) I like that better.

132 Interviewer Why do you like that better?

133 Rebekah So I can extinguish where one stops and the other begins besides just a dot

or a multiplication sign, because it can kind of look like an x and that

would confuse me.

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.

135 Rebekah (writes x^2, 5x and 2 above boxes, x and 3 each left of boxes, partial sums

in boxes)

136 Interviewer What are you multiplying in these boxes?

137 Rebekah I'm doing my x (points to x^2 above box) times the x in the other one

(points to x left of box)

138 Rebekah It's also just like the distribution, but in a different, order, because you can

do it in any order you want on this one, I guess.

139 Rebekah (below boxes writes x^3+5x+2x+3x^2+15x+6, next line

x^3+3x^2+7x+21)

140 Rebekah (points back to boxes) Oh, it's supposed to be 5 x squared, there we go. I'm

just going to redo that, because I wrote it down wrong (scratches previous

two lines)

141 Rebekah (next line writes x^3+5x^2+2x+3x^2+15x+6, next line writes

x^3+8x^2+17x+6 while slashing terms in previous line)

142 Interviewer What are you doing now?

143 Rebekah Well I'm combining my like terms, and I was just crossing them off

because they were all out of order up here (points to

x^3+5x^2+2x+3x^2+15x+6)

144 Rebekah These are my squares so I crossed them off together and got them out of

the way while I wrote them down in my final answer there (points to 8x^2)

145 Rebekah And then the same thing with the, just the plain old x's not the x squares

(points to 17x)

232

146 Interviewer Do you know any other methods for multiplying these factors?

147 Rebekah The distributive one and that stacking one. Do I need to do them too?

148 Interviewer If you would like to, you can…

149 Rebekah I don't (quickly)

150 Interviewer If you rather not, that's fine too.

151

152 Interviewer This task is to multiply x^2+3x+5 and x^2+x+2.

153 Rebekah (writes (x^2+3x+5)(x^2+x+2), below writes 3 by 3 boxes) I'm going to do

the box one again.

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)

156 Rebekah and then to the third (points to boxes 1,2 and 2,1) and then to the second

(points to boxes 1,3 2,2 and 3,1)

157 Rebekah and then the first (points to boxes 2,3 and 3,2) and then your normal

(points to box 3,3 in lower right) or plain integer or whatever

158 Rebekah (writes x^2, 3x and 5 each left of boxes, x^2, x and 2 above boxes, partial

sums in boxes)

159 Rebekah (below boxes writes x^4+4x^3+10x^2+11x+10 while writing diagonal

lines through like terms in boxes, underlines answer)

160 Interviewer Do you know any other methods for multiplying these factors?

161 Rebekah Distributive and stacking.

162 Interviewer Could you at least set those up, even if you don't want to do them?

163 Rebekah Yeah.

164 Interviewer You can tell me no too, if you would rather not do them.

165 Rebekah The distributive one is just the FOIL again (writes curved arrows above

(x^2+3x+5)(x^2+x+2)) and that, and then you do your next one, and the

one after that.

166 Rebekah and then you get (writes x^4+x^3+2x^2+3x^3+3x^2+6x+5x^2+5x+10)

167 Rebekah and then we'll see, it spreads them all out and makes it a little bit harder for

combining terms.

168 Rebekah (next line writes x^4+4x^3+10x^2+11x+10 while scratching terms in

previous line, underlines answer) and that's the same thing

169 Rebekah and then the last one (writes x^2+3x+5, next line x^2+x+2 PV aligned,

equal bar below, next line 2x^2+6x+10 PV's aligned)

170 Rebekah (next line writes x^3+3x^2+5x PV's NOT aligned, next line

x^4+3x^3+5x^2 PV's NOT aligned, equal bar below)

171 Rebekah (next line writes x^4+4x^3+10x^2+11x+10 while scratching partial sums)

172 Rebekah Oh and I guess this is kind of like the box method, how it spreads out the

like terms (points to partial sums in boxes) it just does it backwards (points

to partial sums in stacks)

173 Interviewer Why did you pick this last method last?

174 Rebekah It's my least favorite.

175 Interviewer Why?

233

176 Rebekah I don't know. I guess I don't have a very good reason.

177 Interviewer Ok, you can tell me that.

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,

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.

180

181 Interviewer This task is to multiply x^2+10 and x+4.

182 Rebekah (writes x^2+10)(x+4)) I guess I'll start with the distributive.

183 Interviewer Why did you decide to start with distributive?

184 Rebekah Cause I was thinking that you would ask if I knew any other ways to do it,

and then elaborate, so I'm going to have to do all of them and might as

well start with this.

185 Rebekah (next line writes x^2+4x^2+10x+40, underlines) And it's not too much to

keep track of in my head,

186 Rebekah and I think the distributive, since there's no boxes of distinct numbers, is a

little more to do in your head than otherwise.

187 Interviewer As you mentioned, I'm probably going to ask you, do you know any other

methods for multiplying these factors?

188 Rebekah Yes, sure do (writes 2 by two boxes) box method!

189 Rebekah (writes x^2 and 10 each left of boxes, x and 4 above boxes, partial sums in

boxes, below boxes x^2+4x^2+10x+40, underlines)

190 Rebekah And (writes x^2+10, next line x+4 PV's NOT aligned, equal bar below,

next line writes 4x^2+40, next line x^3+10x PV's NOT aligned, equal bar

below)

191 Rebekah (next line writes x^2+4x^2+10x+40, underlines)

192 Interviewer Do you know any other methods for multiplying these factors?

193 Rebekah No.

194

195 Interviewer This last task is to multiply x+3x^2+2 and 3+x.

196 Rebekah (writes (x+3x^2+2)(3+x))

197 Interviewer And if I only let you use one method, what method would you use to

multiply these?

198 Rebekah I would use the box method because when it's more than just two terms

time two terms, that's my preference.

199 Rebekah (writes two by three boxes, x and 3 each left of boxes, x, 3x^2 and 2 above

boxes, partial sums in boxes)

200 Rebekah (below boxes writes 3x^3+10x^2+5x+6, underlines)

201 Rebekah And others as well.

202 Interviewer What would be the second one you would pick?

203 Rebekah I think I'll do this, the stacking one.

204 Interviewer Why would you pick that one second?

205 Rebekah Because now that I think about it, I think it's more similar to this (points to

boxes) than distributive and I like this one (still pointing to boxes)

234

206 Rebekah So I'll do it this way (writes x+3x^2+2) except for that's kind of weird, I'm

going to switch those (scratches x+3x^2+2)

207 Rebekah (next line writes 3x^2+x+2)

208 Interviewer Why did you find that weird and switch those?

209 Rebekah Because just usually if, um, the variable to the highest power goes furthest

left, I don't think it really matters, it's just a habit.

210 Rebekah (next line writes x 3, PV's aligned, equal bar below, next line writes

9x^2+3x+6 PV's aligned, next line writes 2x directly below 3x)

211 Interviewer Why did you put that 2x there, what is that for?

212 Rebekah It just line up with this one (circles 3x and 2x in partial sums) my other

ones, I guess I could have done it, I just didn't, cause I don't know.

213 Rebekah (writes 0 right of 2x, finishes writing 3x^3+x^2+2x 0 PV's aligned, equal

bar below, next line writes 3x^3+10x^2+5x+6 PV's aligned, underlines)

214 Rebekah And then lastly I'll do distributive (below (x+3x^2+2)(3+x) writes

3x+x^2+9x^2+3x^3+6+2x, scratching terms in (x+3x^2+2)(3+x))

215 Rebekah And I crossed that out because I'm done multiplying it by the one's in the

other parentheses.

216 Rebekah (next line writes 3x^3+10x^2+5x+6) Kind of running into my box method,

but that's ok.

217 Interviewer Do you need more paper?

218 Rebekah No, I just didn't want to write it down there, cause I already had it written

out up here (circles answer)

219 Interviewer Why did you pick this method last for this problem?

220 Rebekah Because it doesn't leave the highest powers in order like it does in the box

method or the stacking one,

221 Rebekah and I feel like I'm more likely to mess up by putting them out of order, and

I'd rather not mess up.

222 Interviewer Ok, thank you.

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APPENDIX N: INTERVIEW WITH SARAH

Interviewer: Cameron Sweet

1 Interviewer The first one is to multiply the factors 26 and 7.

2 Sarah 26 and 7?

3 Interviewer Yes.

4 Sarah (writes 26, next line 7 PV aligned below 6, equal bar below) So do you

want me to like, what times what?

5 Interviewer So why did you set up 26 times 7 like that?

6 Sarah Cause it's hard, right? And 6 times 7 is 42 (writes 2 below equal bar, PV

aligned with factors one's)

7 Sarah and 4 goes there ( writes 4 above equal bar and below 2, PV aligned with

factors ten's)

8 Sarah and 7 times 2 is 14, plus 4 (points to 4) is 18 (writes 18 below equal bar

left of 2) so it's 182.

9 Interviewer Do you know any other methods for multiplying 26 and 7?

10 Sarah No.

11

12 Interviewer The next task is to multiply 53 and 10.

13 Sarah Divide?

14 Interviewer Multiply 53 and 10. These will all be multiplication.

15 Sarah Oh, 53 and 10 (writes 53, next line 10 PV aligned, equal bar below)

16 Sarah I just know that when there's like 10 there's always the zero in the back,

so it's 530 (writes 530 below equal bar, PV's aligned with factors)

17 Interviewer Do you know any other methods for multiplying 53 and 10?

18 Sarah No.

19

20 Interviewer The next task is to multiply 311 and 129.

21 Sarah (writes 311, next line 129 PV aligned, equal bar below) Well I'm gonna

(writes 2799 below equal bar, PV's aligned with factors)

22 Sarah (next line writes 622 below 279, next line writes 311, PV's aligned with

partial sums and factors, equal bar below)

23 Sarah (writes 311, next line 129 PV aligned, equal bar below) Well I'm gonna

(writes 2799 below equal bar, PV's aligned with factors)

24 Sarah (writes 40119 below equal bar, PV's aligned with factors) That's it.

25 Interviewer Do you know any other methods for multiplying 311 and 129?

26 Sarah No.

27

28 Interviewer The last integer multiplication task is to multiply 102 and 97.

29 Sarah (writes 102, next line 97 PV aligned, equal bar below, next line 714, next

line 918, equal bar below, next line 9994 PV's aligned for factors, partial

sums and product)

30 Interviewer Do you know any other methods to multiply those integers?

31 Sarah No.

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32

33 Interviewer This task is to multiply 5 and x+2.

34 Sarah (writes 5(x+2)) x plus 2?

35 Interviewer Yes, and why did you write it this way?

36 Sarah Because since there's x (points to x) I like to like factor it, and just put

parentheses just so I know it's separated.

37 Sarah So (next line writes x+2, next line writes 5 below 2, PV aligned) I like to

do this method, but (writes equal bar below, next line writes 5x+10, PV's

aligned with factors)

38 Interviewer Why do you like to do that method?

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

40 Interviewer Any reason why you find it easier?

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.

42 Interviewer How is this similar to when you're multiplying big numbers, as you said?

43 Sarah It's the how the way you set it up is similar to just multiplying numbers,

like how you write it down, like not in your head. So I find it easier.

44 Interviewer And what parts of this set up are similar to how you multiply numbers?

45 Sarah How you have like both number on the top and the bottom, and you

multiply from the bottom part.

46 Interviewer Why do you multiply from the bottom part?

47 Sarah Like for example from in this problem like 5 is on the bottom (points to 5

factors in PV), so you times it by like 5 on everything on the top (points

from 5 to x+2 in PV)

48 Interviewer Do you know any other method for multiplying 5 and x+2?

49 Sarah I do. I learn to multiply like, like that (writes curved arrows from 5 to x

and 2 in 5(x+2)) and you can get (writes =) 5x+10.

50 Interviewer Why would you choose this after your first method?

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.

52 Sarah But there's like other one, which is like the box, boxing, but I don't know.

53 Interviewer Boxing, ok. Where did you learn how to do this last year?

54 Sarah In algebra one.

55 Interviewer Was that here?

56 Sarah Yes.

57 Interviewer And you said that you don't like doing boxing?

58 Sarah No.

59

60 Interviewer This task is to multiply the square of the quantity x+5.

61 Sarah The square of the quantity? X plus 5.

62 Interviewer x+5, and the square of that quantity.

63 Sarah Oh (writes (x+5)^2) here you go.

64 Sarah I like to do the same method as the last problem (next line writes x+5,

next line writes x+5 directly below, equal bar below)

245

65 Sarah Times it by the bottom so 5 times 5 is 25, 5 times x is 5x (next line writes

x+25 below equal bar with PV's aligned with factors)

66 Sarah (next line writes x^2+5x with PV's aligned, equal bar below, next line

writes x^2+10x+25 with PV's aligned)

67 Interviewer Now on the last problem, when you multiplied you only had one line, and

here you have two. Why do you have two lines and why did you set them

up the way you did?

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.

71 Interviewer Do you know another method for multiplying?

72 Sarah No.

73

74 Interviewer This task is to multiply x+2 and x+5.

75 Sarah (writes (x+2)(x+5), next line x+2, next line x+5, equal bar below, next

line 5x+10, next line x^2+2x, equal bar below, next line x^2+7x+10 with

PV's aligned)

76 Sarah I set it up as the same as the last problem and I multiplied it from the

bottom (points to row x+5) and I line up the same, similar letters, or x

77 Sarah and x 2 plus 7 x plus 10 (points to x^2+7x+10)

78 Interviewer Does your method relate to how you multiplied integers?

79 Sarah No, yes.

80 Interviewer Yes?

81 Sarah Yes. Like, integers?

82 Interviewer So that was the first set of tasks, your numbers. (points to integer

multiplication tasks)

83 Sarah Yeah, yes.

84 Interviewer How does it relate?

85 Sarah The way it's set up like multiplying, this part (points to row x+5) can be

on the top (points to row x+2) and this part can be on the bottom (points

to x+2, back to x+5)

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

91 Interviewer Why?

92 Sarah I don't know, that's how I just learned.

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93 Interviewer And you were thinking of telling me no, they're not the same. Are there

differences in how you would multiply x+2 and x+5 versus how you

would multiply integers?

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.

98 Interviewer Do you know another method for multiplying x+2 and x+5?

99 Sarah No.

100

101 Interviewer This task is to multiply x^2+5x+2 and x+3.

102 Sarah (writes (x^2+5x+2)(x+3), next line x^2+5x+2, next line x+3, equal bar

below, next line 3x^2+15x+6, PV's aligned)

103 Interviewer And what did you do there?

104 Sarah Here I multiply by 3 (points to 3 in row x+3) 3 times 2 is 6, 3 times 5 is

15, but there's an x so I added the x (points to 15x) and 3 times x squared

is 3 x squared.

105 Sarah And now I'm going to add the (points to x in row x+3) I mean multiply

the x by the top numbers (points to row x^2+5x+2)

106 Sarah (next line writes x^3+5x^2+2x, equal bar below, next line writes

x^3+8x^2+17x+6)

107 Interviewer Do you know any other methods for multiplying these factors?

108 Sarah No.

109

110 Interviewer This task is to multiply x^2+3x+5 and x^2+x+2.

111 Sarah (writes (x^2+3x+5)(x^2+x+2), next line x^2+3x+5, next line x^2+x+2

with PV's aligned equal bar below)

112 Sarah (next line writes 2x^2+6x+10, next line writes x^3+3x^2+5x, next line

x^4+3x^3+5x^2, equal bar below, next line x^4+4x^3+10x^2+11x+10,

PV's aligned)

113 Sarah So what I did here was I lined up as the last problem, and this was like

harder because there's more numbers on the bottom, so it took longer,

114 Sarah but I lined up the like terms again and added up (points to partial sums)

and I got x^4+4x^3+10x^2+11x+10.

115 Interviewer Why are there more numbers on the bottom on this one?

116 Sarah Well the problem is longer than the last problem.

117 Interviewer Ok (now I understand what is meant by numbers on the bottom)

118 Sarah So there's more work to do.

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119 Interviewer Does your method for multiplying these factors relate to how you

multiply integers?

120 Sarah Yes.

121 Interviewer How?

122 Sarah Well how lining up is (points to rows of factors) and times-ing, like

multiplying the numbers from the bottom to the top.

123 Interviewer What do you mean by the bottom and the top?

124 Sarah So for this (points to row x^2+3x+5) I times it by 2 first (points to 2 in

row x^2+x+2)

125 Interviewer Times what by 2?

126 Sarah 2 (points to 2 in x^2+x+2) by 5 (points to 5 in x^2+3x+5) and 2 by 3

(points from 2 to 3x) and that's how I got 6 x,

127 Sarah and 2 times x squared (points from 2 to x^2 in x^2+3x+5) is 2 x (points to

2x^2)

128 Sarah and I also did the same for the x and the x 2 (points to x and x^2 in

x^2+x+2)

129 Sarah and I line up the like terms (points to columns of partial sums) and I got

those (points to x^4+4x^3+10x^2+11x+10)

130 Interviewer And how is that similar to how you would multiply integers?

131 Sarah I don't know.

132 Interviewer Are you up for doing two more tasks?

133 Sarah Sure.

134

135 Interviewer So this task is to multiply x^2+10 and x+4.

136 Sarah (writes (x^2+10)(x+4), next line x^2+10, next line x+4 directly below

without PV's aligned, equal bar below)

137 Sarah (next line writes 4x^2+40 with 40 below 4 and 10 of previous rows, x^2

below x^2 and x of previous rows)

138 Sarah (next line leaves space for place holder, writes +10x below + of previous

row)

139 Interviewer And what did you do there?

140 Sarah So I realized there's no x, just only the x, there's x 2 (points to 10x^2)

141 Interviewer There's no x where?

142 Sarah In the, um, answer? So I just like to like remind myself that there's a 10 x

before the 4 x 2 (points right of 4x^2), so I just like to put it there (points

to 10x)

143 Sarah and there's no, anything number with a second power, so I just like to

move it over here (writes x^3 left of 10x)

144 Sarah and put x^3 because x times x again (points from x in x+3 row to x^2 in

x^2+10 row) is x^3 (writes equal bar below partial sums)

145 Sarah So (next line writes x^3+4x^2+10x+40)

146 Interviewer Do you know any other methods for multiplying x^2+10 and x+4 ?

147 Sarah No.

148

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149 Interviewer This task is to multiply x+3x^2+2 and 3+x.

150 Sarah (writes (x+3x)^2+2) 2, just the 2 ?

151 Interviewer x+3x^2+2

152 Sarah Plus 2 (points to +2) and 3 x squared?

153 Interviewer So that's your first factor, and then the second factor is 3+x.

154 Sarah (finishes writing (x+3x)^2+2(3+x)) like that?

155 Interviewer I'm not sure?

156 Sarah Plus 2…

157 Interviewer So it's x plus 3 x squared

158 Sarah 3 x squared.

159 Interviewer Correct.

160 Sarah Ok wait (scratches (x+3x)^2+2(3+x)) I'm going to rewrite that.

161 Interviewer x plus 3 x squared plus 2

162 Sarah (next line writes x+3x^2+2)

163 Interviewer So that's your first factor, your second factor is 3+x)

164 Sarah (finishes writing (x+3x^2+2)(3+x)) Ok. I've gotta rearrange this part

because I like to have the bigger exponents in the front.

165 Sarah (next line writes 3x^2+x+2)

166 Interviewer Why do you like to have the bigger exponents in the front?

167 Sarah It's more easy to line up the numbers. (next line writes 3+x, equal bar

below)

168 Sarah Oh, and then I'm going to switch that too (scratches 3 and x, writes x

above 3 and 3 above x, PV's are now aligned)

169 Sarah (next line writes 9x^2+3x+6, next line writes 3x^3+x^2+2x, equal bar

below, next line 3x^3+10x^2+5x+6)

170 Sarah I also switched the 3+x (points to (x+3x^2+2)(3+x)) just so I like to have

the x in the front, so it's easier to line up the like terms when you multiply.

171 Interviewer Do you know any other method for multiplying these factors?

172 Sarah No.

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APPENDIX O: INTERVIEW WITH JOHN

Interviewer: Cameron Sweet

1 Interviewer The first task I have is to multiply 26 and 7.

2 John (writes 26, next line 7 PV aligned)

3 Interviewer Why did you set these up like this?

4 John So I can multiply 7 by each (points to 2 and 6, writes equal bar below 7)

5 John And then I go down right here, so 7 times 6 is, 42 I think, yeah 42 (writes

2 below equal bar, PV aligned with 7 and 6 above)

6 John And then move it up here (writes 4 above 2 in 26) and then 14, 15, 16, 17,

18 (writes 18 below equal bar left of 2, PV's aligned) so 182.

7 Interviewer Do you know any other methods for multiplying 26 and 7?

8 John I guess I could go backwards and do 7 times 20.

9 Interviewer Could you show me please?

10 John So 7 times 20 (writes 20, next line 7 PV aligned, equal bar below, next

line 140) is 140,

11 John and then add 42 to that (writes +42 right of 140)

12 Interviewer Why did you add 42 to the 140 ?

13 John Cause then you do 7 times 6 (writes 7*6 above 42 with down arrow to 42)

and then they 182 (writes =182 left of 42) is another way.

14 Interviewer Do you know any other methods for multiplying 26 and 7?

15 John Not really. Well, you could write 26 out seven times and then add them.

16 Interviewer Could you show me?

17 John (writes 26+26+26+26+26+26+26=) and then you just add them up, but it

wouldn't be very effective I don't think.

18 Interviewer Why not?

19 John Cause it would take longer and way more math.

20

21 Interviewer The next task I have for you is to multiply 53 and 10.

22 John (writes 53, next line 10, PV aligned, equal bar below) So write it out, just

like that?

23 John So since it's a 10, I can just add a zero to the end of 53 (writes 530 right of

work) and get it that way.

24 Interviewer Why?

25 John Because it's a 10, and when you multiply by 10 it always is going to end in

zero, I guess, or it would be a multiple of 10, so yeah.

26 Interviewer Do you know any other ways to multiply 53 and 10?

27 John Well I could go like I did here (points to PV 26 and 7) so zero times 3 and

zero times 5 (writes 00 below PV 53 and 10)

28 John and add a zero (next line writes 0 in units place) 1 times 3 and 1 times 5

(finishes writing 530 below 00, PV's aligned with factors)

29 John and then you add, but since these are both zeroes (points to 00) that's just

the answer.

30 Interviewer What's just the answer?

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31 John 530

32 Interviewer Why did you choose this method after the adding zero you described

earlier?

33 John Like why'd I choose this one?

34 Interviewer I'm wondering why you chose this one first (points to work on right) or

this one second (points to PV)?

35 Interviewer Why you chose this one (points to work on right) before this one (points to

PV) or this one (points to PV) after that one (points to work on right) ?

36 John Well this one (points to work on right) all you have to do is add it, take the

zero and move it right behind it, so it doesn't take much work,

37 John where as this (points to PV) you actually have to do a little bit of

multiplying, even if it is easy, 1 times zero, but it's a little work than just

adding a zero to it.

38 Interviewer Thank you. Do you know any other methods for multiplying 53 and 10?

39 John Not really.

40

41 Interviewer The next task is to multiply 311 and 129.

42 John (writes 311, next line 129, PV aligned, equal bar below) So for this one,

I'll do what I did over here (points to PV 53 and 10) by multiplying each

one out

43 John So (next line writes 2799) then you add a zero for the place holder (next

line writes 6220) now I have to add another place holder (next line writes

31100)

44 Interviewer What were the place holders for?

45 John Holding these places right here (points to 129) like this one (points to 0 in

6220) was holding that one (points to 9 in 129)

46 John and since you went out three (points to 129) you have to cover those two

(points to 00 in 31100)

47 John then you add all these so (writes equal bar below, next line 40,119, partial

sums and product PV aligned) you get 40,119.

48 Interviewer Do you know any other methods for multiplying 311 and 129?

49 John No.

50

51 Interviewer The last integer multiplication task I have is to multiply 102 and 97.

52 John (writes 102, next line 97, PV aligned, equal bar below, next line 714)

53 John and your place holder (next line writes 0 below 4 in 714, finishes writing

9180) so (next line writes 9894, partial sums and product PV aligned)

9894

54 Interviewer Do you know any other methods for multiply 102 and 97?

55 John Mmm-mm.

56

57 Interviewer The first task I have here is to multiply 5 and x+2.

58 John (writes 5*x+2) Is x plus 2 in parentheses?

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59 Interviewer x+2 is one of the factors and 5 is the other, and you're going to multiply

those two factors. I'm not giving you these in writing so I can see how

you're writing these out.

60 John Ok, I guess I'll do it two ways. (writes 5(x+2)) so if you do it this way, 5

times x is 5 x and 5 times 2 is 10 (next line writes 5x+10, standard dist.)

61 Interviewer Where did you get 5x?

62 John You're going to use, I think it's called FOIL, and you do 5 times x (writes

curve from 5 to x) then times 2 (writes curve from 5 to 2)

63 John Then this way would just be (points to 5*x+2, next line writes 5x+2) 5 x

plus 2 cause you're just times-ing that and that (points to 5 and x)

64 John you're not factoring it out, just times-ing it by that.

65 Interviewer Why did you choose this method?

66 John I chose this one cause it's the only way I know how, and it seemed pretty

easy to do it that way. I was trying to figure it out without…

67 John I guess I could do it like this (writes x+2, next line 5, PV aligned equal bar

below) and then factor it out (next line writes 5x+10, PV's aligned)

68 John You could do it like that. There's another way!

69 Interviewer And why did you choose,… so you told me that your first method was the

only one you knew… ?

70 John Yeah, I just thought about that.

71 Interviewer You just thought about this one?

72 John Yeah, I just remembered this way.

73 Interviewer Why did you choose this one after your first method?

74 John After this method right here? (points to standard dist.)

75 Interviewer Yeah, why did you choose this one (points to 5*x+2) after this one (points

to 5(x+2)) ?

76 John Didn't you ask if we had any other ways?

77 Interviewer I didn't ask that yet. I don't think I asked that (I didn't in this task)

78 John Just showing other possible ways I guess.

79 Interviewer But I was going to ask that. Do you know any other ways to multiply 5

and x+2 ?

80 John The lattice, I think it's called, so the box (writes 1 by 2 boxes, x+2 above,

5 right of boxes)

81 John and you split each letter and number in it's box (points to x and 2 above

boxes) and you do 5 times, or x times 5 (writes 5x in box)

82 John and 2 times 5, 10 (writes 10 in other box) and then you just combine those,

so (writes below boxes) 5x+10, and that'd be the lattice method.

83 Interviewer Why did you choose this one last?

84 John I guess I just remembered it last.

85

86 Interviewer Next task I have for you is to multiply the square of the quantity x+5.

87 John (writes (x+5)^2) And so you multiply the square of the quantity,… so…

88 Interviewer Do you have a question for me, or are you unsure what to do with this?

260

89 John I'm trying to figure out what the question your asking is exactly? Trying to

set it up in my mind, if it's set up like that or something else.

90 Interviewer I think your set up is right.

91 John Ok. Just with the FOIL or factoring, you'd square x (writes curve from x to

2 exponent)

92 John so x squared (left of work writes x^2) and you add, and square 5, which

would be 25 (finishes writing x^2+25)

93 Interviewer And why did you choose this method?

94 John I think it's the only way to do it?

95 Interviewer So you don't know any other ways to go about…

96 John Mmm-mm

97

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.

101 Interviewer Why?

102 John It's more like doing a normal multiplication problem, like 5 times 10 or

something like that.

103 John It's more like that than this (points to (x+2)*(x+5))

104 Interviewer What makes it like doing 5 times 10 ?

105 John Well, that was just an example, but I'm saying that in the way that you set

up a multiplication problem you have, sometimes, not always,

106 John you have one, a number right there (points to x+2 in PV) and a number

under it (points to x+5) and then you multiply it out like that,

107 John I more used to doing that (points to PV) than that (points to (x+2)*(x+5)),

I think it's a bit easier

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

110 John Then you have to put a zero here (next line writes 0 below 10)

111 Interviewer Why do you have to put a zero there?

112 John For your place holder right there. Then (finishes writing x^2 2x 0, PV's

aligned)

113 Interviewer What did you need the place holder for?

114 John To, for not, when I moved to times an x by things (points to x in x+5), so I

had to put, place holder for 5 I guess (points to 5 in x+5).

115 John Then you got to combine them by adding, so (next line writes x^2+7x+10,

PV's aligned)

116 Interviewer So now that you have this all multiplied out, and I'll even let you refer

back to your earlier work with integers if you like,

261

117 Interviewer How is, you said that this was similar to how you normally multiply

numbers.

118 John Mmm-hmm. How is it similar?

119 Interviewer Yes.

120 John Well, mainly in the way that you set it up like this (points to PV)

121 John On here (points to integer multiplication) it's set up with one number

above and one number below,

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)

129 John I can finish this one now (points to (x+2)*(x+5)) right here.

130 John So factoring them out like that (writes curves from first x to second x and

5, and from 2 to second x and 5 in standard dist.)

131 John its x^2 and then 5x (next line right of x^2 writes +5x) and then you're done

cause there's no more

132 John 2 times x, 2x, 2 times 5 is 10 (finishes writing x^2+5x+2x+10)

133 John and then you combine these cause they're all the same, same x I think is

what it's called (next line writes 7x)

134 John So you just drop this down cause these are not the same (writes down

arrow below x^2, next line x^2 below down arrow)

135 John It has a squared, that one doesn't (points to 7x, finishes writing

x^2+7x+10) and that's it.

136 Interviewer And why did you choose to do this method after your first method?

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

one (points to PV)

138 Interviewer Do you know any other methods for multiplying x+2 and x+5?

139 John Yeah, the lattice method that I showed on one of those papers, you set up

your box (writes 2 by 2 boxes, x+2 above, x+5 each right of boxes)

140 John and you have one column for each number, like that, then you do (writes

partial sums in boxes)…

141 John and then you have to combine diagonally (circles boxes with 2x and 5x)

142 Interviewer Why do you combine diagonally?

143 John That's how that method works and these (points to boxes with 2x and 5x)

both have the same, um, I'm forgetting the word,

262

144 John but the same x's, they're not x squared, x cubed or anything, so you can

combine them.

145 John So (writes 7x below boxes) and then you can't combine this with anything

(points to box with 10) and you can't combine that (points to box with x^2)

146 John so it's just (finishes writing x^2+7x+10) that

147 Interviewer Do you know any other methods for multiplying x+2 and x+5?

148 John Mmm-mm.

149

150 Interviewer This next task is to multiply x^2+5x+2 and x+3.

151 John (writes (x^2+5x+2)*(x+3))

152 Interviewer Why did you write these like this?

153 John Well, I wrote them out, just as you said, and you said multiply it, so you

multiply anything like that, so I just wrote it out, like you said it I guess.

154 John This time I'll factor it out before I do the other methods.

155 John So x times, x squared times x is x cubed, x squared time 3x, sorry x

squared plus 3 is 3 x squared (next line writes x^3+3x^2)

156 John And then you go 5 x times x equals 5 x squared (next line writes 5x^2

directly below 3x^2, hybrid of standard dist. and PV)

157 Interviewer Why did you write the 5 x squared here?

158 John You could put it up here (points right os 3x^2) but I prefer right here

(points to 5x^2 on next line) because it matches up all the similar x's I

guess.

159 John So then when you add them it's easier, and this (points to x^3) just drops

down. I guess it's visually more pleasing.

160 John 5x times 3 is 10 x (right of 5x^2 writes 10x) sorry, 15 x (scratches 0,

writes 5, finishing partial sum 5x^2+15x)

161 John And then 2 times x, so 2 x, then 2 times 3, 6 (next line writes 2x+6 with 2x

directly below 15x)

162 John Then this drops down (writes down arrow below x^3, next line x^3)

163 John These combine, so (points to column of 3x^2 and 5x^2, writes +8x^2

below and right of x^3)

164 John These combine, so (points to column of 15x and 2x, writes +17x below

and right of 8x^2)

165 John And this (points to 6) has nothing to combine with, so it's just 6 (writes +6

below and right of 17x, to finish product x^3+8x^2+17x+6)

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.

168 Interviewer More organized than?

169 John The other way being (writes x^2+5x+2, next line x+3, PV aligned, equal

bar below)

170 John the main two ways that I would probably do this would be this equation

(points to PV) or like that (points to hybrid)

171 Interviewer So you're saying that your first method is more organized than your

second?

263

172 John Yeah, it really depends.

173 Interviewer What does it depend on?

174 John Kind of what I'm feeling, I like them both, I think they both do, I think

they're both better than the box. I don't like the box too much.

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.

177 John So this way (points to PV) you do (next line writes 3x^2+15x+6, PV's

aligned)

178 John and then place holder if you want (next line writes 0 below 6, finishes

partial sum? x^3 5x^2 2x 0, PV's aligned)

179 John Then add them (next line writes x^3+8x^2+17x+6)

180 John Then I'll do the box method, I don't think you've asked me to show you

other ways, but…

181 John (writes 2 by 3 boxes, x^2+5x+2 above boxes, x+3 each right of boxes)

182 Interviewer And why do you write these polynomials where you do?

183 John Each number or letter (points to x^2) has its own column, so x has its own

(points to row with x right of boxes)

184 John 3 has its own (points to row with 3 right of boxes) x^2 has its own (points

to column with x^2 above)

185 Interviewer Why?

186 John I'm not sure, I guess that's how I learned to do this method, and it makes it

so that when you combine it, it's always diagonal.

187 John (writes partial sums in boxes) 2 times x, I may have done this wrong, hold

on

188 John (looks back to PV, hybrid) I guess 6 goes here (writes 6 in box row 3,

column 2)

189 Interviewer Why is that 6 ?

190 John Because 2 times 3, and I already put 2 x right there (points to box above 6

in row x, column 2) and you wouldn't need to do it again, I guess.

191 John Combine those (points to diagonals, writes 8x^2+15x below boxes,

finishes writing product x^3+8x^2+17x+6) like that.

192 John And that's all the ways I know.

193

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.

199 John (right of 3x^3 writes 3x^2 6x, next line 5x^2 5x 10, PV's aligned for

partial sums?)

264

200 John Combine them (points to column of x^3 and 3x^2, next line writes 4x^3)

201 John And then add this (points to column of 2x^2, 3x^2 and 5x^2, writes 10x^2

below and right of 4x^3)

202 John And this (points to column of 6x and 5x) would be (writes below column,

right of 10x^2) 11 x

203 John And 10 (writes 10 below 10, right of 11x, writes + between terms of

product)

204 John And you drop this down (writes down arrow below x^4, left of 4x^3,

finishing product x^4+4x^3+10^2+11x+10) like that.

205 Interviewer Does this method relate to how you multiply integers?

206 John On the one? (points to integer multiplication tasks)

207 Interviewer Just in general. If you want to relate it back to how you multiplied

integers, you can do that.

208 John Yeah, it does.

209 Interviewer It does? How?

210 John In this way (points to 311 times 129) you just set it up with it on top and

bottom,

211 John while in this (points to polynomial hybrid) it's just side to side, but you're

doing the same thing I guess,

212 John you're factoring it out like you were on here (points back to 311 times 129)

so when you times it, you factor this (points to 129 digits) to each number

213 John So it's the same thing, just in a different way, I guess.

214 Interviewer How is it different?

215 John They're not above and below each other (points to (x^2+3x+5)(x^2+x+2))

they're right next to each other instead.

216 Interviewer Are there any other ways that it's different?

217 John You don't have to add zero and you don't have to add a place holder

(points to (x^2+3x+5)(x^2+x+2))

218 Interviewer Why not?

219 John I guess you could add a place holder in this, but I don't,

220 John you could add a zero right here (points left of 3x^3 3x^2 6x) or two

zeroes (points left of 5x^2 5x 10), but I don't do that.

221 John Why don't you? I don't know.

222 Interviewer Do you know any other methods for multiplying these factors?

223 John Mmm-hmm, you do the above one like this (writes x^2+3x+5, x^2+x+2,

PV aligned, equal bar below)

224 John And then, like I was saying, you factor it out just like you did in that

(points to hybrid) except you start with these ones (points to 2)

225 John (next line writes 2x^2 6x 10) add a place holder (next line writes 0

directly below 10, x^3 3x^2 5x left of 0)

226 John These are all positive (points to partial sums?) I'm just not adding anything

because, if they were negative I'd add something, but they're positive.

227 John Then two place holders this time (next line writes 0 0 below 5x 0) for

those two (points to x+2)

265

228 John (left of 0 0 writes x^4 3x^3 5x^2) then you combine them all (next line

writes 11x+10)

229 Interviewer How are you combining all?

230 John I'm adding, just adding through all of them (finishes writing product

x^4+4x^3+10^2+11x+10)

231 Interviewer Does this method relate to how you multiply numbers or integers?

232 John Yeah, this is probably more closer since you have place holders (points to

0's)

233 John and you're multiplying (points to 2 in (x^2+x+2)) you're starting from the

right side or, I guess the right side of the equation.

234 John And you're factoring out the same. Instead of going side by side, you're

now above and below,

235 John so it's probably more similar to these (points to integer multiplication

tasks) than this is (points to polynomial hybrid)

236 John And then, I'll do the box or lattice method (writes 3 by 3 boxes, x^2+3x+5

above boxes, x^2+x+2 each right of boxes)

237 John (writes partial sums in boxes) Then you combine diagonally (circles boxes

with 3x^3 and x^3)

238 Interviewer Why are you combining diagonally?

239 John They're all the same, they have the same x's I guess, x squared, x squared,

x squared (circles boxes with 2x^2, 3x^2 and 5x^2)

240 John x cubed, x cubed (points to circled boxes with 3x^3 and x^3)

241 John (circles boxes with 6x and 5x) these ones all have single x's.

242 John Then that one just drops down (points to upper left box with x^4, writes

x^4 below boxes)

243 John (finishes writing x^4+4x^3+10^2+11x+10 below boxes)

244 Interviewer Do you know any other methods for multiplying these?

245 John No.

246 Interviewer This lattice method that you did last, does it relate to how you multiplied

integers?

247 John Yeah, I guess? They're all going to relate to it cause they're all multiplying.

So it's going to relate to it in the way that they're being multiplied.

248 John It's a little bit different.

249 Interviewer How is it different?

250 John You're combining diagonally instead of straight down and there are no

place holders, it's in a box for one, so there's a few.

251

252 Interviewer This task is to multiply x^2+10 and x+4.

253 John (writes (x^2+10)) and x squared plus 4 ?

254 Interviewer x plus 4.

255 John x plus 4 (finishes writing (x^2+10)(x+4)) I'll to use FOIL. You said

multiply, correct?

256 Interviewer Yes.

266

257 John Ok. x^2 times x (writes curve from x^2 to x, next line writes x^3, standard

dist.) x cubed.

258 John x^2 times 4 (writes curve from x^2 to 4, next line writes +4x) 4x.

259 John Then 10 times x (writes curve from 10 to x) is 10 x (line below x^3+4x

writes 10x below 4x, NOT PV aligned)

260 John Then 10 times 4 is 40 (writes +40 right of 10x)

261 John Add those (next line writes 14x below 10x and 4x) drop that down (writes

+40 below +40, right of 14x)

262 John drop that down (writes x^3+ below x^3, left of 14x, product reads

x^3+14x+40)

263 Interviewer Do you know any other methods for multiplying x^2+10 and x+4 ?

264 John Uh-huh (writes x^2+10, next line x+4 directly below, NOT PV aligned,

equal bar below) I'm going to do this way real quick.

265 John 4 times 10 is 40 and 4 times x^2 (next line writes 4x^2 40 directly below

+4) is 4 x^2, there's your place holder (next line writes 0 below 40)

266 John 10 times x was (writes 10x left of 0) this is 10 x, and then x times x

squared, which is x cubed (writes x^3 left of 10x)

267 John So you combine those, which is 40 (next line writes 40 below 0)

268 John and these don't combine (points to 10x and 4x^2) so could just drop it

down, that's 10 x (writes 10x + left of 40)

269 John (mutters about problem, looks at standard dist.) then 4 x squared and x