Analyzing common algebra-related misconceptions and errors of middle school students.

371 Pages·2016·14.08 MB·English

by  Sarah B. Bush

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Electronic Theses and Dissertations
12-2011
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Sarah B. Bush
University of Louisville
Follow this and additional works at: https://ir.library.louisville.edu/etd
Part of the Algebra Commons, and the Junior High, Intermediate, Middle School Education and
Teaching Commons
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Bush, Sarah B., "Analyzing common algebra-related misconceptions and errors of middle school
students." (2011). Electronic Theses and Dissertations. Paper 187.
https://doi.org/10.18297/etd/187
This Doctoral Dissertation is brought to you for free and open access by ThinkIR: The University of Louisville's
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ANAL YZING COMMON ALGEBRA-RELATED MISCONCEPTIONS AND
ERRORS OF MIDDLE SCHOOL STUDENTS
By
Sarah B. Bush
B.S., Indiana University Southeast, 2005
M.ED., Indiana Wesleyan University, 2008
A Dissertation Submitted to the Faculty of the
College of Education and Human Development of the University of Louisville
in Partial Fulfillment of the Requirements
for the Degree of
Doctor of Philosophy
Department of Teaching and Learning
University of Louisville
Louisville, KY
December 2011
-------- ---~----
Copyright 2011 by Sarah B. Bush
All rights reserved
ANALYZING COMMON ALGEBRA-RELATED MISCONCEPTIONS AND
ERRORS OF MIDDLE SCHOOL STUDENTS

By

Sarah B. Bush
B.S., Indiana University Southeast, 2005
M.ED., Indiana Wesleyan University, 2008

A Dissertation Approved on

October 6, 2011

By the following Dissertation Committee:

Karen Karp, Dissertation Director

Maggie McGatha

Sam Stringfield

E. Todd Brown

William Bush
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DEDICATION
In loving memory of my grandmother (Me-Me)
Lenelda June Sieg
a great role model who taught me the value of kindness.
&
To my dad
Norman Anthony Thomas Jr.
who taught me the value of hard work.
iii
ACKNOWLEDGMENTS
This dissertation represents a milestone and stepping stone in my journey as a
lifelong learner and contributor to the field of mathematics education. Without the
guidance, support, patience, and critical advice given to me by those involved in my
doctoral journey, this dissertation would not be complete.
I will start by thanking Dr. William lamski, my undergraduate advisor at Indiana
University Southeast, who inspired and encouraged me to pursue my doctorate in
mathematics education. If it were not for him, my life would not have taken this path nor
would I have been led to the University of Louisville.
Most importantly, I would like to thank Dr. Karen Karp, my advisor and
dissertation chair. In addition to being a uniquely great advisor, Dr. Karp also became a
wonderful role model as I observed her dedication, success, and passion for the field of
mathematics education, my Ph.D. Mom, a colleague I will always feel privileged and
honored to work with, and a wonderful friend. From the beginning stages of my
coursework, Dr. Karp challenged me far beyond the program requirements and as a
result, truly prepared me for a new career in academia through invaluable research,
service, and teaching experiences. While Dr. Karp's expectations were extremely high,
she provided wonderful support, coaching, and mentoring so that I could be successful. I
cannot ever thank her enough for being a rigorous advisor who always took time to
provide me with timely and constructive feedback so that I could reflect and grow as a
teacher and learner of mathematics education.
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I would also like to thank my committee members, Drs. E. Todd Brown, William
Bush, Maggie McGatha and Samuel Stringfield for their help, support, and feedback
throughout my dissertation journey. Specifically, I would like to thank Dr. Brown and Dr.
McGatha for their valuable guidance from the beginning as I chose my topic and for
extremely helpful feedback throughout the entire process. I would like to thank Dr. Bush
for his extra time and help during the initial outline phase of my proposal and for his
detailed editing comments and thoughtful suggestions throughout the remainder of the
journey. I would like to thank Dr. Stringfield for his significant contributions as I drafted,
edited, and fine-tuned my proposal. I will forever be indebted and grateful for their
participation and role in my dissertation journey.
I would like to thank the following people who helped contribute to my
dissertation either through help on a specific task or through their caring support: Robert
Ronau, Amy English Hunter, Ryan Higgins, Elizabeth Popelka, Christopher Rakes,
Namok Choi, and Stefanie Livers.
Finally, I would like to thank my husband and best friend Sam, who was
enthusiastically my biggest supporter and fan on this journey. He is always willing to
support my dreams and goals, listen to my frustrations, sacrifice our time for the sake of
my dissertation, celebrate in my accomplishments, and encourage me to strive for more.
v
ABSTRACT
ANAL YZING COMMON ALGEBRA-RELATED MISCONCEPTIONS AND
ERRORS OF MIDDLE SCHOOL STUDENTS
Sarah B. Bush
October 6, 2011
The purpose of this study was to examine common algebra-related
misconceptions and errors of middle school students. In recent years, success in Algebra I
is often considered the mathematics gateway to graduation from high school and success
beyond. Therefore, preparation for algebra in the middle grades is essential to student
success in Algebra I and high school. This study examines the following research
question: What common algebra-related misconceptions and errors exist among students
in grades six and eight as identified on student responses on an annual statewide
standardized assessment?
In this study, qualitative document analysis of existing data was used in order to
analyze sixth- and eighth-grade student responses on a statewide standardized
assessment. Secondary data sources consisted of Algebra I student responses which were
also analyzed qualitatively using document analysis and follow-up interviews with key
informants.
The primary analysis indicated that (l) numerous misconceptions and errors
identified in the review of literature were present on both the sixth- and eighth-grade
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open-responses; (2) basic computational errors with whole numbers (a secondary skill),
were found consistently throughout the sixth- and eighth-grade open-responses; (3) a
greater number of misconceptions and errors identified in the review of the literature
were present on the eighth-grade items than were found on the sixth-grade items; (4)
students often lost points for reasons other than mathematical misconceptions or errors;
and (5) some refinement and reorganization of Welder's (2007) framework could prove
beneficial when using the framework for data analytic purposes.
The results of this study provided information about the common misconceptions
and errors students possess on prerequisite algebra skills. The findings revealed common
algebra misconceptions and trends that can help guide instruction for middle school
mathematics teachers. The findings have direct implications for classroom practice and
further confirm the need for strong and knowledgeable teachers of mathematics at the
elementary and middle grades.
The researcher suggests that schools, both in the state whose standardized
assessment was examined as well as other states, use this information to help build
awareness of common prerequisite algebra-related misconceptions and errors in
elementary and middle grades mathematics teachers.
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TABLE OF CONTENTS
PAGE
DEDICATION....................................................................... 111
ACKNOWLEDGMENTS........................................... ...............
IV
ABSTRACT.. . .. . ... ... ... . .. . .. .. . ... ... ... ... ..... . ... ... .................. ... .....
VI
LIST OF TABLES.......................... ......................................... X111
LIST OF FIGURES....................... ....................................... .... XIX
CHAPTER I: INTRODUCTION............................................... ... 1
Overview ........................................................................... .
Problem Statement................................................................ 1
Existing Literature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3
This Specific Study... ... ...... . .. ... ... ... ......... ... ............................ ... 7
Research Question............................................................. ... 8
Significance of Study...... . .. .. . ... ... ... ... ... ... .............. ........ ... ... ... 8
Delimitations...... ................................................................. 8
Assumptions...... ...... .. .... . .. .. .. .. .. .. .. ...... .. ......................... ... ... 9
Definition of Terms. ..................................... .......................... 9
Organization of the Dissertation.... .. .. . ... ...... ...... ... ... ......... ... ...... ..... 10
Summary....... ........ ... ... ......... ...... ...... ...... ......... ... ... ............ ..... 11
CHAPTER II: LITERATURE REVIEW.......................................... 12
Introduction............................................................................ 12
Literature Search... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 13
Algebra Curriculum................................................................... 14
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