AN ANALYSIS OF CONTENT KNOWLEDGE AND COGNITIVE ABILITIES AS
FACTORS THAT ARE ASSOCIATED WITH ALGEBRA PERFORMANCE
By
Tamika Ann McLean

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Educational Psychology and Educational Technology – Doctor of Philosophy
2017

ABSTRACT
AN ANALYSIS OF CONTENT KNOWLEDGE AND COGNITIVE ABILITIES AS
FACTORS THAT ARE ASSOCIATED WITH ALGEBRAIC PERFORMANCE
By
Tamika Ann McLean
The current study investigated college students’ content knowledge and cognitive
abilities as factors associated with their algebra performance, and examined how combinations of
content knowledge and cognitive abilities related to their algebra performance. Specifically, the
investigation examined the content knowledge factors of computational fluency, numeracy skills,
fraction knowledge, understanding of equivalence, and algebraic reasoning skills, and the
cognitive abilities of spatial visualization, crystallized intelligence, and fluid intelligence. A
multiple regression analysis found that while controlling for gender, the highest math course
taken, and the number of years since an algebra course, fraction knowledge and the spatial
visualization ability of spatial imagery were statistically significant predictors of algebra
performance along with the control variable identifying whether or not participants had taken at
least one calculus course. In addition, cluster analysis identified six content knowledge and
cognitive ability profiles, with varying levels of both content knowledge and cognitive abilities
observed across the six clusters. The six profiles – characterized as Low All, Moderate-Low All,
Moderate-High MASMI, Moderate-Low Spatial, Moderate-High All, and High Spatial – varied
somewhat in terms of their algebra performance scores. In particular, the participants in the High
Spatial cluster group and participants in the Moderate-High All cluster group had similarly high
algebra performance scores, which were significantly higher than performances scores observed
for participants in the other cluster groups. Additionally, the participants in the other cluster
groups exhibited similar low algebra performance scores to each other except for participants in

the in the Moderate-Low Spatial and Low All cluster groups. Participants in the Moderate-Low
Spatial cluster group had significantly higher algebra performance scores than participants in the
Low All cluster group. The differences in algebra performance scores among cluster groups
suggested that the observation of higher algebra performance occurred when participants had
strong spatial visualization skills, strong fluid intelligence skills, and high content knowledge or
when participants had strong fraction knowledge, numeracy skills, algebraic reasoning skills, and
spatial imagery skills.

Copyright by
TAMIKA ANN MCLEAN
2017

To Mama Hawkins whose simple words of encouragement inspired a little girl to achieve more
than she ever dreamed.

v

ACKNOWLEDGEMENTS

It has taken me more years than I would have liked to finish graduate school, but the journey has
been at times hard and difficult, but mostly fun and enjoyable because of the people around me.
It is these people that I would like to acknowledge and give my heartfelt thanks. First, to my
advisor Lisa, I say thank you for taking me in and nurturing me the last year. I know our research
interests are quite different, but working with you has pushed me to be a better researcher and
scholar. To my undergraduate research assistants Maggie, Kathryn, and Harmony, thanks for
your tireless efforts in collecting and coding data. Without your help, I would never have
completed my dissertation. To the people in the Center for Ethics and Humanities in the Life
Sciences, I say thank you for generously giving your resources, respecting my time, and your
endless words of encouragement and support. To my graduate school friends and colleagues
DaSha, Chiharu, Rani, William, Emily, and Kristy, I say thank you for your listening ears and
endless words of advice and support. Talking with you made me feel less of an imposter, and
showed me I was not alone in this journey. Lastly, to my parents Patricia and Thaddeus, I say
thank you for being my guiding light. You always brought me back when I felt myself drifting
away. Even more thanks to my mother who has been my sounding board throughout this whole
dissertation process. Sometimes it was your random and weird comments that sparked ideas and
regenerated my thinking when it had fizzled out.

vi

TABLE OF CONTENTS

LIST OF TABLES ......................................................................................................................... ix
LIST OF FIGURES ........................................................................................................................ x
CHAPTER 1: Introduction ............................................................................................................. 1
CHAPTER 2: Literature Review .................................................................................................... 5
Cognitive Abilities, Content Knowledge, and Algebra Performance ......................................... 6
Cognitive Abilities Related to Algebra Performance ................................................................ 10
Fluid intelligence. ................................................................................................................. 11
Crystallized intelligence. ....................................................................................................... 13
Spatial abilities. ..................................................................................................................... 14
Content Knowledge Related to Algebra Performance .............................................................. 16
Understanding of numbers and operations............................................................................ 17
Proficiency with fractions. .................................................................................................... 18
Understanding of equivalence. .............................................................................................. 19
Algebraic Reasoning. ............................................................................................................ 20
Current Study ............................................................................................................................ 21
Rationale for design. ............................................................................................................. 22
Research questions and hypotheses ...................................................................................... 24
RQ 1: Which forms of content knowledge and cognitive abilities most strongly predict
algebra performance? ........................................................................................................ 25
RQ 2a: What combination of content knowledge and cognitive abilities naturally occur in
students who have studied algebra? .................................................................................. 26
RQ 2b: How do students with these different content knowledge and cognitive abilities
profiles perform in algebra? .............................................................................................. 26
CHAPTER 3: Method ................................................................................................................... 28
Participants ................................................................................................................................ 28
Assessments and Measures ....................................................................................................... 28
Cognitive abilities. ................................................................................................................ 28
Fluid and crystallized intelligence. ................................................................................... 29
Fluid intelligence index (FII). ....................................................................................... 29
Crystallized intelligence index (CII). ............................................................................ 30
Spatial visualization .......................................................................................................... 30
Content knowledge. .............................................................................................................. 31
Understanding of numbers and operations. ...................................................................... 31
Computational fluency. ................................................................................................. 32
Numeracy. ..................................................................................................................... 33
Fractions. ........................................................................................................................... 34
Equivalence. ...................................................................................................................... 35
Algebraic reasoning. ......................................................................................................... 36
Algebra performance............................................................................................................. 37
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Participant demographic survey ............................................................................................ 37
Data Collection Procedures ....................................................................................................... 38
Data Analysis ............................................................................................................................ 40
Research question 1. ............................................................................................................. 40
Research question 2a ............................................................................................................. 40
Research question 2b. ........................................................................................................... 41
CHAPTER 4: Results ................................................................................................................... 43
Preliminary Analyses ................................................................................................................ 43
Regression Analysis .................................................................................................................. 45
Preliminary test for assumptions. .......................................................................................... 45
Regression model. ................................................................................................................. 47
Cluster Analysis ........................................................................................................................ 47
Cluster labels ......................................................................................................................... 53
Demographic characteristics. ................................................................................................ 58
Cluster Membership and Algebra Performance ........................................................................ 61
CHAPTER 5: Discussion.............................................................................................................. 69
Predictors of Algebra Performance ........................................................................................... 69
Cognitive Abilities and Content Knowledge Profiles and Algebra Performance ..................... 73
Variable-Oriented vs. Person-Oriented: What do the differences mean? ................................. 79
Limitations ................................................................................................................................ 81
Implications ............................................................................................................................... 85
Conclusion ................................................................................................................................. 87
APPENDICES .............................................................................................................................. 88
Appendix A: Tables .................................................................................................................. 89
Appendix B: Figures ............................................................................................................... 105
Appendix C: Assessments and Measures ................................................................................ 113
Appendix D: Research Protocols ............................................................................................ 128
Appendix E: Item Level Analyses .......................................................................................... 134
REFERENCES ........................................................................................................................... 140

viii

LIST OF TABLES

Table 1 Basic Information about Assessments and Measures ...................................................... 89
Table 2 Descriptive Statistics for Assessments and Measures ..................................................... 90
Table 3 Radomized Testing Orders .............................................................................................. 92
Table 4 Bivariate Correlations for Assessments and Measures .................................................... 91
Table 5 Analysis of Item Categories for All Content Knowledge Assessments .......................... 93
Table 6 Summary of Independent Samples T-Test and Mann-Whitney U Test Analysis for
Gender Differences ....................................................................................................................... 94
Table 7 Summary of Multiple Regression Analysis ..................................................................... 95
Table 8 Ward’s Method Agglomoration Schedule for Clusters 1-9 ............................................. 96
Table 9 Indices for Ward’s Method Cluster Solutions ................................................................. 97
Table 10 Indices for K-Means Cluster Solutions.......................................................................... 98
Table 11 Raw Score Mean, Standard Deviation, F-Statistic, and Partial Eta Squared for Cluster
Variables by Cluster Groups ......................................................................................................... 99
Table 12 NAEP Achievement Level Descriptions ..................................................................... 101
Table 13 Frequency Counts and Percentages of Demographic Characteristics by Cluster Group
..................................................................................................................................................... 102
Table 14 Raw Score Mean, Standard Deviation, F-Statistic, and Partial Eta Squared for Algebra
Performance by Cluster Groups .................................................................................................. 103
Table 15 Summary of Multiple Regression Analysis for Demographic Characteristics ad Cluster
Contrasts ..................................................................................................................................... 104

ix

LIST OF FIGURES

Figure 1.Variance explained in algebra performance scores by predictor variables in multiple
regression analysis. ..................................................................................................................... 105
Figure 2. Results of six-cluster solution, showing the average standardized scores on all
clustering variables for each cluster group. ................................................................................ 106
Figure 3. Mean algebra performance scores for participants in each cluster group. .................. 107
Figure 4. Cluster variable scores and algebra performance score comparison of participants in the
Moderate Low All and Moderate High MASMI cluster groups................................................. 108
Figure 5.Cluster variable scores and algebra performance score comparison of participants in the
Moderate Low Spatial and High Spatial cluster groups. ............................................................ 109
Figure 6. Cluster variable scores and algebra performance score comparison of participants in the
Moderate High All and High Spatial cluster groups. .................................................................. 110
Figure 7. Cluster variable scores and algebra performance score comparison of participants in the
Moderate High MASMI and Moderate High All cluster groups. ............................................... 111
Figure 8. Cluster variable scores and algebra performance score comparison of participants in the
Moderate Low Spatial and Moderate High All cluster groups. .................................................. 112

x

CHAPTER 1: Introduction
In a society that places emphasis on students’ ability to reason through quantitative
situations in work, school, and daily life, it is important to understand students’ mathematical
development. One pivotal component of mathematical development is students’ understanding
of algebra. It provides students with the tools in which to reason mathematically about real-life
situations (Lacampagne, Blair, & Kaput, 1995; National Research Council, 1998; RAND
Mathematics Study Panel, 2003). In addition, the completion of an algebra course is seen as a
necessary requirement for learning higher level mathematics, having higher education
opportunities, and getting technically skilled jobs (Kaput, 1998; R. P. Moses & Cobb Jr., 2001;
National Mathematics Advisory Panel, 2008; Vogel, 2008). Yet, students encounter many
obstacles as they learn algebra and exhibit difficulty comprehending a range of algebraic
concepts (Booth, 1988; French, 2002; National Council of Teachers of Mathematics, 1988).
The literature on the teaching and learning of algebra is substantial and centered on
student learning and instructional approaches. It helps to explain the areas with which students
have difficulty (e.g. Booth, 1984; French, 2002; Kuchemann, 1978; National Council of
Teachers of Mathematics, 1988; Stacey & MacGregor, 1997b), the reasons for these difficulties
(e.g. Godino, Neto, Wilhelmi, Ake, & Etchegaray, 2015; National Mathematics Advisory Panel,
2008; Sfard & Linchevski, 1994), and the different ways to help students understand algebra
(e.g. Bednarz, Kieran, & Lee, 1996; Jacobs, Franke, Carpenter, Levi, & Battey, 2007; Nathan,
Kintsch, & Young, 1992; Stacey & MacGregor, 2000). Results from the National Assessment of
Educational Progress (NAEP) studies provide evidence that the focus on student learning and
instructional approaches have increased average students’ scores in algebra (Kloosterman, 2016;
Kloosterman & Lester, 2004; Kloosterman & Lester Jr., 2007; Kloosterman, Moher, & Walcott,

1

2016; National Center for Educational Statistics, 2012, 2015). In fact, Kloosterman (2016) stated
that current 4th and 8th grade students are better prepared now to take a formal algebra course
than 9th grade students were in 1990. Nevertheless, the progress of improvement for 4th and 8th
grade students in algebra has slowed since 2005 and there have been no gains in 12th grade
students’ algebra performance since 2009 (Kloosterman, 2016; Perez, Roach, Creager, &
Kloosterman, 2016). These findings seem to imply that there is a limit to how much
improvement can occur when you focus on student learning and instructional approaches.
One possible reason for this limitation is the narrow focus of the research on the teaching
and learning of algebra, and in particular, the narrow viewpoint research has taken on student
learning. Researchers assumed that students’ weak content knowledge led to weak algebra
performance. Thus, examinations of student learning focused on identifying students’ errors and
misconceptions as well as the fundamental mathematical concepts necessary for solving algebra
problems (e.g. Booth, 1984; Ketterlin-Geller & Chard, 2011). This viewpoint on student learning
ignores the fact that the student is more than what content knowledge they learn. Students are
also a dynamic system that develops based on many factors that interact with one another, and
these factors could be the mental, behavioral, and biological aspects of individual functioning as
well as the social, cultural, or physical nature of the environment (Bergman, Magnusson, & ElKhouri, 2003). Content knowledge as the sole determinant of student learning neglects the
consideration of other aspects of individual functioning such as other factors like cognitive
abilities, working memory, executive functioning, motivation, and personality, which have been
shown to play a role in students learning (e.g. Bailey et al., 2014; Spinath, Spinath, Harlaar, &
Plomin, 2006). Therefore, it begs the question what might be some other factors of individual
functioning that associated with students’ algebra performance.

2

Research on the factors that predict mathematics achievement holds promise for helping
us better understand and investigate additional factors of individual functioning that may shape
students’ algebra performance. Of the many individual functioning predictors of mathematics
achievement, cognitive abilities is frequently identified as the strongest predictor (Colom &
Flores-Mendoza, 2007; Gagne & St. Pere, 2002; Hofer, Kuhnle, Kilian, & Fries, 2012; Karbach,
Gottschling, Spengler, Hegewald, & Spinath, 2013; Kriegbaum, Jansen, & Spinath, 2014;
Kyttälä & Lehto, 2008; Lu, Weber, Spinath, & Shi, 2011; Weber, Lu, Shi, & Spinath, 2013).
This could be because cognitive abilities may limit and/or completely prevent students from
being able to process the information presented to them. For instance, some students can take
away from a course enough knowledge to use it in real life situations and others cannot
(Anderson, Brubaker, Alleman-brooks, & Duffy, 1985; A. L. Brown, Campione, Reeve, Ferrara,
& Palincsar, 1991; Erlwanger, 1973; Loveless, 2008; Stylianides & Stylianides, 2007). This
difference could be because students receive and process information differently. Some process
new information by connecting it to knowledge that they already have. Others take it in as
isolated facts they need to remember for a test, and then forget it. Some may argue that making
connections versus isolated facts depends on the instructional approaches used, but the student
still determines the reception of that information. For example, the teacher may present the
connections between Lesson A and Lesson B multiple times throughout a course session in many
different ways, and still have some students who get it and others who do not. Because of these
differences, it stands to reason that students’ cognitive abilities may influence their mathematics
achievement, and may possibly be another factor that shapes students’ algebra performance.
Some research has shown that an association existed between students’ general cognitive
abilities, content knowledge, and algebra performance (Fuchs et al., 2012, 2016; Geary, Hoard,

3

Nugent, & Rouder, 2015; Lee, Ng, & Ng, 2009; Lee, Ng, Bull, Pe, & Ho, 2011), but a number of
factors limited the finding from this research. With these limitations, it is hard to say how
students’ cognitive abilities and content knowledge relate to their algebra performance.
Therefore, in this study I sought to examine how students’ cognitive abilities and content
knowledge related to their algebra performance by identifying what specific skills,
understandings, and/or abilities were associated with algebra performance as well as how
varying combinations of these characteristics of students relate to algebra performance. Knowing
these specific details provides a deeper understanding of the factors associated with students’
algebra performance.

4

CHAPTER 2: Literature Review
Many have recognized that there are numerous factors besides content knowledge and
instructional experiences that relate to student performance (e.g. Wang, Haertel, & Walberg,
1990, 1993). What these factors are depends upon what perspective you take. The perspective of
this study is the holistic-interactionist perspective (Bergman et al., 2003; Magnusson, 2003). The
holistic –interactionist perspective emphasizes the importance of the individual in the study of
development because all individuals do not function and develop in the same way. It views the
individual as an integrated organism that has mental, behavioral, and biological factors that
influence functioning and development. By taking the holistic-interactionist perspective, I
acknowledge that there are other factors besides content knowledge and instructional experiences
that influence the way students perform, and assume that students are different with respect to
their algebra performance because there are differences in their functioning and development.
Differences in the way an individual functions and develops could be because of many different
factors, but I contend that the differences in students’ cognitive abilities and content knowledge
changes the way students perform. Although there could be many other factors beyond cognitive
abilities and content knowledge that change the way students perform (e.g. students’ motivation
or personality), cognitive abilities are the most consistent and strongest predictors of students’
general academic achievement and mathematics achievement (Colom & Flores-Mendoza, 2007;
Gagne & St. Pere, 2002; Hofer et al., 2012; Karbach et al., 2013; Kriegbaum et al., 2014; Kyttälä
& Lehto, 2008; Lu et al., 2011; Weber et al., 2013). Thus, understanding how cognitive abilities
and content knowledge combine to predict algebra performance is an important first step in
understanding the complexity of variables that are associated with algebra performance. In the
following sections, I give an overview of the research on the relation between cognitive abilities,

5

content knowledge, and algebra performance. Then I identify the specific forms of cognitive
abilities and content knowledge investigated and how a relation exists between them and
students’ algebra performance.
Cognitive Abilities, Content Knowledge, and Algebra Performance
Some researchers would agree that general mathematics performance depends on both
content knowledge and general cognitive abilities (e.g. D. C. Geary, 2004; von Aster & Shalev,
2007). This claim is based on prior research that establishes their effect both separately and
together (G. Brown & Quinn, 2007; Bull, Espy, & Wiebe, 2008; De Smedt et al., 2009; Fuchs,
Geary, Compton, Fuchs, Hamlett, & Bryant, 2010; Fuchs, Geary, Compton, Fuchs, Hamlett,
Seethaler, et al., 2010; Kroesbergen, Van Luit, & Aunio, 2012; Siegler et al., 2012). Much of this
research centered on the elementary grades. There is little research on mathematics performance
beyond the elementary grades (Caviola, Mammarella, Lucangeli, & Cornoldi, 2014; Fuchs,
Geary, Compton, Fuchs, Hamlett, Seethaler, et al., 2010; Jordan et al., 2013; Krajewski &
Schneider, 2009; Lefevre et al., 2010; Seethaler, Fuchs, Star, & Bryant, 2011; Vukovic et al.,
2014; Ye et al., 2016). The focus on the elementary grades was most likely because of the desire
to understand how mathematical competency develops in order to identify precursors to
mathematics difficulties and learning disabilities. The assumption was that if research could
identify the weaknesses in content knowledge and general cognitive abilities from the onset then
interventions can be designed to help alleviate mathematical difficulties and support mathematics
learning disabilities (Cowan & Powell, 2014; Hecht & Vagi, 2010; Hornung, Schiltz, Brunner, &
Martin, 2014; Passolunghi & Lanfranchi, 2012; Peng et al., 2016). The few studies that
examined mathematics performance in higher grade levels have also found that mathematics

6

performance depended upon both content knowledge and cognitive abilities (Cirino, Tolar,
Fuchs, & Huston-Warren, 2016; Geary et al., 2015).
Researchers defined content knowledge in terms of basic numerical competencies such as
counting, computational strategies, quantity comparison, and studied general cognitive abilities
such as intelligence, working memory, processing speed, and executive functioning (Geary,
2011; C. M. Irwin, 2013; Krajewski & Schneider, 2009; Passolunghi & Lanfranchi, 2012;
Passolunghi, Lanfranchi, Altoe, & Sollazzo, 2015). The relation of content knowledge and
cognitive abilities on mathematics performance varied depending upon the type of mathematics
performance investigated. Prior research suggested that cognitive abilities and content
knowledge were associated with students’ mathematics performance for calculations (e.g.
Östergren & Träff, 2013; Peng et al., 2016), word problems (e.g. Cowan & Powell, 2014; Hecht
& Vagi, 2010), fractions (Jordan et al., 2013; Ye et al., 2016), algebra (e.g. Fuchs et al., 2016;
Lee, Ng, Bull, Pe, & Ho, 2011), and general mathematics achievement (e.g. Chu, Van Marle, &
Geary, 2016; Geary, 2011). The research findings suggested that mathematical performance is a
complex association between content knowledge and general cognitive abilities where each
context of mathematics learning comes with its own constellation of important content
knowledge and cognitive abilities that facilitate its development.
The few research studies that have looked at algebra performance in relation to general
cognitive abilities and content knowledge have shown similar results to the studies for the
different types of mathematics learning in the elementary grades. For instance, Fuchs et al.
(2012) and Fuchs et al. (2016) investigated the connection between general cognitive abilities
and content knowledge on algebra performance via students pre-algebraic knowledge. They
defined pre-algebraic knowledge in terms of Pillay, Wilss, and Boulton-Lewis (1998)’s model of

7

algebra development. Students were within the pre-algebra stage of development when they
understood the relational meaning of equivalence (i.e. both sides are the same value) in
nonstandard equations (e.g. 7 = 3 + 4), recognized unknowns and variables in equations and
expressions, and understood the concept of concatenation (i.e. 3x means 3 times x). Findings
from the path analysis demonstrated that second grade word problem solving skills, calculation
skills, approximate representation of numerical magnitudes, nonverbal reasoning, working
memory, and attentive behavior had both direct and indirect effects on fourth grade students’
pre-algebra knowledge. In addition, second and third grade calculation skills and word problem
solving skills mediated the indirect effects of these second grade skills.
Instead of pre-algebraic knowledge, Lee et al. (2009) and Lee et al. (2011) focused on the
contributions of general cognitive abilities and content knowledge on algebraic word problems.
Specifically, they examined the effects of working memory, executive functioning,
computational fluency, pattern recognition, problem formation, and problem representation. Lee
and colleagues found an association between working memory and algebraic word problems. In
particular, working memory was important for problem formation and problem representation.
They also found that executive functioning had no bearing on algebraic word problems solving
performance, but pattern recognition and computational fluency did.
In an attempt to understand a more complex form of algebra performance, Geary et al.
(2015) studied knowledge of the coordinate plane, fluency and accuracy in evaluating algebraic
expressions, and memory for algebraic equations in relation to students’ acuity of the
approximate number system (ANS) and memory for addition facts. While controlling for
parental education, sex, reading achievement, speed of numeral processing, fluency of symbolic
number processing, intelligence, and the central executive component of working memory, they

8

found that ANS acuity was related to knowledge of the coordinate plane and fluency in
evaluating algebraic expression. On the other hand, memory for addition facts related to memory
of algebraic equations only. While not considering how the control variables related to these
algebraic topics, the researchers did examine the external validity of these algebraic topics as
predictors of general algebra achievement. Algebra achievement was participant’s performance
on questions concerning solving for x, systems of equations, factoring, determining equation
slope, and concept questions such as the definition for a vertical line. Results indicated that
knowledge of the coordinate plane, fluency and accuracy in evaluating expressions, and memory
for algebraic equations were all significant predictors of algebraic achievement in both separate
and simultaneous regression analyses.
Even though these studies provided some evidence of the relation between cognitive
abilities, content knowledge, and algebra performance, conclusions about students’ algebra
performance are restricted by the age of the studies population and the measures for algebra
performance. Using elementary grade populations meant that the results from these few studies
did not cover a full range of algebra problems, cognitive abilities, nor types of content
knowledge. This leaves questions concerning how cognitive abilities and content knowledge may
associate with students’ algebra performance after formal algebra instruction. Thus, in this study
I extended the research by considering how cognitive abilities and content knowledge related to
the performance of students who have had formal algebra instruction. Additionally, given the
differences in the knowledge and abilities of an elementary population versus those with formal
algebra instruction, I also extended the research by considering other cognitive abilities and types
of content knowledge that are relevant to algebra performance. In the following sections, I
identify what theses abilities and skills are and how algebra performance relates to them.

9

Cognitive Abilities Related to Algebra Performance
Research on content knowledge, cognitive abilities, and algebra performance has
examined only a few cognitive abilities such as working memory, executive functioning, and
processing speed. This is a limited perspective when there are many more types of cognitive
abilities. Carroll (1993) defined cognitive ability as “any ability that concerns some class of
cognitive tasks” (pg. 10), and cognitive tasks are “any task in which correct or appropriate
processing of mental information is critical for performance” (pg. 10). In this sense, cognitive
abilities are skills needed to perform certain types of tasks appropriately. Research suggested that
cognitive abilities fall within a three stratum hierarchal model called the Carrol-Horn-Cattell
(CHC) theory (Ackerman & Lohman, 2006; McGrew, 2009; McGrew & Wendling, 2010).
Stratum I consisted of about 70 primary/narrow cognitive abilities. Each of these narrow abilities
concerns different cognitive processes that are a greater specialization of the broad abilities
found in Stratum II, and represent specific skills acquired through experience and learning
(Carroll, 1993). Stratum II consisted of the nine broad abilities of fluid reasoning/intelligence,
comprehension knowledge/crystallized intelligence, short-term memory, visual processing,
auditory processing, long-term memory retrieval, processing speed, decision/reaction time, read
and writing, and quantitative reasoning. The abilities in Stratum II are broad domains of behavior
representative of an individual person with emphasis on the process (i.e. skills for reasoning,
memory, and learning), content (i.e. information they know and perceive), and manner of
response (i.e. speediness of response) (Carroll, 1993). Stratum III consisted of the general ability
factor g, which is the idea that all cognitive abilities are independent factors of one main
construct like general intelligence.

10

The research on cognitive abilities and general mathematics achievement has identified
specific cognitive abilities that predict educational success in math. Some of these cognitive
abilities were the broad abilities of crystallized intelligence, fluid intelligence, and processing
speed (Floyd, Evans, & McGrew, 2003; Keith, 1999; McGrew & Hessler, 1995; McGrew &
Wendling, 2010; Taub, Keith, Floyd, & Mcgrew, 2008) as well as the narrow abilities of
phonological processing, working memory, and perceptual speed (McGrew & Wendling, 2010;
Proctor, 2012). From these, the three cognitive abilities most related to algebra were (1)
crystallized intelligence, (2) fluid intelligence, and (3) spatial abilities (Floyd et al., 2003;
McGrew & Wendling, 2010; Proctor, 2012; Taub et al., 2008; Tolar, Lederberg, & Fletcher,
2009). Below is a discussion of both theory and research that suggest how each of these three
cognitive abilities relates to algebra performance.
Fluid intelligence. Fluid intelligence is the ability to solve novel tasks that cannot be
performed automatically using mental operations such as identifying relations, drawing
inferences, concept formation, concept recognition, extrapolating, etc. (Horn, 1989; McGrew,
2009; McGrew & Evans, 2004). Research has shown that fluid intelligence related to many
different types of mathematics achievement such as the measurement of students’ ability to solve
a range of calculations problems (i.e. calculation skills) and the measurement of students’ ability
to analyze and solve problems by comprehending what the problem is asking, recognizing
relevant information, and choosing the appropriate strategy for calculations (i.e. problem
solving) (CalderĂłn-Tena, 2016; Floyd et al., 2003; Geary, 2011; Keith, 1999; McGrew &
Wendling, 2010; Proctor, 2012). This cognitive ability related to algebra performance because
the mental operations of fluid intelligence are similar to some algebraic reasoning skills. For
example, the algebraic reasoning skill of generalization and functional thinking involve the

11

mental operations of drawing inferences, identifying relations, and extrapolation. Specifically,
generalization requires that students draw inferences from a given set of information and
transform that information into another form. This usually occurs when students are asked to
write an algebraic equation in order to find a solution for a pattern of numerical values (Radford,
2006; Rivera, 2010; Rivera & Becker, 2008; Steele, 2008). In addition, functional thinking is
about identifying the relations between quantities (Kaput, 1995; Usiskin, 1999), and
understanding how one quantity changes because of changes in the another. Thus, functional
thinking requires that students must identify the relationship between two quantities from a small
set of given values and be able to apply it for any value not given. Given that both generalization
and functional thinking require the use of the fluid intelligence mental operations of extracting
and applying information, fluid intelligence may be a cognitive ability related to students’
algebra performance.
Research also suggested that fluid intelligence was associated with students’
understanding of algebra because it facilitated students’ ability to connect natural language to
mathematical symbols. Herscovics (1988) and Stacey and MacGregor (1997b) mentioned that
translating from natural language into algebraic symbolism is an area of difficulty for students. A
reason for this difficulty was that students were not able to process the meaning of the words in
conjunction with how mathematical symbols were used (Clement, 1982; MacGregor & Stacey,
1993). The connection between the natural language and mathematical symbols may require that
students develop a conceptual organization for the mathematical symbols, which includes the
individual symbol’s meaning as well as the meaning behind combinations of symbols. Students'
ability to process the connection between the meaning of the words and the use of mathematical
symbols determines the formation of this conceptual organization. For example, Chesney et al.

12

(2014) investigated students’ conceptual organization of addition facts in relation to their
understanding of the equal sign. They found those who mentally organized their addition facts
around equivalent values were more likely to have a better interpretation of the equal sign. Since
students’ poor interpretation of the equal sign is linked to poor algebraic performance (Kieran,
1988, 1992; Knuth, Alibali, McNeil, Weinberg, & Stephens, 2005; Knuth, Stephens, Mcneil, &
Alibali, 2006; Seo & Ginsburg, 2003), the results from Chesney et al. (2014) demonstrated that
students’ fluid intelligence may be related to students’ algebra performance because this
difficulty highlights students’ inability to perform the mental operation of concept formation.
Crystallized intelligence. Crystallized intelligence is the ability to apply the breadth and
depth of acquired knowledge from cultural, educational and life experiences (Floyd, Evans, &
McGrew, 2003; McGrew, 2009; McGrew & Evans, 2004; Proctor, 2012), which emphasizes the
importance of cultural, societal, and everyday knowledge. This is an important idea that has
already been shown to have some bearing on students’ understanding of numbers and operations
(Baranes, Perry, & Stigler, 1989; T. N. Carraher, Carraher, & Schiliemann, 1985; T. N. Carraher,
Carraher, & Schliemann, 1987; Keith, 1999; Schliemann & Carraher, 2002). Carraher, Carraher,
and Schiliemann (1985) in their study about how Brazillian children solved mathematics in and
out of school found that children of street vendors were better able to solve math problems when
they were presented in an out of school context rather than an in school context. Also, Baranes,
Perry, and Stigler (1989) found that when math problems were presented in an out of school
context using numbers that were meaningful to the context, students performed better on number
matched content types of problems than the school type problems. This suggested that
participants’ understanding of numbers and operations was dependent upon their cultural,
societal, and everyday knowledge.

13

Additionally, measures of crystallized intelligence rely heavily on language, which have
a big influence on students’ understanding of mathematical concepts as well as their algebra
performance (Barton, Fairhall, & Trinick, 1998; MacGregor & Stacey, 1997; Philipp, 1992). For
instance, one particular concept that is an area of difficulty is the concept of variable. The main
reason for students’ trouble with the concept of variable was the many different uses of variables
and how it is presented to students (Kieran, 1988; MacGregor & Stacey, 1997; Philipp, 1992).
Philipp (1992) and MacGregor and Stacey (1997) both recognized the importance and influence
language based teaching materials had on students’ understanding of the concept of variable.
MacGregor and Stacey (1997) demonstrated that the initial teaching practice of using letters as
abbreviated words and letters contributed to the misinterpretations students had with variables.
Likewise, Philipp (1992) theorized that a lack of discussion in mathematics leads to syntactically
but semantically weak understanding of concepts. He used two discussion activities on the
concept of variable to demonstrate that students can develop a better understanding of variables
by talking about the many different ways variables are used. Since interactions with teachers and
teaching materials rely on language, this research would suggest that language impacts students’
understanding of the concept of variable, and in turn their algebra performance.
Spatial abilities. There are many different types of spatial abilities, but there is no
broadly accepted definition that defines all spatial abilities. Some researchers have attempted to
categorize the broad set of abilities referred to as spatial abilities based on whether or not spatial
tasks involve a single object (i.e. intrinsic) or group of objects (i.e. extrinsic) and the movement
of these objects in space (i.e. dynamic or static) (Mix & Cheng, 2012; Uttal et al., 2012).
However, most researchers simply define spatial abilities based on a specific type of ability
being assessed. For instance, spatial abilities have been defined in terms of disembedding, spatial

14

visualization, mental rotation, spatial perception, and perspective taking (Mix & Cheng, 2012;
Uttal et al., 2012). In the current work, I focus on the type of spatial ability found to be
associated with algebra performance (Chrysostomou et al., 2013; Logan, 2015; Terao,
Koedinger, Sohn, Qin, & Anderson, 2004; Tolar et al., 2009), namely spatial visualization,
which refers to the mental representation and transformation of an object or groups of objects in
a 2-dimensional and 3-dimentisonal plane.
The empirical findings on the relationship between spatial abilities, broadly defined, and
mathematics achievement are mixed, with some research suggesting that spatial abilities are
important for mathematical development (e.g. Gunderson, Ramirez, Beilock, & Levine, 2012;
Proctor et al., 2005; Skagerlund & Träff, 2016), and others suggesting the opposite (e.g. Floyd et
al., 2003; McGrew & Wendling, 2010). The inconsistency of the connection between spatial
abilities and mathematics achievement may be because of the cognitive ability tests used to
measure spatial abilities. For instance, measures of visual-spatial thinking, which is the ability to
create, recognize, and transform visual images, typically use multiple forms of spatial abilities
including visualization, spatial relations, closure speed, visual memory, spatial scanning, etc.
(Floyd et al., 2003; McGrew, 2009; McGrew & Evans, 2004; Proctor, 2012). It is possible that
including all these spatial abilities as a measure of visual-spatial thinking washes out the relation
that any one of the spatial abilities may have with general mathematics achievement. This is
evident by a detailed analysis of working memory that suggested the spatial component of
working memory was predictive of general mathematics achievement (Kyttälä & Lehto, 2008;
Reuhkala, 2001), and visual spatial representations were important for solving word problems
(Hegarty & Kozhevnikov, 1999; van Garderen, 2006).

15

As for algebra performance, there have been a few studies that identified a relation with
the specific spatial ability of spatial visualization (Chrysostomou et al., 2013; Logan, 2015;
Terao et al., 2004; Tolar et al., 2009). Spatial visualization is the ability to imagine, manipulate,
or transform mental images and identify how they would appear under different conditions
(McGrew & Evans, 2004; Mix & Cheng, 2012; Tartre, 1990), and is only one type of spatial
ability. Measurements of spatial visualization abilities involve mental rotation and mental
transformation tasks (Tartre, 1990). Mental rotation tasks measure the ability to be able to
determine if more than one objects are the same by mentally rotating them while mental
transformation tasks measure the ability to identify the transformation of object by either
mentally putting it together or taking it apart (Tartre, 1990). Studies on spatial visualization and
algebra performance have shown that in a structural equation model it had a direct effect on
algebra achievement (Tolar et al., 2009), was related to student performance on numeracy and
algebraic reasoning (Chrysostomou et al., 2013), and solving algebra word problems activated
the visuospatial regions of the brain (Terao et al., 2004). Spatial visualization was theorized to be
important for algebra performance because in general, elements of mathematics inherently have
a visuospatial component (Logan, 2015; Terao et al., 2004) and specifically, because algebra
requires the ability to represent functional relationships graphically and manipulate visual-spatial
representations mentally (Tolar et al., 2009).
Content Knowledge Related to Algebra Performance
Researchers investigated student learning to determine the reasons why students had
difficulty learning algebra. Some of this research concentrated on eliciting student thinking while
solving algebra problems in order to identify students’ errors and misconceptions about algebra
(Booth, 1984; Greenes & Rubenstein, 2008; Kuchemann, 1978; National Council of Teachers of

16

Mathematics, 1988; M. Russell, O’Dwyer, & Miranda, 2009; Stacey & MacGregor, 1997b;
Welder, 2012). Others tried to determine which foundational mathematical concepts were
necessary for solving algebra problems (Britt & Irwin, 2008; K. C. Irwin & Britt, 2005;
Ketterlin-Geller & Chard, 2011; Ketterlin-Geller, Gifford, & Perry, 2015; National Mathematics
Advisory Panel, 2008; Schifter, 1999; Stacey & MacGregor, 1997a; Wu, 2001). From these
investigations, multiple researchers have proposed many different types of content knowledge
needed for learning algebra. The content knowledge needed for learning algebra consisted of
things that student should be able to do (Ketterlin-Geller & Chard, 2011; Ketterlin-Geller et al.,
2015; Kieran, 1988, 1992; Schifter, 1999; Stacey & MacGregor, 1997a; Wu, 2001) as well as the
necessary understandings needed to do those things (Blanton & Kaput, 2005; Herbert & Brown,
1997; Herscovics, 1988; Jacobs et al., 2007; Kaput, 1999; Kieran, 2004; M. Russell et al., 2009;
Stacey & MacGregor, 1997b; Welder, 2012). Theory and empirical evidence, taken together,
point to the connection between content knowledge and students’ algebra performance as being
dependent upon four key types of content knowledge: (1) understanding numbers and operations,
(2) proficiency with fractions, (3) understanding equivalence, and (4) algebraic reasoning. Below
is a discussion about the empirical evidence that demonstrated the relation between these types
of content knowledge and algebra performance.
Understanding of numbers and operations. Multiple researchers suggested that an
understanding of numbers and operations is important for students’ understanding of algebra
(Britt & Irwin, 2008; K. C. Irwin & Britt, 2005; Ketterlin-Geller & Chard, 2011; Ketterlin-Geller
et al., 2015; National Mathematics Advisory Panel, 2008; Schifter, 1999; Stacey & MacGregor,
1997a). An understanding of numbers and operations involves being able to understand place
value, understand numerical magnitudes, to compose/decompose numbers, understand

17

mathematical operations, as well as understand the properties of numbers. Stacey and
MacGregor (1997a) proposed that an understanding of numbers helps when students fail to
remember the rules of symbolic manipulation. Students can use their knowledge about how to
manipulate numbers to figure out how to work with algebraic expressions. K. C. Irwin and Britt
(2005) theorized that the ability to generalize mental operational strategies and use them
effectively to solve different numerical problems represents algebraic thinking. It was algebraic
thinking because students were using the number as variables instead of letters. An investigation
into the development of students’ generalized mental strategies demonstrated that those who had
instructional experiences that promoted generalized mental strategies exhibited the use of
algebraic operational strategies more often in comparison to others who had not. In a second
study, Britt and Irwin (2008), showed that after developing their mental strategies students were
able to transfer this knowledge to literal symbols. Given that students were able to demonstrate
the same skills for numbers and operations with letters as variables suggested that their
understanding of numbers and operations was associated with their algebra performance.
Proficiency with fractions. Fractions are mathematical quantities that represent parts of
a whole, and researchers theorized that students’ understanding of how to operate and use
fractions relates to their algebra performance. Researchers have theorized various reasons for the
connection between fractional knowledge and algebra performance. Wu (2001) and the National
Mathematics Advisory Panel (2008) suggested that students’ knowledge about fractions provides
a stepping stone for their understanding of rules of symbolic manipulation used in algebra.
Empson, Levi,and Carpenter (2011) suggested that fraction knowledge related to students’
algebra performance because solving problems involving operation on and with fractions uses
the same understanding of numbers and operations that are necessary for algebra. Additionally,

18

Kilpatrick and Izsak (2008) claimed that some big ideas used by fractions are also important for
algebra. Specifically, an understanding of fractions helps to develop students’ multiplicative
structures and understanding of the distributive property, which are important ideas for working
with algebraic equations and expressions. Even with these reasons, only a few studies have
shown that proficiency with fractions does impact algebra performance (e.g. G. Brown & Quinn,
2007; Siegler et al., 2012). G. Brown and Quinn (2007) investigated students’ proficiency with
fractions in relation to their success in algebra. Proficiency, defined by this study, was as the
ability to understand fractional concepts and manipulate fractions for accurate computation
without the aid of a calculator. Success in algebra was student tests scores from an algebra
course. A Pearson Correlation Coefficient calculation determined that there was a significant
relationship between the two test scores. In addition, Siegler et al. (2012) found that fraction
knowledge measured 5 or 6 years before algebra instruction was a stronger predictor of later
algebra achievement than other types of mathematical knowledge, general intelligence, working
memory, and family income and education.
Understanding of equivalence. Mathematics is heavily depended upon the use of
symbols, and one symbol that demonstrated a relation to students’ algebra performance is the
equal sign. The equal sign represents equivalence, which is the understanding that two quantities
on each side of the equals sign are the same. Equivalence is seen as a common issue related to
algebra because misunderstanding equivalence makes it difficult to understand, remember, or
apply algebraic processes and principles (Seo & Ginsburg, 2003). Additionally, the
misunderstanding of equivalence limits students’ understanding of why the process of doing the
same operation on both sides of the equation balances the equation (Kieran, 1988), limits their
understanding of the transformations used to solve algebraic equations (Knuth et al., 2005), and

19

limits their ability to perceive algebraic expressions as mathematical objects rather than
processes (Kieran, 1992). Additionally, researchers have demonstrated that students'
understanding of equal sign as well as how misinterpretations of this symbol led to difficulties
with algebra. In particular, Knuth, Stephens, McNeil, and Alibali (2006) examined middle school
students’ understanding of the equal sign in relation to their performance on solving algebraic
equations. Results demonstrated that participants with a relational understanding of equivalence
(i.e. the equal sign means the same as) were more likely to solve algebraic equations correctly
regardless of grade level and mathematics ability.
Algebraic Reasoning. Some researchers found that the emphasis on teaching symbolism
before the application of mathematical knowledge to word problems was counter to how students
naturally approached problems (Godino, AkĂŠ, Gonzato, & Wilhelmi, 2014; Godino et al., 2015;
Nathan & Koedinger, 2000; Nathan & Koellner, 2007; Nathan & Petrosino, 2003; Sfard &
Linchevski, 1994). The researchers realized that students used verbal reasoning skills before
symbolism, which allowed students to use their mathematical knowledge to solve algebra
problems. As such, some researchers claimed that there are reasoning skills related to algebra.
They suggested that the underdevelopment of these reasoning skills makes some students
unsuccessful. Kaput (1998, 2000, 2008) stated that there are five forms of reasoning that
constitute the reasoning skills needed for algebra: (1) the expression and use of generalizations
based on arithmetic and quantitative reasoning, (2) reasoning with and acting upon symbols
given the rules of manipulation, (3) the understanding of mathematical structures as objects
rather than process based on generalizations built from arithmetic and quantitative reasoning, (4)
the understanding of the relationship between two or more quantities that vary, and (5) the use of
mathematic symbols as a language to express situational behaviors.

20

Beyond the validation studies for early algebra education (e. g. Bastable & Schifter,
2008; M. L. Blanton & Kaput, 2004, 2011; D. W. Carraher, Martinez, & Schliemann, 2008;
Cooper & Warren, 2011; Lannin, 2003, 2005; Moss & McNab, 2011; S. J. Russell et al., 2011),
there has not been much research that has examined the connection between students’ algebra
performance and their algebraic reasoning skills. However, one particular study evaluated the
effect of sustained and comprehensive early algebra instruction. The study was in fact the first
step in trying to determine the influence of algebraic reasoning skills on students’ understanding
of algebraic concepts. Blanton et al. (2015) implemented a yearlong intervention focused on
developing students’ algebraic reasoning alongside the traditional arithmetic focused elementary
mathematics curriculum. Results demonstrated greater improvement in the understanding of
algebraic concepts and practices of intervention students as compared to nonintervention
students. The researchers suggested that traditional arithmetic-focused elementary mathematics
curriculums alone are not enough to prepare students in the future for algebra. The results also
suggested even though some basic mathematical content knowledge like understanding numbers
and operations and fractions may be associated with algebra performance, algebraic reasoning
skills are necessary for strong algebra performance. Thus, it is important to include algebraic
reasoning skills as factors of students’ algebra performance.
Current Study
When trying to understand why students perform differently from expectations, past
algebra research assumed that it was because of students’ content knowledge and the instruction
that they experienced (e.g. Ketterlin-Geller & Chard, 2011; Ketterlin-Geller et al., 2015; Nathan
& Koedinger, 2000; Sfard & Linchevski, 1994). Researchers conducted studies that examined
students’ understanding of multiple mathematical concepts (e.g. variables and equivalence,

21

Knuth, Alibali, McNeil, Weinberg, & Stephens, 2011), but little attention was given to other
factors like cognitive abilities, which are the underlying cognitive processes that can affect what
a person knows and understands. Cognitive abilities are another aspect of the individual that
changes the ways students perform. Thus, in this study I examined both cognitive abilities (i.e.
crystalized intelligence, fluid intelligence, spatial visualization) and content knowledge (i.e.
understanding numbers and operations, fractions, equivalence, algebraic reasoning) in order to
determine the different factors and combinations of factors that have a relation with students’
algebra performance.
Rationale for design. In particular, I studied the cognitive abilities and content
knowledge of undergraduate students. The population is particularly important because previous
cognitive studies have always focused on student who are just beginning to study algebra, and
rarely considered those who are expected to have a good grasp of the mathematical content
(Kieran, 1990, 2006). This is particularly problematic because research has shown that algebraic
misconceptions can linger well after formal algebra instruction (e.g. Bernardo et al., 1994;
McNeil & Alibali, 2005b; Triguero & Ursini, 2003). For instance, multiple studies demonstrated
that even though undergraduate students understand the equal sign as a relational symbol, an
unsophisticated understanding still exists, and when this unsophisticated understanding is
activated students perform in similar ways to beginning algebra students (Chesney & Mcneil,
2014; Chesney, McNeil, Brockmole, & Kelley, 2013; McNeil & Alibali, 2005b; McNeil, RittleJohnson, Hattikudur, & Peterson, 2010). This performance is especially salient under timed
conditions. Since the present study has time constraints it is possible that undergraduate students
will demonstrate a less sophisticated understanding of equivalence that would relate to their
algebra performance. Similarly, prior research has shown that even though undergraduate

22

students have a conceptualization of variables, they lack a rich conceptualization (Trigueros &
Jacobs, 2014; Trigueros & Ursini, 2003; Weinberg, Dresen, & Slater, 2016) and use them
incorrectly (AkgĂźn, 2011; Bernardo & Okagaki, 1994; Clement, 1982; Clement, Lochhead, &
Monk, 1981).
With algebra having a heavy emphasis on both equivalence and variables, having these
sorts of difficulties could influence performance. Thus, studying undergraduate students should
provide an understanding of how cognitive abilities and content knowledge relate to algebra
performance in similar ways that studying beginning algebra students would. It is also possible
that students’ cognitive abilities may have helped them overcome these obstacles. Therefore, by
studying college students, I am able to determine if students’ performance is heavily reliant on
content knowledge or if cognitive abilities could have a supportive impact on students’
performance when content knowledge is limited.
Another aspect that is different about this study is that in investigating cognitive abilities
and content knowledge I am taking both a variable-oriented and person-oriented approach to
algebra performance. Both of these approaches allow me to examine algebra performance in
different ways. The person-oriented approach acknowledges the person as a complex system
with many factors of influence, and seeks to answer questions concerning the differences among
individuals (Bergman et al., 2003; Laursen & Hoff, 2006; Magnusson, 2003). This approach
allows me to understand common patterns of knowledge and examine how these patterns relate
to performance. It can describe what types of students do well or poorly. For example, a
cognitive obstacle students faced in algebra has been the difference in meaning and
interpretations given to operations and symbols in arithmetic as compared to their use in algebra
(Booth, 1984; Kieran, 1990). The person-oriented approach can identify if those who do well

23

have strong vocabulary, reading skills or some other aspect of individual functioning. In contrast,
the variable-oriented approach assumes that each individual will perform in similar ways and
tries to determine how the different variables independently predict algebra performance, on
average for the entire sample (Bergman et al., 2003; Laursen & Hoff, 2006; Magnusson, 2003).
For the same cognitive obstacle mentioned above, the variable-oriented approach revealed that
much of the difficulty students faced can be traced back to the type of instruction they had
received (e.g. McNeil, 2008; McNeil et al., 2006), and changing that instructional experience
helped (e.g. Chesney et al., 2014; McNeil, Fyfe, Petersen, Dunwiddie, & Brletic-Shipley, 2011).
Separately, these two approaches may reveal two different stories, but taken together they
provide new insights into the connection between content knowledge, cognitive abilities, and
algebra performance. Specifically, these new insights could possible provide new avenues for
teaching algebra as well as new ways to prepare students for learning algebra.
Research questions and hypotheses. Much of what we know about the variations in
students’ algebra performance has come from research that has examined how students solve
algebra problems (e.g. Booth, 1984; Kuchemann, 1978). The focus of this research was on how
the individual students made sense of the problems in order to solve them. This research has
provided a wealth of information that has led to new understandings about student knowledge
and the influence of instructional practices (Booth, 1984; Kieran, 1990, 2006). Nevertheless,
recent research into the teaching and learning of algebra has focused less on the differences
across individual students and more on the general tendencies among groups of people. This
approach may have contributed to the slowdown in progress research was making in improving
students’ algebra performance as evident from the recent analysis of NAEP scores (Kloosterman,
2016; Perez et al., 2016). The current study returns the focus back on the differences across

24

individuals in order to determine what other factors besides content knowledge link to students’
algebra performance. Specifically, this study allowed me to answer the following questions:
RQ 1: Which forms of content knowledge and cognitive abilities most strongly predict
algebra performance? One goal of this study is to determine if other factors besides content
knowledge predict algebra performance. Prior research on the factors associated with
mathematics achievement have demonstrated that both content knowledge and general cognitive
abilities (Fuchs, Geary, Compton, Fuchs, Hamlett, Seethaler, et al., 2010; Fuhs, Hornburg, &
McNeil, 2016) predicted mathematics achievement; therefore, I expect that both students’
content knowledge and cognitive abilities will predict their algebra performance. It is uncertain
which specific skills will indicate a significant relation to students’ algebra performance, but I
hypothesize that the assessments of fluid intelligence, algebraic reasoning, equivalence, and
numeracy will be particularly important predictors. These skills are particularly important
predictors of algebra performance because they are measures of abilities that allow students to
reason algebraically. For example, Britt and Irwin (2008) and K. C. Irwin and Britt (2005)
connected students’ numeracy skills to their ability to generalize with variables, which are an
important part of algebra (e.g. Booth, 1984, 1988; Knuth et al., 2005). With fluid intelligence
involving the same general skills as algebraic reasoning, it also seems to represent a more
general reasoning ability that might facilitate algebra performance. In addition, equivalence is
important because those who think in relational terms have the necessary skills to reason
algebraically (M. Stephens, 2007; M. Stephens & Wang, 2008). Moreover, researchers theorized
that algebraic reasoning skills are important for learning algebra because they represent different
ways of thinking based on the different approaches to teaching algebra (Bednarz et al., 1996;
Kaput, 1998, 2000, 2008), and Kloosterman (2016) suggested that even when algebraic content

25

knowledge was lacking, reasoning skills helped to bolster 4th and 8th grade students’ algebra
performance.
RQ 2a: What combination of content knowledge and cognitive abilities naturally occur
in students who have studied algebra? Person-oriented analyses describe groups or types of
individuals that share particular traits, attributes, or relations among variables (Laursen & Hoff,
2006; Magnusson, 2003). These groups can help researchers identify specific reasons why
certain students perform the way that they do. By grouping people who are similar in content
knowledge and cognitive abilities together and examining their algebra performance as a
function of their grouping, one can gain a more natural picture to how different combinations of
predictors are associated with algebra performance. Current studies suggested that linked to
mathematics performance are combinations of content knowledge and cognitive abilities, so I
hypothesize that it is possible that a mixture of content knowledge and cognitive abilities or
content knowledge and cognitive abilities by themselves may characterize groups of participants.
For my mixture hypothesis, there could be combinations of skills and abilities defined by
participants’ performance scores that could be high or low for fluid intelligence, algebraic
reasoning, equivalence, and numeracy because they measure skills related to algebraic reasoning.
Given that there is a lack of research that used a person-oriented approach, this hypothesis just
speculates about what could possible occur.
RQ 2b: How do students with these different content knowledge and cognitive abilities
profiles perform in algebra? It is impossible to know exactly what profiles will emerge, but I
theorize that there will be the typical high and low performance groups characterized by
strengths and weaknesses in both content knowledge and cognitive abilities or in one over the
other that will have different algebra performance. Additionally, there could be unique groups,

26

which would defy expectations. For example, an expected low content knowledge group might
perform well because they have strong cognitive abilities. Similarly, a group expected to do well
(e.g. high content knowledge) may have low performance because they have weak cognitive
abilities. Overall, I hypothesize that participants’ cognitive abilities will preclude better algebra
performance over having a good grasp of content knowledge.

27

CHAPTER 3: Method
Participants
Participants were (N = 141) undergraduate students at Michigan State University from all
school levels (34 freshman, 29 sophomore, 30 junior, 37 senior, 11 5+ year senior) and a variety
of majors (27% Biological Sciences, 19.1% Communication Arts & Sciences, 12.1% Social
Sciences, 9.2% Business & Management, 8.5% Engineering, 8.5% Education, 6.4%
Environmental Sciences, 3.5% Fine Arts & Letters, 2.1% Physical & Mathematical Sciences,
2.8% Undecided, 0.7% Unknown). The mean age of participants was 20.28 years (SD = 1.98).
The majority of participants were female students (110 female, 31 male). The distribution of
race/ethnicity in the sample was as follows: 8.5% Asian, 22% Black/African American, 1.4%
Hispanic/Latino, 61.7% White, and 6.4% Multiracial.
Assessments and Measures
In the sections below, I described each of the assessments and measures used in the
current study. There are three assessments for cognitive abilities, five assessments for content
knowledge, and a single outcome measure for algebra performance. Additionally, see Table 1 for
the number items, time limit, and calculator usage of each assessments and measures.
Cognitive abilities. In the literature, the most common test of human cognitive abilities
is the Woodcock Johnson Tests of Cognitive Abilities (WJ), which is a licensed restricted
assessment in its fourth edition that measures all nine of the broad cognitive abilities as defined
by the CHC theory of cognitive abilities (Mather & Wendling, 2014). Due to the restricted nature
of the WJ, this study will be using alternative assessments for measuring participants’ cognitive
abilities. The assessments for cognitive abilities consisted of a single measure to assess fluid and
crystallized intelligence, and two measures to assess spatial visualization.

28

Fluid and crystallized intelligence. To assess fluid and crystallized intelligence this study
used the Reynolds Adaptable Intelligence Test (RAIT). The RAIT is a good alternative to the WJ
because 1) it is a standardized assessment that does not require a license in psychology, 2) it
measures fluid and crystallized intelligence in a similar way to the WJ, 3) it allows for group or
individual administration, and 4) it has standardized scores for participants 10-75 years old. It
consists of seven subtests for the measurement of fluid intelligence, crystallized intelligence, and
quantitative intelligence. Each subtest has a time limit that allows the RAIT to be a “power” test
instead of a “speeded” test which means that the time limits were set based on 95% of the
participants being able to get the same number of answers correct regardless if they were given a
time limit or not. For each subtest, raw scores were the number of questions correctly answered,
which converted into T-scores with a mean of 50 and a standard deviation of 10. There was no
penalty for incorrect answers. A combination of subtests yielded scaled scores or indexes for
fluid intelligence, crystallized intelligence, quantitative intelligence, total intelligence (i.e. fluid
and crystallized combined), and total battery intelligence (i.e. all three combined). The indexes
are scaled scores set to a mean of 100 and a standard deviation of 15. This study only used the
subtests for the Fluid Intelligence Index and the Crystallized Intelligence Index.
Fluid intelligence index (FII). The FII consisted of the subtest of Sequences (SEQ) and
Nonverbal Analogies (NVA), which measure deductive reasoning using nonverbal reasoning
tasks. The summation of the scaled scores for each subtest yields the FII scaled score. The
median reliability for SEQ and NVA are .86 and .89 respectively, and the FII has a median
composite reliability of .93. The SEQ subtest asked participants to complete a series of pictures
that denoted a change progression by picking which image went next. Participants had 10
minutes to complete the section. The NVA subtest asked participants to complete an analogy of

29

the format ___ is to___ as ___ is to ___ using images instead of words. Participants got three
images and had to choose the fourth that completed the relationship. NVA had a time limit of 7
minutes.
Crystallized intelligence index (CII). Three subtests comprised the CII. All three subtests
assessed crystallized intelligence by using verbal reasoning tasks that invoked inductive
reasoning. The three subtests are General Knowledge (GK), Odd Word Out (OWO), and Word
Opposites (WO). They have median reliabilities of .84, .83, and .81 respectively, and the CII has
a median composite reliability of .93. The GK subtest measured common cultural knowledge,
reasoning skills, and classification skills by asking participants to categorize the names of wellknown people into one of six categories like politics, military, religion, arts, and sciences.
Participants only had 3 minutes to make these categorizations. OWO assessed vocabulary and
verbal reasoning by having participants choose the one word that did not belong in the set. The
set contained a group of five words where four words had a conceptual link to one another and
one word did not. Participants had 5 minutes to complete this subtest. The last subtest WO also
assessed vocabulary and verbal reasoning by having participants choosing the one word out of
five that had the opposite meaning of the target word. Participants had 5 minutes to complete this
subtest.
Spatial visualization. In order to measure spatial visualization, this study used two
assessments that are similar to how WJ-IV measures spatial visualization. The measures were the
Measure of the Ability to form Spatial Mental Imagery (MASMI) and the Measure of the Ability
to Rotate Mental Images (MARMI) (Campos, 2009, 2012), which are companion measures of
spatial visualization that used a net of an unfolded cube with different symbols on each side. In
the WJ-IV, the measurement for spatial visualization used (a) a spatial relations task where

30

participants have to determine which pieces put together form a complete shape, and (b) block
rotation, which requires the participants to identify the same 3-dimensional shape when rotated.
The MASMI is the match for the spatial relations task because it asked participants to mental
reassembly the cube in order to identify which cubes had the correct symbol for the left and right
side of the target cube. Although different from the WJ-IV, the MASMI is measuring a similar
construct because each task asked participants to form a mental image of a shape and answer
questions about that shape. The MARMI is the match for the block rotation because each tasks
had participants identify blocks that are the same when given a different rotation. In both
assessments, participants have to reassemble the cube in their mind in order to answer the
assessment questions. Each question presents the same unfolded cube reassembled and rotated
differently. Each assessment had 23 questions with four options. Of these four options, two
options were correct and two were incorrect. Participants had to pick the two options that they
thought were correct for each question. There was a penalty for incorrect responses where
participants lost one point from the total amount of correct responses for each incorrect response.
The internal sample reliabilities for MASMI and MARMI are .82 and .93 respectively.
Content knowledge. The assessments for content knowledge measured participants’
performance on a number of tasks that related to the important skills and concepts of
understanding of numbers and operations, fractions, equivalence, and algebraic reasoning. See
Appendix C for complete details about all content knowledge assessments, and see Table 2 for
Cronbach’s Alpha internal reliabilities for each assessment based on the given sample.
Understanding of numbers and operations. An understanding of numbers and
operations is demonstrated by: (1) an understanding of place value and magnitude, (2) the ability
to compose and decompose numbers, (3) grasping the meaning of the operations, (4) the ability

31

to use and understand the properties of distributive, commutative, associative, and (5) automatic
recall of addition, subtraction, multiplication, and division facts (Ketterlin-Geller & Chard, 2011;
Ketterlin-Geller et al., 2015; National Mathematics Advisory Panel, 2008). The current study
used two assessments (i.e. computational fluency and numeracy) to measure these different
skills. Together both assessments measured each of the different ways to demonstrate an
understanding of numbers and operations given above. The computational fluency assessment
covers the an understanding of place value and recall of math facts (i.e. points 1 and 5) while the
numeracy assessment covers the understanding of magnitude, the ability to compose and
decompose numbers, and the ability to use the properties of numbers and operations (i.e. points
1, 2, 3, 4).
Computational fluency. This measure demonstrated an understanding of numbers and
operations by assessing students’ understanding of place value and their recall of addition,
subtraction, multiplication, and division facts. It was called computational fluency to describe
the fact that it asked participants to solve a number of computational math problems within a
given time period. To measure computational fluency, this study used the Curriculum Based
Measurement (CBM) Computational Fluency Assessment, which is a different type of
assessment from those used by others (see Chapter 5 for further discussion). The CBMComputational Fluency Assessment is a time based assessment that allows for teachers to
quickly and efficiently assess how accurately students can solve addition, subtraction,
multiplication, and division problems (Wright, 2013). The computational fluency assessment
considers accuracy to be the correct number of digits per problem. For example, in the given
problem 6220 + 3545 the correct answer is 9765 which means that a participant can receive an
accuracy score of 4 or less for this problem. The participant would receive a score of 4 if they

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gave 9765 as their answer, but would receive a score of 3 if they gave 9755 because the second
five is an incorrect digit. The correct digit coding is a more nuanced way of examining
participants’ computational fluency. It gives them credit for being able to add, subtract, multiple,
and divide numbers while also allowing for the occasional error. For this assessment, participants
received a worksheet that included 16 items for addition, subtraction, multiplication, and division
math facts. The worksheet focused on multi-digit calculations with regrouping and no
remainders. Participants solved problems involving 2-6 digit numbers. They had 3 minutes to
complete as many as they could without the use of a calculator.
Numeracy. Numeracy is the ability to reason with numbers and numerical concepts by
using knowledge about mathematical relationships. Numeracy skills include flexibility and
efficiency with strategies, an understanding of algorithms, and using reasoning skills to calculate
instead of performing prescribed steps (Harris, 2011). This assessment consisted of 24 multiplechoice questions taken from the As Close as It Gets activities from Harris (2014). As Close as It
Gets activities are multiple-choice questions with answer choices that do not have the correct
answer. Such questions require students to consider the numbers in the problem to inform which
strategy to use as well as determining the reasonableness of an answer choice based on the
magnitudes of the numbers. These types of questions measure place value concepts, magnitude,
the ability to compose and decompose numbers, meanings of operations, and an understanding of
the properties of numbers. The 24 questions of this assessment consisted of six problems for each
mathematical operation. In addition, the questions included problems with whole numbers,
decimals, and fractions. To encourage students to use their numeracy skills rather than
algorithms, participants had only 4 minutes to complete the assessment without the aid of a

33

calculator. The time limit calculations were a 10-second time limit for each question and a 1minute time limit for each operation.
Fractions. G. Brown and Quinn’s (2006) 25-question fractions assessment was a guide
for the development of the fraction assessment for this study. The assessment used questions
from previous research and questions devised by the researchers. It assessed conceptual
knowledge and computational fluency aimed at developing an understanding of rational
numbers. There were six categories of questions; algorithmic operations, application of basic
fraction concepts in word problems, elementary algebraic concepts, specific arithmetic skills
prerequisite for algebra, comprehension of the structure of rational numbers, and computational
fluency.
In this study, I adapted this assessment to include twelve questions covering all of the six
categories. I kept two questions from each of the six categories. The algorithm operations
category asked participants to find the difference of fractions and write a mixed number as an
improper fraction. The word problem category asked participants to solve two word problems.
The elementary algebraic concepts category question asked participants to solve two algebraic
equations involving fractions. As part of the arithmetic skills prerequisite for algebra category,
participants wrote a fraction in form of a sum and evaluated the value of fraction divided by zero.
The questions from comprehension of the structure of rational numbers asked participants to
order fractions and to compare the values of fractional quotients. For the computational fluency
category, one question asked participants to find the sum of a complex fraction equation, and the
second asked participants to find the remaining fractional component needed to equal one.
Participants got 10 minutes to complete as many questions as they could without the aid of a
calculator.

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Equivalence. The equivalence assessment evaluated participants’ understanding of
equivalence. Using prior research Matthews et al. (2012) created an equivalence assessment
compiled from three different ways of measuring students’ understanding of equivalence; open
equation solving items (e.g. 8 + 4 = ___ + 5), equation structure items (e.g. is 3 + 5 = 5 + 3
true or false), and equal sign definition items. Matthews et al.'s (2012) assessment was for
participants in grades 2-6. Since the participants in this study were much older, it was not
feasible to use the same assessment. Nevertheless, the design of this assessment used similar
items.
This equivalence assessment consisted of six items; two open equation items, two
equation structure items, and two equal sign definition items. All questions on this assessment
required some form of explanation for participants’ answers, so participants received full credit
(e.g. 2 points) for problems if and only if they provided an appropriate explanation. Students’
understanding of equivalence can be separated into two distinct views; a relational view and an
operational view (Alibali, Knuth, Hattikudur, McNeil, & Stephens, 2007; Baroody & Ginsburg,
1983; Kieran, 1981; Knuth, Alibali, Hattikudur, McNeil, & Stephens, 2008; Knuth et al., 2005,
2006; MacGregor & Stacey, 1999; McNeil, 2007; McNeil & Alibali, 2005a). The relational view
of equivalence is the general idea that the equal sign represents the relationship that the two
quantities separated by the symbol are the same. The operational view of equivalence is the
general idea that the equal sign means the answer comes next or to apply the operation to all the
numbers. An appropriate explanation had answers that expressed a relational understanding,
which used terms such as “the same as”, “can mean two numbers are the same”, “same as the
other number”, or “the value on one side is the same as the value on the other side”. Otherwise,
participants received partial credit (e.g. 1 point) for either having the correct answer and a non-

35

relational explanation or an incorrect answer and a relational explanation. Participants received
no credit for incorrect answers and non-relational explanations. Participants received 5 minutes
to complete the assessment without the aid of a calculator.
Algebraic reasoning. The algebraic reasoning assessment evaluated participants’
algebraic reasoning skills. The assessment consisted of two items for each of the five forms of
algebraic reasoning. The five forms of algebraic reasoning are generalization, functional
thinking, modeling, symbolic manipulation, and structure sense (Kaput, 1998, 2000, 2008). The
items were questions used in prior research or were researcher designed based on common errors
identified from prior research. Generalization is the ability to abstract out common relationships
in order to apply rules to any particular instance such as writing equations from patterns. The
generalization items in this assessment asked participants to write an equation or an expression
that represented a functional relationship based on information presented in tabular form.
Functional thinking is the ability to understand the relationship between two or more quantities
such as being able to describe the relationship in multiple forms (e.g. symbolically or in words).
The functional thinking items in this assessment required the participants to determine the
relationship between two or more items using a table format. Then participants had to describe
the relationship using words, an equation, and/or an expression. Modeling is the ability to
represent real world situations with mathematics such as translating a written relationship into
symbolic form. The modeling items in this assessment required participants to translate a written
relationship into an equation. Symbolic manipulation is the ability to understand the rules that
govern how to work with symbols such as knowing how to combine like terms, knowing how to
balance an equation, and being able to evaluate symbolic solutions for accuracy. In this
assessment, symbolic manipulation items required participants to write an equivalent expression

36

for a given expression. Structure sense is the ability to look at the structure of a problem and use
that understanding to help solve the problem. The structure sense items in this assessment asked
participants to solve equations that have cancellable elements that make finding the answer
easier. The ability to see cancellable elements of a problem to make problem solving easier
rather than just solving the problem is an indicator of structure sense, and participants received
credit for these questions if and only if they demonstrated cancellation. Participants received 10
minutes to complete the assessment without the aid of a calculator.
Algebra performance. The algebra performance assessment measured participants’
current algebra performance. Included in this assessment were the following topics: systems of
equations, functions, solving equations, inequalities, graphing, exponents, factoring, complex
numbers, polynomial division, and logarithms. The assessment consisted of 20 questions. The
items were multiple-choice items with five answers choices. Participant had 20 minutes to
complete the entire assessment. Questions for this assessment were from the National
Assessment of Educational Progress (NAEP) mathematics released item database (n = 9) and
supplemented by researcher-designed questions (n = 11). The NAEP release items are algebra
content questions from the years of 1990-2013. The questions range in difficulty from medium to
hard. This assessment used NAEP questions because multiple panels of experts had reviewed
them, and the questions were pilot tested (National Center for Educational Statistics, 2016). The
supplemental questions covered additional topics not found in the NAEP item release database.
The inclusion of these items gave a good sample of questions that covered a range of topics
found in college algebra courses.
Participant demographic survey. The participant demographic survey collected
demographic information about each participant (see Appendix C). The survey asked

37

participants to identify their age, gender, race/ethnicity, school level, and major in school. The
survey also asked about past mathematics education. Since the participants were college
students, time and level of algebraic exposure are confounding variables. These are confounding
variables because memory decays over time, especially for information that is not used on a
regularly basis (Santrock, 2008), and as evident from practice effects the more exposure that a
person has to something the better that can become (Bullard, Griss, Greene, & Gekker, 2013).
Given these circumstances, there was a need to establish the length of time since the participants
have taken a math course or studied algebra as well as the highest level of mathematics taken. So
the survey had each participant identify the types of math courses taken in high school and
college as well as the years taken. Variables for time since the last algebra based course taken
and the highest level of mathematics taken were calculated and used as covariates in data
analysis.
Data Collection Procedures
Participants completed all assessments and measures in paper and pencil format during a
2-hour individually administrated session. Since there are number of measures for this research
study, there existed the possibility of order effects. Order effects are the differences in
participants’ responses due to the order in which the assessments and measures occurred. The
types of order effects that could possible show up in this research study are fatigue effects and
carryover effects. Fatigue affects occur when the data collection procedure is long, repetitive, or
uninteresting. In order to deal with fatigue all assessments had time limits of no longer than 20
minutes (see Table 1). Additionally, researchers told participants that they could take a break at
any time and asked participants if they wanted a break when they looked like they were tired.
Carryover effects occur when participants’ performance on one assessment influence their

38

performance on another. In order to deal with the carryover effects, each participant got a preassigned a randomized order (see Table 3). To create the testing orders, I first grouped the
assessments into equally timed first hour and second hour of testing. The groups each equaled
out to be 55 minutes of testing. Additionally, I made sure to separate any assessments that I
thought would contribute to a carryover effect. For example, I separated MARMI and MASMI
so that participants would know that they were two separate assessments even though they
looked similar. Orders 1 and 3 consisted of the testing order of the first hour group and then the
second hour group while orders 2 and 4 had the second hour group first then the first hour group.
Within each hour group the tests were randomized across each of the four testing order, which
made sure that no assessment or measure occurred in the same testing order (e.g. if fractions
came first in order 1 it did not come first in order 3).
Trained undergraduate research assistants or I gave each research session. Each session
began with participants given time to read the information and consent form, and ask questions.
Before signing the consent, the participant got a short summary of the consent form that
explained what was going happen during the research session that participation was voluntary,
and that they got compensation upon completion. The summary allowed the researchers to make
sure that participant understood what the research process was and understood the information
on the information and consent form. Then participants got the assessments and measures in the
order indicated by their assigned order. The researcher read aloud the printed directions for each
assessment and measure from the front cover. For complete details on the research protocols, see
Appendix D.

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Data Analysis
In this section, I briefly describe the analytical plan for answering each research question.
The results section provides a more detailed description of each analysis.
Research question 1. The first research question asked whether the prediction of
participants’ algebra performance depended upon the degree to which they mastered the content
and/or the strength of their cognitive abilities. To answer this question, I conducted a multiple
regression analysis with participants’ scores on the algebra performance assessment as the
outcome measure and participant scores on the assessments of computational fluency, numeracy,
equivalence, algebraic reasoning, MARMI, MASMI, crystallized intelligence, and fluid
intelligence were predictor variables. Anticipating different lengths of time since participants
had done the types of problems within the algebra performance assessment, I added to the
regression model a control variable for the number of years since an algebra course. I also
anticipated that there would be individual level variability in the degree to which participants had
continued to take higher-level mathematics courses, so I also included a control variable
indicating highest math course taken. Due to the variety of higher-level math courses taken by
the sample population (e.g. one or more algebra courses, pre-calculus/trigonometry, or
multivariate calculus), I created a dichotomous variable that indicated if each participant’s
highest math course was at or above Calculus 1. A control variable for gender was also included
in the regression model due to the differences found for mathematics performance (e.g. ElseQuest, Hyde, & Linn, 2010; Hyde, Fennema, & Lamon, 1990) and the uneven representation of
gender in this sample population.
Research question 2a. A multiple regression analysis assumes a linear relationship
between algebra performance, content knowledge, and cognitive abilities, but development does

40

not necessarily occur in a linear fashion (Bergman et al., 2003). Thus, the second research
question asked whether there existed groups of people who perform similarly on these
assessments of content knowledge and cognitive abilities. I used cluster analysis to group
participants, which identified groups of people that were closely related to one another but
distinct from other groups. The cluster variables were participant scores on the cognitive
assessments of crystallized intelligence, fluid intelligence, MARMI, and MASMI, and the
content knowledge assessments of computational fluency, numeracy, fractions, equivalence, and
algebraic reasoning.
Analyses were conducted using a two-step procedure where I performed both a
hierarchical (Ward’s Method) and a nonhierarchical (k-means) cluster analysis. Lastly, I
examined differences in the clustering variables for the final cluster solution by performing a
multivariate analysis of variance (MANOVA) on cluster variables by cluster membership and
examined basic demographic differences across clusters using Chi-square Tests of Independence
and a one-way analysis of variance (ANOVA).
Research question 2b. The third question asked whether the groups found from the
cluster analysis varied in algebra performance. I used a one-way analysis of variance (ANOVA)
to examine these differences. The outcome variable was participant scores on the algebra
performance assessment, and the independent variable was the cluster memberships. This
ANOVA analysis did not include any control variables because they were categorical variables,
which cannot be included as covariates. Therefore, I also used a separate multiple regression
analyses to determine if there was a difference in algebra performance by cluster group while
accounting for the control variables. The regression model included participant scores on the
algebra performance assessment as the outcome measure, and the predictor variables were the

41

different demographic characteristics of race/ethnicity, major, highest math course taken, and
years since an algebra course as well as planned contrasts representing cluster membership.

42

CHAPTER 4: Results
Preliminary Analyses
See Table 2 and Table 4 for descriptive statistics, bivariate correlations, and internal
reliabilities for all assessments. Participants’ average performance scores for all assessments
were at or above 50% with the exception for algebra performance. The low scores on the algebra
performance measure could be because of test difficulty. After completing the assessment
multiple participants would remark on how hard the assessment was or how long it has been
since they had seen certain types of problems. On the other hand, item level analysis of the
content knowledge assessments (See Table 5) suggested that the low performance scores of
participants were because participants made errors while taking the assessments or they did not
have a complete understanding of the assessed mathematical content. This is evident by the fact
that the majority of participants only received partial credit for most item categories as well as
the low means of certain items on the fraction, equivalence, algebraic reasoning, and algebra
performance assessments (See Appendix E). Another possibility is that the time limits for the
assessment may not have provided enough time for the participants to complete the assessments.
Moreover, there are a number of limitations that could have attributed to participants’ low
performance scores. For more in depth discussion of the impact of these limitations, see the
limitation section in Chapter 5.
Bivariate correlations show that, as expected, all content knowledge assessments had
statistically significant correlations with each other at p < 0.05, but did not have extremely high
correlations (i.e. r > 0.70). All Pearson’s correlations were below 0.70 with the correlations
between fractions and numeracy, fractions and algebraic reasoning, and crystallized and fluid
intelligence being close to 0.70 at .626, .651, and .633 respectively. There were also statistically

43

significant correlations between all content knowledge and cognitive abilities assessments with
the outcome measure of algebra performance between .366 and .657, with most around .400 to
.500.
Additionally, I conducted analyses on the randomized orders and gender to determine if
the necessity of control variables for these issues in subsequent analyses. Multivariate analysis of
variance (MANOVA) showed that were no statistically significant differences between
randomized orders on all assessments, Wilks’s λ (30, 376.381) = .780, p = .319, and therefore
randomized orders were not included as controls in any subsequent analyses. In order to
determine gender differences, I performed both an Independent Samples T-Test and a MannWhitney U Test for those measures that violated the assumptions for the t-test. The assumptions
for the t-test assume that the data has no significant outliers, is approximately normally
distributed, and that there is homogeneity of variance between the groups. To assess for outliers
and normality, I inspected a boxplot for outliers and a Normal Q-Q Plot for the distribution of the
data. There were only three measures (i.e. MARMI, crystallized intelligence, and fluid
intelligence) that violated assumptions of normality with no outliers and normality assumptions
and thus required the use of the Mann-Whitney U Test; all other measures met these
assumptions. In addition, all measures except for algebra performance met the last assumption of
equality of variance as assessed by Levene’s Test.
An independent samples t-test indicated that there was a statistically significant
difference in scores for males and females for numeracy, with males scoring higher than females
(see Table 6). There were no significant differences in the assessments of computational fluency,
fractions, equivalence, algebraic reasoning, MASMI, and algebra performance. The MannWhitney U Test revealed that there were no statistical difference in scores for males and females

44

for the assessments of MARMI, crystallized intelligence, and fluid intelligence (See Table 6).
Since there was a gender difference for one of the assessments, gender was also included as a
control variable in the following analyses.
Regression Analysis
To determine which forms of content knowledge and/or cognitive abilities predicted
algebra performance, a multiple regression analysis was performed (RQ1). Participants’ scores
on the algebra performance assessment was the outcome measure, and participant scores on the
assessments of computational fluency, numeracy, equivalence, algebraic reasoning, MARMI,
MASMI, crystallized intelligence, and fluid intelligence were predictor variables. In addition,
there were control variables for the number of years since an algebra course, the highest math
course taken, and gender.
Preliminary test for assumptions. Tests for all assumptions for multiple regression
analysis confirmed the validity of the regression model. The first assumption was independence
of observations. There was independence of observations as determined by a Durbin-Watson
statistic of 2.229; a value close to two indicates an independence of observations. The next
assumption was that there are linear relations between the dependent variable and independent
variables both separately and collectively. There was linearity as assessed by partial regression
plots and a plot of studentized residuals against predicted values. For the partial regression plots,
no violation of linearity occurs when the plots visually show the points falling in somewhat of a
straight line. All the partial plots visually showed somewhat of a straight line with Highest Math
– Calculus or Above, Fractions, and MASMI showing a defined upward sloping line and all
others a straight horizontal line.

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The third tested assumption was that of constant error variance. The usual evaluation of
constant variance is a visual inspection of the scatterplot of the studentized residuals against the
predicted values. The scatterplot must show that the variance along the line of best fit remains
similar as you move along the x-axis. A visual examination of the scatterplot seemed to meet the
assumption of constant variance, but there was still a level of uncertainty; therefore, a statistical
inference test also determined the assumption of constant variance. The Breusch-Pagan Test for
Heteroscedasticity uses a chi-square test statistic to test a null hypothesis of constant variance
against the alternative hypothesis of no constant variance. A large chi-square value means that
there were no constant variance and returns a small p-value. In this regression model, there was
constant variance, χ2 (1) = 10.149, p = .603. The next assumption was that there are no issues of
multicollinearity. An examination of correlation coefficients and tolerance values assessed
multicollinearity. Correlation coefficients should not have values above 0.70, which
demonstrates a high correlation between variables, and tolerance values should be greater than
0.1. For all predictor variables, there were no correlation coefficients above 0.70 or tolerance
values below 0.1.
Another assumption was that there are no significant outliers, influential points or
leverage points. There were no outliers as assessed by not having any studentized deleted
residuals at ±3. Cook’s distance values measured the influence of each observation by
calculating the amount the data changes by deleting the observed value. The optimal Cook’s
distance values are those less than 1.0. In this regression model, all Cook’s values were less than
1.0. Leverage points are extremely low or high values of the predictor that may exert undue
influence on the statistical analysis. Safe leverage points are those less than 0.20, but risky
leverage points occur between 0.20 and 0.50. All leverage points were safe except one, which

46

had a value of 0.27. Even though this one participant had a risky leverage point, the regression
model included this participant because all other tests for unusual points deemed this participant
within acceptable range. The last assumption test was that the data in the regression model is
approximately normal. There was normality as assessed by the visual examination of a Q-Q Plot.
For the data to be approximately normal, the data points should fall closely along the diagonal
line of the Q-Q Plot. The Q-Q Plot showed that the data points follow the line closely enough to
be approximately normally distributed.
Regression model. The multiple regression model statistically significantly predicted
algebra performance, F(12, 128) = 12.368, p < .001. The predictors explained about 50% of the
variation in algebra performance, R2 = .537, Adjusted R2 = .494 (see Figure1). The only
significant predictors of algebra performance were participant scores on fractions and MASMI,
and having taken at least one calculus course (see Table 7). The squared semi partial correlations
showed that fraction scores accounted for 5.8% of the variation in algebra performance. MASMI
scores accounted for 2.2% of the variation, and having taken at least one calculus course
accounted for 1.8% of the variance in algebra performance. Thus, the results of this regression
model suggested that better algebra performance is more likely for those having taken at least
one calculus course, those with strong fraction knowledge, and those with strong spatial imagery
ability.
Cluster Analysis
The use of cluster analysis in this study provided information about the different
participant profiles as a function of content knowledge and cognitive abilities. The profiles
highlighted the different combinations of content knowledge and cognitive abilities that naturally
occur within participants who have studied algebra (RQ2a). Since both the presence of outliers

47

and the scale of the clustering variables affect measures of similarity, I made adjustments for
each of these issues before identifying cluster groups. By not checking for outliers and adjusting
the scale for each of the clustering variables, the measure of similarity would be unduly
influenced by the variable with the largest standard deviation, which would mask the influence
of any other clustering variable (Hair & Black, 2000; Hair, Black, Babin, & Anderson, 2009). I
used Grubbs' (1969) outlier test to identify any potential outliers for each clustering variable. The
test only identified a single outlier for the crystallized intelligence assessment. Since the
participant had only that one outlier, I changed the crystallized intelligence score to the same
score as its closest neighbor in order to retain the participant. After adjusting for outliers, I
calculated standardized z-scores to change all cluster variable raw scores to the same scale with a
total sample mean of 0 and standard deviation of 1.
The first step in the analysis identified the number of cluster groups within the sample.
Specifically, I conducted a hierarchical clustering procedure called Ward’s Method, with squared
Euclidean distance as the measure of similarity, to develop different numbers of cluster
solutions. Ward’s method systematically combines clusters by joining the clusters that minimizes
the within-cluster variance (Hair & Black, 2000; Hair et al., 2009; Mooi & Sarstedt, 2011; Tan,
Steinbach, & Kumar, 2005). The generated cluster solutions range from individual cases in a
cluster group by themselves to all cases in one single cluster group. An inspection of the
agglomeration schedule for large differences in the fusion coefficients can aid in the
identification of possible cluster solutions. Large differences in the fusion coefficient suggest
that moving to a smaller number of clusters combines more disparate clusters or groups of
participants. Table 8 shows the agglomeration schedule for the first nine cluster solutions where
large differences in the fusion coefficients start to occur. The changes in the fusion coefficients

48

were large, but how similar or different those changes are can pinpoint viable cluster solutions.
As would be expected, a very large change in the fusion coefficient occurred when the entire
sample was one cluster; the change from 1 to 2 clusters resulted in a difference in fusion
coefficient of 445.352. The most similar changes in the fusion coefficients were the differences
between 5-to 6-cluster solution, 6-to 7- cluster solutions, and 7-to 8-cluster solution; the
differences in their fusion coefficients were within 3-5 points of each other. On the other hand,
the most dissimilar fusion coefficient differences were between 4-to 5-cluster solution and 5-to
6- cluster solutions as well as 7-to 8- cluster solutions and 8-to 9- cluster solutions; their fusion
coefficient were more than 10 points apart. Based on these differences, it was possible that a
viable cluster solution could involve anywhere between 4 to 8 cluster groups.
The next step in the analysis was to narrow down the number of possible cluster
solutions. To do this I performed a multivariate analysis of variance (MANOVA) of all
clustering variables for the 4-8 cluster solutions. The MANOVA provided means, standard
deviations, and partial eta squared values for all cluster variables as well as the sample sizes of
each cluster group within each possible cluster solution. See Table 9 for information on the
variance explained and sample sizes for the 4-8 cluster solutions. An examination of all this
information helped to narrow down the number of possible cluster solutions. First, I examined
the sample size of each cluster group for unreasonable numbers. There is no general rule for
what is a reasonable and unreasonable sample size, so I considered a sample sizes less than 10%
of the total sample size as unreasonable. Using a 10% cutoff value, takes into consideration the
use of the final cluster solution in further data analysis where small sample sizes would be
problematic. For this criteria, any cluster solution that had cluster group sample sizes smaller

49

than 14 would not be a viable solution. This eliminated both the 7- and 8-cluster solutions, which
had cluster group sample sizes of 10 and 13.
Next, I considered partial eta squared values. Partial eta squared values are the proportion
of variance explained by the cluster groups on the cluster variables when accounting for the
effect the cluster variables have on each other (Richardson, 2011). Richardson (2011) stated that
partial eta squared measures are comparable to the Cohen’s d measure of effect size which
suggest a good partial eta squared value would be at or above 0.40. All cluster solutions had
partial eta squared value below 0.40 except for the 8-cluster solution, but they increased when
going from a 4 to a 5 to a 6 cluster solution. In particular, in going from the 4 to 5 clusters there
was a substantial increase in the variance explained for numeracy (i.e. Δ = .178) and large
increases (e.g. Δ > .050) for algebraic reasoning and MASMI, and in going from 5 to 6 clusters
there was a substantial increase for MASMI (Δ = .176) and a large increase for fluid intelligence.
Since there was some improvement of the variance explained for each of these cluster solutions,
I did not eliminate the 4-, 5-, or 6-cluster solutions as possible final cluster solutions.
The last thing that I considered was the changes in the means of each cluster variable
when moving from one cluster solution to the next. With Ward’s hierarchical procedure, cluster
groups combined based on preexisting cluster groups, so it is possible to consider the theoretical
and practical implications for the separation or addition of cluster groups. Going from a 4-cluster
solution to a 5-cluster solution split apart a cluster group from the 4-cluster solution that had
performance scores above the mean for all cluster variables except for MARMI, which was
below the mean. Splitting this cluster group into 2 clusters for the 5-cluster solution resulted in
one cluster with scores above the mean on numeracy, algebraic reasoning, and MASMI and the
other cluster had scores at or below the mean. This suggests that there may be potentially

50

important differences in these three variables that were lost within a 4-cluster solution. In
addition, going from a 5-cluster solution to a 6-cluster solution split apart a cluster group from
the 5-cluster solution that had performance scores below the mean for all cluster variables into
two clusters in the 6-cluster solution differentiated by their performance on MASMI. One cluster
had a MASMI performance score below the mean while the other had a MASMI performance
score above the mean. The separation of cluster groups produced high/low MASMI scores or
high/low MASMI scores in conjunction with high/low scores on other variables (i.e. numeracy
and algebraic reasoning). The identification of this shared but different characteristic may
support my hypothesis that both content knowledge and cognitive abilities are associated with
algebra performance, as well as that there are combinations of content knowledge and cognitive
abilities related to changes in algebra performance. In addition, MASMI was one of the
significant predictors found in the regression analysis, which determining if there are differences
in algebra performance for these cluster groups may corroborate the finding from the regression
analysis. On the other hand, it is possible that the 4-cluster solution had enough differentiation
between groups for changes in algebra performance. Thus, I conducted the next step in the
cluster analysis using the 4-, 5-, and 6-cluster solutions.
The third step was to find a final cluster solution by rerunning the cluster analysis using a
nonhierarchical clustering procedure called k-means. The k-means analysis partitions the sample
into a specified number of clusters, using the variable means from the hierarchical cluster
analyses, and iteratively moves cases into and out of clusters in order to maximize the
homogeneity within the cluster and the differences between cluster groups (Hair & Black, 2000;
Hair et al., 2009; Mooi & Sarstedt, 2011; Tan et al., 2005). The k-means analysis has an
advantage over hierarchical (Ward’s) because with Ward’s there is no switching of cases once

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combined. By not switching cases once combined, there is an increased chance that the final
cluster groups obtained with Ward’s method have combined less homogenous cases together due
to the order of the cases in the dataset. Another MANOVA analysis provided the necessary
information for choosing a final cluster solution.
As shown in Table 10, all cluster solutions had reasonable sample sizes for each cluster
group, and moderate to large amount of variance explained. There were some fluctuations in the
variance explained. Specifically, the variance explained for equivalence, fluid intelligence, and
algebraic reasoning decreased slightly when adding another cluster group. On the other hand, the
variance explained for MASMI increased substantially with each addition of a cluster group; the
MASMI variance increased by .123 when going from 4 to 5 clusters and increased by .102 when
going from 5 to 6 clusters. In addition, the changes in means for each cluster variable were
similar to the changes found with Ward’s where the defining characteristic of change was
MASMI scores. In particular, going from a 4 to 5 clusters splits apart a cluster with just a low
MARMI score into two groups with either a high or a low MASMI score and a low MARMI
score. Similarly, going from a 5 to 6 clusters splits a cluster group with low scores on all cluster
variables into two groups with low scores on the cluster variables with either a high or a low
MASMI score. As mentioned above, this distinction based on MASMI performance scores,
which the 4-cluster solution suppresses, was valuable information so the best choice for the final
cluster solution was either the 5-cluster solution or the 6-cluster solution. Given that both cluster
solutions provided conceptually different cluster groups and there were moderate to large
amounts of variance explained, the final cluster solution selected was the 6-cluster solution. The
final 6-cluster solution demonstrated statistically significant differences between cluster groups
on all cluster variables, Wilks’s λ (45, 571.205) = .017, p < .001, partial η2 = .559. See Table 10

52

for the raw score means, standard deviations, and partial eta squared values of the final cluster
solution, and Figure 2 for a graphical representation of the average standardized means for each
cluster variable by cluster group. This final 6-cluster solution is described in greater detailed in
the next section.
The last step was to validate the stability and replicability of the final cluster solution
using the double-split cross-validation procedure. The guidelines for the double-split crossvalidation procedure are: (1) split the sample into two equal halves, (2) perform the two-step
(Ward’s followed by K-means) cluster analysis on both halves, (3) combine the two datasets by
reassigning similar cases into the same cluster group, (4) conduct a nearest neighbor analysis and
(5) compare your assignment with the nearest neighbor analysis. By doing a nearest neighbor
analysis, which reassigns half of the cases in your dataset to the most similar profile of the most
similar case in the other, you can determine how well you did in cluster classification. Cohen’s
kappa, which is a measurement of interrater reliability (Îş > 0.60 indicates acceptable
replicability), was then used as a way to check the stability and reliability of the cluster solution.
For the 6-cluster solution, Cohen’s kappa was 0.536. This is a moderate level of agreement,
which indicates that the cluster solution is somewhat stable and replicable. The lack of strong
stability and replicability could be due to number of cluster variables used for this particular
sample size, which is a limitation of this study.
Cluster labels. I labelled each cluster group to reflect the level of each clustering
variable, with a particularly emphasis on labeling clusters based on the more extreme values
(high or low based on sample averages). Since I used standardized z-scores instead of raw scores
to create the cluster groups, extremely high and low means for the clustering variables were the
variables that had standardized means that were greater than 1 and less than -1, but the

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standardized means that fall between -1 and 1 are moderately high and low. Also included are
further descriptions of each cluster group focusing on raw scores instead of standardized scores
in order to compare this sample’s performance scores across different studies. In particular, I
interpreted all content knowledge scores using the NAEP Achievement Levels for grade 12, see
Table 12 for a description of each achievement level. To interpret spatial visualization
performance scores, I used previously published means, which were 21.49 and 22.49 for
MASMI (Campos, 2009, 2012) and 8.90 for MARMI (Campos, 2012). Additionally, I used
percentile ranks to interpret how well the participants in each cluster group did on the
standardized intelligence measures.
The first cluster group (n = 21, 14.89%) had participants with low average standardized
scores on all clustering variables with seven out of the nine being extremely low. Specifically,
the participants performed extremely low on numeracy, fractions, equivalence, algebraic
reasoning, MASMI, crystallized intelligence, and fluid intelligence, and had low average
performance scores on computational fluency and MARMI. Given these low average scores on
all assessments and their extremely low average standardized scores on seven out of nine
clustering variables, the label for this cluster group was Low All. With these scores, the
participants in this cluster group had statistically significantly lower performance scores than
participants in the other cluster groups on all clustering variables except for algebraic reasoning
and computational fluency. The participants’ scores in the Low All cluster group were similar to
the Moderate-High MASMI cluster groups’ participants’ scores for computational fluency and
the Moderate-Low All and Moderate-High MASMI cluster groups’ participants’ performance
scores for algebraic reasoning. In terms of raw scores, the participants in this cluster group
averaged between 20-45% accuracy on all content knowledge variables, averaged less than 10

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correct problems for both spatial visualization measures, were within the 21st percentile for
crystallized intelligence, and were within the 37th percentile for fluid intelligence. Their content
knowledge accuracy performance placed them below basic level using the NAEP achievement
level classification. This means that participants in the Low All cluster group were not able to
solve the most basic of mathematical problems. In comparison to previously published means for
the measures of spatial visualization, the participants in this cluster group were well below
average for both their MARMI and MASMI scores, which were more than 7 points below the
published means (e.g. 21.49 vs. 7.00, 22.49 vs. 7.00, and 8.90 vs. 1.14).
The second cluster group (n = 22, 15.60%) had participants with moderately low average
standardized scores on all clustering variables, with none being extremely low. Thus, the label
for this cluster group was Moderate-Low All. Even though Low All and Moderately Low-All have
participants with low average standardized scores on all clustering variables, they were
statistically significantly differences on all clustering variables except for all the cognitive
variables and algebraic reasoning. In raw score performance, the participants in this cluster group
averaged between 45-50% accuracy on all content knowledge variables, averaged less than one
correct problem for MARMI, averaged 11 correct problems for MASMI, was within the 37th
percentile for crystallized intelligence, and was within the 47th percentile for fluid intelligence.
NAEP achievement level classifications suggested that participants in the Moderate-Low All
cluster group were below the basic level for algebraic reasoning and equivalence, at or above the
basic level for computational fluency, and at or above the proficient level for numeracy and
fractions. This means that participants in this cluster group could not solve the most basic
algebraic reasoning and equivalence problems, but could solve basic computational fluency
problems as well as apply and integrate mathematical concepts for numeracy and fractions.

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Similar to the participants in the Low All cluster group, the participants in the Moderate-Low All
cluster group was also well below average for spatial visualization; both means were more than
10 points below the published means.
The third cluster group (n = 27, 19.15%) had participants with moderately high average
standardized scores on all clustering variables except for the MARMI and MASMI variables,
which were moderately low average standardized scores. Thus, the label for this cluster group is
Moderate-Low Spatial. The participants in this cluster group averaged 60-78% accuracy on all
content knowledge variables, averaged about two correct problems for MARMI, averaged 20
correct problems for MASMI, was within the 63rd percentile for crystallized intelligence, and
was within the 77th percentile for fluid intelligence. Some of their content knowledge scores were
at or above the proficient level and others were at or above the advanced level. Specifically, the
participants in the Moderate-Low Spatial cluster group were at or above proficient for
computational fluency, numeracy, equivalence, and algebraic reasoning as well as at or above the
advanced level for fractions. Even with moderately low spatial visualization scores, the
participants in this cluster group had similar performance scores on all content knowledge
variables as the participants in the sixth cluster group (i.e. High Spatial) that had extremely high
spatial visualization scores. For spatial visualization comparisons, the participants in the
Moderate-Low Spatial cluster group were below the published mean averages; both the MASMI
and MARMI scores were at least two points below average.
The fourth cluster group (n = 19, 13.48%) had participants with moderately low average
standardized scores for all clustering variables except for one. Specifically, the average
standardized scores for MASMI was moderately high while all other clustering variables were
moderately low. Given the distinctive moderately high average standardized score on MASMI,

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the label for this cluster group was Moderate-High MASMI. Even with a single clustering
variable with a moderately high average standardized score, the participants in Moderate-High
MASMI cluster group had similar performance scores to the participants in the Moderate-Low All
cluster group on the content knowledge variables. The content knowledge raw scores were also
similar to the participants in the Moderate-Low All cluster group with 40-63% accuracy. In
particular, they were below basic for computational fluency and equivalence, at or above basic
for fractions and algebraic reasoning, and at or above proficient for numeracy. As for the
cognitive variables, the participants in Moderate-High MASMI cluster group had similar
performance scores to the participants in the Moderate-Low Spatial cluster group on everything
but MASMI. Their average crystallized intelligence standardized score was within the 47th
percentile and their average fluid intelligence score was within the 70th percentile. Additionally,
their average MARMI raw score was below average with more than three points below the
published mean. Conversely, on MASMI their average raw score was well above average at 14
or more points above the published means.
The fifth and largest cluster group (n = 37, 26.24%) had participants with moderately
high average standardized scores on all clustering variable with no extremely low or high values;
thus I labeled this cluster group Moderate-High All. The participants in this cluster group had
raw score values at or above average for all clustering variables. The averages for their content
knowledge scores were between 70-89%, which means that their performance was at or above
both the proficient and advanced levels. Specifically, their scores were at or above proficient on
computational fluency and equivalence, and were at or above advanced for numeracy, fractions,
and algebraic reasoning. Additionally, their MASMI scores were more than 13 points above the
published means. The only performance score that was not at or above average was MARMI.

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Even though the participants in the Moderate-High All cluster group had an average MARMI
scores that was the second highest among all cluster group averages, it was still about two points
below the published average mean. In overall performance across all clustering variables, the
participants in the Moderate-High All cluster group outperformed all other cluster groups’
participants by having the highest raw scores on five out of nine times.
The sixth cluster group (n = 15, 10.64%) had participants with moderately high average
standardized scores for all cluster variables with two of those scores being extremely high.
Specifically, while the scores for MASMI and MARMI were extremely high, the average
standardized scores on all content knowledge and intelligence cluster variables were moderately
high, which were similar to the performance scores of participants in the Moderate -High All
cluster group. Given that the two extremely high scores were measures of spatial visualization,
the label for this cluster group was High Spatial. The raw score performance for the participants
in this cluster group was 62-86% accuracy for all content knowledge variables, more than 50%
accuracy for MARMI, more than 90% accuracy for MASMI, was within the 75th percentile for
crystallized intelligence, and was within the 90th percentile for fluid intelligence. Similar to the
Moderate-High All cluster group, this cluster group had content knowledge scores at or above
both the proficient and advanced levels, but unlike the Moderate-High All group, their scores on
algebraic reasoning were at or above the proficient level rather than at or above the advanced
level. MASMI scores were also well above average like the Moderate-High All group, but the
participants in the High Spatial group outperformed them on the MARMI with performance
scores more than 20 points above the published means.
Demographic characteristics. In addition to the differences in clustering variables, I
also considered differences in the demographic variables. I performed a Chi-square Test of

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Independence to determine if there was a difference in the proportion of students in each cluster
group based on demographic characteristics. Given the number of participants and the variety of
categories for some of the demographic variables, most demographic variables failed to meet the
expected count assumption for Chi-square analysis, so for all the categorical demographic
variables that were not already dichotomous I recoded them into dichotomous variables. For
instance, the majority of students in this study majored in the natural sciences, so I reclassified
the majors as STEM (Science, Technology, Engineering, and Mathematics) and Non-STEM (e.g.
social sciences, fine arts business, etc.). In addition, the majority of participants in the sample
were White, so I reclassified race/ethnicity to White and Non-White. I also changed school levels
from a five classification to a dichotomous classification where freshmen and sophomores
categorized as lowerclassmen and juniors, seniors, and 5+ seniors as upperclassmen.
As shown in Table 13, Chi-square analyses revealed that there was no statistically
significant difference for gender [χ2 (5) = 4.715, p = .460] and school level [χ2 (5) = 3.042, p =
.693], but there were for race/ethnicity [χ2 (5) = 14.356, p = .013], major [χ2 (5) = 21.364, p =
.001], and highest math course taken [χ2 (5) = 32.954, p < .001]. To identify the difference in
cluster membership for each of the demographic variables I used the adjusted standardized
residuals, which were the difference in the observed frequency and expected frequency. Large
residuals values over 2.0, both positive and negative, identified any associations (Laerd
Statistics, 2016). The race/ethnicity difference in cluster membership was driven by the more
than expected number of Non-White participants in the Low All group (residual = 2.9). The
statistically significant difference in cluster membership for majors was because of the less than
expected number of STEM participants in the Low All cluster group (residual = 2.5), less than
expected number of STEM participants in the Moderate-Low All cluster group (residual = 2.7) ,

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and the more than expected number of STEM participants in the Moderate-High All cluster
group (residual = 3.4). The difference in cluster membership for highest math course taken was
because of the less than expected number of Calculus participants in the Low All cluster group
(residual = 4.2) as well as more than expected number of Calculus participants in the ModerateHigh All cluster group (residual = 4.0).
The only demographic variable that was continuous was the number of years since an
algebra course; therefore, a one-way Analysis of Variance was conducted to determine if the
number of years since an algebra course differed by cluster groups. There were some outliers for
the Moderate-Low Spatial (n = 1), Moderate-Low All, (n = 4), and High Spatial (n = 1) cluster
groups that were changed to the value of the nearest neighbor instead of being removed in order
to keep the composition of each cluster group. There was also a lack of homogeneity of error
variance as identified by Levene’s Test of Homogeneity of Variances (p = .003), but the data was
normally distributed by inspection of Normal Q-Q Plot. Since there was a lack of constant
variance a one-way Welch ANOVA revealed that the number of years since an algebra course
was statistically significantly different between cluster groups, Welch’s F(5, 55.650) = 12.392, p
< .001. The number of years increased from the Moderate-Low All (M = 2.73, SD = .985) to the
Low All (M = 3.76, SD = 2.143), Moderate-High MASMI (M = 3.74, SD = 1.968), Moderate-Low
Spatial (M = 4.11, SD = 2.225), High Spatial (M = 4.93, SD = 1.280), and Moderate-High All (M
= 5.08, SD = 1.479) cluster groups in that order. Games-Howell post-hoc analysis showed that
the statistically significant difference in cluster membership was because of the Moderate-Low
All, High Spatial, Moderate-High All cluster groups. The participants in the Moderate-Low All
cluster group had more than two years difference between the participants in the High Spatial
and the Moderate-High All cluster groups, which mean that the Moderate-Low All cluster group

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had more participants who have had an algebra course more recently as compared to the High
Spatial and the Moderate-High All cluster groups. This could be because they have taken a
college algebra course or they recently graduated from high school.
Cluster Membership and Algebra Performance
The cluster memberships outlined above identified participant profiles with different
combinations of strengths and weaknesses in content knowledge and cognitive abilities. Each
profile had certain skills, understandings, and/or abilities that participants were good at, which
could have a different association with their algebra performance (RQ2b). To determine if cluster
membership made a difference in algebra performance, I conducted a one-way Analysis of
Variance (ANOVA). Before analysis, I checked to ensure the data meet all assumptions such as
no outliers, linearity, and homogeneity of variance. There were no outliers as assessed by
inspecting a boxplot. The inspection of a Normal Q-Q Plot determined that algebra performance
scores were normally distributed. In addition, there was a violation of homogeneity of variance
as assessed by Levene’s Test of Homogeneity of Variances (p = .004), so a Welch ANOVA was
performed instead. As shown in Table 14 and Figure 3, analysis revealed that there was
statistically significant difference in algebra performance as a function of cluster membership,
F(5, 56.603) = 18.896, p < .001. In particular, the participants in the High Spatial and ModerateHigh All cluster groups had similar algebra performance scores to each other but were
significantly different from the algebra performance scores of participants in the rest of the
cluster groups. Additionally, the participants in the Moderate-High MASMI and Moderate-Low
All cluster groups had similar algebra performance scores to the participants in the ModerateLow Spatial and Low All cluster groups, but the participants in the Moderate-Low Spatial and
Low All cluster groups had different algebra performance scores from each other.

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In describing the cluster groups, Chi-squared analyses indicated statistically significant
differences in cluster membership for some demographic variables. The previous ANOVA
analysis did not account for any of the demographic variables because they were categorical
instead of continuous variables, so I performed a multiple regression analysis to determine if
cluster membership made a difference in algebra performance after controlling for demographic
characteristics. In this multiple regression analysis, the outcome measure was participants’ raw
score on the algebra performance measure, and the predictor variables were the dichotomous
demographic variables already shown to have statistically significant association with cluster
membership (i.e., highest math taken, race/ethnicity, major, and the continuous variable of years
since last algebra course). The demographic variables that were not statistically significant (i.e.
gender and school level) were not included because they were not a differences in cluster
membership that could be attributed to differences in algebra performance. In addition, I used
planned contrasts to compare cluster groups while controlling for these demographic variables.
Planned contrast variables are an extended form of dummy coding that allows for
researchers to examine mean differences between groups (Davis, 2010). They are categorical
variables like cluster membership, but instead of the values denoting cluster membership they
were coded to denote the cluster groups being compared. For example, for the comparison of
clusters groups 1 and 3, the planned contrast variable would be (1, 0, -1, 0, 0, 0). Cluster groups
1 and 3 have the values of 1 and -1 because they are the comparison groups. In addition, each
contrast was orthogonal so that each contrast was not affected by other contrasts (Davis, 2010).
To construct orthogonal contrasts, the sum of the cluster membership values should equal to zero
as well as the summation of the cross product of each contrast to another contrast. The advantage
of using orthogonal contrasts is that it accounts for issues of suppression effects when comparing

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groups within a regression model; however, it also restricts cluster group comparisons, which
makes it difficult to do pairwise comparisons.
In this model, I created five planned contrasts, which examined group comparisons
based on theoretical implications for the differences in cluster groups, which the differences in
clustering variables identified. Specifically, the first three contrasts examined differences in
cluster groups based on participants’ levels of content knowledge to address the question of
whether or not there were differences between cluster groups whose participants had different
levels of content knowledge or similar levels of content knowledge. The last two contrasts
investigated differences in participants’ cognitive abilities given participants’ similar levels of
content knowledge (e.g., whether or not there were differences in algebra performance for cluster
groups that had participant performance that varied for spatial visualization and fluid intelligence
abilities but had similar levels of content knowledge). I describe the five contrasts in more detail
below in the presentation of the findings.
See Table 15 for the results of the multiple regression analysis. The multiple regression
model statistically significantly predicted algebra performance, F(9, 131) = 12.273, p < .001, and
met all assumptions. There was independence of observations as determined by a Durbin-Watson
statistic of 2.129. There was linearity as assessed by partial regression plots and a plot of
studentized residuals against predicted values. There was somewhat of a straight horizontal line
for all plots except for Highest Math Taken-Calculus or Above and the orthogonal contrast
between participants within the low and high content knowledge cluster groups, which showed
an upward sloping line. The Breusch-Pagan Test for Heteroscedasticity demonstrated constant
variance, χ2 (1, N = 140) = 11.213, p = .261). There were no issues of multicollinearity as
assessed by all correlation coefficients below 0.70 and all tolerance values greater than 0.10.

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There were no outliers as assessed by not having any studentized deleted residuals at Âą3, and no
influential or leverage points as assessed by no Cook’s distances above 1 and no leverage values
greater than 0.2. Lastly, the Normal Q-Q Plot showed that the data points follow the line closely
enough to be approximately normally distributed. The predictors explained 42% of the variance
in algebra performance, R2 = .457, Adjusted R2 = .420. The only statistically significant
demographic predictor was the highest math course taken. Taking at least one calculus course
predicted a 1.652 point increase in algebra performance, which suggests that the increased
exposure to mathematics improves algebra performance. As for group comparisons, only two of
the five contrasts were statistically significantly different.
The first contrast investigated the differences between participants’ algebra performance
scores of those within low content knowledge cluster groups (i.e. Low All, Moderate-Low All,
and Moderate-High MASMI) and the high content knowledge cluster groups (i.e. Moderate-High
All, High Spatial, and Moderate-Low Spatial). The most basic assumption when examining
group difference is that better performance is associated with being more knowledgeable. The
comparison of participants within these cluster groups helped to provide evidence for this basic
assumption. Even though this contrast was a given, it was included in the regression model
because differences in content knowledge was one of the main distinctive features of cluster
membership, and it was assumed that it would explain a good portion of the variance in algebra
performance. As expected, results revealed that there was a statistically significant difference
between participants’ algebra performance scores for those in the low content knowledge cluster
groups and the high content knowledge cluster groups, which being in the high content
knowledge cluster groups predicted a 5.179 point increase in participants’ algebra performance
scores.

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The next two contrasts compared the differences in algebra performance between
participants in the cluster groups with similar levels of content knowledge. Specifically, I
compared the participants in all of the high content knowledge cluster groups to each other, and
the participants in all of the low content knowledge groups to each other. Rules for orthogonal
contrasts dictated that to compare the participants of three cluster groups I would need to
combine two cluster groups together. Orthogonal contrasts also allowed for the direct
comparison of the participants of the two combined cluster groups. Therefore, the choice of
which two cluster groups to combine had to make sense theoretically.
I chose to combine the participants in the High Spatial and Moderate-Low Spatial cluster
groups and the participants in the Moderate-Low All and Moderate-High MASMI cluster groups.
The reason I chose these particular cluster groups was because both pairs of cluster groups had
participants with similar content knowledge and crystallized intelligence scores but different
spatial visualization and fluid intelligence scores. The only differences between the participants
in the cluster group pairs were their levels of content knowledge. Participants in the High Spatial
and Moderate-Low Spatial cluster groups had moderately high levels of content knowledge
while participants in the Moderate-Low All and Moderate-High MASMI cluster groups had
moderately low levels of content knowledge. By comparing, the participants in these cluster
groups (i.e. High Spatial vs. Moderate-Low Spatial and Moderate-Low All vs. Moderate-High
MASMI), I could make possible conclusions about how content knowledge might interact with
cognitive abilities, and in particular did the differences in participants’ spatial visualization and
fluid intelligence abilities support their algebra performance irrespective of their level of content
knowledge.

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Thus, for the investigation of the differences in algebra performance scores of
participants with similar levels of content knowledge, I combined the two cluster groups that I
wanted to do a pairwise comparison for later on. In particular, I compared the participants in
Moderate-High All cluster group to the combined performance of participants in the ModerateLow Spatial and High Spatial cluster groups as well as the participants in the Low All cluster
group to the combined performance of the participants in the Moderate-Low All and ModerateHigh MASMI cluster groups. The analysis showed that there was no statistically significant
difference in algebra performance scores for participants for either level of content knowledge.
Specifically, the Moderate-High All cluster group had participants’ algebra performance scores
that were similar to the algebra performance scores of participants in the Moderate-Low Spatial
and High Spatial cluster groups combined. In addition, the algebra performance scores of
participants in the Low All cluster group were similar to the algebra performance scores of the
participants in the Moderate-Low All and Moderate-High MASMI cluster groups combined. This
suggests that even with differences in the performance scores for content knowledge and
cognitive abilities, participants with similar levels of content knowledge had similar algebra
performance scores.
The fourth contrast compared cluster groups based on participants with similar levels of
content knowledge but different levels of spatial visualization and fluid intelligence. Specifically,
the contrast examined the differences between the participants in the Moderate-Low Spatial and
High Spatial cluster groups. The comparison of the participants in these two cluster groups
helped to determine if cognitive abilities supported algebra performance irrespective of
moderately high content knowledge, since their participants had similar levels of moderatelyhigh content knowledge but differed in terms of spatial abilities and fluid intelligence. Analysis

66

indicated that the participants in the Moderate-Low Spatial group had statistically different
algebra performance scores from the participants in the High Spatial cluster group. In particular,
being in the High Spatial cluster group increased participants’ algebra performance scores by
1.340 points. This suggests that having stronger spatial visualization and fluid intelligence
abilities was associated with higher algebra performance scores for those students who already
have moderately high content knowledge.
Lastly, the fifth planned contrasts also compared cluster groups who had participants with
similar performance scores on the content knowledge variables but were different on spatial
visualization and fluid intelligence. Instead of examining the differences between two cluster
groups with participants with moderately high content knowledge this contrast examined the
differences between two cluster groups with participants with moderately low content
knowledge. In particular, the contrasts compared the differences between participants in the
Moderate-Low All and the Moderate-High MASMI cluster groups. Thus, the comparison of
Moderate-Low All and Moderate-High MASMI helped to clarify if participants’ differences in
spatial visualization and fluid intelligence related to their algebra performance given the
similarity between their moderately low content knowledge scores. Results indicated that there
were no statistically significant differences between the participants in the Moderate-Low All and
Moderate-High MASMI cluster groups. This suggests that for students with moderately low
content knowledge stronger spatial visualization and fluid intelligence skills are not enough to
show higher algebra performance scores.
Even though the multiple regression analysis did not allow for the same pairwise
comparisons as the one-way ANOVA, the planned contrasts indicated that the analyses shared
similar results. In particular, participants in the low content knowledge cluster groups (i.e. Low

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All, Moderate-Low All, and Moderate-High MASMI) had similar algebra performance scores.
Participants in the high content knowledge cluster groups (i.e. Moderate-High All, High Spatial,
and Moderate-Low Spatial), on the other hand, had varying levels of algebra performance scores.
In particular, the participants in the Moderate-Low Spatial group had different algebra
performance scores than the participants in the Moderate-High All and High Spatial cluster
groups, but had similar algebra performance scores to the participants in the low content
knowledge cluster groups of Moderate-Low All and Moderate-High MASMI. In contrast,
participants’ spatial visualization scores were not classified based on just high or low scores for
the all spatial visualization variables. Instead, there was a high spatial group (i.e. High Spatial
and Moderate-High All), a low spatial group (i.e. Low All and Moderate-Low All), and a mixed
spatial group (i.e. Moderate-Low Spatial and Moderate-High MASMI). The low and high spatial
groups had participants with similar performance scores for both MASMI and MARMI, but the
mixed spatial group had participants with similar MARMI scores but different MASMI scores.
Unlike the content knowledge groups, there was no statistically significant difference in algebra
performance for participants within these groups, but there was similarity in participant
performance scores across cluster groups. Specifically, both mixed spatial cluster groups had
participants with similar algebra performance to the participants in the Moderate-Low All cluster
group, but only participants in the Moderate-High MASMI cluster group had similar algebra
performance scores to participants in the Low ALL cluster group.

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CHAPTER 5: Discussion
Previous research on improving students’ algebra performance has focused on
instructional approaches and students’ content knowledge (e.g. Blanton et al., 2015; KetterlinGeller, Gifford, & Perry, 2015; Sfard & Linchevski, 1994). The assumption was that better
instructional practices and math skills would more likely lead to better performance. Even
though changes in instructional approaches and students’ content knowledge have been
somewhat effective in improving students’ algebra performance, there is still more to understand
(Kloosterman, 2016; Kloosterman & Lester, 2004; Kloosterman & Lester Jr., 2007). Prior
research on pre-algebra knowledge demonstrated that domain general and domain specific
factors such as cognitive abilities and content knowledge related to pre-algebra performance, but
there is a lack of corroboration for these findings with investigations focused on algebra
performance. Thus, this study aimed to extend the research on algebra performance by
examining the connection students’ cognitive abilities and content knowledge may have on
algebra performance. This study also extends previous research by taking a person-oriented
approach, which allows for the identification of different constellations of skills associated with
algebra performance. The research findings provide a more nuanced understanding of algebra
performance as well as new insights into how different combinations of skills, understandings,
and/or abilities related to algebra performance.
Predictors of Algebra Performance
Similar to prior research, the current study suggested that both content knowledge and
general cognitive abilities are associated with algebra performance. However, contrary to my
hypothesis, the skills related to algebra performance were not fluid intelligence, algebraic
reasoning, equivalence, and numeracy. I assumed that given their seemingly close relation to

69

algebra they would be significant predictors. Instead, the regression analysis revealed that
fraction knowledge and spatial imagery better predicted algebra performance along with the
control variable for calculus as the highest math course taken. The finding regarding prior
calculus course completion also seemed to indicate that exposure to higher-level mathematics
may improve algebra performance because algebraic concepts are integrated within higher-level
mathematics. The reason that fractions and spatial imagery were positively associated with
algebra performance could be because they each have something to do with understanding
relationships. In particular, fraction knowledge helps with working with quantitative
relationships while spatial imagery helps with identifying the relationships between knowns and
unknowns variables.
Fraction are another form of numbers that can speak to the value of discrete (e.g. the
number of objects) and continuous (e.g., length, area, volume) quantitates. Reasoning with these
values requires an understanding of the relationship between numbers and operations, which
prior research has connected to algebra performance. Siegler et al. (2012) theorized that fraction
knowledge is important for algebra because understanding fractional magnitudes help with
estimating values of unknowns and evaluating the reasonableness of algebraic equations. As
3

Siegler et al. (2012) explained, when given the equation 4 𝑥 = 6, understanding fractions allows
you to see that the value of the unknown is slightly larger than six because you understand how
multiplication works with fractions. Similarly, linear functions use fractions to denote the ratio
between the independent and dependent variable, which when multiplied by a value of the
independent variable determines a value for the dependent variable. By understanding how
multiplication works with fractions, students can also understand the multiplicative relationship
between the independent and dependent variables of a linear function.

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Spatial imagery focuses on the mental transformation of images such as putting things
together or taking them apart. This is also similar to understanding relationships because Terao et
al. (2004) found that visuospatial regions of the brain activated when participants constructed an
equation from a problem statement, which requires an understanding of the relationships
between the known and unknown elements of a problem. Similarly, Tolar et al. (2009) suggested
that spatial visualization, which includes spatial imagery, was linked to problem solving abilities.
Problem solving abilities also involve identifying the relationship between knowns and
unknowns. Moreover, Chrysostomou et al. (2013) found that those who had higher spatial
imagery had better algebraic reasoning achievement, and connected to the algebraic reasoning
skills of functional thinking, generalization, and modelling is the ability to understand
relationships between quantities as well as known and unknown variables.
My current research findings are both consistent and inconsistent with prior research.
The consistency was mostly for the fact that I examined both content knowledge and general
cognitive abilities as predictors of algebra performance. Only one particular study, Tolar et al.
(2009), closely resembled my current study. Both studies investigated content knowledge and
cognitive abilities as factors associated with algebra performance of college students, and
defined algebra performance in terms of symbolic algebra. Also consistent with Tolar et al.
(2009) was the fact that past algebra education and spatial visualization were directly connected
with algebra performance. As much as the current study and Tolar et al. (2009) had their
similarities in sample population and some findings, there were inconsistencies in the design that
were in turn reflected in the results. In particular, Tolar et al. (2009) examined only four factors
while the current study examined nine different factors. The addition of more content knowledge
and cognitive ability factors may help to explain the differences between the two studies in the

71

relation between computational fluency and algebra performance. Computational fluency was no
longer directly associated with algebra performance; instead, fraction knowledge took precedent.
In addition, the results indicated that a specific form of spatial visualization connected to algebra
performance instead of a composite measure.
The current study was also inconsistent with prior research in terms of the specific factors
theorized to be connected to algebra performance. While past research would consider content
knowledge as basic numerical competencies (e.g. Geary, 2011; Passolunghi & Lanfranchi,
2012), the current study considered content knowledge in similar ways to the algebra readiness
research (e.g. Ketterlin-Geller & Chard, 2011; Ketterlin-Geller et al., 2015). This perspective
allowed for the identification of additional potential content knowledge factors not previously
considered by those who studied the predictors of algebra performance. For example, researchers
studying the predictors of algebra performance have not considered fraction knowledge. Their
focus has been on students’ understanding of whole numbers and their properties as well as
calculation and word problem solving skills (e.g. Fuchs et al., 2012, 2016). This is mostly
because their research populations are in the elementary grades that have not begun to learn
about fractions. The lack of consideration given to fractions neglects the fact that the entire
number system, which includes whole numbers, fractions, and decimals, lays the foundation for
learning algebra as well as many other mathematical subjects.
Another inconsistency was that most studies about the predictors of algebra performance
examined grade levels without formal algebra instruction. By studying the earlier grades on prealgebra knowledge, researchers hoped to find the foundational mathematics skills that would
support the learning of algebra, and alleviate mathematical learning difficulties (e,g, Caviola et
al., 2014; Vukovic et al., 2014; Ye et al., 2016). The present study also wanted to determine what

72

skills would help to alleviate mathematical learning difficulties, but instead of investigating the
foundational skills before instruction, this study investigated the foundational skills after
instruction. Unlike previous studies, I aimed to identify which existing skills would predict
current algebra performance, and more specifically which existing skills currently support strong
algebra performance given that time has elapsed since formal instruction. By taking a top down
approach rather than the bottom up approach, I hoped to determine which developed skills might
alleviate mathematical learning difficulties.
The current findings suggested that, after formal algebra instruction, both fraction
knowledge and spatial imagery statistically significantly predicted algebra performance.
Unfortunately, very few studies have examined fractional knowledge and spatial imagery in
relation to algebra performance. Some have made theoretical claims about fraction knowledge
and spatial imagery (e.g. Kilpatrick & Izsak, 2008; Mix & Cheng, 2012; National Mathematics
Advisory Panel, 2008; Wu, 2001), but few have proven it with empirical evidence. The present
study adds to this literature by showing that even when considering other forms of mathematical
content knowledge (e.g. computational fluency, understanding of equivalence, numeracy) and
cognitive abilities (e.g. crystallized and fluid intelligence), fraction knowledge and spatial
imagery stand out as important factors related to algebra performance. Thus, it may be prudent
for teachers to focus on developing students’ fractional knowledge and spatial imagery skills in
preparation for formal algebra instruction.
Cognitive Abilities and Content Knowledge Profiles and Algebra Performance
Previous research has shown that content knowledge and cognitive abilities are
associated with algebra performance. It has even identified which forms of content knowledge
and cognitive abilities have a strong association. What previous research does not explain is how

73

different combinations of content knowledge and cognitive abilities may change algebra
performance. Thus, the present study sought to identify these different combinations of content
knowledge and cognitive abilities and their relation to algebra performance. Specifically, I
conducted a cluster analysis to identify groups of participants with similar performance on all
predictors within the group, but had different performance on all predictors across groups. The
subsequent one-way ANOVA analyses determined the differences between each group, and
interpretations of these differences explained the different combinations of skills, knowledge,
and/or abilities that can contribute to changes in algebra performance.
The results of the cluster analysis indicated that within the current sample there were six
cluster groups with different levels of content knowledge and cognitive ability performance. The
speculation about the types of groups that would emerge held true for the idea that there would
be separate groups based on strengths and weaknesses for content knowledge or cognitive
abilities, but did not for the idea that there would be defined groups characterized by certain
types of skills sets. The lack of defined skill sets could be because the 6-cluster solution
suppressed any skill set differences. The sample size of the current study limits the number of
possible cluster solutions. With more clusters, it may be possible to find skill set differences, but
with the current study, the more salient differences were for overall performance on both content
knowledge and cognitive ability measures and spatial visualization. Specifically, there were two
groups with low performance on all variables (i.e. Low All and Moderate-Low All), one group
with high performance on all variables (i.e. Moderate-High All), and three groups that had
distinguishable differences in spatial visualization performance in comparison to content
knowledge and intelligence (i.e. Moderate-Low Spatial, Moderate-High MASMI, High Spatial).
Unlike the results of the regression analysis, fractions did not appear to be a key factor in

74

differentiating among clusters. Fractions may not have contributed to the creation of the cluster
groups because there were larger differences in participants’ spatial visualization performance
scores in which to group cases than there were for fractions. In addition, not captured by the
cluster analysis is the fact that fractions have a higher association with algebra performance such
that even when accounting for its association with other predictors it still stood out, and because
the clusters were not based on its association with algebra performance fraction knowledge did
not stand out when clustering participants.
Differences in clustering variable performance suggested that cluster group spatial
visualization differences can vary regardless of the level of content knowledge and/or
intelligence. This was evident by the content knowledge and/or intelligence performance scores
similarities for the participants in cluster groups with differences in spatial visualization
performance scores. For example, the participants in the Moderate-Low All group and
Moderate-High MASMI group had different performance scores on both measures of spatial
visualization, yet they had similar moderately low content knowledge performance scores (see
Figure 4). The participants in the Moderate-Low Spatial group and High Spatial group also had
different performance scores on both measures of spatial visualization, but they had similar
moderately high content knowledge performance scores (see Figure 5). Moreover, the
participants in the Moderate-High All and High Spatial groups had different performance scores
on both measures of spatial visualization, but had similar performance scores on all content
knowledge and intelligence variables (see Figure 6). Also highlighted was the fact that the
cluster group with the highest participants’ raw score averages for five out of nine clustering
variables was the Moderate-High All cluster group, which is not one of the cluster groups with
distinctive participants’ performance scores for spatial visualization. These findings would

75

suggested that even though there were identifiable variations in spatial visualization performance
scores, the participants in the cluster groups were mostly different based on their overall low and
high performance scores on both content knowledge and cognitive abilities measures.
As for algebra performance, interpretations of both the one-way ANOVA and multiple
regression analysis suggested that the same differences between cluster groups existed even
when controlling for demographic characteristics. With the similarities in results for both the
multiple regression and one-way ANOVA analysis, I used the ANOVA post hoc analysis results
as the primary for interpreting the relation between cluster membership and algebra performance,
and the results of the multiple regression analysis were a supplement. The results indicated that
similar to previous research both content knowledge and cognitive abilities were associated with
algebra performance. As shown in Table 14 and Figure 2, the highest algebra performance scores
were for those participants in clusters groups with both high content knowledge and high
cognitive abilities, and lower algebra performance scores accompanied lower cognitive abilities
or content knowledge performance scores.
Additionally, the current findings suggested that higher algebra performance scores
occurred for participants with higher levels of overall content knowledge as indicated by the
difference in algebra performance scores for participants within the Moderate-High All and
Moderate-High MASMI cluster groups (see Figure 7). These two cluster groups shared similar
performance scores on the spatial visualization variables and fluid intelligence, but were
different on all content knowledge variables and crystallized intelligence. Thus, the difference in
algebra performance scores suggested that higher content knowledge and stronger crystallized
intelligence scores demonstrated higher algebra performance scores.

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The research finding also suggested that there were specific combinations of content
knowledge and cognitive abilities that were associated with higher algebra performance scores.
In particular, the combination of strong spatial visualization abilities, strong fluid intelligence,
and high content knowledge was associated with higher algebra performance scores as found by
the comparing participants within the High Spatial and Moderate-Low Spatial cluster groups (see
Figure 5), the participants within the High Spatial and Moderate-High All cluster groups (see
Figure 6), and the participants within the Moderate-High MASMI and Moderate-Low All cluster
groups (see Figure 7). In addition, stronger fraction, numeracy, algebraic reasoning, and spatial
imagery skills appear to be associated with higher algebra performance scores as seen by
comparing participants within the Moderate-High All cluster group and the participants in the
Moderate-Low Spatial cluster group (see Figure 8).
The participants in the High Spatial and Moderate-Low Spatial cluster groups had similar
performance scores for all content knowledge variables, but had different performance scores on
spatial visualization and fluid intelligence (see Figure 5). In particular, the participants in the
High Spatial cluster group had higher mean averages for spatial visualization and fluid
intelligence. Their difference in algebra performance scores showed that the High Spatial cluster
group’s participants had a statistically significant higher scores on algebra performance than the
Moderate-Low Spatial cluster groups’ participants, which suggested that the stronger spatial
visualization and fluid intelligence abilities of the participants in the High Spatial cluster group
might have contributed to better scores on algebra performance.
Additionally, participants within the High Spatial and Moderate-High All were different
only on the measures of spatial visualization (see Figure 6). The similarity between their algebra
performance scores suggested that spatial visualization skills alone are not enough to

77

demonstrate higher algebra performance scores. On the other hand, the participants within the
Moderate-High MASMI and Moderate-Low All cluster groups were very similar to the
participants within the Moderate-Low Spatial and High Spatial cluster groups in differences in
clustering variables (see Figure 4 and Figure 5). They too were only different for their scores on
spatial visualization and fluid intelligence, but the participants in these cluster groups
demonstrated low levels of content knowledge instead. The lack of difference in their algebra
performance scores suggests that for participants with low levels of content knowledge, stronger
spatial visualization, and fluid intelligence abilities may not be enough to yield higher algebra
performance scores. Taken together the results of these three comparisons suggest that it is the
combination of stronger spatial visualization, fluid intelligence abilities, and moderately high
levels of content knowledge that may best support higher algebra performance scores rather than
any of these skills by themselves.
Moreover, the participants in the Moderate-High All cluster group and the Moderate-Low
Spatial cluster group had similar performance scores on all content knowledge and cognitive
ability assessments except for numeracy, fractions, algebraic reasoning, and MASMI (see Figure
8). The statistically significant higher algebra performance scores for the participants in the
Moderate-High All cluster group over the Moderate-Low Spatial cluster group suggested that
stronger skills in numeracy, fractions, algebraic reasoning, and MASMI might have contributed
to the difference in algebra performance.
These research findings confirmed my hypothesis that better cognitive abilities (e.g., fluid
intelligence, spatial visualization) may support better algebra performance scores, but the fact
that the better cognitive abilities occurred in addition to high content knowledge was surprising.
This was a surprising finding because I would logically assume that better cognitive abilities

78

would support better algebra performance scores irrespective of level of content knowledge.
One possible explanation would be that the relative differences in content knowledge are more
salient at this level of knowledge. In particular, the High Spatial cluster group’s advanced
achievement level in numeracy and fractions may have contributed to the differences in algebra
performance. Unfortunately, the information provided by the current study does not allow for
definite conclusions on this matter, but the statistically significant difference in algebra
performance for Moderate-High All and Moderate-Low Spatial suggested that this assumption
could be true.
Even though it is the beyond the scope of this study to uncover why or how these
different combinations of skills, understandings, and/or abilities relate to algebra performance,
the results did provide a more nuanced understanding of how the factors of content knowledge,
spatial visualization, and intelligence could contribute to differences in algebra performance. The
understanding was that there are two circumstances associated with higher algebra performance
scores: students who had strong spatial visualization skills, strong fluid intelligence skills, and
high content knowledge or students who had strong fraction knowledge, numeracy skills,
algebraic reasoning skills, and spatial imagery skills.
Variable-Oriented vs. Person-Oriented: What do the differences mean?
Most studies about the predictors of algebra performance have taken a variable-oriented
approach that emphasizes the independent relation of each predictor variable on the average
performance of the whole sample. Few have considered a person-oriented approach that
emphasizes individual patterns of development. Research that has come close to the personoriented approach were the studies that focused on understanding common errors and
misconceptions of specific topics based on group level performances (e.g. Booth, 1988;

79

Kuchemann, 1978). These studies were individual interviews that drew conclusions about the
whole sample instead of evaluating individual patterns of development for different
combinations of skills, knowledge, and/or abilities related to algebra performance.
Investigating individual patterns of development highlights how combinations of skills,
knowledge, and/or abilities can differentially predict algebra performance that the variableoriented approach may obscure. For instance, findings from the regression analysis suggested
that fraction and MASMI scores predicted algebra performance scores while the cluster analysis
suggested that a noticeable difference in algebra performance occurs when you have strong
content knowledge in addition to strong spatial visualization abilities and fluid intelligence.
Additionally, the cluster analysis suggested that other skills that related to algebra performance
were numeracy and algebraic reasoning. The only findings shared between the regression
analysis and cluster analysis were the importance of taking at least one calculus course and
spatial visualization. In both analyses those who had taken a calculus course had higher
performance scores compared to those who had only taken at least one algebra course,
trigonometry, or pre-calculus course, and strong spatial imagery abilities were defining factors
for high algebra performance.
The differences found between the cluster and regression analysis highlight the fact that
there are certain patterns of relations among the clustering variables that multiple regression may
or may not detect. The multiple regression analysis has limited capacity to identify these
relationships. The regression model does not automatically account for these unless it is a
predetermined addition to the model in terms of an interaction term. Their detection depends
upon whether or not there is sufficient power within the model to detect statistically significant
associations, which depends upon sample size and the number of predictors (Hair et al., 2009).

80

Too many or too little in sample size and the number of predictors can change the outcome of the
multiple regression. While cluster analysis may bring our attention to the sorts of relationships
missed by regression analysis, it is a more subjective research methodology. It requires the
researcher to make decisions based on theoretical implications guided by statistical outcomes.
This means that generalizability of the research findings depends on researchers coming to the
similar conclusions. This may be difficult because each researcher has his or her own theoretical
framework for identifying clusters. Nevertheless, the information gleaned from the cluster
analysis gives new insight to algebra performance given the limitations of the multiple regression
analysis.
Limitations
There are a number of limitations for this study. The first limitation would be for the
sample population, which are college students whose exposure to mathematics goes beyond
algebra. Most prior research on algebra performance has examined grade levels without formal
algebra instruction or those who are just beginning to learn algebra. It is possible that the
exposure to additional math topics altered the association between algebra performance and the
content knowledge variables, and would alter the findings from this study. Therefore, it is
necessary to replicate this study with beginning algebra students in order to be able to generalize.
Similarly, the current study’s small sample size is also a limitation of the study that could
alter the findings. The plan was to conduct the study with 200 participants, which apriori power
analyses suggested would be sufficient for about 80% - 95% power depending on the statistical
analysis. Additionally, these power analysis were made with the assumption that the present
study would have a medium effect size (i.e. f 2 = .15 or f = .25) (Faul, Erdfelder, Lang, &
Buchner, 2007). The actual sample size for the current study was 141 participants, which is

81

smaller the targeted number of participants. Even though I was not able to run 200 participants,
the calculated effect sizes for the multiple regression and ANOVA analyses were large (i.e. f 2 >
.35 or f > .40) instead of the assumed medium effect (Faul et al., 2007) . For the multiple
regression analysis to determine the strongest predictors of algebra performance the calculated
effect size was 1.159, and the one-way ANOVA and multiple regression analysis used to
determine the differences in algebra performance based on cluster membership had calculated
effect sizes of 0.834 and 0.845 respectively. The larger effect sizes meant that I was able to
achieve 95% power with a smaller sample size, so the multiple regression and ANOVA analyses
had sufficient power with only 141 participants instead of the targeted 200. The only analysis
that the small sample size may have affected was the cluster analysis. The recommended sample
size for cluster analysis is 2m, where m is the number of clustering variables. In the present study,
there were 9 clustering variables, so the recommended sample size was 29 = 512. The current
sample size of 141 is much smaller than the suggested 512. This smaller sample size would have
limited the number of cluster groups found, the number of participants within each cluster group,
changed the general make-up of the types of clusters groups found, as well as possible changed
the differences in algebra performance between cluster groups. In the current study, the biggest
impact that the small sample size had was on the number of clusters found. As shown in Table 9,
the amount of explained variance in the cluster variables was higher with more cluster groups,
which suggested that there was more differentiation in participant performance scores on the
cluster variables that may denote more types of participants. This differentiation could have
signified more combinations of content knowledge and cognitive abilities that may relate to
participants’ algebra performance scores. Nevertheless, the results found in the current study

82

have given the first indication that there are certain combinations of content knowledge and
cognitive abilities that are associated with better algebra performance scores.
Another limitation is that each content knowledge assessment had a time limit, which
may or may not have been enough for all participants. Very few participants finished each
assessment, even though pilot testing showed that all assessments could be finished within the
given period. This means that the score of zero for incorrect and blank answers may not be an
accurate representation of the knowledge participants had.
In addition to the time limit, most content knowledge measures were also researcher
designed measures that had some design flaws. One particular design flaw was the issues of
counterbalancing problem types. Not all assessments made sure that participants were able to see
all problem types no matter if they finished or not. Specifically, the numeracy and algebra
performance assessments had issues with counterbalance. For numeracy, the problem was that all
addition problems and subtraction problems came before all the multiplication problems and
division problems. This was an issue when participants focused on using algorithms instead of
their knowledge of the properties of numbers and operations to answer the problem because the
focus on doing the algorithms rarely got them beyond the addition or the subtraction problems
within three minutes, and they missed the multiplication and division problems. Unlike the
numeracy assessment, I tried to counterbalance the algebra performance assessments, but the
procedure that I used did not provide true randomization. The assessment ended up with all the
factoring and exponent questions at the end of the tests, which some students did not reach. A
better way to counterbalance would be to make sure that one question out from each item
category would end up in each half of the test in a randomized order.

83

Another design flaw was for the types of questions used to assess the mathematical
content knowledge. This is an issue because there are many different ways to measure the
mathematical content knowledge. For instance, Tolar et al. (2009) measured computational using
the ‘number facility’ subtest of the ETS Kit of Factored-Referenced Cognitive Tests, which
measured computational fluency with both multi-digit calculation problems and problems that
had participants determine if the suggested as was correct for the given problem. I on the other
hand only measured computational fluency with multiple multi-digit calculation problems, which
could possible account for the differences in the results for computational fluency. Similarly,
Hecht, Close, and Santisi (2003) and Hecht and Vagi (2010) measured fraction knowledge with
assessments for computing fractions, estimating fractions, word problems, comparing fractions,
and identifying fractions while I based my fraction assessment off of the assessment designed by
G. Brown and Quinn (2007). Their assessment measured fraction knowledge for algorithmic
applications, word problems, elementary algebraic concepts, arithmetic skills, structure of
rational numbers, and computational fluency. The inconsistency of assessments between research
studies makes it difficult to know if the questions used are an accurate representation of the
mathematical content knowledge. All that guides us is whether or not the internal consistency of
the assessment has an acceptable Cronbach’s Alpha value of 0.70 or better, but this only
provides evidence that the questions we are using are measuring the same construct, which may
or may not be the same given another set of questions.
The algebra performance assessment was also a limitation of this study. This study
defined algebra performance in similar ways to Tolar et al. (2009), whose definition is the ability
to solve algebra problems using pre-learned symbolic manipulation algorithms. The heavy
emphasis on symbolic algebra may not be an accurate representation of what algebra is. The

84

choice for focusing on symbolic algebra was to make sure that it did not overlap with the
algebraic reasoning assessment. If I could change the algebra performance assessment, I would
probably put more emphasis on application type questions. I think this would have shown
participants ability to both demonstrate symbolic knowledge as well as reasoning skills. The only
downside to the application questions would be that I would have to eliminate algebraic
reasoning as a predictor variable in multiple regression analysis because of possible issue with
being highly related to the outcome measure; however, it would not be a problem with a cluster
analysis, and is worth considering.
Implications
Although more work is needed to have a better understanding of why factors such as
fraction knowledge and spatial imagery (a.k.a. MASMI) predict algebra performance, their
statistically significance in the regression model suggest that, in preparation for formal algebra
instruction, it may be useful for students to develop their fraction knowledge and spatial imagery
skills. Spatial imagery development may be as simple as encouraging students to draw more
pictures when trying to solve problems or using more pictorial based problems solving method
such as the Singapore Model Method (e.g. Lee & Ng, 2009). On the other hand developing
fraction knowledge can be more challenging. Fractions are a mathematical concept with which
students already have difficulty (e.g. G. Brown & Quinn, 2006; Peck & Matassa, 2016). Some
researchers have attributed this difficulty to the lack of personal understanding that teachers have
for fractions (e.g. Siegler et al., 2012), so developing students’ fraction knowledge may involve
developing not only students fractional content knowledge but teachers as well.
Just like with the results found from the multiple regression analysis, more work is need
to understand completely how content knowledge and cognitive abilities profiles are associated

85

with algebra performance. In particular, the why and how the different combinations of content
knowledge and cognitive abilities support algebra performance; however, the current findings
have certain implication for educators. One implication is that educators may scaffold the
development of content knowledge with the use of instructional practices that take advantage of
the spatial imagery and fluid intelligence of students. For example, the use of more discussions
about students’ problem solving strategies that include visual depictions of different strategies
emphasizes both spatial imagery and fluid intelligence. The different strategies make apparent
the different logical steps used to arrive at the same answer, and promote the use of fluid
intelligence skills by getting student to synthesize across methods as to why they both work. The
use of visual depictions of the solution method gets students to use their spatial imagery skills
and makes apparent how students think about the relationship between the known and unknown
variables.
Another implication is that the elementary grades are a place to develop the skills of
necessary for strong algebra performance. In particular, the development of the skills of fractions
and numeracy already occur in the elementary grades while algebraic reasoning and spatial
imagery can be (e.g. Cooper & Warren, 2011; Lannin, 2003; Lee & Ng, 2009; Moss & McNab,
2011). In fact, the identification of these sets of skills as factors that can improve algebra
performance provided credence to the recommendation to teach algebraic concepts starting in the
elementary grades (Blanton et al., 2015; A. Stephens, Blanton, Knuth, Isler, & Gardiner, 2015).
Additionally, with more research we can also identify which content knowledge and cognitive
abilities relates to particular algebraic topics, which may point towards a way to provide extra
help for those who do not have mathematics learning disabilities.

86

Conclusion
The present study investigated content knowledge and cognitive abilities as factors
associated with algebra performance. This association was determined by conducting a multiple
regression and cluster analysis. The two approaches allowed the examination of algebra
performance from both variable-oriented and person-oriented approaches. The variable-oriented
approach (i.e. multiple regression analysis) revealed the independent predictors of algebra
performance, while the person-oriented approach (i.e. cluster analysis) revealed how different
patterns of content knowledge and cognitive abilities related to algebra performance. Each
perspective brings out its own unique understanding of algebra performance that has expanded
our understanding of what factors are associated with it. The finding from the current study
proposed that fraction knowledge and spatial imagery are additional predictors of algebra
performance not covered by previous research, which makes necessary more research in order to
understand how they connect with the other factors already known. Additionally, the claim that
algebra performance is associated with both content knowledge and general cognitive abilities
was supported through not only with the addition of fraction knowledge and spatial imagery as
predictors, but with different combinations of content knowledge and cognitive abilities between
participants in the cluster groups that may have contributed to differences in their algebra
performance. More research is needed to corroborate these findings but they are an important
first step into understanding how content knowledge and cognitive abilities extend our
understanding of the complex array of factors associated with algebra performance.

87

APPENDICES

88

Appendix A: Tables
Table 1
Basic Information about Assessments and Measures
Tests

Time (mins)

Calculator

Number of Items

Nonverbal Analogies

7

N

52

Sequences

10

N

43

General Knowledge

3

N

47

Odd Word Out

5

N

40

Word Opposites

5

N

40

MASMI

10

N

23

MARMI

10

N

23

Computational Fluency

3

N

16

Numeracy

4

N

24

Fractions

10

N

12

Equivalence

5

N

6

Algebraic Reasoning

10

N

7

20

Y

20

Cognitive Abilities
Fluid Intelligence

Crystallized Intelligence

Spatial Visualization

Content Knowledge

Algebra Performance

Note. Y = Yes; N = No; MASMI = Measure of the Ability to Form Spatial Mental Images;
MARMI = Measure of the Ability to Rotate Mental Images.

89

Table 2
Descriptive Statistics for Assessments and Measures
M

SD

Îą

Minimum Score

Maximum Score

Gender

−

−

−

−

−

Calculus

−

−

−

−

−

Years since algebra course

4.20

2.112

−

0

14

Assessments & Measures
Control Variables

Content Knowledge
Computational Fluency

31.57

10.19

0.80

0

55

Numeracy

16.86

4.98

0.87

0

24

Fractions

8.30

2.70

0.76

0

12

Equivalence

6.11

2.70

0.73

0

11

12.73

4.49

0.85

0

22

MARMI

5.99

10.89

0.82

-46

46

MASMI

25.55

15.30

0.93

-46

46

Crystallized Intelligencea

102.22

12.39

−b

35

185

Fluid Intelligencea

108.18

11.12

−b

35

171

Algebra Performance

7.64

3.79

0.76

0

20

Algebraic Reasoning
Cognitive Abilities

Note. MASMI = Measure of the Ability to Form Spatial Mental Images; MARMI = Measure of
the Ability to Rotate Mental Images.
a
Standardized assessments with standard scores of ¾ = 100, σ =15. b Sample internal reliabilities
not calculated because assessments were standardized.
**
p < 0.01. *p < 0.05.

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Table 3
Randomized Testing Orders
Group Order
Test Order

1

2

3
Algebra
Performance

4

1

Fractions

Numeracy

2

Computational
Fluency

RAIT

Equivalence

Algebraic
Reasoning

3

MARMI

MASMI

Computational
Fluency

RAIT

4

Equivalence

Algebraic
Reasoning

MARMI

Numeracy

5

Algebra
Performance

MARMI

Fractions

Computational
Fluency

6

RAIT

Fractions

Algebraic
Reasoning

MARMI

7

Numeracy

MASMI

Equivalence

8

Algebraic
Reasoning

Algebra
Performance
Computational
Fluency

Numeracy

Fractions

9

MASMI

Equivalence

RAIT

Algebra
Performance

MASMI

Note. MASMI = Measure of the Ability to Form Spatial Mental Images; MARMI = Measure of
the Ability to Rotate Mental Images.

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Table 4
Bivariate Correlations for Assessments and Measures
Assessments & Measures
1
2
3
Control Variables
1. Gender
‒
2. Calculus
-.143
‒
**
3. Years since algebra course -.300 .281**
‒
Content Knowledge
4. Computational Fluency
-.045 .312** .128
5. Numeracy
-.219** .394** .239**
6. Fractions
-.118 .415** .218**
7. Equivalence
-.035 .418** .171*
8. Algebraic Reasoning
-.082 .419** .180*
Cognitive Abilities
9. MARMI
-.113 .238** .184*
10. MASMI
.014 .350** .150
11. Crystallized Intelligencea -.004 .228** .222**
12. Fluid Intelligencea
.010 .353** .210*
Algebra Performance
-.110 .471** .200*

4

5

6

7

8

‒
.399**
.496**
.433**
.363**

‒
.626**
.475**
.567**

‒
.536**
.651**

‒
.517**

‒

.202*
.216**
.377**
.350**
.443**

.316**
.446**
.482**
.535**
.470**

.293**
.474**
.605**
.527**
.657**

.300**
.432**
.529**
.589**
.459**

.267**
.499**
.482**
.540**
.544**

9

10

‒
.475**
‒
.264** .412**
.428** .585**
.336** .504**

11

12

‒
.633**
.411**

‒
.425**

Note. MASMI = Measure of the Ability to Form Spatial Mental Images; MARMI = Measure of the Ability to Rotate Mental Images.
a
Standardized assessments with standard scores of ¾ = 100, σ =15.
**
p < 0.01. *p < 0.05.

92

Table 5
Analysis of Item Categories for All Content Knowledge Assessments
Descriptive Statistics
Percent Correct
Total
Full
Partial
Assessment Items Categories
M
SD
Points
Credit
Credit
Computational Fluency
Addition
11.23
2.517
16
10.6
89.4
Subtraction
8.43
3.514
15
13.5
86.5
Multiplication
8.47
4.633
17
7.1
85.1
Division
3.44
2.237
7
8.5
75.9
Numeracy
Addition
5.10
1.110
6
48.2
51.8
Subtraction
4.64
1.091
6
24.8
75.2
Multiplication
3.77
1.843
6
24.8
67.4
Division
3.35
2.208
6
25.5
55.4
Whole Numbers
4.65
1.459
6
41.8
58.2
Decimals
8.08
1.848
10
29.8
70.2
Fractions
4.13
2.507
8
11.3
82.3
Fractions
Algorithmic Operations
1.59
.633
2
66.7
25.5
Word Problems
1.74
.516
2
77.3
19.1
Algebraic Concepts
1.68
.552
2
72.3
23.4
Arithmetic Skills
.98
.788
2
29.8
38.3
Rational Number
1.38
.692
2
49.6
38.3
Computational Fluency
.94
.791
2
28.4
37.6
Equivalence
Interpretation
1.98
1.017
3
38.3
49.6
Structure
1.76
1.242
4
8.5
71.6
Open Equation
2.38
1.251
4
25.5
67.4
Algebraic Reasoning
Functional Thinking
1.55
1.485
4
14.2
51.0
Generalization
2.43
1.431
4
23.4
56.0
Modeling
.55
.671
2
9.9
34.8
Symbolic Manipulation
5.97
1.912
8
15.6
81.6
Structure Sense
2.23
1.155
4
17.7
73.1
Algebra Performance
Systems of Equations
.91
.774
2
25.5
39.7
Functions
1.14
.713
2
33.3
47.5
Solving Equations
1.31
.698
2
44.7
41.8
Inequalities
.72
.658
2
11.3
48.9
Graphing
.98
.751
2
27.0
44.0
Exponents
.42
.611
2
6.4
29.1
Factoring
.84
.733
2
19.9
44.0
Complex Numbers
.28
.468
2
0.7
27.0
Polynomial Division
.49
.683
2
10.6
27.7
Logarithms
.55
.659
2
9.2
36.9
93

No
Credit
0.0
0.0
7.8
15.6
0.0
0.0
7.8
19.1
0.0
0.0
6.4
7.8
3.5
4.3
31.9
12.1
34.0
12.1
19.9
7.1
34.8
20.6
55.3
2.8
9.2
34.8
19.1
13.5
39.7
29.1
64.5
36.2
72.3
61.7
53.9

Table 6
Summary of Independent Samples T-Test and Mann-Whitney U Test Analysis for Gender Differences
Independent Samples T-Test

Assessment

Mean
Differencea

Standard
Error
Difference

Mann-Whitney

t(139)

Mean Rank
Female
Male
(n = 110)
(n = 31)

U

z

p

Content Knowledge
Computational Fluency

-1.092

2.077

-.526

.600

Numeracy

-2.621

.991

-2.645

.009

Fraction

-.767

.546

-1.403

.163

Equivalence

-.227

.549

-.413

.680

Algebraic Reasoning

-.883

.913

-.967

.335

Cognitive Abilities
MARMI
MASMI

68.55
.502

3.122

79.71

1435.00

-1.346

.161

.178
.872

Crystallized Intelligence

70.28

73.55

1626.00

-0.393

.694

Fluid Intelligence

71.09

70.69

1695.50

-0.047

.962

Algebra Performance

-1.001

.843

-1.188

.242

Note. MASMI = Measure of the Ability to Form Spatial Mental Images; MARMI = Measure of the Ability to Rotate Mental Images.
a

Mean Difference calculated by subtracting the mean scores of Males from the mean scores of Females.

94

Table 7
Summary of Multiple Regression Analysis
Variable

B

SEb

β

p

rpartial

r2partial

Intercept

1.876

2.897

Gender

-.229

.601

-.025

.704

-.023

.001

Calculus

1.286

.575

.163

.027

.134

.018

Years Since Algebra Course

.011

.119

.006

.925

.006

< .001

Computational Fluency

.051

.027

.136

.063

.113

.013

-.028

.065

-.037

.671

-.026

.001

Fraction

.553

.138

.393

< .001

.241

.058

Equivalence

.043

.118

.030

.716

.022

< .001

Algebraic Reasoning

.087

.074

.103

.241

.071

.005

MARMI

.023

.025

.067

.349

.057

.003

MASMI

.051

.021

.204

.015

.148

.022

Crystallized Intelligence

-.005

.027

-.016

.857

-.011

< .001

Fluid Intelligence

-.027

.033

-.080

.405

-.050

.003

Numeracy

.518

Note. B = unstandardized regression coefficient; SEb = standard error of unstandardized
regression coefficient; β = standardized regression coefficient; rpartial = semi partial correlation;
r2partial = unique variance of predictors; MARMI = Measure of the Ability to Rotate Mental
Images; MASMI = Measure of the Ability to Form Spatial Mental Images.

95

Table 8
Ward’s Method Agglomeration Schedule for Clusters 1-9
Number of Clusters

Coefficient

Δcoefficient

1

1260.000

−

2

814.648

445.352

3

731.239

83.409

4

666.827

64.412

5

610.524

56.303

6

572.588

37.936

7

539.223

33.365

8

508.884

30.340

9

489.169

19.715

Note. Δcoefficient = absolute difference in coefficients.

96

Table 9
Indices for Ward’s Method Cluster Solutions
Variance Explained
Cluster
Solutions

CF

N

F

E

AR

MARMI

MASMI

CII

FII

Sample Sizes

4

0.271

0.450

0.640

0.554

0.428

0.616

0.427

0.399

0.450

21, 38, 63, 19

5

0.294

0.628

0.649

0.555

0.496

0.623

0.501

0.420

0.472

21, 38, 27, 36, 19

6

0.301

0.628

0.651

0.555

0.497

0.651

0.677

0.426

0.525

21, 21, 27, 17, 36, 19

7

0.373

0.652

0.651

0.556

0.513

0.652

0.734

0.493

0.525

21, 21, 17, 17, 36, 19, 10

8

0.530

0.655

0.663

0.560

0.526

0.656

0.735

0.517

0.525

21, 21, 17, 17, 23, 19, 10, 13

Note. CF = computational fluency; N = numeracy; F = fractions; E = equivalence; AR = algebraic reasoning; MARMI = Measure of
the Ability to Rotate Mental Images; MASMI = Measure of the Ability to Form Spatial Mental Images; CII = crystalized intelligence
index; FII = fluid intelligence index

97

Table 10
Indices for K-Means Cluster Solutions
Variance Explained
Cluster
Solutions

CF

N

F

E

AR

MARMI

MASMI

CII

FII

Sample Sizes

4

0.313

0.514

0.600

0.521

0.539

0.639

0.483

0.423

0.502

23, 44, 51, 23

5

0.354

0.536

0.638

0.512

0.588

0.657

0.606

0.424

0.490

23, 43, 23, 43, 18

6

0.397

0.558

0.654

0.526

0.579

0.683

0.708

0.432

0.526

21, 22, 27, 19, 37, 15

Note. CF = computational fluency; N = numeracy; F = fractions; E = equivalence; AR = algebraic reasoning; MARMI = Measure of
the Ability to Rotate Mental Images; MASMI = Measure of the Ability to Form Spatial Mental Images; CII = crystalized intelligence
index; FII = fluid intelligence index

98

Table 11
Raw Score Mean, Standard Deviation, F-Statistic, and Partial Eta Squared for Cluster Variables by Cluster Groups
Cluster Group
Mod. Low
Mod. Low Mod. High Mod. High
High
Low All
Variable
All
Spatial
MASMI
All
Spatial
F(5,135)
Total n
21
22
27
19
37
15
Computational Fluency
M
21.62d
29.09bc
34.11ab
24.37cd
39.43a
34.27ab
17.799
SD
3.918
5.740
10.207
6.898
9.257
8.884
Numeracy
M
10.24d
14.95c
17.00bc
15.11c
21.35a
19.80ab
34.130
SD
2.644
3.359
3.893
4.202
2.761
3.468
Fractions
M
4.29d
7.27c
9.37b
6.74c
10.43a
10.27ab
50.934
SD
1.875
1.830
1.275
2.023
1.345
1.486
Equivalence
M
2.57c
4.68b
7.52a
5.05b
7.73a
8.00a
29.941
SD
1.469
1.524
1.626
2.248
2.077
2.299
Algebraic Reasoning
M
7.57d
10.18d
12.74bc
10.58cd
17.22a
15.33ab
37.185
SD
3.059
2.889
3.096
3.746
2.573
2.469
MARMI
M
1.14cd
-2.36d
2.44bc
5.16bc
6.81b
30.40a
58.250
SD
4.693
4.489
5.515
5.747
7.222
8.990
MASMI
M
7.00d
11.41cd
19.81c
36.95ab
36.16b
42.00a
65.535
SD
6.550
8.798
11.533
6.169
8.719
3.761
Crystallized Intelligence
M
88.71d
95.23cd
105.19ab
99.21bc
110.81a
109.53a
20.514
SD
9.012
9.507
9.319
7.406
10.298
8.700
Fluid Intelligence
M
95.43c
98.64c
110.89b
107.79b
114.78ab
119.33a
29.903
SD
9.453
6.091
6.247
9.265
7.307
9.131
99

Ρ2

.397

.558

.654

.526

.579

.683

.708

.432

.526

Table 11 cont’d
Note. All F values are significant at p <.001. Uncommon superscripts indicate means that are statistically significantly different at p <
.05. MARMI = Measure of the Ability to Rotate Mental Images; MASMI = Measure of the Ability to Form Spatial Mental Images.

100

Table 12
NAEP Achievement Level Descriptions
Cut Score
Achievement Levels

Points

Percentage

Description

Basic

141

47

The ability to solve problems that are the direct
application of mathematical concepts and
procedures.

Proficient

176

57

The mastery of mathematical concepts
demonstrated by the appropriate application of
concepts and procedures to solve and analyze
problems.

Advanced

216

72

The ability to use mathematical knowledge to
solve unfamiliar and challenging problems, make
mathematical justifications, make justifiable
generalization, and use appropriate mathematical
language and notation.

Note. Cut score is out of 300 possible points.

101

Table 13
Frequency Counts and Percentages of Demographic Characteristics by Cluster Group
Cluster Groups

Low All

Mod. Low
All

Mod. Low
Spatial

Mod.
High
MASMI

Mod.
High All

High
Spatial

(n = 21)

(n = 22)

(n = 27)

(n = 19)

(n = 37)

(n = 15)

Total

χ2

5 (3.5)

11 (7.8)

17 (12.1)

11 (7.8)

34 (24.1)

13 (9.2)

91 (64.5)

32.954**

16 (11.3)

11 (7.8)

10 (7.1)

8 (5.7)

3 (2.1)

2 (1.4)

50 (35.5)

16 (11.3)

19 (13.5)

23 (16.3)

16 (11.3)

25 (17.7)

11 (7.8)

110 (78.0)

5 (3.5)

3 (2.1)

4 (2.8)

3 (2.1)

12 (8.5)

4 (2.8)

31 (22.0)

4 (2.8)

4 (2.8)

11 (7.8)

9 (6.4)

25 (17.7)

9 (6.4)

62 (44.0)

17 (12.1)

18 (12.8)

16 (11.3)

10 (7.1)

12 (8.5)

6 (4.3)

79 (56.0)

White

7 (5.0)

14 (9.9)

13 (9.2)

14 (9.9)

28 (19.9)

11 (7.8)

87 (61.7)

Non-White

14 (9.9)

8 (5.7)

14 (9.9)

5 (3.5)

9 (6.4)

4 (2.8)

54 (38.3)

Lowerclassmen (F, So)

8 (5.7)

8 (5.7)

15 (10.6)

9 (6.4)

15 (10.6)

8 (5.7)

63 (44.7)

Upperclassmen (Jr., Sr., 5+ )

13 (9.2)

14 (9.9)

12 (8.5)

10 (7.1)

22 (15.6)

7 (5.0)

78 (55.3)

Demographic Characteristic
Highest math course takena
Calculus
No Calculus
a

Gender

Female
Male

4.715

Majorsa
STEM
Non-STEM

21.364*

Race/Ethnicitya
14.356*

School Level

Note. F = Freshmen. So = Sophomore. Jr. = Junior. Sr. = Senior 5+. = 5+ year Senior.
*
p < .05. ** p < .001.
102

3.042

Table 14
Raw Score Mean, Standard Deviation, F-Statistic, and Partial Eta Squared for Algebra Performance by Cluster Groups
Cluster Group
Mod. Low
Mod. Low Mod. High Mod. High
High
Low All
Variable
All
Spatial
MASMI
All
Spatial
F(5,56.603)
Ρ2
Total n
21
22
27
19
37
15
Algebra Performance
M
4.33c
5.64bc
7.81b
5.58bc
10.30a
10.93a
18.896
.410
SD
2.153
2.300
2.936
2.735
3.650
3.218
Note. All F values are significant at p <.001. Uncommon superscripts indicate means that are statistically significantly different at p <
.05.

103

Table 15
Summary of Multiple Regression Analysis for Demographic Characteristics and Cluster Contrasts
Variable

b

β

SEb

p

rpartial

r2partial

Intercept

7.148

.867

Calculus

1.652

.621

.209

.009

.171

.029

Years Since Algebra Course

-.076

.146

-.038

.604

-.033

.001

Race/Ethnicity

-.079

.535

-.010

.883

-.009

< .001

Major

-.671

.563

-.088

.236

-.077

.006

MLA, MHM, LA – MHA, HS, MLSa

5.719

.870

.502

< .001

.423

.179

MLS, HS – MHAa

.348

.457

.053

.447

.049

.0024

LA – MLA, MHMa

.410

.550

.051

.457

.048

.0023

1.340

.480

.191

.006

.180

.032

.150

.466

.021

.749

.021

< .001

MLS – HSa
MLA – MHMa

< .001

Note. MHA = moderate high all; HS = high spatial; MLS = moderate low spatial; MHM = moderate high MASMI; MLA = moderate
low all; LA = low all; b = unstandardized regression coefficient; SEb = standard error of unstandardized regression coefficient; β =
standardized regression coefficient; rpartial = semi partial correlation.
a
Planned contrasts for cluster membership comparisons going from cluster group (s) A to cluster group(s) B.

104

Appendix B: Figures

0.053%

1.809%

0.003% 1.274%
0.066%
5.814%

0.048%
0.502% 0.320%
2.192% 0.012%
0.253%

Gender
Calculus
Years Since Algebra Course
Computational Fluency
Numeracy

41.355%

Fraction
Equivalence
Algebraic Reasoning
MARMI
46.300%

MASMI
Crystallized Intelligence

Fluid Intelligence
Unexplained
Shared

Figure 1.Variance explained in algebra performance scores by predictor variables in multiple regression analysis.

Average Standardized Scores for Cluster Variables

2.500
2.000
1.500
1.000

Computational Fluency
0.500

Numeracy
Fractions

0.000

Equivalence
Algebraic Reasoning

-0.500

MARMI
-1.000

MASMI
Crystallized Intelligence

-1.500

Fluid Intelligence
-2.000
-2.500
Low All

Moderate Moderate Moderate
Low All Low Spatial
High
MASMI

Moderate High Spatial
High All

Cluster Groups

Figure 2. Results of six-cluster solution, showing the average standardized scores on all
clustering variables for each cluster group.

Figure 3. Mean algebra performance scores for participants in each cluster group.

*

*
*

Figure 4. Cluster variable scores and algebra performance score comparison of participants in the Moderate Low All and Moderate
High MASMI cluster groups.
*
statistically significant difference at p < .05.

*

*
*

*

Figure 5.Cluster variable scores and algebra performance score comparison of participants in the Moderate Low Spatial and High
Spatial cluster groups.
*
statistically significant difference at p < .05.

*

*

Figure 6. Cluster variable scores and algebra performance score comparison of participants in the Moderate High All and High Spatial
cluster groups.
*
statistically significant difference at p < .05.

*
*

*
*

*
*

*

Figure 7. Cluster variable scores and algebra performance score comparison of participants in the Moderate High MASMI and
Moderate High All cluster groups.
*
statistically significant difference at p < .05.

*
*
*

*

*

Figure 8. Cluster variable scores and algebra performance score comparison of participants in the Moderate Low Spatial and
Moderate High All cluster groups.
*
statistically significant difference at p < .05.

Appendix C: Assessments and Measures
Computational Fluency Assessment
1. 6220 + 3545
2. 104 á 4
3. 4352 − 2311
4. 21 × 13
5. 444 − 387
6. 36 × 48
7. 27531 + 18515
8. 432 á 54
9. 785 × 37
10. 43 + 58 + 28
11. 488 á 8
12. 9101 − 7247
13. 630 á 15
14. 93245 − 71378
15. 2615 × 11
16. 811 + 168 + 237 + 753

Numeracy Assessment
Addition and Subtraction
1. 1,504 + 6.2

2. −7.2 + 6.9

a. 1,560

b. 1,100

a. -14

b. 0

c. 1,510

d. 1,504.62

c. 1

d. 14

3. 11.98 + 6.02

4. 73 + 7.3

a. 7

b. 18

a. 140

b. 150

c. 1,800

d. 1,000

c. 80

d. 70

5. 782 − 83

6. 3,012 − 2,998

a. 700

b. 600

a. 0

b. 100

c. 800

d. 750

c. 1,000

d. 1,999

7. 609 − 0.69

9.

4

8. 0.25 − 0.12

a. 550

b. 500

a. 0.05

b. 0.12

c. 600

d. 609

c. 0.5

d. 0

9

1

+8
5

8

1

10. 2 3 + 4

a. 2

b. 1

a. 2.5

b. 2

c. 3

d. 4

c. 1

d. 3

6

1

11. 6 − 8
a. 0
c.

1

12. 5 5 − 2 2

8
4

b. 0.5
9

d. 4

a.

1

1

b. 1 2

2
1

c. 2 2

1

d. 3 2

Multiplication and Division
13. 15 × 9

14. 26 × 16

a. 150

b. 200

a. 50

b. 400

c. 900

d. 1,500

c. 800

d. 4,000

15. 0.9 × 5

16. 0.3 × 9

a. 0

b. 2

a. 0.1

b. 0.5

c. 5

d. 45

c. 2

d. 3

17. 1,602 á 99

18. 4,942 á 49

a. 0

b. 1

a. 0.5

b. 10

c. 10

d. 16

c. 100

d. 1,000

19. 8.2 á 10

1

20. 61 á 5.9

a. 1

b. 4

a. 1

b. 10

c. 8

d. 82

c. 15

d. 20

7

15

a. 0
d.

1
2

1

b. 10
d. 1

49

a. 0

b. 1

e. 2

d. 2 2

1

11

23. 2 á 100

24. 12 á 3

a. 0
f.

8

22. 8 × 4

21. 2 × 16

3
4

b.

1
2

d. 4

115

1

a. 0

b.

d. 1

d. 3

3

Fractions Assessment

Category I: Algorithmic Operations
3

1. Subtract 5 from 8.
5

2. Write 3 6 as an improper fraction.
Category II: Application of Basic Fraction Concepts in Word Problems
1. If you have a half ball of string and each kite needs an eighth of a ball, how many kites
can you fly?
2. Adrian has conquered only 6 giants in his new video game, Giant Trouble, but it is only
two-fifths of the giants that he must conquer. How many giants are there in the new video
game?
Category III: Elementary Algebraic Concepts
1

1. Solve 𝑥 + 3 = 7.
2.

1
3

× 𝑎 =?

Category IV: Specific Arithmetic Skills that are Prerequisite for Algebra
2

1. Write 5 7 as a sum.
2. Find

18
0

.

Category V: Comprehension of the Structure of Rational Numbers
4 5 3

1. Write fractions 7 , 9 , 5 in order from least to greatest.
2. The quotient of

1
2

1

1

á 3 is greater than (>) or less than (<) 2?

Category VI: Computational Fluency
7+5

5
6
5
3

1. Find the sum 3+5 + .
1

9

2. In an election, candidate A got 3 of the votes, candidate B got 20 of the votes, and
2

candidate C got 15 of the votes. What fraction of the votes did candidate D get?

116

Equivalence Assessment

Equal Sign Interpretation
1. What does the equal sign (=) mean? Can it mean anything else?
2. What does the equal sing mean in this statement? 1 dollar = 100 pennies
Equation Structure
1. Is the number that goes in the blank the same number in the following two number
sentences? Yes, No, How do you know? 2 × ____ = 58 8 × 2 × ____ = 8 × 58
2. Find a number that can go in each blank. Can another number go in these blanks?
Explain. 8 + 2 + ____ = 10 + ____
Open Equation
1. Fill in the blanks with the value that makes the following number sentences true.
a. 4,436 + 2,897 = _____ + 3,000
b. 3,901 − 2,012 = 3,889 − _____
2. Place the four numbers 𝑛 − 1, 𝑛 + 1, 𝑚 + 3, 𝑚 + 1 in the following boxes so that the number
sentence is always true. □ + □ = □ + □

117

Algebraic Reasoning Assessment
Functional Thinking & Generalization
1.
Number of sets

Number of red
Tiles
1
2
3
4
5

1
2
3
4
5

Number of green
tiles
2
4
6
8
10

Total number of
tiles
3
6
9
12
15
33

12
40
Using the information in the table answer:
a. Identify the relationship between the number of red tiles and any given number of
sets (x).
b. Identify the relationship between the number of green tiles and any given number
of sets (x).
c. Identify the relationship between the total number of tiles and any given number
of sets (x).

2.

Number of Squares

Number of
Vertices

1

4

2

7

3

10

4

__________

10

__________

100

__________

a. Suppose you are given the pattern of squares shown above. How would you describe the
relationship between the number of squares and the number of vertices in words?
b. How would you represent the relationship between the number of squares and the number
of vertices using algebra?
118

Modeling
1. Write an equation using the variables C and S to represent the following statement: “At
Mindy’s restaurant, for every four people who ordered cheesecake, there are five people
who ordered strudel.” Let C represent the number of cheesecakes and S represent the
number of strudels ordered.
2. Write an equation using the variables S and P to represent the following statement: "There
are six times as many students as professors at a certain university." Use S for the number
of students and P for the number of professors.
Structure Sense
1. Solve for x.
1

𝑥

1

𝑥

a. (4 − 𝑥−1) − 𝑥 = 6 + (4 − 𝑥−1)
1

1

1

b. 1 − 𝑥+2 − (1 − 𝑥+2) = 110
2. Solve the following number sentences.
a. 237 + 89 − 89 + 267 − 92 + 92 = ?
b. 217 − 59 + 59 + 62 − 28 − 28 = ?
Symbolic Manipulation
1. For each example, write an equivalent expression.
a. 4ℎ + 𝑡 = _______________
b. 𝑢 + 5 + 6 + 5 + 𝑢 = _____________
c. 4(𝑛 + 5) = _______________
d. 𝑝 + 0.05𝑝 = _____________
e. 5(𝑒 + 2) = _____________
f. 𝑥 − 𝑥 + 2 = _______________
g. (15 + 10𝑥) + (35 + 5𝑥) = _______________
h. 3𝑥 + 4 + 6(𝑥 + 5) = _____________

119

Algebra Performance Assessment
Systems of Equations
x + 2 y = 17
x-2y=3
1. The graphs of the two equations shown above intersect at the point
x at the point of intersection?
A.
B. 5
C. 7
D. 10
E. 20
(NAEP Question ID: 2005-12M3 #12 M095201)

2. In the solution of the system of equations above, what is the value of x?
A. - 1
B. 2
C. 3
D. 4
E. 5
(NAEP Question ID: 2005-12M4 #11 M053201)

Functions
1. If

and

, then

A. 3
B. 5
C. 7
D. 7 5/9
E. 16 2/3
(NAEP Question ID: 1992-12M5 #20 M025401)
120

. What is the value of

2. Yvonne has studied the cost of tickets over time for her favorite sports team. She has created a
model to predict the cost of a ticket in the future. Let C represent the cost of a ticket in dollars
and y represent the number of years in the future. Her model is as follows.

Based on this model, how much will the cost of a ticket increase in two years?
A. $5
B. $8
C. $13
D. $18
E. $26
(NAEP Question ID: 2005-12M12 #17 M130101)

Solving Equations
1. If

and

in the formula

, then

A.
B.
C. 5
D.
E. 6
(NAEP Question ID: 1990-12M9 #12 M030231)
2. If

𝟏
𝒂

𝟒

𝟏

+ 𝟕 = 𝟐, then a = (?).
A.

3
5

B. -14
C.

−3
5

D. 14
E.

14
3

121

Inequalities
1. What are all values of x such that

?

A.
B.
C.
D.
E.
(NAEP Question ID: 2005-12M4 #16 M019401)
𝑙

2. For which values of w is 1.25 ≤ 𝑤 ≤ 2.5 if 𝑙 = 4?
A. 𝑤 ≥ 1.6 𝑜𝑟 𝑤 ≤ 3.2
B. 1.6 ≤ 𝑤 ≤ 3.2
C. 𝑤 ≥ 0.16 𝑜𝑟 𝑤 ≤ 0.30
D. 0.16 ≤ 𝑤 ≤ 0.30
E. 5 ≤ 𝑤 ≤ 10
Graphing

1. The graphs of

and

many values of x is the product

for

are shown in the figure above. For how
for

A. Two
B. Four
C. Five
D. Six
E. Seven
(NAEP Question ID: 1990-12M9 #19 M030931)
122

?

2. Which of the following is the graph of

?

A.
B.
C.
D.
E.
(NAEP Question ID: 2005-12M3 #15 M011831)
Exponents
1. For what value of x is

?

A. 3
B. 4
C. 8
D. 9
E. 12
(NAEP Question ID: 1992-12M7 #6 M057701)
3

2. 1212 =
A. 1331
B. 1728
C. 2197
D. 4096
E. 1452

123

Factoring
1. Factor 28𝑛4 + 16𝑛3 − 80𝑛2 . One of the factors is (?).
A. 7𝑛 + 10
B. 𝑛 − 2
C. 𝑛 + 5
D. 4𝑛2
E. 𝑛 − 4
2. 𝑚2 + 10𝑚 + 14 + 𝑏 is a perfect square. Find b.
A. 0
B. 7
C. -4
D. 11
E. 21
Complex Numbers
1

1. If 𝑖 = √−1 then 6+4𝑖 = ?
A.
B.
C.
D.
E.

6−4𝑖
52
6+4𝑖
52
6+4𝑖
20
6−4𝑖
20
6−4𝑖
32

124

2. Let 𝒊 = √−𝟏. After expanding and simplifying (𝟐 + 𝒊)𝟑 =?
A. 8 + 𝑖 3
B. 8 − 𝑖
C. 2 + 11𝑖
D. 6 + 3𝑖
E. 10 + 11𝑖
Polynomial Division
1. If 6𝑥 3 + 5𝑥 − 8 is divided by 𝑥 − 2, the remainder is?
A. 50
B. -64
C. -8
D. 0
E. -16
2. (𝑥 3 + 12𝑥 2 + 47𝑥 + 60) ÷ (𝑥 + 5) = ?
720

A. 𝑥 2 + 17𝑥 + 132 + 𝑥+5
50

B. 𝑥 2 + 17𝑥 − 38 − 𝑥+5
C. 𝑥 2 + 7𝑥 + 12
D. 𝑥 3 + 7𝑥 2 + 12𝑥
215

E. 𝑥 2 + 7𝑥 − 31 + 𝑥+5

125

Logarithms
1. Which one is equivalent to log 4 (16𝑥 6 )?
A. 4 + 6 log 4 𝑥
B. 12 log 4 𝑥
C. 2 + 6 log 4 𝑥
D. 6 log 4 𝑥
E. 2 log 4 𝑥
2. Solve log 𝑥 16 = 2 for x. x = (?).
A.

2
16

B. 4
C. 256
D. 32
E. 8

126

Participant Demographic Survey
Please answer the following questions.
Name (please print): __________________________________
Age: _____________________
Gender:

□ Female

□ Male

□ Other: _______________

Race/Ethnicity (check all that apply):
□ American Indian/Alaska Native
□ Hispanic/Latino

□ Asian

□ Black/African American

□ Native Hawaiian/Other Pacific Islander

□ White

□ Other: _______________
School Level:
□ Freshman

□ Sophomore

□ Junior

□ Senior

□ 5+ year Senior

Major: _________________________________________________
List all mathematics courses taken in high school and the year in which it was taken.
High School Mathematics Course
a) ___________________
b) ___________________
c) ___________________
d) ___________________
e) ___________________

Year Taken
_______________
_______________
_______________
_______________
_______________

List all mathematics courses taken in college and the year in which it was taken.
College Mathematics Course
a) ___________________
b) ___________________
c) ___________________
d) ___________________
e) ___________________

Year Taken
_______________
_______________
_______________
_______________
_______________

127

Appendix D: Research Protocols
Starting Procedures
1. As the participant comes in pass out the participant information and consent form and
say, “Hi my name is [your name}. Welcome to the Factors of Algebra Performance
study. Thank you for coming. Here is the Participant Information and Consent form.
This form explains this study in detail. Take a minute to read it.”
2. Give the participant a few minutes to read, and then say, “Today I am going to be asking
you to do a number of different tasks. Some of them will be mathematical in nature
and some will not. Some questions will be easy and some will be hard. Each task will
have a time limit. Most people do not get to all the questions nor do they get them all
right. We just want you to do your best. Participation is voluntary and you may quit at
any time. You may also take a break when you need to. After the completion of the
study, you will receive $15 dollars as a thank you for participating. Are there any
questions?” Answer any questions.
3. Then say, “If you are willing to continue please fill out and sign the second page of the
Participant Information and Consent Form that I have pass out to you, and hand it
back to me. You may keep the first page.” Collect the signed form.
4. Then say, “Thank you for agreeing to participate in this study. Let’s get started with the
first task.” Use the given order for the group and begin with the first task.
Computational Fluency
1. Distribute the computational fluency packet to the student, and say, “Now we are going
to be doing another math task. Follow along with me as I read the directions aloud.
The sheets in front of you are math facts. There are several types of problems on the
sheet. Some are addition, some are subtraction, some are multiplication, and some are
division. Look at each problem carefully before you answer it. When I say, begin, turn
to the next page and begin answering the problems. Start on the first problem on the
left on the top row [point]. Work across then go to the next row. If you cannot answer a
problem, make an ‘X’ on it and go to the next one. If you finish one page, go to the
next one. You will have only three minutes to complete all problems. It is okay if you do
not complete all the problems. Just try to do your best. Are there any questions?”
Answer any questions.
2. Then say, “If there are not any [more] questions [pause], then you may begin.” Start
the timer. Participants get 3 minutes to complete each worksheet.
3. After three minutes have passed say, “Pencil down. Time is up and we need to move on
to the next task” and collect the computational fluency worksheet.

128

Numeracy
1. Distribute the numeracy packet to the student, and say, “Now we will be doing another
math task. Follow along with me as I read the directions aloud. When I say, ‘Begin’
turn to the next page, and begin with the first problem. Pay close attention to all the
signs indicating positive and negative numbers. For each problem, the exact answer is
not listed, so choose the answer that is as close to the exact answer as possible by
circling your answer. You will only get 4 minutes to complete the whole task, so work
as quickly as you can. It is okay if you do not complete all problems. Just try to do your
best. Again, the exact answer is not listed, so choose an answer that is closest to the
exact answer as possible. Are there any questions?” Answer any questions.
2. Then say, “If there are not any [more] questions [pause], then you may begin.” Start
the timer. Participants get 4 minutes to complete the task.
3. After 4 minutes have passed say, “Pencil down. Time is up and we need to move on to
the next task.” and collect the numeracy packets.
Fractions
1. Distribute the fraction packet to the student, and say, “Now we will be doing another
math task. Follow along with me as I read the directions aloud. When I say, ‘Begin’
turn to the next page, and begin with the first problem. Make sure to show all your
work in the space provided. If you need more space, you may use scratch paper. If you
use scratch paper, please label your work by writing the problem number next to it.
After you solve each problem, circle your answer. Some of these problems will be easy
and some will be hard. If you do not know how to solve a problem, guess or estimate to
the best of your ability or go on to the next one until you have completed as many as
you can or are told to stop. You will only get 10 minutes to complete all the problems. It
is okay if you do not complete all problems. Just try to do your best. Are there any
questions?” Answer any questions.
2. Then say, “If there are not any [more] questions [pause], then you may begin.” Start
the timer. Participants get 10 minutes to complete the task.
3. After 10 minutes have passed say, “Pencil down. Time is up and we need to move on to
the next task.” and collect the fraction packets.
Equivalence
1. Distribute the equivalence packet to the student, and say, “Now we will be doing another
math task. Follow along as I read the directions aloud. When I say, ‘Begin’ turn to the
next page, and begin with the first problem. Make sure to show all your work in the
space provided. If you do not know how to solve a problem, guess or estimate to the
best of your ability or go on to the next one until you have completed as many as you
can or are told to stop. You will only get 10 minutes to complete all the problems. It is
129

okay if you do not complete all problems. Just try to do your best. Are there any
questions?” Answer any questions.
2. Then say, “If there are not any [more] questions [pause], then you may begin.” Start
the timer. Participants get 10 minutes to complete the task.
3. After 10 minutes have passed say, “Pencil down. Time is up and we need to move on to
the next task.” and collect the equivalence packets.
Algebraic Reasoning
1. Distribute the algebraic reasoning packet to the student, and say, “Now we will be doing
another math task. Follow along as I read the directions aloud. When I say, ‘Begin’
turn to the next page, and begin with the first problem. Pay close attention to all the
signs indicating positive and negative numbers. Make sure to show all your work in the
space provided. If you need more space, you may use scratch paper. If you use scratch
paper, please label your work by writing the problem number next to it. Some of these
problems will be easy and some will be hard. If you do not know how to solve a
problem, guess or estimate to the best of your ability or go on to the next one until you
have completed as many as you can or are told to stop. You will only get 10 minutes to
complete all the problems. It is okay if you do not complete all problems. Just try to do
your best. Are there any questions?” Answer any questions.
2. Then say, “If there are not any [more] questions [pause], then you may begin.” Start
the timer. Participants get 10 minutes to complete the task.
3. After 10 minutes have passed say, “Pencil down. Time is up and we need to move on to
the next task.” and collect the algebraic reasoning packets.
Algebra Performance
1. Distribute the algebraic reasoning packet to the student, and say, “Now we will be doing
another math task. Follow along as I read the directions aloud. When I say, ‘Begin’
turn to the next page, and begin with the first problem. Pay close attention to all the
signs indicating positive and negative numbers and exponentials. You may use the
calculator or scratch paper to help you solve the problems. If you use scratch paper,
please label your work by writing the problem number next to it. After you solve each
problem, make sure to circle your answer. Some of these problems will be easy and
some will be hard. If you do not know how to solve a problem, guess or estimate to the
best of your ability or go on to the next one until you have completed as many as you
can or are told to stop. You will only get 20 minutes to complete all the problems. It is
okay if you do not complete all problems. Just try to do your best. Are there any
questions?” Answer any questions.
2. Then say, “If there are not any [more] questions [pause], then you may begin.” Start
the timer. Participants get 20 minutes to complete the task.

130

3. After 20 minutes have passed say, “Pencil down. Time is up and we need to move on to
the next task.” and collect the algebra performance packets.
MASMI: Measure of the Ability to form Spatial Mental Imagery
1. Distribute the MARMI/MASMI packet to the student, and say, “Now we will be doing
another cognitive task. Follow along as I read the directions aloud.” Use one of the
MASMI copies to read the directions on the first page.
2. Then say, “Are there any questions?” Answer any questions.
3. Then say, “If there are not any [more] questions [pause], then you may begin.” Start
the timer. Participants get 10 minutes to complete the task.
4. After 10 minutes have passed say, “Pencil down. Time is up and we need to move on to
the next task.” and collect the MARMI/MASMI packets.
RAIT: Reynolds Adaptable Intelligence Test
1. Distribute the test booklets and answer sheets to all students, and say, “Now we will be
doing another cognitive task. Here is your test booklet and answer sheet. Go ahead and
read the instructions on the front cover of the test booklet. Do not turn to then next
page until I tell you to. Then write your date of birth in the correct spaces on the
answer sheets. Do not worry about filling out the rest. Let me know when you are
finished.”
2. After the student has read the instructions on the front cover of the test booklet and filled
out the answer sheet, say, “We will not do all the sections in the test booklet. I will let
you know which sections we are going to complete. We will do one section at a time to,
so please pay close attention to which page numbers that I give so that you can
complete the right section. It will only take 30 minutes to complete the all the sections.
Are there any questions about the instructions on the cover of the test booklet?”
Answer any questions.
3. Then say, “If there are not any [more] questions, then please turn to page 3 in your test
booklet. We will start with the first section in the test booklet. Take a moment to read
the example for section 1. Let me know when you are finished.”
4. Once the student has read the example, say, “Do you understand what to do for section
1? [wait for affirmation] You will have only 3 minutes to complete this section. Please
do not mark in your test booklet. Mark all your answers on your answer sheet in the
first section on left labeled GK. If you finish before time is up, you may review your
answer for section 1 only. Otherwise, let me know when you are finished so we can
move on to the next section. Are there any questions?” Answer any questions.
5. Then say, “If there are not any [more] questions [pause], then you may begin.” Start
the timer. Participants get 3 minutes to complete section1.
6. After 3 minutes have passed say, “Pencil down. Time is up and we need to move on to
the next section.”
131

7. Then say, “Please turn to page 7 in your test booklet. This should be section 2. Take a
moment and read the example for section 2. Let me know when you are finished”
8. Once the student has read the example, say, “Do you understand what to do for section
2? [wait for affirmation] You will have only 7 minutes to complete this section. Please
do not mark in your test booklet. Mark all your answers on your answer sheet in the
middle section labeled NVA. If you finish before time is up, you may review your
answer for section 2 only. Otherwise, let me know when you are finished so we can
move on to the next section. Are there any questions?” Answer any questions.
9. Then say, “If there are not any [more] questions [pause], then you may begin.” Start
the timer. Participants get 7 minutes to complete section 2.
10. After 7 minutes have passed say, “Pencil down. Time is up and we need to move on to
the next section.”
11. Then say, “Please turn to page 35 in your test booklet. This should be section 3. Take a
moment and read through the example for section 3. Let me know when you are
finished.”
12. Once the student has read the example, say, “Do you understand what to do for section
3? [wait for affirmation] You will have only 10 minutes to complete this section. Please
do not mark in your test booklet. Mark all your answers on your answer sheet in the
section on the right labeled SEQ. If you finish before time is up, you may review your
answer for section 3 only. Otherwise, let me know when you are finished so we can
move on to the next section. Are there any questions?” Answer any questions.
13. Then say, “If there are not any [more] questions [pause], then you may begin.” Start
the timer. Participants get 10 minutes to complete section 3.
14. After 10 minutes have passed say, “Pencil down. Time is up and we need to move on to
the next section.”
15. Then say, “Please turn to page 67 in your test booklet. This should be section 6. Take a
moment and read through the example for section 6. Let me know when you are
finished.”
16. Once the student has read the example, say, “Do you understand what to do for section
6? [wait for affirmation] You will have only 5 minutes to complete this section. Please
do not mark in your test booklet. Mark all your answers on your answer sheet in the
third section from the left section labeled OWO on side 2. If you finish before time is
up, you may review your answer for section 6 only. Otherwise, let me know when you
are finished so we can move on to the next section. Are there any questions?” Answer
any questions.
17. Then say, “If there are not any [more] questions [pause], then you may begin.” Start
the timer. Participants get 5 minutes to complete section 6.
18. After 5 minutes have passed say, “Pencil down. Time is up and we need to move on to
the next section.”

132

19. Then say, “Please turn to page 70 in your test booklet. This should be section 7. Take a
moment and read through the example for section 7.”
20. Once the student has read the example, say, “Do you understand what to do for section
7? [wait for affirmation] You will have only 5 minutes to complete this section. Please
do not mark in your test booklet. Mark all your answers on your answer sheet in the
last section labeled WO on side 2. If you finish before time is up, you may review your
answer for section 7 only. Otherwise, let me know when you are finished so we can
move on to the next section. Are there any questions?” Answer any questions.
21. Then say, “If there are not any [more] questions [pause], then you may begin.” Start
the timer. Participants get 5 minutes to complete section1.
22. After 5 minutes have passed say, “Pencil down. Time is up. This was the last section for
this task. Make sure you have filled out your date of birth on your answer sheet before
handing it back.”
Participant Demographic Survey
1. Say, “We are now at the end of the study. You have completed all the tasks. Now I
would like to ask you to fill out this survey. [pass out survey to the participant] The
survey will ask you a number of demographic questions and about your past
mathematics education. Please answer all questions to the best of your ability. If you
come to a question that you do not wish to answer, please feel free to skip it and move
on to the next one. Once you have completed the survey, place hand it back to me. I
will then give you your appreciation gift and you are free to go. Are there any
questions?” Answer any questions.
2. Collect the complete surveys. As participants hand in their survey, say, “Thank you
again for participating in the Factors of Algebra Performance Study. Have a great
day!”

133

Appendix E: Item Level Analyses
Item Level Analysis for Fraction Assessment
M

SD

Total
Points

.84

.364

1

2. Write 3 6 as an improper fraction.
Word Problems
1. If you have a half ball of string and each kite needs an eighth of a ball, how many kites can
you fly?
2. Adrian has conquered only 6 giants in his new video game, Giant Trouble, but it is only twofifths of the giants that he must conquer. How many giants are there in the new video game?
Algebraic Concepts
1
1. Solve 𝑥 + 3 = 7.

.74

.438

1

.92

.269

1

.82

.389

1

.79

.406

1

2. 3 × 𝑎 =?
Arithmetic Skills
2
1. Write 5 7 as a sum.

.89

.318

1

.43

.497

1

2. Find 0 .
Rational Number
4 5 3
1. Write fractions 7 , 9 , 5 in order from least to greatest.

.55

.500

1

.62

.488

1

.76

.429

1

.51

.502

1

.43

.497

1

Problems
Algorithmic Operations
3
1. Subtract 5 from 8.
5

1

18

1

1

1

2. The quotient of 2 á 3 is greater than (>) or less than (<) 2?
Computational Fluency
7+5

5
6
5
3

1. Find the sum 3+5 + .
1

9

2. In an election, candidate A got 3 of the votes, candidate B got 20 of the votes, and candidate
2

C got 15 of the votes. What fraction of the votes did candidate D get?
134

Item Level Analysis for Equivalence Assessment
M

SD

Total
Points

1. What does the equal sign (=) mean? Can it mean anything else?

1.39

.800

2

2. What does the equal sing mean in this statement? 1 dollar = 100 pennies

.59

.494

2

1. Is the number that goes in the blank the same number in the following two number
sentences? Yes, No, How do you know? 2 × ____ = 58 8 × 2 × ____ = 8 × 58

.99

.815

2

2. Find a number that can go in each blank. Can another number go in these blanks? Explain.
8 + 2 + ____ = 10 + ____

1.02

.882

2

a. 4,436 + 2,897 = _____ + 3,000

.79

.411

1

b. 3,901 − 2,012 = 3,889 − _____

.57

.497

1

.77

.762

2

Problems
Equal Sing Interpretation

Equation Structure

Open Equation
1. Fill in the blanks with the value that makes the following number sentences true.

2. Place the four numbers 𝑛 − 1, 𝑛 + 1, 𝑚 + 3, 𝑚 + 1 in the following boxes so that the number
sentence is always true. □ + □ = □ + □

135

Item Level Analysis for Algebraic Reasoning Assessment
Problems
Functional Thinking & Generalization
1a. Identify the relationship between the number of red tiles and any given number of sets (x).
1b. Identify the relationship between the number of green tiles and any given number of sets (x).
1c. Identify the relationship between the total number of tiles and any given number of sets (x).
2. Table: 4 squares = ___ vertices
2. Table: 10 squares = ___ vertices
2. Table: 100 squares = ___ vertices
2a. Suppose you are given the pattern of squares shown above. How would you describe the
relationship between the number of squares and the number of vertices in words?
2b. How would you represent the relationship between the number of squares and the number of
vertices using algebra?
Modeling
1. Write an equation using the variables C and S to represent the following statement: “At
Mindy’s restaurant, for every four people who ordered cheesecake, there are five people who
ordered strudel.” Let C represent the number of cheesecakes and S represent the number of
strudels ordered.
2. Write an equation using the variables S and P to represent the following statement: "There
are six times as many students as professors at a certain university." Use S for the number of
students and P for the number of professors.
Structure Sense
1. Solve for x.
1

𝑥

4

𝑥−1

a. ( −

1

1

𝑥

4

𝑥−1

)−𝑥 =6+( −
1

)

1

b. 1 − 𝑥+2 − (1 − 𝑥+2) = 110
2. Solve the following number sentences.
a. 237 + 89 − 89 + 267 − 92 + 92 = ?
b. 217 − 59 + 59 + 62 − 28 − 28 = ?
136

M

SD

Total
Points

.72
.74
.70
.62
.41
.35

.449
.438
.459
.486
.494
.478

1
1
1
1
1
1

.17

.377

1

.26

.442

1

.10

.300

1

.45

.499

1

.36

.482

1

.29

.456

1

.87
.71

.343
.456

1
1

Item Level Analysis for Algebraic Reasoning Assessment
M

SD

Total
Points

a. 4ℎ + 𝑡 = _______________

.23

.420

1

b. 𝑢 + 5 + 6 + 5 + 𝑢 = _____________

.82

.389

1

c. 4(𝑛 + 5) = _______________

.91

.290

1

d. 𝑝 + 0.05𝑝 = _____________

.65

.478

1

e. 5(𝑒 + 2) = _____________

.93

.258

1

f. 𝑥 − 𝑥 + 2 = _______________

.89

.318

1

g. (15 + 10𝑥) + (35 + 5𝑥) = _______________

.72

.452

1

h. 3𝑥 + 4 + 6(𝑥 + 5) = _____________

.84

.371

1

Problems
Symbolic Manipulation
1. For each example, write an equivalent expression.

137

Item Level Analysis for Algebra Performance Assessment
Problems
System Equations
1. The graphs of the two equations shown above intersect at the point
. What is the value
of x at the point of intersection?
2. In the solution of the system of equations above, what is the value of x?
Functions
1. If
and
, then
2. Yvonne has studied the cost of tickets over time for her favorite sports team. She has created
a model to predict the cost of a ticket in the future. Let C represent the cost of a ticket in
dollars and y represent the number of years in the future. Her model is as follows.
Based on this model, how much will the cost of a ticket increase in two
years?
Solving Equations
1. If
and
in the formula
, then
𝟏

𝟒

𝟏

2. If 𝒂 + 𝟕 = 𝟐, then a = (?).
Inequalities
1. What are all values of x such that

?
𝑙

2. For which values of w is 1.25 ≤ 𝑤 ≤ 2.5 if 𝑙 = 4?
Graphing
1. The graphs of
and
for
are shown in the figure above. For how
many values of x is the product
for
?
2. Which of the following is the graph of
?
Exponents
1. For what value of x is
?
3
2

2. 121 =

Factoring
1. Factor 28𝑛4 + 16𝑛3 − 80𝑛2 . One of the factors is (?).
2. 𝑚2 + 10𝑚 + 14 + 𝑏 is a perfect square. Find b.
138

M

SD

Total
Points

.37

.484

1

.54

.500

1

.77

.425

1

.38

.486

1

.74

.442

1

.57

.496

1

.26

.438

1

.46

.500

1

.55

.499

1

.43

.496

1

.12
.30

.327
.459

1
1

.57
.27

.497
.445

1
1

Item Level Analysis for Algebra Performance Assessment
Problems
Complex Numbers
1
1. If 𝑖 = √−1 then 6+4𝑖 = ?
𝟑

2. Let 𝒊 = √−𝟏. After expanding and simplifying (𝟐 + 𝒊) =?
Polynomial Division
1. If 6𝑥 3 + 5𝑥 − 8 is divided by 𝑥 − 2, the remainder is?
2. (𝑥 3 + 12𝑥 2 + 47𝑥 + 60) ÷ (𝑥 + 5) = ?
Logarithms
1. Which one is equivalent to log 4 (16𝑥 6 )?
2. Solve log 𝑥 16 = 2 for x. x = (?).

139

M

SD

Total
Points

.07
.21

.258
.411

1
1

.13
.36

.335
.482

1
1

.14
.41

.350
.494

1
1

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